Commit ee9c9be6 authored by Leonard Guetta's avatar Leonard Guetta
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Dodo ! I need to finish section 4.6

parent f2905a9e
......@@ -894,13 +894,13 @@ The previous proposition admits the following corollary, which will be of great
\[
K_0 \overset{\partial}{\longleftarrow} K_1 \overset{\partial}{\longleftarrow} K_2 \overset{\partial}{\longleftarrow} \cdots \overset{\partial}{\longleftarrow} K_n,
\]
where $\partial \circ \partial =0$. There is a canonical functor $\Ch^{\leq n} \to \Ch$ where an object $K$ of $\Ch^{\leq n}$ is sent to the chain complex
where $\partial \circ \partial =0$, and morphisms of $\Ch^{\leq n}$ are defined the expected way. Write $\iota_n : \Ch^{\leq n} \to \Ch$ for the canonical functor that sends an object $K$ of $\Ch^{\leq n}$ to the chain complex
\[
K_0 \overset{\partial}{\longleftarrow} K_1 \overset{\partial}{\longleftarrow} K_2 \overset{\partial}{\longleftarrow} \cdots \overset{\partial}{\longleftarrow} K_n \longleftarrow 0 \longleftarrow 0 \longleftarrow \cdots.
\]
This functor is fully faithful and $\Ch^{\leq n}$ may be identified with the full subcategory of $\Ch$ spanned by chain complexes $K$ such that $K_k = 0$ for every $k >n$.
Notice now that for an $n$\nbd-category, seen as an $\oo$\nbd-category $C$ via the canonical inclusion $n\Cat \to \oo\Cat$, the chain complex $\lambda(C)$ is such that
Notice now that for an $n$\nbd-category $C$, seen as an $\oo$\nbd-category via the canonical inclusion $\iota_n : n\Cat \to \oo\Cat$, the chain complex $\lambda(C)$ is such that
\[
\lambda(C)_k=0
\]
......@@ -918,20 +918,20 @@ The previous proposition admits the following corollary, which will be of great
%% \]
%% This functor being
Recall from \ref{paragr:defncat} that for every $n \geq 0$, the canonical inclusion $n\Cat \to \oo\Cat$ has a left adjoint $\tau^{i}_{\leq n} : \oo\Cat \to n\Cat$, where for an $\oo$\nbd-category $C$, $\tau_{\leq n }^{i}(C)$ is the $n$\nbd-category whose set of $k$\nbd-cells is $C_k$ for $k<n$ and whose set of $n$\nbd-cells is the quotient of $C_n$ by the equivalence relation $\sim$ generated by
Recall from \ref{paragr:defncat} that for every $n \geq 0$, the canonical inclusion $\iota_n : n\Cat \to \oo\Cat$ has a left adjoint $\tau^{i}_{\leq n} : \oo\Cat \to n\Cat$, where for an $\oo$\nbd-category $C$, $\tau_{\leq n }^{i}(C)$ is the $n$\nbd-category whose set of $k$\nbd-cells is $C_k$ for $k<n$ and whose set of $n$\nbd-cells is the quotient of $C_n$ by the equivalence relation $\sim$ generated by
\[
x \sim y \text{ when there exists } z : x \to y \text{ in } C_{n+1}.
\]
Note that when $C$ is an $n$\nbd-category, seen as an $\oo$\nbd-category with only unit cells in dimension higher that $n$ via the canonical inclusion $n\Cat \to \oo\Cat$, then $\tau^{i}_{\leq n}(C) = C$.
Similarly, for a chain complex $K$, write $\tau^{\leq n}(K)$ for the object of $\Ch^{\leq n}$ defined as
Note that when $C$ is an $n$\nbd-category, seen as an $\oo$\nbd-category with only unit cells in dimension higher that $n$ via the canonical inclusion $\iota_n : n\Cat \to \oo\Cat$, then $\tau^{i}_{\leq n}(C) = C$.
