Commit 777544a7 authored by Leonard Guetta's avatar Leonard Guetta
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Edited a lot of typos

parent a120b417
......@@ -22,7 +22,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego
\[
L : \Rgrph \to \Cat.
\]
For a reflexive graph $G$, the objects of $L(G)$ are exactly the objects of $G$ and an arrow $f$ of of $L(G)$ is a chain
For a reflexive graph $G$, the objects of $L(G)$ are exactly the objects of $G$ and an arrow $f$ of $L(G)$ is a chain
\[
\begin{tikzcd}
X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1} \ar[r,"f_n"]& X_{n}
......@@ -42,7 +42,7 @@ In this section, we review some homotopical results concerning free ($1$-)catego
Conversely, if $C \simeq L(G)$ for some reflexive graph $G$, then the description of the arrows of $L(G)$ given in the previous paragraph shows that $C$ is free and that its set of generating $1$-cells is (isomorphic to) the non unital $1$-cells of $G$.
\end{proof}
\begin{remark}
In other words, a category is free on a graph if and only if it is free on a reflexive graph. The difference between these two notions is at the level of morphisms: there are more morphisms of reflexive graphs because (generating) $1$\nbd{}cells may be sent to units. Hence, for a morphism of reflexing graphs $f : G \to G'$, the induced functor $L(f)$ is not necessarily rigid in the sense of Definition \ref{def:rigidmorphism}.
In other words, a category is free on a graph if and only if it is free on a reflexive graph. The difference between these two notions is at the level of morphisms: there are more morphisms of reflexive graphs because (generating) $1$\nbd{}cells may be sent to units. Hence, for a morphism of reflexive graphs $f : G \to G'$, the induced functor $L(f)$ is not necessarily rigid in the sense of Definition \ref{def:rigidmorphism}.
\end{remark}
\begin{paragr}
There is another important description of the category $\Rgrph$. Consider $\Delta_{\leq 1}$ the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$. Then, the category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq 1}}$, the category of pre-sheaves on $\Delta_{\leq 1}$. In particular, the canonical inclusion $i : \Delta_{\leq 1} \rightarrow \Delta$ induces by pre-composition a functor
......@@ -152,7 +152,7 @@ In other words, a category is free on a graph if and only if it is free on a ref
\]
which proves that the right vertical arrow is a trivial cofibration.
\end{proof}
From this lemma, we deduce the following propositon.
From this lemma, we deduce the following proposition.
\begin{proposition}
Let $G$ be a reflexive graph. The map
\[
......@@ -203,7 +203,7 @@ From the previous proposition, we deduce the following very useful corollary.
This square is cocartesian because $i_!$ is a left adjoint. Since $i_!$ preserves monomorphisms (Lemma \ref{lemma:monopreserved}), the result follows from the fact that monomorphisms are cofibrations of simplicial sets for the standard Quillen model structure and Lemma \ref{lemma:hmtpycocartesianreedy}.
\end{proof}
\begin{paragr}
Actually, by working a little more, we obtain a more general result, which is stated in the propositon below. Let us say that a morphism of reflexive graphs, $\alpha : A \to B$, is \emph{quasi-injective on arrows} when for all $f$ and $g$ arrows of $A$, if
Actually, by working a little more, we obtain a more general result, which is stated in the proposition below. Let us say that a morphism of reflexive graphs, $\alpha : A \to B$, is \emph{quasi-injective on arrows} when for all $f$ and $g$ arrows of $A$, if
\[
\alpha(f)=\alpha(g),
\]
......@@ -287,7 +287,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp
\begin{remark}
Since every free category is obtained by recursively adding generators starting from a set of objects (seen as a $0$-category), the previous example yields another proof that \emph{free} (1-)categories are \good{} (which we already knew since we have seen that \emph{all} (1-)categories are \good{}).
\end{remark}
\begin{example}[Identyfing two generators]
\begin{example}[Identifying two generators]
Let $C$ be a free category and $f,g : A \to B$ parallel generating arrows of $C$ such that $f\neq g$. Now consider the category $C'$ obtained from $C$ by ``identifying'' $f$ and $g$, i.e. defined with the following cocartesian square
\[
\begin{tikzcd}
......@@ -341,7 +341,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp
\end{align*}
The category of bisimplicial sets is denoted by $\Psh{\Delta\times\Delta}$.
\iffalse Moreover for every $n \geq 0$, if we fix the first variable to $n$, we obtain a simplical set
\iffalse Moreover for every $n \geq 0$, if we fix the first variable to $n$, we obtain a simplicial set
\begin{align*}
X_{n,\bullet} : \Delta^{\op} &\to \Set \\
[m] &\mapsto X_{n,m}.
......@@ -351,7 +351,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp
X_{\bullet,n} : \Delta^{\op} &\to \Set \\
[m] &\mapsto X_{m,n}.
\end{align*}
The correspondances
The correspondences
\[
n \mapsto X_{n,\bullet} \,\text{ and }\, n\mapsto X_{\bullet,n}
\]
......@@ -499,7 +499,7 @@ Diagonal weak equivalences are not the only interesting weak equivalences for bi
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}) \to \Ho(\Psh{\Delta})$ is an
equivalence of op-prederivators.
\end{proof}
In practise, we will use the following corollary.
In practice, we will use the following corollary.
\begin{corollary}\label{cor:bisimplicialsquare}
Let
\[
......@@ -603,7 +603,7 @@ The \emph{bisimplicial nerve} of a $2$-category $C$ is the bisimplicial set $\bi
for all $n,m \geq 0$.
\end{definition}
\begin{paragr}\label{paragr:formulabisimplicialnerve}
In other words, the bisimplicial nerve of $C$ is obtained by ``uncurryfying'' the functor $NH(C) : \Delta^{op} \to \Psh{\Delta}$.
In other words, the bisimplicial nerve of $C$ is obtained by ``un-curryfying'' the functor $NH(C) : \Delta^{op} \to \Psh{\Delta}$.
Since the nerve $N$ commutes with products and sums, we obtain the formula
\begin{equation}\label{fomulabinerve}
......@@ -621,7 +621,8 @@ More intuitively, an element of $\binerve(C)_{n,m}$ consists of a ``pasting sche
\end{tikzcd}\right.}_{ n }.
\]
\end{paragr}
In the definition of the bisimplicial nerve of a $2$\nbd{}category we gave, we have priviledged one direction of the bisimplicial set over the other. We now give another definition of the bisimplicial nerve using the other direction.
