Commit 24f2551e authored by Leonard Guetta's avatar Leonard Guetta
Browse files

edited typos and did some small corrections suggested by Garner

parent beb43c6a
......@@ -312,20 +312,17 @@ From the previous proposition, we deduce the following very useful corollary.
is quasi-injective on arrows; the remaining case being symmetric.
Let use denote by $E$ the set of objects of $B$ that are in the image of
$\beta$. For each element $x$ of $E$, we denote by $F_x$ the ``fiber'' of $x$,
that is the set of objects of $A$ that $\beta$ sends to $x$. We consider the
set $E$ and each $F_x$ as discrete reflexive graphs, i.e.\ reflexive graphs
$\beta$. We consider this set as well as the set $A_0$ of objects of $A$ as discrete reflexive graphs, i.e.\ reflexive graphs
with no non-unit arrows. Now, let $G$ be the reflexive graph defined by the
following cocartesian square
\[
\begin{tikzcd}
\displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[d]\\
A_0\ar[r] \ar[d] & E \ar[d]\\
A \ar[r] & G, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
\end{tikzcd}
\]
where the morphism \[ \coprod_{x \in E}F_x \to A\] is induced by the inclusion
of each $F_x$ in $A$, and the morphism \[\coprod_{x \in E}F_x \to E\] is the
only one that sends an element $a \in F_x$ to $x$. In other words, $G$ is
where the morphism \[ A_0 \to A\] is the canonical inclusion, and the
morphism \[A_0 \to E\] is induced by the restriction of $\beta$ on objects. In other words, $G$ is
obtained from $A$ by collapsing the objects that are identified through
$\beta$. It admits the following explicit description: $G_0$ is (isomorphic
to) $E$ and the set of non-unit arrows of $G$ is (isomorphic to) the set of
......@@ -337,7 +334,7 @@ From the previous proposition, we deduce the following very useful corollary.
Now, we have the following solid arrow commutative diagram
\[
\begin{tikzcd}
\displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[ddr,bend left]\ar[d]&\\
A_0 \ar[r] \ar[d] & E \ar[ddr,bend left]\ar[d]&\\
A \ar[drr,bend right,"\beta"'] \ar[r] & G \ar[dr, dotted]&\\
&&B, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
\end{tikzcd}
......@@ -351,7 +348,7 @@ From the previous proposition, we deduce the following very useful corollary.
obtained earlier, we obtain a diagram of three cocartesian squares:
\[
\begin{tikzcd}[row sep = large]
\displaystyle\coprod_{x \in E}F_x \ar[r] \ar[d] & E \ar[d]&\\
A_0\ar[r] \ar[d] & E \ar[d]&\\
A \ar[d,"\alpha"] \ar[r] & G \ar[d] \ar[r] & B \ar[d,"\delta"]\\
C \ar[r] & H \ar[r] & D. \ar[from=1-1,to=2-2,phantom,"\ulcorner" very near
end,"\text{\textcircled{\tiny \textbf{1}}}" near start, description]
......@@ -368,15 +365,19 @@ From the previous proposition, we deduce the following very useful corollary.
$L$ of square \textcircled{\tiny \textbf{3}} is homotopy cocartesian. Hence,
in virtue of Lemma \ref{lemma:pastinghmtpycocartesian}, all we have to show is
that the image by $L$ of square \textcircled{\tiny \textbf{2}} is homotopy
cocartesian. On the other hand, the morphism
cocartesian. On the other hand, the morphisms
\[
\coprod_{x \in E}F_x \to A
A_0 \to A
\]
is a monomorphism and thus, using Corollary
and
\[
A_0 \to C
\]
are monomorphisms and thus, using Corollary
\ref{cor:hmtpysquaregraph}, we deduce that the image by $L$ of square
\textcircled{\tiny \textbf{1}} and of the pasting of squares
\textcircled{\tiny \textbf{1}} and \textcircled{\tiny \textbf{2}}
are homotopy cocartesian. This proves that the image by $L$ of
are homotopy cocartesian. By Lemma \ref{lemma:pastinghmtpycocartesian} again, this proves that the image by $L$ of
square \textcircled{\tiny \textbf{2}} is homotopy cocartesian.
\end{proof}
We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
......@@ -1105,7 +1106,7 @@ of $2$-categories.
\begin{proof}
Let $r : \Delta_n \to \Delta_1$ the unique functor such that
\[
r(n)=1\text{ and } r(k)=1 \text{ for } k\neq n.
r(0)=0\text{ and } r(k)=1 \text{ for } k>0.