Similarly, for a chain complex $K$, write $\tau^{i}_{\leq n}(K)$ for the object of $\Ch^{\leq n}$ defined as
\[
K_0 \overset{\partial}{\longleftarrow} K_1 \overset{\partial}{\longleftarrow} K_2 \overset{\partial}{\longleftarrow} \cdots \overset{\partial}{\longleftarrow} K_{n-1} \overset{\partial}{\longleftarrow} K_{n}/{\partial(K_{n-1})}.
K_0 \overset{\partial}{\longleftarrow} K_1 \overset{\partial}{\longleftarrow} K_2 \overset{\partial}{\longleftarrow} \cdots \overset{\partial}{\longleftarrow} K_{n-1} \overset{\partial}{\longleftarrow} K_{n}/{\partial(K_{n+1})}.
\]
This defines a functor
\[
\tau^{i}_{\leq n} : \Ch^{\leq n} \mapsto \Ch
\]
which is left adjoint to the canonical inclusion $\Ch^{\leq n} \to \Ch$.
which is left adjoint to the canonical inclusion $\iota_n : \Ch^{\leq n} \to \Ch$.
\end{paragr}
\begin{lemma}
The square
......@@ -946,7 +946,7 @@ The previous proposition admits the following corollary, which will be of great
\begin{proof}
\todo{À écrire}
\end{proof}
With this lemma at hand we can prove the important following proposition which basically says that if an $\oo$\nbd-category $C$ is free up to dimension $n-1$, then there is no need to find a cofibrant replacement in order to compute $H^{\pol}_k(C)$ provided that $0 \leq k \leq n$.
With this lemma at hand we can prove the important following proposition which basically says that if an $\oo$\nbd-category $C$ is free up to dimension $n-1$, then for any $k$ such that $0 \leq k \leq n$ there is no need to find a cofibrant replacement in order to compute $H^{\pol}_k(C)$.
\begin{proposition}
Let $n \geq 0$ and $C$ be an $\oo$\nbd-category. If $C$ has a $k$\nbd-basis for every $ 0 \leq k < n$, then the canonical map of $\ho(\Ch)$
\[
......@@ -962,7 +962,7 @@ With this lemma at hand we can prove the important following proposition which b
\todo{À écrire}
\end{proof}
\begin{paragr}
Since every $\oo$\nbd-category admits trivially $C_0$ as a $0$\nbd-base, it follows from the previous proposition that $H^{\pol}_0(C)$ and $H^{\pol}_1(C)$ respectively are (canonically isomorphic to) the $0$-th and first homology groups of the chain complex $\lambda(C)$. This means that no cofibrant resolution of $C$ is needed to compute its first two polygraphic homology groups.
Since every $\oo$\nbd-category $C$ admits $C_0$ as a $0$\nbd-base, it follows from the previous proposition that $H^{\pol}_0(C)$ and $H^{\pol}_1(C)$ respectively are (canonically isomorphic to) the $0$-th and first homology groups of the chain complex $\lambda(C)$. This means that no cofibrant resolution of $C$ is needed to compute its first two polygraphic homology groups.
\end{paragr}
Somewhat related is the following proposition.
\begin{proposition}
......@@ -979,5 +979,33 @@ Somewhat related is the following proposition.
\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
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
\[
N_n : n\Cat \overset{\iota_n}{\longrightarrow} \oo\Cat \overset{N_{\oo}}{\longrightarrow} \Psh{\Delta},
\]
and for $n \in \mathbb{N}\cup \{\oo\}$ we write $c_n : \Psh{\Delta} \to n\Cat$ for the left adjoint of $N_n$.
\begin{lemma}
For every $n \in \mathbb{N}$, the following triangle of functors
\[
\begin{tikzcd}
\Psh{\Delta} \ar[r,"c_{\oo}"] \ar[d,"c_n"] & \oo\Cat \ar[dl,"\tau^{i}_{\leq n}"] \\
n\Cat&
\end{tikzcd}
\]
is commutative (up to an isomorphism).
