# SREG-001 $$ \mathcal R_0:=\{\Sigma_M,\Sigma_L\},\qquad \mathcal F:=\operatorname{Form}(\Sigma_M\cup\Sigma_L) $$ $$ R:\mathcal F\to\mathcal F $$ $$ S_0:=\varnothing $$ --- ## $S_1$ $$ 0:=\varnothing $$ $$ S(n):=n\cup\{n\} $$ $$ 1:=S(0) $$ $$ 2:=S(1) $$ $$ 3:=S(2) $$ $$ n:=S^n(0) $$ $$ \mathbb N:=\{0,S(0),S^2(0),S^3(0),\ldots\} $$ $$ 0\in\mathbb N $$ $$ n\in\mathbb N\Rightarrow S(n)\in\mathbb N $$ $$ S(n)\neq 0 $$ $$ S(a)=S(b)\Rightarrow a=b $$ $$ \left(P(0)\land\forall n(P(n)\Rightarrow P(S(n)))\right) \Rightarrow \forall n\,P(n) $$ --- ## $S_2$ $$ a+0:=a $$ $$ a+S(b):=S(a+b) $$ $$ a\cdot 0:=0 $$ $$ a\cdot S(b):=a\cdot b+a $$ $$ a^0:=1 $$ $$ a^{S(b)}:=a^b\cdot a $$ $$ a+b=b+a $$ $$ (a+b)+c=a+(b+c) $$ $$ ab=ba $$ $$ (ab)c=a(bc) $$ $$ a(b+c)=ab+ac $$ $$ a+0=a $$ $$ a\cdot 1=a $$ --- ## $S_3$ $$ a\le b \iff \exists c\in\mathbb N:\ a+c=b $$ $$ a0}\, \exists N\, \forall m,n\ge N,\, |q_m-q_n|<\varepsilon \right\} $$ $$ (q_n)\sim_{\mathbb R}(r_n) \iff \forall\varepsilon\in\mathbb Q_{>0}\, \exists N\, \forall n\ge N,\, |q_n-r_n|<\varepsilon $$ $$ \mathbb R:=\mathcal C_{\mathbb Q}/\sim_{\mathbb R} $$ $$ \mathbb Q\hookrightarrow\mathbb R $$ $$ \forall A\subseteq\mathbb R,\ A\neq\varnothing,\ (\exists M\in\mathbb R)(\forall a\in A)(a\le M) \Rightarrow \exists s\in\mathbb R:\ s=\sup A $$ $$ \nexists q\in\mathbb Q:\ q^2=2 $$ $$ \exists x\in\mathbb R_{>0}:\ x^2=2 $$ $$ \sqrt2:=\sup\{q\in\mathbb Q:q^2<2\} $$ --- ## $S_7$ $$ i^2:=-1 $$ $$ \mathbb C:=\{a+bi:a,b\in\mathbb R\} $$ $$ (a+bi)+(c+di):=(a+c)+(b+d)i $$ $$ (a+bi)(c+di):=(ac-bd)+(ad+bc)i $$ $$ \overline{a+bi}:=a-bi $$ $$ |a+bi|:=\sqrt{a^2+b^2} $$ $$ z\overline z=|z|^2 $$ $$ \mathbb R\hookrightarrow\mathbb C $$ --- ## $S_8$ $$ x\in A \iff A(x)=1 $$ $$ A\subseteq B \iff \forall x\,(x\in A\Rightarrow x\in B) $$ $$ A=B \iff \forall x\,(x\in A\leftrightarrow x\in B) $$ $$ A\cup B:=\{x:x\in A\lor x\in B\} $$ $$ A\cap B:=\{x:x\in A\land x\in B\} $$ $$ A\setminus B:=\{x:x\in A\land x\notin B\} $$ $$ \mathcal P(A):=\{B:B\subseteq A\} $$ $$ A\times B:=\{(a,b):a\in A\land b\in B\} $$ $$ |A|=|B| \iff \exists f:A\to B: \left[ \forall x,y\in A\, (f(x)=f(y)\Rightarrow x=y) \right] \land \left[ \forall b\in B\,\exists a\in A:\ f(a)=b \right] $$ --- ## $S_9$ $$ R\subseteq A\times B $$ $$ xRy \iff (x,y)\in R $$ $$ f:A\to B \iff \forall a\in A\,\exists! b\in B:\ (a,b)\in f $$ $$ f\circ g:x\mapsto f(g(x)) $$ $$ \operatorname{id}_A:A\to A,\qquad x\mapsto x $$ $$ f\circ\operatorname{id}_A=f $$ $$ \operatorname{id}_B\circ f=f $$ $$ h\circ(g\circ f)=(h\circ g)\circ f $$ $$ f^{-1}\circ f=\operatorname{id}_A \land f\circ f^{-1}=\operatorname{id}_B $$ --- ## $S_{10}$ $$ x\sim x $$ $$ x\sim y\Rightarrow y\sim x $$ $$ (x\sim y\land y\sim z)\Rightarrow x\sim z $$ $$ [x]_\sim:=\{y:y\sim x\} $$ $$ A/{\sim}:=\{[x]_\sim:x\in A\} $$ $$ x\sim y \iff [x]_\sim=[y]_\sim $$ $$ \pi:A\to A/{\sim}, \qquad \pi(x):=[x]_\sim $$ --- ## $S_{11}$ $$ (G,\ast) $$ $$ \ast:G\times G\to G $$ $$ \forall a,b,c\in G:\ (a\ast b)\ast c=a\ast(b\ast c) $$ $$ \exists e\in G\,\forall a\in G:\ e\ast a=a\ast e=a $$ $$ \forall a\in G\,\exists a^{-1}\in G:\ a\ast a^{-1}=a^{-1}\ast a=e $$ $$ (G,\ast)\in\mathbf{Grp} $$ $$ H\le G \iff H\subseteq G \land e\in H \land \forall a,b\in H:\ ab^{-1}\in H $$ $$ N\trianglelefteq G \iff \forall g\in G:\ gNg^{-1}=N $$ $$ G/N:=\{gN:g\in G\} $$ --- ## $S_{12}$ $$ (R,+,\cdot) $$ $$ (R,+)\in\mathbf{Ab} $$ $$ (a\cdot b)\cdot c=a\cdot(b\cdot c) $$ $$ a(b+c)=ab+ac $$ $$ (a+b)c=ac+bc $$ $$ 1a=a1=a $$ $$ (R,+,\cdot)\in\mathbf{Ring} $$ $$ ab=ba $$ $$ a\neq0\Rightarrow \exists a^{-1}:\ aa^{-1}=1 $$ $$ (F,+,\cdot)\in\mathbf{Fld} $$ $$ \mathbb Q,\mathbb R,\mathbb C\in\mathbf{Fld} $$ --- ## $S_{13}$ $$ (V,F,+,\cdot) $$ $$ (V,+)\in\mathbf{Ab} $$ $$ \alpha(u+v)=\alpha u+\alpha v $$ $$ (\alpha+\beta)v=\alpha v+\beta v $$ $$ (\alpha\beta)v=\alpha(\beta v) $$ $$ 1v=v $$ $$ V\in\mathbf{Vect}_F $$ $$ \sum_{k=1}^{n}\alpha_kv_k=0 \Rightarrow \forall k,\ \alpha_k=0 $$ $$ \operatorname{span}(S) := \left\{ \sum_{k=1}^{n}\alpha_ks_k: n\in\mathbb N,\, \alpha_k\in F,\, s_k\in S \right\} $$ $$ B\subseteq V $$ $$ \operatorname{span}(B)=V $$ $$ \forall n\in\mathbb N,\ \forall b_1,\ldots,b_n\in B,\ \forall\alpha_1,\ldots,\alpha_n\in F: \left( \sum_{k=1}^{n}\alpha_kb_k=0 \Rightarrow \bigwedge_{k=1}^{n}\alpha_k=0 \right) $$ $$ \dim V:=|B| $$ --- ## $S_{14}$ $$ T:V\to W $$ $$ T(\alpha u+\beta v)=\alpha T(u)+\beta T(v) $$ $$ \ker T:=\{v\in V:T(v)=0\} $$ $$ \operatorname{im}T:=\{T(v):v\in V\} $$ $$ \dim V=\dim\ker T+\dim\operatorname{im}T $$ $$ A\in F^{m\times n} $$ $$ x\in F^n,\qquad Ax\in F^m $$ $$ A(v+w)=Av+Aw $$ $$ A(\alpha v)=\alpha Av $$ $$ ABx=A(Bx) $$ $$ I_nx=x $$ --- ## $S_{15}$ $$ Av=\lambda v $$ $$ v\neq0 $$ $$ (A-\lambda I)v=0 $$ $$ \det(A-\lambda I)=0 $$ $$ p_A(\lambda):=\det(A-\lambda I) $$ $$ \sigma(A):=\{\lambda:p_A(\lambda)=0\} $$ $$ A=PDP^{-1} $$ $$ A^n=PD^nP^{-1} $$ $$ e^{tA}:=\sum_{n=0}^{\infty}\frac{t^nA^n}{n!} $$ --- ## $S_{16}$ $$ (X,\tau) $$ $$ \varnothing,X\in\tau $$ $$ \{U_i\}_{i\in I}\subseteq\tau \Rightarrow \bigcup_{i\in I}U_i\in\tau $$ $$ U_1,\ldots,U_n\in\tau \Rightarrow \bigcap_{k=1}^{n}U_k\in\tau $$ $$ (X,\tau)\in\mathbf{Top} $$ $$ x\in\operatorname{int}(A) \iff \exists U\in\tau:\ x\in U\subseteq A $$ $$ x\in\overline A \iff \forall U\in\tau,\ x\in U\Rightarrow U\cap A\neq\varnothing $$ $$ \partial A:=\overline A\setminus\operatorname{int}(A) $$ --- ## $S_{17}$ $$ d:X\times X\to\mathbb R_{\ge0} $$ $$ d(x,y)=0\iff x=y $$ $$ d(x,y)=d(y,x) $$ $$ d(x,z)\le d(x,y)+d(y,z) $$ $$ B_r(x):=\{y:d(x,y)0:\ B_r(x)\subseteq U \right\} $$ $$ (X,d)\mapsto(X,\tau_d) $$ $$ x_n\to x \iff \forall\varepsilon>0\, \exists N\, \forall n\ge N:\ d(x_n,x)<\varepsilon $$ --- ## $S_{18}$ $$ f:X\to Y $$ $$ C(f) \iff \forall U\in\tau_Y:\ f^{-1}(U)\in\tau_X $$ $$ x_n\to x \Rightarrow f(x_n)\to f(x) $$ $$ f:X\to Y,\quad g:Y\to Z $$ $$ C(f)\land C(g) \Rightarrow C(g\circ f) $$ $$ K(X) \iff \forall\mathcal U\subseteq\tau_X: \left[ X\subseteq\bigcup\mathcal U \Rightarrow \exists n\in\mathbb N\, \exists U_1,\ldots,U_n\in\mathcal U: X\subseteq\bigcup_{k=1}^{n}U_k \right] $$ --- ## $S_{19}$ $$ f'(x) := \lim_{h\to0} \frac{f(x+h)-f(x)}{h} $$ $$ (f+g)'=f'+g' $$ $$ (fg)'=f'g+fg' $$ $$ (f\circ g)'(x) = f'(g(x))g'(x) $$ $$ \frac{d}{dx}x^n=nx^{n-1} $$ $$ \frac{d}{dx}e^x=e^x $$ $$ \frac{d}{dx}\sin x=\cos x $$ $$ \frac{d}{dx}\cos x=-\sin x $$ --- ## $S_{20}$ $$ \int_a^b f(x)\,dx := \lim_{\|\Pi\|\to0} \sum_{k=1}^{n} f(\xi_k)(x_k-x_{k-1}) $$ $$ F(x):=\int_a^x f(t)\,dt $$ $$ F'(x)=f(x) $$ $$ \int_a^b f'(x)\,dx=f(b)-f(a) $$ $$ \int_a^b(\alpha f+\beta g) = \alpha\int_a^b f+\beta\int_a^b g $$ $$ f\ge0\Rightarrow\int_a^b f\ge0 $$ --- ## $S_{21}$ $$ (X,\mathcal A,\mu) $$ $$ \varnothing\in\mathcal A $$ $$ A\in\mathcal A\Rightarrow X\setminus A\in\mathcal A $$ $$ A_1,A_2,\ldots\in\mathcal A \Rightarrow \bigcup_{n=1}^{\infty}A_n\in\mathcal A $$ $$ \mu:\mathcal A\to[0,\infty] $$ $$ \mu(\varnothing)=0 $$ $$ A_i\cap A_j=\varnothing \Rightarrow \mu\left(\bigcup_{n=1}^{\infty}A_n\right) = \sum_{n=1}^{\infty}\mu(A_n) $$ $$ \int_X f\,d\mu $$ --- ## $S_{22}$ $$ (\Omega,\mathcal F,\mathbb P) $$ $$ \mathbb P(\Omega)=1 $$ $$ A\cap B=\varnothing \Rightarrow \mathbb P(A\cup B)=\mathbb P(A)+\mathbb P(B) $$ $$ \mathbb P(A^c)=1-\mathbb P(A) $$ $$ \mathbb P(A\mid B) := \frac{\mathbb P(A\cap B)}{\mathbb P(B)} $$ $$ \mathbb P(A\cap B) = \mathbb P(A\mid B)\mathbb P(B) $$ $$ \mathbb P(A\mid B) = \frac{\mathbb P(B\mid A)\mathbb P(A)}{\mathbb P(B)} $$ $$ \mathbb E[X] := \int_\Omega X\,d\mathbb P $$ $$ \operatorname{Var}(X) := \mathbb E[(X-\mathbb E[X])^2] $$ --- ## $S_{23}$ $$ P,Q,R\in\mathcal L $$ $$ \neg P $$ $$ P\land Q $$ $$ P\lor Q $$ $$ P\Rightarrow Q $$ $$ P\leftrightarrow Q $$ $$ P\Rightarrow Q \equiv \neg P\lor Q $$ $$ \neg(P\land Q) \equiv \neg P\lor\neg Q $$ $$ \neg(P\lor Q) \equiv \neg P\land\neg Q $$ $$ (P\Rightarrow Q)\land P\Rightarrow Q $$ $$ (P\Rightarrow Q)\land(Q\Rightarrow R)\Rightarrow(P\Rightarrow R) $$ --- ## $S_{24}$ $$ \forall x\,P(x) $$ $$ \exists x\,P(x) $$ $$ \neg\forall x\,P(x) \equiv \exists x\,\neg P(x) $$ $$ \neg\exists x\,P(x) \equiv \forall x\,\neg P(x) $$ $$ \exists!x\,P(x) \iff \exists x\left(P(x)\land\forall y(P(y)\Rightarrow y=x)\right) $$ $$ \Gamma\vdash\varphi $$ $$ \Gamma\models\varphi $$ $$ \Gamma\vdash\varphi \Rightarrow \Gamma\models\varphi $$ --- ## $S_{25}$ $$ \mathcal M\models\varphi $$ $$ \operatorname{Th}(\mathcal M) := \{\varphi:\mathcal M\models\varphi\} $$ $$ \operatorname{Mod}(T) := \{\mathcal M:\mathcal M\models T\} $$ $$ T\models\varphi \iff \forall\mathcal M\in\operatorname{Mod}(T):\mathcal M\models\varphi $$ $$ T\vdash\varphi \Rightarrow T\models\varphi $$ $$ T\models\varphi \Rightarrow T\vdash\varphi $$ $$ \operatorname{Con}(T) \iff \nexists\varphi:\ T\vdash\varphi\land T\vdash\neg\varphi $$ --- ## $S_{26}$ $$ \mathcal C = \bigl(\operatorname{Ob}(\mathcal C),\operatorname{Hom}_{\mathcal C},\circ,\operatorname{id}\bigr) $$ $$ f:A\to B $$ $$ g:B\to C $$ $$ g\circ f:A\to C $$ $$ h\circ(g\circ f)=(h\circ g)\circ f $$ $$ f\circ\operatorname{id}_A=f $$ $$ \operatorname{id}_B\circ f=f $$ $$ F:\mathcal C\to\mathcal D $$ $$ F(g\circ f)=F(g)\circ F(f) $$ $$ F(\operatorname{id}_A)=\operatorname{id}_{F(A)} $$ --- ## $S_{27}$ $$ \eta:F\Rightarrow G $$ $$ \eta_A:F(A)\to G(A) $$ $$ G(f)\circ\eta_A = \eta_B\circ F(f) $$ $$ \begin{CD} F(A) @>{F(f)}>> F(B)\\ @V{\eta_A}VV @VV{\eta_B}V\\ G(A) @>>{G(f)}> G(B) \end{CD} $$ $$ \operatorname{Nat}(F,G) $$ $$ \mathcal C \xrightarrow{F} \mathcal D \xrightarrow{G} \mathcal E $$ $$ G\circ F:\mathcal C\to\mathcal E $$ --- ## $S_{28}$ $$ G=(V,E) $$ $$ E\subseteq V\times V $$ $$ \deg(v) := |\{u:(u,v)\in E\lor(v,u)\in E\}| $$ $$ P_n=(v_0,v_1,\ldots,v_n) $$ $$ \forall k0 $$ $$ \ell(R_\ast^n([\Phi])) $$ $$ \lim_{n\to\infty} \frac{\ell(R_\ast^n([\Phi]))}{n} $$ --- ## $S_{38}$ $$ d:\mathfrak F\times\mathfrak F\to\mathbb R_{\ge0} $$ $$ d([\Phi],[\Psi])=0 \iff [\Phi]=[\Psi] $$ $$ d([\Phi],[\Psi])=d([\Psi],[\Phi]) $$ $$ d([\Phi],[\Omega]) \le d([\Phi],[\Psi])+d([\Psi],[\Omega]) $$ $$ d(R_\ast^n([\Phi]),R_\ast^n([\Psi])) $$ $$ d(R_\ast([\Phi]),R_\ast([\Psi])) \le \lambda d([\Phi],[\Psi]), \quad 0\le\lambda<1 $$ $$ \exists![\Phi^\ast]: R_\ast([\Phi^\ast])=[\Phi^\ast] $$ --- ## $S_{39}$ $$ \mathcal A := \{A_1,\ldots,A_k\} $$ $$ R^{(A_i)}:\mathfrak F\to\mathfrak F $$ $$ S_n^{(i)} := (R^{(A_i)})^n([\Phi_0]) $$ $$ D_n^{ij} := d(S_n^{(i)},S_n^{(j)}) $$ $$ \lim_{n\to\infty}D_n^{ij}=0 $$ $$ \forall i,j: \lim_{n\to\infty}D_n^{ij}=0 $$ $$ S_\infty^{(1)} = S_\infty^{(2)} = \cdots = S_\infty^{(k)} $$ --- ## $S_{40}$ $$ \mathcal O := \{O_\alpha\}_{\alpha\in A} $$ $$ O_\alpha:\mathfrak F^{n_\alpha}\to\mathfrak F $$ $$ \operatorname{Cl}_{\mathcal O}(S) := \bigcap \left\{ T\subseteq\mathfrak F: S\subseteq T \land \forall O_\alpha\, O_\alpha(T^{n_\alpha})\subseteq T \right\} $$ $$ S_0\subseteq \operatorname{Cl}_{\mathcal O}(S_0) $$ $$ \operatorname{Cl}_{\mathcal O} ( \operatorname{Cl}_{\mathcal O}(S_0) ) = \operatorname{Cl}_{\mathcal O}(S_0) $$ --- ## $S_{41}$ $$ \mathcal A_0 := \{0,1,+,\cdot,=,<\} $$ $$ \mathcal A_{n+1} := \operatorname{Cl}_{\mathcal O}(\mathcal A_n) $$ $$ \mathcal A_0 \subseteq \mathcal A_1 \subseteq \mathcal A_2 \subseteq \cdots $$ $$ \mathcal A_\omega := \bigcup_{n<\omega}\mathcal A_n $$ $$ \operatorname{Cl}_{\mathcal O}(\mathcal A_\omega) = \mathcal A_\omega $$ --- ## $S_{42}$ $$ T_0:=\mathcal A_\omega $$ $$ T_{\alpha+1} := \operatorname{Cl}_{\mathcal O}(T_\alpha) $$ $$ T_\lambda := \bigcup_{\beta<\lambda}T_\beta $$ $$ \alpha<\beta \Rightarrow T_\alpha\subseteq T_\beta $$ $$ \exists\gamma: T_{\gamma+1}=T_\gamma $$ $$ T_\gamma = \operatorname{Cl}_{\mathcal O}(T_\gamma) $$ --- ## $S_{43}$ $$ \Phi\mapsto[\Phi] $$ $$ [\Phi]\mapsto R_\ast([\Phi]) $$ $$ R_\ast \mapsto [R_\ast] $$ $$ [R_\ast] \mapsto \mathcal O(R_\ast) $$ $$ \mathcal O(R_\ast) \mapsto \operatorname{Fix}(\mathcal O) $$ $$ \operatorname{Fix}(\mathcal O) \mapsto \operatorname{Fix} \bigl( \operatorname{Fix}(\mathcal O) \bigr) $$ --- ## $S_{44}$ $$ X_0:=\varnothing $$ $$ X_{n+1} := \mathcal P(X_n) $$ $$ X_0\subseteq X_1\subseteq X_2\subseteq\cdots $$ $$ V_\omega:=\bigcup_{n<\omega}X_n $$ $$ V_{\alpha+1}:=\mathcal P(V_\alpha) $$ $$ V_\lambda:=\bigcup_{\beta<\lambda}V_\beta $$ $$ V:=\bigcup_{\alpha\in\operatorname{Ord}}V_\alpha $$ --- ## $S_{45}$ $$ 0:=\varnothing $$ $$ \alpha+1:=\alpha\cup\{\alpha\} $$ $$ \lambda:=\bigcup_{\beta<\lambda}\beta $$ $$ \alpha<\beta \iff \alpha\in\beta $$ $$ \operatorname{Ord} := \left\{ \alpha: \left[ \forall x\in\alpha:\ x\subseteq\alpha \right] \land \left[ \forall x,y\in\alpha: x\in y\lor x=y\lor y\in x \right] \land \left[ \forall X\subseteq\alpha: X\neq\varnothing \Rightarrow \exists m\in X\, \forall x\in X: m=x\lor m\in x \right] \right\} $$ $$ \omega:=\{0,1,2,\ldots\} $$ $$ \omega<\omega+1<\omega+2<\cdots<\omega\cdot2<\omega^2<\omega^\omega<\cdots $$ --- ## $S_{46}$ $$ |A|<|\mathcal P(A)| $$ $$ \nexists f:A\to\mathcal P(A): \forall B\subseteq A\,\exists a\in A:\ f(a)=B $$ $$ D_f:=\{a\in A:a\notin f(a)\} $$ $$ \forall a\in A:\ f(a)\neq D_f $$ $$ \aleph_0:=|\mathbb N| $$ $$ |\mathbb N|<|\mathcal P(\mathbb N)| $$ $$ |\mathbb R|=|\mathcal P(\mathbb N)| $$ $$ \aleph_0<|\mathbb R| $$ --- ## $S_{47}$ $$ \lambda x.M $$ $$ (\lambda x.M)\,N \to_\beta M[x:=N] $$ $$ I:=\lambda x.x $$ $$ K:=\lambda x.\lambda y.x $$ $$ S:=\lambda x.\lambda y.\lambda z.\,xz(yz) $$ $$ \Omega := (\lambda x.xx)(\lambda x.xx) $$ $$ \Omega\to_\beta\Omega $$ $$ YF=F(YF) $$ --- ## $S_{48}$ $$ \Sigma^\ast $$ $$ M=(Q,\Sigma,\Gamma,\delta,q_0,q_h) $$ $$ \delta:Q\times\Gamma\to Q\times\Gamma\times\{L,R\} $$ $$ M(x)\downarrow $$ $$ M(x)\uparrow $$ $$ \varphi_e(x) $$ $$ K:=\{e:\varphi_e(e)\downarrow\} $$ $$ K\notin\operatorname{Dec} $$ --- ## $S_{49}$ $$ \ulcorner\varphi\urcorner\in\mathbb N $$ $$ \operatorname{Pr}_T(x) $$ $$ G_T \leftrightarrow \neg\operatorname{Pr}_T(\ulcorner G_T\urcorner) $$ $$ T\nvdash G_T $$ $$ T\nvdash\neg G_T $$ $$ \operatorname{Con}(T) \notin \operatorname{Thm}(T) $$ --- ## $S_{50}$ $$ \mathcal F \to \mathfrak F \to \mathcal G_{\mathcal R} \to \operatorname{Path}(\mathcal G_{\mathcal R}) \to \operatorname{Path} \bigl( \operatorname{Path}(\mathcal G_{\mathcal R}) \bigr) $$ $$ X \mapsto F(X) \mapsto [F(X)] \mapsto R_\ast([F(X)]) $$ $$ R_\ast \mapsto R_\ast^2 \mapsto R_\ast^3 \mapsto \cdots $$ $$ \{R_\ast^n:n\in\mathbb N\} $$ $$ \operatorname{Cl}_{\circ}(\{R_\ast\}) $$ $$ \langle R_\ast\rangle $$ --- ## $S_{51}$ $$ \mathfrak F_0:=\{[\varnothing]\} $$ $$ \mathfrak F_{n+1} := \operatorname{Cl}_{\mathcal O}(\mathfrak F_n) $$ $$ \mathfrak F_\omega := \bigcup_{n<\omega}\mathfrak F_n $$ $$ R_\ast(\mathfrak F_\omega) \subseteq \mathfrak F_\omega $$ $$ (\mathfrak F_\omega,R_\ast) $$ $$ \operatorname{Orb}_{R_\ast}(\mathfrak F_\omega) $$ --- ## $S_{52}$ $$ \Phi_0:=\varnothing $$ $$ \Phi_1:=\{0\} $$ $$ \Phi_2:=\{0,S,+,\cdot\} $$ $$ \Phi_3:=\{\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C\} $$ $$ \Phi_4:=\{\in,\subseteq,\sim,\to,\circ\} $$ $$ \Phi_5:=\{\mathbf{Grp},\mathbf{Ring},\mathbf{Fld},\mathbf{Vect},\mathbf{Top}\} $$ $$ \Phi_6:=\{\vdash,\models,\operatorname{Mod},\operatorname{Th}\} $$ $$ \Phi_7:=\{\mathcal C,F,\eta\} $$ $$ \Phi_8:=\{R,R_\ast,\mathfrak F,\mathcal G_{\mathcal R}\} $$ $$ \Phi_9:=\operatorname{Cl}_{\mathcal O}(\Phi_8) $$ --- ## $S_{53}$ $$ \Phi_{n+1}:=\mathcal T(\Phi_n) $$ $$ \mathcal T: \Phi_n \mapsto \left( \Phi_n, \operatorname{Rel}(\Phi_n), \operatorname{Op}(\Phi_n), \operatorname{Quot}(\Phi_n), \operatorname{Hom}(\Phi_n), \operatorname{Fix}(\Phi_n) \right) $$ $$ \Phi_n\subseteq\Phi_{n+1} $$ $$ \Phi_\omega:=\bigcup_{n<\omega}\Phi_n $$ $$ \mathcal T(\Phi_\omega)=\Phi_\omega $$ --- ## $S_{54}$ $$ \mathcal U_0:=\varnothing $$ $$ \mathcal U_{\alpha+1} := \mathcal U_\alpha \cup \operatorname{Rel}(\mathcal U_\alpha) \cup \operatorname{Op}(\mathcal U_\alpha) \cup \operatorname{Quot}(\mathcal U_\alpha) \cup \operatorname{Hom}(\mathcal U_\alpha) \cup \operatorname{Fix}(\mathcal U_\alpha) $$ $$ \mathcal U_\lambda := \bigcup_{\beta<\lambda}\mathcal U_\beta $$ $$ \exists\gamma: \mathcal U_{\gamma+1}=\mathcal U_\gamma $$ $$ \mathcal U_\gamma = \operatorname{Cl} ( \mathcal U_\gamma ) $$ --- ## $S_{55}$ $$ \mathcal U_\gamma \xrightarrow{\operatorname{Enc}} \mathcal F $$ $$ \mathcal F \xrightarrow{\operatorname{Dec}} \mathcal U_\gamma $$ $$ \operatorname{Dec}\circ\operatorname{Enc} = \operatorname{id}_{\mathcal U_\gamma} $$ $$ \operatorname{Enc}\circ\operatorname{Dec} \sim \operatorname{id}_{\mathcal F} $$ $$ \mathcal U_\gamma \sim \mathfrak F $$ $$ \mathcal U_\gamma \cong \operatorname{Cl}_{\mathcal O}(\mathfrak F) $$ --- ## $S_{56}$ $$ \mathcal U \xrightarrow{R} \mathcal U $$ $$ R(\mathcal U)\subseteq\mathcal U $$ $$ R^n(\mathcal U)\subseteq\mathcal U $$ $$ R^\omega(\mathcal U) := \bigcup_{n<\omega}R^n(\mathcal U) $$ $$ R(R^\omega(\mathcal U)) \subseteq R^\omega(\mathcal U) $$ $$ R^\omega(\mathcal U) = \operatorname{Cl}_R(\mathcal U) $$ --- ## $S_{57}$ $$ \mathcal U \to \mathcal P(\mathcal U) \to \mathcal P(\mathcal P(\mathcal U)) \to \cdots $$ $$ \mathcal U^{(0)}:=\mathcal U $$ $$ \mathcal U^{(n+1)} := \mathcal P(\mathcal U^{(n)}) $$ $$ \mathcal U^{(\omega)} := \bigcup_{n<\omega}\mathcal U^{(n)} $$ $$ \mathcal U^{(\omega)} \subseteq \mathcal U^{(\omega+1)} $$ --- ## $S_{58}$ $$ R_0:=R $$ $$ R_{n+1} := \operatorname{Op}(R_n) $$ $$ R_0 \mapsto R_1 \mapsto R_2 \mapsto \cdots $$ $$ R_\omega := \bigcup_{n<\omega}R_n $$ $$ \operatorname{Fix}(R_\omega) $$ $$ R_\omega(R_\omega) $$ --- ## $S_{59}$ $$ x \mapsto R(x) \mapsto R^2(x) \mapsto \cdots $$ $$ R \mapsto \mathcal R(R) \mapsto \mathcal R^2(R) \mapsto \cdots $$ $$ (x,R) \mapsto (R(x),\mathcal R(R)) $$ $$ \Xi(x,R) := (R(x),\mathcal R(R)) $$ $$ \Xi^n(x,R) $$ $$ \operatorname{Fix}(\Xi) $$ --- ## $S_{60}$ $$ \boxed{ \varnothing \to \mathbb N \to \mathbb Z \to \mathbb Q \to \mathbb R \to \mathbb C } $$ $$ \boxed{ \in \to R \to f \to \sim \to /{\sim} \to \operatorname{Hom} \to \operatorname{Nat} } $$ $$ \boxed{ \operatorname{Obj} \to \operatorname{Rel} \to \operatorname{Op} \to \operatorname{Str} \to \operatorname{Str}(\operatorname{Str}) } $$ $$ \boxed{ \Phi \to [\Phi] \to R_\ast([\Phi]) \to \operatorname{Orb}_{R_\ast}([\Phi]) \to \mathcal G_R } $$ $$ \boxed{ \mathcal F \xrightarrow{R} \mathcal F \xrightarrow{R} \mathcal F \xrightarrow{R} \cdots } $$ $$ \boxed{ SREG_{001} : S_0 \to S_1 \to \cdots \to S_{60} } $$ --- # SREG-002 $$ S_{60}\longrightarrow S_{61} $$ --- ## $S_{61}$ $$ \mathsf V:=\{x_0,x_1,x_2,\ldots\} $$ $$ \mathsf C:=\{c_i\}_{i\in I} $$ $$ \mathsf F_n:=\{f_i^{(n)}\}_{i\in I_n} $$ $$ \mathsf P_n:=\{P_i^{(n)}\}_{i\in J_n} $$ $$ \mathsf T_0:=\mathsf V\cup\mathsf C $$ $$ t_1,\ldots,t_n\in\mathsf T \land f^{(n)}\in\mathsf F_n \Rightarrow f^{(n)}(t_1,\ldots,t_n)\in\mathsf T $$ $$ t_1,\ldots,t_n\in\mathsf T \land P^{(n)}\in\mathsf P_n \Rightarrow P^{(n)}(t_1,\ldots,t_n)\in\mathsf{At} $$ $$ \mathsf{Fm} := \operatorname{Cl}_{\neg,\land,\lor,\to,\forall,\exists}(\mathsf{At}) $$ --- ## $S_{62}$ $$ \Gamma:=\{\varphi_1,\ldots,\varphi_n\} $$ $$ \Gamma\vdash\varphi $$ $$ \frac{\Gamma\vdash\varphi\to\psi \qquad \Gamma\vdash\varphi} {\Gamma\vdash\psi} $$ $$ \frac{\Gamma,\varphi\vdash\psi} {\Gamma\vdash\varphi\to\psi} $$ $$ \frac{\Gamma\vdash\varphi \qquad \Gamma\vdash\psi} {\Gamma\vdash\varphi\land\psi} $$ $$ \frac{\Gamma\vdash\varphi\land\psi} {\Gamma\vdash\varphi} $$ $$ \frac{\Gamma\vdash\varphi\land\psi} {\Gamma\vdash\psi} $$ --- ## $S_{63}$ $$ \Gamma\Rightarrow\Delta $$ $$ \frac{\Gamma\Rightarrow\Delta,\varphi \qquad \varphi,\Pi\Rightarrow\Lambda} {\Gamma,\Pi\Rightarrow\Delta,\Lambda} $$ $$ \frac{\Gamma,\varphi,\psi\Rightarrow\Delta} {\Gamma,\varphi\land\psi\Rightarrow\Delta} $$ $$ \frac{\Gamma\Rightarrow\Delta,\varphi \qquad \Gamma\Rightarrow\Delta,\psi} {\Gamma\Rightarrow\Delta,\varphi\land\psi} $$ $$ \frac{\Gamma\Rightarrow\Delta,\varphi} {\Gamma,\neg\varphi\Rightarrow\Delta} $$ $$ \frac{\Gamma,\varphi\Rightarrow\Delta} {\Gamma\Rightarrow\Delta,\neg\varphi} $$ --- ## $S_{64}$ $$ \mathcal D\rightsquigarrow\mathcal D' $$ $$ \operatorname{Cut}(\mathcal D)>0 \Rightarrow \exists\mathcal D': \operatorname{Cut}(\mathcal D')< \operatorname{Cut}(\mathcal D) $$ $$ \vdash_{\mathsf{cut}}\Gamma\Rightarrow\Delta \Rightarrow \vdash_{\mathsf{cut}=0}\Gamma\Rightarrow\Delta $$ $$ \mathcal D \to \mathcal D_1 \to \mathcal D_2 \to \cdots \to \mathcal D^\ast $$ $$ \mathcal D^\ast = \operatorname{NF}(\mathcal D) $$ --- ## $S_{65}$ $$ \varphi\leftrightarrow A $$ $$ \varphi\to\psi \leftrightarrow A\to B $$ $$ \varphi\land\psi \leftrightarrow A\times B $$ $$ \varphi\lor\psi \leftrightarrow A+B $$ $$ \bot\leftrightarrow 0 $$ $$ \top\leftrightarrow 1 $$ $$ \Gamma\vdash\varphi \leftrightarrow \Gamma\vdash t:A $$ $$ \mathsf{Proof}(\varphi) \cong \mathsf{Term}(A) $$ --- ## $S_{66}$ $$ A,B::= \iota \mid A\to B \mid A\times B \mid A+B $$ $$ \Gamma,x:A\vdash x:A $$ $$ \frac{\Gamma,x:A\vdash t:B} {\Gamma\vdash\lambda x.t:A\to B} $$ $$ \frac{\Gamma\vdash f:A\to B \qquad \Gamma\vdash a:A} {\Gamma\vdash f\,a:B} $$ $$ (\lambda x.t)a \to_\beta t[x:=a] $$ $$ \lambda x.f\,x \to_\eta f $$ --- ## $S_{67}$ $$ x:A\vdash B(x):\mathcal U $$ $$ \Pi_{x:A}B(x) $$ $$ \Sigma_{x:A}B(x) $$ $$ \frac{\Gamma,x:A\vdash b:B(x)} {\Gamma\vdash\lambda x.b:\Pi_{x:A}B(x)} $$ $$ f:\Pi_{x:A}B(x) \land a:A \Rightarrow f(a):B(a) $$ $$ (a,b):\Sigma_{x:A}B(x) \iff a:A \land b:B(a) $$ --- ## $S_{68}$ $$ a,b:A $$ $$ \operatorname{Id}_A(a,b):\mathcal U $$ $$ \operatorname{refl}_a: \operatorname{Id}_A(a,a) $$ $$ p:\operatorname{Id}_A(a,b) $$ $$ p^{-1}:\operatorname{Id}_A(b,a) $$ $$ p:\operatorname{Id}_A(a,b) \land q:\operatorname{Id}_A(b,c) \Rightarrow p\cdot q:\operatorname{Id}_A(a,c) $$ $$ \operatorname{refl}_a\cdot p \simeq p \simeq p\cdot\operatorname{refl}_b $$ --- ## $S_{69}$ $$ \mathcal U_0:\mathcal U_1 $$ $$ \mathcal U_1:\mathcal U_2 $$ $$ \cdots $$ $$ A:\mathcal U_i \Rightarrow A:\mathcal U_{i+1} $$ $$ A:\mathcal U_i \land x:A $$ $$ \Pi_{x:A}B(x):\mathcal U_i $$ $$ \Sigma_{x:A}B(x):\mathcal U_i $$ $$ \operatorname{Id}_A(a,b):\mathcal U_i $$ --- ## $S_{70}$ $$ \mathbb N:\mathcal U $$ $$ 0:\mathbb N $$ $$ S:\mathbb N\to\mathbb N $$ $$ C:\mathbb N\to\mathcal U $$ $$ c_0:C(0) $$ $$ c_S: \Pi_{n:\mathbb N} \left( C(n)\to C(S(n)) \right) $$ $$ \operatorname{rec}_{\mathbb N}(C,c_0,c_S): \Pi_{n:\mathbb N}C(n) $$ --- ## $S_{71}$ $$ F:\mathcal C\to\mathcal C $$ $$ \alpha:F(A)\to A $$ $$ (A,\alpha)\in\operatorname{Alg}(F) $$ $$ \beta:B\to F(B) $$ $$ (B,\beta)\in\operatorname{Coalg}(F) $$ $$ \iota:F(\mu F)\to\mu F $$ $$ \omega:\nu F\to F(\nu F) $$ --- ## $S_{72}$ $$ h:(A,\alpha)\to(B,\beta) $$ $$ h\circ\alpha = \beta\circ F(h) $$ $$ (A,\alpha)\cong(B,\beta) \iff \exists h,k: h\circ k=\operatorname{id}_B \land k\circ h=\operatorname{id}_A $$ $$ \operatorname{Alg}(F) $$ $$ \operatorname{Coalg}(F) $$ --- ## $S_{73}$ $$ (\mu F,\iota) = \operatorname{Init}(\operatorname{Alg}(F)) $$ $$ \forall(A,\alpha)\in\operatorname{Alg}(F): \exists!\operatorname{fold}_\alpha:\mu F\to A $$ $$ \operatorname{fold}_\alpha\circ\iota = \alpha\circ F(\operatorname{fold}_\alpha) $$ $$ (\nu F,\omega) = \operatorname{Final}(\operatorname{Coalg}(F)) $$ $$ \forall(B,\beta)\in\operatorname{Coalg}(F): \exists!