Commit aca3dd48 authored by Leonard Guetta's avatar Leonard Guetta
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parent 8dda3bc3
......@@ -19,10 +19,10 @@
s_i &:= X(\sigma^i): X_{n+1} \to X_n.
\end{aligned}
\]
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}
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}
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{steiner2004omega}, \cite[chapitre ?]{ara2016joint} \todo{finir mettre références}) 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 litterature 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}
......@@ -43,10 +43,9 @@
for $i \in \{0,\cdots,n\}$.
\end{itemize}
\begin{description}
\item[(OR2)] For each $n>0$, the weight of the $(n-1)$-cell corresponding to $\partial_i$ in the \emph{source} of the principal cell of $\Or_n$ (see \ref{paragr:weight}) is $1$ if $i$ is odd and $0$ if $i$ is even and the other way around for the \emph{target} of the principal cell of $\Or_n$.
\item[(OR2)] For $n>0$, the source (resp.\ target) of the principal cell of $\Or_n$ can be expressed as a composite of all the generating $(n-1)$\nbd-cells corresponding to $\delta^i$ with $i$ odd (resp.\ even); each of these generating $(n-1)$\nbd-cell appearing exactly once in the composite.
\end{description}
Intuitively speaking, this last property means that the source (resp.\ target) of the principal cell of $\Or_n$ can be expressed as a composite of all the generating $(n-1)$-cells corresponding to $\delta^i$ with $i$ odd (resp.\ even); each of these generating $(n-1)$-cell appearing exactly once in the composite. Note that this way of formulating things
Another way of formulating \textbf{(OR2)} is: for $n>0$ the weight of the $(n-1)$-cell corresponding to $\delta_i$ in the \emph{source} of the principal cell of $\Or_n$ (see \ref{paragr:weight}) is $1$ if $i$ is odd and $0$ if $i$ is even and the other way around for the \emph{target} of the principal cell of $\Or_n$.
Here are some pictures in low dimension:
\[
\Or_0 = \langle 0 \rangle,
......@@ -65,22 +64,34 @@
\end{tikzcd},
\]
\[
\Or_3=
\Or_3=\begin{tikzcd}[column sep=huge,row sep=huge]
\langle 0 \rangle \ar[r,"\langle 01 \rangle"description] \ar[d,"\langle 13 \rangle"description] & \langle 1 \rangle \ar[d,"\langle 02 \rangle"description]\\
\langle 3 \rangle & \langle 3 \rangle \ar[l,"\langle 23 \rangle" description]
\ar[from=1-1,to=2-2,"\langle 02 \rangle" description,""{name=A,above}]
\ar[from=1-2,to=A,Rightarrow,"\langle 013 \rangle"description]
\end{tikzcd}
\overset{\langle 0123 \rangle}{\Rrightarrow}
\begin{tikzcd}[column sep=huge,row sep=huge]
\langle 0 \rangle \ar[r,"\langle 01 \rangle"description] \ar[d,"\langle 13 \rangle"description] & \langle 1 \rangle \ar[d,"\langle 02 \rangle"description]\\
\langle 3 \rangle & \langle 3 \rangle \ar[l,"\langle 23 \rangle" description]
\ar[from=1-2,to=2-1,"\langle 12 \rangle" description]
\end{tikzcd}
\]
\[
\begin{tikzcd}
& \langle 1 \rangle \ar[rd,"\langle 12 \rangle"]& \\
\langle 0 \rangle \ar[ru,"\langle 01 \rangle"] \ar[rd,"\langle 03 \rangle"',""{name=B,above}] \ar[rr,"\langle 02 \rangle" description,""{name=A,above}]& & \langle 2 \rangle \ar[ld,"\langle 23 \rangle"]\\
& \langle 3 \rangle &
\ar[from=A,to=1-2,Rightarrow,"\langle 012 \rangle"]
\ar[from=B,to=2-3,Rightarrow,"\langle 023 \rangle" description]
\ar[from=B,to=2-3,Rightarrow,"\langle 023 \rangle" description, shorten <= 1em, shorten >= 1em]
\end{tikzcd}
\overset{\langle 0123 \rangle}{\Rrightarrow}
\begin{tikzcd}
& \langle 1 \rangle \ar[rd,"\langle 12 \rangle"] \ar[dd,"\langle 13 \rangle"' description] & \\
\langle 0 \rangle \ar[ru,"\langle 01 \rangle"] \ar[rd,"\langle 03 \rangle"'] & & \langle 2 \rangle \ar[ld,"\langle 23 \rangle"]\\
& \langle 3 \rangle &
\end{tikzcd}
\end{tikzcd}
\]
\todo{Refaire les diagrammes...}
\end{paragr}
\begin{paragr}\label{paragr:nerve}
For every $\omega$-category $X$, the \emph{nerve of $X$} is the simplicial set $N_{\omega}(X)$ defined as
......
