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\chapter{Homology of contractible \texorpdfstring{$\oo$}{ω}-categories its consequences}
\section{Contractible \texorpdfstring{$\oo$}{ω}-categories}
Recall that for any $\oo$\nbd{}category $C$, we write $p_C : C \to \sD_0$ the canonical morphism to the terminal object of $\sD_0$.
Recall that for every $\oo$\nbd{}category $C$, we write $p_C : C \to \sD_0$ the canonical morphism to the terminal object of $\sD_0$.
\begin{definition}\label{def:contractible}
An $\oo$\nbd{}category $C$ is \emph{oplax contractible} when the canonical morphism $p_C : C \to \sD_0$ is an oplax homotopy equivalence (Definition \ref{def:oplaxhmtpyequiv}).
......@@ -57,11 +57,11 @@ Recall that for any $\oo$\nbd{}category $C$, we write $p_C : C \to \sD_0$ the ca
Consider the commutative square
\[
\begin{tikzcd}
\sH^{\pol}(C) \ar[d,"\sH^{\pol}"] \ar[r,"\pi_C"] & \sH(C) \ar[d,"\sH^{\sing}(C)"] \\
\sH^{\pol}(\sD_0) \ar[r,"\pi_{\sS_0}"] & \sH(\sD_0).
\sH^{\pol}(C) \ar[d,"\sH^{\pol}(p_C)"] \ar[r,"\pi_C"] & \sH^{\sing}(C) \ar[d,"\sH^{\sing}(p_C)"] \\
\sH^{\pol}(\sD_0) \ar[r,"\pi_{\sS_0}"] & \sH^{\sing}(\sD_0).
\end{tikzcd}
\]
It follows respectively from Proposition \ref{prop:oplaxhmtpyisthom} and Proposition \ref{prop:oplaxhmtpypolhmlgy} that the right and left morphisms of the above square are isomorphisms. Then, a simple explicit computation shows that $\sD_0$ is \good{} and that $\sH^{\pol}(\sD_0)\simeq \sH^{\sing}(\sD_0)\simeq \mathbb{Z}$. By a 2-out-of-3 property, we deduce that $\pi_C : \sH^{\sing}(C)\to \sH^{\pol}(C)$ is an isomorphism.
It follows respectively from Proposition \ref{prop:oplaxhmtpyisthom} and Proposition \ref{prop:oplaxhmtpypolhmlgy} that the right and left morphisms of the above square are isomorphisms. Then, an immediate computation left to the reader shows that $\sD_0$ is \good{} and that $\sH^{\pol}(\sD_0)\simeq \sH^{\sing}(\sD_0)\simeq \mathbb{Z}$. By a 2-out-of-3 property, we deduce that $\pi_C : \sH^{\sing}(C)\to \sH^{\pol}(C)$ is an isomorphism.
\end{proof}
\begin{remark}
Definition \ref{def:contractible} admits an obvious ``lax'' variation and Proposition \ref{prop:contractibleisgood} is also true for lax contractible $\oo$\nbd{}categories.
......@@ -80,7 +80,7 @@ We end this section with an important result on slices $\oo$\nbd{}category (Para
\begin{proof}
Recall that we write $e_n$ for the unique non-trivial $n$\nbd{}cell of $\sD_n$ and that by definition $\sD_n$ has exactly two non-trivial $k$\nbd{}cells for every $k$ such that $0\leq k<n$. These two $k$\nbd{}cells are parallel and are given by $\src_k(e_n)$ and $\trgt_k(e_n)$.
Let $r : \sD_0 \to \sD_n$ be the $\oo$\nbd{}functor that points to $\trgt_0(e_n)$ (which means that $r=\langle \trgt_0(e_n) \rangle$ with the notations of \ref{paragr:defglobe}). Hence for every $k$\nbd{}cell $x$ of $\sD_n$, we have
Let $r : \sD_0 \to \sD_n$ be the $\oo$\nbd{}functor that points to $\trgt_0(e_n)$ (which means that $r=\langle \trgt_0(e_n) \rangle$ with the notations of \ref{paragr:defglobe}). For every $k$\nbd{}cell $x$ of $\sD_n$, we have
\[
r(p(x))=\1^k_{\trgt_0(e_n)},
\]
......@@ -135,11 +135,11 @@ In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}. Recall from \re
\sD_n \ar[r,"j_n^-"'] & \sS_{n}.
