# SREG-018 $$ S_{1020}\longrightarrow S_{1021} $$ --- ## $S_{1021}$ $$ \mathcal C_{\operatorname{rec}} = \operatorname{Fix}(E_\Theta) $$ $$ E_\Theta^2 = E_\Theta $$ $$ X\in\mathcal C_{\operatorname{rec}} $$ $$ E_\Theta(X)=X $$ --- ## $S_{1022}$ $$ \mathcal V_i \qquad (i\in I) $$ $$ R_i: \mathcal C_{\operatorname{rec}} \to \mathcal V_i $$ $$ R_i(X) $$ $$ R_i(f) $$ --- ## $S_{1023}$ $$ R := \prod_{i\in I}R_i $$ $$ R: \mathcal C_{\operatorname{rec}} \to \prod_{i\in I}\mathcal V_i $$ $$ R(X) = (R_i(X))_{i\in I} $$ --- ## $S_{1024}$ $$ X\sim_RY \iff R(X)\cong R(Y) $$ $$ [X]_R $$ $$ \mathcal C_R := \mathcal C_{\operatorname{rec}}/{\sim_R} $$ $$ \Pi_R: \mathcal C_{\operatorname{rec}} \to \mathcal C_R $$ --- ## $S_{1025}$ $$ \Pi_R(X)=\Pi_R(Y) \iff X\sim_RY $$ $$ \Pi_R^2 = \Pi_R $$ $$ \operatorname{im}(\Pi_R) = \mathcal C_R $$ --- ## $S_{1026}$ $$ \mathcal S_R: \mathcal C_R \to \mathcal C_{\operatorname{rec}} $$ $$ \Pi_R\mathcal S_R = \operatorname{id}_{\mathcal C_R} $$ $$ \mathcal E_R := \mathcal S_R\Pi_R $$ $$ \mathcal E_R^2 = \mathcal E_R $$ --- ## $S_{1027}$ $$ \operatorname{Fix}(\mathcal E_R) = \operatorname{im}(\mathcal S_R) $$ $$ \mathcal C_{\operatorname{rec}}^{R} := \operatorname{Fix}(\mathcal E_R) $$ $$ X\in\mathcal C_{\operatorname{rec}}^{R} \Rightarrow \mathcal E_R(X)=X $$ --- ## $S_{1028}$ $$ (\mathcal C_{\operatorname{rec}},\otimes,\mathbb 1) $$ $$ R_i(X\otimes Y) \cong R_i(X)\otimes R_i(Y) $$ $$ R_i(\mathbb 1) \cong \mathbb 1_i $$ $$ R_i \in \operatorname{Fun}^{\otimes} $$ --- ## $S_{1029}$ $$ X^\vee $$ $$ \operatorname{coev}_X: \mathbb 1 \to X\otimes X^\vee $$ $$ \operatorname{ev}_X: X^\vee\otimes X \to \mathbb 1 $$ $$ X^{\vee\vee} \cong X $$ --- ## $S_{1030}$ $$ f:X\to X $$ $$ \operatorname{tr}(f) : \mathbb 1\to\mathbb 1 $$ $$ R_i(\operatorname{tr}(f)) = \operatorname{tr}(R_i(f)) $$ $$ \operatorname{tr}(fg) = \operatorname{tr}(gf) $$ --- ## $S_{1031}$ $$ Z_f(t) := \exp \left( \sum_{n\ge1} \frac{\operatorname{tr}(f^n)}{n}t^n \right) $$ $$ R_i(Z_f) = Z_{R_i(f)} $$ $$ Z_{f\oplus g} = Z_fZ_g $$ --- ## $S_{1032}$ $$ K_0(\mathcal C_{\operatorname{rec}}) $$ $$ [X] $$ $$ [X\oplus Y] = [X]+[Y] $$ $$ [X\otimes Y] = [X][Y] $$ $$ K_0(R_i) $$ --- ## $S_{1033}$ $$ K(\mathcal C_{\operatorname{rec}}) $$ $$ \pi_nK(\mathcal C_{\operatorname{rec}}) = K_n(\mathcal C_{\operatorname{rec}}) $$ $$ K: \mathbf{Cat}_{\operatorname{st}} \to \mathbf{Sp} $$ $$ K(\mathcal C\oplus\mathcal D) \simeq K(\mathcal C)\oplus K(\mathcal D) $$ --- ## $S_{1034}$ $$ \operatorname{THH}: \mathbf{Cat}_{\operatorname{st}} \to \mathbf{Sp} $$ $$ \operatorname{TC}: \mathbf{Cat}_{\operatorname{st}} \to \mathbf{Sp} $$ $$ K \to \operatorname{TC} $$ $$ \operatorname{THH}(\mathcal C) $$ --- ## $S_{1035}$ $$ \mathcal U_{\operatorname{loc}} : \mathbf{Cat}_{\operatorname{st}} \to \mathbf{Mot}_{\operatorname{loc}} $$ $$ \mathcal C'\to\mathcal C\to\mathcal C'' $$ $$ \mathcal U_{\operatorname{loc}}(\mathcal C') \to \mathcal U_{\operatorname{loc}}(\mathcal C) \to \mathcal U_{\operatorname{loc}}(\mathcal C'') $$ --- ## $S_{1036}$ $$ \operatorname{Mot} := \mathbf{Mot}_{\operatorname{loc}} $$ $$ M_X := \mathcal U_{\operatorname{loc}} ( \operatorname{Perf}(X) ) $$ $$ X\mapsto M_X $$ $$ \mathcal M: \mathcal C_{\operatorname{rec}} \to \operatorname{Mot} $$ --- ## $S_{1037}$ $$ X\sim_M Y \iff M_X\cong M_Y $$ $$ \mathcal C_M := \mathcal C_{\operatorname{rec}}/{\sim_M} $$ $$ \Pi_M: \mathcal C_{\operatorname{rec}} \to \mathcal C_M $$ --- ## $S_{1038}$ $$ \operatorname{Hom}_{\operatorname{Mot}}(M_X,M_Y) $$ $$ \pi_n \operatorname{Map}_{\operatorname{Mot}}(M_X,M_Y) $$ $$ \operatorname{End}_{\operatorname{Mot}}(M_X) $$ $$ \operatorname{Aut}_{\operatorname{Mot}}(M_X) $$ --- ## $S_{1039}$ $$ R_i = \widetilde R_i \circ \mathcal M $$ $$ \widetilde R_i: \operatorname{Mot}\to\mathcal V_i $$ $$ R_i(X) = \widetilde R_i(M_X) $$ $$ \operatorname{Real}(M_X) $$ --- ## $S_{1040}$ $$ M_X\cong M_Y \Rightarrow R_i(X)\cong R_i(Y) $$ $$ \sim_M \subseteq \sim_R $$ $$ \mathcal C_M \to \mathcal C_R $$ --- ## $S_{1041}$ $$ \mathcal S_M: \mathcal C_M \to \mathcal C_{\operatorname{rec}} $$ $$ \Pi_M\mathcal S_M = \operatorname{id} $$ $$ E_M := \mathcal S_M\Pi_M $$ $$ E_M^2=E_M $$ --- ## $S_{1042}$ $$ \operatorname{Fix}(E_M) $$ $$ \mathcal C_{\operatorname{mot}} := \operatorname{Fix}(E_M) $$ $$ X\in\mathcal C_{\operatorname{mot}} \Rightarrow X = E_M(X) $$ --- ## $S_{1043}$ $$ M_{X\otimes Y} \cong M_X\otimes M_Y $$ $$ M_{\mathbb 1} \cong \mathbb 1_{\operatorname{Mot}} $$ $$ M_{X^\vee} \cong M_X^\vee $$ $$ \mathcal M \in \operatorname{Fun}^{\otimes} $$ --- ## $S_{1044}$ $$ \operatorname{tr}(M_f) = \mathcal M(\operatorname{tr}(f)) $$ $$ Z_{M_f}(t) = \mathcal M(Z_f(t)) $$ $$ \dim(M_X) = \mathcal M(\dim X) $$ --- ## $S_{1045}$ $$ \omega: \mathcal C_{\operatorname{mot}} \to \mathbf{Vect}_k $$ $$ \omega \in \operatorname{Fun}^{\otimes} $$ $$ \omega(X^\vee) \cong \omega(X)^\vee $$ $$ \omega(\mathbb 1) \cong k $$ --- ## $S_{1046}$ $$ H_\omega := \int^{X\in\mathcal C_{\operatorname{mot}}} \omega(X)^\vee\otimes\omega(X) $$ $$ \Delta: H_\omega \to H_\omega\otimes H_\omega $$ $$ \varepsilon: H_\omega \to k $$ --- ## $S_{1047}$ $$ m: H_\omega\otimes H_\omega \to H_\omega $$ $$ u:k\to H_\omega $$ $$ S:H_\omega\to H_\omega $$ $$ H_\omega \in \mathbf{Hopf}_k $$ --- ## $S_{1048}$ $$ G_\omega := \operatorname{Spec}(H_\omega) $$ $$ G_\omega \simeq \operatorname{Aut}^{\otimes}(\omega) $$ $$ \mathcal C_{\operatorname{mot}} \to \operatorname{Rep}(G_\omega) $$ $$ X \mapsto \omega(X) $$ --- ## $S_{1049}$ $$ \mathcal C_{\operatorname{mot}} \simeq \operatorname{Rep}_{\operatorname{fd}}(G_\omega) $$ $$ G_\omega \simeq \operatorname{Aut}^{\otimes}(\omega) $$ $$ \operatorname{TanRec} $$ --- ## $S_{1050}$ $$ \Omega_{\omega_1,\omega_2} := \operatorname{Isom}^{\otimes}(\omega_1,\omega_2) $$ $$ \Omega_{\omega,\omega} = G_\omega $$ $$ \Omega_{\omega_1,\omega_2} $$ $$ \Omega_{\omega_1,\omega_2} \times \Omega_{\omega_2,\omega_3} \to \Omega_{\omega_1,\omega_3} $$ --- ## $S_{1051}$ $$ \mathfrak F_{\operatorname{fib}} $$ $$ \operatorname{Ob}(\mathfrak F_{\operatorname{fib}}) = \{\omega\} $$ $$ \operatorname{Mor}(\omega_1,\omega_2) = \Omega_{\omega_1,\omega_2} $$ $$ \mathfrak F_{\operatorname{fib}} \in \mathbf{Gpd} $$ --- ## $S_{1052}$ $$ B\mathfrak F_{\operatorname{fib}} $$ $$ \pi_0(B\mathfrak F_{\operatorname{fib}}) $$ $$ \pi_1(B\mathfrak F_{\operatorname{fib}},\omega) \cong G_\omega $$ $$ \operatorname{FibMod} $$ --- ## $S_{1053}$ $$ U\to X $$ $$ U_\bullet $$ $$ U_n = \underbrace{ U\times_X\cdots\times_XU }_{n+1} $$ $$ \check C(U/X) $$ --- ## $S_{1054}$ $$ \mathcal F(X) \to \operatorname{Tot} \mathcal F(U_\bullet) $$ $$ \mathcal F(X) \simeq \operatorname{Tot} \mathcal F(U_\bullet) $$ $$ \operatorname{Desc}(\mathcal F) $$ --- ## $S_{1055}$ $$ U_\bullet \to V_\bullet $$ $$ \operatorname{Tot}\mathcal F(U_\bullet) \to \operatorname{Tot}\mathcal F(V_\bullet) $$ $$ \operatorname{HyperCov} $$ $$ \operatorname{HyperDesc} $$ --- ## $S_{1056}$ $$ \omega_\bullet $$ $$ G_{\omega_\bullet} $$ $$ \operatorname{Tot}(G_{\omega_\bullet}) $$ $$ G_X := \operatorname{Tot}(G_{\omega_\bullet}) $$ --- ## $S_{1057}$ $$ \{R_i\}_{i\in I} $$ $$ \mathcal R_{\operatorname{all}}(X) := (R_i(X))_{i\in I} $$ $$ \mathcal R_{\operatorname{all}} : \mathcal C_{\operatorname{rec}} \to \prod_{i\in I}\mathcal V_i $$ --- ## $S_{1058}$ $$ I_0\subseteq I_1\subseteq\cdots $$ $$ \mathcal R_n := \prod_{i\in I_n}R_i $$ $$ \sim_{n+1} \subseteq \sim_n $$ $$ X\sim_nY \iff \mathcal R_n(X)\cong\mathcal R_n(Y) $$ --- ## $S_{1059}$ $$ \sim_\omega := \bigcap_{n<\omega}\sim_n $$ $$ \mathcal C_\omega := \mathcal C_{\operatorname{rec}}/{\sim_\omega} $$ $$ \Pi_\omega $$ $$ \mathcal R_\omega $$ --- ## $S_{1060}$ $$ I_\lambda := \bigcup_{\beta<\lambda}I_\beta $$ $$ \mathcal R_\lambda := \prod_{i\in I_\lambda}R_i $$ $$ \sim_\lambda = \bigcap_{\beta<\lambda}\sim_\beta $$ --- ## $S_{1061}$ $$ \exists\Gamma: \sim_{\Gamma+1} = \sim_\Gamma $$ $$ \mathcal C_\Gamma := \mathcal C_{\operatorname{rec}}/{\sim_\Gamma} $$ $$ \Pi_\Gamma : \mathcal C_{\operatorname{rec}} \to \mathcal C_\Gamma $$ --- ## $S_{1062}$ $$ \mathcal S_\Gamma: \mathcal C_\Gamma \to \mathcal C_{\operatorname{rec}} $$ $$ \Pi_\Gamma\mathcal S_\Gamma = \operatorname{id} $$ $$ E_\Gamma := \mathcal S_\Gamma\Pi_\Gamma $$ $$ E_\Gamma^2 = E_\Gamma $$ --- ## $S_{1063}$ $$ \mathcal C_{\operatorname{obs}} := \operatorname{Fix}(E_\Gamma) $$ $$ \mathcal C_{\operatorname{obs}} = \operatorname{im}(\mathcal S_\Gamma) $$ $$ X\in\mathcal C_{\operatorname{obs}} \Rightarrow E_\Gamma(X)=X $$ --- ## $S_{1064}$ $$ \mathcal M: \mathcal C_{\operatorname{obs}} \to \operatorname{Mot} $$ $$ \omega: \mathcal C_{\operatorname{obs}} \to \operatorname{Rep}(G_\omega) $$ $$ \mathcal R_\Gamma: \mathcal C_{\operatorname{obs}} \to \prod_{i\in I_\Gamma}\mathcal V_i $$ --- ## $S_{1065}$ $$ X \mapsto \left( M_X, \omega(X), (R_i(X))_{i\in I_\Gamma} \right) $$ $$ \mathcal J_\Gamma(X) $$ $$ \mathcal J_\Gamma: \mathcal C_{\operatorname{obs}} \to \mathcal W_\Gamma $$ --- ## $S_{1066}$ $$ X\sim_{\mathcal J}Y \iff \mathcal J_\Gamma(X) \cong \mathcal J_\Gamma(Y) $$ $$ \operatorname{Ker}(\mathcal J_\Gamma) $$ $$ \mathcal C_{\operatorname{obs}}/{\sim_{\mathcal J}} $$ --- ## $S_{1067}$ $$ \mathcal R_{\mathcal J}: \mathcal W_\Gamma \to \mathcal C_{\operatorname{obs}} $$ $$ \mathcal R_{\mathcal J}\mathcal J_\Gamma \simeq \operatorname{id}_{\mathcal C_{\operatorname{obs}}} $$ $$ \mathcal J_\Gamma\mathcal R_{\mathcal J} \simeq \operatorname{id}_{\mathcal W_\Gamma} $$ --- ## $S_{1068}$ $$ \mathcal C_{\operatorname{obs}} \simeq \mathcal W_\Gamma $$ $$ \mathcal W_\Gamma := \operatorname{im}(\mathcal J_\Gamma) $$ $$ \operatorname{ObsRec} $$ --- ## $S_{1069}$ $$ \operatorname{End}(\mathcal C_{\operatorname{obs}}) $$ $$ \Phi: \mathcal C_{\operatorname{obs}} \to \mathcal C_{\operatorname{obs}} $$ $$ \widehat\Phi := \mathcal J_\Gamma \Phi \mathcal R_{\mathcal J} $$ $$ \widehat\Phi: \mathcal W_\Gamma \to \mathcal W_\Gamma $$ --- ## $S_{1070}$ $$ \widehat{\Phi\Psi} = \widehat\Phi\widehat\Psi $$ $$ \widehat{\operatorname{id}} = \operatorname{id} $$ $$ \operatorname{End}(\mathcal C_{\operatorname{obs}}) \cong \operatorname{End}(\mathcal W_\Gamma) $$ --- ## $S_{1071}$ $$ \Phi^2=\Phi $$ $$ \widehat\Phi^2 = \widehat\Phi $$ $$ \operatorname{Fix}(\Phi) \simeq \operatorname{Fix}(\widehat\Phi) $$ --- ## $S_{1072}$ $$ \mathcal P_0 := \mathcal C_{\operatorname{obs}} $$ $$ \mathcal P_{n+1} := \operatorname{Fix}(\Phi_n) $$ $$ \Phi_n: \mathcal P_n \to \mathcal P_n $$ $$ \mathcal P_{n+1} \subseteq \mathcal P_n $$ --- ## $S_{1073}$ $$ \mathcal P_\lambda := \bigcap_{\beta<\lambda}\mathcal P_\beta $$ $$ \Phi_\lambda := \varprojlim_{\beta<\lambda}\Phi_\beta $$ $$ \mathcal P_{\lambda+1} = \operatorname{Fix}(\Phi_\lambda) $$ --- ## $S_{1074}$ $$ \exists\Theta: \mathcal P_{\Theta+1} = \mathcal P_\Theta $$ $$ \Phi_\Theta \simeq \operatorname{id}_{\mathcal P_\Theta} $$ $$ \mathcal P_\infty := \mathcal P_\Theta $$ --- ## $S_{1075}$ $$ \mathcal P_\infty \subseteq \mathcal C_{\operatorname{obs}} \subseteq \mathcal C_{\operatorname{rec}} $$ $$ \forall X\in\mathcal P_\infty: E_\Theta(X)=X $$ $$ \forall X\in\mathcal P_\infty: \Phi_\Theta(X)=X $$ --- ## $S_{1076}$ $$ \mathcal J_\Gamma : \mathcal P_\infty \to \mathcal W_\infty $$ $$ \mathcal R_{\mathcal J} : \mathcal W_\infty \to \mathcal P_\infty $$ $$ \mathcal R_{\mathcal J}\mathcal J_\Gamma = \operatorname{id}_{\mathcal P_\infty} $$ $$ \mathcal J_\Gamma\mathcal R_{\mathcal J} = \operatorname{id}_{\mathcal W_\infty} $$ --- ## $S_{1077}$ $$ \mathcal P_\infty \simeq \mathcal W_\infty $$ $$ \operatorname{Mot} \leftrightarrow \operatorname{Rep} \leftrightarrow \operatorname{Real} $$ $$ \operatorname{Obj} \leftrightarrow \operatorname{Obs} $$ --- ## $S_{1078}$ $$ \mathfrak R_\infty := \mathcal R_{\mathcal J}\mathcal J_\Gamma $$ $$ \mathfrak R_\infty^2 = \mathfrak R_\infty $$ $$ \operatorname{Fix}(\mathfrak R_\infty) = \mathcal P_\infty $$ --- ## $S_{1079}$ $$ X \in \mathcal P_\infty $$ $$ X = \mathfrak R_\infty(X) $$ $$ X \simeq \mathcal R_{\mathcal J} \left( M_X, \omega(X), (R_i(X))_{i\in I_\Gamma} \right) $$ $$ \operatorname{SelfRec}(X) $$ --- ## $S_{1080}$ $$ S_{1020} \to S_{1021} \to \cdots \to S_{1080} $$ $$ \operatorname{Real} \to \operatorname{Mot} \to \operatorname{LocalInv} \to \operatorname{Tensor} \to \operatorname{TanRec} \to \operatorname{Desc} \to \operatorname{ObsTower} \to \operatorname{ObsRec} \to \operatorname{TransFix} \to \operatorname{SelfRec} $$ $$ \mathcal C_{\operatorname{rec}} \to \mathcal C_{\operatorname{mot}} \to \mathcal C_{\operatorname{obs}} \to \mathcal P_\infty $$ $$ E_\Gamma^2 = E_\Gamma $$ $$ \mathfrak R_\infty^2 = \mathfrak R_\infty $$ $$ \operatorname{Fix}(\mathfrak R_\infty) = \mathcal P_\infty $$ $$ \boxed{ SREG_{018} : S_{1021} \to S_{1022} \to \cdots \to S_{1080} } $$ $$ \boxed{ S_{1080}\to S_{1081} } $$