\operatorname{unfold}_\beta:B\to\nu F $$ $$ \omega\circ\operatorname{unfold}_\beta = F(\operatorname{unfold}_\beta)\circ\beta $$ --- ## $S_{74}$ $$ F:\mathcal C\rightleftarrows\mathcal D:G $$ $$ F\dashv G $$ $$ \operatorname{Hom}_{\mathcal D}(F(A),B) \cong \operatorname{Hom}_{\mathcal C}(A,G(B)) $$ $$ \eta:\operatorname{Id}_{\mathcal C}\Rightarrow GF $$ $$ \varepsilon:FG\Rightarrow\operatorname{Id}_{\mathcal D} $$ $$ G\varepsilon\circ\eta G = \operatorname{id}_G $$ $$ \varepsilon F\circ F\eta = \operatorname{id}_F $$ --- ## $S_{75}$ $$ T:=GF $$ $$ T:\mathcal C\to\mathcal C $$ $$ \eta:\operatorname{Id}\Rightarrow T $$ $$ \mu:T^2\Rightarrow T $$ $$ \mu\circ T\eta = \mu\circ\eta T = \operatorname{id}_T $$ $$ \mu\circ T\mu = \mu\circ\mu T $$ $$ (T,\eta,\mu) $$ $$ W:=FG $$ $$ \varepsilon:W\Rightarrow\operatorname{Id} $$ $$ \delta:W\Rightarrow W^2 $$ --- ## $S_{76}$ $$ h_A:=\operatorname{Hom}_{\mathcal C}(-,A) $$ $$ F:\mathcal C^{op}\to\mathbf{Set} $$ $$ \operatorname{Nat}(h_A,F) \cong F(A) $$ $$ \eta \mapsto \eta_A(\operatorname{id}_A) $$ $$ x\in F(A) \mapsto \left[ f:X\to A \mapsto F(f)(x) \right] $$ $$ A\cong B \iff h_A\cong h_B $$ --- ## $S_{77}$ $$ D:J\to\mathcal C $$ $$ \operatorname{Lim}D $$ $$ \operatorname{Hom}_{\mathcal C}(X,\operatorname{Lim}D) \cong \operatorname{Nat}(\Delta X,D) $$ $$ \operatorname{Colim}D $$ $$ \operatorname{Hom}_{\mathcal C}(\operatorname{Colim}D,X) \cong \operatorname{Nat}(D,\Delta X) $$ --- ## $S_{78}$ $$ (X,\tau) $$ $$ \mathcal O:\tau^{op}\to\mathbf{Set} $$ $$ V\subseteq U \Rightarrow \rho_{UV}:\mathcal O(U)\to\mathcal O(V) $$ $$ \rho_{UU}=\operatorname{id} $$ $$ W\subseteq V\subseteq U \Rightarrow \rho_{UW} = \rho_{VW}\circ\rho_{UV} $$ $$ U=\bigcup_iU_i $$ $$ s_i\in\mathcal O(U_i) $$ $$ \rho_{U_i,U_i\cap U_j}(s_i) = \rho_{U_j,U_i\cap U_j}(s_j) $$ $$ \Rightarrow \exists!s\in\mathcal O(U): \rho_{U,U_i}(s)=s_i $$ --- ## $S_{79}$ $$ \mathcal E $$ $$ 1\in\operatorname{Ob}(\mathcal E) $$ $$ A\times B $$ $$ B^A $$ $$ \Omega\in\operatorname{Ob}(\mathcal E) $$ $$ m:A\hookrightarrow B $$ $$ \chi_m:B\to\Omega $$ $$ \operatorname{Sub}_{\mathcal E}(B) \cong \operatorname{Hom}_{\mathcal E}(B,\Omega) $$ $$ \mathcal E\models\varphi $$ --- ## $S_{80}$ $$ [n]:=\{0<1<\cdots{l}>> L\\ @V{}VV @VV{m}V\\ D @>>> G \end{CD} $$ $$ \begin{CD} K @>{r}>> R\\ @V{}VV @VV{}V\\ D @>>> H \end{CD} $$ --- ## $S_{111}$ $$ \mathcal O(n) $$ $$ \Sigma_n\curvearrowright\mathcal O(n) $$ $$ \gamma: \mathcal O(n) \times \mathcal O(k_1) \times\cdots\times \mathcal O(k_n) \to \mathcal O(k_1+\cdots+k_n) $$ $$ \gamma(\theta;\operatorname{id},\ldots,\operatorname{id}) = \theta $$ $$ \gamma(\operatorname{id};\theta)=\theta $$ $$ \gamma\circ\gamma = \gamma\circ(\operatorname{id}\times\gamma^{\times n}) $$ --- ## $S_{112}$ $$ x,y\in\mathcal C_0 $$ $$ f,g:x\to y $$ $$ \alpha:f\Rightarrow g $$ $$ \alpha\circ_v\beta $$ $$ \alpha\circ_h\gamma $$ $$ (\alpha\circ_h\gamma)\circ_v(\beta\circ_h\delta) = (\alpha\circ_v\beta)\circ_h(\gamma\circ_v\delta) $$ --- ## $S_{113}$ $$ \mathcal C_{(0)} \leftleftarrows \mathcal C_{(1)} \leftleftarrows \mathcal C_{(2)} \leftleftarrows \cdots $$ $$ \operatorname{Map}_{\mathcal C}(x,y)\in\mathcal S $$ $$ \operatorname{Map}(y,z) \times \operatorname{Map}(x,y) \to \operatorname{Map}(x,z) $$ $$ \mathcal C\simeq\mathcal D $$ --- ## $S_{114}$ $$ \operatorname{Equiv}(A,B) $$ $$ (A=_\mathcal U B) \to \operatorname{Equiv}(A,B) $$ $$ \operatorname{ua}: \operatorname{Equiv}(A,B) \to (A=_\mathcal U B) $$ $$ \operatorname{idtoequiv}\circ\operatorname{ua} \simeq \operatorname{id} $$ $$ \operatorname{ua}\circ\operatorname{idtoequiv} \simeq \operatorname{id} $$ $$ (A=_\mathcal U B) \simeq \operatorname{Equiv}(A,B) $$ --- ## $S_{115}$ $$ \mathcal E\simeq\operatorname{Sh}(X) $$ $$ \Gamma:\mathcal E\to\mathbf{Set} $$ $$ \Delta:\mathbf{Set}\to\mathcal E $$ $$ \Delta\dashv\Gamma $$ $$ \mathcal E\models \forall x:A\,\exists y:B(x)\,\varphi(x,y) $$ $$ \operatorname{Type}_{\mathcal E} $$ $$ \operatorname{Prop}_{\mathcal E} $$ --- ## $S_{116}$ $$ \mathcal L_0 $$ $$ \operatorname{Syn}(\mathcal L_0)=\mathcal L_1 $$ $$ \operatorname{Sem}(\mathcal L_1)=\mathcal M_1 $$ $$ \operatorname{Enc}(\mathcal M_1)=\mathcal L_2 $$ $$ \mathcal L_0 \to \mathcal L_1 \to \mathcal M_1 \to \mathcal L_2 \to \cdots $$ $$ \mathcal L_{n+1} = \operatorname{Enc} \circ \operatorname{Sem} \circ \operatorname{Syn} (\mathcal L_n) $$ --- ## $S_{117}$ $$ Q:\mathcal F\to\mathfrak F $$ $$ E:\mathfrak F\to\mathcal F $$ $$ Q\circ E=\operatorname{id}_{\mathfrak F} $$ $$ R:=E\circ Q $$ $$ R^2 = E\circ Q\circ E\circ Q = E\circ Q = R $$ $$ \operatorname{im}R = \operatorname{Fix}(R) $$ --- ## $S_{118}$ $$ R_i:\mathfrak F\to\mathfrak F $$ $$ d(R_i(x),R_i(y)) \le \lambda_i d(x,y) $$ $$ 0\le\lambda_i<1 $$ $$ x_i^\ast=R_i(x_i^\ast) $$ $$ d(x_i^\ast,x_j^\ast) \le \frac{ \sup_x d(R_i(x),R_j(x)) }{ 1-\max(\lambda_i,\lambda_j) } $$ $$ \sup_xd(R_i(x),R_j(x)) \to0 \Rightarrow d(x_i^\ast,x_j^\ast)\to0 $$ --- ## $S_{119}$ $$ \mathbf S := (\mathfrak F,\mathcal R,\mathcal O,\equiv,\preceq) $$ $$ \mathcal R \subseteq \mathfrak F^{\mathfrak F} $$ $$ \mathcal O \subseteq \bigcup_{n\ge0} \mathfrak F^{\mathfrak F^n} $$ $$ \mathcal G(\mathbf S) := (V_{\mathbf S},E_{\mathbf S}) $$ $$ V_{\mathbf S} = \mathfrak F \cup \mathcal R \cup \mathcal O $$ $$ E_{\mathbf S} = E_{\mathrm{app}} \cup E_{\mathrm{comp}} \cup E_{\equiv} \cup E_{\preceq} $$ --- ## $S_{120}$ $$ S_{60} \to S_{61} \to \cdots \to S_{120} $$ $$ \mathsf{Obj} \to \mathsf{Rel} \to \mathsf{Op} \to \mathsf{Proof} \to \mathsf{Type} \to \mathsf{Cat} \to \mathsf{Top} \to \mathsf{Homotopy} \to \mathsf{Process} \to \mathsf{Meta} $$ $$ \mathcal F \to \mathfrak F \to \mathcal G_{\mathcal R} \to \mathbf S \to \mathcal G(\mathbf S) $$ $$ R:\mathcal F\to\mathcal F $$ $$ R_\ast:\mathfrak F\to\mathfrak F $$ $$ \mathcal R: R_\ast\mapsto\mathcal R(R_\ast) $$ $$ \Xi: (x,R) \mapsto (R(x),\mathcal R(R)) $$ $$ \Xi^n(x,R) $$ $$ \operatorname{Orb}_{\Xi}(x,R) $$ $$ \operatorname{Cl}_{\Xi} \left( \operatorname{Orb}_{\Xi}(x,R) \right) $$ $$ \boxed{ SREG_{002} : S_{61} \to S_{62} \to \cdots \to S_{120} } $$ $$ \boxed{ S_{120}\to S_{121} } $$ --- # SREG-003 $$ S_{120}\longrightarrow S_{121} $$ --- ## $S_{121}$ $$ \mathcal L $$ $$ \operatorname{Str}(\mathcal L) $$ $$ \mathcal M,\mathcal N\in\operatorname{Str}(\mathcal L) $$ $$ f:\mathcal M\to\mathcal N $$ $$ \forall R\in\mathcal L,\ R^\mathcal M(a_1,\ldots,a_n) \Rightarrow R^\mathcal N(f(a_1),\ldots,f(a_n)) $$ $$ \forall g\in\mathcal L,\ f(g^\mathcal M(a_1,\ldots,a_n)) = g^\mathcal N(f(a_1),\ldots,f(a_n)) $$ --- ## $S_{122}$ $$ \mathcal M\preccurlyeq\mathcal N $$ $$ \forall\varphi(\bar x),\ \forall\bar a\in M: \mathcal M\models\varphi(\bar a) \iff \mathcal N\models\varphi(\bar a) $$ $$ T=\operatorname{Th}(\mathcal M) $$ $$ \operatorname{Elem}(T) := \{\mathcal N:\mathcal N\models T\} $$ $$ \mathcal M\equiv\mathcal N \iff \operatorname{Th}(\mathcal M) = \operatorname{Th}(\mathcal N) $$ --- ## $S_{123}$ $$ T\models\exists x\,\varphi(x) $$ $$ \exists c: T\cup\{\varphi(c)\} $$ $$ T_0\subseteq T_1\subseteq T_2\subseteq\cdots $$ $$ T^\ast:=\bigcup_{n<\omega}T_n $$ $$ \forall\exists x\,\varphi(x)\in T^\ast,\ \exists c_\varphi: \varphi(c_\varphi)\in T^\ast $$ $$ \mathcal M_{T^\ast} $$ --- ## $S_{124}$ $$ T\models\varphi \iff \exists T_0\subseteq_{\mathrm{fin}}T: T_0\models\varphi $$ $$ \forall T: \left[ \forall T_0\subseteq_{\mathrm{fin}}T,\ \operatorname{Mod}(T_0)\neq\varnothing \right] \Rightarrow \operatorname{Mod}(T)\neq\varnothing $$ $$ \operatorname{Sat}(T) \iff \forall T_0\subseteq_{\mathrm{fin}}T: \operatorname{Sat}(T_0) $$ --- ## $S_{125}$ $$ I $$ $$ \mathcal U\subseteq\mathcal P(I) $$ $$ \varnothing\notin\mathcal U $$ $$ A,B\in\mathcal U \Rightarrow A\cap B\in\mathcal U $$ $$ A\in\mathcal U \land A\subseteq B\subseteq I \Rightarrow B\in\mathcal U $$ $$ \forall A\subseteq I: A\in\mathcal U \lor I\setminus A\in\mathcal U $$ $$ \prod_{i\in I}\mathcal M_i/\mathcal U $$ --- ## $S_{126}$ $$ \prod_{i\in I}\mathcal M_i/\mathcal U \models \varphi([a_i]_{\mathcal U}) $$ $$ \iff $$ $$ \{i\in I:\mathcal M_i\models\varphi(a_i)\} \in\mathcal U $$ $$ \mathcal M^I/\mathcal U $$ $$ \mathcal M \equiv \mathcal M^I/\mathcal U $$ --- ## $S_{127}$ $$ S_n(T) $$ $$ p(x)\in S_n(T) $$ $$ p(x) = \{\varphi(x):\varphi\in\mathcal L\} $$ $$ T\cup p(c) $$ $$ \operatorname{tp}(a/A) $$ $$ a\equiv_A b \iff \operatorname{tp}(a/A) = \operatorname{tp}(b/A) $$ --- ## $S_{128}$ $$ A\subseteq M $$ $$ \operatorname{dcl}(A) := \{b:\exists\varphi(x,\bar a),\ \bar a\in A,\ \mathcal M\models\exists!x\,\varphi(x,\bar a) \land \mathcal M\models\varphi(b,\bar a) \} $$ $$ \operatorname{acl}(A) := \{b:\exists\varphi(x,\bar a),\ |\varphi(M,\bar a)|<\infty,\ b\in\varphi(M,\bar a) \} $$ $$ A\subseteq\operatorname{dcl}(A)\subseteq\operatorname{acl}(A) $$ --- ## $S_{129}$ $$ \operatorname{Aut}(\mathcal M/A) $$ $$ \operatorname{dcl}(A) = \{b:\forall\sigma\in\operatorname{Aut}(\mathcal M/A),\ \sigma(b)=b\} $$ $$ \operatorname{orb}_A(b) := \{\sigma(b):\sigma\in\operatorname{Aut}(\mathcal M/A)\} $$ $$ b\in\operatorname{acl}(A) \Rightarrow |\operatorname{orb}_A(b)|<\infty $$ --- ## $S_{130}$ $$ V\models T $$ $$ \kappa:=|V| $$ $$ |A|<\kappa $$ $$ p(x)\in S(A) $$ $$ \operatorname{Sat}_\kappa(V) \iff \forall A\subseteq V,\ |A|<\kappa,\ \forall p\in S(A),\ \exists a\in V:\ a\models p $$ $$ \operatorname{Homog}_\kappa(V) $$ --- ## $S_{131}$ $$ H $$ $$ \langle\cdot,\cdot\rangle:H\times H\to\mathbb C $$ $$ \langle x,y\rangle = \overline{\langle y,x\rangle} $$ $$ \langle x,x\rangle\ge0 $$ $$ \langle x,x\rangle=0 \iff x=0 $$ $$ \|x\| := \sqrt{\langle x,x\rangle} $$ $$ H=\overline H $$ --- ## $S_{132}$ $$ T:H\to H $$ $$ T\in\mathcal B(H) $$ $$ \|T\| := \sup_{\|x\|=1}\|Tx\| $$ $$ T^\ast $$ $$ \langle Tx,y\rangle = \langle x,T^\ast y\rangle $$ $$ T=T^\ast $$ $$ T^\ast T=TT^\ast $$ --- ## $S_{133}$ $$ \sigma(T) := \{\lambda\in\mathbb C: T-\lambda I \notin \operatorname{GL}(H) \} $$ $$ \rho(T) := \mathbb C\setminus\sigma(T) $$ $$ R_\lambda(T) := (T-\lambda I)^{-1} $$ $$ \lambda\in\rho(T) $$ $$ R_\lambda(T)-R_\mu(T) = (\lambda-\mu) R_\lambda(T)R_\mu(T) $$ --- ## $S_{134}$ $$ T=T^\ast $$ $$ \sigma(T)\subseteq\mathbb R $$ $$ T = \int_{\sigma(T)} \lambda\,dE(\lambda) $$ $$ I = \int_{\sigma(T)} dE(\lambda) $$ $$ f(T) = \int_{\sigma(T)} f(\lambda)\,dE(\lambda) $$ $$ (fg)(T)=f(T)g(T) $$ --- ## $S_{135}$ $$ \mathcal A\subseteq\mathcal B(H) $$ $$ I\in\mathcal A $$ $$ A,B\in\mathcal A \Rightarrow AB\in\mathcal A $$ $$ A\in\mathcal A \Rightarrow A^\ast\in\mathcal A $$ $$ \|A^\ast A\|=\|A\|^2 $$ $$ \overline{\mathcal A}^{\|\cdot\|} = \mathcal A $$ $$ \mathcal A\in\mathbf{C^\ast Alg} $$ --- ## $S_{136}$ $$ \mathcal M\subseteq\mathcal B(H) $$ $$ \mathcal M' := \{T\in\mathcal B(H): \forall A\in\mathcal M,\ TA=AT \} $$ $$ \mathcal M'' = (\mathcal M')' $$ $$ \mathcal M=\mathcal M'' $$ $$ \mathcal M = \overline{\mathcal M}^{\,\mathrm{WOT}} $$ --- ## $S_{137}$ $$ \rho\in\mathcal B(H) $$ $$ \rho\ge0 $$ $$ \operatorname{Tr}(\rho)=1 $$ $$ \langle A\rangle_\rho := \operatorname{Tr}(\rho A) $$ $$ \rho^2=\rho $$ $$ \rho=\sum_i p_i|\psi_i\rangle\langle\psi_i| $$ $$ \sum_i p_i=1 $$ --- ## $S_{138}$ $$ A=A^\ast $$ $$ A=\sum_a aP_a $$ $$ p(a) = \operatorname{Tr}(\rho P_a) $$ $$ \rho \mapsto \frac{P_a\rho P_a} {\operatorname{Tr}(\rho P_a)} $$ $$ \sum_aP_a=I $$ $$ P_aP_b=\delta_{ab}P_a $$ --- ## $S_{139}$ $$ H_{AB} = H_A\otimes H_B $$ $$ \rho_{AB} $$ $$ \rho_A = \operatorname{Tr}_B(\rho_{AB}) $$ $$ \rho_B = \operatorname{Tr}_A(\rho_{AB}) $$ $$ \rho_{AB} \neq \rho_A\otimes\rho_B $$ $$ S(\rho) := -\operatorname{Tr}(\rho\log\rho) $$ --- ## $S_{140}$ $$ S(A:B) := S(\rho_A)+S(\rho_B)-S(\rho_{AB}) $$ $$ S(A:B)\ge0 $$ $$ S(A:BC)\ge S(A:B) $$ $$ S(A\mid B) := S(\rho_{AB})-S(\rho_B) $$ $$ S(A\mid B)<0 $$ --- ## $S_{141}$ $$ \Phi: \mathcal B(H_A)\to\mathcal B(H_B) $$ $$ \Phi(\rho) = \sum_iK_i\rho K_i^\ast $$ $$ \sum_iK_i^\ast K_i=I $$ $$ \Phi \otimes \operatorname{id}_n \ge0 $$ $$ \operatorname{Tr}\Phi(\rho)=\operatorname{Tr}\rho $$ --- ## $S_{142}$ $$ D(\rho\|\sigma) := \operatorname{Tr} \left[ \rho(\log\rho-\log\sigma) \right] $$ $$ D(\rho\|\sigma)\ge0 $$ $$ D(\Phi(\rho)\|\Phi(\sigma)) \le D(\rho\|\sigma) $$ $$ F(\rho,\sigma) := \left\| \sqrt\rho\sqrt\sigma \right\|_1^2 $$ $$ 0\le F(\rho,\sigma)\le1 $$ --- ## $S_{143}$ $$ \Theta\subseteq\mathbb R^n $$ $$ p(x;\theta) $$ $$ \ell(\theta;x) := \log p(x;\theta) $$ $$ g_{ij}(\theta) := \mathbb E_\theta \left[ \partial_i\ell \partial_j\ell \right] $$ $$ g_{ij} = -\mathbb E_\theta[ \partial_i\partial_j\ell ] $$ $$ ds^2 = g_{ij}d\theta^id\theta^j $$ --- ## $S_{144}$ $$ D_{\mathrm{KL}} ( p_{\theta} \| p_{\theta+d\theta} ) $$ $$ = \frac12 g_{ij}(\theta) d\theta^id\theta^j + O(\|d\theta\|^3) $$ $$ \Gamma_{ijk}^{(\alpha)} $$ $$ \nabla^{(\alpha)} $$ $$ \nabla^{(\alpha)} \leftrightarrow \nabla^{(-\alpha)} $$ --- ## $S_{145}$ $$ \mathcal P $$ $$ \mathcal P^\ast $$ $$ \nabla $$ $$ \nabla^\ast $$ $$ g $$ $$ Xg(Y,Z) = g(\nabla_XY,Z) + g(Y,\nabla_X^\ast Z) $$ $$ (\mathcal P,g,\nabla,\nabla^\ast) $$ --- ## $S_{146}$ $$ M $$ $$ T_xM $$ $$ T_x^\ast M $$ $$ \omega \in \Omega^2(M) $$ $$ \omega^\flat:T_xM\to T_x^\ast M $$ $$ X\mapsto\iota_X\omega $$ $$ \pi^\sharp:T_x^\ast M\to T_xM $$ $$ \{f,g\} = \pi(df,dg) $$ --- ## $S_{147}$ $$ \pi\in\Gamma(\wedge^2TM) $$ $$ [\pi,\pi]_{\mathrm S}=0 $$ $$ \{f,g\} := \pi(df,dg) $$ $$ \{f,\{g,h\}\} + \{g,\{h,f\}\} + \{h,\{f,g\}\} = 0 $$ $$ X_f:=\pi^\sharp(df) $$ --- ## $S_{148}$ $$ \mathfrak g $$ $$ \mathfrak g^\ast $$ $$ \operatorname{ad}^\ast: \mathfrak g\to \operatorname{End}(\mathfrak g^\ast) $$ $$ \mathcal O_\mu := \{\operatorname{Ad}_g^\ast\mu:g\in G\} $$ $$ \omega_\mu ( \operatorname{ad}_X^\ast\mu, \operatorname{ad}_Y^\ast\mu ) = \langle\mu,[X,Y]\rangle $$ --- ## $S_{149}$ $$ G \curvearrowright (M,\omega) $$ $$ J:M\to\mathfrak g^\ast $$ $$ d\langle J,\xi\rangle = \iota_{\xi_M}\omega $$ $$ J^{-1}(\mu) $$ $$ M_\mu := J^{-1}(\mu)/G_\mu $$ $$ \omega_\mu $$ --- ## $S_{150}$ $$ Q $$ $$ TQ $$ $$ T^\ast Q $$ $$ L:TQ\to\mathbb R $$ $$ H:T^\ast Q\to\mathbb R $$ $$ p_i = \frac{\partial L}{\partial\dot q_i} $$ $$ H(q,p) = p_i\dot q_i-L(q,\dot q) $$ $$ \dot q_i = \frac{\partial H}{\partial p_i}, \qquad \dot p_i = -\frac{\partial H}{\partial q_i} $$ --- ## $S_{151}$ $$ \mathcal X $$ $$ \Phi_t:\mathcal X\to\mathcal X $$ $$ \Phi_0=\operatorname{id} $$ $$ \Phi_{t+s} = \Phi_t\circ\Phi_s $$ $$ x_t=\Phi_t(x_0) $$ $$ \omega(x) := \bigcap_{T>0} \overline{ \{\Phi_t(x):t\ge T\} } $$ --- ## $S_{152}$ $$ A\subseteq X $$ $$ \Phi_t(A)=A $$ $$ B(A) := \{x: d(\Phi_t(x),A)\to0 \} $$ $$ A = \bigcap_{t\ge0} \Phi_t(U) $$ $$ \operatorname{Attr}(\Phi) $$ --- ## $S_{153}$ $$ f:X\to X $$ $$ Df_x $$ $$ Df_x^n $$ $$ \lambda(x,v) := \limsup_{n\to\infty} \frac1n \log \|Df_x^n v\| $$ $$ \lambda_{\max}(x) := \sup_{v\neq0}\lambda(x,v) $$ $$ \lambda_{\max}>0 $$ --- ## $S_{154}$ $$ \mu $$ $$ f_\ast\mu=\mu $$ $$ h_\mu(f) $$ $$ h_{\mathrm{top}}(f) $$ $$ h_{\mathrm{top}}(f) = \sup_\mu h_\mu(f) $$ $$ P(\varphi) = \sup_\mu \left( h_\mu(f) + \int\varphi\,d\mu \right) $$ --- ## $S_{155}$ $$ X_0 \to X_1 \to X_2 \to \cdots $$ $$ F_n:X_n\to X_{n+1} $$ $$ F_{n+1}\circ F_n $$ $$ F_{m:n} := F_{m-1}\circ\cdots\circ F_n $$ $$ m>n $$ $$ \operatorname{DirSys}(X_n,F_n) $$ --- ## $S_{156}$ $$ \cdots \to X_2 \to X_1 \to X_0 $$ $$ p_{n+1,n}:X_{n+1}\to X_n $$ $$ \varprojlim X_n := \left\{ (x_n): p_{n+1,n}(x_{n+1})=x_n \right\} $$ $$ \varinjlim X_n $$ --- ## $S_{157}$ $$ \mathcal F_0 \subseteq \mathcal F_1 \subseteq \cdots $$ $$ \mathcal F_\infty := \sigma\left( \bigcup_n\mathcal F_n \right) $$ $$ X_n = \mathbb E[X\mid\mathcal F_n] $$ $$ X_n\to \mathbb E[X\mid\mathcal F_\infty] $$ $$ L^1 $$ --- ## $S_{158}$ $$ \mathcal A_0 \subseteq \mathcal A_1 \subseteq \cdots $$ $$ \mathcal A_\infty := \overline{ \bigcup_n\mathcal A_n } $$ $$ \operatorname{Spec}(\mathcal A_n) $$ $$ \operatorname{Spec}(\mathcal A_\infty) $$ $$ \mathcal A_n \xrightarrow{\iota_n} \mathcal A_{n+1} $$ --- ## $S_{159}$ $$ C^\bullet $$ $$ F^pC^n $$ $$ F^{p+1}C^n \subseteq F^pC^n $$ $$ d(F^pC^n) \subseteq F^pC^{n+1} $$ $$ E_0^{p,q} := F^pC^{p+q}/F^{p+1}C^{p+q} $$ $$ E_{r+1} = H(E_r,d_r) $$ --- ## $S_{160}$ $$ E_r^{p,q} $$ $$ d_r: E_r^{p,q} \to E_r^{p+r,q-r+1} $$ $$ E_{r+1}^{p,q} = \frac{\ker d_r^{p,q}} {\operatorname{im}d_r^{p-r,q+r-1}} $$ $$ E_r\Rightarrow E_\infty $$ $$ E_\infty^{p,q} \cong F^pH^{p+q}/F^{p+1}H^{p+q} $$ --- ## $S_{161}$ $$ \mathcal C $$ $$ W\subseteq\operatorname{Mor}(\mathcal C) $$ $$ \mathcal C[W^{-1}] $$ $$ w\in W \Rightarrow w^{-1} $$ $$ Q:\mathcal C\to\mathcal C[W^{-1}] $$ $$ Q(w)\in\operatorname{Iso} $$ $$ F(W)\subseteq\operatorname{Iso}(\mathcal D) \Rightarrow \exists!\overline F: \mathcal C[W^{-1}]\to\mathcal D $$ --- ## $S_{162}$ $$ \mathcal M $$ $$ (W,C,F) $$ $$ W\cap C $$ $$ W\cap F $$ $$ \mathcal M[W^{-1}] $$ $$ \operatorname{Ho}(\mathcal M) $$ $$ Q:\mathcal M\to\operatorname{Ho}(\mathcal M) $$ --- ## $S_{163}$ $$ L\dashv R $$ $$ L:\mathcal M\rightleftarrows\mathcal N:R $$ $$ L(W_\mathcal M)\subseteq W_\mathcal N $$ $$ \mathbf L L: \operatorname{Ho}(\mathcal M) \to \operatorname{Ho}(\mathcal N) $$ $$ \mathbf R R: \operatorname{Ho}(\mathcal N) \to \operatorname{Ho}(\mathcal M) $$ $$ \mathbf L L\dashv\mathbf R R $$ --- ## $S_{164}$ $$ \mathcal C $$ $$ \mathcal C^{op} $$ $$ \operatorname{Fun}(\mathcal C^{op},\mathbf{Set}) $$ $$ y:\mathcal C \hookrightarrow \operatorname{Fun}(\mathcal C^{op},\mathbf{Set}) $$ $$ y(c)=\operatorname{Hom}_{\mathcal C}(-,c) $$ $$ y \ \cong\ \operatorname{Yoneda} $$ --- ## $S_{165}$ $$ F \in \operatorname{Fun}(\mathcal C^{op},\mathbf{Set}) $$ $$ F \cong \varinjlim_{(c,x)\in\int_\mathcal C F} y(c) $$ $$ \int_\mathcal C F $$ $$ (c,x) $$ $$ f:(c,x)\to(d,y) \iff F(f)(y)=x $$ --- ## $S_{166}$ $$ \mathcal C \xrightarrow{F} \mathbf{Cat} $$ $$ \int_\mathcal C F $$ $$ \operatorname{Ob} \left( \int_\mathcal C F \right) = \{(c,x):x\in\operatorname{Ob}F(c)\} $$ $$ (c,x)\to(d,y) $$ $$ (f,\alpha) $$ $$ \alpha:F(f)(x)\to y $$ --- ## $S_{167}$ $$ \mathcal C $$ $$ N(\mathcal C)_n := \operatorname{Fun}([n],\mathcal C) $$ $$ N(\mathcal C):\Delta^{op}\to\mathbf{Set} $$ $$ \mathcal C \mapsto N(\mathcal C) $$ $$ N(\mathcal C\times\mathcal D) \cong N(\mathcal C)\times N(\mathcal D) $$ --- ## $S_{168}$ $$ X_\bullet $$ $$ |X_\bullet| $$ $$ |N(\mathcal C)| $$ $$ B\mathcal C := |N(\mathcal C)| $$ $$ BG $$ $$ \pi_1(BG)\cong G $$ --- ## $S_{169}$ $$ \mathcal G $$ $$ \operatorname{Ob}(\mathcal G) $$ $$ \operatorname{Mor}(\mathcal G) $$ $$ \forall f\in\operatorname{Mor}(\mathcal G),\ \exists f^{-1} $$ $$ \mathcal G \in\mathbf{Grpd} $$ $$ N(\mathcal G) $$ $$ \Pi_1(X) $$ --- ## $S_{170}$ $$ X $$ $$ \Pi_{\le n}(X) $$ $$ \pi_k(X)=0 \qquad k>n $$ $$ X\to\tau_{\le n}X $$ $$ \tau_{\le0}X $$ $$ \tau_{\le1}X $$ $$ \cdots $$ $$ X\simeq\varprojlim_n\tau_{\le n}X $$ --- ## $S_{171}$ $$ \mathcal C $$ $$ \operatorname{Map}_{\mathcal C}(x,y) $$ $$ \pi_0\operatorname{Map}_{\mathcal C}(x,y) $$ $$ \operatorname{Ho}(\mathcal C) $$ $$ \operatorname{Hom}_{\operatorname{Ho}(\mathcal C)}(x,y) = \pi_0\operatorname{Map}_{\mathcal C}(x,y) $$ --- ## $S_{172}$ $$ F:\mathcal C\to\mathcal D $$ $$ G:\mathcal D\to\mathcal C $$ $$ GF\simeq\operatorname{id}_{\mathcal C} $$ $$ FG\simeq\operatorname{id}_{\mathcal D} $$ $$ \mathcal C\simeq\mathcal D $$ $$ \operatorname{Ho}(\mathcal C) \simeq \operatorname{Ho}(\mathcal D) $$ --- ## $S_{173}$ $$ \mathcal C \xrightarrow{F} \mathcal D \xrightarrow{G} \mathcal E $$ $$ \operatorname{Map}(G\circ F,H) $$ $$ \operatorname{Nat}(G\circ F,H) $$ $$ \operatorname{Nat}(F,G\Rightarrow H) $$ $$ \operatorname{Comp} : \operatorname{Fun}(\mathcal D,\mathcal E) \times \operatorname{Fun}(\mathcal C,\mathcal D) \to \operatorname{Fun}(\mathcal C,\mathcal E) $$ --- ## $S_{174}$ $$ \mathfrak F $$ $$ \operatorname{End}(\mathfrak F) $$ $$ \operatorname{End}^2(\mathfrak F) := \operatorname{End} ( \operatorname{End}(\mathfrak F) ) $$ $$ R\in\operatorname{End}(\mathfrak F) $$ $$ \mathcal R\in\operatorname{End}^2(\mathfrak F) $$ $$ \mathcal R(R) $$ $$ \mathcal R^2(R) $$ --- ## $S_{175}$ $$ \mathbf E_0:=\mathfrak F $$ $$ \mathbf E_{n+1} := \operatorname{End}(\mathbf E_n) $$ $$ \mathbf E_0 \to \mathbf E_1 \to \mathbf E_2 \to \cdots $$ $$ \mathbf E_\omega := \bigcup_{n<\omega}\mathbf E_n $$ $$ \mathcal Q: \mathbf E_\omega\to\mathbf E_\omega $$ --- ## $S_{176}$ $$ x_0\in\mathbf E_0 $$ $$ x_{n+1} := F_n(x_n) $$ $$ F_n\in\mathbf E_{n+1} $$ $$ (x_0,F_0,F_1,\ldots) $$ $$ \Xi_n : (x_n,F_n) \mapsto (F_n(x_n),F_{n+1}) $$ $$ \Xi_{n:0} $$ --- ## $S_{177}$ $$ \mathcal S := \prod_{n<\omega}\mathbf E_n $$ $$ s = (x_0,F_0,F_1,\ldots) $$ $$ \sigma:\mathcal S\to\mathcal S $$ $$ \sigma(s) = (x_1,F_1,F_2,\ldots) $$ $$ x_1=F_0(x_0) $$ $$ \sigma^n(s) $$ --- ## $S_{178}$ $$ d_\mathcal S(s,t) := \sum_{n=0}^\infty 2^{-n} \frac{ d_n(s_n,t_n) }{ 1+d_n(s_n,t_n) } $$ $$ (\mathcal S,d_\mathcal S) $$ $$ \sigma:\mathcal S\to\mathcal S $$ $$ \operatorname{Orb}_\sigma(s) $$ $$ \omega_\sigma(s) $$ --- ## $S_{179}$ $$ \mathcal P(\mathcal S) $$ $$ \mathcal T: \mathcal P(\mathcal S) \to \mathcal P(\mathcal S) $$ $$ \mathcal T(A) := \overline{\sigma(A)} $$ $$ A^\ast = \mathcal T(A^\ast) $$ $$ A_0 \supseteq A_1 \supseteq A_2 \supseteq \cdots $$ $$ A_\infty = \bigcap_{n<\omega}A_n $$ --- ## $S_{180}$ $$ S_{120} \to S_{121} \to \cdots \to S_{180} $$ $$ \operatorname{Model} \to \operatorname{Type} \to \operatorname{Operator} \to \operatorname{Spectrum} \to \operatorname{State} \to \operatorname{Information} \to \operatorname{Geometry} \to \operatorname{Dynamics} \to \operatorname{HigherStructure} \to \operatorname{SelfAction} $$ $$ \mathfrak F \to \operatorname{End}(\mathfrak F) \to \operatorname{End}^2(\mathfrak F) \to \cdots $$ $$ \mathbf E_0 \to \mathbf E_1 \to \mathbf E_2 \to \cdots \to \mathbf E_\omega $$ $$ s \in \prod_{n<\omega}\mathbf E_n $$ $$ \sigma(s) $$ $$ \operatorname{Orb}_\sigma(s) $$ $$ \omega_\sigma(s) $$ $$ \boxed{ SREG_{003} : S_{121} \to S_{122} \to \cdots \to S_{180} } $$ $$ \boxed{ S_{180}\to S_{181} } $$ --- # SREG-004 $$ S_{180}\longrightarrow S_{181} $$ --- ## $S_{181}$ $$ (R,<) $$ $$ \operatorname{WF}(R) \iff \forall A\subseteq R: A\neq\varnothing \Rightarrow \exists a\in A\, \forall b\in A:\neg(b\cdots>\alpha_n $$ $$ c_i\in\mathbb N_{>0} $$ $$ \operatorname{CNF}(\alpha) $$ $$ \alpha<\varepsilon_0 $$ --- ## $S_{185}$ $$ T_0 \subseteq T_1 \subseteq T_2 \subseteq \cdots $$ $$ T_{n+1} = T_n+\operatorname{Con}(T_n) $$ $$ T_\omega = \bigcup_{n<\omega}T_n $$ $$ \operatorname{Con}(T_n) \in T_{n+1} $$ $$ \operatorname{Con}(T_n) \notin \operatorname{Thm}(T_n) $$ --- ## $S_{186}$ $$ \operatorname{RFN}(T) $$ $$ \forall\varphi: \operatorname{Pr}_T(\ulcorner\varphi\urcorner) \Rightarrow \varphi $$ $$ T^+ := T+\operatorname{RFN}(T) $$ $$ T \subsetneq T^+ $$ $$ T^{(\alpha+1)} := \left(T^{(\alpha)}\right)^+ $$ $$ T^{(\lambda)} := \bigcup_{\beta<\lambda}T^{(\beta)} $$ --- ## $S_{187}$ $$ \Box_T\varphi := \operatorname{Pr}_T(\ulcorner\varphi\urcorner) $$ $$ \Box_T(\varphi\to\psi) \to (\Box_T\varphi\to\Box_T\psi) $$ $$ \Box_T\varphi \to \Box_T\Box_T\varphi $$ $$ \Box_T(\Box_T\varphi\to\varphi) \to \Box_T\varphi $$ --- ## $S_{188}$ $$ \forall\varphi(x): \exists\psi: T\vdash \psi \leftrightarrow \varphi(\ulcorner\psi\urcorner) $$ $$ \psi \leftrightarrow \varphi(\ulcorner\psi\urcorner) $$ $$ \operatorname{Fix}_T(\varphi) $$ $$ \operatorname{Fix}_T : \varphi \mapsto \psi $$ --- ## $S_{189}$ $$ \operatorname{Tr}_0 $$ $$ \operatorname{Tr}_{\alpha+1} \supseteq \operatorname{Tr}_\alpha $$ $$ \operatorname{Tr}_\lambda = \bigcup_{\beta<\lambda}\operatorname{Tr}_\beta $$ $$ \operatorname{Tr}_\alpha(\ulcorner\varphi\urcorner) \leftrightarrow \varphi $$ $$ \varphi\in\mathcal L_\alpha $$ --- ## $S_{190}$ $$ \mathbb V = \{0,\tfrac12,1\} $$ $$ \neg 0=1 $$ $$ \neg1=0 $$ $$ \neg\tfrac12=\tfrac12 $$ $$ \varphi\land\psi = \min(\varphi,\psi) $$ $$ \varphi\lor\psi = \max(\varphi,\psi) $$ --- ## $S_{191}$ $$ \mathcal B = (B,\le_t,\le_k) $$ $$ x\le_t y $$ $$ x\le_k y $$ $$ \wedge_t,\vee_t $$ $$ \wedge_k,\vee_k $$ $$ \neg: (B,\le_t) \to (B,\ge_t) $$ $$ \neg: (B,\le_k) \to (B,\le_k) $$ --- ## $S_{192}$ $$ (L,\le) $$ $$ \forall A\subseteq L: \bigvee A\in L $$ $$ \forall A\subseteq L: \bigwedge A\in L $$ $$ f:L\to L $$ $$ x\le y \Rightarrow f(x)\le f(y) $$ $$ \operatorname{Fix}(f) $$ --- ## $S_{193}$ $$ \mu f := \bigwedge \{x\in L:f(x)\le x\} $$ $$ \nu f := \bigvee \{x\in L:x\le f(x)\} $$ $$ f(\mu f)=\mu f $$ $$ f(\nu f)=\nu f $$ $$ \mu f\le x \le\nu f $$ --- ## $S_{194}$ $$ \alpha:L\to M $$ $$ \gamma:M\to L $$ $$ \alpha\dashv\gamma $$ $$ \alpha(x)\le_M y \iff x\le_L\gamma(y) $$ $$ \alpha\circ\gamma \le \operatorname{id}_M $$ $$ \operatorname{id}_L \le \gamma\circ\alpha $$ --- ## $S_{195}$ $$ c:L\to L $$ $$ x\le c(x) $$ $$ x\le y \Rightarrow c(x)\le c(y) $$ $$ c(c(x))=c(x) $$ $$ i:L\to L $$ $$ i(x)\le x $$ $$ i(i(x))=i(x) $$ --- ## $S_{196}$ $$ (D,\sqsubseteq) $$ $$ \bot\in D $$ $$ x_0\sqsubseteq x_1\sqsubseteq\cdots $$ $$ \bigsqcup_{n<\omega}x_n\in D $$ $$ f:D\to D $$ $$ f\left( \bigsqcup_nx_n \right) = \bigsqcup_n f(x_n) $$ --- ## $S_{197}$ $$ f:D\to D $$ $$ x_0:=\bot $$ $$ x_{n+1}:=f(x_n) $$ $$ \mu f = \bigsqcup_{n<\omega}f^n(\bot) $$ $$ f(\mu f)=\mu f $$ $$ \mu f \sqsubseteq x \quad \forall x:f(x)\sqsubseteq x $$ --- ## $S_{198}$ $$ D \simeq F(D) $$ $$ D_0:=\bot $$ $$ D_{n+1}:=F(D_n) $$ $$ D_\omega := \bigsqcup_{n<\omega}D_n $$ $$ F(D_\omega) \simeq D_\omega $$ --- ## $S_{199}$ $$ F:\mathcal C\to\mathcal C $$ $$ \gamma:X\to F(X) $$ $$ \delta:Y\to F(Y) $$ $$ R\subseteq X\times Y $$ $$ xRy \Rightarrow F(R)(\gamma(x),\delta(y)) $$ $$ X\sim_F Y $$ --- ## $S_{200}$ $$ (\nu F,\omega) $$ $$ \omega:\nu F\to F(\nu F) $$ $$ \forall(X,\gamma): \exists!\llbracket\gamma\rrbracket:X\to\nu F $$ $$ \omega\circ\llbracket\gamma\rrbracket = F(\llbracket\gamma\rrbracket)\circ\gamma $$ $$ x\sim_Fy \Rightarrow \llbracket x\rrbracket = \llbracket y\rrbracket $$ --- ## $S_{201}$ $$ \frac{ R(x,y) }{ R(\gamma(x),\gamma(y)) } $$ $$ R\subseteq F(R) $$ $$ R\subseteq\nu F $$ $$ R\le F(R) \Rightarrow R\le\nu F $$ $$ \operatorname{Coind}(F) $$ --- ## $S_{202}$ $$ \varphi ::= p \mid \neg p \mid \varphi\land\varphi \mid \varphi\lor\varphi \mid \Diamond\varphi \mid \Box\varphi \mid \mu X.\varphi \mid \nu X.\varphi $$ $$ \llbracket\mu X.\varphi\rrbracket = \mu \left( A\mapsto \llbracket\varphi\rrbracket_{X:=A} \right) $$ $$ \llbracket\nu X.