\end{tikzcd}
\]
is cartesian and all of the four morphisms are monomorphisms. Since the
is cartesian all of the four morphisms are monomorphisms. Since the
functor $\Hom_{\oo\Cat}(\Or_k,-)$ preserves limits, the square
\eqref{squarenervesphere} is a cartesian square of $\Set$ and all of whose four morphisms are
monomorphisms.
Hence, what we need to show is that for every $k \geq 0$ and
Hence, what we need to show is that for every $k \geq 0$ and every
$\oo$\nbd{}functor $\varphi : \Or_k \to \sS_{n}$, there exists an
$\oo$\nbd{}functor $\varphi' : \Or_k \to \sD_n$ such that either $j_n^+ \circ \varphi ' = \varphi$ or $j_n^- \circ \varphi' = \varphi$.
......@@ -206,7 +206,7 @@ is a homotopy cocartesian square of simplicial sets. Since $N_{\oo}$ induces an
\]
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
square is cocartesian and since $\sS_1$, $\sD_0$ and $\sD_2$ are free and
$i_2$ is a cofibration for the canonical model structure, the square is also
homotopy cocartesian with respect to folk weak equivalences. If $\J$ was
homotopy cocontinuous, then this square would also be homotopy cocartesian
......@@ -214,8 +214,7 @@ is a homotopy cocartesian square of simplicial sets. Since $N_{\oo}$ induces an
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 practice and we will only use it for theoretical purposes.
already know that they are free.
\end{paragr}
\begin{proposition}
Let $C$ be a free $\oo$\nbd{}category and for every $k \in \mathbb{N}$ let
......@@ -247,7 +246,7 @@ $(1-)$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 with respect to Thomason equivalences.
with respect to the folk weak equivalences and with respect to the 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
......@@ -275,7 +274,7 @@ higher than $1$.
the canonical forgetful functor.
Recall also that given an $\oo$\nbd{}functor $f : X \to A$ and an object $a_0$
of $A$, we defined the $\oo$\nbd{}category $A/a_0$ and the $\oo$\nbd{}functor
of $A$, we have defined the $\oo$\nbd{}category $A/a_0$ and the $\oo$\nbd{}functor
\[
f/a_0 : X/a_0 \to A//a_0
\]
......@@ -300,7 +299,7 @@ higher than $1$.
\]
\end{paragr}
\begin{paragr}\label{paragr:unfolding}
Let $f : X \to A$ be an $\oo$\nbd{}functor with $A$ a $1$-category. Any arrow $\beta : a_0 \to a_0'$ induces an $\oo$\nbd{}functor
Let $f : X \to A$ be an $\oo$\nbd{}functor with $A$ a $1$-category. Every arrow $\beta : a_0 \to a_0'$ induces an $\oo$\nbd{}functor
\begin{align*}
X/\beta : X/a_0 &\to X/{a_0'} \\
(x,p) & \mapsto (x,\beta \circ p),
......@@ -317,7 +316,7 @@ higher than $1$.
X/{-} : A &\to \oo\Cat\\
a_0 &\mapsto X/a_0
\end{align*}
and a canonical map
and a canonical $\oo$\nbd{}functor
\[
\colim_{a_0 \in A} (X/{a_0}) \to X.
\]
......@@ -367,7 +366,7 @@ higher than $1$.
\[
(x,1_{f(\trgt_0(x))})
\]
is a $n$\nbd{}arrow of $X/f(\trgt_0(x))$. We leave it to the reader to prove that the formula
is a $n$\nbd{}arrow of $X/f(\trgt_0(x))$. We leave it to the reader to check that the formula
\begin{align*}
\phi : X &\to C \\
x &\mapsto g_{f(\trgt_0(x))}(x,1_{f(\trgt_0(x))}).
......@@ -416,11 +415,11 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd{}func
It is immediate to check that for every object $a_0$ of $A$, the canonical
forgetful functor $\pi_{a_0} : A/a_0 \to A$ is a Conduché functor (see Section
\ref{section:conduche}). Hence, from Lemma \ref{lemma:pullbackconduche} we
know that $X/a_0 \to X$ is a discrete $\oo$\nbd{}functor. The result follows
know that $X/a_0 \to X$ is a discrete Conduché $\oo$\nbd{}functor. The result follows
then from Theorem \ref{thm:conduche}.
