Commit 339b0b39 by Leonard Guetta

### security commit

parent e1734e74
 ... ... @@ -250,9 +250,9 @@ From the previous proposition, we deduce the following very useful corollary. \begin{tikzcd} \displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[ddr,bend left]\ar[d]&\\ A \ar[drr,bend right,"\beta"'] \ar[r] & G \ar[dr, dotted]&\\ &&B &&B, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"] \end{tikzcd}, \end{tikzcd} \] where the arrow $E \to B$ is the canonical inclusion. Hence, by universal property, the dotted arrow exists and makes the whole diagram commutes. A thorough verification easily shows that the morphism $G \to B$ is a monomorphism of $\Rgrph$. ... ... @@ -261,11 +261,11 @@ From the previous proposition, we deduce the following very useful corollary. \begin{tikzcd}[row sep = large] \displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[d]&\\ A \ar[d,"\alpha"] \ar[r] & G \ar[d] \ar[r] & B \ar[d,"\delta"]\\ C \ar[r] & H \ar[r] & D C \ar[r] & H \ar[r] & D. \ar[from=1-1,to=2-2,phantom,"\ulcorner" very near end,"\text{\textcircled{\tiny \textbf{1}}}" near start, description] \ar[from=2-1,to=3-2,phantom,"\ulcorner" very near end,"\text{\textcircled{\tiny \textbf{2}}}", description] \ar[from=2-2,to=3-3,phantom,"\ulcorner" very near end,"\text{\textcircled{\tiny \textbf{3}}}", description] \end{tikzcd}. \end{tikzcd} \] What we want to prove is that the image by the functor $L$ of the pasting of squares \textcircled{\tiny \textbf{2}} and \textcircled{\tiny \textbf{3}} is homotopy cocartesian. Since the morphism $G \to B$ is a monomorphism, we deduce from Corollary \ref{cor:hmtpysquaregraph} that the image by the functor $L$ of square \textcircled{\tiny \textbf{3}} is homotopy cocartesian. Hence, in virtue of Lemma \ref{lemma:pastinghmtpycocartesian}, all we have to show is that the image by $L$ of square \textcircled{\tiny \textbf{2}} is homotopy cocartesian. On the other hand, we know that both morphisms $... ... @@ -818,11 +818,16 @@ 2\nbd{}categories, let us recall the following particular case of Corollary \end{cases} \item generating 2-cell:  \alpha : f_{m}\circ \cdots \circ f_1 \Rightarrow g_n \circ \cdots \circ g_1. \end{itemize} Notice that A_{(1,1)} is nothing but \sD_2. We are going to prove that if n\neq 0 or m\neq 0, then A_{(m,n)} is \good{} and has the homotopy type of a point. When m\neq0 \emph{and} n\neq0, this result is not surprising, but when n=0 or m=0 (but not both), it is \emph{a priori} less clear what the homotopy type of A_{(m,n)} is and whether it is \good{} or not. For example, A_{(1,0)} can be pictured as follows Notice that A_{(1,1)} is nothing but \sD_2. We are going to prove that if n\neq 0 or m\neq 0, then A_{(m,n)} is \good{} and has the homotopy type of a point. When m\neq0 \emph{and} n\neq0, this result is not surprising, but when n=0 or m=0 (but not both), it is \emph{a priori} less clear what the homotopy type of A_{(m,n)} is and whether it is \good{} or not. For example, A_{(1,0)} can be pictured as \[ %% \begin{tikzcd} %% A \ar[r, bend left=70, "f",""{name=A,below}] \ar[r,bend right=70,"1_A"',""{name=B,above}] & A, \ar[from=A,to=B,Rightarrow,"\alpha"] %% \end{tikzcd} %% \text{ or } \begin{tikzcd} A \ar[r, bend left=70, "f",""{name=A,below}] \ar[r,bend right=70,"1_A"',""{name=B,above}] & A \ar[from=A,to=B,Rightarrow,"\alpha"] \end{tikzcd}, A \ar[loop,in=50,out=130,distance=1.5cm,"f",""{name=A,below}] \ar[from=A,to=1-1,Rightarrow,"\alpha"] \end{tikzcd}$ and has many non trivial $2$-cells, such as $f\comp_0 \alpha \comp_0 f$. ... ... @@ -856,9 +861,9 @@ For any $n \geq 0$, consider the following cocartesian square $\begin{tikzcd} \Delta_1 \ar[r,"i"] \ar[d,"\tau"] & \Delta_n \ar[d] \\ A_{(1,1)} \ar[r] & A_{(1,n)} A_{(1,1)} \ar[r] & A_{(1,n)}, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"] \end{tikzcd}, \end{tikzcd}$ where $\tau : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique non-trivial $1$\nbd{}cell of $\Delta_1$ to the target of the generating $2$-cell of $A_{(1,1)}$. It is not hard to check that $\tau$ is strong deformation retract and thus, a co-universal Thomason equivalence (Lemma \ref{lemma:pushoutstrngdefrtract}). Hence, the morphism $A_{(1,1)} \to A_{(1,n)}$ is also a (co-universal) Thomason equivalence and the square is Thomason homotopy cocartesian (Lemma \ref{lemma:hmtpycocartsquarewe}). Now, the morphism $\tau : \Delta_1 \to A_{(1,1)}$ is also a folk cofibration and since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ are \good{}, it follows from what we said in \ref{paragr:criterion2cat} that $A_{(1,n)}$ is \good{}. Finally, since $\Delta_1$, $\Delta_n$ and $A_{(1,1)}$ have the homotopy type of a point, the fact that the previous square is Thomason homotopy cocartesian implies that $A_{(1,n)}$ has the homotopy type of a point. ... ... @@ -866,9 +871,9 @@ For any $n \geq 0$, consider the following cocartesian square $\begin{tikzcd} \Delta_1 \ar[r,"i"] \ar[d,"\sigma"] & \Delta_m \ar[d] \\ A_{(1,1)} \ar[r] & A_{(m,1)} A_{(1,1)} \ar[r] & A_{(m,1)}, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"] \end{tikzcd}, \end{tikzcd}$ where $\sigma : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique non trivial $1$\nbd{}cell of $\Delta_1$ the source of the generating $2$\nbd{}cell of $A_{(1,1)}$, we can prove that $A_{(m,1)}$ is \good{} and has the homotopy type of a point. ... ... @@ -1171,14 +1176,17 @@ isomorphisms, which means by definition that $P$, $P'$ and $P''$ are \good{}. \begin{itemize}[label=-] \item generating $0$\nbd{}cell: $A$, \item generating $1$\nbd{}cell: $f,g: A \to A$, \item generating $2$\nbd{}cell: $\alpha,\beta : f \to g$. \item generating $2$\nbd{}cell: $\alpha,\beta : f \Rightarrow g$. \end{itemize} In picture, this gives: %% $%% \begin{tikzcd} %% \end{tikzcd} %%$ $\begin{tikzcd}[column sep=huge] A \ar[r,bend left=75,"f",""{name=A,below,pos=9/20},""{name=C,below,pos=11/20}] \ar[r,bend right=75,"g"',""{name=B,above,pos=9/20},""{name=D,above,pos=11/20}] & A. \ar[from=C,to=D,bend left,"\alpha",Rightarrow] \ar[from=A,to=B,bend right,"\beta"',Rightarrow] \end{tikzcd}$ \todo{À finir} \end{paragr} \begin{paragr} Let $P$ be the free $2$\nbd{}category defined as follows: ... ... @@ -1197,9 +1205,9 @@ isomorphisms, which means by definition that $P$, $P'$ and $P''$ are \good{}. \ar[from=E,to=F,Rightarrow,"\gamma",bend left] \end{tikzcd} \] Now let $P'$ be the sub-$2$\nbd{}category of $P$ spanned by $A$,$B$, $\alpha$ and $beta$, and let $P''$ be the sub-$2$\nbd{}category of $P$ spanned by $A$,$B$,$\beta$ and $\gamma$. These $2$\nbd{}categories are Now let $P'$ be the sub-$2$\nbd{}category of $P$ spanned by $A$, $B$, $\alpha$ and $\beta$, and let $P''$ be the sub-$2$\nbd{}category of $P$ spanned by $A$, $B$, $\beta$ and $\gamma$. These $2$\nbd{}categories are simply copies of $\sS_2$. Notice that we have a cocartesian square \label{square:bouquet} ... ... @@ -1242,30 +1250,41 @@ isomorphisms, which means by definition that $P$, $P'$ and $P''$ are \good{}. \ar[from=G,to=H,Rightarrow,"\delta",bend left] \end{tikzcd} \] Let us prove that this $2$\nbd{}category is \good{}. Let $P'$ be the sub-$2$\nbd{}category of $P$ spanned by $A$, $B$, $f$, $g$, $\alpha$ and $\beta$, let $P''$ be the sub-$2$\nbd{}category of $P$ spanned by $A$, $B$, $g$, $h$, $\gamma$ and $\delta$ and let $P''$ be the sub-$2$\nbd{}category of $P$ spanned by $A$, $B$ and $g$. These two $2$\nbd{}categories are copies of $\sS_2$ and we have a cocartesian square Let us prove that this $2$\nbd{}category is \good{}. Let $P_0$ be the sub-$1$category of $P$ spanned by $A$, $B$ and $g$, let $P_1$ be the sub-$2$\nbd{}category of $P$ spanned by $A$, $B$, $g$, $h$, $\gamma$ and $\delta$ and let $P_2$ be the sub-$2$\nbd{}category of $P$ spanned by $A$, $B$, $f$, $g$, $\alpha$ and $\beta$. The $2$\nbd{}categories $P_1$ and $P_2$ are copies of $\sS_2$, and $P_0$ is a copy of $\sS_1$. Moreover, we have a cocartesian square of inclusions $\begin{tikzcd} \sD_1 \ar[d,"\langle g \rangle"] \ar[r,"\langle g \rangle"] & P' \ar[d] \\\ P'' \ar[r] & P. P_0 \ar[d,hook] \ar[r,hook] & P_2 \ar[d,hook] \\\ P_1 \ar[r,hook] & P. \end{tikzcd}$ Let us prove that this square is Thomason homotopy cocartesian using the second part of Corollary \ref{prop:critverthorThomhmtpysquare}. First, all the morphisms of the previous square are isomorphisms on objects and thus, the image by $V_0$ of the above square is obviously cocartesian. Now, notice that the categories $P(A,B)$, $P'(A,B)$ and $P''(A,B)$ are respectively free on the graphs Let us prove that this square is Thomason homotopy cocartesian using the second part of Corollary \ref{prop:critverthorThomhmtpysquare}. This means that we have to show that for every $k \geq 0$, the induced square $\begin{tikzcd} f \ar[r,shift left,"\alpha"] \ar[r, shift right, "\beta"'] & g \ar[r,shift left,"\gamma"'] \ar[r,shift right,"\delta"] & h, \end{tikzcd}$ $\begin{tikzcd} f \ar[r,shift left,"\alpha"] \ar[r, shift right, "\beta"'] & g, \end{tikzcd}$ and $\begin{tikzcd} g \ar[r,shift left,"\gamma"] \ar[r,shift right,"\delta"'] & h. H_k(P_0) \ar[d] \ar[r] & H_k(P_2) \ar[d] \\\ H_k(P_1) \ar[r] & H_k(P) \end{tikzcd}$ is cocartesian. First, all the morphisms of the previous square are isomorphisms on objects and thus, the image by $V_0$ of the above square is obviously cocartesian. Now, notice that all the categories $P_i(A,A)$ and $P_i(B,B)$ for $0 \leq i \leq 2$ are all isormophic to the terminal category $\sD_0$ and the categories $P_i(B,A)$ for $0 \leq i \leq 2$ are all the empty category. It follows that for %% notice that the categories $P(A,B)$, $P'(A,B)$ and $P''(A,B)$ are respectively free on the graphs %% $%% \begin{tikzcd} %% f \ar[r,shift left,"\alpha"] \ar[r, shift right, "\beta"'] & g \ar[r,shift left,"\gamma"'] \ar[r,shift right,"\delta"] & h, %% \end{tikzcd} %%$ %% $%% \begin{tikzcd} %% f \ar[r,shift left,"\alpha"] \ar[r, shift right, "\beta"'] & g, %% \end{tikzcd} %%$ %% and %% $%% \begin{tikzcd} %% g \ar[r,shift left,"\gamma"] \ar[r,shift right,"\delta"'] & h. %% \end{tikzcd} %%$ Besides, all the categories $P(A,A)$, $P(B,B)$, $P'(A,A)$, $P'(B,B)$, $P''(A,A)$ and $P''(B,B)$ are all isormophic to the terminal category $\sD_0$. \todo{À finir} \end{paragr} ... ...
 ... ... @@ -309,8 +309,8 @@ higher than $1$. $\begin{tikzcd}[column sep=tiny] X/{a_0} \ar[rr,"X/{\beta}"] \ar[dr] && X/{a'_0} \ar[dl] \\ &X& \end{tikzcd}. &X&. \end{tikzcd}$ This defines a functor \begin{align*} ... ... @@ -319,7 +319,7 @@ higher than $1$. \end{align*} and a canonical map $\colim_{a_0 \in A} X/{a_0} \to X. \colim_{a_0 \in A} (X/{a_0}) \to X.$ This map is natural in $X$ in the following sense. Let $f' : X' \to A$ be another $\oo$\nbd{}functor and let $... ... @@ -341,8 +341,8 @@ higher than 1. \[ \begin{tikzcd} \displaystyle \colim_{a_0 \in A}(X/a_0) \ar[d] \ar[r] & X \ar[d,"g"] \\ \displaystyle\colim_{a_0 \in A}(X'/a_0) \ar[r] & X' \end{tikzcd}, \displaystyle\colim_{a_0 \in A}(X'/a_0) \ar[r] & X', \end{tikzcd}$ is commutative. \end{paragr} ... ... @@ -481,7 +481,7 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd{}func \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end] \end{tikzcd} \] From the second part of Proposition \ref{prop:modprs}, we deduce that $\sk_{n-1}(X/-) \to \sk_{n}(X/-)$ is a cofibration for the From the second part of Proposition \ref{prop:modprs}, we deduce that $\sk_{n-1}(X/-) \to \sk_{n}(X/-)$ is a cofibration for the projective model structure on $\oo\Cat(A)$. Thus, the transfinite composition $\emptyset \to \sk_{0}(X/-) \to \sk_{1}(X/) \to \cdots \to \sk_{n}(X/-) \to \cdots, ... ...  ... ... @@ -52,7 +52,7 @@ The functor \kappa : \Psh{\Delta} \to \Ch is left Quillen and sends weak equiv as a generating trivial cofibrations (see for example \cite[Section I.1]{goerss2009simplicial} for the notations). A quick computation, which we leave to the reader, shows that the image by \kappa of \partial\Delta_n \hookrightarrow \Delta_n is a monomorphism with projective cokernel and the image by \kappa of \Lambda^i_n \hookrightarrow \Delta_n is a quasi-isomorphism. This proves that \kappa is left Quillen. Since all simplicial sets are cofibrant, it follows from Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model} that \kappa also preserves weak equivalences. \end{proof} \begin{remark} The previous lemma admits also as more conceptual proof as follows. From the Dold-Kan equivalence, we know that \Ch is equivalent to the category \Ab(\Delta) of simplicial abelian groups and with this identification the functor \kappa : \Psh{\Delta} \to \Ch is left adjoint of the canonical forgetful functor The previous lemma admits also a more conceptual proof as follows. From the Dold-Kan equivalence, we know that \Ch is equivalent to the category \Ab(\Delta) of simplicial abelian groups and with this identification the functor \kappa : \Psh{\Delta} \to \Ch is left adjoint of the canonical forgetful functor \[ U : \Ch \simeq \Ab(\Delta) \to \Psh{\Delta}$ ... ... @@ -641,7 +641,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins $H^{\pol}_k(C)=\begin{cases} \mathbb{Z} \text{ if } k=0,2\\ 0 \text{ in other cases.}\end{cases}$ On the other hand, it was proven in \cite[Theorem 4.9 and Example 4.10]{ara2019quillen} that (the nerve of) $C$ is a $K(\mathbb{Z},2)$-space. In particular, it has non-trivial singular homology groups in every even dimension. This proves that $\sH^{\pol}(C)$ is \emph{not} isomorphic to $\sH^{\sing}(C)$; which means that triangle \eqref{cmprisontrngle} cannot be commutative (up to an isomorphism). On the other hand, it is proven in \cite[Theorem 4.9 and Example 4.10]{ara2019quillen} that (the nerve of) $C$ is a $K(\mathbb{Z},2)$-space. In particular, it has non-trivial singular homology groups in every even dimension. This proves that $\sH^{\pol}(C)$ is \emph{not} isomorphic to $\sH^{\sing}(C)$; which means that triangle \eqref{cmprisontrngle} cannot be commutative (up to an isomorphism). \end{paragr} Another consequence of the above counter-example is the following result, which we claimed in \ref{paragr:polhmlgythomeq}. Recall that given a morphism $u : C \to D$ of $\oo\Cat$, we write $\sH^{\pol}(u)$ instead of $\sH^{\pol}(\gamma^{\folk}(u))$. \begin{proposition}\label{prop:polhmlgynotinvariant} ... ... @@ -687,14 +687,14 @@ Another consequence of the above counter-example is the following result, which \begin{tikzcd} \oo\Cat \ar[r,"\lambda"] \ar[d,"\gamma^{\folk}"] & \Ch \ar[d,"\gamma^{\Ch}"] \\ \ho(\oo\Cat^{\folk}) \ar[d,"\J"] \ar[r,"\sH^{\pol}"'] & \ho(\Ch)\\ \ho(\oo\Cat^{\Th}) \ar[ru,"G"',bend right] & \ho(\oo\Cat^{\Th}) \ar[ru,"G"',bend right] &. \ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >=1em,Rightarrow]\\ \ar[from=3-1,to=2-2,"\delta"',shorten <= 1em, shorten >= 1em,Rightarrow] \end{tikzcd}. \end{tikzcd} \] But