\]
By definition we have $r \circ i = 1_{\Delta_1}$. Now, the natural order on
$\Delta_n$ induces a natural transformation
......@@ -1217,12 +1218,12 @@ results at the end of the previous section.
\begin{equation}\label{square:lemniscate}
\begin{tikzcd}
\sD_0 \ar[r] \ar[d] & A_{(1,0)} \ar[d]\\
A_{(0,1)} \ar[r] & P. \ar[from=1-1,to=2-2,very near
A_{(1,0)} \ar[r] & P. \ar[from=1-1,to=2-2,very near
end,"\ulcorner",phantom]
\end{tikzcd}
\end{equation}
Since $\sD_0$, $A_{(1,0)}$ and $A_{(0,1)}$ are all free and \good{} and since
$\sD_0 \to A_{(1,0)}$ and $\sD_0 \to A_{(0,1)}$ are folk cofibrations, all we
Since $\sD_0$, $A_{(1,0)}$ are free and \good{} and since
$\sD_0 \to A_{(1,0)}$ is a folk cofibration, all we
have to show to prove that $P$ is \good{} is that the above square is Thomason
homotopy cocartesian. Notice that the $2$\nbd{}category $A_{(1,0)}$ is
obtained as the following amalgamated sum
......@@ -1252,7 +1253,7 @@ Let us now get into more sophisticated examples.
\item generating $2$\nbd{}cells: $\alpha : f \Rightarrow g$, $\beta: g
\Rightarrow f$.
\end{itemize}
In picture, this gives
In pictures, this gives
\[
\begin{tikzcd}
A \ar[r,bend left=75,"f",""{name=A,below}] \ar[r,bend
......@@ -1287,13 +1288,27 @@ Let us now get into more sophisticated examples.
\item $F(\alpha)=\gamma$ and $F(\beta)=1_h$.
\end{itemize}
We wish to prove that this $2$\nbd{}functor is a Thomason equivalence. Since
it is an isomorphism on objects, it suffices to prove that the functor induced
it is an isomorphism on objects, it suffices to prove that the functors induced
by $F$
\[
F_{A,A} : P(A,A) \to P'(F(A),F(A)),
\]
\[
F_{B,B} : P(B,B) \to P'(F(B),F(B)),
\]
\[
F_{B,A} : P(B,A) \to P'(F(B),F(A))
\]
and
\[
F_{A,B} : P(A,B) \to P'(F(A),F(B))
\]
is a Thomason equivalence of categories (Corollary
\ref{cor:criterionThomeqI}). The category $P(A,B)$ is the free category on the
are Thomason equivalences of categories (Corollary
\ref{cor:criterionThomeqI}). For the first two ones, this follows trivially
from the fact that the categories $P(A,A)$, $P'(A',A')$, $P(B,B)$ and
$P'(B',B')$ are all isomorphic to $\sD_0$. For the third one, this follows
trivially from the fact that the categories $P(B,A)$ and $P'(B',A')$ are the
empty category. For the fourth one, this can be seen as follows. The category $P(A,B)$ is the free category on the
graph
\[
\begin{tikzcd}
......@@ -1313,7 +1328,7 @@ Let us now get into more sophisticated examples.
\ref{example:killinggenerator}). In particular, the square
\[
\begin{tikzcd}
\sD_1 \ar[r,"\langle \beta \rangle"] \ar[d]& P(A,B) \ar[d,"F_{(A,B)}"] \\
\sD_1 \ar[r,"\langle \beta \rangle"] \ar[d]& P(A,B) \ar[d,"F_{A,B}"] \\
\sD_0 \ar[r,"\langle h \rangle" ] & P'(A',B')
\end{tikzcd}
\]
......@@ -1346,7 +1361,7 @@ Let us now get into more sophisticated examples.
\item generating $2$\nbd{}cells: $\lambda : l \Rightarrow 1_{A''}$ and $\mu: l
\Rightarrow 1_{A''}$.
\end{itemize}
In picture, this gives
In pictures, this gives
\[
\begin{tikzcd}[column sep=huge]
A'' \ar[r,bend left=75,"l",""{name=A'',below}]\ar[r,bend
......@@ -1447,7 +1462,7 @@ Let us now get into more sophisticated examples.
\item generating $2$\nbd{}cells: $\alpha : f \Rightarrow 1_A$ and $\beta: 1_A
\Rightarrow f$.