In the definition of the bisimplicial nerve of a $2$\nbd{}category we gave, one
direction of the bisimplicial set is privileged over the other. We now give another definition of the bisimplicial nerve using the other direction.
\begin{paragr}
Let $C$ be a $2$\nbd{}category. For every $k \geq 1$, we define a $1$\nbd{}category $V(C)_k$ in the following fashion:
\begin{itemize}[label=-]
......@@ -650,7 +651,7 @@ In the definition of the bisimplicial nerve of a $2$\nbd{}category we gave, we h
\end{itemize}
For $k=0$, we define $V(C)_0$ to be the category obtained from $C$ by simply
forgetting the $2$\nbd{}cells (which is nothing but $\tau^{s}_{\leq 1}(C)$
with the notations of \ref{paragr:defncat}). The correspondance $n \mapsto V(C)_n$ defines to a simplicial object in $\Cat$
with the notations of \ref{paragr:defncat}). The correspondence $n \mapsto V(C)_n$ defines to a simplicial object in $\Cat$
\[
V(C) : \Delta^{\op} \to \Cat,
\]
......@@ -677,7 +678,7 @@ In the definition of the bisimplicial nerve of a $2$\nbd{}category we gave, we h
\end{lemma}
\begin{proof}
It follows from what is shown in \cite[Section 2]{bullejos2003geometry} that
there is a zig-zag of weak equivalence of simplicial sets
there is a zigzag of weak equivalence of simplicial sets
\[
\delta^*(\binerve(C)) \leftarrow \cdots \rightarrow N(C)
\]
......@@ -755,7 +756,7 @@ It also follows from Lemma \ref{lemma:binervthom} that the bisimplicial nerve in
\]
is an \emph{equivalence} of op-prederivators.
From Proposition \ref{prop:streetvsbisimplicial}, we deduce the proposition below which contains two useful critera to detect Thomason homotopy cocartesian square of $2\Cat$.
From Proposition \ref{prop:streetvsbisimplicial}, we deduce the proposition below which contains two useful criteria to detect Thomason homotopy cocartesian square of $2\Cat$.
\end{paragr}
\begin{proposition}
Let
......@@ -859,7 +860,7 @@ For any $n \geq 0$, consider the following cocartesian square
\ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
\end{tikzcd},
\]
where $\tau : \Delta_1 \to A_{(1,1)}$ is the $2$-functor that sends the unique non-trival $1$\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.
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.
Similarly, for any $m \geq 0$, by considering the cocartesian square
\[
......@@ -1066,7 +1067,7 @@ the category $V(P'')$ is the free category on the graph
and $V(H)_0$ comes from a morphism of reflexive graphs obtained by ``killing the
generator $j$''. Hence, it is a Thomason equivalence of categories. For $k>0$,
the category $V(\sS_2)_k$ has two objects $\overline{A}$ and $\overline{B}$ and
an arrow $\overline{A} \to \overline{B}$ is a $k$\nbd{}uple of either one of the
an arrow $\overline{A} \to \overline{B}$ is a $k$\nbd{}tuple of either one of the
following form
\begin{itemize}[label=-]
\item $(1_i,\cdots,1_i,\delta,1_j,\cdots,1_j)$
......
......@@ -201,7 +201,7 @@ From these two lemmas, follows the important proposition below.
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\]
where the map $\sD_2 \to B^2\mathbb{N}$ points the unique genarating $2$-cell of $B^2\mathbb{N}$ and
where the map $\sD_2 \to B^2\mathbb{N}$ points the unique generating $2$-cell of $B^2\mathbb{N}$ and
$\sD_0 \to B^2\mathbb{N}$ points to the only object of $B^2\mathbb{N}$. It is easily checked that this
square is cocartesian and since $\sD_2$, $\sD_0$ and $\sD_2$ are free and
$i_2$ is a cofibration for the canonical model structure, the square is also
......@@ -212,7 +212,7 @@ From these two lemmas, follows the important proposition below.
From Proposition \ref{prop:spheresaregood}, we also deduce the proposition
below which gives a criterion to detect \good{} $\oo$\nbd{}category when we
already know that they are free. Note that it seems hard
to use in practise and we will only use it for theoretical purposes.
to use in practice and we will only use it for theoretical purposes.
\end{paragr}
\begin{proposition}
Let $C$ be a free $\oo$\nbd{}category and for every $k \in \mathbb{N}$ let
......@@ -244,7 +244,7 @@ $(1-)$categories as particular cases of $\oo$\nbd{}categories.
The goal of what follows is to show that every $1$-category is \good{}. In order to do that, we will prove that every 1-category
is a canonical colimit of contractible $1$-categories and that this colimit is
homotopic both
with respect to folk weak equivalences and and with respect to Thomason equivalences.
with respect to folk weak equivalences and with respect to Thomason equivalences.
We call the reader's attention to an important subtlety here: even though the
desired result only refers to $1$\nbd{}categories, we have to work in the setting
of $\oo$\nbd{}categories. This can be explained from the fact that if we take a
......@@ -462,7 +462,7 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd{}func
\]
is a set a generating folk cofibrations.
From Lemmas \ref{lemma:sliceisfree} and \ref{lemma:basisofslice} we deduce
that for every objet $a_0$ of $A$ and every $n \in \mathbb{N}$, the canonical square
that for every object $a_0$ of $A$ and every $n \in \mathbb{N}$, the canonical square
\[
\begin{tikzcd}
\displaystyle\coprod_{x \in \Sigma^X_n}\coprod_{\Hom_A(f(\trgt_0(x)),a_0)}\sS_{n-1} \ar[r] \ar[d] & \sk_{n-1}(X/a_0) \ar[d]\\
......@@ -583,7 +583,7 @@ We now recall an important Theorem due to Thomason.
A thorough analysis of all the isomorphisms involved (or see ) shows that this last isomorphism is indeed induced by the co-cone $(A/a \to A)_{a \in \Ob(A)}$.
\end{proof}
\begin{remark}
It is possible to extend the previous corollary to prove that for any functor $f : X \to A$ ($X$ and $A$ being $1$-categories), we have \[\hocolim^{\Th}_{a \in A} (X/a) \simeq X.\] However, to prove that it is also the case when $X$ is an $\oo$\nbd{}category and $f$ an $\oo$\nbd{}functor, as in Corollary \ref{cor:folkhmtpycol}, one would need to extend the Grothendieck construction to functors with value in $\oo\Cat$ and to prove an $\oo$\nbd{}categorical analoguous of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go way beyond the scope of this dissertation.
It is possible to extend the previous corollary to prove that for any functor $f : X \to A$ ($X$ and $A$ being $1$-categories), we have \[\hocolim^{\Th}_{a \in A} (X/a) \simeq X.\] However, to prove that it is also the case when $X$ is an $\oo$\nbd{}category and $f$ an $\oo$\nbd{}functor, as in Corollary \ref{cor:folkhmtpycol}, one would need to extend the Grothendieck construction to functors with value in $\oo\Cat$ and to prove an $\oo$\nbd{}categorical analogous of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go way beyond the scope of this dissertation.
\end{remark}
Putting all the pieces together, we are now able to prove the awaited Theorem.
\begin{theorem}
......@@ -615,7 +615,6 @@ Putting all the pieces together, we are now able to prove the awaited Theorem.
\end{itemize}
Thus, Proposition \ref{prop:criteriongoodcat} applies and this proves that $A$ is \good{}.
\end{proof}
\todo{Section supplémentaire qui parle de la généralisation aux ncat?}
%%% Local Variables:
%%% mode: latex
......