\end{proof}
\begin{paragr}
When $X$ is free, any arrow $\beta : a_0 \to a'_0$ of $A$ induces a map
When $X$ is free, every arrow $\beta : a_0 \to a'_0$ of $A$ induces a map
\begin{align*}
\Sigma^{X/a_0}_n &\to \Sigma^{X/a'_0}_n \\
(x,p) &\mapsto (x,\beta\circ p).
......@@ -438,7 +437,7 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd{}func
\]
\end{lemma}
\begin{proof}
For every object $a_0$ of $A$ and every $x \in \Sigma_n^X$, there is a canonical map
For every object $a_0$ of $A$ and every $x \in \Sigma_n^X$, we have a canonical map
\begin{align*}
\Hom_A\left(f(\trgt_0(x)),a_0\right) &\to \Sigma^{X/a_0}_n \\
p &\mapsto (x,p).
......@@ -450,7 +449,7 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd{}func
which is natural in $a_0$. A simple verification shows that it is a bijection.
\end{proof}
\begin{proposition}\label{prop:sliceiscofibrant}
Let $A$ a $1$\nbd{}category, $X$ a free $\oo$\nbd{}category and $f : X \to A$ be an $\oo$\nbd{}functor. The functor
Let $A$ be a $1$\nbd{}category, $X$ be a free $\oo$\nbd{}category and $f : X \to A$ be an $\oo$\nbd{}functor. The functor
\begin{align*}
A &\to \oo\Cat \\
a_0 &\mapsto X/a_0
......@@ -476,12 +475,12 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd{}func
$a_0$ in an obvious sense, which means that we have a cocartesian square in $\oo\Cat(A)$:
\[
\begin{tikzcd}
\displaystyle\coprod_{x \in \Sigma^X_n}\coprod_{\Hom_A(f(\trgt_0(x)),-)}\sS_{n-1} \ar[r] \ar[d] & \sk_{n-1}(X/-) \ar[d]\\
\displaystyle\coprod_{x \in \Sigma^X_n}\coprod_{\Hom_A(f(\trgt_0(x)),-)}\sD_n \ar[r]& \sk_n{(X/-)}.
\displaystyle\coprod_{x \in \Sigma^X_n}\sS_{n-1}\otimes f(\trgt_0(x)) \ar[r] \ar[d] & \sk_{n-1}(X/-) \ar[d]\\
\displaystyle\coprod_{x \in \Sigma^X_n}\sD_n\otimes f(\trgt_0(x)) \ar[r]& \sk_n{(X/-)}
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\]
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
(see \ref{paragr:cofprojms} for notations). From the second part of Proposition \ref{prop:modprs}, we deduce that for every $n\geq 0$, \[\sk_{n-1}(X/-) \to \sk_{n}(X/-)\] 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,
......@@ -489,7 +488,7 @@ Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd{}func
which is canonically isomorphic to $\emptyset \to X/-$ (see Lemma \ref{lemma:filtration}), is also a cofibration for the projective model structure.
\end{proof}
\begin{corollary}\label{cor:folkhmtpycol}
Let $A$ be a $1$\nbd{}category and $f : X \to A$ an $\oo$\nbd{}functor. The canonical arrow of $\ho(\oo\Cat^{\folk})$
Let $A$ be a $1$\nbd{}category and $f : X \to A$ be an $\oo$\nbd{}functor. The canonical arrow of $\ho(\oo\Cat^{\folk})$
\[
\hocolim^{\folk}_{a_0 \in A}(X/a_0) \to X,
\]
......@@ -531,7 +530,7 @@ We now move on to the next step needed to prove that every $1$-category is \good
k : F(f)(x) \to x'.
\]
\end{itemize}
The identity arrow on $(a,x)$ is the pair $(1_a,1_x)$ and the composition of $(f,k) : (a,x) \to (a',x')$ and $(f',k') : (a',x') \to (a'',x'')$ is given by:
The unit on $(a,x)$ is the pair $(1_a,1_x)$ and the composition of $(f,k) : (a,x) \to (a',x')$ and $(f',k') : (a',x') \to (a'',x'')$ is given by:
\[
(f',k')\circ(f,k)=(f'\circ f,k'\circ F(f')(k)).
\]
......@@ -596,7 +595,7 @@ 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 analogous 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 every 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 analogue 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}\label{thm:categoriesaregood}
......
......@@ -353,7 +353,7 @@ For the definition of \emph{homotopy of chain complexes} see for example \cite[D
\]
is a monomorphism with projective cokernel. Hence $\lambda$ sends folk cofibrations to cofibrations of chain complexes.