\end{lemma}
\begin{proof}
\todo{À écrire (facile)}
\end{proof}
Slightly less trivial is the following lemma.
\begin{lemma}
The following triangle of functors
\[
\begin{tikzcd}
\oo\Cat \ar[r,"\tau^{i}_{\leq 1}"] \ar[d,"N_{\oo}"] & \Cat \\
\Psh{\Delta} \ar[ru,"c_1"'] &
\end{tikzcd}
\]
is commutative (up to an isomorphism).
\end{lemma}
\section{Homology and Homotopy of $\oo$-categories in low dimension}
......@@ -115,29 +115,30 @@ This way of understanding polygraphic homology as a left derived functor has bee
As such, this result is only a small generalization of Lafont and Métayer's result concerning monoids (although this new result, even restricted to monoids, is more precise because it means that the \emph{canonical comparison map} is an isomorphim). But the novelty lies in the proof which is more conceptual that the one of Lafont and Métayer and of which we now give a outline.
\end{named}
\begin{named}[The big picture]
Let us end this introduction with another point of view on the comparison of Street and polygraphic homologies. This point of view is not adressed at all in the rest of the dissertation because it is higly conjectural. It ought to be thought of as a guideline for future work.
\todo{À finir !!}
%% \begin{named}[The big picture]
%% Let us end this introduction with another point of view on the comparison of Street and polygraphic homologies. This point of view is not adressed at all in the rest of the dissertation because it is higly conjectural. It ought to be thought of as a guideline for future work.
Let us come back to the canonical $2$-triangle
\[
\begin{tikzcd}
\ho(\oo\Cat^{\folk}) \ar[rd,"\sH^{\folk}=\LL \lambda^{\folk}"',""{name=B,above}] \ar[r,"\J"] & \ho(\oo\Cat^{\Th}) \ar[d,"\LL \lambda^{\Th}=\sH^{\St}"] \\
& \ho(\Ch)
\ar[from=1-2,to=B,Rightarrow,"\pi"]
\end{tikzcd}
\]
and ask the question:
\begin{center}
What \emph{would} it mean that the natural transformation $\pi$ be an isomorphism (i.e.\ that all $\oo$-categories be homologically coherent) ?
\end{center}
For simplification, let us assume that the conjectured Thomason-like model structure on $\oo\Cat$ was established and that $\lambda$ was left Quillen with respect to this model structure (which is also conjectured).
Now, the conjectured cofibrations of the Thomason-like model structure (see \cite{ara2014vers}) are particular cases of folk cofibrations and thus, all Thomason cofibrant objects are folk cofibrant objects. The converse, on the other hand, is not true. Consequently, Quillen's theory of derived functors tells us that for a \emph{Thomason} cofibrant object $P$, we have
\begin{equation}\tag{$\ast$}\label{equationintro}
\LL \lambda^{\Th}(P) \simeq \lambda(P) \simeq \LL \lambda^{\folk}(P),
\end{equation}
(and the resulting isomorphism is obviously the canonical comparison map). Now, \emph{if} the natural transformation $\pi$ were an isomorphism, then a quick 2-out-of-3 reasoning would show that \eqref{equationintro} would also be true when $P$ is only \emph{folk} cofibrant. Hence, intuitively speaking, if $\pi$ were an isomorphism, then folk cofibrant objects would be \emph{sufficiently cofibrant} for the homology, even though there are not Thomason cofibrant. (And in fact, using cofibrant replacements, it can be shown that this condition is sufficient to ensure that $\pi$ be an isomorphism).
%% Let us come back to the canonical $2$-triangle
%% \[
%% \begin{tikzcd}
%% \ho(\oo\Cat^{\folk}) \ar[rd,"\sH^{\folk}=\LL \lambda^{\folk}"',""{name=B,above}] \ar[r,"\J"] & \ho(\oo\Cat^{\Th}) \ar[d,"\LL \lambda^{\Th}=\sH^{\St}"] \\
%% & \ho(\Ch)
%% \ar[from=1-2,to=B,Rightarrow,"\pi"]
%% \end{tikzcd}
%% \]
%% and ask the question:
%% \begin{center}
%% What \emph{would} it mean that the natural transformation $\pi$ be an isomorphism (i.e.\ that all $\oo$-categories be homologically coherent) ?