......@@ -94,7 +94,7 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equiv
\section{Abelianization}
We write $\Ab$ for category of abelian groups and for an abelian group $G$, we write $\vert G \vert$ for the underlying set of $G$.
\begin{paragr}
Let $C$ be an $\oo$-category. For every $n\geq 0$, we define $\lambda_n(C)$ as the abelian group obtained by quotienting out $\mathbb{Z}C_n$, the free abelian group on $C_n$, by the congruence generated by the relations
Let $C$ be an $\oo$-category. For every $n\geq 0$, we define $\lambda_n(C)$ as the abelian group obtained by quotienting $\mathbb{Z}C_n$, the free abelian group on $C_n$, by the congruence generated by the relations
\[
x \comp_k y \sim x+y
\]
......@@ -133,7 +133,7 @@ We write $\Ab$ for category of abelian groups and for an abelian group $G$, we w
which we call the \emph{abelianization functor}.
\end{paragr}
\begin{lemma}\label{lemma:adjlambda}
The functor $\lambda$ is a left ajdoint.
The functor $\lambda$ is a left adjoint.
\end{lemma}
\begin{proof}
The category of chain complexes is equivalent to the category $\omega\Cat(\Ab)$ of $\omega$-categories internal to abelian groups (see \cite[Theorem 3.3]{bourn1990another}) and with this identification, the functor $\lambda : \omega\Cat \to \omega\Cat(\Ab)$ is nothing but the left adjoint of the canonical forgetful functor $\omega\Cat(\Ab) \to \omega\Cat$.
......@@ -165,7 +165,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c
\Hom_{n\Cat}(C,B^nG) &\to \Hom_{\Set}(C_n,\vert G \vert)\\
F &\mapsto F_n,
\end{align*}
where $\vert G \vert$ is the underlying set of $G$, is injective and its image consists of those functiorns $f : C_n \to \vert G \vert$ such that:
where $\vert G \vert$ is the underlying set of $G$, is injective and its image consists of those functions $f : C_n \to \vert G \vert$ such that:
\begin{enumerate}[label=(\roman*)]
\item\label{cond:comp} for every $0 \leq k <n $ and every pair $(x,y)$ of $k$\nbd{}composable $n$\nbd{}cells of $C$, we have
\[
......@@ -188,7 +188,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c
\[
\Hom_{\Set}(C_n,\vert G \vert) \simeq \Hom_{\Ab}(\mathbb{Z}C_n,G),
\]
we have that $\Hom_{n\Cat}(C,B^nG)$ is naturally isomorphic to the set of morphisms of abelian groups $g : \mathbb{Z}C_n \to G$ such that for every pair $(x,y)$ of $k$\nbd{}composable elments of $C_n$ for some $k<n$, we have
we have that $\Hom_{n\Cat}(C,B^nG)$ is naturally isomorphic to the set of morphisms of abelian groups $g : \mathbb{Z}C_n \to G$ such that for every pair $(x,y)$ of $k$\nbd{}composable elements of $C_n$ for some $k<n$, we have
\[
g(x\comp_ky)=g(x)+g(y).
\]
......@@ -219,7 +219,7 @@ As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain c
%% \item[-] for all $x$ and $y$ in $G$ and $i<n$,
%% \[x \ast_i y := x +y.\]
%% \end{itemize}
%% It it straightforward to check that this defines an $n$-category. Note that the previous definition would still make sense with $G$ an abelian \emph{monoid}. Moreover, when $n=1$, we didn't even need it to be abelian, but for $n\geq 2$ this hypothesis is necessary because of the Eckmann-Hilton argument. For $n=0$, we only needed that $G$ was a set.
%% It is straightforward to check that this defines an $n$-category. Note that the previous definition would still make sense with $G$ an abelian \emph{monoid}. Moreover, when $n=1$, we didn't even need it to be abelian, but for $n\geq 2$ this hypothesis is necessary because of the Eckmann-Hilton argument. For $n=0$, we only needed that $G$ was a set.
%% This defines a functor
%% \[
......@@ -351,13 +351,13 @@ In the next Lemma, recall the definition of \emph{homotopy of chain complexes} (
is the total left derived functor of $\lambda : \oo\Cat \to \Ch$ (where $\oo\Cat$ is equipped with folk weak equivalences). For an $\oo$\nbd{}category $C$, $\sH^{\pol}(C)$ is the \emph{polygraphic homology of $C$}.
\end{definition}
\begin{paragr}
Similarly to singular homology groups, for $k\geq0$ the $k$-th polygraphic homology group of $C$ is defined as
Similarly to singular homology groups, for $k\geq0$ the $k$\nbd{}th polygraphic homology group of $C$ is defined as
\[
H^{\pol}_k(C):=H_k(\sH^{\pol}(C))
\]
where $H_k : \ho(\Ch) \to \Ab$ is the usual functor that associate to an object of $\ho(\Ch)$ its $k$-th homology group.
In practise, this means that one has to find a cofibrant replacement of $C$, that is to say a free $\oo$\nbd{}category $P$ and a folk trivial fibration
In practice, this means that one has to find a cofibrant replacement of $C$, that is to say a free $\oo$\nbd{}category $P$ and a folk trivial fibration
\[
P \to C,
\]
......@@ -430,7 +430,7 @@ From now on, when given an $\oo$\nbd{}functor $u$, we write $\sH^{\pol}(u)$ inst
\[
q : D' \to D
\]
be trivial fibrations for the canonical model structure with $C'$ and $D'$ cofibrant. Using that $q$ is a trival fibration and $C'$ is cofibrant, we know that there exists $u' : C' \to D'$ and $v' : C' \to D'$ such that the squares
be trivial fibrations for the canonical model structure with $C'$ and $D'$ cofibrant. Using that $q$ is a trivial fibration and $C'$ is cofibrant, we know that there exists $u' : C' \to D'$ and $v' : C' \to D'$ such that the squares
\[
\begin{tikzcd}
C' \ar[d,"p"] \ar[r,"u'"] & D' \ar[d,"q"] \\
......@@ -466,7 +466,7 @@ The following proposition is an immediate consequence of the previous lemma.
\[
\sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch).
\]
Moreover, we also have a universal $2$-morphism wich we again denote by $\alpha^{\pol}$:
Moreover, we also have a universal $2$-morphism which we again denote by $\alpha^{\pol}$:
\[
\begin{tikzcd}
\oo\Cat \ar[d,"\gamma^{\folk}"'] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
......@@ -575,7 +575,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins
\ar[from=2-1,to=1-2,"\alpha^{\sing}",shorten <= 1em, shorten >= 1em, Rightarrow]
\end{tikzcd}
\]
A throrough reading of the proofs of Proposition \ref{prop:gonzalezcritder} and Proposition \ref{prop:hmlgyderived} enables us to give the following descritption of $\alpha^{\sing}$. By post-composing the co-unit of the adjunction $c_{\oo} \dashv N_{\oo}$ with the abelianization functor, we obtain $2$-morphism
A thorough reading of the proofs of Proposition \ref{prop:gonzalezcritder} and Proposition \ref{prop:hmlgyderived} enables us to give the following description of $\alpha^{\sing}$. By post-composing the co-unit of the adjunction $c_{\oo} \dashv N_{\oo}$ with the abelianization functor, we obtain $2$-morphism
\[
\lambda c_{\oo} N_{\oo} \Rightarrow \lambda.
\]
......@@ -612,7 +612,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins
\]
As it happens, this is not possible as the following counter-example, due to Ara and Maltsiniotis, shows.