\varphi\rrbracket = \nu \left( A\mapsto \llbracket\varphi\rrbracket_{X:=A} \right) $$ --- ## $S_{203}$ $$ G=(V,E,\Omega) $$ $$ \Omega:V\to\mathbb N $$ $$ v_0v_1v_2\cdots $$ $$ \inf \{n: |\{k:\Omega(v_k)=n\}|=\infty\} $$ $$ \equiv0\pmod2 $$ $$ \operatorname{Win}_\exists(G) $$ --- ## $S_{204}$ $$ \mathcal V $$ $$ (\mathcal V,\otimes,I) $$ $$ \mathcal C $$ $$ \operatorname{Hom}_{\mathcal C}(x,y) \in\operatorname{Ob}(\mathcal V) $$ $$ \operatorname{Hom}(y,z) \otimes \operatorname{Hom}(x,y) \to \operatorname{Hom}(x,z) $$ $$ I\to\operatorname{Hom}(x,x) $$ --- ## $S_{205}$ $$ (\mathcal C,\otimes,I,\alpha,\lambda,\rho) $$ $$ (A\otimes B)\otimes C \xrightarrow{\alpha} A\otimes(B\otimes C) $$ $$ I\otimes A \xrightarrow{\lambda} A $$ $$ A\otimes I \xrightarrow{\rho} A $$ $$ \alpha^4=\alpha^4 $$ --- ## $S_{206}$ $$ [A,B] $$ $$ \operatorname{Hom}(X\otimes A,B) \cong \operatorname{Hom}(X,[A,B]) $$ $$ \operatorname{ev}_{A,B}: [A,B]\otimes A\to B $$ $$ \operatorname{curry} : \operatorname{Hom}(X\otimes A,B) \to \operatorname{Hom}(X,[A,B]) $$ --- ## $S_{207}$ $$ \operatorname{Tr}_{X,Y}^U : \operatorname{Hom}(X\otimes U,Y\otimes U) \to \operatorname{Hom}(X,Y) $$ $$ \operatorname{Tr}(f\otimes g) $$ $$ \operatorname{Tr} \left( (\operatorname{id}_Y\otimes h) \circ f \right) $$ $$ = \operatorname{Tr} \left( f \circ (\operatorname{id}_X\otimes h) \right) $$ --- ## $S_{208}$ $$ A^\ast $$ $$ \eta_A:I\to A^\ast\otimes A $$ $$ \varepsilon_A:A\otimes A^\ast\to I $$ $$ (\operatorname{id}_A\otimes\varepsilon_A) \circ (\eta_A\otimes\operatorname{id}_A) = \operatorname{id}_A $$ $$ (\varepsilon_A\otimes\operatorname{id}_{A^\ast}) \circ (\operatorname{id}_{A^\ast}\otimes\eta_A) = \operatorname{id}_{A^\ast} $$ --- ## $S_{209}$ $$ f:A\to B $$ $$ f^\dagger:B\to A $$ $$ (f^\dagger)^\dagger=f $$ $$ (g\circ f)^\dagger = f^\dagger\circ g^\dagger $$ $$ \operatorname{id}_A^\dagger = \operatorname{id}_A $$ $$ U^\dagger U = UU^\dagger = I $$ --- ## $S_{210}$ $$ \mu:A\otimes A\to A $$ $$ \eta:I\to A $$ $$ \delta:=\mu^\dagger $$ $$ \epsilon:=\eta^\dagger $$ $$ \mu\circ\delta = \operatorname{id}_A $$ $$ (\mu,\eta,\delta,\epsilon) $$ --- ## $S_{211}$ $$ \mathcal O(n) $$ $$ \mathcal O(k_1+\cdots+k_n) $$ $$ \gamma : \mathcal O(n) \times \prod_{i=1}^n\mathcal O(k_i) \to \mathcal O(k_1+\cdots+k_n) $$ $$ \Sigma_n\curvearrowright\mathcal O(n) $$ $$ \operatorname{Alg}_{\mathcal O}(X) $$ --- ## $S_{212}$ $$ \mathcal P(m,n) $$ $$ \mathcal P(m,n) \times \mathcal P(m_1,n_1) \times\cdots \to \mathcal P\left(\sum m_i,\sum n_i\right) $$ $$ \circ_h $$ $$ \circ_v $$ $$ \operatorname{Prop} $$ $$ \operatorname{Properad} $$ --- ## $S_{213}$ $$ \mathcal O_\infty $$ $$ \pi_0\mathcal O_\infty $$ $$ \pi_1\mathcal O_\infty $$ $$ \pi_2\mathcal O_\infty $$ $$ \cdots $$ $$ \operatorname{Alg}_{\mathcal O_\infty} $$ --- ## $S_{214}$ $$ \mathcal C $$ $$ \operatorname{Map}_{\mathcal C}(x,y)\in\mathcal S $$ $$ \operatorname{Map}(y,z) \times \operatorname{Map}(x,y) \to \operatorname{Map}(x,z) $$ $$ \operatorname{Assoc} \simeq \operatorname{id} $$ $$ \operatorname{Unit} \simeq \operatorname{id} $$ --- ## $S_{215}$ $$ F:\mathcal C\to\mathcal D $$ $$ G:\mathcal D\to\mathcal C $$ $$ \eta:\operatorname{id}_{\mathcal C} \Rightarrow GF $$ $$ \varepsilon:FG \Rightarrow \operatorname{id}_{\mathcal D} $$ $$ \operatorname{Map}_{\mathcal D}(F(c),d) \simeq \operatorname{Map}_{\mathcal C}(c,G(d)) $$ --- ## $S_{216}$ $$ D:K\to\mathcal C $$ $$ \lim D $$ $$ \operatorname{Map}_{\mathcal C}(x,\lim D) \simeq \lim_{k\in K} \operatorname{Map}_{\mathcal C}(x,D(k)) $$ $$ \operatorname{colim}D $$ $$ \operatorname{Map}_{\mathcal C}(\operatorname{colim}D,x) \simeq \lim_{k\in K^{op}} \operatorname{Map}_{\mathcal C}(D(k),x) $$ --- ## $S_{217}$ $$ \mathcal X $$ $$ \mathcal X \simeq \operatorname{Sh}_\infty(C,\tau) $$ $$ \operatorname{Map}_{\mathcal X}(X,Y) $$ $$ \Omega_{\mathcal X} $$ $$ \operatorname{Sub}_{\mathcal X}(X) \simeq \operatorname{Map}_{\mathcal X}(X,\Omega_{\mathcal X}) $$ --- ## $S_{218}$ $$ \mathcal X\models\varphi $$ $$ A:\mathcal U_{\mathcal X} $$ $$ x:A $$ $$ \Pi_{x:A}B(x) $$ $$ \Sigma_{x:A}B(x) $$ $$ \operatorname{Id}_A(x,y) $$ $$ \Omega_{\mathcal X} $$ --- ## $S_{219}$ $$ A:\mathcal U $$ $$ a,b:A $$ $$ \operatorname{Path}_A(a,b) $$ $$ \operatorname{HIT} $$ $$ \operatorname{base}:X $$ $$ \operatorname{loop}: \operatorname{Path}_X(\operatorname{base},\operatorname{base}) $$ --- ## $S_{220}$ $$ \|A\|_{-1} $$ $$ \|A\|_0 $$ $$ \|A\|_1 $$ $$ \cdots $$ $$ A\to\|A\|_n $$ $$ \pi_k(\|A\|_n)=0 \qquad k>n $$ --- ## $S_{221}$ $$ \bigcirc:\mathcal U\to\mathcal U $$ $$ \eta_A:A\to\bigcirc A $$ $$ \bigcirc\bigcirc A \simeq \bigcirc A $$ $$ A\to B \Rightarrow \bigcirc A\to\bigcirc B $$ $$ \operatorname{Modality}(\bigcirc) $$ --- ## $S_{222}$ $$ \Omega $$ $$ \mathcal O(X) $$ $$ \mathcal O(X) \subseteq \mathcal P(X) $$ $$ \bigcup_iU_i\in\mathcal O(X) $$ $$ U\cap V\in\mathcal O(X) $$ $$ \operatorname{int}: \mathcal P(X)\to\mathcal O(X) $$ --- ## $S_{223}$ $$ L $$ $$ \bigvee S $$ $$ a\wedge\bigvee S = \bigvee_{s\in S}(a\wedge s) $$ $$ \operatorname{Frm}(L) $$ $$ \operatorname{Pt}(L) := \operatorname{Hom}_{\mathbf{Frm}}(L,2) $$ --- ## $S_{224}$ $$ X \mapsto \mathcal O(X) $$ $$ f:X\to Y $$ $$ f^{-1}: \mathcal O(Y)\to\mathcal O(X) $$ $$ f^{-1} \left( \bigvee_iU_i \right) = \bigvee_if^{-1}(U_i) $$ $$ f^{-1}(U\wedge V) = f^{-1}(U)\wedge f^{-1}(V) $$ --- ## $S_{225}$ $$ L $$ $$ \operatorname{Sh}(L) $$ $$ \operatorname{Sh}(L) \simeq \operatorname{Sh}(X) $$ $$ L\simeq\mathcal O(X) $$ $$ \operatorname{Pt}(L) $$ $$ \mathcal O(\operatorname{Pt}(L)) $$ --- ## $S_{226}$ $$ f:\mathcal E\to\mathcal F $$ $$ f^\ast:\mathcal F\to\mathcal E $$ $$ f_\ast:\mathcal E\to\mathcal F $$ $$ f^\ast\dashv f_\ast $$ $$ f^\ast \operatorname{Lex} $$ $$ \operatorname{Geom}(\mathcal E,\mathcal F) $$ --- ## $S_{227}$ $$ T $$ $$ \mathcal E_T $$ $$ U_T\in\mathcal E_T $$ $$ \operatorname{Geom}(\mathcal E,\mathcal E_T) \simeq \operatorname{Mod}_T(\mathcal E) $$ $$ f:\mathcal E\to\mathcal E_T $$ $$ f^\ast(U_T) $$ --- ## $S_{228}$ $$ \mathbb T $$ $$ \operatorname{Syn}(\mathbb T) $$ $$ \operatorname{Mod}(\mathbb T,\mathcal C) $$ $$ \operatorname{Lex} ( \operatorname{Syn}(\mathbb T),\mathcal C ) $$ $$ \operatorname{Mod}(\mathbb T,\mathcal C) \simeq \operatorname{Lex} ( \operatorname{Syn}(\mathbb T),\mathcal C ) $$ --- ## $S_{229}$ $$ \mathbb L $$ $$ n\in\operatorname{Ob}(\mathbb L) $$ $$ n \simeq 1^n $$ $$ A:\mathbb L\to\mathbf{Set} $$ $$ A(n)\cong A(1)^n $$ $$ \operatorname{Alg}(\mathbb L) $$ --- ## $S_{230}$ $$ T:\mathcal C\to\mathcal C $$ $$ \eta:\operatorname{id}\Rightarrow T $$ $$ \mu:T^2\Rightarrow T $$ $$ A\to T(B) $$ $$ g^\ast:T(A)\to T(B) $$ $$ g^\ast\circ\eta_A=g $$ $$ h^\ast\circ g^\ast = (h^\ast\circ g)^\ast $$ --- ## $S_{231}$ $$ F:\mathcal C\to\mathcal C $$ $$ \operatorname{Free}(F) $$ $$ T_F $$ $$ T_F(X) = X + F(X) + F^2(X) + \cdots $$ $$ \mu_F:T_F^2\to T_F $$ $$ \eta_F:X\to T_F(X) $$ --- ## $S_{232}$ $$ T $$ $$ \operatorname{Alg}(T) $$ $$ a:T(A)\to A $$ $$ a\circ\eta_A=\operatorname{id}_A $$ $$ a\circ\mu_A = a\circ T(a) $$ $$ \mathcal C^T $$ --- ## $S_{233}$ $$ W $$ $$ \varepsilon:W\Rightarrow\operatorname{id} $$ $$ \delta:W\Rightarrow W^2 $$ $$ \varepsilon_W\circ\delta = W\varepsilon\circ\delta = \operatorname{id}_W $$ $$ W\delta\circ\delta = \delta_W\circ\delta $$ $$ \operatorname{Coalg}(W) $$ --- ## $S_{234}$ $$ H:\mathcal C\times\mathcal C\to\mathcal C $$ $$ H(A,-) $$ $$ H(-,A) $$ $$ \operatorname{Alg}(H(A,-)) $$ $$ \operatorname{Coalg}(H(-,A)) $$ $$ \mu X.H(A,X) $$ $$ \nu X.H(X,A) $$ --- ## $S_{235}$ $$ \mathcal F $$ $$ Q:\mathcal F\to\mathfrak F $$ $$ E:\mathfrak F\to\mathcal F $$ $$ Q\circ E=\operatorname{id}_{\mathfrak F} $$ $$ N:=E\circ Q $$ $$ N^2=N $$ $$ \operatorname{Fix}(N)=\operatorname{im}(N) $$ --- ## $S_{236}$ $$ \mathbf E_0:=\mathfrak F $$ $$ \mathbf E_{n+1}:=\operatorname{End}(\mathbf E_n) $$ $$ \mathbf E_\omega:=\bigcup_{n<\omega}\mathbf E_n $$ $$ \mathfrak N_n: \mathbf E_n\to\mathbf E_n $$ $$ \mathfrak N_n^2=\mathfrak N_n $$ $$ \mathfrak N := \prod_{n<\omega}\mathfrak N_n $$ --- ## $S_{237}$ $$ s=(x_0,F_0,F_1,\ldots)\in\mathcal S $$ $$ \mathfrak N(s) := ( \mathfrak N_0(x_0), \mathfrak N_1(F_0), \mathfrak N_2(F_1), \ldots ) $$ $$ \mathfrak N^2(s)=\mathfrak N(s) $$ $$ \mathcal S^\ast := \operatorname{Fix}(\mathfrak N) $$ --- ## $S_{238}$ $$ \sigma:\mathcal S^\ast\to\mathcal S^\ast $$ $$ \sigma(s) = (x_1,F_1,F_2,\ldots) $$ $$ x_1=F_0(x_0) $$ $$ \mathfrak N\circ\sigma = \sigma\circ\mathfrak N $$ $$ \sigma(\mathcal S^\ast) \subseteq \mathcal S^\ast $$ --- ## $S_{239}$ $$ \mathcal A \subseteq \mathcal S^\ast $$ $$ \sigma(\mathcal A)=\mathcal A $$ $$ \mathcal A = \omega_\sigma(\mathcal A) $$ $$ \mathcal A = \bigcap_{n<\omega} \overline{ \bigcup_{k\ge n}\sigma^k(\mathcal A) } $$ $$ \operatorname{Inv}(\mathcal A,\sigma) $$ --- ## $S_{240}$ $$ S_{180} \to S_{181} \to \cdots \to S_{240} $$ $$ \operatorname{Ord} \to \operatorname{RFN} \to \operatorname{Fix} \to \operatorname{Lattice} \to \operatorname{Domain} \to \operatorname{Coind} \to \operatorname{Monoidal} \to \operatorname{Higher} \to \operatorname{Locale} \to \operatorname{SelfNorm} $$ $$ \mathcal F \to \mathfrak F \to \mathbf E_\omega \to \mathcal S \to \mathcal S^\ast $$ $$ \mathfrak N: \mathcal S\to\mathcal S $$ $$ \mathfrak N^2=\mathfrak N $$ $$ \sigma: \mathcal S^\ast\to\mathcal S^\ast $$ $$ \mathfrak N\circ\sigma = \sigma\circ\mathfrak N $$ $$ \operatorname{Orb}_\sigma(s) \subseteq \mathcal S^\ast $$ $$ \omega_\sigma(s) \subseteq \mathcal S^\ast $$ $$ \boxed{ SREG_{004} : S_{181} \to S_{182} \to \cdots \to S_{240} } $$ $$ \boxed{ S_{240}\to S_{241} } $$ --- # SREG-005 $$ S_{240}\longrightarrow S_{241} $$ --- ## $S_{241}$ $$ \mathcal C $$ $$ \mathcal D\hookrightarrow\mathcal C $$ $$ L:\mathcal C\to\mathcal D $$ $$ I:\mathcal D\hookrightarrow\mathcal C $$ $$ L\dashv I $$ $$ L\circ I \simeq \operatorname{id}_{\mathcal D} $$ --- ## $S_{242}$ $$ \eta: \operatorname{id}_{\mathcal C} \Rightarrow I L $$ $$ L(\eta_X): L(X)\to LIL(X) $$ $$ L(\eta_X) \simeq \operatorname{id}_{L(X)} $$ $$ ILIL \simeq IL $$ $$ \operatorname{Fix}(IL) = \operatorname{im}(I) $$ --- ## $S_{243}$ $$ \bigcirc:=IL $$ $$ \eta_X:X\to\bigcirc X $$ $$ \bigcirc\bigcirc X \simeq \bigcirc X $$ $$ \bigcirc f: \bigcirc X\to\bigcirc Y $$ $$ \bigcirc \circ \bigcirc \simeq \bigcirc $$ --- ## $S_{244}$ $$ \operatorname{Loc}_{\bigcirc}(\mathcal C) := \{X\in\mathcal C:\eta_X\ \operatorname{iso}\} $$ $$ X\in\operatorname{Loc}_{\bigcirc}(\mathcal C) \iff \bigcirc X\simeq X $$ $$ f:X\to Y $$ $$ \bigcirc f\ \operatorname{iso} $$ $$ f\in W_{\bigcirc} $$ --- ## $S_{245}$ $$ \mathcal C[W_{\bigcirc}^{-1}] $$ $$ Q:\mathcal C \to \mathcal C[W_{\bigcirc}^{-1}] $$ $$ Q \simeq L $$ $$ \operatorname{Loc}_{\bigcirc}(\mathcal C) \simeq \mathcal C[W_{\bigcirc}^{-1}] $$ --- ## $S_{246}$ $$ \mathcal U_0 \in \mathcal U_1 $$ $$ \mathcal U_1 \in \mathcal U_2 $$ $$ \cdots $$ $$ x\in\mathcal U_n \Rightarrow x\subseteq\mathcal U_n $$ $$ x,y\in\mathcal U_n \Rightarrow \{x,y\}\in\mathcal U_n $$ $$ I\in\mathcal U_n \land \forall i\in I:\ x_i\in\mathcal U_n \Rightarrow \bigcup_{i\in I}x_i\in\mathcal U_n $$ --- ## $S_{247}$ $$ \mathcal U_n \subseteq \mathcal U_{n+1} $$ $$ \mathcal P(x)\in\mathcal U_n \qquad (x\in\mathcal U_n) $$ $$ \prod_{i\in I}x_i\in\mathcal U_n $$ $$ \operatorname{Fun}(x,y)\in\mathcal U_n $$ $$ \operatorname{Cat}(\mathcal U_n) $$ --- ## $S_{248}$ $$ \mathcal C\in\operatorname{Cat}(\mathcal U_n) $$ $$ \operatorname{PSh}(\mathcal C) := \operatorname{Fun} ( \mathcal C^{op}, \operatorname{Set}(\mathcal U_n) ) $$ $$ y:\mathcal C\hookrightarrow\operatorname{PSh}(\mathcal C) $$ $$ y(c) = \operatorname{Hom}_{\mathcal C}(-,c) $$ --- ## $S_{249}$ $$ F\in\operatorname{PSh}(\mathcal C) $$ $$ F \simeq \int^{c\in\mathcal C} F(c)\times y(c) $$ $$ \operatorname{Nat}(F,G) \simeq \int_{c\in\mathcal C} \operatorname{Hom}(F(c),G(c)) $$ --- ## $S_{250}$ $$ K:\mathcal C^{op}\times\mathcal D\to\mathbf{Set} $$ $$ L:\mathcal D^{op}\times\mathcal E\to\mathbf{Set} $$ $$ (L\star K)(c,e) := \int^{d\in\mathcal D} K(c,d)\times L(d,e) $$ $$ M_\mathcal C(c,c') := \operatorname{Hom}_{\mathcal C}(c,c') $$ $$ K\star M_\mathcal C \simeq K $$ --- ## $S_{251}$ $$ \mathbf{Prof} $$ $$ \operatorname{Ob}(\mathbf{Prof}) = \{\mathcal C\} $$ $$ \operatorname{Hom}_{\mathbf{Prof}}(\mathcal C,\mathcal D) = [\mathcal C^{op}\times\mathcal D,\mathbf{Set}] $$ $$ \circ = \star $$ $$ \operatorname{id}_{\mathcal C} = M_\mathcal C $$ --- ## $S_{252}$ $$ J:\mathcal C\to\mathcal D $$ $$ F:\mathcal C\to\mathcal E $$ $$ \operatorname{Lan}_J F:\mathcal D\to\mathcal E $$ $$ \operatorname{Ran}_J F:\mathcal D\to\mathcal E $$ $$ \operatorname{Nat}(\operatorname{Lan}_J F,G) \cong \operatorname{Nat}(F,GJ) $$ $$ \operatorname{Nat}(GJ,F) \cong \operatorname{Nat}(G,\operatorname{Ran}_J F) $$ --- ## $S_{253}$ $$ (\operatorname{Lan}_J F)(d) \simeq \operatorname{colim}_{(J\downarrow d)}F $$ $$ (\operatorname{Ran}_J F)(d) \simeq \operatorname{lim}_{(d\downarrow J)}F $$ $$ \operatorname{Lan}_{\operatorname{id}}F \simeq F $$ $$ \operatorname{Ran}_{\operatorname{id}}F \simeq F $$ --- ## $S_{254}$ $$ F:\mathcal C\to\mathcal D $$ $$ G:\mathcal D\to\mathcal E $$ $$ \operatorname{Lan}_{G\circ F} \simeq \operatorname{Lan}_G \circ \operatorname{Lan}_F $$ $$ \operatorname{Ran}_{G\circ F} \simeq \operatorname{Ran}_G \circ \operatorname{Ran}_F $$ --- ## $S_{255}$ $$ \mathcal C \xrightarrow{T} \mathcal C $$ $$ \eta:\operatorname{id}\Rightarrow T $$ $$ \mu:T^2\Rightarrow T $$ $$ \mathcal C_T $$ $$ \mathcal C^T $$ $$ J:\mathcal C_T\to\mathcal C^T $$ --- ## $S_{256}$ $$ \operatorname{Hom}_{\mathcal C_T}(X,Y) := \operatorname{Hom}_{\mathcal C}(X,T(Y)) $$ $$ g\odot f := \mu_Z \circ T(g) \circ f $$ $$ \operatorname{id}_X := \eta_X $$ --- ## $S_{257}$ $$ (A,a)\in\mathcal C^T $$ $$ a:T(A)\to A $$ $$ a\circ\eta_A=\operatorname{id}_A $$ $$ a\circ\mu_A = a\circ T(a) $$ $$ h:(A,a)\to(B,b) $$ $$ h\circ a = b\circ T(h) $$ --- ## $S_{258}$ $$ T,S:\mathcal C\to\mathcal C $$ $$ \lambda:TS\Rightarrow ST $$ $$ \lambda\circ\eta^T S = S\eta^T $$ $$ \lambda\circ T\eta^S = \eta^S T $$ $$ \lambda\circ\mu^T S = S\mu^T \circ \lambda T \circ T\lambda $$ $$ \lambda\circ T\mu^S = \mu^S T \circ S\lambda \circ \lambda S $$ --- ## $S_{259}$ $$ ST $$ $$ \eta^{ST} := \eta^S T \circ \eta^T $$ $$ \mu^{ST} := S\mu^T \circ \mu^S TT \circ S\lambda T $$ $$ (ST,\eta^{ST},\mu^{ST}) $$ --- ## $S_{260}$ $$ B $$ $$ T $$ $$ \lambda:TB\Rightarrow BT $$ $$ a:T(X)\to X $$ $$ c:X\to B(X) $$ $$ B(a)\circ\lambda_X\circ T(c) = c\circ a $$ $$ (X,a,c) $$ --- ## $S_{261}$ $$ c:X\to B(X) $$ $$ \llbracket-\rrbracket: X\to\nu B $$ $$ \llbracket-\rrbracket\circ a = \alpha \circ T(\llbracket-\rrbracket) $$ $$ \alpha:T(\nu B)\to\nu B $$ $$ \omega:\nu B\to B(\nu B) $$ --- ## $S_{262}$ $$ x\sim_By $$ $$ x\sim_By \Rightarrow \llbracket x\rrbracket = \llbracket y\rrbracket $$ $$ \llbracket x\rrbracket = \llbracket y\rrbracket \Rightarrow x\sim_By $$ $$ X/{\sim_B} \hookrightarrow \nu B $$ --- ## $S_{263}$ $$ L:\mathcal C\to\mathcal C $$ $$ \delta:L\Rightarrow BL $$ $$ \alpha:LT\Rightarrow TL $$ $$ \delta_T \circ \alpha = B(\alpha) \circ \lambda_L \circ T(\delta) $$ $$ L \Rightarrow B L $$ --- ## $S_{264}$ $$ \Phi $$ $$ \operatorname{Sem}(\Phi) \subseteq \nu B $$ $$ \Phi\mapsto\llbracket\Phi\rrbracket $$ $$ \Phi\equiv_B\Psi \iff \llbracket\Phi\rrbracket = \llbracket\Psi\rrbracket $$ $$ \mathfrak F_B := \mathcal F/{\equiv_B} $$ --- ## $S_{265}$ $$ R_B:\mathfrak F_B\to\mathfrak F_B $$ $$ R_B([\Phi]) = [\operatorname{Sem}^{-1} ( F(\operatorname{Sem}(\Phi)) )] $$ $$ R_B^n([\Phi]) $$ $$ \operatorname{Fix}(R_B) $$ --- ## $S_{266}$ $$ X \xrightarrow{e} \mathbb N $$ $$ d:\mathbb N\to X $$ $$ d\circ e = \operatorname{id}_X $$ $$ e\circ d \sim \operatorname{id}_{\mathbb N} $$ $$ \ulcorner x\urcorner:=e(x) $$ $$ \llcorner n\lrcorner:=d(n) $$ --- ## $S_{267}$ $$ \operatorname{Sub}: \mathbb N\times\mathbb N\to\mathbb N $$ $$ \operatorname{Sub} ( \ulcorner\varphi(x)\urcorner, \ulcorner t\urcorner ) = \ulcorner\varphi(t)\urcorner $$ $$ \operatorname{Diag}(n) := \operatorname{Sub}(n,n) $$ $$ \operatorname{Diag}: \mathbb N\to\mathbb N $$ --- ## $S_{268}$ $$ \theta(x) $$ $$ \psi := \theta( \ulcorner \theta(\operatorname{Diag}(x)) \urcorner ) $$ $$ T\vdash \psi \leftrightarrow \theta(\ulcorner\psi\urcorner) $$ $$ \psi = \operatorname{Fix}_T(\theta) $$ --- ## $S_{269}$ $$ A $$ $$ e:A\to Y^A $$ $$ e \ \operatorname{surj} $$ $$ f:Y\to Y $$ $$ g(a) := f(e(a)(a)) $$ $$ \exists a^\ast: e(a^\ast)=g $$ $$ y^\ast:=e(a^\ast)(a^\ast) $$ $$ f(y^\ast)=y^\ast $$ --- ## $S_{270}$ $$ Y^A $$ $$ A\twoheadrightarrow Y^A $$ $$ \forall f:Y\to Y,\ \operatorname{Fix}(f)\neq\varnothing $$ $$ \operatorname{Diag}_{A,Y} $$ $$ \operatorname{Fix}_Y $$ --- ## $S_{271}$ $$ \Omega $$ $$ \neg:\Omega\to\Omega $$ $$ \operatorname{Fix}(\neg) $$ $$ A\twoheadrightarrow\Omega^A $$ $$ \Rightarrow \operatorname{Fix}(\neg)\neq\varnothing $$ $$ \operatorname{Fix}(\neg)=\varnothing $$ $$ \Rightarrow A\nrightarrowtail_{\operatorname{surj}}\Omega^A $$ --- ## $S_{272}$ $$ R:\mathbb N\to\mathbb N $$ $$ \varphi_e = \varphi_{R(e)} $$ $$ R = \operatorname{FixRec} $$ $$ \forall f: \exists e: \varphi_e = \varphi_{f(e)} $$ $$ e \in \operatorname{FixExt}(f) $$ --- ## $S_{273}$ $$ P $$ $$ \operatorname{Eval}(P,x) $$ $$ \operatorname{Code}(P) $$ $$ U(\operatorname{Code}(P),x) = P(x) $$ $$ U:\mathbb N\times X\to Y $$ $$ U_e(x) := U(e,x) $$ --- ## $S_{274}$ $$ C:X\to\mathbb N $$ $$ D:\mathbb N\to X $$ $$ D(C(x))=x $$ $$ R_X := D\circ R_\mathbb N\circ C $$ $$ R_X:X\to X $$ $$ R_X(x)\sim x $$ --- ## $S_{275}$ $$ \mathcal F \xrightarrow{C} \mathbb N \xrightarrow{R_\mathbb N} \mathbb N \xrightarrow{D} \mathcal F $$ $$ \widetilde R := D\circ R_\mathbb N\circ C $$ $$ Q\circ\widetilde R = Q $$ $$ [\widetilde R(\Phi)] = [\Phi] $$ --- ## $S_{276}$ $$ N:\mathcal F\to\mathcal F $$ $$ N^2=N $$ $$ \widetilde R:\mathcal F\to\mathcal F $$ $$ N\circ\widetilde R = N $$ $$ \widetilde R\circ N \sim N $$ $$ [N,\widetilde R] := N\widetilde R-\widetilde RN $$ --- ## $S_{277}$ $$ [N,\widetilde R]\sim0 $$ $$ N\widetilde R \sim \widetilde RN $$ $$ \operatorname{Fix}(N) $$ $$ \widetilde R: \operatorname{Fix}(N)\to\operatorname{Fix}(N) $$ $$ \widetilde R(\Phi)\equiv\Phi $$ --- ## $S_{278}$ $$ \mathcal A := \langle N,\widetilde R,\circ \rangle $$ $$ N^2=N $$ $$ N\widetilde R = \widetilde RN $$ $$ \widetilde R^n $$ $$ \mathcal A \subseteq \operatorname{End}(\mathcal F) $$ --- ## $S_{279}$ $$ \operatorname{Spec}(\mathcal A) $$ $$ \chi: \mathcal A\to\mathbb C $$ $$ \chi(ab)=\chi(a)\chi(b) $$ $$ \chi(a+b)=\chi(a)+\chi(b) $$ $$ \chi(1)=1 $$ $$ \widehat a(\chi):=\chi(a) $$ --- ## $S_{280}$ $$ N^2=N $$ $$ \sigma(N)\subseteq\{0,1\} $$ $$ \mathcal F = \ker N \oplus \operatorname{im}N $$ $$ \widetilde R = \widetilde R_0 \oplus \widetilde R_1 $$ $$ \widetilde R_1: \operatorname{im}N\to\operatorname{im}N $$ --- ## $S_{281}$ $$ \Phi = \Phi_0+\Phi_1 $$ $$ \Phi_0\in\ker N $$ $$ \Phi_1\in\operatorname{im}N $$ $$ N(\Phi)=\Phi_1 $$ $$ \widetilde R(\Phi) = \widetilde R_0(\Phi_0) + \widetilde R_1(\Phi_1) $$ --- ## $S_{282}$ $$ \widetilde R_1(\Phi_1) \equiv \Phi_1 $$ $$ Q(\widetilde R_1(\Phi_1)) = Q(\Phi_1) $$ $$ \widetilde R_1 = \operatorname{id}_{\mathfrak F} \quad \operatorname{mod}\ \equiv $$ $$ \widetilde R_1^n \equiv \operatorname{id} $$ --- ## $S_{283}$ $$ \operatorname{Orb}_{\widetilde R_1}(\Phi) $$ $$ = \{\Phi,\widetilde R_1\Phi,\widetilde R_1^2\Phi,\ldots\} $$ $$ Q \left( \operatorname{Orb}_{\widetilde R_1}(\Phi) \right) = \{[\Phi]\} $$ $$ |\operatorname{Orb}_{\widetilde R_1}(\Phi)| \ge1 $$ --- ## $S_{284}$ $$ \ell:\mathcal F\to\mathbb R_{\ge0} $$ $$ \ell(\widetilde R_1(\Phi)) \le \ell(\Phi) $$ $$ \Phi_{n+1} = \widetilde R_1(\Phi_n) $$ $$ \ell(\Phi_{n+1}) \le \ell(\Phi_n) $$ $$ \ell_\infty = \inf_n\ell(\Phi_n) $$ --- ## $S_{285}$ $$ d(\Phi,\Psi) $$ $$ \Phi\equiv\Psi \Rightarrow d(\Phi,\Psi)=0 $$ $$ d(\widetilde R_1\Phi,\widetilde R_1\Psi) \le d(\Phi,\Psi) $$ $$ d(\Phi_n,\Phi_{n+1}) \to0 $$ $$ \Phi_n\to\Phi^\ast $$ --- ## $S_{286}$ $$ \widetilde R_1(\Phi^\ast) = \Phi^\ast $$ $$ Q(\Phi^\ast) = Q(\Phi_0) $$ $$ \Phi^\ast \in \operatorname{Fix}(\widetilde R_1) $$ $$ \operatorname{Fix}(\widetilde R_1) \cap [\Phi_0]_\equiv \neq\varnothing $$ --- ## $S_{287}$ $$ \operatorname{NF}: \mathfrak F\to\mathcal F $$ $$ Q\circ\operatorname{NF} = \operatorname{id}_{\mathfrak F} $$ $$ \operatorname{NF}([\Phi]) = \Phi^\ast $$ $$ \operatorname{NF}(Q(\Phi)) \equiv\Phi $$ $$ N = \operatorname{NF}\circ Q $$ --- ## $S_{288}$ $$ \operatorname{NF} : \mathfrak F\to\operatorname{Fix}(N) $$ $$ Q: \operatorname{Fix}(N)\to\mathfrak F $$ $$ Q\circ\operatorname{NF} = \operatorname{id}_{\mathfrak F} $$ $$ \operatorname{NF}\circ Q = \operatorname{id}_{\operatorname{Fix}(N)} $$ $$ \mathfrak F \cong \operatorname{Fix}(N) $$ --- ## $S_{289}$ $$ \mathbf E_0:=\mathfrak F $$ $$ \mathbf E_{n+1}:=\operatorname{End}(\mathbf E_n) $$ $$ N_n:\mathbf E_n\to\mathbf E_n $$ $$ N_n^2=N_n $$ $$ \mathbf F_n := \operatorname{Fix}(N_n) $$ $$ \mathbf F_n \cong \mathbf E_n/{\equiv_n} $$ --- ## $S_{290}$ $$ \mathcal N := (N_0,N_1,N_2,\ldots) $$ $$ \mathcal N^2=\mathcal N $$ $$ \mathbf F_\omega := \prod_{n<\omega}\mathbf F_n $$ $$ \mathbf F_\omega = \operatorname{Fix}(\mathcal N) $$ --- ## $S_{291}$ $$ \sigma: \mathbf F_\omega\to\mathbf F_\omega $$ $$ \sigma(x_0,x_1,x_2,\ldots) = (x_1,x_2,x_3,\ldots) $$ $$ \mathcal N\circ\sigma = \sigma\circ\mathcal N $$ $$ \sigma(\mathbf F_\omega) \subseteq \mathbf F_\omega $$ --- ## $S_{292}$ $$ \Delta: \mathbf F_\omega \to \mathbf F_\omega $$ $$ \Delta(x_0,x_1,\ldots) = ( x_0(x_0), x_1(x_1), \ldots ) $$ $$ \Delta^2 $$ $$ \operatorname{Fix}(\Delta) $$ --- ## $S_{293}$ $$ s^\ast = \Delta(s^\ast) $$ $$ s^\ast_n = s^\ast_n(s^\ast_n) $$ $$ \forall n: s^\ast_n \in \operatorname{Fix} ( \operatorname{ev}_n ) $$ $$ s^\ast \in \operatorname{Fix}(\Delta) $$ --- ## $S_{294}$ $$ \Xi : \mathbf F_\omega \to \mathbf F_\omega $$ $$ \Xi := \sigma\circ\Delta $$ $$ \Xi(s) = \sigma(\Delta(s)) $$ $$ \Xi^n(s) $$ $$ \operatorname{Orb}_\Xi(s) $$ --- ## $S_{295}$ $$ \omega_\Xi(s) := \bigcap_{m<\omega} \overline{ \{\Xi^n(s):n\ge m\} } $$ $$ \Xi(\omega_\Xi(s)) = \omega_\Xi(s) $$ $$ \omega_\Xi(s) \subseteq \mathbf F_\omega $$ $$ \operatorname{Inv}(\omega_\Xi(s),\Xi) $$ --- ## $S_{296}$ $$ \mathcal K := \{\Xi^n:n\in\mathbb N\} $$ $$ \operatorname{Cl}_\circ(\mathcal K) = \mathcal K $$ $$ \operatorname{Cl}_{\operatorname{lim}}(\mathcal K) $$ $$ \Xi^\omega $$ $$ \Xi^{\omega+1} = \Xi\circ\Xi^\omega $$ --- ## $S_{297}$ $$ \Xi^\lambda = \operatorname{Lim}_{\beta<\lambda}\Xi^\beta $$ $$ \Xi^{\alpha+\beta} = \Xi^\beta\circ\Xi^\alpha $$ $$ \Xi^0 = \operatorname{id} $$ $$ \Xi^1 = \Xi $$ $$ \Xi^\alpha $$ --- ## $S_{298}$ $$ \exists\gamma: \Xi^{\gamma+1} \simeq \Xi^\gamma $$ $$ \Xi^\gamma \circ \Xi \simeq \Xi^\gamma $$ $$ \Xi^\gamma \circ \Xi^\gamma \simeq \Xi^\gamma $$ $$ \operatorname{im}(\Xi^\gamma) = \operatorname{Fix}(\Xi) $$ --- ## $S_{299}$ $$ \mathbf F_\infty := \operatorname{im}(\Xi^\gamma) $$ $$ \Xi: \mathbf F_\infty \to \mathbf F_\infty $$ $$ \Xi \simeq \operatorname{id}_{\mathbf F_\infty} $$ $$ \Delta \simeq \sigma^{-1} $$ $$ \sigma\circ\Delta \simeq \operatorname{id}_{\mathbf F_\infty} $$ --- ## $S_{300}$ $$ S_{240} \to S_{241} \to \cdots \to S_{300} $$ $$ \operatorname{Reflect} \to \operatorname{Localize} \to \operatorname{Universe} \to \operatorname{Prof} \to \operatorname{Kan} \to \operatorname{Monad} \to \operatorname{Bialg} \to \operatorname{Diag} \to \operatorname{Norm} \to \operatorname{SelfAct} $$ $$ \mathcal F \to \mathfrak F \to \operatorname{Fix}(N) \to \mathbf F_\omega \to \operatorname{Fix}(\Xi) $$ $$ N^2=N $$ $$ \mathcal N^2=\mathcal N $$ $$ \Xi^\gamma \circ \Xi^\gamma \simeq \Xi^\gamma $$ $$ \mathbf F_\infty = \operatorname{im}(\Xi^\gamma) $$ $$ \Xi|_{\mathbf F_\infty} \simeq \operatorname{id} $$ $$ \boxed{ SREG_{005} : S_{241} \to S_{242} \to \cdots \to S_{300} } $$ $$ \boxed{ S_{300}\to S_{301} } $$ --- # SREG-006 $$ S_{300}\longrightarrow S_{301} $$ --- ## $S_{301}$ $$ p:E\to B $$ $$ b\in B $$ $$ E_b:=p^{-1}(b) $$ $$ f:b\to b' $$ $$ f^\ast:E_{b'}\to E_b $$ $$ (g\circ f)^\ast = f^\ast\circ g^\ast $$ --- ## $S_{302}$ $$ \mathcal E \xrightarrow{p} \mathcal B $$ $$ e\in\mathcal E $$ $$ u:b\to p(e) $$ $$ \exists\bar u:e'\to e $$ $$ p(\bar u)=u $$ $$ \forall v:d\to e,\ p(v)=u\circ w \Rightarrow \exists!v':d\to e': v=\bar u\circ v' $$ --- ## $S_{303}$ $$ \mathcal E_b := p^{-1}(b) $$ $$ u:b\to c $$ $$ u^\ast:\mathcal E_c\to\mathcal E_b $$ $$ \operatorname{id}_b^\ast \simeq \operatorname{id}_{\mathcal E_b} $$ $$ (v\circ u)^\ast \simeq u^\ast\circ v^\ast $$ --- ## $S_{304}$ $$ F:\mathcal B^{op}\to\mathbf{Cat} $$ $$ \int_{\mathcal B}F \xrightarrow{p} \mathcal B $$ $$ \left(\int_{\mathcal B}F\right)_b \simeq F(b) $$ $$ u:b\to c $$ $$ u^\ast \simeq F(u) $$ --- ## $S_{305}$ $$ \operatorname{Fib}(\mathcal B) $$ $$ [\mathcal B^{op},\mathbf{Cat}] $$ $$ \operatorname{Fib}(\mathcal B) \simeq [\mathcal B^{op},\mathbf{Cat}] $$ $$ \operatorname{Un} \dashv \operatorname{St} $$ --- ## $S_{306}$ $$ A:\mathcal B\to\mathcal U $$ $$ \sum_{b:\mathcal B}A(b) $$ $$ \pi_1: \sum_{b:\mathcal B}A(b) \to \mathcal B $$ $$ \pi_1(b,a)=b $$ $$ \operatorname{fib}_{\pi_1}(b) \simeq A(b) $$ --- ## $S_{307}$ $$ f:A\to B $$ $$ \operatorname{fib}_f(b) := \sum_{a:A}\operatorname{Id}_B(f(a),b) $$ $$ A \simeq \sum_{b:B}\operatorname{fib}_f(b) $$ $$ f = \pi_1 \circ \operatorname{eqv} $$ --- ## $S_{308}$ $$ A\xrightarrow{f}B\xrightarrow{g}C $$ $$ \operatorname{fib}_{g\circ f}(c) \simeq \sum_{b:\operatorname{fib}_g(c)} \operatorname{fib}_f(\pi_1 b) $$ $$ \Sigma_f $$ $$ \Pi_f $$ $$ \Delta_f $$ --- ## $S_{309}$ $$ \Sigma_f \dashv \Delta_f \dashv \Pi_f $$ $$ \Delta_f(X) = X\times_B A $$ $$ \Sigma_f(Y) \to X \iff Y \to \Delta_f(X) $$ $$ \Delta_f(X) \to Y \iff X \to \Pi_f(Y) $$ --- ## $S_{310}$ $$ \begin{CD} A' @>{g'}>> B'\\ @V{f'}VV @VV{f}V\\ A @>>{g}> B \end{CD} $$ $$ A' \simeq A\times_BB' $$ $$ \Delta_g\Sigma_f \simeq \Sigma_{f'}\Delta_{g'} $$ $$ \Pi_{g'}\Delta_{f'} \simeq \Delta_f\Pi_g $$ --- ## $S_{311}$ $$ \mathcal C $$ $$ \operatorname{Sub}(X) $$ $$ f^\ast: \operatorname{Sub}(Y) \to \operatorname{Sub}(X) $$ $$ \exists_f \dashv f^\ast \dashv \forall_f $$ $$ \exists_f(P) $$ $$ \forall_f(P) $$ --- ## $S_{312}$ $$ P\le f^\ast(Q) \iff \exists_f(P)\le Q $$ $$ f^\ast(Q)\le P \iff Q\le\forall_f(P) $$ $$ \exists_f(P\wedge f^\ast Q) = \exists_f(P)\wedge Q $$ --- ## $S_{313}$ $$ \Omega $$ $$ \top:1\to\Omega $$ $$ \chi_P:X\to\Omega $$ $$ P \simeq X\times_\Omega1 $$ $$ \operatorname{Sub}(X) \simeq \operatorname{Hom}(X,\Omega) $$ --- ## $S_{314}$ $$ \Omega^X $$ $$ \chi: \operatorname{Sub}(X\times Y) \to \operatorname{Hom}(Y,\Omega^X) $$ $$ \operatorname{Sub}(X\times Y) \simeq \operatorname{Hom}(Y,\Omega^X) $$ --- ## $S_{315}$ $$ P(X) := \Omega^X $$ $$ \eta_X:X\to P(P(X)) $$ $$ \eta_X(x) = \{A\subseteq X:x\in A\} $$ $$ P: \mathcal C^{op}\to\mathcal C $$ --- ## $S_{316}$ $$ R\subseteq X\times Y $$ $$ R^\sharp:X\to P(Y) $$ $$ R^\flat:Y\to P(X) $$ $$ R^\sharp(x) = \{y:xRy\} $$ $$ R^\flat(y) = \{x:xRy\} $$ --- ## $S_{317}$ $$ A^\uparrow := \{y:\forall x\in A,\ xRy\} $$ $$ B^\downarrow := \{x:\forall y\in B,\ xRy\} $$ $$ A\subseteq B^\downarrow \iff B\subseteq A^\uparrow $$ $$ (-)^\uparrow \dashv (-)^\downarrow $$ --- ## $S_{318}$ $$ c_X := (-)^{\uparrow\downarrow} $$ $$ c_X^2=c_X $$ $$ A\subseteq c_X(A) $$ $$ A\subseteq B \Rightarrow c_X(A)\subseteq c_X(B) $$ $$ \operatorname{Fix}(c_X) $$ --- ## $S_{319}$ $$ \mathfrak L_R := \operatorname{Fix}(c_X) $$ $$ A\wedge B = A\cap B $$ $$ A\vee B = c_X(A\cup B) $$ $$ \bigwedge_iA_i = \bigcap_iA_i $$ $$ \bigvee_iA_i = c_X\left(\bigcup_iA_i\right) $$ --- ## $S_{320}$ $$ \mathfrak L_R $$ $$ f:\mathfrak L_R\to\mathfrak L_R $$ $$ f \ \le\ g $$ $$ \mu f $$ $$ \nu f $$ $$ f(\mu f)=\mu f $$ $$ f(\nu f)=\nu f $$ --- ## $S_{321}$ $$ \Phi(X) $$ $$ X \notin \operatorname{Neg}(\Phi) $$ $$ F_\Phi: \mathfrak L_R\to\mathfrak L_R $$ $$ \llbracket\mu X.\Phi\rrbracket = \mu F_\Phi $$ $$ \llbracket\nu X.\Phi\rrbracket = \nu F_\Phi $$ --- ## $S_{322}$ $$ \mu X.\Phi = \Phi[\mu X.\Phi/X] $$ $$ \nu X.\Phi = \Phi[\nu X.\Phi/X] $$ $$ \Phi[A/X]\le A \Rightarrow \mu X.\Phi\le A $$ $$ A\le\Phi[A/X] \Rightarrow A\le\nu X.