\end{itemize}
In picture, this gives
In pictures, this gives
\[
\begin{tikzcd}[column sep=huge]
A \ar[r,bend left=75,"f",""{name=A,below}]\ar[r,bend
......@@ -1551,7 +1566,7 @@ Let us now move on to bouquets of spheres.
\item generating $1$\nbd{}cells: $f,g: A \to A$,
\item generating $2$\nbd{}cells: $\alpha,\beta : f \Rightarrow g$.
\end{itemize}
In picture, this gives:
In pictures, this gives:
\[
\begin{tikzcd}[column sep=huge]
A \ar[r,bend
......@@ -1578,8 +1593,8 @@ Now let $\sS_2$ be labelled as
\begin{equation}
\begin{tikzcd}
\sS_0 \ar[d,"i_1"] \ar[r,"{\langle C,D \rangle}"] & \sS_2 \ar[d]\\
\sD_1 \ar[r] & P.
\sS_0 \ar[d,"p"] \ar[r,"{\langle C,D \rangle}"] & \sS_2 \ar[d]\\
\sD_0 \ar[r] & P.
\ar[from=1-1,to=2-2,"\ulcorner",phantom,very near end]
\end{tikzcd}\label{eq:squarebouquethybrid}
\end{equation}
......@@ -1589,12 +1604,13 @@ Now let $\sS_2$ be labelled as
have to show that the induced square of $\Cat$
\begin{equation}
\begin{tikzcd}
V_k(\sS_0) \ar[r,"{V_k(\langle C,D \rangle)}"] \ar[d,"V_k(i_1)"] & V_k(\sS_2) \ar[d]\\
V_k(\sD_1) \ar[r] & V_k(P)
V_k(\sS_0) \ar[r,"{V_k(\langle C,D \rangle)}"] \ar[d,"V_k(p)"] & V_k(\sS_2) \ar[d]\\
V_k(\sD_0) \ar[r] & V_k(P)
\end{tikzcd}\label{eq:squarebouquethybridvertical}
\end{equation}
is Thomason homotopy cocartesian for every $k \geq 0$. Notice first that we
trivially have $V_0(\sS_0)\simeq \sS_0$ and that $V_0(\sS_2)$ is the free category on the
trivially have $V_k(\sS_0)\simeq \sS_0$ and $V_k(\sD_0)\simeq \sD_0$ for every
$k\geq 0$ and that $V_0(\sS_2)$ is the free category on the
graph
\[
\begin{tikzcd}
......@@ -1628,7 +1644,9 @@ Now let $\sS_2$ be labelled as
In particular, square \eqref{eq:squarebouquethybridvertical} is again a
cocartesian square of identification of two objects of a free category, and
thus, it is Thomason homotopy cocartesian. This implies that square
\eqref{eq:squarebouquethybrid} is Thomason homotopy cocartesian. Since $\sS_0$, $\sD_1$ and $\sS_2$ are \good{} and since the morphisms $\langle C, D \rangle : \sS_0 \to \sS_2$ and $i_1 : \sS_0 \to \sD_1$ are folk cofibrations, this proves that $P$ is \good{}
\eqref{eq:squarebouquethybrid} is Thomason homotopy cocartesian. Since $\sS_0$,
$\sD_0$ and $\sS_2$ are \good{} and since $\langle C, D \rangle : \sS_0 \to
\sS_2$ is a folk cofibration, this proves that $P$ is \good{}
and has the homotopy type of the bouquet of a $1$\nbd{}sphere with a $2$\nbd{}sphere.
\end{paragr}
......@@ -1639,7 +1657,7 @@ Now let $\sS_2$ be labelled as
\item generating $1$\nbd{}cells: $f,g : A \to B$,
\item generating $2$\nbd{}cells: $\alpha,\beta,\gamma : f \Rightarrow g$.
\end{itemize}
In picture, this gives
In pictures, this gives
\[
\begin{tikzcd}[column sep=huge]
A \ar[r,bend
......@@ -1689,7 +1707,7 @@ Now let $\sS_2$ be labelled as
\item generating $2$\nbd{}cells: $\alpha,\beta:f \Rightarrow g$ and
$\delta,\gamma:g \Rightarrow h$.