\end{paragr}
\begin{paragr}[Ara and Maltisiniotis' counter-example]\label{paragr:bubble}
\begin{paragr}[Ara and Maltsiniotis' counter-example]\label{paragr:bubble}
Write $\mathbb{N}=(\mathbb{N},+,0)$ for the commutative monoid of non-negative integers and let $C$ be the $2$\nbd{}category defined as
\[
C:=B^2\mathbb{N}
......@@ -802,7 +802,7 @@ The rest of this dissertation is devoted to the study of \good{} $\oo$\nbd{}cate
%% \[
%% \sH : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)
%% \]
%% are homotopy cocontinous (Definition \ref{def:cocontinuous}). In the first case, this follows from Theorem \ref{thm:cisinskiII} and the fact that $\lambda : \oo\Cat \to \Ch$ is left Quillen with respect to the folk model structure on $\oo\Cat$. In the second case, this follows from the fact that $\overline{N_{\oo}} : \Ho(\oo\Cat^{\Th} \to \Ho(\Ch)$ induces an equivalence of op-prederivators and that $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen. Besides, the construction of the canonical comparison map from Paragraph \ref{paragr:cmparisonmap} can be reproduced \emph{mutatis mutandis} in the $2$\nbd{}category of op-prederivators, yielding a $2$-morphism of op-prederivators
%% are homotopy cocontinuous (Definition \ref{def:cocontinuous}). In the first case, this follows from Theorem \ref{thm:cisinskiII} and the fact that $\lambda : \oo\Cat \to \Ch$ is left Quillen with respect to the folk model structure on $\oo\Cat$. In the second case, this follows from the fact that $\overline{N_{\oo}} : \Ho(\oo\Cat^{\Th} \to \Ho(\Ch)$ induces an equivalence of op-prederivators and that $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen. Besides, the construction of the canonical comparison map from Paragraph \ref{paragr:cmparisonmap} can be reproduced \emph{mutatis mutandis} in the $2$\nbd{}category of op-prederivators, yielding a $2$-morphism of op-prederivators
%% \[
%% \begin{tikzcd}
%% \Ho(\oo\Cat^{\folk}) \ar[r,"\J"] \ar[rd,"\sH^{\pol}"',""{name=A,above}] & \Ho(\oo\Cat^{\Th}) \ar[d,"\sH"] \\
......@@ -845,7 +845,7 @@ The rest of this dissertation is devoted to the study of \good{} $\oo$\nbd{}cate
\Ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"'] & \Ho(\Ch)\ar[from=2-1,to=A,"\pi",Rightarrow].
\end{tikzcd}
\]
In particular, by naturality, we have a commutative diagram in $\ho(\Ch)$: \todo{Être plus soigneux ?}
In particular, by naturality, we have a commutative diagram in $\ho(\Ch)$:
\[
\begin{tikzcd}
\displaystyle\hocolim_{i\in I}\sH^{\sing}(d_i) \ar[d] \ar[r] & \displaystyle\hocolim_{i \in I}\sH^{\pol}(d_i) \ar[d] \\
......@@ -899,7 +899,7 @@ The previous proposition admits the following corollary, which will be of great
\begin{proposition}\label{prop:fmsncat}
There exists a model structure on $n\Cat$ such that:
\begin{itemize}[label=-]
\item weak equivalences are exactely those morphisms $f : C \to D$ such that $\iota_n(f)$ is a weak equivalence for the folk model structure on $\oo\Cat$,
\item weak equivalences are exactly those morphisms $f : C \to D$ such that $\iota_n(f)$ is a weak equivalence for the folk model structure on $\oo\Cat$,
\item fibrations are exactly those morphisms $f : C \to D$ such that $\iota_n(f)$ is a fibrations for the folk model structure on $\oo\Cat$.
\end{itemize}
Moreover, there exists a set $I$ of generating cofibrations (resp.\ a set $J$ of generating trivial cofibrations) for the folk model structure on $\oo\Cat$ such that the image by $\tau^{i}_{\leq n}$ of $I$ (resp.\ $J$) is a set of generating cofibrations (resp.\ generating trivial cofibrations) of the above model structure on $n\Cat$.
......@@ -922,7 +922,7 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati
Since every $\oo$\nbd{}category is fibrant for the folk model structure on
$\oo\Cat$ \cite[Proposition 9]{lafont2010folk}, it suffices to show that
$\tau^{i}_{\leq n}$ sends folk trivial fibrations of $\oo\Cat$ to weak
equivalences of $n\Cat$ (in vertue of
equivalences of $n\Cat$ (in virtue of
Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model}).
% Let $C$ be an $\oo$-category and let $\eta_C : C \to \iota_n\tau^{i}_{\leq
% n}(C)$ be the unit morphism of the adjunction $\tau^{i}_{\leq n} \dashv
......@@ -986,7 +986,7 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati
$\eta_C(x')=x$ and $\eta_C(y')=y$ (this is always possible by definition
of $T(C)$). We have $\eta_{D}(f(x'))=f(x)=f(y)=\eta_{D}(f(y'))$. By
definition of the functor $\tau^{i}_{\leq n}$, this means that there
exists a zig-zag of $(n+1)$\nbd{}cells of $D$ from $f(x')$ to $f(y')$.
exists a zigzag of $(n+1)$\nbd{}cells of $D$ from $f(x')$ to $f(y')$.
More precisely, this means that there exists a sequence
\[
(z_0,\beta_1,z_1,\cdots,z_{p-1},\beta_p,z_p)
......@@ -995,7 +995,7 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati
and $z_p=f(y')$, and each $\beta_i$ is $(n+1)$\nbd{}cell of $C$ either
from $z_{i-1}$ to $z_i$ or from $z_{i}$ to $z_{i-1}$. Using the fact that
$f$ is a folk trivial fibration, it is easy to prove the existence of a
zig-zag from $x'$to $y'$, which implies in particular that $x=\eta_C(x')=\eta_C(y')=y$.
zigzag from $x'$to $y'$, which implies in particular that $x=\eta_C(x')=\eta_C(y')=y$.
\item[Case $k>n$ :] Since all $k$\nbd{}cells of $T(C)$ and $T(D)$ with $k>n$
are units, we trivially have $f(x)=f(y)$ and $x=y$.
\end{description}
......@@ -1082,8 +1082,7 @@ We now turn to truncations of chain complexes.
What is left to show then is that for every $k > 0$ and every object $X$ of $\Ch^{\leq n}$, the canonical inclusion map
\[
X \to X \oplus \tau^{i}_{\leq n}(D_k)
\]
is sent by $\iota_n$ to a weak equivalence of $\Ch$. This follows immediatly from the fact that homology groups commute with direct sums.
\]zigzag is sent by $\iota_n$ to a weak equivalence of $\Ch$. This follows immediately from the fact that homology groups commute with direct sums.
\end{proof}
\begin{paragr}
We refer to the model structure of the previous proposition as the \emph{projective model structure on $\Ch^{\leq n}$}.
......@@ -1269,7 +1268,7 @@ A useful consequence of Proposition \ref{prop:polhmlgytruncation} is the followi
\[
\LL \lambda_{\leq n}(\tau^{i}_{\leq n}(C)) \to \lambda_{\leq n}(\tau^{i}_{\leq n}(C)).