%% \end{center}
%% For simplification, let us assume that the conjectured Thomason-like model structure on $\oo\Cat$ was established and that $\lambda$ was left Quillen with respect to this model structure (which is also conjectured).
%% Now, the conjectured cofibrations of the Thomason-like model structure (see \cite{ara2014vers}) are particular cases of folk cofibrations and thus, all Thomason cofibrant objects are folk cofibrant objects. The converse, on the other hand, is not true. Consequently, Quillen's theory of derived functors tells us that for a \emph{Thomason} cofibrant object $P$, we have
%% \begin{equation}\tag{$\ast$}\label{equationintro}
%% \LL \lambda^{\Th}(P) \simeq \lambda(P) \simeq \LL \lambda^{\folk}(P),
%% \end{equation}
%% (and the resulting isomorphism is obviously the canonical comparison map). Now, \emph{if} the natural transformation $\pi$ were an isomorphism, then a quick 2-out-of-3 reasoning would show that \eqref{equationintro} would also be true when $P$ is only \emph{folk} cofibrant. Hence, intuitively speaking, if $\pi$ were an isomorphism, then folk cofibrant objects would be \emph{sufficiently cofibrant} for the homology, even though there are not Thomason cofibrant. (And in fact, using cofibrant replacements, it can be shown that this condition is sufficient to ensure that $\pi$ be an isomorphism).
Yet, as we have already seen, such property is not true: there are folk cofibrant objects that are \emph{not} enough cofibrant to compute (Street) homology. The archetypal example being the ``bubble'' of Ara and Maltsiniotis. However, even if false, the idea that folk cofibrant objects are sufficiently cofibrants for homology is seducing and I conjecturally believe that this defect is a mere consequence of working in a too narrow setting, as I shall now explain.
%% Yet, as we have already seen, such property is not true: there are folk cofibrant objects that are \emph{not} enough cofibrant to compute (Street) homology. The archetypal example being the ``bubble'' of Ara and Maltsiniotis. However, even if false, the idea that folk cofibrant objects are sufficiently cofibrants for homology is seducing and I conjecturally believe that this defect is a mere consequence of working in a too narrow setting, as I shall now explain.
In the same way that bicategories and tricategories are ``weak'' variations of the notions of (strict) $2$-categories and $3$-categories, there exists a general notion of \emph{weak $\oo$-categories}. These objects can be defined, for example, using the formalism of Grothendieck's coherators \cite{maltsiniotis2010grothendieck}, or of Batanin's globular operads \cite{batanin1998monoidal}. (In fact, each of these formalism give rise to many different possible notions of weak $\oo$-categories, which are conjectured to be all equivalent, at least in some higher categorical sense.)
\end{named}
%% In the same way that bicategories and tricategories are ``weak'' variations of the notions of (strict) $2$-categories and $3$-categories, there exists a general notion of \emph{weak $\oo$-categories}. These objects can be defined, for example, using the formalism of Grothendieck's coherators \cite{maltsiniotis2010grothendieck}, or of Batanin's globular operads \cite{batanin1998monoidal}. (In fact, each of these formalism give rise to many different possible notions of weak $\oo$-categories, which are conjectured to be all equivalent, at least in some higher categorical sense.)