Then, we know from \cite[Sections 4.6 and 4.7]{lafont2010folk} and \cite[Remarque B.1.16]{ara2016joint} (see also \cite[Paragraph 3.11]{ara2019folk}) that there exists a set of generating trivial cofibrations $J$ of the folk model structure on $\omega\Cat$ such that any $j : X \to Y$ in $J$ is a deformation retract (see Paragraph \ref{paragr:defrtract}).
Then, we know from \cite[Sections 4.6 and 4.7]{lafont2010folk} and \cite[Remarque B.1.16]{ara2016joint} (see also \cite[Paragraph 3.11]{ara2019folk}) that there exists a set of generating trivial cofibrations $J$ of the folk model structure on $\omega\Cat$ such that every $j : X \to Y$ in $J$ is a deformation retract (see Paragraph \ref{paragr:defrtract}).
From Lemma \ref{lemma:abeloplax}, we conclude that $\lambda$ sends folk trivial cofibrations to trivial cofibrations of chain complexes.
\end{proof}
In particular, $\lambda$ is totally left derivable (when $\oo\Cat$ is equipped with folk weak equivalences). This motivates the following definition.
......@@ -398,7 +398,7 @@ For the definition of \emph{homotopy of chain complexes} see for example \cite[D
\]
be commutative squares in $\omega\Cat$ for $\epsilon\in\{0,1\}$.
If $C'$ is a free $\omega$-category and $v$ folk trivial fibration, then for any oplax transformation \[\alpha : f_0 \Rightarrow f_1,\] there exists an oplax transformation \[\alpha' : f_0' \Rightarrow f_1'\] such that
If $C'$ is a free $\omega$-category and $v$ folk trivial fibration, then for every oplax transformation \[\alpha : f_0 \Rightarrow f_1,\] there exists an oplax transformation \[\alpha' : f_0' \Rightarrow f_1'\] such that
\[
v \star \alpha' = \alpha \star u.
\]
......@@ -469,7 +469,7 @@ The following proposition is an immediate consequence of the previous lemma.
is an isomorphism.
\end{proposition}
\begin{paragr}\label{paragr:polhmlgythomeq}
Oplax homotopy equivalences being particular cases of Thomason equivalences, one may wonder whether it is true that \emph{any} Thomason equivalence induce an isomorphism in polygraphic homology. As we shall see later (Proposition \ref{prop:polhmlgynotinvariant}), it is not the case.
Oplax homotopy equivalences being particular cases of Thomason equivalences, one may wonder whether it is true that \emph{every} Thomason equivalence induce an isomorphism in polygraphic homology. As we shall see later (Proposition \ref{prop:polhmlgynotinvariant}), it is not the case.
\end{paragr}
\begin{remark}
Proposition \ref{prop:oplaxhmtpypolhmlgy} is also true if we replace ``oplax'' by ``lax''.
......@@ -522,7 +522,7 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins
\begin{proof}
All the functors involved are cocontinuous, hence it suffices to prove that the triangle is commutative when pre-composed by the Yoneda embedding $\Delta \to \Psh{\Delta}$. This follows immediately from the description of the orientals in \cite{steiner2004omega}.
\end{proof}
Recall now that the notion of adjunction and equivalence is valid in any $2$-category and in particular in the $2$\nbd{}category of pre-derivators (see \ref{paragr:prederequivadjun}). We omit the proof of the following lemma, which is the same as when the ambient $2$-category is the $2$-category of categories.
Recall now that the notion of adjunction and equivalence is valid in every $2$-category and in particular in the $2$\nbd{}category of pre-derivators (see \ref{paragr:prederequivadjun}). We omit the proof of the following lemma, which is the same as when the ambient $2$-category is the $2$-category of categories.
\begin{lemma}\label{lemma:adjeq}
Let $\begin{tikzcd} f : y \ar[r,shift left]&z :g\ar[l,shift left] \end{tikzcd}$ be an adjunction and $h : x \to y$ an equivalence with quasi-inverse $k : y \to x$. Then $fh$ is left adjoint to $kg$.
\end{lemma}
......@@ -904,7 +904,7 @@ The previous proposition admits the following corollary, which will be of great
Then, the $\oo$\nbd{}category $D$ is \good{}.
\end{corollary}
\begin{proof}
The fact that $A$,$B$ and $C$ are free and one of the morphism $u$ or $f$ is a folk cofibration ensure that the square is folk homotopy cocartesian. The conclusion follows then from Proposition \ref{prop:criteriongoodcat}.
The fact that $A$,$B$ and $C$ are free and one of the morphism $u$ or $f$ is a folk cofibration ensure that the square is folk homotopy cocartesian (Lemma \ref{lemma:hmtpycocartesianreedy}). The conclusion follows then from Proposition \ref{prop:criteriongoodcat}.
\end{proof}
\section{Equivalence of homologies in low dimension}
\begin{paragr}
......@@ -987,7 +987,7 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati
Let us first treat the case of $0$\nbd{}cells. Let $y$ be a $0$\nbd{}cell of $T(D)$. The map
$\eta_D$ being surjective on $0$\nbd{}cells (even if $n=0$), there exists
$y'$ such that $\eta_D(y')=y$. Since $f$ is a folk trivial fibration, there
exists $x'$, a $0$\nbd{}cell of $C$, such that $f(x')=y'$ and then if we set $x:=\eta_C(x')$,
exists a $0$\nbd{}cell $x'$ of $C$ such that $f(x')=y'$ and then if we set $x:=\eta_C(x')$,
we have $T(f)(x)=y$.
Now let $x,y$ be parallel $k$\nbd{}cells of $T(C)$ and suppose given a
......@@ -1001,7 +1001,7 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati
\item[Case $k=n-1$ :] By definition of $T(D)$, there exists an $n$\nbd{}cell
of $D$, $\beta ' : f(x) \to f(y)$, such that
$\eta_{D}(\beta')=\beta$. Since $f$ is a folk trivial fibration, there
exists a $n$\nbd{}cell of $C$, $\alpha' : x \to y$ such that
exists a $n$\nbd{}cell $\alpha' : x \to y$ of $C$ such that
$f(\alpha')=\beta'$. If we set $\alpha:=\eta_{C}(\alpha')$, we have
$T(f)(\alpha)=\beta$.
\item[Case $k=n$ :] Since all $l$\nbd{}cells of $T(C)$ and $T(D)$ with $l>n$
......@@ -1106,7 +1106,7 @@ 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
\[
X \to X \oplus \tau^{i}_{\leq n}(D_k)
\]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.
\]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 $\Ch^{\leq n}$}.
......@@ -1123,7 +1123,7 @@ We now turn to truncations of chain complexes.
for every $0 \leq k \leq n$.
\end{lemma}
\begin{proof}
For $0 \leq k < n-1$, this is trivial. For $k = n-1$, this follows easily from the fact that the image of $\partial : K_k/{\partial(K_{k+1})}\to K_{k-1}$ is equal to the image of $\partial : K_k \to K_{k+1}$. Finally for $k = n$, it is straightforward to check that
For $0 \leq k < n-1$, this is trivial. For $k = n-1$, this follows easily from the fact that the image of $\partial : K_k/{\partial(K_{k+1})}\to K_{k-1}$ is equal to the image of $\partial : K_k \to K_{k-1}$. Finally for $k = n$, it is straightforward to check that
\[
H_n(K)=\frac{\mathrm{Ker}(\partial : K_n \to K_{n-1})}{\mathrm{Im}(\partial : K_{n+1} \to K_n)}
\]
......@@ -1145,7 +1145,7 @@ As a consequence of this lemma, we have the analogous of Proposition \ref{prop:t
\iota_n\tau^{i}_{\leq n}(K) \ar[r,"\iota_n\tau^{i}_{\leq n}(f)"] & \iota_n\tau^{i}_{\leq n}(K'),
\end{tikzcd}
\]
where $\eta$ is the unit map of the adjunction $\tau^{i}_{\leq n} \dashv \iota_n$. Now it follows from Lemma \ref{lemma:unitajdcomp} that
where $\eta$ is the unit map of the adjunction $\tau^{i}_{\leq n} \dashv \iota_n$. It follows from Lemma \ref{lemma:unitajdcomp} that
\[
H_k(\iota_n\tau^{i}_{\leq n}(f)) : H_k(\iota_n\tau^{i}_{\leq n}(K)) \to \iota_n\tau^{i}_{\leq n}(K'))
\]
......@@ -1157,7 +1157,7 @@ We now investigate the relation between truncation and abelianization.
\[
\lambda(\iota_n(C))_k=0
\]
for every $k > n$ and hence $\lambda(\iota_n(C))$ can be seen as an object of $\Ch^{\leq n}$. Thus, we can define a functor $\lambda_{\leq n } : n\Cat \to \Ch^{\leq n}$ as
for every $k > n$. Hence, $\lambda(\iota_n(C))$ can be seen as an object of $\Ch^{\leq n}$ and we can define a functor $\lambda_{\leq n } : n\Cat \to \Ch^{\leq n}$ as
\begin{align*}
\lambda_{\leq n} : n\Cat &\to \Ch^{\leq n}\\
C&\mapsto \lambda(\iota_n(C)).