\]
From Lemma \ref{lemma:cofncatfolk}, we have that $\tau^{i}_{\leq n}(C)$ is cofibrant for the folk model structure on $n\Cat$, and the result follows immediatly from the fact that $\lambda_{\leq n}$ is left Quillen.
From Lemma \ref{lemma:cofncatfolk}, we have that $\tau^{i}_{\leq n}(C)$ is cofibrant for the folk model structure on $n\Cat$, and the result follows immediately from the fact that $\lambda_{\leq n}$ is left Quillen.
\end{proof}
\begin{paragr}\label{paragr:polhmlgylowdimension}
Since every $\oo$\nbd{}category trivially admits its set of $0$\nbd{}cells as a $0$\nbd{}base, it follows from the previous proposition that for every $\oo$\nbd{}category $C$ we have
......@@ -1313,7 +1312,7 @@ and for $n \in \mathbb{N}\cup \{\oo\}$ we write $c_n : \Psh{\Delta} \to n\Cat$ f
is commutative (up to an isomorphism).
\end{lemma}
\begin{proof}
Straighforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and the fact that the composition of left adjoints if the left adjoint of the composition.
Straightforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and the fact that the composition of left adjoints if the left adjoint of the composition.
\end{proof}
\begin{paragr}
In particular, it follows from the previous lemma that the co-unit of the adjunction $c_{\oo} \dashv N_{\oo}$ induces, for every $\oo$\nbd{}category $C$ and every $n \geq 0$, a canonical morphism of $n\Cat$
......@@ -1334,7 +1333,7 @@ Straighforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and th
\begin{equation}
\Hom_{\Cat}(c_1N_{\oo}(C),D) \simeq \Hom_{\Psh{\Delta}}(N_{\oo}(C),N_1(D)).
\end{equation}
Now let $\Delta_{\leq 2}$ be the full subcategory of $\Delta$ spanned by $[0]$, $[1]$ and $[2]$ and $i : \Delta_{\leq 2} \to \Delta$ the canonical inclusion. This inclusion induces by pre-composition a functor $i^* : \Psh{\Delta} \to \Psh{\Delta_{\leq 2}}$ which has a right-adjoint $i_* : \Psh{\Delta} \to \Psh{\Delta_{\leq 2}}$. Recall that the nerve of a (small) category is $2$-coskelettal (see for example \cite[Theorem 5.2]{street1987algebra}), which means that for every category $D$, the unit morphism $ N_1(D) \to i_* i^* (N_1(D))$ is an isomorphism of simplicial sets. In particular, we have
Now let $\Delta_{\leq 2}$ be the full subcategory of $\Delta$ spanned by $[0]$, $[1]$ and $[2]$ and $i : \Delta_{\leq 2} \to \Delta$ the canonical inclusion. This inclusion induces by pre-composition a functor $i^* : \Psh{\Delta} \to \Psh{\Delta_{\leq 2}}$ which has a right-adjoint $i_* : \Psh{\Delta} \to \Psh{\Delta_{\leq 2}}$. Recall that the nerve of a (small) category is $2$-coskeletal (see for example \cite[Theorem 5.2]{street1987algebra}), which means that for every category $D$, the unit morphism $ N_1(D) \to i_* i^* (N_1(D))$ is an isomorphism of simplicial sets. In particular, we have
\[
\Hom_{\oo\Cat}(N_{\oo}(C),N_1(D)) \simeq \Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))).
\]
......@@ -1436,7 +1435,7 @@ Finally, we obtain the result we were aiming for.
\[
H^{\sing}_k(C) \simeq H^{\pol}_k(C).
\]
The only missing part to adapt the proof of Proposition \ref{prop:comphmlgylowdimension} for these values of $k$ is the analoguous of Lemma \ref{lemma:truncationcounit}. But contrary to the case $k=1$, it is not generally true that the canonical morphism $c_k N_{\oo}(C) \to \tau^{i}_{\leq k}(C)$ is an isomorphism when $k \geq 2$. However, what we really need is that the image by $\lambda$ of this morphism be a quasi-isomorphism. In the case $k=2$, it seems that this canonical morphism admits an oplax $2$\nbd{}functor as an inverse up to oplax transformation which could be an hint towards the conjecture that $H^{\sing}_2(C) \simeq H^{\pol}_2(C)$ for every $\oo$\nbd{}category $C$.
The only missing part to adapt the proof of Proposition \ref{prop:comphmlgylowdimension} for these values of $k$ is the analogous of Lemma \ref{lemma:truncationcounit}. But contrary to the case $k=1$, it is not generally true that the canonical morphism $c_k N_{\oo}(C) \to \tau^{i}_{\leq k}(C)$ is an isomorphism when $k \geq 2$. However, what we really need is that the image by $\lambda$ of this morphism be a quasi-isomorphism. In the case $k=2$, it seems that this canonical morphism admits an oplax $2$\nbd{}functor as an inverse up to oplax transformation which could be an hint towards the conjecture that $H^{\sing}_2(C) \simeq H^{\pol}_2(C)$ for every $\oo$\nbd{}category $C$.
\end{paragr}
%% Slightly less trivial is the following lemma.
%% \begin{lemma}
......
......@@ -9,8 +9,8 @@
\[
\sigma^i : [n+1] \to [n]
\]
the only sujerctive non-decreasing map such that the pre-image of $i \in [n]$ contains exactly two elements.
the only surjective non-decreasing map such that the pre-image of $i \in [n]$ contains exactly two elements.
dualized
The category $\Psh{\Delta}$ of simplicial sets is the category of presheaves on $\Delta$. For a simplicial set $X$, we use the notations
\[
\begin{aligned}
......@@ -22,7 +22,7 @@
Elements of $X_n$ are referred to as \emph{$n$-simplices of $X$}, the maps $\partial_i$ are the \emph{face maps} and the maps $s_i$ are the \emph{degeneracy maps}.
\end{paragr}
\begin{paragr}\label{paragr:orientals}
We denote by $\Or : \Delta \to \omega\Cat $ the cosimplicial object introduced by Street in \cite{street1987algebra}. The $\omega$-category $\Or_n$ is called the \emph{$n$-oriental}. There are various ways to give a precise definition of the orientals, but each of them needs some machinery that we don't want to introduce here. Instead, we only recall some important facts on orientals that we shall need in the sequel and refer to the litterature on the subject (such as \cite{street1987algebra}, \cite{street1991parity,street1994parity}, \cite{steiner2004omega}, \cite{buckley2016orientals} or \cite[chapitre 7]{ara2016joint}) for details.
We denote by $\Or : \Delta \to \omega\Cat $ the cosimplicial object introduced by Street in \cite{street1987algebra}. The $\omega$-category $\Or_n$ is called the \emph{$n$-oriental}. There are various ways to give a precise definition of the orientals, but each of them needs some machinery that we don't want to introduce here. Instead, we only recall some important facts on orientals that we shall need in the sequel and refer to the literature on the subject (such as \cite{street1987algebra}, \cite{street1991parity,street1994parity}, \cite{steiner2004omega}, \cite{buckley2016orientals} or \cite[chapitre 7]{ara2016joint}) for details.