since $\J$ acts as localization functor, $\delta$ also factorizes uniquely as $\begin{tikzcd}[column sep=small] \ho(\oo\Cat^{\folk}) \ar[r,"\J"] & \ho(\oo\Cat^{\Th}) \ar[r,bend left,"\overline{\sH^{\pol}}",""{name=A,below}] \ar[r,bend right, "G"',pos=16/30,""{name=B,above}] & \ho(\Ch) \ar[from=B,to=A,Rightarrow,"\delta'"]\end{tikzcd}. \begin{tikzcd}[column sep=small] \ho(\oo\Cat^{\folk}) \ar[r,"\J"] & \ho(\oo\Cat^{\Th}) \ar[r,bend left,"\overline{\sH^{\pol}}",""{name=A,below}] \ar[r,bend right, "G"',pos=16/30,""{name=B,above}] & \ho(\Ch). \ar[from=B,to=A,Rightarrow,"\delta'"]\end{tikzcd}$ Altogether we have that $\beta$ factorizes as $... ... @@ -703,7 +703,7 @@ Another consequence of the above counter-example is the following result, which \ho(\oo\Cat^{\Th}) \ar[r,"\overline{\sH^{\pol}}",""{name=B,below}] & \ho(\Ch). \ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >=1em,Rightarrow] \ar[from=2-1,to=2-2,"G"',pos=16/30,bend right,""{name=A,above}] \ar[from=A,to=B,Rightarrow,"\delta'"]. \ar[from=A,to=B,Rightarrow,"\delta'"] \end{tikzcd}$ The uniqueness of such a factorization follows from a similar argument which is left to the reader. This proves that $\overline{\sH^{\pol}}$ is the left derived functor of $\lambda$ when $\oo\Cat$ is equipped with Thomason equivalences and in particular we have ... ... @@ -976,8 +976,8 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati $\begin{tikzcd} C \ar[d,"\eta_C"] \ar[r,"f"] & D \ar[d,"\eta_D"] \\ T(C) \ar[r,"T(f)"] & T(D) \end{tikzcd}, T(C) \ar[r,"T(f)"] & T(D), \end{tikzcd}$ where $\eta$ is the unit of the adjunction $\tau^{i}_{\leq n} \dashv \iota_n$. ... ... @@ -1088,9 +1088,9 @@ We now turn to truncations of chain complexes. $\begin{tikzcd} \tau^{i}_{\leq n}(A) \ar[r] \ar[d,"\tau^{i}_{\leq n}(j)"'] & X \ar[d,"g"] \\ \tau^{i}_{\leq n}(B) \ar[r] & Y \tau^{i}_{\leq n}(B) \ar[r] & Y, \ar[from=1-1, to=2-2, phantom, "\ulcorner",very near end] \end{tikzcd}, \end{tikzcd}$ the morphism $\iota_n(g)$ is a weak equivalence of $\Ch$. As explained in \cite[Proposition 7.19]{dwyer1995homotopy}, there exists a set of generating trivial cofibrations of the projective model structure on $\Ch$ consisting of the maps $... ... @@ -1187,10 +1187,10 @@ is commutative. \[ \begin{tikzcd} \oo\Cat \ar[r,"\tau_{\leq n}^{i}"] \ar[rd,"\mathrm{id}"',""{name=A,right}] & n\Cat \ar[d,"\iota_n"] \ar[r,"\lambda_{\leq n}"] & \Ch^{\leq n} \ar[d,"\iota_n"'] \ar[dr,"\mathrm{id}",""{name=B,left}] & \\ &\oo\Cat \ar[r,"\lambda"] & \Ch \ar[r,"\tau^{i}_{\leq n}"'] & \Ch^{\leq n} &\oo\Cat \ar[r,"\lambda"] & \Ch \ar[r,"\tau^{i}_{\leq n}"'] & \Ch^{\leq n}. \ar[from=A, to=1-2,Rightarrow,"\eta"] \ar[from=2-3,to=B,Rightarrow,"\epsilon"] \end{tikzcd}. \end{tikzcd}$ Since for every $\oo$\nbd{}category $C$ and every $k  ... ... @@ -59,9 +59,9 @@ \Or_2= \begin{tikzcd} &\langle 1 \rangle \ar[rd,"\langle 12 \rangle"]& \\ \langle 0 \rangle \ar[ru,"\langle 01 \rangle"]\ar[rr,"\langle 02 \rangle"',""{name=A,above}]&&\langle 2 \rangle \langle 0 \rangle \ar[ru,"\langle 01 \rangle"]\ar[rr,"\langle 02 \rangle"',""{name=A,above}]&&\langle 2 \rangle, \ar[Rightarrow,from=A,to=1-2,"\langle 012 \rangle"] \end{tikzcd}, \end{tikzcd} \] $\Or_3= ... ... @@ -123,9 +123,9 @@ \[ \begin{tikzcd} & X_j \ar[rd,"{X_{j,k}}"]& \\ X_i \ar[ru,"X_{i,j}"]\ar[rr,"X_{i,k}"',""{name=A,above}]&&X_k X_i \ar[ru,"X_{i,j}"]\ar[rr,"X_{i,k}"',""{name=A,above}]&&X_k, \ar[Rightarrow,from=A,to=1-2,"X_{i,j,k}"] \end{tikzcd}, \end{tikzcd}$ \end{itemize} subject to the following axiom: ... ... @@ -235,8 +235,8 @@ From now on, we will consider that the category$\Psh{\Delta}$is equipped with $\begin{tikzcd}[column sep=tiny] \Ho(n\Cat^{\Th}) \ar[rr] \ar[rd,"\overline{N_n}"'] & & \Ho(m\Cat) \ar[dl,"\overline{N_m}"] \\ &\Ho(\Psh{\Delta})& \end{tikzcd}. &\Ho(\Psh{\Delta})&. \end{tikzcd}$ \end{proof} \section{Tensor product and oplax transformations} ... ... @@ -271,9 +271,9 @@ From now on, we will consider that the category$\Psh{\Delta}$is equipped with $\begin{tikzcd} X\ar[rd,"u"] \ar[d,"i_0^X"']& \\ \sD_1\otimes X \ar[r,"\alpha"] & Y \\ X \ar[ru,"v"'] \ar[u,"i_1^X"] \end{tikzcd}, \sD_1\otimes X \ar[r,"\alpha"] & Y, \\ X \ar[ru,"v"'] \ar[u,"i_1^X"]& \end{tikzcd}$ where$i_0^X$and$i_1^X$are induced by the two$\oo$-functors$\sD_0 \to \sD_1$and where we implicitly used the isomorphism$\sD_0 \otimes X \simeq X$, is commutative. \item As an$\oo$-functor$\alpha : X \to \homlax(\sD_1,Y)$such that the following diagram ... ... @@ -410,15 +410,15 @@ From now on, we will consider that the category$\Psh{\Delta}$is equipped with \label{diagramtransf}\tag{ii} \begin{tikzcd} B\ar[rd,"\mathrm{id}_B"] \ar[d,"i_0^B"']& \\ \sD_1\otimes B \ar[r,"\alpha"] & B \\ B \ar[ru,"i\circ r"'] \ar[u,"i_1^B"] \end{tikzcd}, \sD_1\otimes B \ar[r,"\alpha"] & B, \\ B \ar[ru,"i\circ r"'] \ar[u,"i_1^B"]& \end{tikzcd} and \label{diagramstrong}\tag{iii} \begin{tikzcd} \sD_1 \otimes A \ar[rr, bend right,"p\otimes i"']\ar[r,"\sD_1 \otimes i"] & \sD_1 \otimes B \ar[r,"\alpha"] & B \end{tikzcd}, \sD_1 \otimes A \ar[rr, bend right,"p\otimes i"']\ar[r,"\sD_1 \otimes i"] & \sD_1 \otimes B \ar[r,"\alpha"] & B, \end{tikzcd} where$p$is the unique morphism$\sD_1 \to \sD_0$, are commutative. ... ... @@ -427,9 +427,9 @@ From now on, we will consider that the category$\Psh{\Delta}$is equipped with \begin{tikzcd} A \ar[r,"u"] \ar[d,"i"] & A' \ar[d,"i'"] \ar[dd,bend left=75,"\mathrm{id}_{B'}"] \\ B \ar[d,"r"] \ar[r,"v"] & B' \ar[d,"r'",dashed ] \\ A \ar[r,"u"] & A' A \ar[r,"u"] & A', \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end] \end{tikzcd}, \end{tikzcd} \] we deduce the existence of$r' : B' \to A'$that makes the whole diagram commutes. In particular, we have$r' \circ i' = \mathrm{id}_{B'}$. ... ... @@ -438,8 +438,8 @@ From now on, we will consider that the category$\Psh{\Delta}$is equipped with \begin{tikzcd} \sD_1\otimes A \ar[r,"\sD_1\otimes u"] \ar[d,"\sD_1\otimes i"] & \sD_1 \otimes A' \ar[d,"\sD_1 \otimes i'"] \ar[dd,bend left=75,"p\otimes i'"] \\ \sD_1\otimes B \ar[d,"\alpha"] \ar[r,"\sD_1 \otimes v"] & \sD_1 \otimes B' \ar[d,"\alpha'",dashed ] \\ \sD_1 \otimes B \ar[r,"v"] & \sD_1 \otimes B' \end{tikzcd}. \sD_1 \otimes B \ar[r,"v"] & \sD_1 \otimes B'. \end{tikzcd} \] The existence of$\alpha' : \sD_1 \otimes B' \to B'$that makes the whole diagram commutes follows from the fact that the functor$\sD_1 \otimes \shortminus$preserves colimits. In particular, we have$\alpha' \circ (\sD_1 \otimes i') = p \otimes i'$. ... ... @@ -812,8 +812,8 @@ The nerve functor$N_{\omega} : \omega\Cat \to \Psh{\Delta}$sends equivalences $\begin{tikzcd}[column sep=small] A \ar[rr,"u"] \ar[dr,"v"'] & &B \ar[dl,"w"] \\ &C& \end{tikzcd}, &C&, \end{tikzcd}$ then for any object$c_0$of$C$, we have a functor$u/c_0 : A/c_0 \to B/c_0defined as \begin{align*} ... ...  ... ... @@ -104,11 +104,11 @@ To end this section, we recall a derivability criterion due to Gonzalez, which w $\begin{tikzcd} & \C' \ar[rr,"\gamma'"]\ar[rd,"G"] & &\ho(\C') \ar[rr,"\mathrm{id}",""{name=B,below}]\ar[rd,"\RR G"'] & &\ho(\C') \\ \C\ar[ru,"F"] \ar[rr,"\mathrm{id}"',""{name=A,above}] && \C \ar[rr,"\gamma"'] &&\ho(\C)\ar[ru,"F'"'] & \C\ar[ru,"F"] \ar[rr,"\mathrm{id}"',""{name=A,above}] && \C \ar[rr,"\gamma"'] &&\ho(\C)\ar[ru,"F'"'] &. \ar[from=A,to=1-2,"\eta",Rightarrow, shorten <= 0.5em, shorten >= 0.5em] \ar[from=2-3,to=1-4,Rightarrow,"\beta",shorten <= 1em, shorten >= 1em] \ar[from=2-5,to=B,Rightarrow,"\epsilon'"',shorten <= 0.5em, shorten >= 0.5em] \end{tikzcd}. \end{tikzcd}$ \end{paragr} \begin{proposition}[{\cite[Theorem 3.1]{gonzalez2012derivability}}]\label{prop:gonz} ... ... @@ -200,8 +200,8 @@ We now turn to the most important way of obtaining op-prederivators. and every\begin{tikzcd}A \ar[r,bend left,"u",""{name=A,below}] \ar[r,bend right, "v"',""{name=B,above}] & B \ar[from=A,to=B,Rightarrow,"\alpha"]\end{tikzcd}$induces by pre-composition a$2$-morphism of localizers $\begin{tikzcd} (\C(B),\W_B) \ar[r,bend left,"u^*",""{name=A,below}] \ar[r,bend right, "v^*"',""{name=B,above}] & (\C(A),\W_A) \ar[from=A,to=B,Rightarrow,"\alpha^*"] \end{tikzcd}. (\C(B),\W_B) \ar[r,bend left,"u^*",""{name=A,below}] \ar[r,bend right, "v^*"',""{name=B,above}] & (\C(A),\W_A). \ar[from=A,to=B,Rightarrow,"\alpha^*"] \end{tikzcd}$ (This last property is trivial since a$2$-morphism of localizers is simply a natural transformation between the underlying functors.) Then, by the universal property of localization, we have for every$u : A \to B$an induced functor, which we still denote$u^*, $... ... @@ -210,8 +210,8 @@ We now turn to the most important way of obtaining op-prederivators. and for every \begin{tikzcd}A \ar[r,bend left,"u",""{name=A,below}] \ar[r,bend right, "v"',""{name=B,above}] & B \ar[from=A,to=B,Rightarrow,"\alpha"]\end{tikzcd}, an induced natural transformation \[ \begin{tikzcd} \ho(\C(B)) \ar[r,bend left,"u^*",""{name=A,below}] \ar[r,bend right, "v^*"',""{name=B,above}] & \ho(\C(A)) \ar[from=A,to=B,Rightarrow,"\alpha^*"] \end{tikzcd}. \ho(\C(B)) \ar[r,bend left,"u^*",""{name=A,below}] \ar[r,bend right, "v^*"',""{name=B,above}] & \ho(\C(A)). \ar[from=A,to=B,Rightarrow,"\alpha^*"] \end{tikzcd}$ Altogether, this defines an op-prederivator \begin{align*} ... ... @@ -426,11 +426,11 @@ We now turn to the most important way of obtaining op-prederivators. $\begin{tikzcd} \sD(A) \ar[dr,"\mathrm{id}"',""{name=A,above}] \ar[r,"u_!"] &\sD(B) \ar[d,"u^*"] \ar[r,"F_B"] & \sD'(B)\ar[d,"u^*"] \ar[dr,"\mathrm{id}",""{name=B,below}]\\ & \sD(A) \ar[r,"F_A"'] & \sD'(A) \ar[r,"u_!"'] & \sD'(B) & \sD(A) \ar[r,"F_A"'] & \sD'(A) \ar[r,"u_!"'] & \sD'(B). \ar[from=2-2,to=1-3,Rightarrow,"F_u"',"\sim"] \ar[from=A,to=1-2,Rightarrow,"\eta"] \ar[from=2-3,to=B,Rightarrow,"\epsilon"] \end{tikzcd}. \end{tikzcd}$ For example, when\sD$is the homotopy op-prederivator of a localizer,$B$is the terminal category$e$, for any$X$object of$\sD(A)$the previous canonical morphism reads $... ... @@ -527,8 +527,8 @@ We now turn to the most important way of obtaining op-prederivators. \[ \begin{tikzcd} (0,0) \ar[d] \ar[r] & (0,1) \\ (1,0) & \end{tikzcd}. (1,0) &. \end{tikzcd}$ Finally, we write$i_{\ulcorner} : \ulcorner \to \square$for the canonical inclusion functor. \end{paragr} ... ... @@ -559,17 +559,17 @@ We now turn to the most important way of obtaining op-prederivators. $\begin{tikzcd} \ulcorner \ar[rr,"i_{\ulcorner}",""{name=A,below}] \ar[rd,"p"'] && \square\\ &e\ar[ru,"{(1,1)}"']& &e\ar[ru,"{(1,1)}"']&, \ar[from=A,to=2-2,Rightarrow,"\alpha"] \end{tikzcd}, \end{tikzcd}$ where we wrote$p$instead of$p_{\ulcorner}$for short. Hence, we have a$2$-triangle $\begin{tikzcd} \sD(\ulcorner) && \sD(\square) \ar[ll,"i_{\ulcorner}^*"',""{name=A,below}] \ar[dl,"{(1,1)}^*"] \\ & \sD(e) \ar[ul,"p^*"]& & \sD(e) \ar[ul,"p^*"]&. \ar[from=A,to=2-2,Rightarrow,"\alpha^*"] \end{tikzcd}. \end{tikzcd}$ Suppose now that$\sD$has left Kan extensions. For$X$an object of$\sD(\square)$, we have a canonical morphism$p_!