Commit 339b0b39 authored by Leonard Guetta's avatar Leonard Guetta
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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
\begin{equation}\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 $k <n$, we have
\[
......@@ -1430,8 +1430,8 @@ Finally, we obtain the result we were aiming for.
\[
\begin{tikzcd}[column sep=tiny]
\sH^{\sing}(C) \ar[rd,"\alpha^{\sing}"'] \ar[rr,"\pi_C"] & & \sH^{\pol}(C) \ar[dl,"\alpha^{\pol}"] \\
&\lambda(C)&
\end{tikzcd}.
&\lambda(C)&.
\end{tikzcd}
\]
From Proposition \ref{prop:singhmlgylowdimension}, we know that $\alpha^{\sing}$ induces isomorphisms \[H_k^{\sing}(C) \simeq H_k(\lambda(C))\] for $k \in \{0,1\}$ and from Corollary \ref{cor:polhmlgycofibrant} and Paragraph \ref{paragr:polhmlgylowdimension} we know that $\alpha^{\pol}$ induces isomorphisms $H_k^{\pol}(C) \simeq H_k(\lambda(C))$ for $k \in \{0,1\}$. The result follows then from an immediate 2-out-of-3 property.
\end{proof}
......
......@@ -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
\begin{equation}\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}
\end{equation}
and
\begin{equation}\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}
\end{equation}
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 $u/c_0 : A/c_0 \to B/c_0$ defined 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}
......
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......@@ -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}