......@@ -104,7 +104,7 @@ This way of understanding polygraphic homology as a left derived functor has bee
\begin{center}
Polygraphic homology is a way of computing Street homology groups of a homologically coherent $\oo$-category.
\end{center}
The point is that given a \emph{free} $\oo$-category $P$ (which is thus its own polygraphic resolution), the chain complex $\lambda(P)$ is much less ``bigger'' than the chain complex associated to the nerve of $P$ and hence the polygraphic homology groups of $P$ are much easier to compute than its Street homology groups. The situation is much comparable to using cellular homology for computing singular homology of a CW-complex. The difference is that in this last case, such thing is always possible while in the case of $\oo$-categories, one must ensure that the (free) $\oo$-category is homologically coherent. %Intuitively speaking, this means that some free $\oo$-categories are not ``cofibrant enough'' for homology.
The point is that given a \emph{free} $\oo$-category $P$ (which is thus its own polygraphic resolution), the chain complex $\lambda(P)$ is much ``smaller'' than the chain complex associated to the nerve of $P$ and hence the polygraphic homology groups of $P$ are much easier to compute than its Street homology groups. The situation is much comparable to using cellular homology for computing singular homology of a CW-complex. The difference is that in this last case, such thing is always possible while in the case of $\oo$-categories, one must ensure that the (free) $\oo$-category is homologically coherent. %Intuitively speaking, this means that some free $\oo$-categories are not ``cofibrant enough'' for homology.
\end{named}
\begin{named}[Finding homologically coherent $\oo$-categories]
One of the main result presented in this dissertation is:
......
......@@ -16,7 +16,7 @@
\maketitle
\tableofcontents
%\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. }
......
......@@ -20,8 +20,9 @@
@article{ara2016joint,
title={Joint et tranches pour les $\infty$-cat{\'e}gories strictes},
author={Ara, Dimitri and Maltsiniotis, Georges},
journal={arXiv preprint arXiv:1607.00668},
year={2016}
journal={M{\'e}moires de la Soci{\'e}t{\'e} Math{\'e}matique de France},
year={2020},
note={To appear}
}
@article{ara2018theoreme,
title={Un th{\'e}or{\`e}me {A} de {Q}uillen pour les $\infty$-cat{\'e}gories strictes {I} : la preuve simpliciale},
......@@ -44,8 +45,10 @@ year={2020}
@article{ara2019folk,
title={The folk model category structure on strict $\omega$-categories is monoidal},
author={Ara, Dimitri and Lucas, Maxime},
journal={arXiv preprint arXiv:1909.13564},
year={2019}
journal={Theory and Applications of Categories},
volume={35},
pages={742--808},
year={2020}
}
@article{ara2019quillen,
title={A Quillen Theorem B for strict $\infty$-categories},
......@@ -77,6 +80,20 @@ year={2020}
year={1990},
publisher={Elsevier}
}
@article{buckley2016orientals,
title = "Orientals and cubes, inductively",
abstract = "We provide direct inductive constructions of the orientals and the cubes, exhibiting them as the iterated cones, respectively, the iterated cylinders, of the terminal strict globular ω-category.",
author = "Mitchell Buckley and Richard Garner",
year = "2016",
doi = "10.1016/j.aim.2016.07.026",
language = "English",
volume = "303",
pages = "175--191",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Elsevier"
}
@article{bullejos2003geometry,
title={On the geometry of 2-categories and their classifying spaces},
author={Bullejos, Manuel and Cegarra, Antonio M},
......@@ -456,6 +473,24 @@ note={In preparation}
year={1987},
publisher={Elsevier}
}
@article{street1991parity,
title={Parity complexes},
author={Street, Ross},
journal={Cahiers de topologie et g{\'e}om{\'e}trie diff{\'e}rentielle cat{\'e}goriques},
volume={32},
number={4},
pages={315--343},
year={1991}
}
@article{street1994parity,
title={Parity complexes: corrigenda},
author={Street, Ross},
journal={Cahiers de topologie et g{\'e}om{\'e}trie diff{\'e}rentielle cat{\'e}goriques},
volume={35},
number={4},
pages={359--361},
year={1994}
}
@article{thomason1980cat,
title={Cat as a closed model category},
author={Thomason, Robert W},
......