(i_{\ulcorner}^*(X)) \to X_{(1,1)}$defined as the composition $... ... @@ -582,8 +582,8 @@ We now turn to the most important way of obtaining op-prederivators. X= \begin{tikzcd} A \ar[r,"u"] \ar[d,"f"]& B \ar[d,"g"] \\ C \ar[r,"v"]&D \end{tikzcd}, C \ar[r,"v"]&D, \end{tikzcd}$ this previous morphism reads $... ... @@ -622,8 +622,8 @@ We now turn to the most important way of obtaining op-prederivators. \[ \begin{tikzcd} A \ar[r,"u"] \ar[d,"1_A"]& B \ar[d,"1_B"] \\ A \ar[r,"u"]&B \end{tikzcd}. A \ar[r,"u"]&B. \end{tikzcd}$ The result follows then from \cite[Proposition 3.12(2)]{groth2013derivators}. \end{proof} ... ... This diff is collapsed. No preview for this file type  ... ... @@ -213,21 +213,21 @@ A \emph{morphism of$\oo$-magmas}$f : X \to Y$is a morphism of underlying$\oo \end{itemize} This completely describes the $n$-category $\sD_n$ as no non-trivial composition can occur. Here are pictures in low dimension: $\sD_0= \begin{tikzcd}\bullet\end{tikzcd}, \sD_0= \begin{tikzcd}\bullet,\end{tikzcd}$ $\sD_1 = \begin{tikzcd} \bullet \ar[r] &\bullet \end{tikzcd}, \sD_1 = \begin{tikzcd} \bullet \ar[r] &\bullet, \end{tikzcd}$ $\sD_2 = \begin{tikzcd} \bullet \ar[r,bend left=50,""{name = U,below}] \ar[r,bend right=50,""{name=D}]&\bullet \ar[Rightarrow, from=U,to=D] \end{tikzcd}, \bullet \ar[r,bend left=50,""{name = U,below}] \ar[r,bend right=50,""{name=D}]&\bullet, \ar[Rightarrow, from=U,to=D] \end{tikzcd}$ $\sD_3 = \begin{tikzcd} \bullet \ar[r,bend left=50,""{name = U,below,near start},""{name = V,below,near end}] \ar[r,bend right=50,""{name=D,near start},""{name = E,near end}]&\bullet \ar[Rightarrow, from=U,to=D, bend right,""{name= L,above}]\ar[Rightarrow, from=V,to=E, bend left,""{name= R,above}] \bullet \ar[r,bend left=50,""{name = U,below,near start},""{name = V,below,near end}] \ar[r,bend right=50,""{name=D,near start},""{name = E,near end}]&\bullet. \ar[Rightarrow, from=U,to=D, bend right,""{name= L,above}]\ar[Rightarrow, from=V,to=E, bend left,""{name= R,above}] \arrow[phantom,"\Rrightarrow",from=L,to=R] \end{tikzcd}. \end{tikzcd}$ For any $\oo$-category $C$, the map \begin{align*} ... ... @@ -271,15 +271,15 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo Here are some pictures of the$n$-spheres in low dimension: $\sS_0= \begin{tikzcd}\bullet & \bullet \end{tikzcd}, \sS_0= \begin{tikzcd}\bullet & \bullet, \end{tikzcd}$ $\sS_1 = \begin{tikzcd} \bullet \ar[r,bend left=50,""{name = U,below}] \ar[r,bend right=50,""{name=D}]&\bullet \end{tikzcd}, \sS_1 = \begin{tikzcd} \bullet \ar[r,bend left=50,""{name = U,below}] \ar[r,bend right=50,""{name=D}]&\bullet, \end{tikzcd}$ $\sS_2 = \begin{tikzcd} \bullet \ar[r,bend left=50,""{name = U,below,near start},""{name = V,below,near end}] \ar[r,bend right=50,""{name=D,near start},""{name = E,near end}]&\bullet \ar[Rightarrow, from=U,to=D, bend right,""{name= L,above}]\ar[Rightarrow, from=V,to=E, bend left,""{name= R,above}] \end{tikzcd}. \bullet \ar[r,bend left=50,""{name = U,below,near start},""{name = V,below,near end}] \ar[r,bend right=50,""{name=D,near start},""{name = E,near end}]&\bullet. \ar[Rightarrow, from=U,to=D, bend right,""{name= L,above}]\ar[Rightarrow, from=V,to=E, bend left,""{name= R,above}] \end{tikzcd}$ For an$\oo$-category$C$and$n\geq 0$, an$\oo$-functor ... ... @@ -405,11 +405,11 @@ So far, we have not yet seen examples of free$\oo$-categories. In order to do t $(a \ast b) \ast (c \ast d) = (a \ast c ) \ast (b \ast d).$ It is straightforward to see that a necessary and sufficient condition for this equation to hold is to require that$M$be commutative. Hence, we have proven the following lemma. It is straightforward to see that a necessary and sufficient condition for this equation to hold is to require that$M$be commutative. Hence, we have proved the following lemma. \end{paragr} \begin{lemma} Let$M$be a monoid and$n \in \mathbb{N}\$. Then: \begin{itemize}