%% \end{named}
......@@ -161,9 +161,9 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
\]
that simply discards all the cells of dimension strictly higher than $n$. This functor has a left adjoint
\[
\iota : n\Cat \to \oo\Cat,
\iota_n : n\Cat \to \oo\Cat,
\]
where for an $n$-category $C$, the $\oo$-category $\iota(C)$ has the same $k$-cells as $C$ for $k\leq n$ and only unit cells in dimension strictly higher than $n$. This functor itself has a left adjoint
where for an $n$-category $C$, the $\oo$-category $\iota_n(C)$ has the same $k$-cells as $C$ for $k\leq n$ and only unit cells in dimension strictly higher than $n$. This functor itself has a left adjoint
\[
\tau_{\leq n }^i : \oo\Cat \to n\Cat,
\]
......@@ -173,24 +173,24 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
\]
The functor $\tau_{\leq n}^s$ also have a right adjoint
\[
\kappa : n\Cat \to \oo\Cat,
\kappa_n : n\Cat \to \oo\Cat,
\]
where for an $n$-category $C$, the $\oo$-category $\kappa(C)$ has the same $k$-cells as $C$ for $k \leq n$ and has exactly one $m$-cell $x \to y$ for every pair of parallel $(m-1)$-cells $(x,y)$ with $m>n$.
where for an $n$-category $C$, the $\oo$-category $\kappa_n(C)$ has the same $k$-cells as $C$ for $k \leq n$ and has exactly one $m$-cell $x \to y$ for every pair of parallel $(m-1)$-cells $(x,y)$ with $m>n$.
The sequence of adjunctions
\[
\kappa \dashv \tau^s_{\leq n} \dashv \iota \dashv \tau^{i}_{\leq n}
\kappa_n \dashv \tau^s_{\leq n} \dashv \iota_n \dashv \tau^{i}_{\leq n}
\]
is maximal in that $\kappa$ doesn't have a left adjoint and $\tau^{i}_{\leq n}$ doesn't have right adjoint.
is maximal in that $\kappa_n$ doesn't have a left adjoint and $\tau^{i}_{\leq n}$ doesn't have right adjoint.
The functors $\tau^{s}_{\leq n}$ and $\tau^{i}_{\leq n}$ are respectively referred to as the \emph{stupid truncation functor} and the \emph{intelligent truncation functor}.
The functor $\iota$ is fully faithful and preserves both limits and colimits; in regards to these properties, we often identify $n\Cat$ with the essential image of $\iota$, which is the full subcategory of $\oo\Cat$ spanned by $\oo$-categories whose $k$-cells for $k >n$ are all units on lower dimensional cells.
The functor $\iota_n$ is fully faithful and preserves both limits and colimits; in regards to these properties, we often identify $n\Cat$ with the essential image of $\iota_n$, which is the full subcategory of $\oo\Cat$ spanned by $\oo$-categories whose $k$-cells for $k >n$ are all units on lower dimensional cells.
\end{paragr}
\begin{paragr}
For $n \geq 0$, we define the $n$-skeleton functor $\sk_n : \oo\Cat \to \oo\Cat$ as
\[
\sk_n := \iota \circ \tau^{s}_{\leq n}.
\sk_n := \iota_n \circ \tau^{s}_{\leq n}.
\]
This functor preserves both limits and colimits. For an $\oo$-category $C$, $\sk_n(C)$ is the sub-$\oo$-category of $C$ generated by the $k$-cells of $C$ with $k\leq n$, in an obvious sense. In particular, we have a canonical filtration
\[
......@@ -203,7 +203,7 @@ A \emph{morphism of $\oo$-magmas} $f : X \to Y$ is a morphism of underlying $\oo
\[
\sk_{0}(C) \hookrightarrow \sk_{1}(C) \hookrightarrow \cdots \hookrightarrow\sk_{n}(C) \hookrightarrow\cdots
\]
is $C$ and the universal arrow $\sk_{n}(C) \to C$ is given by the co-unit of the adjunction $\tau^{s}_{\leq n} \dashv \iota$.
is $C$ and the universal arrow $\sk_{n}(C) \to C$ is given by the co-unit of the adjunction $\tau^{s}_{\leq n} \dashv \iota_n$.
\end{lemma}
\begin{paragr}\label{paragr:defglobe}
For $n \in \mathbb{N}$, the \emph{$n$-globe} $\sD_n$ is the $n$-category that has:
......
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