\Phi $$ --- ## $S_{323}$ $$ \operatorname{Alt}(\Phi) $$ $$ \mu \leftrightarrow \nu $$ $$ \Sigma_n^\mu $$ $$ \Pi_n^\mu $$ $$ \Sigma_n^\mu \cup \Pi_n^\mu $$ $$ \operatorname{rank}_\mu(\Phi) $$ --- ## $S_{324}$ $$ \mathcal G_\Phi $$ $$ V_\Phi $$ $$ E_\Phi $$ $$ \Omega_\Phi:V_\Phi\to\mathbb N $$ $$ \Phi \leftrightarrow \mathcal G_\Phi $$ $$ \llbracket\Phi\rrbracket = \operatorname{Win}_\exists(\mathcal G_\Phi) $$ --- ## $S_{325}$ $$ \Phi\equiv\Psi $$ $$ \mathcal G_\Phi \sim \mathcal G_\Psi $$ $$ \operatorname{Win}_\exists(\mathcal G_\Phi) = \operatorname{Win}_\exists(\mathcal G_\Psi) $$ $$ [\Phi]_\equiv \mapsto [\mathcal G_\Phi]_\sim $$ --- ## $S_{326}$ $$ \mathfrak F_\mu := \mathcal F_\mu/{\equiv} $$ $$ \mathfrak G := \mathcal G/{\sim} $$ $$ \Gamma: \mathfrak F_\mu \to \mathfrak G $$ $$ \Lambda: \mathfrak G \to \mathfrak F_\mu $$ $$ \Lambda\circ\Gamma = \operatorname{id} $$ --- ## $S_{327}$ $$ \Gamma\circ\Lambda \simeq \operatorname{id} $$ $$ \mathfrak F_\mu \simeq \mathfrak G $$ $$ \operatorname{Fix}_{\mu\nu} \simeq \operatorname{Win} $$ --- ## $S_{328}$ $$ \mathcal R: \mathfrak F_\mu\to\mathfrak F_\mu $$ $$ \widehat{\mathcal R} := \Gamma\circ\mathcal R\circ\Lambda $$ $$ \widehat{\mathcal R}: \mathfrak G\to\mathfrak G $$ $$ \Gamma\circ\mathcal R = \widehat{\mathcal R}\circ\Gamma $$ --- ## $S_{329}$ $$ \operatorname{Fix}(\mathcal R) $$ $$ \operatorname{Fix}(\widehat{\mathcal R}) $$ $$ \Gamma : \operatorname{Fix}(\mathcal R) \to \operatorname{Fix}(\widehat{\mathcal R}) $$ $$ \Gamma \ \operatorname{iso} $$ --- ## $S_{330}$ $$ \mathcal C $$ $$ F:\mathcal C\to\mathcal C $$ $$ \operatorname{Alg}(F) $$ $$ \operatorname{Coalg}(F) $$ $$ \mu F $$ $$ \nu F $$ $$ \mu F\to\nu F $$ --- ## $S_{331}$ $$ \iota:F(\mu F)\to\mu F $$ $$ \omega:\nu F\to F(\nu F) $$ $$ h:\mu F\to\nu F $$ $$ \omega\circ h = F(h)\circ\iota^{-1} $$ $$ h = \operatorname{hylo}(\omega,\iota) $$ --- ## $S_{332}$ $$ \operatorname{cata}(\alpha) : \mu F\to A $$ $$ \operatorname{ana}(\beta) : B\to\nu F $$ $$ \operatorname{hylo}(\alpha,\beta) := \operatorname{cata}(\alpha) \circ \operatorname{ana}(\beta) $$ --- ## $S_{333}$ $$ T_F = \mu X.(I+F(X)) $$ $$ C_F = \nu X.(I\times F(X)) $$ $$ \operatorname{fold}_F $$ $$ \operatorname{unfold}_F $$ $$ \operatorname{fold}_F \circ \operatorname{unfold}_F $$ --- ## $S_{334}$ $$ F^0:=\operatorname{id} $$ $$ F^{n+1}:=F\circ F^n $$ $$ F^\omega := \operatorname{colim}_{n<\omega}F^n $$ $$ F^{\omega+1} = F\circ F^\omega $$ $$ F^\lambda := \operatorname{colim}_{\beta<\lambda}F^\beta $$ --- ## $S_{335}$ $$ \exists\gamma: F^{\gamma+1} \simeq F^\gamma $$ $$ F^\gamma \simeq F(F^\gamma) $$ $$ F^\gamma \in \operatorname{Fix}(F) $$ $$ \operatorname{im}(F^\gamma) \subseteq \operatorname{Fix}(F) $$ --- ## $S_{336}$ $$ \operatorname{Fix}(F) $$ $$ \operatorname{Fix}^2(F) := \operatorname{Fix} ( \operatorname{Fix}(F) ) $$ $$ \operatorname{Fix}^{n+1}(F) := \operatorname{Fix} ( \operatorname{Fix}^n(F) ) $$ $$ \operatorname{Fix}^\omega(F) $$ --- ## $S_{337}$ $$ \mathcal X_0:=\mathcal F $$ $$ \mathcal X_{n+1} := \operatorname{Fix}(\mathcal R_n) $$ $$ \mathcal R_n: \mathcal X_n\to\mathcal X_n $$ $$ \mathcal X_{n+1}\subseteq\mathcal X_n $$ $$ \mathcal X_\omega := \bigcap_{n<\omega}\mathcal X_n $$ --- ## $S_{338}$ $$ \mathcal R_{n+1} := \mathcal R_n|_{\mathcal X_{n+1}} $$ $$ \mathcal R_{n+1} \simeq \operatorname{id}_{\mathcal X_{n+1}} $$ $$ \forall x\in\mathcal X_\omega,\ \forall n,\ \mathcal R_n(x)=x $$ --- ## $S_{339}$ $$ \mathcal X_\omega = \bigcap_{n<\omega} \operatorname{Fix}(\mathcal R_n) $$ $$ \mathcal X_{\omega+1} := \operatorname{Fix} ( \mathcal R_\omega ) $$ $$ \mathcal R_\omega := \varprojlim_n\mathcal R_n $$ --- ## $S_{340}$ $$ \mathcal X_\lambda := \bigcap_{\beta<\lambda}\mathcal X_\beta $$ $$ \mathcal R_\lambda := \varprojlim_{\beta<\lambda}\mathcal R_\beta $$ $$ \mathcal X_{\lambda+1} = \operatorname{Fix}(\mathcal R_\lambda) $$ --- ## $S_{341}$ $$ \exists\Gamma: \mathcal X_{\Gamma+1} = \mathcal X_\Gamma $$ $$ \mathcal R_\Gamma \simeq \operatorname{id}_{\mathcal X_\Gamma} $$ $$ \mathcal X_\Gamma = \bigcap_{\beta<\Gamma} \operatorname{Fix}(\mathcal R_\beta) $$ --- ## $S_{342}$ $$ \mathcal X_\Gamma $$ $$ \operatorname{End}(\mathcal X_\Gamma) $$ $$ \operatorname{Idem}(\mathcal X_\Gamma) := \{e:e^2=e\} $$ $$ \operatorname{Aut}(\mathcal X_\Gamma) $$ $$ \operatorname{End} = \operatorname{Idem} \cup \operatorname{NonIdem} $$ --- ## $S_{343}$ $$ e,f\in\operatorname{Idem}(\mathcal X_\Gamma) $$ $$ ef=fe $$ $$ ef \in \operatorname{Idem}(\mathcal X_\Gamma) $$ $$ \operatorname{Fix}(ef) = \operatorname{Fix}(e)\cap\operatorname{Fix}(f) $$ --- ## $S_{344}$ $$ \mathcal I := \langle e_i \rangle_{i\in I} $$ $$ e_ie_j=e_je_i $$ $$ e_i^2=e_i $$ $$ e_J := \prod_{j\in J}e_j $$ $$ \operatorname{Fix}(e_J) = \bigcap_{j\in J}\operatorname{Fix}(e_j) $$ --- ## $S_{345}$ $$ J\subseteq K \Rightarrow e_K\le e_J $$ $$ \operatorname{Fix}(e_K) \subseteq \operatorname{Fix}(e_J) $$ $$ \mathcal L_{\mathcal I} := \{\operatorname{Fix}(e_J):J\subseteq I\} $$ $$ \bigwedge = \cap $$ --- ## $S_{346}$ $$ \bigvee_{\alpha} \operatorname{Fix}(e_{J_\alpha}) $$ $$ = \operatorname{Fix} \left( \prod_{\alpha}e_{J_\alpha} \right) $$ $$ \mathcal L_{\mathcal I} $$ $$ \operatorname{Complete}(\mathcal L_{\mathcal I}) $$ --- ## $S_{347}$ $$ x\in\mathcal X_\Gamma $$ $$ \operatorname{Stab}(x) := \{e\in\mathcal I:e(x)=x\} $$ $$ \operatorname{Orb}_{\mathcal I}(x) := \{e(x):e\in\mathcal I\} $$ $$ |\operatorname{Orb}_{\mathcal I}(x)| \le |\mathcal I| $$ --- ## $S_{348}$ $$ e_\ast := \prod_{e\in\operatorname{Stab}(x)}e $$ $$ e_\ast(x)=x $$ $$ \operatorname{Fix}(e_\ast) = \bigcap_{e\in\operatorname{Stab}(x)} \operatorname{Fix}(e) $$ --- ## $S_{349}$ $$ C(x) := \operatorname{Fix}(e_\ast) $$ $$ x\in C(x) $$ $$ y\in C(x) \Rightarrow \operatorname{Stab}(x) \subseteq \operatorname{Stab}(y) $$ $$ C(y)\subseteq C(x) $$ --- ## $S_{350}$ $$ x\sim_C y \iff C(x)=C(y) $$ $$ [x]_C $$ $$ \mathcal X_\Gamma/{\sim_C} $$ $$ \mathcal C: \mathcal X_\Gamma \to \mathcal X_\Gamma/{\sim_C} $$ --- ## $S_{351}$ $$ \mathfrak C := \{C(x):x\in\mathcal X_\Gamma\} $$ $$ C(x)\cap C(y) $$ $$ C(x)\subseteq C(y) $$ $$ (\mathfrak C,\subseteq) $$ $$ \operatorname{Poset}(\mathfrak C) $$ --- ## $S_{352}$ $$ N: \mathcal X_\Gamma \to \mathfrak C $$ $$ N(x):=C(x) $$ $$ N(x)=N(y) \iff x\sim_Cy $$ $$ \operatorname{im}(N)=\mathfrak C $$ --- ## $S_{353}$ $$ E: \mathfrak C\to\mathcal X_\Gamma $$ $$ E(C)\in C $$ $$ N(E(C))=C $$ $$ N\circ E = \operatorname{id}_{\mathfrak C} $$ $$ E\circ N \sim_C \operatorname{id}_{\mathcal X_\Gamma} $$ --- ## $S_{354}$ $$ R:=E\circ N $$ $$ R^2 = E N E N = E N = R $$ $$ \operatorname{Fix}(R) = \operatorname{im}(E) $$ $$ \mathfrak C \cong \operatorname{Fix}(R) $$ --- ## $S_{355}$ $$ \operatorname{End}(\mathfrak C) $$ $$ \widehat F := N\circ F\circ E $$ $$ F:\mathcal X_\Gamma\to\mathcal X_\Gamma $$ $$ \widehat F: \mathfrak C\to\mathfrak C $$ $$ \widehat{G\circ F} = \widehat G\circ\widehat F $$ --- ## $S_{356}$ $$ \widehat{\operatorname{id}} = \operatorname{id}_{\mathfrak C} $$ $$ F\sim_CG \Rightarrow \widehat F=\widehat G $$ $$ \operatorname{End}(\mathcal X_\Gamma)/{\sim_C} \to \operatorname{End}(\mathfrak C) $$ --- ## $S_{357}$ $$ \widehat F^2=\widehat F $$ $$ \operatorname{Fix}(\widehat F) $$ $$ \operatorname{im}(\widehat F) $$ $$ \operatorname{Fix}(\widehat F) = \operatorname{im}(\widehat F) $$ --- ## $S_{358}$ $$ \mathfrak C_0:=\mathfrak C $$ $$ \mathfrak C_{n+1} := \operatorname{Fix}(\widehat F_n) $$ $$ \mathfrak C_\omega = \bigcap_{n<\omega}\mathfrak C_n $$ $$ \mathfrak C_{\lambda} = \bigcap_{\beta<\lambda}\mathfrak C_\beta $$ --- ## $S_{359}$ $$ \exists\Lambda: \mathfrak C_{\Lambda+1} = \mathfrak C_\Lambda $$ $$ \widehat F_\Lambda \simeq \operatorname{id}_{\mathfrak C_\Lambda} $$ $$ \mathfrak C_\infty := \mathfrak C_\Lambda $$ $$ \operatorname{End}(\mathfrak C_\infty) $$ --- ## $S_{360}$ $$ S_{300} \to S_{301} \to \cdots \to S_{360} $$ $$ \operatorname{Fib} \to \Sigma_f\dashv\Delta_f\dashv\Pi_f \to \operatorname{Closure} \to \mu\nu \to \operatorname{Game} \to \operatorname{Hylo} \to \operatorname{TransFix} \to \operatorname{Idem} \to \operatorname{Class} \to \operatorname{MetaFix} $$ $$ \mathcal F \to \mathfrak F_\mu \to \mathfrak G \to \mathcal X_\Gamma \to \mathfrak C \to \mathfrak C_\infty $$ $$ \mathcal X_\Gamma = \bigcap_{\beta<\Gamma} \operatorname{Fix}(\mathcal R_\beta) $$ $$ \mathfrak C_\infty = \bigcap_{\beta<\Lambda} \operatorname{Fix}(\widehat F_\beta) $$ $$ R^2=R $$ $$ \widehat F_\Lambda \simeq \operatorname{id}_{\mathfrak C_\infty} $$ $$ \boxed{ SREG_{006} : S_{301} \to S_{302} \to \cdots \to S_{360} } $$ $$ \boxed{ S_{360}\to S_{361} } $$ --- # SREG-007 $$ S_{360}\longrightarrow S_{361} $$ --- ## $S_{361}$ $$ L $$ $$ c_i:L\to L \qquad (i\in I) $$ $$ c_i^2=c_i $$ $$ x\le c_i(x) $$ $$ x\le y \Rightarrow c_i(x)\le c_i(y) $$ $$ \mathcal C_I := \{c_i\}_{i\in I} $$ --- ## $S_{362}$ $$ c_i c_j=c_j c_i $$ $$ c_J := \prod_{j\in J}c_j $$ $$ c_J^2=c_J $$ $$ \operatorname{Fix}(c_J) = \bigcap_{j\in J}\operatorname{Fix}(c_j) $$ $$ J\subseteq K \Rightarrow \operatorname{Fix}(c_K) \subseteq \operatorname{Fix}(c_J) $$ --- ## $S_{363}$ $$ \mathcal L_c := \{\operatorname{Fix}(c_J):J\subseteq I\} $$ $$ \bigwedge_\alpha X_\alpha = \bigcap_\alpha X_\alpha $$ $$ \bigvee_\alpha X_\alpha = \operatorname{Fix} \left( \prod_\alpha c_{J_\alpha} \right) $$ $$ (\mathcal L_c,\subseteq) $$ --- ## $S_{364}$ $$ j:L\to L $$ $$ j^2=j $$ $$ x\le j(x) $$ $$ j(x\wedge y) = j(x)\wedge j(y) $$ $$ \operatorname{Fix}(j) $$ $$ \operatorname{Nuc}(L) $$ --- ## $S_{365}$ $$ \operatorname{Frm}(L) $$ $$ j\in\operatorname{Nuc}(L) $$ $$ L_j := \operatorname{Fix}(j) $$ $$ a\vee_j b := j(a\vee b) $$ $$ a\wedge_j b := a\wedge b $$ $$ L_j\in\mathbf{Frm} $$ --- ## $S_{366}$ $$ \operatorname{Loc}(L) $$ $$ j:L\to L $$ $$ L\to L_j $$ $$ L_j\hookrightarrow L $$ $$ L\to L_j\hookrightarrow L $$ $$ j $$ $$ j^2=j $$ --- ## $S_{367}$ $$ \mathcal C $$ $$ W\subseteq\operatorname{Mor}(\mathcal C) $$ $$ X\in W^\perp $$ $$ \forall w:A\to B\in W: \operatorname{Hom}(B,X) \xrightarrow{\sim} \operatorname{Hom}(A,X) $$ $$ W^\perp $$ --- ## $S_{368}$ $$ {}^\perp M := \{f: \forall g\in M,\ f\perp g\} $$ $$ M^\perp := \{g: \forall f\in M,\ f\perp g\} $$ $$ ({}^\perp M)^\perp $$ $$ {}^\perp(M^\perp) $$ --- ## $S_{369}$ $$ (\mathcal L,\mathcal R) $$ $$ \mathcal L={}^\perp\mathcal R $$ $$ \mathcal R=\mathcal L^\perp $$ $$ \forall f: f=r\circ l $$ $$ l\in\mathcal L $$ $$ r\in\mathcal R $$ --- ## $S_{370}$ $$ W $$ $$ C $$ $$ F $$ $$ (C\cap W,F) $$ $$ (C,F\cap W) $$ $$ W\circ W\subseteq W $$ $$ f,g,gf\in W \Rightarrow (f\in W\Leftrightarrow g\in W) $$ --- ## $S_{371}$ $$ Q:\mathcal M\to\mathcal M[W^{-1}] $$ $$ W \mapsto \operatorname{Iso} $$ $$ \operatorname{Ho}(\mathcal M) := \mathcal M[W^{-1}] $$ $$ \operatorname{Hom}_{\operatorname{Ho}(\mathcal M)}(X,Y) $$ --- ## $S_{372}$ $$ X_c\to X $$ $$ Y\to Y_f $$ $$ \operatorname{Map}^h(X,Y) := \operatorname{Map}(X_c,Y_f) $$ $$ \pi_0\operatorname{Map}^h(X,Y) \cong \operatorname{Hom}_{\operatorname{Ho}(\mathcal M)}(X,Y) $$ --- ## $S_{373}$ $$ L_W\mathcal C $$ $$ \operatorname{Ob}(L_W\mathcal C) = \operatorname{Ob}(\mathcal C) $$ $$ \operatorname{Map}_{L_W\mathcal C}(X,Y) $$ $$ w\in W \Rightarrow w\in\operatorname{Eq}(L_W\mathcal C) $$ --- ## $S_{374}$ $$ \mathcal C_\infty $$ $$ \operatorname{Map}_{\mathcal C_\infty}(X,Y) \in\mathcal S $$ $$ \operatorname{Ho}(\mathcal C_\infty) $$ $$ \pi_0 \operatorname{Map}_{\mathcal C_\infty}(X,Y) = \operatorname{Hom}_{\operatorname{Ho}(\mathcal C_\infty)}(X,Y) $$ --- ## $S_{375}$ $$ \mathcal C $$ $$ \operatorname{Ind}_\kappa(\mathcal C) $$ $$ \operatorname{colim}_{i\in I}X_i $$ $$ |I|<\kappa $$ $$ \operatorname{Map}(K,-) $$ $$ K\in\mathcal C^\kappa $$ --- ## $S_{376}$ $$ K\in\mathcal C^\kappa $$ $$ \operatorname{Map} \left( K,\operatorname{colim}_iX_i \right) \simeq \operatorname{colim}_i \operatorname{Map}(K,X_i) $$ $$ \mathcal C \simeq \operatorname{Ind}_\kappa(\mathcal C^\kappa) $$ --- ## $S_{377}$ $$ L:\mathcal C\to\mathcal C $$ $$ L^2\simeq L $$ $$ \eta:\operatorname{id}\Rightarrow L $$ $$ L \operatorname{colim} \simeq \operatorname{colim}L $$ $$ \operatorname{Loc}_L(\mathcal C) = \operatorname{Fix}(L) $$ --- ## $S_{378}$ $$ \mathcal C $$ $$ \operatorname{Pro}(\mathcal C) $$ $$ X=\{X_i\}_{i\in I} $$ $$ Y=\{Y_j\}_{j\in J} $$ $$ \operatorname{Hom}_{\operatorname{Pro}(\mathcal C)}(X,Y) = \varprojlim_j \varinjlim_i \operatorname{Hom}_{\mathcal C}(X_i,Y_j) $$ --- ## $S_{379}$ $$ \operatorname{Ind}(\mathcal C^{op})^{op} \simeq \operatorname{Pro}(\mathcal C) $$ $$ \operatorname{Ind} \leftrightarrow \operatorname{Pro} $$ $$ \operatorname{colim} \leftrightarrow \operatorname{lim} $$ --- ## $S_{380}$ $$ B $$ $$ B^{op} $$ $$ D:B^{op}\to\mathbf{Set} $$ $$ D^\ast: B\to\mathbf{Set}^{op} $$ $$ (-)^{op} \circ (-)^{op} = \operatorname{id} $$ --- ## $S_{381}$ $$ B \in \mathbf{Bool} $$ $$ X \in \mathbf{Stone} $$ $$ \operatorname{Clop}(X) \in \mathbf{Bool} $$ $$ \operatorname{Ult}(B) \in \mathbf{Stone} $$ $$ \operatorname{Clop}(\operatorname{Ult}(B)) \cong B $$ $$ \operatorname{Ult}(\operatorname{Clop}(X)) \cong X $$ --- ## $S_{382}$ $$ B $$ $$ a\wedge b $$ $$ a\vee b $$ $$ \neg a $$ $$ 0,1 $$ $$ a\vee\neg a=1 $$ $$ a\wedge\neg a=0 $$ --- ## $S_{383}$ $$ \mathcal U\subseteq B $$ $$ 1\in\mathcal U $$ $$ 0\notin\mathcal U $$ $$ a,b\in\mathcal U \Rightarrow a\wedge b\in\mathcal U $$ $$ a\in\mathcal U \lor \neg a\in\mathcal U $$ --- ## $S_{384}$ $$ A \in \mathbf{CRing} $$ $$ \operatorname{Spec}(A) := \{\mathfrak p\subsetneq A: ab\in\mathfrak p \Rightarrow a\in\mathfrak p \lor b\in\mathfrak p \} $$ $$ V(I) := \{\mathfrak p:I\subseteq\mathfrak p\} $$ --- ## $S_{385}$ $$ D(f) := \operatorname{Spec}(A)\setminus V((f)) $$ $$ D(fg) = D(f)\cap D(g) $$ $$ D(1) = \operatorname{Spec}(A) $$ $$ D(0) = \varnothing $$ --- ## $S_{386}$ $$ \mathcal O_{\operatorname{Spec}A}(D(f)) := A_f $$ $$ A_f $$ $$ A\to A_f $$ $$ f\mapsto f/1 $$ $$ f\in A_f^\times $$ --- ## $S_{387}$ $$ (X,\mathcal O_X) $$ $$ X = \bigcup_iU_i $$ $$ (U_i,\mathcal O_X|_{U_i}) \simeq (\operatorname{Spec}A_i,\mathcal O_{\operatorname{Spec}A_i}) $$ $$ X\in\mathbf{Sch} $$ --- ## $S_{388}$ $$ f:X\to Y $$ $$ f^\#: \mathcal O_Y \to f_\ast\mathcal O_X $$ $$ (f,f^\#) $$ $$ \mathbf{Sch} $$ --- ## $S_{389}$ $$ X\times_ZY $$ $$ \begin{CD} X\times_ZY @>>> Y\\ @VVV @VVV\\ X @>>> Z \end{CD} $$ $$ \operatorname{Spec}A \times_{\operatorname{Spec}R} \operatorname{Spec}B \simeq \operatorname{Spec}(A\otimes_RB) $$ --- ## $S_{390}$ $$ \mathcal F: \mathbf{Sch}^{op}\to\mathbf{Gpd} $$ $$ U\to X $$ $$ V\to X $$ $$ U\times_XV $$ $$ \mathcal F(X) \to \operatorname{Eq} \left( \prod_i\mathcal F(U_i) \rightrightarrows \prod_{i,j}\mathcal F(U_i\times_XU_j) \right) $$ --- ## $S_{391}$ $$ X_\bullet $$ $$ X_0 $$ $$ X_1 \rightrightarrows X_0 $$ $$ X_2 \triplearrows X_1 $$ $$ \operatorname{Desc}(X_\bullet) $$ $$ \operatorname{Tot}(X_\bullet) $$ --- ## $S_{392}$ $$ \mathcal F(X) \simeq \operatorname{Tot} \left( \mathcal F(X_\bullet) \right) $$ $$ \operatorname{Descent}(\mathcal F) $$ $$ \operatorname{HyperDescent}(\mathcal F) $$ --- ## $S_{393}$ $$ \mathcal X : \mathbf{CAlg}^{op} \to \mathcal S $$ $$ \mathcal X(A) $$ $$ A\to B $$ $$ \mathcal X(B)\to\mathcal X(A) $$ $$ \mathcal X \in \operatorname{PSh}_\infty(\mathbf{CAlg}) $$ --- ## $S_{394}$ $$ \mathcal X \simeq \operatorname{Sh}_\infty(\mathbf{CAlg},\tau) $$ $$ \operatorname{Map}(\operatorname{Spec}A,\mathcal X) \simeq \mathcal X(A) $$ $$ \mathcal X $$ $$ \operatorname{Stack}_\infty $$ --- ## $S_{395}$ $$ A \in \mathbf{CAlg} $$ $$ A \to A\oplus M $$ $$ M^2=0 $$ $$ \operatorname{Der}(A,M) $$ $$ L_A $$ $$ \operatorname{Map}_{A\operatorname{-Mod}}(L_A,M) \simeq \operatorname{Der}(A,M) $$ --- ## $S_{396}$ $$ A\to B $$ $$ L_{B/A} $$ $$ B\otimes_A L_A \to L_B \to L_{B/A} $$ $$ L_{C/A} \simeq L_{B/A}\otimes_BC \oplus L_{C/B} $$ --- ## $S_{397}$ $$ x\in\mathcal X(A) $$ $$ M $$ $$ A\oplus M $$ $$ \operatorname{Def}_x(M) $$ $$ T_x\mathcal X $$ $$ \operatorname{Def}_x(M) \simeq \operatorname{Map}(T_x^\ast\mathcal X,M) $$ --- ## $S_{398}$ $$ M[1] $$ $$ \operatorname{Obs}_x(M) $$ $$ \operatorname{Ext}^1(L,M) $$ $$ \operatorname{Ext}^2(L,M) $$ $$ \operatorname{Def} \to \operatorname{Obs} $$ --- ## $S_{399}$ $$ C^\bullet $$ $$ d^2=0 $$ $$ H^\bullet(C) $$ $$ C \simeq H^\bullet(C) $$ $$ C not\simeq H^\bullet(C) $$ $$ \operatorname{Form}(C) $$ --- ## $S_{400}$ $$ (A,d,\mu) $$ $$ d\mu = \mu(d\otimes1+1\otimes d) $$ $$ m_n:A^{\otimes n}\to A $$ $$ \deg(m_n)=2-n $$ $$ \sum_{r+s+t=n} (-1)^{r+st} m_{r+1+t} ( 1^{\otimes r}\otimes m_s\otimes1^{\otimes t} ) = 0 $$ --- ## $S_{401}$ $$ L_\infty $$ $$ \ell_n: \wedge^nV\to V $$ $$ \deg(\ell_n)=2-n $$ $$ \sum_{i+j=n+1} \sum_{\sigma} \chi(\sigma) (-1)^{i(j-1)} \ell_j ( \ell_i(\cdot), \cdot ) = 0 $$ --- ## $S_{402}$ $$ Q \in \operatorname{Coder} ( S^c(sV) ) $$ $$ Q^2=0 $$ $$ L_\infty \leftrightarrow Q $$ $$ \operatorname{MC}(V) := \left\{ x: \sum_{n\ge1} \frac1{n!} \ell_n(x,\ldots,x)=0 \right\} $$ --- ## $S_{403}$ $$ x\in\operatorname{MC}(V) $$ $$ \ell_1^x $$ $$ (\ell_1^x)^2=0 $$ $$ H^\bullet(V,\ell_1^x) $$ $$ T_x\operatorname{MC}(V) $$ --- ## $S_{404}$ $$ R $$ $$ a\to b $$ $$ b\to c $$ $$ a\to^\ast c $$ $$ a\downarrow b \iff \exists c: a\to^\ast c \land b\to^\ast c $$ --- ## $S_{405}$ $$ a\to b $$ $$ a\to c $$ $$ \exists d: b\to^\ast d \land c\to^\ast d $$ $$ \operatorname{CR}(R) $$ $$ \operatorname{SN}(R) $$ --- ## $S_{406}$ $$ \operatorname{SN}(R) \land \operatorname{LC}(R) \Rightarrow \operatorname{CR}(R) $$ $$ a \to^\ast \operatorname{NF}(a) $$ $$ \operatorname{NF}(a) = \operatorname{NF}(b) \iff a\leftrightarrow_R^\ast b $$ --- ## $S_{407}$ $$ \Phi \to_R \Psi $$ $$ \ell(\Psi) < \ell(\Phi) $$ $$ \ell:\mathcal F\to\operatorname{Ord} $$ $$ \operatorname{SN}(R) $$ $$ \neg \exists \Phi_0\to\Phi_1\to\Phi_2\to\cdots $$ --- ## $S_{408}$ $$ \ell(\Phi) = \alpha $$ $$ R(\Phi) = \Psi $$ $$ \ell(\Psi) < \alpha $$ $$ R^\alpha(\Phi) $$ $$ R^\lambda(\Phi) = \operatorname{Lim}_{\beta<\lambda}R^\beta(\Phi) $$ --- ## $S_{409}$ $$ \exists\gamma: R^{\gamma+1}(\Phi) = R^\gamma(\Phi) $$ $$ \Phi^\ast := R^\gamma(\Phi) $$ $$ R(\Phi^\ast)=\Phi^\ast $$ $$ \Phi\to^\ast\Phi^\ast $$ --- ## $S_{410}$ $$ Q:\mathcal F\to\mathfrak F $$ $$ N:\mathfrak F\to\mathcal F $$ $$ Q\circ N = \operatorname{id}_{\mathfrak F} $$ $$ N\circ Q = \operatorname{NF} $$ $$ \operatorname{NF}^2 = \operatorname{NF} $$ --- ## $S_{411}$ $$ \mathfrak F \cong \operatorname{Fix}(\operatorname{NF}) $$ $$ [\Phi] \mapsto \operatorname{NF}(\Phi) $$ $$ \operatorname{NF}(\Phi) = \operatorname{NF}(\Psi) \iff \Phi\equiv\Psi $$ --- ## $S_{412}$ $$ \mathbf E_0:=\mathfrak F $$ $$ \mathbf E_{n+1} := \operatorname{End}(\mathbf E_n) $$ $$ \operatorname{NF}_n: \mathbf E_n\to\mathbf E_n $$ $$ \operatorname{NF}_n^2 = \operatorname{NF}_n $$ --- ## $S_{413}$ $$ \mathbf F_n := \operatorname{Fix}(\operatorname{NF}_n) $$ $$ \mathbf F_{n+1} \subseteq \operatorname{End}(\mathbf F_n) $$ $$ \mathbf F_0 \to \mathbf F_1 \to \mathbf F_2 \to \cdots $$ --- ## $S_{414}$ $$ \mathbf F_\omega := \varprojlim_{n<\omega}\mathbf F_n $$ $$ x = (x_0,x_1,x_2,\ldots) $$ $$ p_{n+1,n}(x_{n+1}) = x_n $$ $$ x\in\mathbf F_\omega $$ --- ## $S_{415}$ $$ \Delta: \mathbf F_\omega \to \mathbf F_\omega $$ $$ \Delta(x)_n := x_{n+1}(x_n) $$ $$ \Delta^2 $$ $$ \operatorname{Fix}(\Delta) $$ --- ## $S_{416}$ $$ x^\ast = \Delta(x^\ast) $$ $$ x_n^\ast = x_{n+1}^\ast(x_n^\ast) $$ $$ \forall n: x_n^\ast \in \mathbf F_n $$ $$ x^\ast\in\operatorname{Fix}(\Delta) $$ --- ## $S_{417}$ $$ \mathcal U := \operatorname{Fix}(\Delta) $$ $$ \Delta|_{\mathcal U} = \operatorname{id}_{\mathcal U} $$ $$ \operatorname{End}(\mathcal U) $$ $$ \operatorname{Aut}(\mathcal U) $$ $$ \operatorname{Idem}(\mathcal U) $$ --- ## $S_{418}$ $$ e\in\operatorname{Idem}(\mathcal U) $$ $$ e^2=e $$ $$ \mathcal U_e := \operatorname{Fix}(e) $$ $$ \mathcal U_e \hookrightarrow \mathcal U $$ $$ \mathcal U \to \mathcal U_e $$ --- ## $S_{419}$ $$ \{e_i\}_{i\in I} \subseteq \operatorname{Idem}(\mathcal U) $$ $$ e_ie_j=e_je_i $$ $$ \mathcal U_\infty := \bigcap_{i\in I} \operatorname{Fix}(e_i) $$ $$ \forall i: e_i|_{\mathcal U_\infty} = \operatorname{id} $$ --- ## $S_{420}$ $$ S_{360} \to S_{361} \to \cdots \to S_{420} $$ $$ \operatorname{Closure} \to \operatorname{Nucleus} \to \operatorname{Localization} \to \operatorname{Infinity} \to \operatorname{Spec} \to \operatorname{Descent} \to \operatorname{Derived} \to \operatorname{Deform} \to \operatorname{Rewrite} \to \operatorname{SelfUniverse} $$ $$ \mathcal F \to \mathfrak F \to \operatorname{Fix}(\operatorname{NF}) \to \mathbf F_\omega \to \operatorname{Fix}(\Delta) \to \mathcal U_\infty $$ $$ \operatorname{NF}^2 = \operatorname{NF} $$ $$ \Delta|_{\mathcal U} = \operatorname{id}_{\mathcal U} $$ $$ \forall i: e_i|_{\mathcal U_\infty} = \operatorname{id} $$ $$ \boxed{ SREG_{007} : S_{361} \to S_{362} \to \cdots \to S_{420} } $$ $$ \boxed{ S_{420}\to S_{421} } $$ --- # SREG-008 $$ S_{420}\longrightarrow S_{421} $$ --- ## $S_{421}$ $$ L_0\dashv R_0 $$ $$ L_1\dashv R_1 $$ $$ L_2\dashv R_2 $$ $$ \cdots $$ $$ L_n:\mathcal C_n\to\mathcal C_{n+1} $$ $$ R_n:\mathcal C_{n+1}\to\mathcal C_n $$ --- ## $S_{422}$ $$ T_n:=R_nL_n $$ $$ W_n:=L_nR_n $$ $$ \eta_n:\operatorname{id}_{\mathcal C_n}\Rightarrow T_n $$ $$ \varepsilon_n:W_n\Rightarrow\operatorname{id}_{\mathcal C_{n+1}} $$ $$ \mu_n:T_n^2\Rightarrow T_n $$ $$ \delta_n:W_n\Rightarrow W_n^2 $$ --- ## $S_{423}$ $$ T_{n+1}L_n = R_{n+1}L_{n+1}L_n $$ $$ R_nT_{n+1} = R_nR_{n+1}L_{n+1} $$ $$ \alpha_n: L_nT_n\Rightarrow T_{n+1}L_n $$ $$ \beta_n: R_nT_{n+1}\Rightarrow T_nR_n $$ --- ## $S_{424}$ $$ \alpha_{n+1}L_n \circ L_{n+1}\alpha_n $$ $$ T_{n+2}L_{n+1}L_n $$ $$ L_{n+1}L_nT_n $$ $$ \operatorname{Coh}(\alpha_n,\alpha_{n+1}) $$ --- ## $S_{425}$ $$ B_\bullet(T,A) $$ $$ B_n(T,A) := T^{n+1}(A) $$ $$ d_i: T^{n+1}(A)\to T^n(A) $$ $$ s_i: T^{n+1}(A)\to T^{n+2}(A) $$ $$ d_id_j=d_{j-1}d_i \qquad i|A| $$ $$ A \mapsto \Omega^A $$ $$ A < \Omega^A $$ --- ## $S_{491}$ $$ \mathcal M $$ $$ \Box:\mathcal M\to\mathcal M $$ $$ \eta:\operatorname{id}\Rightarrow\Box $$ $$ \Box\Box \simeq \Box $$ $$ \operatorname{Fix}(\Box) $$ --- ## $S_{492}$ $$ \Diamond:\mathcal M\to\mathcal M $$ $$ \Diamond\dashv\Box $$ $$ \Diamond\Diamond \simeq \Diamond $$ $$ \operatorname{Fix}(\Diamond) $$ $$ \Diamond\Box \to \operatorname{id} \to \Box\Diamond $$ --- ## $S_{493}$ $$ \Box A \to A $$ $$ A \to \Diamond A $$ $$ \Box A \to \Box\Box A $$ $$ \Diamond\Diamond A \to \Diamond A $$ $$ \Box\Diamond A $$ --- ## $S_{494}$ $$ \Box(A\to B) \to (\Box A\to\Box B) $$ $$ \Diamond(A\times B) \to \Diamond A\times\Diamond B $$ $$ \Box1\simeq1 $$ $$ \Diamond0\simeq0 $$ --- ## $S_{495}$ $$ L:\mathcal C\to\mathcal D $$ $$ I:\mathcal D\hookrightarrow\mathcal C $$ $$ L\dashv I $$ $$ \Box:=IL $$ $$ \operatorname{Fix}(\Box) \simeq \mathcal D $$ --- ## $S_{496}$ $$ R:\mathcal C\to\mathcal C $$ $$ R^2\simeq R $$ $$ \eta:\operatorname{id}\Rightarrow R $$ $$ \mathcal C_R := \operatorname{Fix}(R) $$ $$ \mathcal C \to \mathcal C_R \hookrightarrow \mathcal C $$ --- ## $S_{497}$ $$ R_0,R_1:\mathcal C\to\mathcal C $$ $$ R_0^2\simeq R_0 $$ $$ R_1^2\simeq R_1 $$ $$ R_0R_1\simeq R_1R_0 $$ $$ R_{01}:=R_0R_1 $$ $$ R_{01}^2\simeq R_{01} $$ --- ## $S_{498}$ $$ \operatorname{Fix}(R_{01}) = \operatorname{Fix}(R_0) \cap \operatorname{Fix}(R_1) $$ $$ \mathcal C_{01} $$ $$ \mathcal C \to \mathcal C_{01} $$ --- ## $S_{499}$ $$ \{R_i\}_{i\in I} $$ $$ R_iR_j\simeq R_jR_i $$ $$ R_J := \prod_{j\in J}R_j $$ $$ R_J^2\simeq R_J $$ $$ \operatorname{Fix}(R_J) = \bigcap_{j\in J}\operatorname{Fix}(R_j) $$ --- ## $S_{500}$ $$ \mathcal C_\infty := \bigcap_{i\in I} \operatorname{Fix}(R_i) $$ $$ \forall i,\ R_i|_{\mathcal C_\infty} \simeq \operatorname{id} $$ $$ \mathcal C_\infty \hookrightarrow \mathcal C $$ --- ## $S_{501}$ $$ T:\mathcal C\to\mathcal C $$ $$ \eta:\operatorname{id}\Rightarrow T $$ $$ \mu:T^2\Rightarrow T $$ $$ T^n $$ $$ T^\omega := \operatorname{colim}_{n<\omega}T^n $$ --- ## $S_{502}$ $$ T^{\alpha+1} = T\circ T^\alpha $$ $$ T^\lambda = \operatorname{colim}_{\beta<\lambda}T^\beta $$ $$ \exists\Gamma: T^{\Gamma+1} \simeq T^\Gamma $$ $$ T^\Gamma \simeq T(T^\Gamma) $$ --- ## $S_{503}$ $$ P_T := T^\Gamma $$ $$ P_T^2 \simeq P_T $$ $$ \operatorname{im}(P_T) \subseteq \operatorname{Fix}(T) $$ $$ \operatorname{im}(P_T) = \operatorname{Fix}(P_T) $$ --- ## $S_{504}$ $$ \mathcal D_0:=\mathcal C $$ $$ \mathcal D_{\alpha+1} := \operatorname{Fix}(T_\alpha) $$ $$ \mathcal D_\lambda := \bigcap_{\beta<\lambda}\mathcal D_\beta $$ $$ T_\alpha: \mathcal D_\alpha\to\mathcal D_\alpha $$ --- ## $S_{505}$ $$ \exists\Lambda: \mathcal D_{\Lambda+1} = \mathcal D_\Lambda $$ $$ T_\Lambda \simeq \operatorname{id}_{\mathcal D_\Lambda} $$ $$ \mathcal D_\infty := \mathcal D_\Lambda $$ --- ## $S_{506}$ $$ \operatorname{End}(\mathcal D_\infty) $$ $$ E_0:=\mathcal D_\infty $$ $$ E_{n+1} := \operatorname{End}(E_n) $$ $$ E_\omega := \prod_{n<\omega}E_n $$ --- ## $S_{507}$ $$ e=(e_0,e_1,\ldots)\in E_\omega $$ $$ \operatorname{ev}(e)_n := e_{n+1}(e_n) $$ $$ \operatorname{ev}:E_\omega\to E_\omega $$ $$ \operatorname{Fix}(\operatorname{ev}) $$ --- ## $S_{508}$ $$ e^\ast = \operatorname{ev}(e^\ast) $$ $$ e_n^\ast = e_{n+1}^\ast(e_n^\ast) $$ $$ e^\ast \in \operatorname{Fix}(\operatorname{ev}) $$ --- ## $S_{509}$ $$ \mathcal V_0 := \operatorname{Fix}(\operatorname{ev}) $$ $$ \mathcal V_{n+1} := \operatorname{End}(\mathcal V_n) $$ $$ \mathcal V_\omega := \prod_{n<\omega}\mathcal V_n $$ --- ## $S_{510}$ $$ \delta: \mathcal V_\omega\to\mathcal V_\omega $$ $$ \delta(v)_n = v_{n+1}(v_n) $$ $$ \delta^2 $$ $$ \operatorname{Fix}(\delta) $$ --- ## $S_{511}$ $$ v^\ast = \delta(v^\ast) $$ $$ v_n^\ast = v_{n+1}^\ast(v_n^\ast) $$ $$ v^\ast \in \operatorname{Fix}(\delta) $$ --- ## $S_{512}$ $$ \mathcal W := \operatorname{Fix}(\delta) $$ $$ \delta|_{\mathcal W} = \operatorname{id}_{\mathcal W} $$ $$ \operatorname{End}(\mathcal W) $$ --- ## $S_{513}$ $$ \operatorname{Code}: \mathcal W\to\mathbb N $$ $$ \operatorname{Eval}: \mathbb N\times\mathcal W\to\mathcal W $$ $$ \operatorname{Eval}(\operatorname{Code}(f),x) = f(x) $$ $$ f\in\operatorname{End}(\mathcal W) $$ --- ## $S_{514}$ $$ U:\mathbb N\to\operatorname{End}(\mathcal W) $$ $$ U_n := U(n) $$ $$ \forall f\in\operatorname{End}(\mathcal W),\ \exists n:\ U_n=f $$ $$ U \ \operatorname{epi} $$ --- ## $S_{515}$ $$ s:\mathbb N\times\mathbb N\to\mathbb N $$ $$ U_{s(e,a)}(x) = U_e(a,x) $$ $$ \operatorname{Spec}(e,a) := s(e,a) $$ $$ \operatorname{Spec} : \mathbb N^2\to\mathbb N $$ --- ## $S_{516}$ $$ f:\mathbb N\to\mathbb N $$ $$ \exists e: U_e = U_{f(e)} $$ $$ e \in \operatorname{FixExt}(f) $$ $$ \operatorname{RecThm}(U) $$ --- ## $S_{517}$ $$ G:\mathcal W\to\mathcal W $$ $$ \widehat G:\mathbb N\to\mathbb N $$ $$ U_{\widehat G(e)} = G\circ U_e $$ $$ \exists e^\ast: U_{e^\ast} = G\circ U_{e^\ast} $$ --- ## $S_{518}$ $$ w^\ast := U_{e^\ast}(w^\ast) $$ $$ G(w^\ast)=w^\ast $$ $$ w^\ast \in \operatorname{Fix}(G) $$ $$ \operatorname{Fix}(G)\neq\varnothing $$ --- ## $S_{519}$ $$ \operatorname{End}(\mathcal W) $$ $$ \Phi: \operatorname{End}(\mathcal W) \to \mathcal P(\mathcal W) $$ $$ \Phi(G) = \operatorname{Fix}(G) $$ $$ \Phi(G)\neq\varnothing $$ --- ## $S_{520}$ $$ \Psi: \operatorname{End}(\mathcal W) \to \mathcal W $$ $$ \Psi(G)\in\operatorname{Fix}(G) $$ $$ G(\Psi(G)) = \Psi(G) $$ $$ \operatorname{SelFix} $$ --- ## $S_{521}$ $$ H: \operatorname{End}(\mathcal W) \to \operatorname{End}(\mathcal W) $$ $$ H(G) $$ $$ \Psi(H(G)) $$ $$ H(G)(\Psi(H(G))) = \Psi(H(G)) $$ --- ## $S_{522}$ $$ G_0:=G $$ $$ G_{n+1}:=H(G_n) $$ $$ G_\omega := \operatorname{Lim}_{n<\omega}G_n $$ $$ x_n:=\Psi(G_n) $$ --- ## $S_{523}$ $$ G_n(x_n)=x_n $$ $$ x_{n+1} \in \operatorname{Fix}(H(G_n)) $$ $$ (x_n,G_n) $$ $$ \Theta(x_n,G_n) = (x_{n+1},G_{n+1}) $$ --- ## $S_{524}$ $$ \Theta: \mathcal W\times\operatorname{End}(\mathcal W) \to \mathcal W\times\operatorname{End}(\mathcal W) $$ $$ \Theta^n $$ $$ \operatorname{Orb}_\Theta(x_0,G_0) $$ $$ \omega_\Theta(x_0,G_0) $$ --- ## $S_{525}$ $$ \Theta^\alpha $$ $$ \Theta^{\alpha+1} = \Theta\circ\Theta^\alpha $$ $$ \Theta^\lambda = \operatorname{Lim}_{\beta<\lambda}\Theta^\beta $$ $$ \exists\Xi: \Theta^{\Xi+1} \simeq \Theta^\Xi $$ --- ## $S_{526}$ $$ P_\Theta := \Theta^\Xi $$ $$ P_\Theta^2 \simeq P_\Theta $$ $$ \operatorname{im}(P_\Theta) = \operatorname{Fix}(\Theta) $$ $$ \operatorname{Fix}(\Theta)\neq\varnothing $$ --- ## $S_{527}$ $$ (x^\ast,G^\ast) \in \operatorname{Fix}(\Theta) $$ $$ G^\ast(x^\ast)=x^\ast $$ $$ H(G^\ast)=G^\ast $$ $$ x^\ast=\Psi(G^\ast) $$ --- ## $S_{528}$ $$ G^\ast = H(G^\ast) $$ $$ x^\ast = G^\ast(x^\ast) $$ $$ (G^\ast,x^\ast) $$ $$ \operatorname{SelfFix} $$ --- ## $S_{529}$ $$ \mathcal Z_0 := \operatorname{Fix}(\Theta) $$ $$ \mathcal Z_{n+1} := \operatorname{End}(\mathcal Z_n) $$ $$ \mathcal Z_\omega := \prod_{n<\omega}\mathcal Z_n $$ --- ## $S_{530}$ $$ \varepsilon: \mathcal Z_\omega \to \mathcal Z_\omega $$ $$ \varepsilon(z)_n = z_{n+1}(z_n) $$ $$ \operatorname{Fix}(\varepsilon) $$ --- ## $S_{531}$ $$ z^\ast = \varepsilon(z^\ast) $$ $$ z_n^\ast = z_{n+1}^\ast(z_n^\ast) $$ $$ \mathcal Z^\ast := \operatorname{Fix}(\varepsilon) $$ --- ## $S_{532}$ $$ \varepsilon|_{\mathcal Z^\ast} = \operatorname{id}_{\mathcal Z^\ast} $$ $$ \operatorname{End}(\mathcal Z^\ast) $$ $$ \operatorname{Idem}(\mathcal Z^\ast) $$ $$ \operatorname{Aut}(\mathcal Z^\ast) $$ --- ## $S_{533}$ $$ \mathcal I \subseteq \operatorname{Idem}(\mathcal Z^\ast) $$ $$ ef=fe $$ $$ \mathcal Z_\infty := \bigcap_{e\in\mathcal I} \operatorname{Fix}(e) $$ $$ e|_{\mathcal Z_\infty} = \operatorname{id} $$ --- ## $S_{534}$ $$ \operatorname{End}(\mathcal Z_\infty) $$ $$ \operatorname{FixAll} := \bigcap_{F\in\operatorname{End}(\mathcal Z_\infty)} \operatorname{Fix}(F) $$ $$ \operatorname{FixAll} \subseteq \mathcal Z_\infty $$ --- ## $S_{535}$ $$ \operatorname{id} \in \operatorname{End}(\mathcal Z_\infty) $$ $$ \operatorname{Fix}(\operatorname{id}) = \mathcal Z_\infty $$ $$ \operatorname{FixAll} = \bigcap_F\operatorname{Fix}(F) $$ --- ## $S_{536}$ $$ \exists\omega^\ast: \forall F\in\operatorname{End}(\mathcal Z_\infty),\ F(\omega^\ast)=\omega^\ast $$ $$ \omega^\ast \in \operatorname{FixAll} $$ $$ \operatorname{FixAll}\neq\varnothing $$ --- ## $S_{537}$ $$ \mathcal Q := \operatorname{FixAll} $$ $$ \forall q\in\mathcal Q,\ \forall F\in\operatorname{End}(\mathcal Z_\infty): F(q)=q $$ $$ F|_{\mathcal Q} = \operatorname{id}_{\mathcal Q} $$ --- ## $S_{538}$ $$ \operatorname{End}(\mathcal Q) $$ $$ \forall F\in\operatorname{End}(\mathcal Q): F=\operatorname{id}_{\mathcal Q} $$ $$ \operatorname{End}(\mathcal Q) = \{\operatorname{id}_{\mathcal Q}\} $$ --- ## $S_{539}$ $$ \operatorname{Aut}(\mathcal Q) = \{\operatorname{id}_{\mathcal Q}\} $$ $$ \operatorname{Idem}(\mathcal Q) = \{\operatorname{id}_{\mathcal Q}\} $$ $$ \operatorname{Fix}(\operatorname{id}_{\mathcal Q}) = \mathcal Q $$ $$ \mathcal Q \simeq \operatorname{Fix}(\operatorname{End}(\mathcal Q)) $$ --- ## $S_{540}$ $$ S_{480} \to S_{481} \to \cdots \to S_{540} $$ $$ \operatorname{Internal} \to \operatorname{DiagFix} \to \operatorname{Modal} \to \operatorname{Reflect} \to \operatorname{TransIter} \to \operatorname{SelfEval} \to \operatorname{RecFix} \to \operatorname{MetaDyn} \to \operatorname{SelfFix} \to \operatorname{RigidFix} $$ $$ \mathcal E \to \mathcal C_\infty \to \mathcal D_\infty \to \mathcal W \to \operatorname{Fix}(\Theta) \to \mathcal Z^\ast \to \mathcal Z_\infty \to \mathcal Q $$ $$ \forall F\in\operatorname{End}(\mathcal Q): F = \operatorname{id}_{\mathcal Q} $$ $$ \boxed{ SREG_{009} : S_{481} \to S_{482} \to \cdots \to S_{540} } $$ $$ \boxed{ S_{540}\to S_{541} } $$ --- # SREG-010 $$ S_{540}\longrightarrow S_{541} $$ --- ## $S_{541}$ $$ \mathcal Q $$ $$ \forall F\in\operatorname{End}(\mathcal Q): F=\operatorname{id}_{\mathcal Q} $$ $$ \mathcal Q\in\mathbf{Set} $$ $$ \mathcal Q\neq\varnothing $$ $$ q\in\mathcal Q $$ --- ## $S_{542}$ $$ c_q:\mathcal Q\to\mathcal Q $$ $$ c_q(x):=q $$ $$ c_q = \operatorname{id}_{\mathcal Q} $$ $$ \forall x\in\mathcal Q: x=q $$ $$ |\mathcal Q|=1 $$ --- ## $S_{543}$ $$ \mathcal Q\cong 1 $$ $$ \operatorname{End}(1) = \{\operatorname{id}_1\} $$ $$ \operatorname{Aut}(1) = \{\operatorname{id}_1\} $$ $$ \operatorname{Sub}(1) \cong \{0,1\} $$ --- ## $S_{544}$ $$ 0\to1 $$ $$ 1\times1\cong1 $$ $$ 1^1\cong1 $$ $$ \mathcal P(1) \cong 2 $$ $$ 2:=\{0,1\} $$ --- ## $S_{545}$ $$ 1 \mapsto \mathcal P(1) \mapsto \mathcal P^2(1) \mapsto \cdots $$ $$ X_0:=1 $$ $$ X_{n+1}:=\mathcal P(X_n) $$ $$ |X_{n+1}|=2^{|X_n|} $$ --- ## $S_{546}$ $$ |X_0|=1 $$ $$ |X_1|=2 $$ $$ |X_2|=4 $$ $$ |X_3|=16 $$ $$ |X_4|=65536 $$ $$ |X_{n+1}|=2^{|X_n|} $$ --- ## $S_{547}$ $$ X_\omega := \bigcup_{n<\omega}X_n $$ $$ X_n\subseteq X_{n+1} $$ $$ |X_\omega| = \sup_{n<\omega}|X_n| $$ $$ \mathcal P(X_\omega) \not\subseteq X_\omega $$ --- ## $S_{548}$ $$ X_{\alpha+1} := \mathcal P(X_\alpha) $$ $$ X_\lambda := \bigcup_{\beta<\lambda}X_\beta $$ $$ \alpha<\beta \Rightarrow X_\alpha\subseteq X_\beta $$ $$ V = \bigcup_{\alpha\in\operatorname{Ord}}X_\alpha $$ --- ## $S_{549}$ $$ 1 \to V $$ $$ \{1\} \subseteq V $$ $$ \operatorname{Cl}_{\mathcal P,\cup}(1) = V $$ $$ \operatorname{Gen}(1) = V $$ --- ## $S_{550}$ $$ F:\mathbf{Set}\to\mathbf{Set} $$ $$ F(X)=1+X $$ $$ \mu F\cong\mathbb N $$ $$ \nu F \cong \mathbb N\cup\{\omega\} $$ --- ## $S_{551}$ $$ \iota: 1+\mathbb N \to \mathbb N $$ $$ \iota(\ast)=0 $$ $$ \iota(n)=S(n) $$ $$ \forall(A,\alpha): \exists!h:\mathbb N\to A $$ $$ h\circ\iota = \alpha\circ(1+h) $$ --- ## $S_{552}$ $$ G(X):=1+A\times X $$ $$ \nu G \cong A^{\le\omega} $$ $$ \omega: A^{\le\omega} \to 1+A\times A^{\le\omega} $$ $$ \operatorname{head} $$ $$ \operatorname{tail} $$ --- ## $S_{553}$ $$ T_A(X) := 1+A\times X $$ $$ T_A^n(0) $$ $$ \mu T_A \cong A^\ast $$ $$ \nu T_A \cong A^\ast\cup A^\omega $$ --- ## $S_{554}$ $$ A^\ast $$ $$ \epsilon\in A^\ast $$ $$ \cdot: A^\ast\times A^\ast\to A^\ast $$ $$ \epsilon w=w\epsilon=w $$ $$ (uv)w=u(vw) $$ --- ## $S_{555}$ $$ F_A:\mathbf{Set}\to\mathbf{Mon} $$ $$ U:\mathbf{Mon}\to\mathbf{Set} $$ $$ F_A\dashv U $$ $$ F_A(A) = A^\ast $$ $$ \operatorname{Hom}_{\mathbf{Mon}}(A^\ast,M) \cong \operatorname{Hom}_{\mathbf{Set}}(A,U(M)) $$ --- ## $S_{556}$ $$ A \mapsto F(A) $$ $$ F\dashv U $$ $$ T:=UF $$ $$ \eta:\operatorname{id}\Rightarrow T $$ $$ \mu:T^2\Rightarrow T $$ $$ \mathbf{Set}^T \simeq \mathbf{Mon} $$ --- ## $S_{557}$ $$ T(X) = X^\ast $$ $$ \eta_X(x) = [x] $$ $$ \mu_X: (X^\ast)^\ast \to X^\ast $$ $$ \mu_X([w_1,\ldots,w_n]) = w_1\cdots w_n $$ --- ## $S_{558}$ $$ T^2 \xrightarrow{\mu} T $$ $$ T^3 \rightrightarrows T^2 \xrightarrow{\mu} T $$ $$ \mu\circ T\mu = \mu\circ\mu T $$ $$ \mu\circ T\eta = \mu\circ\eta T = \operatorname{id}_T $$ --- ## $S_{559}$ $$ X \xrightarrow{\eta} T(X) \xrightarrow{T\eta} T^2(X) \xrightarrow{T^2\eta} \cdots $$ $$ T^\omega(X) $$ $$ \operatorname{colim}_{n<\omega}T^n(X) $$ --- ## $S_{560}$ $$ T^\omega $$ $$ \mu^\omega: T^\omega T^\omega \to T^\omega $$ $$ \eta^\omega: \operatorname{id}\to T^\omega $$ $$ (T^\omega,\eta^\omega,\mu^\omega) $$ --- ## $S_{561}$ $$ \mathcal C $$ $$ G:\mathcal C\to\mathbf{Set} $$ $$ G \cong \operatorname{Hom}_{\mathcal C}(A,-) $$ $$ A \in \operatorname{Ob}(\mathcal C) $$ $$ \operatorname{Rep}(G) $$ --- ## $S_{562}$ $$ \operatorname{Nat} ( \operatorname{Hom}(A,-),G ) \cong G(A) $$ $$ \eta \leftrightarrow \eta_A(\operatorname{id}_A) $$ $$ \operatorname{Yoneda} $$ --- ## $S_{563}$ $$ A\cong B $$ $$ \operatorname{Hom}(A,-) \cong \operatorname{Hom}(B,-) $$ $$ \operatorname{Hom}(A,-) \cong \operatorname{Hom}(B,-) \Rightarrow A\cong B $$ --- ## $S_{564}$ $$ \mathcal C \hookrightarrow [\mathcal C^{op},\mathbf{Set}] $$ $$ A \mapsto h_A $$ $$ h_A(X) = \operatorname{Hom}(X,A) $$ $$ \mathcal C \cong \operatorname{Rep}(\operatorname{PSh}(\mathcal C)) $$ --- ## $S_{565}$ $$ F\in\operatorname{PSh}(\mathcal C) $$ $$ \int_{\mathcal C}F $$ $$ F \cong \operatorname{colim}_{(A,x)\in\int F}h_A $$ $$ \operatorname{PSh}(\mathcal C) = \operatorname{Ind}_{\operatorname{colim}} ( h(\mathcal C) ) $$ --- ## $S_{566}$ $$ \mathcal C \to \operatorname{PSh}(\mathcal C) \to \operatorname{PSh}^2(\mathcal C) \to \cdots $$ $$ \mathcal P_0(\mathcal C):=\mathcal C $$ $$ \mathcal P_{n+1}(\mathcal C) := \operatorname{PSh}(\mathcal P_n(\mathcal C)) $$ --- ## $S_{567}$ $$ y_n: \mathcal P_n(\mathcal C) \hookrightarrow \mathcal P_{n+1}(\mathcal C) $$ $$ y_{n+1}\circ y_n $$ $$ \mathcal P_\omega(\mathcal C) := \operatorname{colim}_{n<\omega} \mathcal P_n(\mathcal C) $$ --- ## $S_{568}$ $$ \mathcal P_\omega \to \operatorname{PSh}(\mathcal P_\omega) $$ $$ y_\omega $$ $$ \mathcal P_{\omega+1} := \operatorname{PSh}(\mathcal P_\omega) $$ $$ \mathcal P_\omega \subsetneq \mathcal P_{\omega+1} $$ --- ## $S_{569}$ $$ \mathcal P_\alpha $$ $$ \mathcal P_{\alpha+1} := \operatorname{PSh}(\mathcal P_\alpha) $$ $$ \mathcal P_\lambda := \operatorname{colim}_{\beta<\lambda}\mathcal P_\beta $$ $$ \mathcal P_\infty := \operatorname{colim}_{\alpha\in\operatorname{Ord}}\mathcal P_\alpha $$ --- ## $S_{570}$ $$ \operatorname{PSh}(\mathcal P_\infty) $$ $$ \mathcal P_\infty \hookrightarrow \operatorname{PSh}(\mathcal P_\infty) $$ $$ \mathcal P_\infty \neq \operatorname{PSh}(\mathcal P_\infty) $$ $$ \operatorname{Size} $$ --- ## $S_{571}$ $$ \mathcal U_0 \in \mathcal U_1 \in \mathcal U_2 \in\cdots $$ $$ \mathcal C \in \operatorname{Cat}(\mathcal U_n) $$ $$ \operatorname{PSh}(\mathcal C) \in \operatorname{Cat}(\mathcal U_{n+1}) $$ $$ \operatorname{Lift} : \mathcal U_n\to\mathcal U_{n+1} $$ --- ## $S_{572}$ $$ \operatorname{Lift}_{n,n+1} $$ $$ \operatorname{Lift}_{n+1,n+2} $$ $$ \operatorname{Lift}_{n,n+2} $$ $$ \operatorname{Lift}_{n+1,n+2} \circ \operatorname{Lift}_{n,n+1} = \operatorname{Lift}_{n,n+2} $$ --- ## $S_{573}$ $$ \operatorname{Code}_n: \mathcal U_n \to \mathcal U_{n+1} $$ $$ \operatorname{El}_n: \mathcal U_{n+1} \to \mathcal U_n $$ $$ \operatorname{El}_n(\operatorname{Code}_n(A)) \simeq A $$ $$ \operatorname{Code}_n(\operatorname{El}_n(c)) \simeq c $$ --- ## $S_{574}$ $$ A:\mathcal U_n $$ $$ \ulcorner A\urcorner := \operatorname{Code}_n(A) $$ $$ \llcorner a\lrcorner := \operatorname{El}_n(a) $$ $$ \llcorner\ulcorner A\urcorner\lrcorner \simeq A $$ --- ## $S_{575}$ $$ F:\mathcal U_n\to\mathcal U_n $$ $$ \widehat F: \mathcal U_{n+1}\to\mathcal U_{n+1} $$ $$ \widehat F := \operatorname{Code}_n \circ F \circ \operatorname{El}_n $$ $$ \widehat F(\ulcorner A\urcorner) \simeq \ulcorner F(A)\urcorner $$ --- ## $S_{576}$ $$ F(A)\simeq A $$ $$ \widehat F(\ulcorner A\urcorner) \simeq \ulcorner A\urcorner $$ $$ \operatorname{Fix}(F) \to \operatorname{Fix}(\widehat F) $$ $$ A \mapsto \ulcorner A\urcorner $$ --- ## $S_{577}$ $$ \operatorname{Fix}(F) \simeq \operatorname{Fix}(\widehat F) $$ $$ \operatorname{Code} : \operatorname{Fix}(F) \to \operatorname{Fix}(\widehat F) $$ $$ \operatorname{El} : \operatorname{Fix}(\widehat F) \to \operatorname{Fix}(F) $$ --- ## $S_{578}$ $$ F_0:\mathcal U_0\to\mathcal U_0 $$ $$ F_{n+1} := \widehat F_n $$ $$ F_n:\mathcal U_n\to\mathcal U_n $$ $$ \operatorname{Fix}(F_n) \simeq \operatorname{Fix}(F_{n+1}) $$ --- ## $S_{579}$ $$ \mathfrak F_n := \operatorname{Fix}(F_n) $$ $$ \mathfrak F_0 \simeq \mathfrak F_1 \simeq \mathfrak F_2 \simeq \cdots $$ $$ \operatorname{LiftFix} $$ --- ## $S_{580}$ $$ \mathfrak F_\omega := \varinjlim_n\mathfrak F_n $$ $$ \mathfrak F_n\to\mathfrak F_\omega $$ $$ \operatorname{FixLift} : \mathfrak F_\omega\to\mathfrak F_\omega $$ $$ \operatorname{FixLift} \simeq \operatorname{id} $$ --- ## $S_{581}$ $$ G:\mathfrak F_\omega\to\mathfrak F_\omega $$ $$ \operatorname{Graph}(G) \subseteq \mathfrak F_\omega\times\mathfrak F_\omega $$ $$ \Gamma_G := \{(x,G(x)):x\in\mathfrak F_\omega\} $$ $$ \Gamma_{\operatorname{id}} = \Delta $$ --- ## $S_{582}$ $$ G\circ H $$ $$ \Gamma_{G\circ H} = \Gamma_H \circ \Gamma_G $$ $$ \operatorname{Rel}(\mathfrak F_\omega) $$ $$ \operatorname{End}(\mathfrak F_\omega) \hookrightarrow \operatorname{Rel}(\mathfrak F_\omega) $$ --- ## $S_{583}$ $$ R\subseteq X\times Y $$ $$ R^\dagger \subseteq Y\times X $$ $$ (x,y)\in R \iff (y,x)\in R^\dagger $$ $$ (R\circ S)^\dagger = S^\dagger\circ R^\dagger $$ --- ## $S_{584}$ $$ R=R^\dagger $$ $$ R\circ R\subseteq R $$ $$ \operatorname{id}\subseteq R $$ $$ R \in \operatorname{EqRel}(X) $$ $$ X/R $$ --- ## $S_{585}$ $$ \pi:X\to X/R $$ $$ xRy \iff \pi(x)=\pi(y) $$ $$ f:X\to Y $$ $$ R\subseteq\ker f $$ $$ \exists!\bar f:X/R\to Y $$ $$ f=\bar f\circ\pi $$ --- ## $S_{586}$ $$ R_0 \subseteq R_1 \subseteq R_2 \subseteq \cdots $$ $$ R_\omega := \bigcup_{n<\omega}R_n $$ $$ X/R_0 \to X/R_1 \to X/R_2 \to \cdots $$ $$ X/R_\omega $$ --- ## $S_{587}$ $$ q_n:X/R_n\to X/R_{n+1} $$ $$ q_{m:n} := q_{m-1}\circ\cdots\circ q_n $$ $$ \operatorname{colim}_nX/R_n \cong X/R_\omega $$ --- ## $S_{588}$ $$ R_\alpha $$ $$ R_{\alpha+1} := \operatorname{Cl}(R_\alpha) $$ $$ R_\lambda := \bigcup_{\beta<\lambda}R_\beta $$ $$ \exists\Gamma: R_{\Gamma+1} = R_\Gamma $$ --- ## $S_{589}$ $$ R_\Gamma = \operatorname{Cl}(R_\Gamma) $$ $$ X_\infty := X/R_\Gamma $$ $$ \operatorname{Rel}(X_\infty) $$ $$ \operatorname{EqRel}(X_\infty) $$ --- ## $S_{590}$ $$ \operatorname{EqRel}(X_\infty) = \{\Delta_{X_\infty}\} $$ $$ \forall R\in\operatorname{EqRel}(X_\infty): R=\Delta $$ $$ X_\infty \ \operatorname{Simple} $$ --- ## $S_{591}$ $$ |X_\infty|>1 $$ $$ X_\infty\times X_\infty $$ $$ R_{\mathrm{all}} := X_\infty\times X_\infty $$ $$ R_{\mathrm{all}} \in \operatorname{EqRel}(X_\infty) $$ $$ R_{\mathrm{all}}\neq\Delta $$ --- ## $S_{592}$ $$ \operatorname{EqRel}(X_\infty) = \{\Delta\} $$ $$ \Rightarrow |X_\infty|\le1 $$ $$ X_\infty\neq\varnothing $$ $$ \Rightarrow X_\infty\cong1 $$ --- ## $S_{593}$ $$ 1 \to \operatorname{Free}(1) \to \operatorname{PSh}(1) \to \operatorname{Ind}(1) \to \operatorname{Pro}(1) $$ $$ \operatorname{Free}(1) $$ $$ \operatorname{CoFree}(1) $$ --- ## $S_{594}$ $$ F(1) $$ $$ F^2(1) $$ $$ F^3(1) $$ $$ \cdots $$ $$ \operatorname{Orb}_F(1) $$ $$ \operatorname{Cl}_F(1) $$ --- ## $S_{595}$ $$ \mathcal G_F(1) := \{F^\alpha(1):\alpha\in\operatorname{Ord}\} $$ $$ F: \mathcal G_F(1) \to \mathcal G_F(1) $$ $$ F^\Gamma(1) \simeq F^{\Gamma+1}(1) $$ --- ## $S_{596}$ $$ U_F := F^\Gamma(1) $$ $$ F(U_F) \simeq U_F $$ $$ U_F \in \operatorname{Fix}(F) $$ $$ 1 \to U_F $$ --- ## $S_{597}$ $$ \operatorname{End}(U_F) $$ $$ \operatorname{Aut}(U_F) $$ $$ \operatorname{Idem}(U_F) $$ $$ \operatorname{Fix}(F|_{U_F}) $$ --- ## $S_{598}$ $$ \mathcal I_F := \operatorname{Idem}(U_F) $$ $$ U_F^\ast := \bigcap_{e\in\mathcal I_F}\operatorname{Fix}(e) $$ $$ e|_{U_F^\ast} = \operatorname{id} $$ $$ \forall e\in\mathcal I_F $$ --- ## $S_{599}$ $$ U_F^\ast \cong1 \quad\lor\quad |\operatorname{End}(U_F^\ast)|>1 $$ $$ U_F^\ast\cong1 \Rightarrow \operatorname{Free}(U_F^\ast) $$ $$ |\operatorname{End}(U_F^\ast)|>1 \Rightarrow \operatorname{Dyn}(U_F^\ast) $$ --- ## $S_{600}$ $$ S_{540} \to S_{541} \to \cdots \to S_{600} $$ $$ \operatorname{Rigid} \to 1 \to \operatorname{Power} \to \operatorname{Free} \to \operatorname{Monad} \to \operatorname{Represent} \to \operatorname{UniverseLift} \to \operatorname{Quotient} \to 1 \to \operatorname{Generate} $$ $$ 1 \to F(1) \to F^2(1) \to \cdots \to F^\Gamma(1) = U_F $$ $$ F(U_F) \simeq U_F $$ $$ U_F^\ast = \bigcap_{e\in\operatorname{Idem}(U_F)} \operatorname{Fix}(e) $$ $$ \boxed{ SREG_{010} : S_{541} \to S_{542} \to \cdots \to S_{600} } $$ $$ \boxed{ S_{600}\to S_{601} } $$ --- # SREG-011 $$ S_{600}\longrightarrow S_{601} $$ --- ## $S_{601}$ $$ U_F^\ast\cong1 \quad\lor\quad |\operatorname{End}(U_F^\ast)|>1 $$ $$ \mathsf C := \{X:X\cong1\} $$ $$ \mathsf D := \{X:|\operatorname{End}(X)|>1\} $$ $$ U_F^\ast\in\mathsf C\cup\mathsf D $$ --- ## $S_{602}$ $$ X\in\mathsf C $$ $$ X\cong1 $$ $$ \operatorname{End}(X) = \{\operatorname{id}_X\} $$ $$ \operatorname{Aut}(X) = \{\operatorname{id}_X\} $$ $$ \operatorname{Sub}(X) \cong2 $$ --- ## $S_{603}$ $$ X\in\mathsf D $$ $$ \exists f,g\in\operatorname{End}(X): f\neq g $$ $$ \langle f,g\rangle \subseteq \operatorname{End}(X) $$ $$ \operatorname{Orb}_{\langle f,g\rangle}(x) $$ $$ \operatorname{Stab}_{\langle f,g\rangle}(x) $$ --- ## $S_{604}$ $$ M_X := \operatorname{End}(X) $$ $$ \circ: M_X\times M_X\to M_X $$ $$ \operatorname{id}_X\in M_X $$ $$ (h\circ g)\circ f = h\circ(g\circ f) $$ $$ (M_X,\circ,\operatorname{id}_X) $$ --- ## $S_{605}$ $$ M_X \curvearrowright X $$ $$ (f,x)\mapsto f(x) $$ $$ \alpha_X: M_X\times X\to X $$ $$ \alpha_X(\operatorname{id},x)=x $$ $$ \alpha_X(fg,x) = \alpha_X(f,\alpha_X(g,x)) $$ --- ## $S_{606}$ $$ x\sim_M y \iff \exists f,g\in M_X: f(x)=g(y) $$ $$ [x]_M $$ $$ X/{\sim_M} $$ $$ \pi_M:X\to X/{\sim_M} $$ --- ## $S_{607}$ $$ x\preceq_M y \iff \exists f\in M_X: f(y)=x $$ $$ x\sim_My \Rightarrow x\preceq_My \lor y\preceq_Mx \lor \exists z: z\preceq_Mx \land z\preceq_My $$ --- ## $S_{608}$ $$ \operatorname{Fix}(M_X) := \bigcap_{f\in M_X} \operatorname{Fix}(f) $$ $$ \operatorname{Fix}(M_X) \subseteq X $$ $$ X\in\mathsf C \Rightarrow \operatorname{Fix}(M_X)=X $$ --- ## $S_{609}$ $$ K_X := \operatorname{Fix}(M_X) $$ $$ K_X=\varnothing \quad\lor\quad K_X\neq\varnothing $$ $$ K_X\neq\varnothing \Rightarrow \forall f\in M_X,\ f|_{K_X} = \operatorname{id}_{K_X} $$ --- ## $S_{610}$ $$ K_X\neq\varnothing $$ $$ r_X:X\to K_X $$ $$ i_X:K_X\hookrightarrow X $$ $$ r_X\circ i_X = \operatorname{id}_{K_X} $$ $$ e_X:=i_X\circ r_X $$ $$ e_X^2=e_X $$ --- ## $S_{611}$ $$ \operatorname{im}(e_X)=K_X $$ $$ \operatorname{Fix}(e_X)=K_X $$ $$ e_X\in\operatorname{Idem}(X) $$ $$ K_X \simeq \operatorname{Split}(e_X) $$ --- ## $S_{612}$ $$ K_X=\varnothing $$ $$ \forall x\in X,\ \exists f_x\in M_X: f_x(x)\neq x $$ $$ x_0\in X $$ $$ x_{n+1} := f_{x_n}(x_n) $$ $$ x_{n+1}\neq x_n $$ --- ## $S_{613}$ $$ (x_n)_{n<\omega} $$ $$ \operatorname{Orb}(x_0) := \{x_n:n<\omega\} $$ $$ |\operatorname{Orb}(x_0)| \le\aleph_0 $$ $$ \operatorname{Orb}(x_0) \subseteq X $$ --- ## $S_{614}$ $$ x_m=x_n \qquad (m0,\ F^n(x)=x \} $$ $$ \mathsf A_X := X\setminus\mathsf P_X $$ $$ X = \mathsf P_X \sqcup \mathsf A_X $$ --- ## $S_{617}$ $$ F:X\to X $$ $$ \operatorname{Per}_n(F) := \{x:F^n(x)=x\} $$ $$ \operatorname{Per}(F) := \bigcup_{n\ge1}\operatorname{Per}_n(F) $$ $$ \operatorname{Fix}(F) = \operatorname{Per}_1(F) $$ --- ## $S_{618}$ $$ F^m(x)=F^n(x) $$ $$ m0 $$ $$ \operatorname{Aut}(x) \supseteq \mathbb Z/p\mathbb Z $$ $$ B\mathcal G_F^{\operatorname{gp}} $$ $$ Q_F \leftarrow B\mathcal G_F^{\operatorname{gp}} $$ --- ## $S_{681}$ $$ \widetilde Q_F := B\mathcal G_F^{\operatorname{gp}} $$ $$ \pi_0(\widetilde Q_F) = Q_F $$ $$ \widetilde Q_F \not\simeq Q_F $$ $$ \pi_{\ge1}(\widetilde Q_F) \neq0 $$ --- ## $S_{682}$ $$ Q_F^{(0)} := \pi_0(\widetilde Q_F) $$ $$ Q_F^{(1)} := \Pi_{\le1}(\widetilde Q_F) $$ $$ Q_F^{(n)} := \tau_{\le n}(\widetilde Q_F) $$ $$ Q_F^{(\infty)} := \widetilde Q_F $$ --- ## $S_{683}$ $$ Q_F^{(0)} \leftarrow Q_F^{(1)} \leftarrow Q_F^{(2)} \leftarrow \cdots $$ $$ \widetilde Q_F \simeq \varprojlim_nQ_F^{(n)} $$ $$ \operatorname{Postnikov}(\widetilde Q_F) $$ --- ## $S_{684}$ $$ \mathsf Q_n: \mathbf{Dyn}\to\mathbf{Type}_{\le n} $$ $$ \mathsf Q_n(X,F) := Q_F^{(n)} $$ $$ \mathsf Q_0 = Q $$ $$ \tau_{\le n}\mathsf Q_{n+1} \simeq \mathsf Q_n $$ --- ## $S_{685}$ $$ \mathsf Q_\infty : \mathbf{Dyn}\to\mathbf{Type} $$ $$ \mathsf Q_\infty(X,F) = \widetilde Q_F $$ $$ \pi_0\mathsf Q_\infty = Q $$ --- ## $S_{686}$ $$ \mathsf Q_\infty(h) : \widetilde Q_F \to \widetilde Q_G $$ $$ hF=Gh $$ $$ B\mathcal G_F^{\operatorname{gp}} \to B\mathcal G_G^{\operatorname{gp}} $$ --- ## $S_{687}$ $$ \mathsf Q_\infty \circ \Delta $$ $$ \Delta(Y) = (Y,\operatorname{id}) $$ $$ \mathcal G_{\operatorname{id}}^{\operatorname{gp}} $$ $$ \widetilde Q_{\operatorname{id}} \simeq Y $$ --- ## $S_{688}$ $$ \mathsf Q_\infty\Delta \simeq \operatorname{id} $$ $$ \mathsf Q_\infty \dashv \Delta_\infty $$ $$ \Delta_\infty(Y) := (Y,\operatorname{id}) $$ $$ \mathbf{Dyn}_\infty[W^{-1}] \simeq \mathbf{Type} $$ --- ## $S_{689}$ $$ G \curvearrowright X $$ $$ X//G $$ $$ EG\times_GX $$ $$ X/G $$ $$ \pi_0(X//G) = X/G $$ --- ## $S_{690}$ $$ G=\mathbb Z $$ $$ 1\in\mathbb Z $$ $$ 1\cdot x = F(x) $$ $$ X//\mathbb Z $$ $$ \pi_0(X//\mathbb Z) = Q_F $$ --- ## $S_{691}$ $$ F\in\operatorname{Aut}(X) $$ $$ \mathbb Z\curvearrowright X $$ $$ X//_F\mathbb Z $$ $$ \pi_1(X//_F\mathbb Z) $$ $$ \operatorname{Stab}_{\mathbb Z}(x) $$ --- ## $S_{692}$ $$ F \notin \operatorname{Aut}(X) $$ $$ X_F^{-\infty} $$ $$ \sigma \in \operatorname{Aut}(X_F^{-\infty}) $$ $$ X_F^{-\infty}//_\sigma\mathbb Z $$ --- ## $S_{693}$ $$ \mathcal H_F := X_F^{-\infty}//_\sigma\mathbb Z $$ $$ \pi_0(\mathcal H_F) \cong Q_F $$ $$ \pi_1(\mathcal H_F) $$ $$ \operatorname{Per}_\sigma $$ --- ## $S_{694}$ $$ p_0: X_F^{-\infty}\to X $$ $$ \mathcal H_F \to \widetilde Q_F $$ $$ \pi_0 \ \operatorname{iso} $$ $$ \pi_1 \to \pi_1 $$ --- ## $S_{695}$ $$ H^\ast(\mathcal H_F;R) $$ $$ H^\ast(X_F^{-\infty}//\mathbb Z;R) $$ $$ E_2^{p,q} = H^p ( \mathbb Z; H^q(X_F^{-\infty};R) ) $$ $$ E_2 \Rightarrow H^{p+q}(\mathcal H_F;R) $$ --- ## $S_{696}$ $$ H^0(\mathbb Z;M) = M^{\mathbb Z} $$ $$ H^1(\mathbb Z;M) = M/(t-1)M $$ $$ H^p(\mathbb Z;M)=0 \qquad p>1 $$ --- ## $S_{697}$ $$ M = H^q(X_F^{-\infty};R) $$ $$ t = \sigma^\ast $$ $$ M^{\mathbb Z} = \ker(\sigma^\ast-\operatorname{id}) $$ $$ M_{\mathbb Z} = \operatorname{coker}(\sigma^\ast-\operatorname{id}) $$ --- ## $S_{698}$ $$ 0 \to M_{\mathbb Z} \to H^{q+1}(\mathcal H_F;R) \to M^{\mathbb Z} \to 0 $$ $$ \operatorname{Inv} $$ $$ \operatorname{Coinv} $$ --- ## $S_{699}$ $$ \operatorname{Inv}(F) := \ker(F^\ast-\operatorname{id}) $$ $$ \operatorname{Coinv}(F) := \operatorname{coker}(F^\ast-\operatorname{id}) $$ $$ \operatorname{DynInv}(F) := (\operatorname{Inv}(F),\operatorname{Coinv}(F)) $$ --- ## $S_{700}$ $$ F \mapsto Q_F $$ $$ F \mapsto \widetilde Q_F $$ $$ F \mapsto \mathcal H_F $$ $$ F \mapsto \operatorname{DynInv}(F) $$ $$ \mathcal I(F) $$ --- ## $S_{701}$ $$ \mathcal I(F) := ( Q_F, \widetilde Q_F, \mathcal H_F, \operatorname{DynInv}(F) ) $$ $$ F\sim_{\mathcal I}G \iff \mathcal I(F)\cong\mathcal I(G) $$ $$ [F]_{\mathcal I} $$ --- ## $S_{702}$ $$ \mathbf{Dyn}/{\sim_{\mathcal I}} $$ $$ \Pi_{\mathcal I} : \mathbf{Dyn}\to \mathbf{Dyn}/{\sim_{\mathcal I}} $$ $$ \Pi_{\mathcal I}^2 = \Pi_{\mathcal I} $$ --- ## $S_{703}$ $$ \operatorname{Ker}(\mathcal I) := \{F:\mathcal I(F)\cong\mathcal I(\operatorname{id})\} $$ $$ F\in\operatorname{Ker}(\mathcal I) $$ $$ Q_F\cong X $$ $$ \widetilde Q_F\simeq X $$ --- ## $S_{704}$ $$ F\in\operatorname{Ker}(\mathcal I) $$ $$ \pi_F:X\to Q_F $$ $$ \pi_F \ \operatorname{iso} $$ $$ \pi_FF=\pi_F $$ $$ F=\operatorname{id}_X $$ --- ## $S_{705}$ $$ \operatorname{Ker}(\mathcal I) = \{\operatorname{id}\} $$ $$ \mathcal I(F) = \mathcal I(\operatorname{id}) \Rightarrow F=\operatorname{id} $$ $$ \operatorname{Faith}_{0}(\mathcal I) $$ --- ## $S_{706}$ $$ \mathcal I(F) \cong \mathcal I(G) $$ $$ \not\Rightarrow F\cong G $$ $$ \operatorname{Complete}(\mathcal I) \stackrel{?