\end{itemize}
In picture, this gives:
In pictures, this gives:
\[
\begin{tikzcd}[column sep=huge]
A \ar[r,bend
......@@ -1708,7 +1726,7 @@ Now let $\sS_2$ be labelled as
sub-$2$\nbd{}category of $P$ spanned by $A$, $B$, $g$, $h$, $\gamma$ and
$\delta$ and let $P_2$ be the sub-$2$\nbd{}category of $P$ spanned by $A$,
$B$, $f$, $g$, $\alpha$ and $\beta$. The $2$\nbd{}categories $P_1$ and $P_2$
are copies of $\sS_2$, and $P_0$ is a copy of $\sS_1$. Moreover, we have a
are copies of $\sS_2$, and $P_0$ is a copy of $\sD_1$. Moreover, we have a
cocartesian square of inclusions
\begin{equation}\label{squarebouquet}
\begin{tikzcd}
......@@ -1834,18 +1852,23 @@ Now let $\sS_2$ be labelled as
\]
This implies that square \eqref{squarebouquetvertical} is cocartesian for $k=0$ and in
virtue of Corollary \ref{cor:hmtpysquaregraph} it is also Thomason homotopy
cocartesian for this value of $k$. For $k>0$, the category $V_k(P')$ has two objects $A$
and $B$ and an arrow $A \to B$ is a $k$\nbd{}tuple of one of the following form
\begin{itemize}[label=-]
\item $(1_f,\cdots,1_f,\alpha,1_g,\cdots,1_g)$,
\item $(1_f,\cdots,1_f,\beta,1_g,\cdots,1_g)$,
\item $(1_f,\cdots,1_f)$,
\item $(1_g,\cdots,1_g)$,
\end{itemize}
and these are the only non-trivial arrows of $V_k(P')$. This means that $V_k(P')$
is the free category on the graph that has two objects and $2k+2$ parallel
arrows between these two objects. The same goes for $V_k(P'')$ since $P'$ and
$P''$ are isomorphic. Similarly, the category $V_k(P)$ is free on the graph
cocartesian for this value of $k$. For $k>0$, since $P'$ and $P''$ are both
isomorphic to $\sS_2$, we have already seen in \ref{paragr:variationsphere} that $V_k(P')$ and $V_k(P'')$ are
(isomorphic) to the free category on the graph that has two objects and $2k+2$ parallel
arrows between these two objects.
% the category $V_k(P')$ has two objects $A$
% and $B$ and an arrow $A \to B$ is a $k$\nbd{}tuple of one of the following form
% \begin{itemize}[label=-]
% \item $(1_f,\cdots,1_f,\alpha,1_g,\cdots,1_g)$,
% \item $(1_f,\cdots,1_f,\beta,1_g,\cdots,1_g)$,
% \item $(1_f,\cdots,1_f)$,
% \item $(1_g,\cdots,1_g)$,
% \end{itemize}
% and these are the only non-trivial arrows of $V_k(P')$. This means that $V_k(P')$
% is the free category on the graph that has two objects and $2k+2$ parallel
% arrows between these two objects. The same goes for $V_k(P'')$ since $P'$ and
% $P''$ are isomorphic.
Similarly, the category $V_k(P)$ is free on the graph
that has three objects $A$, $B$, $C$, whose arrows from $A$ to $B$
are $k$\nbd{}tuples of one of the following form
\begin{itemize}[label=-]
......@@ -1889,8 +1912,8 @@ homotopy type of the torus.
From now on, we will use concatenation instead of the symbol $\comp_0$ for the
$0$\nbd{}composition. For example, $fg$ will stand for $f \comp_0 g$. With
this notation, the set $1$\nbd{}cells of $P$ is canonically isomorphic to the
set of finite words on the alphabet $\{f,g\}$ and the set of $2$\nbd{}cells of
$P$ is canonically isomorphic to the set of finite words on the alphabet
set of finite words in the alphabet $\{f,g\}$ and the set of $2$\nbd{}cells of
$P$ is canonically isomorphic to the set of finite words in the alphabet
$\{f,g,\alpha\}$. For a $1$\nbd{}cell $w$ such that $f$ appears $n$ times in
$w$ and $g$ appears $m$ times in $w$, it is a simple exercise left to the
reader to show that there exists a unique $2$\nbd{}cell of $P$ from $w$ to the
......@@ -2026,7 +2049,7 @@ homotopy type of the torus.
\end{itemize}
\end{definition}
\begin{paragr}
In picture, a bubble $x$ is represented as
In pictures, a bubble $x$ is represented as
\[
\begin{tikzcd}
A \ar[r,bend left=75,"1_A",""{name=A,below}] \ar[r,bend
......