......@@ -1211,7 +1211,7 @@ is commutative.
\[
\beta_n : \tau^{i}_{\leq n}(\lambda(C))_n \to \lambda_{\leq n}(\tau^{i}_{\leq n}(C))_n
\]
for every $\oo$\nbd{}category $C$.
is an isomorphism for every $\oo$\nbd{}category $C$.
Recall from Lemma \ref{lemma:adjlambdasusp} that $\lambda_n \circ \iota_n : n\Cat \to \Ab$ (which we abusively wrote as $\lambda_n$) is left adjoint to the functor $B^n : \Ab \to n\Cat$. In particular, for every $\oo$\nbd{}category $C$ and every abelian group $G$, we have
\[
......@@ -1243,7 +1243,7 @@ In the following lemma, $n\Cat$ is equipped with the folk model structure and $\
The functor $\lambda_{\leq n} : n\Cat \to \Ch^{\leq n}$ is left Quillen.
\end{lemma}
\begin{proof}
Let $I$ and $J$ respectively be sets of generating cofibrations and generating trivial cofibrations of the folk model structure on $\oo\Cat$ as in the second part of such that $\tau^{i}_{\leq n}(I)$ and $\tau^{i}_{\leq n}(J)$ respectively are sets of generating cofibrations and generating trivial cofibrations of the projective model structure on $\Ch^{\leq n}$ (which we know exist by the second part of Proposition \ref{prop:fmsncat}). What we have to show is that for every $f$ in $I$ (resp. $J$), $\lambda_{\leq n}(\tau^{i}_{\leq n}(f))$ is a cofibration (resp.\ generating cofibration) for the folk model structure on $n\Cat$.
Let $I$ and $J$ respectively be sets of generating cofibrations and generating trivial cofibrations of the folk model structure on $\oo\Cat$ such that $\tau^{i}_{\leq n}(I)$ and $\tau^{i}_{\leq n}(J)$ respectively are sets of generating cofibrations and generating trivial cofibrations of the projective model structure on $\Ch^{\leq n}$ (which we know exist by the second part of Proposition \ref{prop:fmsncat}). What we have to show is that for every $f$ in $I$ (resp. $J$), $\lambda_{\leq n}(\tau^{i}_{\leq n}(f))$ is a cofibration (resp.\ generating cofibration) for the folk model structure on $n\Cat$.
% Notice that we have $\tau^{i}_{\leq n } \circ \iota_n = \mathrm{id}_{n \Cat}$ and hence
From Lemma \ref{lemma:abelianizationtruncation}, we have
\[
......@@ -1320,7 +1320,7 @@ A useful consequence of Proposition \ref{prop:polhmlgytruncation} is the followi
%% \begin{proof}
%% Let $f : P \to C$ be a cofibrant replacement for $C$. \todo{À finir}.
%% \end{proof}
We now turn to the relation between truncation and singular homology of $\oo$\nbd{}categories. Recall that for any $n \geq 0$, the nerve functor $N_n : n\Cat \to \Psh{\Delta}$ is defined as the following composition
We now turn to the relation between truncation and singular homology of $\oo$\nbd{}categories. Recall that for every $n \geq 0$, the nerve functor $N_n : n\Cat \to \Psh{\Delta}$ is defined as the following composition
\[
N_n : n\Cat \overset{\iota_n}{\longrightarrow} \oo\Cat \overset{N_{\oo}}{\longrightarrow} \Psh{\Delta},
\]
......@@ -1353,7 +1353,7 @@ Straightforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and t
is an isomorphism.
\end{lemma}
\begin{proof}
Let $C$ be an $\oo$\nbd{}category and $D$ a (small) category. By adjunction, we have
Let $C$ be an $\oo$\nbd{}category and $D$ be a (small) category. By adjunction, we have
\begin{equation}
\Hom_{\Cat}(c_1N_{\oo}(C),D) \simeq \Hom_{\Psh{\Delta}}(N_{\oo}(C),N_1(D)).