The two main points to retain are:
\begin{description}
......@@ -108,7 +108,6 @@
Let $X$ be a simplicial set. The $\oo$-category $c_{\oo}(X)$ is free and the set of generating $k$-cells of $c_{\oo}(X)$ is canonically isomorphic the to set of non-degenerate $k$-simplices of $X$.
\end{lemma}
\fi
\todo{Mettre lemme qui dit que la realisation oo-categorique d'un ensemble simplicial est libre ?}
\begin{paragr}
For $n=1$, the functor $N_1$ is the usual nerve of categories. Recall that for a (small) category $C$, an $m$-simplex $X$ of $N_1(C)$ is a sequence of composable arrows of $C$
\[
......@@ -192,7 +191,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
For every $1 \leq n \leq \oo$, the morphism \[{\overline{N}_n : \Ho(n\Cat^\Th) \to \Ho(\Psh{\Delta})}\] is an equivalence of op-prederivators.
\end{theorem}
\begin{proof}
In \cite{gagna2018strict}, Gagna proves that there exists a functor $Q : \Psh{\Delta} \to \Psh{\Delta}$, as well as a zig-zag of morphisms of functors
In \cite{gagna2018strict}, Gagna proves that there exists a functor $Q : \Psh{\Delta} \to \Psh{\Delta}$, as well as a zigzag of morphisms of functors
\[
N_{n}c_{n}Q \overset{\alpha}{\longleftarrow} Q \overset{\gamma}{\longrightarrow} \mathrm{id}_{\Psh{\Delta}}
\]
......@@ -218,10 +217,10 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
For every $1 \leq n \leq \oo$, the class $\W_n^{\Th}$ is saturated (\ref{paragr:loc}).
\end{corollary}
\begin{proof}
This follows immediatly from the fact that $\overline{N_n} : \ho(n\Cat^{\Th}) \to \ho(\Psh{\Delta})$ is an equivalence of categories and the fact that weak equivalences of simplicial sets are saturated (because they are the weak equivalences of a model structure).
This follows immediately from the fact that $\overline{N_n} : \ho(n\Cat^{\Th}) \to \ho(\Psh{\Delta})$ is an equivalence of categories and the fact that weak equivalences of simplicial sets are saturated (because they are the weak equivalences of a model structure).
\end{proof}
\begin{remark}
Corollaries \ref{cor:thomhmtpycocomplete} and \ref{cor:thomsaturated} would also follow from the existence of a model structure on $n\Cat$ with $\W^{\Th}_n$ as the weak equivalences. For $n=1$, this was established by Thomason \cite{thomason1980cat}, and for $n=2$ by Ara and Maltisiniotis \cite{ara2014vers}. For $n>3$, the existence of such a model structure is conjectured but not yet established.
Corollaries \ref{cor:thomhmtpycocomplete} and \ref{cor:thomsaturated} would also follow from the existence of a model structure on $n\Cat$ with $\W^{\Th}_n$ as the weak equivalences. For $n=1$, this was established by Thomason \cite{thomason1980cat}, and for $n=2$ by Ara and Maltsiniotis \cite{ara2014vers}. For $n>3$, the existence of such a model structure is conjectured but not yet established.
\end{remark}
By definition, for all $1 \leq n \leq m \leq \omega$, the canonical inclusion ${n\Cat \hookrightarrow m\Cat}$ sends Thomason equivalences of $n\Cat$ to Thomason equivalences of $m\Cat$. Hence, it induces a morphism of localizers and then a morphism of op-prederivator $\Ho(n\Cat^\Th) \to \Ho(m\Cat^{\Th})$.
\begin{proposition}
......@@ -291,7 +290,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\end{paragr}
\begin{paragr}\label{paragr:formulasoplax}[Formulas for oplax tranformations] We now give a third way of describing oplax transformations based on explicit formulas. The proof that this description is equivalent to those given in the previous paragraph can be found in \cite[Appendice B.2]{ara2016joint}.
\begin{paragr}\label{paragr:formulasoplax}[Formulas for oplax transformations] We now give a third way of describing oplax transformations based on explicit formulas. The proof that this description is equivalent to those given in the previous paragraph can be found in \cite[Appendice B.2]{ara2016joint}.
Let $u, v : X \to Y$ two $\oo$-functors. An oplax transformation $\alpha : u \Rightarrow v$ is given by the data of:
\begin{itemize}[label=-]
......@@ -525,7 +524,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
\item When $n=2$, $x \sim_{\omega} y$ means that $x$ and $y$ are equivalent, i.e.\ there exists $f : x \to y$ and $g : y \to x$ such that $fg$ is isomorphic to $1_y$ and $gf$ is isomorphic to $1_x$.
\end{itemize}
\end{example}
For later reference, we put here the following trivial but important lemma, whose proof is ommited.
For later reference, we put here the following trivial but important lemma, whose proof is omitted.
\begin{lemma}
Let $F : C \to D$ be an $\oo$\nbd{}functor and $x$,$y$ be $n$-cells of $C$ for some $n \geq 0$. If $x \sim_{\oo} y$, then $F(x) \sim_{\oo} F(y)$.
\end{lemma}
......@@ -595,8 +594,7 @@ For later reference, we put here the following trivial but important lemma, whos
\]
For the converse, see \cite{metayer2008cofibrant}.
\end{proof}
\todo{Dire que le structure folk est monoidale ?}
\iffalse
\iffalse
\begin{proposition}
Let $f : A \to B$ and $g : C \to D$ be morphisms of $\oo\Cat$. If $f$ and $g$ are cofibrations for the folk model structure, then so is
\[
......@@ -667,7 +665,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
\]
Let us now give an alternative definition of the $\oo$\nbd{}category $A/a_0$ using explicit formulas. The equivalence with the previous definition follows from the dual of \cite[Proposition B.5.2]{ara2016joint}
\begin{itemize}[label=-]
\item An $n$-cell of $A/a_0$ is a matrix \todo{le mot ``matrix'' est-il maladroit ?}
\item An $n$-cell of $A/a_0$ is a table
\[
(x,a)=\begin{pmatrix}
\begin{matrix}
......@@ -727,7 +725,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
\end{pmatrix}
\]
% It is understood that when $n=1$, the source is simply $(x_0,a_1)$ and the target $(x_0,a_1')$
\item The unit of the $n$-cell $(a,x)$ is given by the matrix:
\item The unit of the $n$-cell $(a,x)$ is given by the table:
\[
1_{(x,a)}=\begin{pmatrix}
\begin{matrix}
......@@ -737,7 +735,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
& (1_{x_n},1_{a_{n+1}})
\end{pmatrix}
\]
\item The composition of $n$-cells $(x,a)$ and $(y,b)$ such that $s_k(y,b)=t_k(a,x)$, is given by the matrix:
\item The composition of $n$-cells $(x,a)$ and $(y,b)$ such that $s_k(y,b)=t_k(a,x)$, is given by the table:
\[
(y,b)\comp_k (x,a)=\begin{pmatrix}
\begin{matrix}
......@@ -778,7 +776,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
\ar[from=1-1,to=2-2,phantom,description,very near start,"\lrcorner"]
\end{tikzcd}
\]
More explicitly, an $n$-cell $(x,b)$ of $A/b_0$ is a matrix
More explicitly, an $n$-cell $(x,b)$ of $A/b_0$ is a table
\[
(x,b)=\begin{pmatrix}
\begin{matrix}
......