......@@ -780,7 +780,7 @@ We can now prove the following proposition, which is the key result of this sect
\begin{example}\label{example:freemonoid}
Let $n \geq 1$ and $M$ be a monoid (commutative if $n>1$). The $n$-category $B^nM$ is free if and only if $M$ is a free monoid (free commutative monoid if $n>1$). If so, it has exactly one generating cell of dimension $0$, no generating cells of dimension $0 <k< n$, and the set of generators of the monoid (which is unique) as generating $n$-cells.
\end{example}
\section{Cells of free $\oo$-categories as words}
\section{Cells of free $\oo$-categories as words}\label{section:cellsaswords}
In this section, we undertake to give a more explicit construction of the $(n+1)$\nbd-category $\E^*$ generated by an $n$-cellular cellular extension $\E=(C,\Sigma,\sigma,\tau)$. By definition of $\E^*$, this amounts to give an explicit description of a particular type of colimit in $\oo\Cat$. Note also that since $\tau_{\leq n}(\E^*)=C$, all we need to do is to describe the $(n+1)$-cells of $\E^*$. This will take place in two steps: first we construct what ought to be called the \emph{free $(n+1)$-magma generated by $\E$}, for which the $(n+1)$-cells are really easy to describe, and then we quotient out these cells as to obtain an $(n+1)$-category, which will be $\E^*$.
......@@ -842,13 +842,8 @@ The \emph{size} of a well-formed word $w$, denoted by $\vert w \vert$, is the nu
\[
w \comp_k w' := (w\fcomp_k w')
\]
for $w$ and $w'$ $k$-composable $(n+1)$-cells of $\E^+$, and we ought to be careful not to confuse the ``real'' composition symbol
\[ \comp_k\]
with the ``formal'' composition symbol
\[
\fcomp_k.\]
As a rule of thumb, it is better not to use both symbols in the same expression. Note also that, since we use the usual symbols
\[(\] and \[)\] as ``formal'' symbols of opening and closing parenthesis, things can get really messy if we don't apply the previous rule because it would be hard to distinguish a formal parenthesis from an ``unformal'' one.
for $w$ and $w'$ $k$-composable $(n+1)$-cells of $\E^+$, and we ought to be careful not to confuse the ``real'' composition symbol ``$\; \comp_k \;$''
with the ``formal'' composition symbol ``$\; \fcomp_k\; $''. As a rule of thumb, it is better not to use both symbols in the same expression. Note also that, since we use the usual symbols ``(`` and ``)'' as formal symbols of opening and closing parenthesis, things can get really messy if we don't apply the previous rule because it would be hard to distinguish a formal parenthesis from an ``unformal'' one.
\end{remark}
In the following definition, we consider that a binary relation $\R$ on a set $E$ is nothing but a subset of $E \times E$, and we write $x \; \R \; x'$ to say $(x,x') \in \R$.
\begin{definition}\label{def:congruence}
......
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