\end{equation}
......@@ -1361,10 +1361,11 @@ Straightforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and t
\[
\Hom_{\oo\Cat}(N_{\oo}(C),N_1(D)) \simeq \Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))).
\]
Using the description of $\Or_0$, $\Or_1$ and $\Or_2$ from \ref{paragr:orientals}, we deduce that a morphism $F : i^*(N_{\oo}(C)) \to i^*(N_1(D))$ in $\Psh{\Delta_{\leq 2}}$ consists of a function $F_0 : C_0 \to D_0$ and a function $F_1 : C_1 \to D_1$ such that
Using the description of $\Or_0$, $\Or_1$ and $\Or_2$ from \ref{paragr:orientals}, we deduce that a morphism $F : i^*(N_{\oo}(C)) \to i^*(N_1(D))$ of $\Psh{\Delta_{\leq 2}}$ consists of a function $F_0 : C_0 \to D_0$ and a function $F_1 : C_1 \to D_1$ such that
\begin{itemize}[label=-]
\item for every $x \in C_0$, we have $F_1(1_x)=1_{F_0(x)}$,
\item for every $x \in C_1$, we have $\src(F_1(x))=F_0(\src(x)))$ and $\trgt(F_1(x))=F_0(\trgt(x)))$,
\item for every $x \in C_1$, we have
\[\src(F_1(x))=F_0(\src(x))) \text{ and }\trgt(F_1(x))=F_0(\trgt(x))),\]
\item for every $2$\nbd{}triangle
\[
\begin{tikzcd}
......@@ -1459,7 +1460,7 @@ 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 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 $C$.
The only missing part to adapt the proof of Proposition \ref{prop:comphmlgylowdimension} for these values of $k$ is the analogue 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}
......
......@@ -38,7 +38,7 @@ is poorly behaved. For example, \fi
\gamma : \C \to \ho(\C)
\]
the localization functor \cite[1.1]{gabriel1967calculus}. Recall the universal
property of the localization: for any category $\D$, the functor induced by
property of the localization: for every category $\D$, the functor induced by
pre-composition
\[
\gamma^* : \underline{\Hom}(\ho(\C),\D) \to \underline{\Hom}(\C,\D)
......@@ -343,7 +343,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\end{definition}
\begin{example}
Let $\C$ be a category. The op\nbd{}prederivator represented by $\C$ has left Kan
extensions if and only if the category $\C$ has left Kan extensions along any
extensions if and only if the category $\C$ has left Kan extensions along every
morphism $u : A \to B$ of $\Cat$ in the usual sense. By a standard categorical
argument, this means that the op\nbd{}prederivator represented by $\C$ has left Kan
extensions if and only if $\C$ is cocomplete. Note that for every small category
......@@ -572,7 +572,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
Let $F : \C \to \D$ be a functor. It induces a strict morphism at
the level of op\nbd{}prederivators, again denoted by $F$, where
for every small category $A$, the functor $F_A : \C(A) \to \C'(A)$
is induced by post-composition. Similarly, any natural
is induced by post-composition. Similarly, every natural
transformation induces a $2$\nbd{}morphism at the level of
represented op\nbd{}prederivators.
\end{example}
......@@ -618,7 +618,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\end{tikzcd}
\]
For example, when $\sD$ is the homotopy op\nbd{}prederivator of a
localizer and $B$ is the terminal category $e$, for any $X$ object
localizer and $B$ is the terminal category $e$, for every $X$ object
of $\sD(A)$ the previous canonical morphism reads
\[
\hocolim_{A}(F_A(X))\to F_e(\hocolim_A(X)).
......@@ -771,7 +771,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\end{paragr}
\begin{definition}\label{def:cocartesiansquare}
Let $\sD$ be an op\nbd{}prederivator. An object $X$ of $\sD(\square)$ is
\emph{cocartesian} if for any $Y$ object of $\sD(\square)$, the canonical
\emph{cocartesian} if for every $Y$ object of $\sD(\square)$, the canonical
map
\[
\Hom_{\sD(\square)}(X,Y) \to
......@@ -972,9 +972,9 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
homotopy left Kan extensions (and in particular homotopy colimits). However,
in practice all the model categories that we shall encounter are
\emph{cofibrantly generated}, in which case the theory is much simpler because $\M(A)$ does admit a model structure with the pointwise weak equivalences as its weak equivalences.