\chapter{Homotopical algebra}
The present chapter sticks out from the others as it contains no original results. Its goal is simply to introduce the langage and tools of homotopical algebra that we shall need in the rest of the dissertation. In consequence, most of the results are simply asserted and the reader will find references to litterature for the proofs. The main notion of homotopical algebra we are aiming for is the one of \emph{homotopy colimits} and the language we chose to express this notion is the one given by the theory of Grothendieck's \emph{derivators} \cite{grothendieckderivators}. We do not assume that the reader is familiar with this theory and will quickly recall the basics. If needed, gentle introductions can be found in \cite{maltsiniotis2001introduction} and in a letter from Grothendieck to Thomason \cite{grothendieck1991letter}; more detailed introductions can be found in \cite{groth2013derivators} and in the first section of \cite{cisinski2003images}; finally, a rather complete (yet unfinished and unpublished) textbook on the subject is \cite{groth2013book}.
The present chapter sticks out from the others as it contains no original results. Its goal is simply to introduce the language and tools of homotopical algebra that we shall need in the rest of the dissertation. In consequence, most of the results are simply asserted and the reader will find references to literature for the proofs. The main notion of homotopical algebra we are aiming for is the one of \emph{homotopy colimits} and the language we chose to express this notion is the one given by the theory of Grothendieck's \emph{derivators} \cite{grothendieckderivators}. We do not assume that the reader is familiar with this theory and will quickly recall the basics. If needed, gentle introductions can be found in \cite{maltsiniotis2001introduction} and in a letter from Grothendieck to Thomason \cite{grothendieck1991letter}; more detailed introductions can be found in \cite{groth2013derivators} and in the first section of \cite{cisinski2003images}; finally, a rather complete (yet unfinished and unpublished) textbook on the subject is \cite{groth2013book}.
\todo{Expliquer le choix des dérivateurs ?} %This theory has the advantage of being much more elementary than the theory of \emph{weak $(\infty,1)$-category}
\iffalse Let us quickly motive this choice for the reader unfamiliar with this theory.
From an elementary point of view, a homotopy theory is given (or rather \emph{presented by}) by a category $\C$ and a class $\W$ of arrows of $\C$, which we traditionnaly refer to as \emph{weak equivalences}. The point of homotopy theory is to consider that the objects of $\C$ connected by a zig-zag of weak equivalences should be indistinguishable. From a category theorist perspective, a most natural One of the most basic invariant associated to such a data is the localisation of $\C$ with respect to $\W$. That is to say, the category $\ho^{\W}(\C)$ obtained from $\C$ by forcing the arrows of $\W$ to become isomorphisms. WhileThe ``problem'' is that the category $\ho^{\W}(\C)$ is poorly behaved. For example, \fi
From an elementary point of view, a homotopy theory is given (or rather \emph{presented by}) by a category $\C$ and a class $\W$ of arrows of $\C$, which we traditionally refer to as \emph{weak equivalences}. The point of homotopy theory is to consider that the objects of $\C$ connected by a zigzag of weak equivalences should be indistinguishable. From a category theorist perspective, a most natural One of the most basic invariant associated to such a data is the localisation of $\C$ with respect to $\W$. That is to say, the category $\ho^{\W}(\C)$ obtained from $\C$ by forcing the arrows of $\W$ to become isomorphisms. While the ``problem'' is that the category $\ho^{\W}(\C)$ is poorly behaved. For example, \fi
\section{Localization, derivation}
\begin{paragr}\label{paragr:loc}
......@@ -101,7 +101,7 @@ A \emph{morphism of localizers} $F : (\C,\W) \to (\C',\W')$ is a functor $F:\C\t
\end{example}
To end this section, we recall a derivability criterion due to Gonzalez, which we shall use in the sequel.
\begin{paragr}\label{paragr:prelimgonzalez}
Let $(\C,\W)$ and $(\C',\W')$ be two localizers and let $\begin{tikzcd} F : \C \ar[r,shift left] & \C' \ar[l,shift left] : G \end{tikzcd}$ be an adjunction whose unit is denoted by $\eta$. Suppose that $G$ is totally right derivable with $(\RR G,\beta)$ its total right derived functor and suppose that $\RR G$ has a left adjoint $F' : \ho(\C) \to \ho(\C')$; the co-unit of this last adjunction being denoted by $\epsilon'$. All this data induce a natural transformation $\alpha : F' \circ \gamma \Rightarrow \gamma' \circ F$ defined as the following compositon
Let $(\C,\W)$ and $(\C',\W')$ be two localizers and let $\begin{tikzcd} F : \C \ar[r,shift left] & \C' \ar[l,shift left] : G \end{tikzcd}$ be an adjunction whose unit is denoted by $\eta$. Suppose that $G$ is totally right derivable with $(\RR G,\beta)$ its total right derived functor and suppose that $\RR G$ has a left adjoint $F' : \ho(\C) \to \ho(\C')$; the co-unit of this last adjunction being denoted by $\epsilon'$. All this data induce a natural transformation $\alpha : F' \circ \gamma \Rightarrow \gamma' \circ F$ defined as the following composition
\[
\begin{tikzcd}
& \C' \ar[rr,"\gamma'"]\ar[rd,"G"] & &\ho(\C') \ar[rr,"\mathrm{id}",""{name=B,below}]\ar[rd,"\RR G"'] & &\ho(\C') \\
......@@ -124,7 +124,7 @@ To end this section, we recall a derivability criterion due to Gonzalez, which w
\section{(op)Derivators and homotopy colimits}
\begin{notation}We denote by $\CCat$ the $2$-category of small categories and $\CCAT$ the $2$-category of big categories. For a $2$-category $\underline{A}$, the $2$-category obtained from $\underline{A}$ by switching the source and targets of $1$-cells is denoted by $\underline{A}^{op}$.
The terminal category, i.e.\ the category with only one object and no non-trival arrows, is canonically denoted by $e$. For a (small) category $A$, the canonical morphism from $A$ to $e$ is denoted by
The terminal category, i.e.\ the category with only one object and no non-trivial arrows, is canonically denoted by $e$. For a (small) category $A$, the canonical morphism from $A$ to $e$ is denoted by
\[
p_A : A \to e.
\]
......@@ -178,7 +178,7 @@ To end this section, we recall a derivability criterion due to Gonzalez, which w
Note that some authors call \emph{prederivator} what we have called \emph{op-prederivator}. The terminology we chose in the above definition is compatible with the original one of Grothendieck, who called \emph{prederivator} a $2$-functor from $\CCat$ to $\CCAT$ that is contravariant at the level of $1$-cells \emph{and} at the level of $2$-cells.