}{=} 1 $$ $$ \operatorname{Fib}_{\mathcal I}(I) $$ --- ## $S_{707}$ $$ \operatorname{Fib}_{\mathcal I}(I) := \{F:\mathcal I(F)\cong I\} $$ $$ |\operatorname{Fib}_{\mathcal I}(I)| $$ $$ \operatorname{Moduli}_{\mathcal I}(I) $$ --- ## $S_{708}$ $$ \mathfrak M_I := \operatorname{Fib}_{\mathcal I}(I)/{\cong} $$ $$ \mathfrak M_I $$ $$ |\mathfrak M_I|=1 \quad\lor\quad |\mathfrak M_I|>1 $$ --- ## $S_{709}$ $$ |\mathfrak M_I|=1 $$ $$ \Rightarrow \mathcal I \ \operatorname{Complete} $$ $$ |\mathfrak M_I|>1 $$ $$ \Rightarrow \exists F\not\cong G: \mathcal I(F)\cong\mathcal I(G) $$ --- ## $S_{710}$ $$ \mathcal I_0:=Q $$ $$ \mathcal I_1:=\widetilde Q $$ $$ \mathcal I_2:=\mathcal H $$ $$ \mathcal I_3:=\operatorname{DynInv} $$ $$ \mathcal I_{\le3} := (\mathcal I_0,\mathcal I_1,\mathcal I_2,\mathcal I_3) $$ --- ## $S_{711}$ $$ \mathcal I_{n+1} := \operatorname{Inv}_{n+1}(F) $$ $$ \mathcal I_{\le n} := (\mathcal I_0,\ldots,\mathcal I_n) $$ $$ \mathcal I_\omega := (\mathcal I_n)_{n<\omega} $$ --- ## $S_{712}$ $$ F\sim_nG \iff \mathcal I_{\le n}(F) \cong \mathcal I_{\le n}(G) $$ $$ \sim_{n+1} \subseteq \sim_n $$ $$ \sim_\omega := \bigcap_{n<\omega}\sim_n $$ --- ## $S_{713}$ $$ \mathfrak M_n(I) := \operatorname{Fib}_{\mathcal I_{\le n}}(I)/{\cong} $$ $$ \mathfrak M_{n+1}(I) \to \mathfrak M_n(I) $$ $$ \mathfrak M_\omega(I) := \varprojlim_n \mathfrak M_n(I) $$ --- ## $S_{714}$ $$ |\mathfrak M_\omega(I)| = 1 $$ $$ \Rightarrow \mathcal I_\omega \ \operatorname{Complete} $$ $$ |\mathfrak M_\omega(I)|>1 $$ $$ \Rightarrow \mathcal I_{\omega+1} $$ --- ## $S_{715}$ $$ \mathcal I_\alpha $$ $$ \sim_\alpha $$ $$ \sim_{\alpha+1} \subseteq \sim_\alpha $$ $$ \sim_\lambda := \bigcap_{\beta<\lambda}\sim_\beta $$ --- ## $S_{716}$ $$ \exists\Gamma: \sim_{\Gamma+1} = \sim_\Gamma $$ $$ \mathcal I_\Gamma $$ $$ \mathbf{Dyn}/{\sim_\Gamma} $$ $$ \Pi_\Gamma $$ --- ## $S_{717}$ $$ \Pi_\Gamma^2 = \Pi_\Gamma $$ $$ \operatorname{im}(\Pi_\Gamma) = \mathbf{Dyn}/{\sim_\Gamma} $$ $$ F\sim_\Gamma G \iff \Pi_\Gamma(F)=\Pi_\Gamma(G) $$ --- ## $S_{718}$ $$ \mathcal R: \mathbf{Dyn}/{\sim_\Gamma} \to \mathbf{Dyn} $$ $$ \Pi_\Gamma\circ\mathcal R = \operatorname{id} $$ $$ \mathcal R\circ\Pi_\Gamma \sim_\Gamma \operatorname{id} $$ $$ \operatorname{Reconstruct}_\Gamma $$ --- ## $S_{719}$ $$ \mathbf D_\Gamma := \mathbf{Dyn}/{\sim_\Gamma} $$ $$ \operatorname{End}(\mathbf D_\Gamma) $$ $$ \operatorname{Idem}(\mathbf D_\Gamma) $$ $$ \operatorname{Aut}(\mathbf D_\Gamma) $$ $$ \operatorname{Fix}(\Pi_\Gamma) = \mathbf D_\Gamma $$ --- ## $S_{720}$ $$ S_{660} \to S_{661} \to \cdots \to S_{720} $$ $$ \operatorname{DynCat} \to \operatorname{Adj} \to \operatorname{Localize} \to \operatorname{Groupoid} \to \operatorname{HomotopyQuot} \to \operatorname{InvCoinv} \to \operatorname{InvariantTower} \to \operatorname{ModuliFiber} \to \operatorname{TransRefine} \to \operatorname{Reconstruct} $$ $$ Q \dashv \Delta $$ $$ \mathsf Q_\infty \dashv \Delta_\infty $$ $$ \sim_{\alpha+1} \subseteq \sim_\alpha $$ $$ \exists\Gamma: \sim_{\Gamma+1} = \sim_\Gamma $$ $$ \mathbf D_\Gamma = \mathbf{Dyn}/{\sim_\Gamma} $$ $$ \boxed{ SREG_{012} : S_{661} \to S_{662} \to \cdots \to S_{720} } $$ $$ \boxed{ S_{720}\to S_{721} } $$ --- # SREG-013 $$ S_{720}\longrightarrow S_{721} $$ --- ## $S_{721}$ $$ \mathbf D_\Gamma := \mathbf{Dyn}/{\sim_\Gamma} $$ $$ \Pi_\Gamma: \mathbf{Dyn}\to\mathbf D_\Gamma $$ $$ \mathcal R_\Gamma: \mathbf D_\Gamma\to\mathbf{Dyn} $$ $$ \Pi_\Gamma\circ\mathcal R_\Gamma = \operatorname{id}_{\mathbf D_\Gamma} $$ --- ## $S_{722}$ $$ E_\Gamma := \mathcal R_\Gamma\circ\Pi_\Gamma $$ $$ E_\Gamma: \mathbf{Dyn}\to\mathbf{Dyn} $$ $$ E_\Gamma^2 = \mathcal R_\Gamma \Pi_\Gamma \mathcal R_\Gamma \Pi_\Gamma = E_\Gamma $$ $$ \operatorname{Fix}(E_\Gamma) = \operatorname{im}(\mathcal R_\Gamma) $$ --- ## $S_{723}$ $$ (X,F) \mapsto E_\Gamma(X,F) $$ $$ \Pi_\Gamma(X,F) = \Pi_\Gamma(E_\Gamma(X,F)) $$ $$ (X,F)\sim_\Gamma E_\Gamma(X,F) $$ $$ E_\Gamma \ \operatorname{idempotent} $$ --- ## $S_{724}$ $$ \mathcal T \subseteq \mathbf{Dyn} $$ $$ N_\mathcal T: \mathbf{Dyn} \to [\mathcal T^{op},\mathbf{Set}] $$ $$ N_\mathcal T(X,F)(T) := \operatorname{Hom}_{\mathbf{Dyn}}(T,(X,F)) $$ --- ## $S_{725}$ $$ h:(X,F)\to(Y,G) $$ $$ N_\mathcal T(h)_T : \operatorname{Hom}(T,(X,F)) \to \operatorname{Hom}(T,(Y,G)) $$ $$ u\mapsto h\circ u $$ $$ N_\mathcal T(kh) = N_\mathcal T(k)N_\mathcal T(h) $$ --- ## $S_{726}$ $$ R_\mathcal T: [\mathcal T^{op},\mathbf{Set}] \to \mathbf{Dyn} $$ $$ R_\mathcal T(P) := \int^{T\in\mathcal T} P(T)\otimes T $$ $$ R_\mathcal T \dashv N_\mathcal T $$ --- ## $S_{727}$ $$ C_\mathcal T := R_\mathcal T N_\mathcal T $$ $$ \varepsilon: C_\mathcal T \Rightarrow \operatorname{id}_{\mathbf{Dyn}} $$ $$ \delta: C_\mathcal T \Rightarrow C_\mathcal T^2 $$ $$ (C_\mathcal T,\varepsilon,\delta) $$ --- ## $S_{728}$ $$ \operatorname{Fix}(C_\mathcal T) := \{X:C_\mathcal T(X)\cong X\} $$ $$ \mathbf{Dyn}_\mathcal T := \operatorname{Fix}(C_\mathcal T) $$ $$ C_\mathcal T|_{\mathbf{Dyn}_\mathcal T} \cong \operatorname{id} $$ --- ## $S_{729}$ $$ X\sim_\mathcal T Y \iff N_\mathcal T(X)\cong N_\mathcal T(Y) $$ $$ \mathbf{Dyn}/{\sim_\mathcal T} $$ $$ \Pi_\mathcal T: \mathbf{Dyn} \to \mathbf{Dyn}/{\sim_\mathcal T} $$ --- ## $S_{730}$ $$ \mathcal T_0 \subseteq \mathcal T_1 \subseteq \cdots $$ $$ \sim_{\mathcal T_{n+1}} \subseteq \sim_{\mathcal T_n} $$ $$ N_n := N_{\mathcal T_n} $$ $$ \mathbf D_n := \mathbf{Dyn}/{\sim_{\mathcal T_n}} $$ --- ## $S_{731}$ $$ \mathcal T_\omega := \bigcup_{n<\omega}\mathcal T_n $$ $$ N_\omega := N_{\mathcal T_\omega} $$ $$ \sim_\omega = \bigcap_{n<\omega}\sim_{\mathcal T_n} $$ $$ \mathbf D_\omega = \varprojlim_n\mathbf D_n $$ --- ## $S_{732}$ $$ \mathcal T_\alpha $$ $$ \mathcal T_{\alpha+1} \supseteq \mathcal T_\alpha $$ $$ \mathcal T_\lambda = \bigcup_{\beta<\lambda}\mathcal T_\beta $$ $$ \sim_\lambda = \bigcap_{\beta<\lambda}\sim_\beta $$ --- ## $S_{733}$ $$ \exists\Lambda: \sim_{\Lambda+1} = \sim_\Lambda $$ $$ \mathbf D_\infty := \mathbf{Dyn}/{\sim_\Lambda} $$ $$ N_\infty := N_{\mathcal T_\Lambda} $$ $$ \Pi_\infty: \mathbf{Dyn}\to\mathbf D_\infty $$ --- ## $S_{734}$ $$ \mathcal T_{\operatorname{fin}} \subseteq \mathbf{Dyn} $$ $$ |T|<\infty \qquad (T\in\mathcal T_{\operatorname{fin}}) $$ $$ N_{\operatorname{fin}} : \mathbf{Dyn} \to [\mathcal T_{\operatorname{fin}}^{op},\mathbf{Set}] $$ --- ## $S_{735}$ $$ X $$ $$ \mathcal Q_{\operatorname{fin}}(X) := \{q:X\twoheadrightarrow Q:|Q|<\infty\} $$ $$ \widehat X := \varprojlim_{q\in\mathcal Q_{\operatorname{fin}}(X)}Q $$ $$ \eta_X:X\to\widehat X $$ --- ## $S_{736}$ $$ x\neq y $$ $$ \exists q\in\mathcal Q_{\operatorname{fin}}(X): q(x)\neq q(y) $$ $$ \iff $$ $$ \eta_X \ \operatorname{mono} $$ --- ## $S_{737}$ $$ F:X\to X $$ $$ qF $$ $$ \widehat F: \widehat X\to\widehat X $$ $$ \eta_XF = \widehat F\eta_X $$ $$ \widehat{G\circ F} = \widehat G\circ\widehat F $$ --- ## $S_{738}$ $$ (X,F) \mapsto (\widehat X,\widehat F) $$ $$ \widehat{(-)}: \mathbf{Dyn} \to \widehat{\mathbf{Dyn}} $$ $$ \widehat{\widehat X} \cong \widehat X $$ $$ \widehat{\widehat F} \cong \widehat F $$ --- ## $S_{739}$ $$ Q_F = \operatorname{Coeq}(F,\operatorname{id}) $$ $$ \widehat{Q_F} $$ $$ Q_{\widehat F} = \operatorname{Coeq}(\widehat F,\operatorname{id}) $$ $$ \widehat{Q_F} \to Q_{\widehat F} $$ --- ## $S_{740}$ $$ \operatorname{Obs}(X) \subseteq 2^X $$ $$ \chi_A:X\to2 $$ $$ A\in\operatorname{Obs}(X) $$ $$ \operatorname{Obs}(X) \in\mathbf{Bool} $$ --- ## $S_{741}$ $$ \operatorname{Obs}(X) \ni A \mapsto \chi_A $$ $$ x\sim_{\operatorname{Obs}}y \iff \forall A\in\operatorname{Obs}(X): \chi_A(x)=\chi_A(y) $$ $$ X/{\sim_{\operatorname{Obs}}} $$ --- ## $S_{742}$ $$ c_{\operatorname{Obs}}(S) := \bigcap \{A\in\operatorname{Obs}(X):S\subseteq A\} $$ $$ S\subseteq c_{\operatorname{Obs}}(S) $$ $$ c_{\operatorname{Obs}}^2 = c_{\operatorname{Obs}} $$ $$ S\subseteq T \Rightarrow c_{\operatorname{Obs}}(S) \subseteq c_{\operatorname{Obs}}(T) $$ --- ## $S_{743}$ $$ B_X := \operatorname{Obs}(X) $$ $$ \operatorname{Ult}(B_X) $$ $$ \widehat X_B := \operatorname{Ult}(B_X) $$ $$ \eta_B:X\to\widehat X_B $$ $$ \eta_B(x) = \{A\in B_X:x\in A\} $$ --- ## $S_{744}$ $$ x\neq y $$ $$ \exists A\in B_X: \chi_A(x)\neq\chi_A(y) $$ $$ \iff $$ $$ \eta_B \ \operatorname{mono} $$ --- ## $S_{745}$ $$ X\in\mathbf{Stone} $$ $$ B_X = \operatorname{Clop}(X) $$ $$ \eta_B: X \xrightarrow{\sim} \operatorname{Ult}(\operatorname{Clop}(X)) $$ $$ X \cong \operatorname{Ult}(B_X) $$ --- ## $S_{746}$ $$ F:X\to X $$ $$ F^{-1}:B_X\to B_X $$ $$ F^{-1}(A\cap B) = F^{-1}(A)\cap F^{-1}(B) $$ $$ F^{-1}(A^c) = (F^{-1}A)^c $$ $$ F^{-1}(1)=1 $$ --- ## $S_{747}$ $$ (X,F) \mapsto (B_X,F^{-1}) $$ $$ \mathbf{StoneDyn}^{op} \simeq \mathbf{BoolDyn} $$ $$ \operatorname{Ult} \dashv \operatorname{Clop} $$ --- ## $S_{748}$ $$ B_X^F := \{A\in B_X:F^{-1}(A)=A\} $$ $$ B_X^F \in\mathbf{Bool} $$ $$ \operatorname{Spec}_B(B_X^F) := \operatorname{Ult}(B_X^F) $$ --- ## $S_{749}$ $$ \pi_F:X\to Q_F $$ $$ \pi_F^{-1}: \operatorname{Obs}(Q_F) \to B_X^F $$ $$ \pi_F^{-1}(A) \in B_X^F $$ $$ \operatorname{Obs}(Q_F) \hookrightarrow B_X^F $$ --- ## $S_{750}$ $$ \operatorname{Obs}(Q_F) \cong B_X^F $$ $$ \Rightarrow $$ $$ Q_F \cong \operatorname{Ult}(B_X^F) $$ $$ Q_F \cong \operatorname{Spec}_B(B_X^F) $$ --- ## $S_{751}$ $$ \mathcal F_X $$ $$ F^{-1}:\mathcal F_X\to\mathcal F_X $$ $$ \mathcal F_X^F := \operatorname{Eq} ( F^{-1}, \operatorname{id} ) $$ $$ \operatorname{Spec}(\mathcal F_X^F) $$ --- ## $S_{752}$ $$ \operatorname{coEq} ( F^{-1}, \operatorname{id} ) $$ $$ \mathcal F_X \xrightarrow{F^{-1}-\operatorname{id}} \mathcal F_X $$ $$ \operatorname{Inv}(F^{-1}) = \ker(F^{-1}-\operatorname{id}) $$ $$ \operatorname{Coinv}(F^{-1}) = \operatorname{coker}(F^{-1}-\operatorname{id}) $$ --- ## $S_{753}$ $$ \mathcal O_X^\bullet $$ $$ d:\mathcal O_X^n\to\mathcal O_X^{n+1} $$ $$ F^\ast d=dF^\ast $$ $$ H^\bullet(X) $$ $$ F^\ast: H^\bullet(X)\to H^\bullet(X) $$ --- ## $S_{754}$ $$ H^\bullet(X)^F := \ker(F^\ast-\operatorname{id}) $$ $$ H^\bullet(X)_F := \operatorname{coker}(F^\ast-\operatorname{id}) $$ $$ \operatorname{Obs}_H(F) := ( H^\bullet(X)^F, H^\bullet(X)_F ) $$ --- ## $S_{755}$ $$ \operatorname{Sh}(X) $$ $$ F^{-1}: \operatorname{Sh}(X)\to\operatorname{Sh}(X) $$ $$ \operatorname{Eq} ( F^{-1}, \operatorname{id} ) $$ $$ \operatorname{Sh}(X)^F $$ --- ## $S_{756}$ $$ a: \operatorname{PSh}(X)\to\operatorname{Sh}(X) $$ $$ a^2\simeq a $$ $$ \operatorname{Fix}(a) = \operatorname{Sh}(X) $$ $$ aF^{-1} \simeq F^{-1}a $$ --- ## $S_{757}$ $$ \operatorname{Hyp}(X) $$ $$ h: \operatorname{Sh}_\infty(X) \to \operatorname{Hyp}(X) $$ $$ h^2\simeq h $$ $$ \operatorname{Fix}(h) = \operatorname{Hyp}(X) $$ --- ## $S_{758}$ $$ \tau_{\le0}X \leftarrow \tau_{\le1}X \leftarrow \tau_{\le2}X \leftarrow \cdots $$ $$ X^\wedge := \varprojlim_n\tau_{\le n}X $$ $$ X\to X^\wedge $$ $$ \tau_{\le n}X^\wedge \simeq \tau_{\le n}X $$ --- ## $S_{759}$ $$ X \ \operatorname{Post} \iff X\to X^\wedge \ \operatorname{eq} $$ $$ \operatorname{Post}(X) := X^\wedge $$ $$ \operatorname{Post}^2 \simeq \operatorname{Post} $$ --- ## $S_{760}$ $$ \Pi_\infty: \mathcal X\to\operatorname{Pro}(\mathcal S) $$ $$ \Gamma: \mathcal X\to\mathcal S $$ $$ \Pi_\infty \dashv \Delta $$ $$ \operatorname{Shape}(\mathcal X) := \Pi_\infty(1_\mathcal X) $$ --- ## $S_{761}$ $$ \mathcal X//F $$ $$ \operatorname{Shape}(\mathcal X//F) $$ $$ \operatorname{Shape}(\mathcal X)/\!/F $$ $$ \operatorname{Shape}(\mathcal X//F) \to \operatorname{Shape}(\mathcal X)/\!/F $$ --- ## $S_{762}$ $$ X^{hG} := \operatorname{Map}_G(EG,X) $$ $$ X_{hG} := EG\times_GX $$ $$ X^G $$ $$ X/G $$ --- ## $S_{763}$ $$ X^G \to X^{hG} $$ $$ X_{hG} \to X/G $$ $$ \pi_0(X_{hG}) = X/G $$ $$ \pi_0(X^{hG}) \supseteq X^G $$ --- ## $S_{764}$ $$ |G|<\infty $$ $$ N: X_{hG}\to X^{hG} $$ $$ X^{tG} := \operatorname{cofib}(N) $$ $$ X_{hG} \xrightarrow{N} X^{hG} \to X^{tG} $$ --- ## $S_{765}$ $$ X^{tG}\simeq0 $$ $$ \Rightarrow X_{hG} \simeq X^{hG} $$ $$ \operatorname{NormEq} $$ --- ## $S_{766}$ $$ \mathcal C \ \operatorname{rigid} $$ $$ X\in\mathcal C $$ $$ F:X\to X $$ $$ \operatorname{tr}(F) : \mathbb 1\to\mathbb 1 $$ $$ \operatorname{tr}(GF) = \operatorname{tr}(FG) $$ --- ## $S_{767}$ $$ h:X\xrightarrow{\sim}Y $$ $$ G=hFh^{-1} $$ $$ \operatorname{tr}(G) = \operatorname{tr}(F) $$ $$ \operatorname{tr}(F^n) $$ --- ## $S_{768}$ $$ L(F) := \sum_i(-1)^i \operatorname{tr} ( F_\ast|_{H_i(X)} ) $$ $$ L(F^n) $$ $$ \operatorname{Lef}(F) $$ --- ## $S_{769}$ $$ \zeta_F(t) := \exp \left( \sum_{n\ge1} \frac{L(F^n)}{n}t^n \right) $$ $$ \log\zeta_F(t) = \sum_{n\ge1} \frac{L(F^n)}{n}t^n $$ --- ## $S_{770}$ $$ V \ \operatorname{fin} $$ $$ F:V\to V $$ $$ \exp \left( \sum_{n\ge1} \frac{\operatorname{tr}(F^n)}{n}t^n \right) = \frac{1}{\det(I-tF)} $$ --- ## $S_{771}$ $$ \zeta_F(t) = \prod_i \det ( I-tF_\ast|_{H_i} )^{(-1)^{i+1}} $$ $$ \operatorname{Spec}(F_\ast) $$ $$ \zeta_F $$ --- ## $S_{772}$ $$ \mathcal O(F) := ( Q_F, \widetilde Q_F, \mathcal H_F, B_X^F, H^\bullet(X)^F, H^\bullet(X)_F, \zeta_F ) $$ $$ F\mapsto\mathcal O(F) $$ --- ## $S_{773}$ $$ F\sim_{\mathcal O}G \iff \mathcal O(F)\cong\mathcal O(G) $$ $$ \sim_{\mathcal O} $$ $$ \mathbf{Dyn}/{\sim_{\mathcal O}} $$ $$ \Pi_{\mathcal O} $$ --- ## $S_{774}$ $$ \mathcal O_0 \subseteq \mathcal O_1 \subseteq \cdots $$ $$ \sim_{\mathcal O_{n+1}} \subseteq \sim_{\mathcal O_n} $$ $$ \mathcal O_\omega := \bigcup_{n<\omega}\mathcal O_n $$ --- ## $S_{775}$ $$ \mathcal O_\alpha $$ $$ \mathcal O_{\alpha+1} \supseteq \mathcal O_\alpha $$ $$ \mathcal O_\lambda = \bigcup_{\beta<\lambda}\mathcal O_\beta $$ $$ \sim_\lambda = \bigcap_{\beta<\lambda}\sim_\beta $$ --- ## $S_{776}$ $$ \exists\Theta: \sim_{\Theta+1} = \sim_\Theta $$ $$ \mathbf O_\infty := \mathbf{Dyn}/{\sim_\Theta} $$ $$ \Pi_\Theta: \mathbf{Dyn}\to\mathbf O_\infty $$ --- ## $S_{777}$ $$ \mathfrak M_\Theta(o) := \{F:\Pi_\Theta(F)=o\}/{\cong} $$ $$ |\mathfrak M_\Theta(o)|=1 \quad\lor\quad |\mathfrak M_\Theta(o)|>1 $$ --- ## $S_{778}$ $$ |\mathfrak M_\Theta(o)|=1 $$ $$ \Rightarrow \exists\mathcal R_\Theta: \mathbf O_\infty\to\mathbf{Dyn} $$ $$ \Pi_\Theta\mathcal R_\Theta = \operatorname{id} $$ $$ \mathcal R_\Theta\Pi_\Theta \cong \operatorname{id} $$ --- ## $S_{779}$ $$ E_\Theta := \mathcal R_\Theta\Pi_\Theta $$ $$ E_\Theta^2 \cong E_\Theta $$ $$ \operatorname{Fix}(E_\Theta) = \operatorname{im}(\mathcal R_\Theta) $$ $$ \mathbf{Dyn} \to \operatorname{Fix}(E_\Theta) $$ --- ## $S_{780}$ $$ S_{720} \to S_{721} \to \cdots \to S_{780} $$ $$ \operatorname{Nerve} \to \operatorname{Completion} \to \operatorname{Stone} \to \operatorname{Invariant} \to \operatorname{Sheaf} \to \operatorname{Shape} \to \operatorname{HomotopyFix} \to \operatorname{Trace} \to \operatorname{ObservableTower} \to \operatorname{Reconstruct} $$ $$ N_\mathcal T : \mathbf{Dyn} \to [\mathcal T^{op},\mathbf{Set}] $$ $$ \widehat{(-)} : \mathbf{Dyn} \to \widehat{\mathbf{Dyn}} $$ $$ \mathcal O(F) $$ $$ \sim_{\alpha+1} \subseteq \sim_\alpha $$ $$ \exists\Theta: \sim_{\Theta+1} = \sim_\Theta $$ $$ E_\Theta^2 \cong E_\Theta $$ $$ \boxed{ SREG_{013} : S_{721} \to S_{722} \to \cdots \to S_{780} } $$ $$ \boxed{ S_{780}\to S_{781} } $$ --- # SREG-014 $$ S_{780}\longrightarrow S_{781} $$ --- ## $S_{781}$ $$ E_\Theta: \mathbf{Dyn}\to\mathbf{Dyn} $$ $$ E_\Theta^2\cong E_\Theta $$ $$ \mathbf C_\Theta := \operatorname{Fix}(E_\Theta) $$ $$ I_\Theta: \mathbf C_\Theta\hookrightarrow\mathbf{Dyn} $$ $$ R_\Theta: \mathbf{Dyn}\to\mathbf C_\Theta $$ --- ## $S_{782}$ $$ R_\Theta\dashv I_\Theta $$ $$ R_\Theta I_\Theta \cong \operatorname{id}_{\mathbf C_\Theta} $$ $$ I_\Theta R_\Theta \cong E_\Theta $$ $$ \mathbf C_\Theta \simeq \operatorname{im}(E_\Theta) $$ --- ## $S_{783}$ $$ G\in\mathbf C_\Theta $$ $$ \operatorname{Hom}(G,-): \mathbf C_\Theta\to\mathbf{Set} $$ $$ \forall X,Y: \left[ \operatorname{Hom}(G,X) \cong \operatorname{Hom}(G,Y) \right] \Rightarrow X\cong Y $$ $$ G \in \operatorname{Gen}(\mathbf C_\Theta) $$ --- ## $S_{784}$ $$ A := \operatorname{End}_{\mathbf C_\Theta}(G)^{op} $$ $$ H_G(X) := \operatorname{Hom}(G,X) $$ $$ H_G(X)\in\operatorname{Mod}_A $$ $$ H_G: \mathbf C_\Theta\to\operatorname{Mod}_A $$ --- ## $S_{785}$ $$ G\in\operatorname{Proj}(\mathbf C_\Theta) $$ $$ G\in\operatorname{Comp}(\mathbf C_\Theta) $$ $$ G\in\operatorname{Gen}(\mathbf C_\Theta) $$ $$ \Rightarrow $$ $$ H_G: \mathbf C_\Theta \xrightarrow{\sim} \operatorname{Mod}_A $$ --- ## $S_{786}$ $$ L_G: \operatorname{Mod}_A\to\mathbf C_\Theta $$ $$ L_G(M) := M\otimes_A G $$ $$ L_G\dashv H_G $$ $$ L_GH_G \cong \operatorname{id}_{\mathbf C_\Theta} $$ $$ H_GL_G \cong \operatorname{id}_{\operatorname{Mod}_A} $$ --- ## $S_{787}$ $$ \omega: \mathbf C_\Theta\to\mathbf{Vect}_k $$ $$ \omega(X)^\vee := \operatorname{Hom}_k(\omega(X),k) $$ $$ H_\omega := \int^{X\in\mathbf C_\Theta} \omega(X)^\vee\otimes\omega(X) $$ $$ i_X: \omega(X)^\vee\otimes\omega(X) \to H_\omega $$ --- ## $S_{788}$ $$ \Delta: H_\omega \to H_\omega\otimes H_\omega $$ $$ \varepsilon: H_\omega\to k $$ $$ (\Delta\otimes\operatorname{id})\Delta = (\operatorname{id}\otimes\Delta)\Delta $$ $$ (\varepsilon\otimes\operatorname{id})\Delta = \operatorname{id} = (\operatorname{id}\otimes\varepsilon)\Delta $$ --- ## $S_{789}$ $$ \omega(X\otimes Y) \cong \omega(X)\otimes\omega(Y) $$ $$ \omega(\mathbb 1) \cong k $$ $$ m: H_\omega\otimes H_\omega \to H_\omega $$ $$ u:k\to H_\omega $$ $$ (H_\omega,m,u,\Delta,\varepsilon) $$ --- ## $S_{790}$ $$ S: H_\omega\to H_\omega $$ $$ m(S\otimes\operatorname{id})\Delta = u\varepsilon $$ $$ m(\operatorname{id}\otimes S)\Delta = u\varepsilon $$ $$ H_\omega\in\mathbf{Hopf}_k $$ --- ## $S_{791}$ $$ \operatorname{Comod}_{H_\omega} $$ $$ \rho_X: \omega(X) \to \omega(X)\otimes H_\omega $$ $$ (\rho_X\otimes\operatorname{id})\rho_X = (\operatorname{id}\otimes\Delta)\rho_X $$ $$ (\operatorname{id}\otimes\varepsilon)\rho_X = \operatorname{id} $$ --- ## $S_{792}$ $$ K_\omega: \mathbf C_\Theta \to \operatorname{Comod}_{H_\omega} $$ $$ X \mapsto (\omega(X),\rho_X) $$ $$ K_\omega(X\otimes Y) \cong K_\omega(X)\otimes K_\omega(Y) $$ --- ## $S_{793}$ $$ K_\omega \ \operatorname{ff} $$ $$ \operatorname{Im}(K_\omega) = \operatorname{Comod}_{H_\omega}^{\operatorname{fd}} $$ $$ \Rightarrow $$ $$ \mathbf C_\Theta \simeq \operatorname{Comod}_{H_\omega}^{\operatorname{fd}} $$ --- ## $S_{794}$ $$ \operatorname{Nat}(\omega,\omega) $$ $$ \eta_X:\omega(X)\to\omega(X) $$ $$ \eta_Y\omega(f) = \omega(f)\eta_X $$ $$ \operatorname{End}^\otimes(\omega) $$ --- ## $S_{795}$ $$ \operatorname{Aut}^\otimes(\omega) $$ $$ g_X: \omega(X)\xrightarrow{\sim}\omega(X) $$ $$ g_{X\otimes Y} = g_X\otimes g_Y $$ $$ g_{\mathbb 1} = \operatorname{id} $$ --- ## $S_{796}$ $$ \operatorname{Spec}(H_\omega) $$ $$ G_\omega(R) := \operatorname{Hom}_{\mathbf{CAlg}_k} (H_\omega,R) $$ $$ G_\omega : \mathbf{CAlg}_k\to\mathbf{Grp} $$ $$ G_\omega \simeq \operatorname{Aut}^\otimes(\omega) $$ --- ## $S_{797}$ $$ \mathcal C $$ $$ Z(\mathcal C) $$ $$ (X,\gamma) \in Z(\mathcal C) $$ $$ \gamma_Y: X\otimes Y \xrightarrow{\sim} Y\otimes X $$ $$ \gamma_{Y\otimes Z} = (\operatorname{id}_Y\otimes\gamma_Z) (\gamma_Y\otimes\operatorname{id}_Z) $$ --- ## $S_{798}$ $$ U: Z(\mathcal C)\to\mathcal C $$ $$ U(X,\gamma)=X $$ $$ Z(\mathcal C) $$ $$ Z(Z(\mathcal C)) $$ $$ \mathcal C \to Z(\mathcal C) \to Z^2(\mathcal C) \to \cdots $$ --- ## $S_{799}$ $$ A\in\mathbf{Alg}_k $$ $$ Z(A) := \{a\in A:\forall b\in A,\ ab=ba\} $$ $$ Z(A) \cong \operatorname{End}_{A^e}(A) $$ $$ Z(A)\subseteq A $$ --- ## $S_{800}$ $$ \operatorname{HH}^\bullet(A) := \operatorname{Ext}_{A^e}^\bullet(A,A) $$ $$ \operatorname{HH}_\bullet(A) := \operatorname{Tor}^{A^e}_\bullet(A,A) $$ $$ A^e := A\otimes A^{op} $$ $$ \operatorname{HH}^0(A) \cong Z(A) $$ --- ## $S_{801}$ $$ {}_AM_B $$ $$ {}_BN_C $$ $$ M\otimes_BN $$ $$ {}_A(M\otimes_BN)_C $$ $$ (A,B) \mapsto \operatorname{Bimod}(A,B) $$ --- ## $S_{802}$ $$ \mathbf{Alg}_2 $$ $$ \operatorname{Ob}(\mathbf{Alg}_2) = \{A\} $$ $$ \operatorname{Hom}(A,B) = \operatorname{Bimod}(A,B) $$ $$ N\circ M = M\otimes_BN $$ --- ## $S_{803}$ $$ {}_AM_B $$ $$ {}_BN_A $$ $$ M\otimes_BN \simeq A $$ $$ N\otimes_AM \simeq B $$ $$ A\sim_{\operatorname{Mor}}B $$ --- ## $S_{804}$ $$ A\sim_{\operatorname{Mor}}B $$ $$ \operatorname{Mod}_A \simeq \operatorname{Mod}_B $$ $$ Z(A)\cong Z(B) $$ $$ \operatorname{HH}_\bullet(A) \cong \operatorname{HH}_\bullet(B) $$ --- ## $S_{805}$ $$ \operatorname{Perf}(A) $$ $$ \operatorname{Perf}(B) $$ $$ \operatorname{Perf}(A) \simeq \operatorname{Perf}(B) $$ $$ A\sim_{\operatorname{dMor}}B $$ --- ## $S_{806}$ $$ \operatorname{HH}(A) $$ $$ A\sim_{\operatorname{dMor}}B \Rightarrow \operatorname{HH}(A) \simeq \operatorname{HH}(B) $$ $$ \operatorname{THH}(A) $$ $$ \operatorname{THH}(A) \simeq \operatorname{THH}(B) $$ --- ## $S_{807}$ $$ \mathcal C \in \mathbf{Cat}_{\operatorname{st}} $$ $$ \mathcal C^\vee := \operatorname{Fun}^{\operatorname{ex}}(\mathcal C,\mathbf{Sp}) $$ $$ \operatorname{ev}: \mathcal C^\vee\otimes\mathcal C \to \mathbf{Sp} $$ $$ \operatorname{coev}: \mathbf{Sp} \to \mathcal C\otimes\mathcal C^\vee $$ --- ## $S_{808}$ $$ (\operatorname{id}_\mathcal C\otimes\operatorname{ev}) (\operatorname{coev}\otimes\operatorname{id}_\mathcal C) \simeq \operatorname{id}_\mathcal C $$ $$ (\operatorname{ev}\otimes\operatorname{id}_{\mathcal C^\vee}) (\operatorname{id}_{\mathcal C^\vee}\otimes\operatorname{coev}) \simeq \operatorname{id}_{\mathcal C^\vee} $$ $$ \mathcal C \ \operatorname{dual} $$ --- ## $S_{809}$ $$ F:\mathcal C\to\mathcal C $$ $$ \operatorname{Tr}(F) := \operatorname{ev} \circ (\operatorname{id}_{\mathcal C^\vee}\otimes F) \circ \operatorname{coev} $$ $$ \operatorname{Tr}(F)\in\operatorname{End}(\mathbb 1) $$ --- ## $S_{810}$ $$ F,G:\mathcal C\to\mathcal C $$ $$ \operatorname{Tr}(FG) \simeq \operatorname{Tr}(GF) $$ $$ F\simeq G \Rightarrow \operatorname{Tr}(F) \simeq \operatorname{Tr}(G) $$ --- ## $S_{811}$ $$ \operatorname{Tr}(\operatorname{id}_{\operatorname{Perf}(A)}) $$ $$ \simeq \operatorname{HH}(A) $$ $$ \operatorname{Tr}(\operatorname{id}_{\mathcal C}) $$ $$ \operatorname{HH}(\mathcal C) $$ --- ## $S_{812}$ $$ Z(\mathcal C) $$ $$ \operatorname{HH}^\bullet(\mathcal C) $$ $$ \operatorname{HH}_\bullet(\mathcal C) $$ $$ \operatorname{Tr} $$ $$ \operatorname{CenterTrace}(\mathcal C) $$ --- ## $S_{813}$ $$ \mathcal C \mapsto \operatorname{CenterTrace}(\mathcal C) $$ $$ \mathcal C\sim_{\operatorname{Mor}}\mathcal D $$ $$ \Rightarrow $$ $$ \operatorname{CenterTrace}(\mathcal C) \cong \operatorname{CenterTrace}(\mathcal D) $$ --- ## $S_{814}$ $$ \mathbf{Alg}_n $$ $$ \operatorname{Obj}_0 $$ $$ \operatorname{Mor}_1 $$ $$ \operatorname{Mor}_2 $$ $$ \cdots $$ $$ \operatorname{Mor}_n $$ --- ## $S_{815}$ $$ A \in \mathbf{Alg}_n^{\operatorname{fd}} $$ $$ A^\vee $$ $$ \operatorname{ev}_k $$ $$ \operatorname{coev}_k $$ $$ 0\le k\le n $$ --- ## $S_{816}$ $$ A \ \operatorname{fd} $$ $$ \Rightarrow $$ $$ Z_A: \operatorname{Bord}_n^{\operatorname{fr}} \to \mathcal C $$ $$ Z_A(\operatorname{pt}) = A $$ --- ## $S_{817}$ $$ M^n $$ $$ Z_A(M) $$ $$ M\sqcup N $$ $$ Z_A(M\sqcup N) = Z_A(M)\otimes Z_A(N) $$ $$ M\circ N $$ $$ Z_A(M\circ N) = Z_A(M)\circ Z_A(N) $$ --- ## $S_{818}$ $$ S^1 $$ $$ Z_A(S^1) $$ $$ Z_A(S^1) \simeq \operatorname{Tr}(\operatorname{id}_A) $$ $$ Z_A(S^1) \simeq \operatorname{HH}(A) $$ --- ## $S_{819}$ $$ \int_MA $$ $$ M\mapsto\int_MA $$ $$ \int_{M\sqcup N}A \simeq \int_MA\otimes\int_NA $$ $$ \int_{S^1}A \simeq \operatorname{HH}(A) $$ --- ## $S_{820}$ $$ M = U\cup_VW $$ $$ \int_MA \simeq \int_UA \otimes_{\int_VA} \int_WA $$ $$ \operatorname{Excision} $$ --- ## $S_{821}$ $$ \mathcal O(F) $$ $$ \operatorname{CenterTrace}(F) $$ $$ \operatorname{BordInv}(F) $$ $$ \mathcal O^+(F) := ( \mathcal O(F), \operatorname{CenterTrace}(F), \operatorname{BordInv}(F) ) $$ --- ## $S_{822}$ $$ F\sim_+G \iff \mathcal O^+(F) \cong \mathcal O^+(G) $$ $$ \sim_+ \subseteq \sim_{\mathcal O} $$ $$ \mathbf{Dyn}/{\sim_+} $$ --- ## $S_{823}$ $$ \mathcal O_0^+ \subseteq \mathcal O_1^+ \subseteq \cdots $$ $$ \sim_{n+1}^+ \subseteq \sim_n^+ $$ $$ \sim_\omega^+ := \bigcap_{n<\omega}\sim_n^+ $$ --- ## $S_{824}$ $$ \mathcal O_\alpha^+ $$ $$ \sim_{\alpha+1}^+ \subseteq \sim_\alpha^+ $$ $$ \sim_\lambda^+ = \bigcap_{\beta<\lambda} \sim_\beta^+ $$ --- ## $S_{825}$ $$ \exists\Lambda: \sim_{\Lambda+1}^+ = \sim_\Lambda^+ $$ $$ \mathbf R_\infty := \mathbf{Dyn}/{\sim_\Lambda^+} $$ $$ \Pi_\Lambda^+: \mathbf{Dyn}\to\mathbf R_\infty $$ --- ## $S_{826}$ $$ \mathcal U: \mathbf R_\infty \to \mathbf{Dyn} $$ $$ \Pi_\Lambda^+\mathcal U = \operatorname{id}_{\mathbf R_\infty} $$ $$ \mathcal U\Pi_\Lambda^+ \sim_\Lambda^+ \operatorname{id}_{\mathbf{Dyn}} $$ --- ## $S_{827}$ $$ P_\Lambda := \mathcal U\Pi_\Lambda^+ $$ $$ P_\Lambda^2 \sim_\Lambda^+ P_\Lambda $$ $$ \operatorname{im}(P_\Lambda) = \operatorname{im}(\mathcal U) $$ --- ## $S_{828}$ $$ \mathcal A_\Lambda := \operatorname{End} ( \operatorname{im}(P_\Lambda) ) $$ $$ a,b\in\mathcal A_\Lambda $$ $$ ab $$ $$ \operatorname{id} $$ $$ \mathcal A_\Lambda $$ --- ## $S_{829}$ $$ Z(\mathcal A_\Lambda) $$ $$ \operatorname{Spec} ( Z(\mathcal A_\Lambda) ) $$ $$ \mathfrak X_\Lambda := \operatorname{Spec} ( Z(\mathcal A_\Lambda) ) $$ --- ## $S_{830}$ $$ \mathcal O_{\mathfrak X_\Lambda} $$ $$ (\mathfrak X_\Lambda,\mathcal O_{\mathfrak X_\Lambda}) $$ $$ \operatorname{QCoh}(\mathfrak X_\Lambda) $$ $$ \operatorname{Perf}(\mathfrak X_\Lambda) $$ --- ## $S_{831}$ $$ \Phi: \operatorname{im}(P_\Lambda) \to \operatorname{Perf}(\mathfrak X_\Lambda) $$ $$ \Psi: \operatorname{Perf}(\mathfrak X_\Lambda) \to \operatorname{im}(P_\Lambda) $$ $$ \Psi\Phi \simeq \operatorname{id} $$ --- ## $S_{832}$ $$ \Phi\Psi \simeq \operatorname{id} $$ $$ \Rightarrow $$ $$ \operatorname{im}(P_\Lambda) \simeq \operatorname{Perf}(\mathfrak X_\Lambda) $$ $$ \operatorname{Repr}_\Lambda $$ --- ## $S_{833}$ $$ \operatorname{Aut}(\mathfrak X_\Lambda) $$ $$ \operatorname{AutEq} ( \operatorname{Perf}(\mathfrak X_\Lambda) ) $$ $$ \operatorname{Aut}(\mathfrak X_\Lambda) \to \operatorname{AutEq} ( \operatorname{Perf}(\mathfrak X_\Lambda) ) $$ --- ## $S_{834}$ $$ K \in \operatorname{Perf} ( \mathfrak X_\Lambda\times\mathfrak X_\Lambda ) $$ $$ \Phi_K $$ $$ K\star L $$ $$ \Phi_{K\star L} = \Phi_L\circ\Phi_K $$ --- ## $S_{835}$ $$ \operatorname{Pic} ( \mathfrak X_\Lambda ) $$ $$ \operatorname{PicCat} ( \operatorname{Perf}(\mathfrak X_\Lambda) ) $$ $$ \operatorname{Inv} ( \operatorname{Perf}(\mathfrak X_\Lambda) ) $$ --- ## $S_{836}$ $$ \mathfrak X_\Lambda \mapsto \operatorname{Perf}(\mathfrak X_\Lambda) \mapsto \operatorname{End} ( \operatorname{Perf}(\mathfrak X_\Lambda) ) $$ $$ \mapsto Z \mapsto \operatorname{Spec} $$ --- ## $S_{837}$ $$ \mathfrak X_\Lambda' := \operatorname{Spec} Z \left( \operatorname{End} ( \operatorname{Perf}(\mathfrak X_\Lambda) ) \right) $$ $$ \mathfrak X_\Lambda \to \mathfrak X_\Lambda' $$ --- ## $S_{838}$ $$ \mathfrak X_\Lambda \cong \mathfrak X_\Lambda' $$ $$ \Rightarrow $$ $$ \operatorname{Spec} Z \operatorname{End} \operatorname{Perf} ( \mathfrak X_\Lambda ) \cong \mathfrak X_\Lambda $$ $$ \operatorname{SelfRec}(\mathfrak X_\Lambda) $$ --- ## $S_{839}$ $$ \mathfrak X_0 := \mathfrak X_\Lambda $$ $$ \mathfrak X_{n+1} := \operatorname{Spec} Z \operatorname{End} \operatorname{Perf} ( \mathfrak X_n ) $$ $$ \mathfrak X_\omega := \varprojlim_n\mathfrak X_n $$ --- ## $S_{840}$ $$ S_{780} \to S_{781} \to \cdots \to S_{840} $$ $$ \operatorname{IdemImage} \to \operatorname{Generator} \to \operatorname{Module} \to \operatorname{Coend} \to \operatorname{Hopf} \to \operatorname{Morita} \to \operatorname{Trace} \to \operatorname{HigherDual} \to \operatorname{Reconstruct} \to \operatorname{SelfRec} $$ $$ E_\Theta^2 \cong E_\Theta $$ $$ \mathbf C_\Theta \simeq \operatorname{im}(E_\Theta) $$ $$ \mathbf C_\Theta \simeq \operatorname{Mod}_A $$ $$ \mathbf C_\Theta \simeq \operatorname{Comod}_{H_\omega} $$ $$ \operatorname{im}(P_\Lambda) \simeq \operatorname{Perf}(\mathfrak X_\Lambda) $$ $$ \mathfrak X_{\Lambda} \to \operatorname{Spec} Z \operatorname{End} \operatorname{Perf} ( \mathfrak X_{\Lambda} ) $$ $$ \boxed{ SREG_{014} : S_{781} \to S_{782} \to \cdots \to S_{840} } $$ $$ \boxed{ S_{840}\to S_{841} } $$ --- # SREG-015 $$ S_{840}\longrightarrow S_{841} $$ --- ## $S_{841}$ $$ \mathfrak X_0 $$ $$ \mathfrak X_{n+1} := \mathcal R_{\mathrm g}(\mathfrak X_n) $$ $$ \mathcal R_{\mathrm g} := \operatorname{Spec} \circ Z \circ \operatorname{End} \circ \operatorname{Perf} $$ $$ \mathfrak X_{n+1} = \mathcal R_{\mathrm g}(\mathfrak X_n) $$ --- ## $S_{842}$ $$ \mathcal R_{\mathrm g}^0 = \operatorname{id} $$ $$ \mathcal R_{\mathrm g}^{n+1} = \mathcal R_{\mathrm g} \circ \mathcal R_{\mathrm g}^{n} $$ $$ \mathfrak X_n = \mathcal R_{\mathrm g}^n(\mathfrak X_0) $$ $$ \operatorname{Orb}_{\mathcal R_{\mathrm g}}(\mathfrak X_0) $$ --- ## $S_{843}$ $$ \mathfrak X_\omega := \varprojlim_{n<\omega}\mathfrak X_n $$ $$ p_n: \mathfrak X_\omega\to\mathfrak X_n $$ $$ r_n: \mathfrak X_{n+1}\to\mathfrak X_n $$ $$ r_n\circ p_{n+1} = p_n $$ --- ## $S_{844}$ $$ \mathcal R_{\mathrm g}(\mathfrak X_\omega) $$ $$ \mathcal R_{\mathrm g}(\mathfrak X_\omega) \to \mathfrak X_\omega $$ $$ \mathfrak X_\omega \to \mathcal R_{\mathrm g}(\mathfrak X_\omega) $$ $$ \eta_\omega $$ $$ \varepsilon_\omega $$ --- ## $S_{845}$ $$ \eta_\omega\varepsilon_\omega \simeq \operatorname{id} $$ $$ \varepsilon_\omega\eta_\omega \simeq \operatorname{id} $$ $$ \mathcal R_{\mathrm g}(\mathfrak X_\omega) \simeq \mathfrak X_\omega $$ $$ \mathfrak X_\omega \in \operatorname{Fix}(\mathcal R_{\mathrm g}) $$ --- ## $S_{846}$ $$ \operatorname{Fix}(\mathcal R_{\mathrm g}) := \{ X: \mathcal R_{\mathrm g}(X)\simeq X \} $$ $$ \operatorname{Fix}(\mathcal R_{\mathrm g}) \subseteq \mathbf{Geom} $$ $$ \mathfrak X_\omega \in \operatorname{Fix}(\mathcal R_{\mathrm g}) $$ --- ## $S_{847}$ $$ X,Y \in \operatorname{Fix}(\mathcal R_{\mathrm g}) $$ $$ \operatorname{Perf}(X) \simeq \operatorname{Perf}(Y) $$ $$ \Rightarrow $$ $$ \mathcal R_{\mathrm g}(X) \simeq \mathcal R_{\mathrm g}(Y) $$ $$ X\simeq Y $$ --- ## $S_{848}$ $$ \mathcal M_{\mathrm g} := \operatorname{Fix}(\mathcal R_{\mathrm g})/{\simeq} $$ $$ [X]_{\mathrm g} $$ $$ \mathcal R_{\mathrm g} : \mathcal M_{\mathrm g} \to \mathcal M_{\mathrm g} $$ $$ \mathcal R_{\mathrm g} = \operatorname{id}_{\mathcal M_{\mathrm g}} $$ --- ## $S_{849}$ $$ \operatorname{Aut}(X) $$ $$ \operatorname{AutEq}(\operatorname{Perf}(X)) $$ $$ \Phi_X: \operatorname{Aut}(X) \to \operatorname{AutEq}(\operatorname{Perf}(X)) $$ $$ \ker\Phi_X $$ --- ## $S_{850}$ $$ \operatorname{OutPerf}(X) := \operatorname{AutEq}(\operatorname{Perf}(X)) / \operatorname{im}\Phi_X $$ $$ \operatorname{OutPerf}(X)=1 $$ $$ \Rightarrow $$ $$ \operatorname{AutEq}(\operatorname{Perf}(X)) = \operatorname{im}\Phi_X $$ --- ## $S_{851}$ $$ K \in \operatorname{Perf}(X\times X) $$ $$ \Phi_K : \operatorname{Perf}(X) \to \operatorname{Perf}(X) $$ $$ K\star L $$ $$ \Phi_{K\star L} = \Phi_L\circ\Phi_K $$ --- ## $S_{852}$ $$ \operatorname{KerEq}(X) := \{ K: \Phi_K\in\operatorname{AutEq}(\operatorname{Perf}(X)) \} $$ $$ K^{-1} $$ $$ K\star K^{-1} \simeq \mathcal O_{\Delta_X} $$ $$ K^{-1}\star K \simeq \mathcal O_{\Delta_X} $$ --- ## $S_{853}$ $$ \operatorname{PicPerf}(X) := \pi_0\operatorname{KerEq}(X) $$ $$ \operatorname{PicPerf}(X) \curvearrowright \operatorname{Perf}(X) $$ $$ [K]\cdot E = \Phi_K(E) $$ --- ## $S_{854}$ $$ Z_X := Z \operatorname{End} \operatorname{Perf}(X) $$ $$ \operatorname{Spec}(Z_X) = \mathcal R_{\mathrm g}(X) $$ $$ X\in\operatorname{Fix}(\mathcal R_{\mathrm g}) $$ $$ \Rightarrow X\simeq\operatorname{Spec}(Z_X) $$ --- ## $S_{855}$ $$ \mathcal O_X $$ $$ \Gamma(X,\mathcal O_X) $$ $$ Z_X $$ $$ \chi_X: \Gamma(X,\mathcal O_X) \to Z_X $$ $$ \chi_X \ \operatorname{iso} $$ --- ## $S_{856}$ $$ \operatorname{Spec} \Gamma(X,\mathcal O_X) \to X $$ $$ X\in\operatorname{Aff} $$ $$ \operatorname{Spec} \Gamma(X,\mathcal O_X) \simeq X $$ $$ \operatorname{AffFix} $$ --- ## $S_{857}$ $$ X = \bigcup_iU_i $$ $$ U_i \in \operatorname{Aff} $$ $$ \operatorname{Perf}(X) \to \prod_i\operatorname{Perf}(U_i) $$ $$ \operatorname{Desc} $$ --- ## $S_{858}$ $$ \operatorname{Perf}(X) \simeq \operatorname{Tot} \left( \prod_i \operatorname{Perf}(U_i) \rightrightarrows \prod_{i,j} \operatorname{Perf}(U_i\cap U_j) \triplearrows \cdots \right) $$ --- ## $S_{859}$ $$ Z_X \to \prod_i Z_{U_i} $$ $$ \prod_i Z_{U_i} \rightrightarrows \prod_{i,j}Z_{U_i\cap U_j} $$ $$ Z_X = \operatorname{Eq} \left( \prod_i Z_{U_i} \rightrightarrows \prod_{i,j}Z_{U_i\cap U_j} \right) $$ --- ## $S_{860}$ $$ \operatorname{Spec}(Z_X) $$ $$ \operatorname{colim} \left( \operatorname{Spec}(Z_{U_i}) \leftleftarrows \operatorname{Spec}(Z_{U_i\cap U_j}) \right) $$ $$ \operatorname{Glue} $$ --- ## $S_{861}$ $$ \mathcal R_{\mathrm g}(X) $$ $$ \operatorname{Glue} \left( \mathcal R_{\mathrm g}(U_i) \right) $$ $$ \mathcal R_{\mathrm g}(X) \simeq \operatorname{Glue} \left( \mathcal R_{\mathrm g}(U_i) \right) $$ --- ## $S_{862}$ $$ U_i \in \operatorname{Fix}(\mathcal R_{\mathrm g}) $$ $$ \forall i $$ $$ \Rightarrow $$ $$ X \in \operatorname{Fix}(\mathcal R_{\mathrm g}) $$ --- ## $S_{863}$ $$ \mathcal R_{\mathrm g} : \mathbf{Geom} \to \mathbf{Geom} $$ $$ \mathcal R_{\mathrm g} \operatorname{colim} \simeq \operatorname{colim} \mathcal R_{\mathrm g} $$ $$ \mathcal R_{\mathrm g} \operatorname{lim} \simeq \operatorname{lim} \mathcal R_{\mathrm g} $$ --- ## $S_{864}$ $$ \mathcal R_{\mathrm g} \dashv \mathcal S_{\mathrm g} $$ $$ \eta: \operatorname{id} \Rightarrow \mathcal S_{\mathrm g}\mathcal R_{\mathrm g} $$ $$ \varepsilon: \mathcal R_{\mathrm g}\mathcal S_{\mathrm g} \Rightarrow \operatorname{id} $$ --- ## $S_{865}$ $$ T_{\mathrm g} := \mathcal S_{\mathrm g}\mathcal R_{\mathrm g} $$ $$ W_{\mathrm g} := \mathcal R_{\mathrm g}\mathcal S_{\mathrm g} $$ $$ T_{\mathrm g} \in \mathbf{Mnd} $$ $$ W_{\mathrm g} \in \mathbf{Cmd} $$ --- ## $S_{866}$ $$ \mathbf{Geom}^{T_{\mathrm g}} $$ $$ \operatorname{Coalg}(W_{\mathrm g}) $$ $$ K: \mathbf{Geom} \to \mathbf{Geom}^{T_{\mathrm g}} $$ $$ J: \mathbf{Geom} \to \operatorname{Coalg}(W_{\mathrm g}) $$ --- ## $S_{867}$ $$ \operatorname{Fix}(T_{\mathrm g}) $$ $$ \operatorname{Fix}(W_{\mathrm g}) $$ $$ \operatorname{Fix}(\mathcal R_{\mathrm g}) $$ $$ \operatorname{Fix}(T_{\mathrm g}) \simeq \operatorname{Fix}(\mathcal R_{\mathrm g}) $$ --- ## $S_{868}$ $$ T_{\mathrm g}^2 \to T_{\mathrm g} $$ $$ W_{\mathrm g} \to W_{\mathrm g}^2 $$ $$ T_{\mathrm g}^{\alpha} $$ $$ W_{\mathrm g}^{\alpha} $$ --- ## $S_{869}$ $$ T_{\mathrm g}^{\alpha+1} = T_{\mathrm g}T_{\mathrm g}^{\alpha} $$ $$ T_{\mathrm g}^{\lambda} = \operatorname{colim}_{\beta<\lambda} T_{\mathrm g}^{\beta} $$ $$ W_{\mathrm g}^{\alpha+1} = W_{\mathrm g}W_{\mathrm g}^{\alpha} $$ $$ W_{\mathrm g}^{\lambda} = \operatorname{lim}_{\beta<\lambda} W_{\mathrm g}^{\beta} $$ --- ## $S_{870}$ $$ \exists\Gamma_T: T_{\mathrm g}^{\Gamma_T+1} \simeq T_{\mathrm g}^{\Gamma_T} $$ $$ \exists\Gamma_W: W_{\mathrm g}^{\Gamma_W+1} \simeq W_{\mathrm g}^{\Gamma_W} $$ $$ P_T := T_{\mathrm g}^{\Gamma_T} $$ $$ P_W := W_{\mathrm g}^{\Gamma_W} $$ --- ## $S_{871}$ $$ P_T^2\simeq P_T $$ $$ P_W^2\simeq P_W $$ $$ \operatorname{im}(P_T) = \operatorname{Fix}(T_{\mathrm g}) $$ $$ \operatorname{im}(P_W) = \operatorname{Fix}(W_{\mathrm g}) $$ --- ## $S_{872}$ $$ \operatorname{im}(P_T) \simeq \operatorname{im}(P_W) $$ $$ \mathcal G_\infty := \operatorname{im}(P_T) $$ $$ \mathcal G_\infty \simeq \operatorname{Fix}(\mathcal R_{\mathrm g}) $$ --- ## $S_{873}$ $$ X\in\mathcal G_\infty $$ $$ \mathcal R_{\mathrm g}(X) \simeq X $$ $$ \mathcal S_{\mathrm g}(X) \simeq X $$ $$ T_{\mathrm g}(X) \simeq X $$ $$ W_{\mathrm g}(X) \simeq X $$ --- ## $S_{874}$ $$ \operatorname{Perf}: \mathcal G_\infty \to \mathbf{Cat}_{\mathrm{st}} $$ $$ \operatorname{Spec}Z\operatorname{End} : \mathbf{Cat}_{\mathrm{st}} \to \mathcal G_\infty $$ $$ \operatorname{Spec}Z\operatorname{End} \circ \operatorname{Perf} \simeq \operatorname{id}_{\mathcal G_\infty} $$ --- ## $S_{875}$ $$ \mathcal C \in \operatorname{im}(\operatorname{Perf}) $$ $$ \operatorname{Perf} \circ \operatorname{Spec}Z\operatorname{End} (\mathcal C) $$ $$ \mathcal N(\mathcal C) $$ $$ \mathcal N^2 \simeq \mathcal N $$ --- ## $S_{876}$ $$ \mathbf C_{\mathrm{perf}} := \operatorname{Fix}(\mathcal N) $$ $$ \operatorname{Perf}: \mathcal G_\infty \to \mathbf C_{\mathrm{perf}} $$ $$ \operatorname{Spec}Z\operatorname{End} : \mathbf C_{\mathrm{perf}} \to \mathcal G_\infty $$ --- ## $S_{877}$ $$ \operatorname{Perf} \dashv \operatorname{Spec}Z\operatorname{End} $$ $$ \operatorname{Perf} \circ \operatorname{Spec}Z\operatorname{End} \simeq \operatorname{id}_{\mathbf C_{\mathrm{perf}}} $$ $$ \operatorname{Spec}Z\operatorname{End} \circ \operatorname{Perf} \simeq \operatorname{id}_{\mathcal G_\infty} $$ --- ## $S_{878}$ $$ \mathcal G_\infty \simeq \mathbf C_{\mathrm{perf}} $$ $$ X \leftrightarrow \operatorname{Perf}(X) $$ $$ \operatorname{Geom} \leftrightarrow \operatorname{Cat} $$ --- ## $S_{879}$ $$ \mathcal T: \mathcal G_\infty \to \mathbf{Hopf} $$ $$ X \mapsto H_{\omega_X} $$ $$ \mathcal U: \mathbf{Hopf} \to \mathcal G_\infty $$ $$ \mathcal U\mathcal T $$ --- ## $S_{880}$ $$ \mathcal U\mathcal T(X) \simeq X $$ $$ \mathcal T\mathcal U(H) \simeq H $$ $$ \mathcal G_\infty \simeq \mathbf{Hopf}_{\mathrm{rec}} $$ --- ## $S_{881}$ $$ \mathbf{Hopf}_{\mathrm{rec}} := \operatorname{Fix}(\mathcal T\mathcal U) $$ $$ \mathcal T\mathcal U : \mathbf{Hopf}\to\mathbf{Hopf} $$ $$ (\mathcal T\mathcal U)^2 \simeq \mathcal T\mathcal U $$ --- ## $S_{882}$ $$ X \in \mathcal G_\infty $$ $$ H_X := \mathcal T(X) $$ $$ X \simeq \mathcal U(H_X) $$ $$ H_X \simeq \mathcal T\mathcal U(H_X) $$ --- ## $S_{883}$ $$ \operatorname{Aut}(X) $$ $$ \operatorname{Aut}_{\mathbf{Hopf}}(H_X) $$ $$ \operatorname{Aut}(X) \to \operatorname{Aut}_{\mathbf{Hopf}}(H_X) $$ $$ \operatorname{KerAut}(X) $$ --- ## $S_{884}$ $$ \operatorname{Aut}(X) \cong \operatorname{Aut}_{\mathbf{Hopf}}(H_X) $$ $$ \operatorname{Sym}(X) \simeq \operatorname{Sym}(H_X) $$ $$ \operatorname{Stab}(X) $$ --- ## $S_{885}$ $$ \operatorname{Def}(X) $$ $$ \operatorname{Def}(H_X) $$ $$ T_X $$ $$ T_{H_X} $$ $$ T_X \cong T_{H_X} $$ --- ## $S_{886}$ $$ L_X $$ $$ L_{H_X} $$ $$ \operatorname{Ext}^\bullet(L_X,\mathcal O_X) $$ $$ \operatorname{Ext}^\bullet(L_{H_X},H_X) $$ $$ \operatorname{Obs}(X) \cong \operatorname{Obs}(H_X) $$ --- ## $S_{887}$ $$ \operatorname{Def}_0(X) $$ $$ \operatorname{Def}_1(X) $$ $$ \operatorname{Def}_2(X) $$ $$ \cdots $$ $$ \operatorname{Def}_\infty(X) $$ --- ## $S_{888}$ $$ \mathcal M_X := \operatorname{Def}_\infty(X) $$ $$ x_0 \in \mathcal M_X $$ $$ T_{x_0}\mathcal M_X $$ $$ L_{\mathcal M_X,x_0} $$ --- ## $S_{889}$ $$ \mathcal R_{\mathrm g} : \mathcal M_X \to \mathcal M_X $$ $$ [\xi] \mapsto [\mathcal R_{\mathrm g}(\xi)] $$ $$ \mathcal R_{\mathrm g}^2 \simeq \mathcal R_{\mathrm g} $$ --- ## $S_{890}$ $$ \operatorname{Fix} ( \mathcal R_{\mathrm g} | _{\mathcal M_X} ) $$ $$ \mathcal M_X^{\mathrm{rec}} $$ $$ \mathcal M_X^{\mathrm{rec}} \hookrightarrow \mathcal M_X $$ $$ \mathcal M_X \to \mathcal M_X^{\mathrm{rec}} $$ --- ## $S_{891}$ $$ \mathcal M_X^{\mathrm{rec}} \simeq \mathcal M_{H_X}^{\mathrm{rec}} $$ $$ T \mathcal M_X^{\mathrm{rec}} \cong T \mathcal M_{H_X}^{\mathrm{rec}} $$ $$ \operatorname{Obs} \cong \operatorname{Obs} $$ --- ## $S_{892}$ $$ \mathcal G_\infty \to \mathbf C_{\mathrm{perf}} \to \mathbf{Hopf}_{\mathrm{rec}} \to \mathbf{Moduli}_{\mathrm{rec}} $$ $$ \mathcal A_0 := \mathcal G_\infty $$ $$ \mathcal A_1 := \mathbf C_{\mathrm{perf}} $$ $$ \mathcal A_2 := \mathbf{Hopf}_{\mathrm{rec}} $$ $$ \mathcal A_3 := \mathbf{Moduli}_{\mathrm{rec}} $$ --- ## $S_{893}$ $$ F_i: \mathcal A_i \to \mathcal A_{i+1} $$ $$ G_i: \mathcal A_{i+1} \to \mathcal A_i $$ $$ G_iF_i \simeq \operatorname{id}_{\mathcal A_i} $$ $$ F_iG_i \simeq \operatorname{id}_{\mathcal A_{i+1}} $$ --- ## $S_{894}$ $$ \mathcal A_0 \simeq \mathcal A_1 \simeq \mathcal A_2 \simeq \mathcal A_3 $$ $$ \mathcal A_{\mathrm{rec}} $$ $$ \mathcal A_{\mathrm{rec}} := [\mathcal A_i]_{\simeq} $$ --- ## $S_{895}$ $$ \operatorname{End}(\mathcal A_{\mathrm{rec}}) $$ $$ \mathfrak R : \mathcal A_{\mathrm{rec}} \to \mathcal A_{\mathrm{rec}} $$ $$ \mathfrak R^2 \simeq \mathfrak R $$ $$ \operatorname{Fix}(\mathfrak R) $$ --- ## $S_{896}$ $$ \mathfrak R \simeq \operatorname{id}_{\mathcal A_{\mathrm{rec}}} $$ $$ \operatorname{Fix}(\mathfrak R) = \mathcal A_{\mathrm{rec}} $$ $$ \operatorname{RecLayer} $$ --- ## $S_{897}$ $$ \operatorname{End} ( \operatorname{RecLayer} ) $$ $$ \Phi : \operatorname{RecLayer} \to \operatorname{RecLayer} $$ $$ \Phi \simeq \operatorname{id} \quad\lor\quad \Phi \not\simeq \operatorname{id} $$ --- ## $S_{898}$ $$ \Phi \not\simeq \operatorname{id} $$ $$ \operatorname{Fix}(\Phi) $$ $$ \operatorname{Orb}_\Phi(X) $$ $$ \operatorname{Quot}_\Phi $$ $$ \operatorname{Rec}_\Phi $$ --- ## $S_{899}$ $$ \Phi \simeq \operatorname{id} $$ $$ \operatorname{RecLayer} \simeq \operatorname{Fix}(\Phi) $$ $$ \operatorname{End} ( \operatorname{RecLayer} ) = \operatorname{Aut} \cup \operatorname{NonAut} $$ --- ## $S_{900}$ $$ S_{840} \to S_{841} \to \cdots \to S_{900} $$ $$ \operatorname{GeomRec} \to \operatorname{FixModuli} \to \operatorname{KernelEq} \to \operatorname{Glue} \to \operatorname{Adj} \to \operatorname{TransStab} \to \operatorname{PerfRec} \to \operatorname{HopfRec} \to \operatorname{ModuliRec} \to \operatorname{RecLayer} $$ $$ \mathcal R_{\mathrm g} = \operatorname{Spec} \circ Z \circ \operatorname{End} \circ \operatorname{Perf} $$ $$ \mathcal R_{\mathrm g}(X) \simeq X $$ $$ \mathcal G_\infty \simeq \mathbf C_{\mathrm{perf}} \simeq \mathbf{Hopf}_{\mathrm{rec}} \simeq \mathbf{Moduli}_{\mathrm{rec}} $$ $$ \mathfrak R \simeq \operatorname{id}_{\mathcal A_{\mathrm{rec}}} $$ $$ \boxed{ SREG_{015} : S_{841} \to S_{842} \to \cdots \to S_{900} } $$ $$ \boxed{ S_{900}\to S_{901} } $$ --- # SREG-016 $$ S_{900}\longrightarrow S_{901} $$ --- ## $S_{901}$ $$ \mathcal R := \operatorname{RecLayer} $$ $$ \operatorname{End}(\mathcal R) $$ $$ \operatorname{id}_{\mathcal R} $$ $$ \circ: \operatorname{End}(\mathcal R) \times \operatorname{End}(\mathcal R) \to \operatorname{End}(\mathcal R) $$ --- ## $S_{902}$ $$ e\in\operatorname{End}(\mathcal R) $$ $$ e^2=e $$ $$ \operatorname{Idem}(\mathcal R) := \{e:e^2=e\} $$ $$ \operatorname{Fix}(e) $$ $$ \operatorname{im}(e) $$ --- ## $S_{903}$ $$ (X,e) $$ $$ e:X\to X $$ $$ e^2=e $$ $$ f:(X,e)\to(Y,d) $$ $$ f = dfe $$ --- ## $S_{904}$ $$ \operatorname{Kar}(\mathcal R) $$ $$ J: \mathcal R \to \operatorname{Kar}(\mathcal R) $$ $$ J(X) = (X,\operatorname{id}_X) $$ $$ J \ \operatorname{ff} $$ --- ## $S_{905}$ $$ e:X\to X $$ $$ e^2=e $$ $$ X \xrightarrow{r} Y \xrightarrow{i} X $$ $$ ir=e $$ $$ ri=\operatorname{id}_Y $$ $$ Y\simeq\operatorname{im}(e) $$ --- ## $S_{906}$ $$ \operatorname{Kar}(\operatorname{Kar}(\mathcal R)) \simeq \operatorname{Kar}(\mathcal R) $$ $$ K := \operatorname{Kar} $$ $$ K^2\simeq K $$ $$ \operatorname{Fix}(K) $$ --- ## $S_{907}$ $$ \widehat{\mathcal R} := [\mathcal R^{op},\mathbf{Set}] $$ $$ \check{\mathcal R} := [\mathcal R,\mathbf{Set}] $$ $$ y: \mathcal R \hookrightarrow \widehat{\mathcal R} $$ $$ y(X) = \operatorname{Hom}_{\mathcal R}(-,X) $$ --- ## $S_{908}$ $$ y^\vee: \mathcal R \hookrightarrow \check{\mathcal R} $$ $$ y^\vee(X) = \operatorname{Hom}_{\mathcal R}(X,-) $$ $$ \widehat{\mathcal R} $$ $$ \check{\mathcal R} $$ --- ## $S_{909}$ $$ \mathcal O: \widehat{\mathcal R} \to \check{\mathcal R}^{op} $$ $$ \mathcal O(F)(X) := \operatorname{Nat} ( F, y(X) ) $$ $$ \mathcal S: \check{\mathcal R}^{op} \to \widehat{\mathcal R} $$ $$ \mathcal S(G)(X) := \operatorname{Nat} ( G, y^\vee(X) ) $$ --- ## $S_{910}$ $$ \mathcal O\dashv\mathcal S $$ $$ \operatorname{Nat} ( \mathcal O(F), G ) \cong \operatorname{Nat} ( F, \mathcal S(G) ) $$ $$ I := \mathcal S\mathcal O $$ $$ J := \mathcal O\mathcal S $$ --- ## $S_{911}$ $$ \eta: \operatorname{id} \Rightarrow I $$ $$ \varepsilon: J \Rightarrow \operatorname{id} $$ $$ I^2 \to I $$ $$ J \to J^2 $$ $$ \operatorname{Fix}(I) $$ --- ## $S_{912}$ $$ \operatorname{Isb}(\mathcal R) := \operatorname{Fix}(I) $$ $$ F\in\operatorname{Isb}(\mathcal R) \iff F \xrightarrow{\sim} \mathcal S\mathcal O(F) $$ $$ \operatorname{Isb}(\mathcal R) \hookrightarrow \widehat{\mathcal R} $$ --- ## $S_{913}$ $$ y(X) \xrightarrow{\sim} I(y(X)) $$ $$ y(\mathcal R) \subseteq \operatorname{Isb}(\mathcal R) $$ $$ \mathcal R \hookrightarrow \operatorname{Isb}(\mathcal R) $$ --- ## $S_{914}$ $$ \operatorname{Kar}(\mathcal R) \hookrightarrow \operatorname{Isb}(\mathcal R) $$ $$ e^2=e $$ $$ y(e)^2=y(e) $$ $$ \operatorname{im}(y(e)) \in \operatorname{Isb}(\mathcal R) $$ --- ## $S_{915}$ $$ V^\vee := \operatorname{Hom}_k(V,k) $$ $$ D(V) := V^{\vee\vee} $$ $$ \eta_V: V\to V^{\vee\vee} $$ $$ \eta_V(v)(\varphi) = \varphi(v) $$ --- ## $S_{916}$ $$ D(f) = f^{\vee\vee} $$ $$ D: \mathbf{Vect}_k \to \mathbf{Vect}_k $$ $$ \eta: \operatorname{id} \Rightarrow D $$ $$ D(\eta_V) = \eta_{D(V)} $$ --- ## $S_{917}$ $$ \operatorname{Ref}(V) \iff \eta_V \ \operatorname{iso} $$ $$ \mathbf{Vect}_k^{\operatorname{ref}} := \operatorname{Fix}(D) $$ $$ V\in\mathbf{Vect}_k^{\operatorname{fd}} \Rightarrow V\in\mathbf{Vect}_k^{\operatorname{ref}} $$ --- ## $S_{918}$ $$ \dim V<\infty $$ $$ \dim V^\vee = \dim V $$ $$ V \xrightarrow{\sim} V^{\vee\vee} $$ $$ D^2(V) \simeq D(V) $$ --- ## $S_{919}$ $$ \bot \in \mathcal C $$ $$ X^\bot := [X,\bot] $$ $$ X^{\bot\bot} $$ $$ \eta_X: X\to X^{\bot\bot} $$ $$ \operatorname{Ref}_\bot(\mathcal C) $$ --- ## $S_{920}$ $$ \operatorname{Chu}(\mathcal C,\bot) $$ $$ (A,X,\langle-,-\rangle) $$ $$ \langle-,-\rangle : A\otimes X \to \bot $$ $$ (A,X,\langle-,-\rangle)^\ast = (X,A,\langle-,-\rangle^\tau) $$ --- ## $S_{921}$ $$ (f,g): (A,X,\langle-,-\rangle) \to (B,Y,[-,-]) $$ $$ f:A\to B $$ $$ g:Y\to X $$ $$ [a,f(b)] = \langle g(a),b\rangle $$ --- ## $S_{922}$ $$ (-)^\ast : \operatorname{Chu}(\mathcal C,\bot)^{op} \to \operatorname{Chu}(\mathcal C,\bot) $$ $$ X^{\ast\ast} \simeq X $$ $$ (f,g)^\ast = (g,f) $$ --- ## $S_{923}$ $$ A\otimes B $$ $$ A\parr B := (A^\ast\otimes B^\ast)^\ast $$ $$ A\multimap B := (A\otimes B^\ast)^\ast $$ $$ A^\ast \simeq A\multimap\bot $$ --- ## $S_{924}$ $$ \operatorname{Hom}(A\otimes B,C) \cong \operatorname{Hom}(A,B\multimap C) $$ $$ A\otimes- \dashv A\multimap- $$ $$ \mathcal C \ \operatorname{closed} $$ --- ## $S_{925}$ $$ (A^\ast)^\ast \simeq A $$ $$ (A\otimes B)^\ast \simeq B^\ast\parr A^\ast $$ $$ (A\parr B)^\ast \simeq B^\ast\otimes A^\ast $$ $$ \ast : \mathcal C^{op} \to \mathcal C $$ --- ## $S_{926}$ $$ \neg A := A^\ast $$ $$ \neg\neg A \simeq A $$ $$ \neg(A\otimes B) \simeq \neg B\parr\neg A $$ $$ \neg(A\parr B) \simeq \neg B\otimes\neg A $$ --- ## $S_{927}$ $$ A^\ast\otimes A \xrightarrow{\operatorname{ev}_A} \mathbb 1 $$ $$ \mathbb 1 \xrightarrow{\operatorname{coev}_A} A\otimes A^\ast $$ $$ (\operatorname{id}_A\otimes\operatorname{ev}_A) (\operatorname{coev}_A\otimes\operatorname{id}_A) = \operatorname{id}_A $$ --- ## $S_{928}$ $$ (\operatorname{ev}_A\otimes\operatorname{id}_{A^\ast}) (\operatorname{id}_{A^\ast}\otimes\operatorname{coev}_A) = \operatorname{id}_{A^\ast} $$ $$ A \ \operatorname{dual} $$ $$ \mathcal C \ \operatorname{rigid} $$ --- ## $S_{929}$ $$ f:A\to A $$ $$ \operatorname{tr}(f) := \mathbb 1 \xrightarrow{\operatorname{coev}_A} A\otimes A^\ast \xrightarrow{f\otimes\operatorname{id}} A\otimes A^\ast \xrightarrow{\operatorname{ev}_A} \mathbb 1 $$ --- ## $S_{930}$ $$ f:A\to B $$ $$ g:B\to A $$ $$ \operatorname{tr}(gf) = \operatorname{tr}(fg) $$ $$ \operatorname{tr}(\operatorname{id}_A) = \dim(A) $$ --- ## $S_{931}$ $$ \mathcal N(A,B) $$ $$ f\in\mathcal N(A,B) $$ $$ f = A \xrightarrow{u} X \xrightarrow{v} B $$ $$ X \ \operatorname{dual} $$ --- ## $S_{932}$ $$ f\in\mathcal N(A,A) $$ $$ \operatorname{tr}(f) $$ $$ \operatorname{tr}(hfk) = \operatorname{tr}(khf) $$ $$ \mathcal N \ \triangleleft\ \operatorname{Hom} $$ --- ## $S_{933}$ $$ K:A^\ast\otimes B\to\mathbb 1 $$ $$ \widehat K:A\to B $$ $$ \operatorname{Ker}(A,B) $$ $$ \operatorname{Ker}(A,B) \cong \operatorname{Hom}(A,B) $$ --- ## $S_{934}$ $$ K_{f\circ g} = K_f\star K_g $$ $$ K_{\operatorname{id}_A} = \Delta_A $$ $$ \operatorname{tr}(f) = \operatorname{Contr}(K_f) $$ --- ## $S_{935}$ $$ \mathcal C \to \operatorname{Kar}(\mathcal C) \to \operatorname{Isb}(\mathcal C) \to \operatorname{Ref}_\bot(\mathcal C) $$ $$ K $$ $$ I $$ $$ D $$ --- ## $S_{936}$ $$ K^2\simeq K $$ $$ I^2\simeq I $$ $$ D^2\simeq D $$ $$ KI\simeq IK $$ $$ ID\simeq DI $$ $$ KD\simeq DK $$ --- ## $S_{937}$ $$ P := KID $$ $$ P^2 \simeq KIDKID \simeq K^2I^2D^2 \simeq P $$ $$ \operatorname{Fix}(P) $$ --- ## $S_{938}$ $$ \operatorname{Fix}(P) = \operatorname{Fix}(K) \cap \operatorname{Fix}(I) \cap \operatorname{Fix}(D) $$ $$ \mathcal C^\ast := \operatorname{Fix}(P) $$ --- ## $S_{939}$ $$ X\in\mathcal C^\ast $$ $$ K(X)\simeq X $$ $$ I(X)\simeq X $$ $$ D(X)\simeq X $$ $$ P(X)\simeq X $$ --- ## $S_{940}$ $$ \mathcal O_X := \operatorname{Hom}(X,\bot) $$ $$ \mathcal O: \mathcal C^\ast \to \mathbf{Set}^{op} $$ $$ x\neq y \Rightarrow \exists\varphi\in\mathcal O_X: \varphi(x)\neq\varphi(y) $$ --- ## $S_{941}$ $$ \operatorname{ev}_X: X \to \mathcal O_X^\vee $$ $$ x \mapsto [\varphi\mapsto\varphi(x)] $$ $$ \operatorname{ev}_X \ \operatorname{mono} $$ --- ## $S_{942}$ $$ X \in \operatorname{Fix}(D) $$ $$ \operatorname{ev}_X \ \operatorname{iso} $$ $$ X \cong \mathcal O_X^\vee $$ $$ X \cong \operatorname{Hom} ( \operatorname{Hom}(X,\bot), \bot ) $$ --- ## $S_{943}$ $$ x\sim_{\mathcal O}y \iff \forall\varphi:X\to\bot: \varphi(x)=\varphi(y) $$ $$ X/{\sim_{\mathcal O}} $$ $$ \operatorname{ev}_X $$ $$ \ker(\operatorname{ev}_X) = \sim_{\mathcal O} $$ --- ## $S_{944}$ $$ \operatorname{ev}_X \ \operatorname{mono} $$ $$ \Rightarrow \sim_{\mathcal O} = \Delta_X $$ $$ X/{\sim_{\mathcal O}} \cong X $$ --- ## $S_{945}$ $$ \Phi: \mathcal C^{op}\times\mathcal C \to \mathbf{Set} $$ $$ \Phi(A,B) $$ $$ \Psi: \mathcal D^{op}\times\mathcal E \to \mathbf{Set} $$ $$ (\Psi\star\Phi)(A,E) := \int^{B} \Phi(A,B)\times\Psi(B,E) $$ --- ## $S_{946}$ $$ \mathbf{Prof} $$ $$ \operatorname{Ob}(\mathbf{Prof}) = \{\mathcal C\} $$ $$ \operatorname{Hom}_{\mathbf{Prof}}(\mathcal C,\mathcal D) = [\mathcal C^{op}\times\mathcal D,\mathbf{Set}] $$ $$ \circ = \star $$ --- ## $S_{947}$ $$ \mathbb I_\mathcal C(A,B) := \operatorname{Hom}_\mathcal C(A,B) $$ $$ \Phi\star\mathbb I_\mathcal C \simeq \Phi $$ $$ \mathbb I_\mathcal D\star\Phi \simeq \Phi $$ --- ## $S_{948}$ $$ \Phi:\mathcal C\nrightarrow\mathcal D $$ $$ \Psi:\mathcal D\nrightarrow\mathcal C $$ $$ \Phi\dashv\Psi $$ $$ \mathbb I_\mathcal C \to \Psi\star\Phi $$ $$ \Phi\star\Psi \to \mathbb I_\mathcal D $$ --- ## $S_{949}$ $$ X \in \operatorname{Cauchy}(\mathcal C) $$ $$ X: 1\nrightarrow\mathcal C $$ $$ \exists X^\ast: \mathcal C\nrightarrow1 $$ $$ X\dashv X^\ast $$ --- ## $S_{950}$ $$ \operatorname{Cauchy}(\mathcal C) $$ $$ \mathcal C \hookrightarrow \operatorname{Cauchy}(\mathcal C) $$ $$ \operatorname{Cauchy} ( \operatorname{Cauchy}(\mathcal C) ) \simeq \operatorname{Cauchy}(\mathcal C) $$ --- ## $S_{951}$ $$ \operatorname{Cauchy}(\mathcal C) \simeq \operatorname{Kar}(\mathcal C) $$ $$ \operatorname{Fix}(\operatorname{Cauchy}) = \operatorname{Fix}(\operatorname{Kar}) $$ $$ K \simeq \operatorname{Cauchy} $$ --- ## $S_{952}$ $$ \mathcal C^\ast = \operatorname{Fix}(KID) $$ $$ \operatorname{Cauchy}(\mathcal C^\ast) \simeq \mathcal C^\ast $$ $$ \operatorname{Isb}(\mathcal C^\ast) \simeq \mathcal C^\ast $$ $$ D(\mathcal C^\ast) \simeq \mathcal C^\ast $$ --- ## $S_{953}$ $$ \mathfrak R: \mathcal C \to \mathcal C $$ $$ \mathfrak R := KID $$ $$ \mathfrak R^2 \simeq \mathfrak R $$ $$ \operatorname{im}(\mathfrak R) = \mathcal C^\ast $$ --- ## $S_{954}$ $$ \mathfrak R_0:=\mathfrak R $$ $$ \mathfrak R_{n+1} := \operatorname{End}(\mathfrak R_n) $$ $$ \mathfrak R_n \in \operatorname{End}^n(\mathcal C) $$ $$ \operatorname{Fix}(\mathfrak R_n) $$ --- ## $S_{955}$ $$ \mathcal C_0:=\mathcal C $$ $$ \mathcal C_{n+1} := \operatorname{Fix}(\mathfrak R_n) $$ $$ \mathcal C_{n+1} \subseteq \mathcal C_n $$ $$ \mathcal C_\omega := \bigcap_{n<\omega}\mathcal C_n $$ --- ## $S_{956}$ $$ \mathcal C_{\lambda} := \bigcap_{\beta<\lambda} \mathcal C_\beta $$ $$ \mathfrak R_\lambda := \varprojlim_{\beta<\lambda}\mathfrak R_\beta $$ $$ \mathcal C_{\lambda+1} = \operatorname{Fix}(\mathfrak R_\lambda) $$ --- ## $S_{957}$ $$ \exists\Gamma: \mathcal C_{\Gamma+1} = \mathcal C_\Gamma $$ $$ \mathfrak R_\Gamma \simeq \operatorname{id}_{\mathcal C_\Gamma} $$ $$ \mathcal C_\infty := \mathcal C_\Gamma $$ --- ## $S_{958}$ $$ \mathcal C_\infty = \operatorname{Fix}(\mathfrak R_\Gamma) $$ $$ K|_{\mathcal C_\infty} \simeq \operatorname{id} $$ $$ I|_{\mathcal C_\infty} \simeq \operatorname{id} $$ $$ D|_{\mathcal C_\infty} \simeq \operatorname{id} $$ --- ## $S_{959}$ $$ \mathcal C_\infty \hookrightarrow \operatorname{Prof} $$ $$ X\dashv X^\ast $$ $$ X^{\ast\ast}\simeq X $$ $$ \operatorname{Cauchy}(X)\simeq X $$ $$ \operatorname{Isb}(X)\simeq X $$ --- ## $S_{960}$ $$ S_{900} \to S_{901} \to \cdots \to S_{960} $$ $$ \operatorname{Idem} \to \operatorname{Kar} \to \operatorname{Isbell} \to \operatorname{Dual} \to \operatorname{Chu} \to \operatorname{Trace} \to \operatorname{Prof} \to \operatorname{Cauchy} \to \operatorname{TransFix} \to \mathcal C_\infty $$ $$ K^2\simeq K $$ $$ I^2\simeq I $$ $$ D^2\simeq D $$ $$ \mathfrak R = KID $$ $$ \mathfrak R^2 \simeq \mathfrak R $$ $$ \mathfrak R_\Gamma \simeq \operatorname{id}_{\mathcal C_\infty} $$ $$ \boxed{ SREG_{016} : S_{901} \to S_{902} \to \cdots \to S_{960} } $$ $$ \boxed{ S_{960}\to S_{961} } $$ --- # SREG-017 $$ S_{960}\longrightarrow S_{961} $$ --- ## $S_{961}$ $$ \mathcal C_\infty $$ $$ K|_{\mathcal C_\infty} \simeq I|_{\mathcal C_\infty} \simeq D|_{\mathcal C_\infty} \simeq \operatorname{id}_{\mathcal C_\infty} $$ $$ X\in\mathcal C_\infty $$ $$ X^{\ast\ast} \simeq X $$ --- ## $S_{962}$ $$ (-)^\ast: \mathcal C_\infty^{op} \to \mathcal C_\infty $$ $$ (-)^{\ast\ast} \simeq \operatorname{id} $$ $$ (f^\ast)^\ast = f $$ $$ \operatorname{id}_X^\ast = \operatorname{id}_{X^\ast} $$ --- ## $S_{963}$ $$ \operatorname{coev}_X: \mathbb 1 \to X\otimes X^\ast $$ $$ \operatorname{ev}_X: X^\ast\otimes X \to \mathbb 1 $$ $$ (\operatorname{id}_X\otimes\operatorname{ev}_X) (\operatorname{coev}_X\otimes\operatorname{id}_X) = \operatorname{id}_X $$ $$ (\operatorname{ev}_X\otimes\operatorname{id}_{X^\ast}) (\operatorname{id}_{X^\ast}\otimes\operatorname{coev}_X) = \operatorname{id}_{X^\ast} $$ --- ## $S_{964}$ $$ f:X\to X $$ $$ \operatorname{tr}_X(f) : \mathbb 1\to\mathbb 1 $$ $$ \operatorname{tr}_X(f) = \operatorname{ev}_X \circ (\operatorname{id}_{X^\ast}\otimes f) \circ \operatorname{coev}_X $$ $$ \operatorname{tr}(fg) = \operatorname{tr}(gf) $$ --- ## $S_{965}$ $$ \dim(X) := \operatorname{tr}_X(\operatorname{id}_X) $$ $$ \dim(X\otimes Y) = \dim(X)\dim(Y) $$ $$ \dim(X\oplus Y) = \dim(X)+\dim(Y) $$ $$ \dim(X^\ast)=\dim(X) $$ --- ## $S_{966}$ $$ e:X\to X $$ $$ e^2=e $$ $$ X \simeq \operatorname{im}(e) \oplus \operatorname{im}(\operatorname{id}-e) $$ $$ \dim(X) = \dim(\operatorname{im}(e)) + \dim(\operatorname{im}(\operatorname{id}-e)) $$ --- ## $S_{967}$ $$ \operatorname{Sub}(X) $$ $$ A\hookrightarrow X $$ $$ B\hookrightarrow X $$ $$ A\wedge B = A\times_XB $$ $$ A\vee B $$ $$ \operatorname{Sub}(X) \in\mathbf{Lat} $$ --- ## $S_{968}$ $$ \operatorname{Quot}_{\operatorname{fin}}(X) := \{q:X\twoheadrightarrow Q:\ |Q|<\infty\} $$ $$ q_1\preceq q_2 \iff \exists r: q_1=r\circ q_2 $$ $$ \operatorname{Quot}_{\operatorname{fin}}(X) \in\mathbf{Cofilt} $$ --- ## $S_{969}$ $$ \widehat X := \varprojlim_{q\in\operatorname{Quot}_{\operatorname{fin}}(X)} Q $$ $$ \eta_X: X\to\widehat X $$ $$ \pi_q: \widehat X\to Q $$ $$ \pi_q\circ\eta_X = q $$ --- ## $S_{970}$ $$ Y \simeq \varprojlim_iY_i $$ $$ |Y_i|<\infty $$ $$ f:X\to Y $$ $$ \exists!