\begin{paragr}
Let $\C$ be a category with coproducts and $A$ a small category. For any
object $X$ of $\C$ and any object $a$ of $A$, we define $X\otimes a$ as the
\begin{paragr}\label{paragr:cofprojms}
Let $\C$ be a category with coproducts and $A$ a small category. For every
object $X$ of $\C$ and every object $a$ of $A$, we define $X\otimes a$ as the
functor
\[
\begin{aligned}
......@@ -1023,7 +1023,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\end{paragr}
\begin{proposition}
Let $(\M,\W,\Cof,\Fib)$ be a cofibrantely generated model
category. For any $u : A \to B$, the adjunction
category. For every $u : A \to B$, the adjunction
\[
\begin{tikzcd}
u_! : \M(A) \ar[r,shift left]& \ar[l,shift left] \M(B) : u^*
......@@ -1097,6 +1097,7 @@ We now turn to the most important way of obtaining op\nbd{}prederivators.
\begin{tikzcd}
A \ar[r,"u"] \ar[d,"f"]& B \ar[d,"g"] \\
C \ar[r,"v"]&D
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\]
be a \emph{cocartesian} square in $\M$. If either $u$ or $f$ is a
......
......@@ -25,7 +25,7 @@ for the category of (strict) $\oo$\nbd{}categories.
\[
N_{\omega} : \oo\Cat \to \Psh{\Delta}
\]
that associates to any $\oo$\nbd{}category $C$ a simplicial set $N_{\oo}(C)$
that associates to every $\oo$\nbd{}category $C$ a simplicial set $N_{\oo}(C)$
called the \emph{nerve of $C$}, generalizing the usual nerve of (small)
categories. Using this functor, we can transfer the homotopy theory of
simplicial sets to $\oo$\nbd{}categories, as it is done for example in the
......@@ -90,7 +90,7 @@ for the category of (strict) $\oo$\nbd{}categories.
\end{center}
\iffalse
\begin{equation}\label{naivequestion}\tag{\textbf{Q}}
\text{Do we have }H_k^{\pol}(C) \simeq H_k^{\sing}(C)\text{ for any }\oo\text{-category }C\text{ ? }
\text{Do we have }H_k^{\pol}(C) \simeq H_k^{\sing}(C)\text{ for every }\oo\text{-category }C\text{ ? }
\end{equation}
\fi A partial answer to this question is given by Lafont and Métayer in
\cite{lafont2009polygraphic}: for a monoid $M$ (seen as category with one
......@@ -402,7 +402,7 @@ for the category of (strict) $\oo$\nbd{}categories.
inverting all the cells of $C$. Then, if we believe in
Grothendieck's conjecture (see \cite{grothendieck1983pursuing} and
\cite[Section 2]{maltsiniotis2010grothendieck}), the category of
weak $\oo$\nbd{}groupoids equipped with weak equivalences of weak
weak $\oo$\nbd{}groupoids equipped with the weak equivalences of weak
$\oo$\nbd{}groupoids (see Paragraph 2.2 of
\cite{maltsiniotis2010grothendieck}) is a model for the homotopy
theory of spaces. In particular, every weak $\oo$\nbd{}groupoid has
......@@ -445,8 +445,7 @@ for the category of (strict) $\oo$\nbd{}categories.
A]{quillen1973higher} and its $\oo$\nbd{}categorical generalization by Ara and
Maltsiniotis \cite{ara2018theoreme,ara2020theoreme}.
The fourth chapter is certainly the most important of the dissertation as it
is there that we define the polygraphic and singular homologies of
In the fourth chapter is certainly we define the polygraphic and singular homologies of
$\oo$\nbd{}categories and properly formulate the problem of their comparison.
Up to Section \ref{section:polygraphichmlgy} included, all the results were
known prior to this thesis (at least in the folklore), but starting from
......
No preview for this file type
\documentclass[12pt,a4paper,draft]{report}
\documentclass[12pt,a4paper]{report}
\usepackage[unicode,psdextra,final]{hyperref}
......@@ -13,8 +13,9 @@
\fi
%%% For line numbering (used for proodreading purposes)
\usepackage[pagewise,displaymath, mathlines]{lineno}
\linenumbers
%% \usepackage[pagewise,displaymath, mathlines]{lineno}
%% \linenumbers
\title{Homology of strict $\omega$-categories}
......
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