\end{remark}
\begin{example}\label{ex:repder}
Let $\C$ be a category. For a small category $A$, we use the notation $\C(A)$ for the category $\underline{\Hom}(A,\C)$ of functors $A \to \C$ and natural transformations between them. The correspondance $A \mapsto \C(A)$ is $2$-functorial in an obvious sense and thus defines an op-prederivator
Let $\C$ be a category. For a small category $A$, we use the notation $\C(A)$ for the category $\underline{\Hom}(A,\C)$ of functors $A \to \C$ and natural transformations between them. The correspondence $A \mapsto \C(A)$ is $2$-functorial in an obvious sense and thus defines an op-prederivator
\begin{align*}
\C : \CCat^{op} &\to \CCAT \\
A &\mapsto \C(A)
......@@ -194,7 +194,7 @@ We now turn to the most important way of obtaining op-prederivators.
\begin{paragr}\label{paragr:homder}
Let $(\C,\W)$ be a localizer. For every small category $A$, there is a localizer $(\C(A),\W_A)$, where $\W_A$ the class of \emph{pointwise weak equivalences}, i.e. arrows $\alpha : d \to d'$ of $\C(A)$ such that $\alpha_a : d(a) \to d'(a)$ belongs to $\W$ for every $a \in \Ob(A)$.
The correspondance $A \mapsto (\C(A),\W_A)$ is $2$-functorial in that every $u : A \to B$ induces by pre-composition a morphism of localizers
The correspondence $A \mapsto (\C(A),\W_A)$ is $2$-functorial in that every $u : A \to B$ induces by pre-composition a morphism of localizers
\[
u^* : (\C(B),\W_B) \to (\C(A),\W_A)
\]
......@@ -255,7 +255,7 @@ We now turn to the most important way of obtaining op-prederivators.
\[
\hocolim_{a \in A}X(a).
\]
When $\C$ is also cocomplete (which we always be the case in practise), it follows from Remark \ref{rem:homotopicalisder} and Proposition \ref{prop:gonz} that the functor \[
When $\C$ is also cocomplete (which we always be the case in practice), it follows from Remark \ref{rem:homotopicalisder} and Proposition \ref{prop:gonz} that the functor \[
\colim_A : \C(A) \to \C
\]
is left derivable and $\hocolim_A$ is the left derived functor of $\colim_A$:
......@@ -356,7 +356,7 @@ We now turn to the most important way of obtaining op-prederivators.
\end{paragr}
\begin{example}
Let $\C$ be a category. The op-predericator represented by $\C$ always satisfy axioms \textbf{Der 1} and \textbf{Der 2}. We have already seen that axioms \textbf{Der 3d} means exactly that $\C$ admis left Kan extensions in the classical sense, in which case axiom \textbf{Der 4d} is automatically satisfied. Hence, the op-prederivator represented by $\C$ is a right op-prederivator if and only if $\C$ is cocomplete.
Let $\C$ be a category. The op-prederivator represented by $\C$ always satisfy axioms \textbf{Der 1} and \textbf{Der 2}. We have already seen that axioms \textbf{Der 3d} means exactly that $\C$ admits left Kan extensions in the classical sense, in which case axiom \textbf{Der 4d} is automatically satisfied. Hence, the op-prederivator represented by $\C$ is a right op-prederivator if and only if $\C$ is cocomplete.
\end{example}
\begin{remark}
Beware not to generalize the previous example too hastily. It is not true in general that axiom \textbf{Der 3d} imply \textbf{Der 4d}; even in the case of the homotopy op-prederivator of a localizer.
......@@ -366,7 +366,7 @@ We now turn to the most important way of obtaining op-prederivators.
A localizer $(\C,\W)$ is \emph{homotopy cocomplete} if the op-prederivator $\Ho(\C)$ is a right op-derivator.
\end{definition}
\begin{paragr}
Axioms \textbf{Der 3d} and \textbf{Der 4d} can be dualized to obtain axioms \textbf{Der 3g} and \textbf{Der 4g}, which informally say that the op-prederivator has right Kan extensions and that they are computed poinwise. An op-prederivator satisfying axioms \textbf{Der 1}, \textbf{Der 2}, \textbf{Der 3g} and \textbf{Der 4g} is a \emph{left op-derivator}. In fact, an op-prederivator $\sD$ is a left op-derivator if and only if the op-prederivator
Axioms \textbf{Der 3d} and \textbf{Der 4d} can be dualized to obtain axioms \textbf{Der 3g} and \textbf{Der 4g}, which informally say that the op-prederivator has right Kan extensions and that they are computed pointwise. An op-prederivator satisfying axioms \textbf{Der 1}, \textbf{Der 2}, \textbf{Der 3g} and \textbf{Der 4g} is a \emph{left op-derivator}. In fact, an op-prederivator $\sD$ is a left op-derivator if and only if the op-prederivator
\begin{align*}
\CCat &\to \CCAT \\
A &\mapsto (\sD(A^{\op}))^{\op}
......@@ -449,7 +449,7 @@ We now turn to the most important way of obtaining op-prederivators.
When $\sD$ and $\sD'$ are homotopy op-prederivators we will often say that a morphism $F : \sD \to \sD'$ is \emph{homotopy cocontinuous} instead of \emph{cocontinuous} to emphasize the fact that it preserves homotopy Kan extensions.
\end{remark}
\begin{example}
Let $F : \C \to \C'$ be a functor and suppose that $\C$ and $\C'$ are cocomplete. The morphism induced by $F$ at the level of represented op-ederivators is cocontinuous if and only if $F$ is cocontinuous in the usual sense.
Let $F : \C \to \C'$ be a functor and suppose that $\C$ and $\C'$ are cocomplete. The morphism induced by $F$ at the level of represented op-prederivators is cocontinuous if and only if $F$ is cocontinuous in the usual sense.
\end{example}
\begin{paragr}\label{paragr:prederequivadjun}
As in any $2$-category, the notions of equivalence and adjunction make sense in $\PPder$. Precisely, we have that:
......@@ -463,7 +463,7 @@ We now turn to the most important way of obtaining op-prederivators.
Let $F : \sD \to \sD'$ be a morphism of op-prederivators. If $F$ is an equivalence then $\sD$ is a right op-derivator (resp.\ left op-derivator, resp.\ op-derivator) if and only if $\sD'$ is one.
\end{lemma}
\begin{lemma}\label{lemma:eqisadj}
Let $F : \sD \to \sD'$ be an equivalence and $G : \sD' \to \sD$ be a quasi-iverse of $G$. Then, $F$ is left adjoint to $G$.
Let $F : \sD \to \sD'$ be an equivalence and $G : \sD' \to \sD$ be a quasi-inverse of $G$. Then, $F$ is left adjoint to $G$.
\end{lemma}
\begin{lemma}\label{lemma:ladjcocontinuous}
Let $\sD$ and $\sD'$ be op-prederivators that admit left Kan extensions and $F : \sD \to \sD'$ a morphism of op-prederivators. If $F$ is left adjoint (of a morphism $G : \sD' \to \sD$), then it is cocontinuous.
......@@ -543,7 +543,7 @@ We now turn to the most important way of obtaining op-prederivators.
\begin{example}
Let $\C$ be a category. An object of $\C(\square)$ is nothing but a commutative square in $\C$ and it is cocartesian in the sense of the previous definition if and only if it is cocartesian in the usual sense.
\end{example}
For the following definition to make sense, recall that for a localizer $(\C,\W)$ and a small category $A$, the objects of $\Ho(\C)(A)=\ho(\C(A))$ are identified with the objects of $