......@@ -72,7 +72,7 @@
\vfill
{\slshape Présentée et soutenue publiquement le 15 janvier 2021 \\
{\slshape Présentée et soutenue publiquement le 28 janvier 2021 \\
devant un jury composé de : \par}
\vspace{0.8cm}
......@@ -96,26 +96,67 @@
% \includegraphics[width=0.6\textwidth]{license.png}
% \vspace*{-3cm}
\end{titlepage}
\abstract{In this dissertation, we study the homology of strict
$\oo$\nbd{}categories. More precisely, we intend to compare the ``classical''
\end{titlepage}
\selectlanguage{french}
\begin{abstract}
Dans cette thèse, on compare l'homologie \og classique \fg{} d'une
$\oo$\nbd{}catégorie (définie comme l'homologie de son nerf de Street) avec
son homologie polygraphique. Plus précisément, on prouve que les deux
homologies ne coïncident pas en général et qualifions d'\emph{homologiquement
cohérente} les $\oo$\nbd{}catégories particulières pour lesquelles l'homologie
polygraphique coïncide effectivement avec l'homologie du nerf. Le but poursuivi
est de trouver des critères abstraits et concrets permettant de détecter les
$\oo$\nbd{}catégories homologiquement cohérentes. Par exemple, on démontre que
toutes les (petites) catégories, que l'on considère comme des
$\oo$\nbd{}catégories strictes dont toutes les cellules au-delà de la dimension
$1$ sont des unités, sont homologiquement cohérente. On introduit également la
notion de $2$\nbd{}catégorie \emph{sans bulles} et on conjecture qu'une
$2$\nbd{}catégorie cofibrante est homologiquement cohérente si et seulement si
elle est sans bulles. On démontre également des résultats importants concernant
les $\oo$\nbd{}catégories strictes qui sont libres sur un polygraphique, comme
le fait que si $F : C \to D$ est un $\oo$\nbd{}foncteur discret de Conduché et
si $D$ est libre sur un polygraphe alors $C$ l'est aussi. Dans son ensemble,
cette thèse établit un cadre général dans lequel étudier l'homologie des
$\oo$\nbd{}catégories en faisant appels à des outils d'algèbre homotopique
abstraite, tels que la théorie des catégorie de modèles de Quillen ou la théorie
des dérivateurs de Grothendieck.
\end{abstract}
\selectlanguage{english}
\begin{abstract}
In this dissertation, we compare the ``classical''
homology of an $\oo$\nbd{}category (defined as the homology of its Street
nerve) with its polygraphic homology. Along the way, we prove several
important results concerning free strict $\oo$\nbd{}categories on polygraphs
(also known as computads) and concerning the homotopy theory of strict $\oo$\nbd{}categories. }
nerve) with its polygraphic homology. More precisely, we prove that both
homologies generally do not coincide and call \emph{homologically coherent} the
particular strict $\oo$\nbd{}categories for which polygraphic homology and
homology of the nerve do coincide. The goal pursued is to find abstract
and concrete criteria to detect homologically coherent $\oo$\nbd{}categories. For
example, we prove that all (small) categories, considered as strict
$\oo$\nbd{}categories with unit cells above dimension $1$, are homologically
coherent. We also introduce the notion of \emph{bubble-free} $2$\nbd{}category
and conjecture that a cofibrant $2$\nbd{}category is homologically
coherent if and only if it is bubble-free.
We also prove important results concerning free strict
$\oo$\nbd{}categories on polygraphs (also known as computads), such as the
fact that if $F : C \to D$ is a discrete Conduché $\oo$\nbd{}functor and $D$
is a free strict $\oo$\nbd{}category on a polygraph, then so is $C$.
Overall, this thesis achieves to build a general framework in which to study the
homology of strict $\oo$\nbd{}categories using tools of abstract homotopical
algebra such as Quillen's theory of model categories or Grothendieck's theory
of derivators.
\end{abstract}
\tableofcontents
\include{remerciements}
% \include{introduction}
% \include{introduction_fr}
% \include{omegacat}
% \include{homtheo}
% \include{hmtpy}
% \include{hmlgy}
% \include{contractible}
% \include{2cat}
%\include{remerciements}
\include{introduction}
\include{introduction_fr}
\include{omegacat}
\include{homtheo}
\include{hmtpy}
\include{hmlgy}
\include{contractible}
\include{2cat}
\addcontentsline{toc}{chapter}{Bibliography}
\bibliographystyle{alpha}
......
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