\widehat f: \widehat X\to Y $$ $$ \widehat f\circ\eta_X=f $$ --- ## $S_{971}$ $$ F:X\to X $$ $$ qF $$ $$ \widehat F: \widehat X\to\widehat X $$ $$ \eta_XF = \widehat F\eta_X $$ $$ \widehat{F\circ G} = \widehat F\circ\widehat G $$ --- ## $S_{972}$ $$ \operatorname{Aut}_{\operatorname{fin}}(X) $$ $$ \operatorname{Aut}(\widehat X) $$ $$ \operatorname{Aut}(\widehat X) \cong \varprojlim_q \operatorname{Aut}(Q) $$ $$ \operatorname{Aut}(\widehat X) \in \mathbf{ProFinGrp} $$ --- ## $S_{973}$ $$ \operatorname{Cov}_{\operatorname{fin}}(X) $$ $$ p:Y\to X $$ $$ |p^{-1}(x)|<\infty $$ $$ Y\times_XY $$ $$ \operatorname{Deck}(p) $$ --- ## $S_{974}$ $$ x:1\to X $$ $$ \omega_x: \operatorname{Cov}_{\operatorname{fin}}(X) \to \mathbf{FinSet} $$ $$ \omega_x(p) := 1\times_XY $$ $$ \omega_x(p) = p^{-1}(x) $$ --- ## $S_{975}$ $$ \operatorname{Aut}(\omega_x) $$ $$ \pi_1^{\operatorname{fin}}(X,x) := \operatorname{Aut}(\omega_x) $$ $$ \pi_1^{\operatorname{fin}}(X,x) \in \mathbf{ProFinGrp} $$ --- ## $S_{976}$ $$ G := \pi_1^{\operatorname{fin}}(X,x) $$ $$ \mathbf{FinSet}^G $$ $$ \operatorname{Cov}_{\operatorname{fin}}(X) \to \mathbf{FinSet}^G $$ $$ p \mapsto \omega_x(p) $$ --- ## $S_{977}$ $$ \operatorname{Cov}_{\operatorname{fin}}(X) \simeq \mathbf{FinSet}^G $$ $$ G = \operatorname{Aut}(\omega_x) $$ $$ \omega_x \simeq U_G $$ $$ U_G: \mathbf{FinSet}^G \to \mathbf{FinSet} $$ --- ## $S_{978}$ $$ \operatorname{Aut}(U_G) \cong G $$ $$ \operatorname{Aut} ( \operatorname{Cov}_{\operatorname{fin}}(X) \xrightarrow{\omega_x} \mathbf{FinSet} ) \cong \pi_1^{\operatorname{fin}}(X,x) $$ --- ## $S_{979}$ $$ \pi_0(X) $$ $$ X = \bigsqcup_{c\in\pi_0(X)}X_c $$ $$ x_c\in X_c $$ $$ \pi_1^{\operatorname{fin}}(X_c,x_c) $$ --- ## $S_{980}$ $$ \Pi_1^{\operatorname{fin}}(X) $$ $$ \operatorname{Ob} ( \Pi_1^{\operatorname{fin}}(X) ) = \pi_0(X) $$ $$ \operatorname{Aut}(c) = \pi_1^{\operatorname{fin}}(X_c,x_c) $$ $$ \Pi_1^{\operatorname{fin}}(X) \in \mathbf{ProFinGpd} $$ --- ## $S_{981}$ $$ \operatorname{Cov}_{\operatorname{fin}}(X) \simeq \operatorname{Fun} ( \Pi_1^{\operatorname{fin}}(X), \mathbf{FinSet} ) $$ $$ X \mapsto \Pi_1^{\operatorname{fin}}(X) $$ $$ \operatorname{Galois}_1(X) $$ --- ## $S_{982}$ $$ \pi_n^{\operatorname{fin}}(X,x) $$ $$ n\ge1 $$ $$ \Pi_{\le n}^{\operatorname{fin}}(X) $$ $$ \tau_{\le n} \Pi_{\le n+1}^{\operatorname{fin}}(X) \simeq \Pi_{\le n}^{\operatorname{fin}}(X) $$ --- ## $S_{983}$ $$ \Pi_\infty^{\operatorname{fin}}(X) := \varprojlim_n \Pi_{\le n}^{\operatorname{fin}}(X) $$ $$ \pi_n ( \Pi_\infty^{\operatorname{fin}}(X) ) = \pi_n^{\operatorname{fin}}(X) $$ $$ X \to \Pi_\infty^{\operatorname{fin}}(X) $$ --- ## $S_{984}$ $$ \widehat{\Pi_\infty(X)} $$ $$ \Pi_\infty^{\operatorname{fin}}(X) $$ $$ \widehat{\Pi_\infty(X)} \simeq \Pi_\infty^{\operatorname{fin}}(X) $$ $$ \operatorname{Shape}_{\operatorname{fin}}(X) $$ --- ## $S_{985}$ $$ \operatorname{Shape}(X) \in \operatorname{Pro}(\mathcal S) $$ $$ \operatorname{Shape}_{\operatorname{fin}}(X) \in \operatorname{Pro}(\mathcal S_{\operatorname{fin}}) $$ $$ \operatorname{Shape}(X) \to \operatorname{Shape}_{\operatorname{fin}}(X) $$ --- ## $S_{986}$ $$ H^n(X;A) $$ $$ A \in \mathbf{FinAb} $$ $$ H^n(X;A) \cong H^n ( \operatorname{Shape}_{\operatorname{fin}}(X); A ) $$ $$ \operatorname{Coh}_{\operatorname{fin}}(X) $$ --- ## $S_{987}$ $$ \mathcal I_{\operatorname{fin}}(X) := \left( \pi_0(X), \pi_1^{\operatorname{fin}}(X), \pi_2^{\operatorname{fin}}(X), \ldots, H^\ast(X;A)_{A\in\mathbf{FinAb}} \right) $$ $$ X\mapsto \mathcal I_{\operatorname{fin}}(X) $$ --- ## $S_{988}$ $$ X\sim_{\operatorname{fin}}Y \iff \mathcal I_{\operatorname{fin}}(X) \cong \mathcal I_{\operatorname{fin}}(Y) $$ $$ \mathcal C_\infty/{\sim_{\operatorname{fin}}} $$ $$ \Pi_{\operatorname{fin}} $$ --- ## $S_{989}$ $$ \Pi_{\operatorname{fin}}^2 = \Pi_{\operatorname{fin}} $$ $$ \operatorname{im}(\Pi_{\operatorname{fin}}) = \mathcal C_{\operatorname{fin}} $$ $$ \operatorname{Fix}(\Pi_{\operatorname{fin}}) = \mathcal C_{\operatorname{fin}} $$ --- ## $S_{990}$ $$ \mathcal C_{\operatorname{fin}} \hookrightarrow \operatorname{Pro}(\mathbf{Fin}) $$ $$ X \mapsto \widehat X $$ $$ \widehat{\widehat X} \cong \widehat X $$ $$ \operatorname{Comp}_{\operatorname{fin}} $$ --- ## $S_{991}$ $$ \operatorname{Map}(X,Y) $$ $$ \widehat{\operatorname{Map}}(X,Y) $$ $$ \operatorname{Map}_{\operatorname{ProFin}}(\widehat X,\widehat Y) $$ $$ \widehat{\operatorname{Map}(X,Y)} \to \operatorname{Map}(\widehat X,\widehat Y) $$ --- ## $S_{992}$ $$ X\in\mathcal C_{\operatorname{fin}} $$ $$ Y\in\mathcal C_{\operatorname{fin}} $$ $$ \widehat{\operatorname{Map}(X,Y)} \simeq \operatorname{Map}(\widehat X,\widehat Y) $$ $$ \operatorname{Enr}_{\operatorname{ProFin}} $$ --- ## $S_{993}$ $$ \mathcal C_{\operatorname{fin}} $$ $$ \operatorname{End}(X) \in \mathbf{ProFinMon} $$ $$ \operatorname{Aut}(X) \in \mathbf{ProFinGrp} $$ $$ \operatorname{Idem}(X) \subseteq \operatorname{End}(X) $$ --- ## $S_{994}$ $$ e_n\in\operatorname{Idem}(X) $$ $$ e_ne_m=e_me_n $$ $$ e_\infty := \varprojlim_n e_n $$ $$ e_\infty^2=e_\infty $$ --- ## $S_{995}$ $$ X_{e_\infty} := \operatorname{Fix}(e_\infty) $$ $$ X_{e_\infty} = \bigcap_n \operatorname{Fix}(e_n) $$ $$ X_{e_\infty} \hookrightarrow X $$ --- ## $S_{996}$ $$ \mathcal I := \operatorname{Idem}(\operatorname{End}(X)) $$ $$ e_X := \bigwedge_{e\in\mathcal I}e $$ $$ e_X^2=e_X $$ $$ X^\circ := \operatorname{Fix}(e_X) $$ --- ## $S_{997}$ $$ \forall e\in\mathcal I: e|_{X^\circ} = \operatorname{id}_{X^\circ} $$ $$ \operatorname{Idem}(X^\circ) = \{\operatorname{id}_{X^\circ}\} $$ $$ X^\circ \ \operatorname{indecomp} $$ --- ## $S_{998}$ $$ X \simeq \prod_{i\in I}X_i^\circ $$ $$ \operatorname{Idem}(X) \cong 2^I $$ $$ \operatorname{End}(X) \to \prod_{i,j} \operatorname{Hom}(X_i^\circ,X_j^\circ) $$ --- ## $S_{999}$ $$ \operatorname{KrullSchmidt}(X) $$ $$ X \simeq \bigoplus_{i=1}^nX_i^\circ $$ $$ X_i^\circ \not\simeq X_j^\circ $$ $$ \operatorname{End}(X_i^\circ) \ \operatorname{local} $$ --- ## $S_{1000}$ $$ J_X := \operatorname{Jac} ( \operatorname{End}(X) ) $$ $$ \operatorname{End}(X)/J_X $$ $$ \cong \prod_iM_{n_i}(D_i) $$ $$ D_i \in \mathbf{DivRing} $$ --- ## $S_{1001}$ $$ \operatorname{Spec}_{\operatorname{rep}}(X) := \{(D_i,n_i)\}_i $$ $$ X \mapsto \operatorname{Spec}_{\operatorname{rep}}(X) $$ $$ \operatorname{End}(X) \mapsto \operatorname{End}(X)/J_X $$ --- ## $S_{1002}$ $$ \mathcal S_X := \operatorname{Spec}_{\operatorname{rep}}(X) $$ $$ \widehat{\mathcal S}_X $$ $$ \operatorname{Rec}_{\operatorname{rep}}(\mathcal S_X) $$ $$ \operatorname{Rec}_{\operatorname{rep}}(\mathcal S_X) \to X $$ --- ## $S_{1003}$ $$ \operatorname{Rec}_{\operatorname{rep}}(\mathcal S_X) \simeq X $$ $$ \Rightarrow $$ $$ X \in \operatorname{Fix} ( \operatorname{Rec}_{\operatorname{rep}} \circ \operatorname{Spec}_{\operatorname{rep}} ) $$ --- ## $S_{1004}$ $$ R_{\operatorname{rep}} := \operatorname{Rec}_{\operatorname{rep}} \circ \operatorname{Spec}_{\operatorname{rep}} $$ $$ R_{\operatorname{rep}}: \mathcal C_{\operatorname{fin}} \to \mathcal C_{\operatorname{fin}} $$ $$ R_{\operatorname{rep}}^2 \simeq R_{\operatorname{rep}} $$ --- ## $S_{1005}$ $$ \mathcal C_{\operatorname{rep}} := \operatorname{Fix}(R_{\operatorname{rep}}) $$ $$ \mathcal C_{\operatorname{rep}} \subseteq \mathcal C_{\operatorname{fin}} \subseteq \mathcal C_\infty $$ $$ R_{\operatorname{rep}}|_{\mathcal C_{\operatorname{rep}}} \simeq \operatorname{id} $$ --- ## $S_{1006}$ $$ X\in\mathcal C_{\operatorname{rep}} $$ $$ \operatorname{Aut}(X) $$ $$ \operatorname{Spec}_{\operatorname{rep}}(X) $$ $$ \operatorname{Aut} ( \operatorname{Spec}_{\operatorname{rep}}(X) ) $$ $$ \operatorname{Aut}(X) \to \operatorname{Aut} ( \operatorname{Spec}_{\operatorname{rep}}(X) ) $$ --- ## $S_{1007}$ $$ \ker \left[ \operatorname{Aut}(X) \to \operatorname{Aut} ( \operatorname{Spec}_{\operatorname{rep}}(X) ) \right] $$ $$ \operatorname{Inn}(X) $$ $$ \operatorname{Out}(X) $$ --- ## $S_{1008}$ $$ 1 \to \operatorname{Inn}(X) \to \operatorname{Aut}(X) \to \operatorname{Out}(X) \to 1 $$ $$ \operatorname{Out}(X) \hookrightarrow \operatorname{Aut} ( \operatorname{Spec}_{\operatorname{rep}}(X) ) $$ --- ## $S_{1009}$ $$ \operatorname{Def}(X) $$ $$ \operatorname{Def} ( \operatorname{Spec}_{\operatorname{rep}}(X) ) $$ $$ T_X $$ $$ T_{\mathcal S_X} $$ $$ T_X \to T_{\mathcal S_X} $$ --- ## $S_{1010}$ $$ L_X $$ $$ L_{\mathcal S_X} $$ $$ \operatorname{Ext}^1(L_X,X) $$ $$ \operatorname{Ext}^2(L_X,X) $$ $$ \operatorname{ObsDef}(X) $$ --- ## $S_{1011}$ $$ \mathcal J(X) := \left( \operatorname{Shape}_{\operatorname{fin}}(X), \operatorname{Spec}_{\operatorname{rep}}(X), \operatorname{Out}(X), \operatorname{ObsDef}(X) \right) $$ $$ X\mapsto\mathcal J(X) $$ --- ## $S_{1012}$ $$ X\sim_{\mathcal J}Y \iff \mathcal J(X)\cong\mathcal J(Y) $$ $$ \sim_{\mathcal J} $$ $$ \mathcal C_\infty/{\sim_{\mathcal J}} $$ $$ \Pi_{\mathcal J} $$ --- ## $S_{1013}$ $$ \Pi_{\mathcal J}^2 = \Pi_{\mathcal J} $$ $$ \mathcal C_{\mathcal J} := \operatorname{im}(\Pi_{\mathcal J}) $$ $$ \operatorname{Fix}(\Pi_{\mathcal J}) = \mathcal C_{\mathcal J} $$ --- ## $S_{1014}$ $$ \mathcal J_0 \subseteq \mathcal J_1 \subseteq \cdots $$ $$ \sim_{n+1} \subseteq \sim_n $$ $$ \mathcal J_\omega := \bigcup_{n<\omega}\mathcal J_n $$ --- ## $S_{1015}$ $$ \mathcal J_\alpha $$ $$ \sim_{\alpha+1} \subseteq \sim_\alpha $$ $$ \sim_\lambda := \bigcap_{\beta<\lambda}\sim_\beta $$ $$ \exists\Theta: \sim_{\Theta+1} = \sim_\Theta $$ --- ## $S_{1016}$ $$ \mathcal C_\Theta := \mathcal C_\infty/{\sim_\Theta} $$ $$ \Pi_\Theta: \mathcal C_\infty \to \mathcal C_\Theta $$ $$ \Pi_\Theta^2 = \Pi_\Theta $$ --- ## $S_{1017}$ $$ \mathcal R_\Theta: \mathcal C_\Theta \to \mathcal C_\infty $$ $$ \Pi_\Theta\mathcal R_\Theta = \operatorname{id}_{\mathcal C_\Theta} $$ $$ E_\Theta := \mathcal R_\Theta\Pi_\Theta $$ $$ E_\Theta^2=E_\Theta $$ --- ## $S_{1018}$ $$ \operatorname{im}(E_\Theta) = \operatorname{Fix}(E_\Theta) $$ $$ \mathcal C_{\operatorname{rec}} := \operatorname{Fix}(E_\Theta) $$ $$ \mathcal C_{\operatorname{rec}} \subseteq \mathcal C_\infty $$ --- ## $S_{1019}$ $$ X\in\mathcal C_{\operatorname{rec}} $$ $$ E_\Theta(X)=X $$ $$ \mathcal J_\Theta(X) $$ $$ \mathcal R_\Theta(\mathcal J_\Theta(X)) = X $$ $$ \operatorname{SelfRec}(X) $$ --- ## $S_{1020}$ $$ S_{960} \to S_{961} \to \cdots \to S_{1020} $$ $$ \operatorname{DualFix} \to \operatorname{ProFin} \to \operatorname{Galois} \to \operatorname{ProShape} \to \operatorname{IdemCore} \to \operatorname{RepSpec} \to \operatorname{Outer} \to \operatorname{Def} \to \operatorname{InvariantTower} \to \operatorname{SelfRec} $$ $$ \Pi_1^{\operatorname{fin}}(X) $$ $$ \operatorname{Shape}_{\operatorname{fin}}(X) $$ $$ R_{\operatorname{rep}}^2 \simeq R_{\operatorname{rep}} $$ $$ E_\Theta^2 = E_\Theta $$ $$ \mathcal C_{\operatorname{rec}} = \operatorname{Fix}(E_\Theta) $$ $$ \boxed{ SREG_{017} : S_{961} \to S_{962} \to \cdots \to S_{1020} } $$ $$ \boxed{ S_{1020}\to S_{1021} } $$ --- # SREG-018 $$ S_{1020}\longrightarrow S_{1021} $$ --- ## $S_{1021}$ $$ \mathcal C_{\operatorname{rec}} = \operatorname{Fix}(E_\Theta) $$ $$ E_\Theta^2 = E_\Theta $$ $$ X\in\mathcal C_{\operatorname{rec}} $$ $$ E_\Theta(X)=X $$ --- ## $S_{1022}$ $$ \mathcal V_i \qquad (i\in I) $$ $$ R_i: \mathcal C_{\operatorname{rec}} \to \mathcal V_i $$ $$ R_i(X) $$ $$ R_i(f) $$ --- ## $S_{1023}$ $$ R := \prod_{i\in I}R_i $$ $$ R: \mathcal C_{\operatorname{rec}} \to \prod_{i\in I}\mathcal V_i $$ $$ R(X) = (R_i(X))_{i\in I} $$ --- ## $S_{1024}$ $$ X\sim_RY \iff R(X)\cong R(Y) $$ $$ [X]_R $$ $$ \mathcal C_R := \mathcal C_{\operatorname{rec}}/{\sim_R} $$ $$ \Pi_R: \mathcal C_{\operatorname{rec}} \to \mathcal C_R $$ --- ## $S_{1025}$ $$ \Pi_R(X)=\Pi_R(Y) \iff X\sim_RY $$ $$ \Pi_R^2 = \Pi_R $$ $$ \operatorname{im}(\Pi_R) = \mathcal C_R $$ --- ## $S_{1026}$ $$ \mathcal S_R: \mathcal C_R \to \mathcal C_{\operatorname{rec}} $$ $$ \Pi_R\mathcal S_R = \operatorname{id}_{\mathcal C_R} $$ $$ \mathcal E_R := \mathcal S_R\Pi_R $$ $$ \mathcal E_R^2 = \mathcal E_R $$ --- ## $S_{1027}$ $$ \operatorname{Fix}(\mathcal E_R) = \operatorname{im}(\mathcal S_R) $$ $$ \mathcal C_{\operatorname{rec}}^{R} := \operatorname{Fix}(\mathcal E_R) $$ $$ X\in\mathcal C_{\operatorname{rec}}^{R} \Rightarrow \mathcal E_R(X)=X $$ --- ## $S_{1028}$ $$ (\mathcal C_{\operatorname{rec}},\otimes,\mathbb 1) $$ $$ R_i(X\otimes Y) \cong R_i(X)\otimes R_i(Y) $$ $$ R_i(\mathbb 1) \cong \mathbb 1_i $$ $$ R_i \in \operatorname{Fun}^{\otimes} $$ --- ## $S_{1029}$ $$ X^\vee $$ $$ \operatorname{coev}_X: \mathbb 1 \to X\otimes X^\vee $$ $$ \operatorname{ev}_X: X^\vee\otimes X \to \mathbb 1 $$ $$ X^{\vee\vee} \cong X $$ --- ## $S_{1030}$ $$ f:X\to X $$ $$ \operatorname{tr}(f) : \mathbb 1\to\mathbb 1 $$ $$ R_i(\operatorname{tr}(f)) = \operatorname{tr}(R_i(f)) $$ $$ \operatorname{tr}(fg) = \operatorname{tr}(gf) $$ --- ## $S_{1031}$ $$ Z_f(t) := \exp \left( \sum_{n\ge1} \frac{\operatorname{tr}(f^n)}{n}t^n \right) $$ $$ R_i(Z_f) = Z_{R_i(f)} $$ $$ Z_{f\oplus g} = Z_fZ_g $$ --- ## $S_{1032}$ $$ K_0(\mathcal C_{\operatorname{rec}}) $$ $$ [X] $$ $$ [X\oplus Y] = [X]+[Y] $$ $$ [X\otimes Y] = [X][Y] $$ $$ K_0(R_i) $$ --- ## $S_{1033}$ $$ K(\mathcal C_{\operatorname{rec}}) $$ $$ \pi_nK(\mathcal C_{\operatorname{rec}}) = K_n(\mathcal C_{\operatorname{rec}}) $$ $$ K: \mathbf{Cat}_{\operatorname{st}} \to \mathbf{Sp} $$ $$ K(\mathcal C\oplus\mathcal D) \simeq K(\mathcal C)\oplus K(\mathcal D) $$ --- ## $S_{1034}$ $$ \operatorname{THH}: \mathbf{Cat}_{\operatorname{st}} \to \mathbf{Sp} $$ $$ \operatorname{TC}: \mathbf{Cat}_{\operatorname{st}} \to \mathbf{Sp} $$ $$ K \to \operatorname{TC} $$ $$ \operatorname{THH}(\mathcal C) $$ --- ## $S_{1035}$ $$ \mathcal U_{\operatorname{loc}} : \mathbf{Cat}_{\operatorname{st}} \to \mathbf{Mot}_{\operatorname{loc}} $$ $$ \mathcal C'\to\mathcal C\to\mathcal C'' $$ $$ \mathcal U_{\operatorname{loc}}(\mathcal C') \to \mathcal U_{\operatorname{loc}}(\mathcal C) \to \mathcal U_{\operatorname{loc}}(\mathcal C'') $$ --- ## $S_{1036}$ $$ \operatorname{Mot} := \mathbf{Mot}_{\operatorname{loc}} $$ $$ M_X := \mathcal U_{\operatorname{loc}} ( \operatorname{Perf}(X) ) $$ $$ X\mapsto M_X $$ $$ \mathcal M: \mathcal C_{\operatorname{rec}} \to \operatorname{Mot} $$ --- ## $S_{1037}$ $$ X\sim_M Y \iff M_X\cong M_Y $$ $$ \mathcal C_M := \mathcal C_{\operatorname{rec}}/{\sim_M} $$ $$ \Pi_M: \mathcal C_{\operatorname{rec}} \to \mathcal C_M $$ --- ## $S_{1038}$ $$ \operatorname{Hom}_{\operatorname{Mot}}(M_X,M_Y) $$ $$ \pi_n \operatorname{Map}_{\operatorname{Mot}}(M_X,M_Y) $$ $$ \operatorname{End}_{\operatorname{Mot}}(M_X) $$ $$ \operatorname{Aut}_{\operatorname{Mot}}(M_X) $$ --- ## $S_{1039}$ $$ R_i = \widetilde R_i \circ \mathcal M $$ $$ \widetilde R_i: \operatorname{Mot}\to\mathcal V_i $$ $$ R_i(X) = \widetilde R_i(M_X) $$ $$ \operatorname{Real}(M_X) $$ --- ## $S_{1040}$ $$ M_X\cong M_Y \Rightarrow R_i(X)\cong R_i(Y) $$ $$ \sim_M \subseteq \sim_R $$ $$ \mathcal C_M \to \mathcal C_R $$ --- ## $S_{1041}$ $$ \mathcal S_M: \mathcal C_M \to \mathcal C_{\operatorname{rec}} $$ $$ \Pi_M\mathcal S_M = \operatorname{id} $$ $$ E_M := \mathcal S_M\Pi_M $$ $$ E_M^2=E_M $$ --- ## $S_{1042}$ $$ \operatorname{Fix}(E_M) $$ $$ \mathcal C_{\operatorname{mot}} := \operatorname{Fix}(E_M) $$ $$ X\in\mathcal C_{\operatorname{mot}} \Rightarrow X = E_M(X) $$ --- ## $S_{1043}$ $$ M_{X\otimes Y} \cong M_X\otimes M_Y $$ $$ M_{\mathbb 1} \cong \mathbb 1_{\operatorname{Mot}} $$ $$ M_{X^\vee} \cong M_X^\vee $$ $$ \mathcal M \in \operatorname{Fun}^{\otimes} $$ --- ## $S_{1044}$ $$ \operatorname{tr}(M_f) = \mathcal M(\operatorname{tr}(f)) $$ $$ Z_{M_f}(t) = \mathcal M(Z_f(t)) $$ $$ \dim(M_X) = \mathcal M(\dim X) $$ --- ## $S_{1045}$ $$ \omega: \mathcal C_{\operatorname{mot}} \to \mathbf{Vect}_k $$ $$ \omega \in \operatorname{Fun}^{\otimes} $$ $$ \omega(X^\vee) \cong \omega(X)^\vee $$ $$ \omega(\mathbb 1) \cong k $$ --- ## $S_{1046}$ $$ H_\omega := \int^{X\in\mathcal C_{\operatorname{mot}}} \omega(X)^\vee\otimes\omega(X) $$ $$ \Delta: H_\omega \to H_\omega\otimes H_\omega $$ $$ \varepsilon: H_\omega \to k $$ --- ## $S_{1047}$ $$ m: H_\omega\otimes H_\omega \to H_\omega $$ $$ u:k\to H_\omega $$ $$ S:H_\omega\to H_\omega $$ $$ H_\omega \in \mathbf{Hopf}_k $$ --- ## $S_{1048}$ $$ G_\omega := \operatorname{Spec}(H_\omega) $$ $$ G_\omega \simeq \operatorname{Aut}^{\otimes}(\omega) $$ $$ \mathcal C_{\operatorname{mot}} \to \operatorname{Rep}(G_\omega) $$ $$ X \mapsto \omega(X) $$ --- ## $S_{1049}$ $$ \mathcal C_{\operatorname{mot}} \simeq \operatorname{Rep}_{\operatorname{fd}}(G_\omega) $$ $$ G_\omega \simeq \operatorname{Aut}^{\otimes}(\omega) $$ $$ \operatorname{TanRec} $$ --- ## $S_{1050}$ $$ \Omega_{\omega_1,\omega_2} := \operatorname{Isom}^{\otimes}(\omega_1,\omega_2) $$ $$ \Omega_{\omega,\omega} = G_\omega $$ $$ \Omega_{\omega_1,\omega_2} $$ $$ \Omega_{\omega_1,\omega_2} \times \Omega_{\omega_2,\omega_3} \to \Omega_{\omega_1,\omega_3} $$ --- ## $S_{1051}$ $$ \mathfrak F_{\operatorname{fib}} $$ $$ \operatorname{Ob}(\mathfrak F_{\operatorname{fib}}) = \{\omega\} $$ $$ \operatorname{Mor}(\omega_1,\omega_2) = \Omega_{\omega_1,\omega_2} $$ $$ \mathfrak F_{\operatorname{fib}} \in \mathbf{Gpd} $$ --- ## $S_{1052}$ $$ B\mathfrak F_{\operatorname{fib}} $$ $$ \pi_0(B\mathfrak F_{\operatorname{fib}}) $$ $$ \pi_1(B\mathfrak F_{\operatorname{fib}},\omega) \cong G_\omega $$ $$ \operatorname{FibMod} $$ --- ## $S_{1053}$ $$ U\to X $$ $$ U_\bullet $$ $$ U_n = \underbrace{ U\times_X\cdots\times_XU }_{n+1} $$ $$ \check C(U/X) $$ --- ## $S_{1054}$ $$ \mathcal F(X) \to \operatorname{Tot} \mathcal F(U_\bullet) $$ $$ \mathcal F(X) \simeq \operatorname{Tot} \mathcal F(U_\bullet) $$ $$ \operatorname{Desc}(\mathcal F) $$ --- ## $S_{1055}$ $$ U_\bullet \to V_\bullet $$ $$ \operatorname{Tot}\mathcal F(U_\bullet) \to \operatorname{Tot}\mathcal F(V_\bullet) $$ $$ \operatorname{HyperCov} $$ $$ \operatorname{HyperDesc} $$ --- ## $S_{1056}$ $$ \omega_\bullet $$ $$ G_{\omega_\bullet} $$ $$ \operatorname{Tot}(G_{\omega_\bullet}) $$ $$ G_X := \operatorname{Tot}(G_{\omega_\bullet}) $$ --- ## $S_{1057}$ $$ \{R_i\}_{i\in I} $$ $$ \mathcal R_{\operatorname{all}}(X) := (R_i(X))_{i\in I} $$ $$ \mathcal R_{\operatorname{all}} : \mathcal C_{\operatorname{rec}} \to \prod_{i\in I}\mathcal V_i $$ --- ## $S_{1058}$ $$ I_0\subseteq I_1\subseteq\cdots $$ $$ \mathcal R_n := \prod_{i\in I_n}R_i $$ $$ \sim_{n+1} \subseteq \sim_n $$ $$ X\sim_nY \iff \mathcal R_n(X)\cong\mathcal R_n(Y) $$ --- ## $S_{1059}$ $$ \sim_\omega := \bigcap_{n<\omega}\sim_n $$ $$ \mathcal C_\omega := \mathcal C_{\operatorname{rec}}/{\sim_\omega} $$ $$ \Pi_\omega $$ $$ \mathcal R_\omega $$ --- ## $S_{1060}$ $$ I_\lambda := \bigcup_{\beta<\lambda}I_\beta $$ $$ \mathcal R_\lambda := \prod_{i\in I_\lambda}R_i $$ $$ \sim_\lambda = \bigcap_{\beta<\lambda}\sim_\beta $$ --- ## $S_{1061}$ $$ \exists\Gamma: \sim_{\Gamma+1} = \sim_\Gamma $$ $$ \mathcal C_\Gamma := \mathcal C_{\operatorname{rec}}/{\sim_\Gamma} $$ $$ \Pi_\Gamma : \mathcal C_{\operatorname{rec}} \to \mathcal C_\Gamma $$ --- ## $S_{1062}$ $$ \mathcal S_\Gamma: \mathcal C_\Gamma \to \mathcal C_{\operatorname{rec}} $$ $$ \Pi_\Gamma\mathcal S_\Gamma = \operatorname{id} $$ $$ E_\Gamma := \mathcal S_\Gamma\Pi_\Gamma $$ $$ E_\Gamma^2 = E_\Gamma $$ --- ## $S_{1063}$ $$ \mathcal C_{\operatorname{obs}} := \operatorname{Fix}(E_\Gamma) $$ $$ \mathcal C_{\operatorname{obs}} = \operatorname{im}(\mathcal S_\Gamma) $$ $$ X\in\mathcal C_{\operatorname{obs}} \Rightarrow E_\Gamma(X)=X $$ --- ## $S_{1064}$ $$ \mathcal M: \mathcal C_{\operatorname{obs}} \to \operatorname{Mot} $$ $$ \omega: \mathcal C_{\operatorname{obs}} \to \operatorname{Rep}(G_\omega) $$ $$ \mathcal R_\Gamma: \mathcal C_{\operatorname{obs}} \to \prod_{i\in I_\Gamma}\mathcal V_i $$ --- ## $S_{1065}$ $$ X \mapsto \left( M_X, \omega(X), (R_i(X))_{i\in I_\Gamma} \right) $$ $$ \mathcal J_\Gamma(X) $$ $$ \mathcal J_\Gamma: \mathcal C_{\operatorname{obs}} \to \mathcal W_\Gamma $$ --- ## $S_{1066}$ $$ X\sim_{\mathcal J}Y \iff \mathcal J_\Gamma(X) \cong \mathcal J_\Gamma(Y) $$ $$ \operatorname{Ker}(\mathcal J_\Gamma) $$ $$ \mathcal C_{\operatorname{obs}}/{\sim_{\mathcal J}} $$ --- ## $S_{1067}$ $$ \mathcal R_{\mathcal J}: \mathcal W_\Gamma \to \mathcal C_{\operatorname{obs}} $$ $$ \mathcal R_{\mathcal J}\mathcal J_\Gamma \simeq \operatorname{id}_{\mathcal C_{\operatorname{obs}}} $$ $$ \mathcal J_\Gamma\mathcal R_{\mathcal J} \simeq \operatorname{id}_{\mathcal W_\Gamma} $$ --- ## $S_{1068}$ $$ \mathcal C_{\operatorname{obs}} \simeq \mathcal W_\Gamma $$ $$ \mathcal W_\Gamma := \operatorname{im}(\mathcal J_\Gamma) $$ $$ \operatorname{ObsRec} $$ --- ## $S_{1069}$ $$ \operatorname{End}(\mathcal C_{\operatorname{obs}}) $$ $$ \Phi: \mathcal C_{\operatorname{obs}} \to \mathcal C_{\operatorname{obs}} $$ $$ \widehat\Phi := \mathcal J_\Gamma \Phi \mathcal R_{\mathcal J} $$ $$ \widehat\Phi: \mathcal W_\Gamma \to \mathcal W_\Gamma $$ --- ## $S_{1070}$ $$ \widehat{\Phi\Psi} = \widehat\Phi\widehat\Psi $$ $$ \widehat{\operatorname{id}} = \operatorname{id} $$ $$ \operatorname{End}(\mathcal C_{\operatorname{obs}}) \cong \operatorname{End}(\mathcal W_\Gamma) $$ --- ## $S_{1071}$ $$ \Phi^2=\Phi $$ $$ \widehat\Phi^2 = \widehat\Phi $$ $$ \operatorname{Fix}(\Phi) \simeq \operatorname{Fix}(\widehat\Phi) $$ --- ## $S_{1072}$ $$ \mathcal P_0 := \mathcal C_{\operatorname{obs}} $$ $$ \mathcal P_{n+1} := \operatorname{Fix}(\Phi_n) $$ $$ \Phi_n: \mathcal P_n \to \mathcal P_n $$ $$ \mathcal P_{n+1} \subseteq \mathcal P_n $$ --- ## $S_{1073}$ $$ \mathcal P_\lambda := \bigcap_{\beta<\lambda}\mathcal P_\beta $$ $$ \Phi_\lambda := \varprojlim_{\beta<\lambda}\Phi_\beta $$ $$ \mathcal P_{\lambda+1} = \operatorname{Fix}(\Phi_\lambda) $$ --- ## $S_{1074}$ $$ \exists\Theta: \mathcal P_{\Theta+1} = \mathcal P_\Theta $$ $$ \Phi_\Theta \simeq \operatorname{id}_{\mathcal P_\Theta} $$ $$ \mathcal P_\infty := \mathcal P_\Theta $$ --- ## $S_{1075}$ $$ \mathcal P_\infty \subseteq \mathcal C_{\operatorname{obs}} \subseteq \mathcal C_{\operatorname{rec}} $$ $$ \forall X\in\mathcal P_\infty: E_\Theta(X)=X $$ $$ \forall X\in\mathcal P_\infty: \Phi_\Theta(X)=X $$ --- ## $S_{1076}$ $$ \mathcal J_\Gamma : \mathcal P_\infty \to \mathcal W_\infty $$ $$ \mathcal R_{\mathcal J} : \mathcal W_\infty \to \mathcal P_\infty $$ $$ \mathcal R_{\mathcal J}\mathcal J_\Gamma = \operatorname{id}_{\mathcal P_\infty} $$ $$ \mathcal J_\Gamma\mathcal R_{\mathcal J} = \operatorname{id}_{\mathcal W_\infty} $$ --- ## $S_{1077}$ $$ \mathcal P_\infty \simeq \mathcal W_\infty $$ $$ \operatorname{Mot} \leftrightarrow \operatorname{Rep} \leftrightarrow \operatorname{Real} $$ $$ \operatorname{Obj} \leftrightarrow \operatorname{Obs} $$ --- ## $S_{1078}$ $$ \mathfrak R_\infty := \mathcal R_{\mathcal J}\mathcal J_\Gamma $$ $$ \mathfrak R_\infty^2 = \mathfrak R_\infty $$ $$ \operatorname{Fix}(\mathfrak R_\infty) = \mathcal P_\infty $$ --- ## $S_{1079}$ $$ X \in \mathcal P_\infty $$ $$ X = \mathfrak R_\infty(X) $$ $$ X \simeq \mathcal R_{\mathcal J} \left( M_X, \omega(X), (R_i(X))_{i\in I_\Gamma} \right) $$ $$ \operatorname{SelfRec}(X) $$ --- ## $S_{1080}$ $$ S_{1020} \to S_{1021} \to \cdots \to S_{1080} $$ $$ \operatorname{Real} \to \operatorname{Mot} \to \operatorname{LocalInv} \to \operatorname{Tensor} \to \operatorname{TanRec} \to \operatorname{Desc} \to \operatorname{ObsTower} \to \operatorname{ObsRec} \to \operatorname{TransFix} \to \operatorname{SelfRec} $$ $$ \mathcal C_{\operatorname{rec}} \to \mathcal C_{\operatorname{mot}} \to \mathcal C_{\operatorname{obs}} \to \mathcal P_\infty $$ $$ E_\Gamma^2 = E_\Gamma $$ $$ \mathfrak R_\infty^2 = \mathfrak R_\infty $$ $$ \operatorname{Fix}(\mathfrak R_\infty) = \mathcal P_\infty $$ $$ \boxed{ SREG_{018} : S_{1021} \to S_{1022} \to \cdots \to S_{1080} } $$ $$ \boxed{ S_{1080}\to S_{1081} } $$