Commit 48fd8f1e authored by Leonard Guetta's avatar Leonard Guetta
Browse files

security commit

parent d565932a
......@@ -771,3 +771,5 @@ For any $n \geq 0$, consider the following cocartesian square
\begin{proposition}
Let $m,n \geq 0$ and consider the $2$-category $A_{(m,n)}$. If $m\neq 0$ or $n\neq 0$, then $A_{(m,n)}$ is \good{} and has the homotopy type of a point. If $n=m=0$, then $A_{(0,0)}$ is not \good{} and has the homotopy type of a $K(\mathbb{Z},2)$.
\end{proposition}
\section{Zoology of $2$-categories: A stub of a criterion}
\section{``Bubble-free'' $2$-categories}
......@@ -56,7 +56,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
For an $\oo$-category $X$, $\sH(X)$ is the \emph{homology of $X$}.
\end{definition}
\begin{paragr}
In simpler words, the homology of an $\oo$-category $X$ is, by definition, the homology of its nerve. We will sometimes call this the \emph{``true'' homology of $X$} in order to distinguish it from other homological invariants that we shall introduce later.
In simpler words, the homology of an $\oo$-category $X$ is, by definition, the homology of its nerve. We will sometimes call this the \emph{Street homology of $X$} in order to distinguish it from other homological invariants that we shall introduce later.
Recall that we consider localization functors as identity on objects (see \ref{paragr:loc}). Hence, the homology of $X$ is simply $\kappa(N_{\oo}(X))$, only considered as a chain complex up to quasi-isomorphism, i.e. an object of $\ho(\Ch)$. The homology groups of $X$ are the homology groups of the chain complex $\kappa(N_{\oo}(X))$. However, with our definition, the \emph{homology of $X$} means something more precise than the mere sequence of homology groups. An alternative terminology would be to call $\sH(X)$ the \emph{homology type of $X$}, in reference to the homotopy type of a topological space.
\end{paragr}
\begin{remark}
......@@ -214,8 +214,8 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
From Theorem \ref{thm:cisinskiII}, this morphism of op-prederivators is cocontinuous. This property will be extremely important in the sequel.
\end{paragr}
\todo{Expliquer concrètement comment calculer l'homologie polygraphique ?}
\section{``True'' homology as derived abelianization}
We have seen in the previous section that the polygraphic homology is the left derived of the abelianization functor with respect to the canonical weak equivalences on $\oo\Cat$.\todo{Uniformiser appellations} As it turns out, the abelianization functor is also left derivable with respect to the Thomason weak equivalences and the left derived functor is the ``true'' homology functor for $\oo$-categories. In order to prove this result, we need a few lemmas.
\section{Street homology as derived abelianization}
We have seen in the previous section that the polygraphic homology is the left derived of the abelianization functor with respect to the canonical weak equivalences on $\oo\Cat$.\todo{Uniformiser appellations} As it turns out, the abelianization functor is also left derivable with respect to the Thomason weak equivalences and the left derived functor is the Street homology functor for $\oo$-categories. In order to prove this result, we need a few lemmas.
\begin{lemma}\label{lemma:nuhomotopical}
Let $\nu : \Ch \to \oo\Cat$ be the right adjoint to the abelianization functor (see Lemma \ref{lemma:adjlambda}). This functor sends weak equivalences of chain complexes to Thomason weak equivalences.
\end{lemma}
......@@ -250,7 +250,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
\[
\LL \lambda : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)
\]
is isomorphic to the ``true'' homology
is isomorphic to the Street homology
\[
\sH : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch).
\]
......@@ -293,7 +293,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{
\overline{\kappa} \overline{N_{\oo}} : \Ho(\oo\Cat^{\Th}) \ar[r,shift left]& \ar[l,shift left] \Ho(\Ch) : M \overline{N_{\oo}} \overline{\nu} \simeq \overline{\nu}.
\end{tikzcd}
\]
From \todo{Critère de Gonzalez version dérivateur à écrire !}, we conclude that $\lambda : \oo\Cat \to \Ch$ is strongly left derivable and that $ \LL^{\Th} \lambda \simeq \overline{\kappa} \overline{N_{\oo}}$, which is, by definition, the ``true'' homology.
From \todo{Critère de Gonzalez version dérivateur à écrire !}, we conclude that $\lambda : \oo\Cat \to \Ch$ is strongly left derivable and that $ \LL^{\Th} \lambda \simeq \overline{\kappa} \overline{N_{\oo}}$, which is, by definition, the Street homology.
\end{proof}
\begin{remark}
Beware that neither $c_{\oo} : \Psh{\Delta} \to \oo\Cat$ nor $\lambda : \oo\Cat \to \Ch$ preserve weak equivalences \todo{Uniformiser appellations}, but this does not contradict the fact that $\lambda c_{\oo} : \Psh{\Delta} \to \Ch$ does.
......@@ -437,7 +437,7 @@ We shall now proceed to give an abstract criterion to find \good{} $\oo$-categor
\[
\sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch)
\]
and the ``true'' homology
and the Street homology
\[
\sH : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)
\]
......@@ -496,3 +496,5 @@ The previous proposition admits the following corollary, which will be of great
\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}.
\end{proof}
\section{Polygraphic homology and truncation}
\section{Homology and Homotopy of $\oo$-categories in low dimension}
......@@ -2,6 +2,12 @@
\usepackage{mystyle}
%%% Watermark
\usepackage{draftwatermark}
\SetWatermarkText{DRAFT}
\SetWatermarkScale{2}
%%%
\title{Homology of strict $\omega$-categories}
\author{Léonard Guetta}
\begin{document}
......@@ -9,7 +15,7 @@
\maketitle
\tableofcontents
\abstract{In this dissertation, we study the homology of strict $\oo$-categories. More precisely, we intend to compare the ``classical'' homology of an $\oo$-category (defined as the homology of its Street nerve) with its polygraphic homology. Along the way, we prove several important result concerning free strict $\oo$\nbd-categories over polygraphs-or-computad and concerning the homotopy theory of strict $\oo$\nbd-categories. }
\include{omegacat}
\include{homtheo}
\include{hmtpy}
......
This diff is collapsed.
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment