Commit 83986a13 authored by Leonard Guetta's avatar Leonard Guetta
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Still translating the introduction into French...

parent 8eb2617c
......@@ -200,7 +200,7 @@ for the category of (strict) $\oo$\nbd{}categories.
\end{tikzcd}
\]
This triangle is \emph{not} commutative (even up to isomorphism), since this
would imply that the Street and polygraphic homology groups coincide for every
would imply that the singular and polygraphic homology groups coincide for every
$\oo$\nbd{}category. However, since both functors $\LL \lambda^{\folk}$ and
$\LL \lambda^{\Th}$ are left derived functors of the same functor $\lambda$,
the existence of a natural transformation $\pi : \LL \lambda^{\Th} \circ \J
......@@ -295,7 +295,7 @@ for the category of (strict) $\oo$\nbd{}categories.
f(x')=y',\, f(x'')=y'' \text{ and } x=x'\comp_k x''.
\]
The main result that we prove concerning discrete Conduché $\oo$\nbd{}functors
is that for a discrete $\oo$\nbd{}functor $f : C \to D$, if the
is that for a discrete Conduché $\oo$\nbd{}functor $f : C \to D$, if the
$\oo$\nbd{}category $D$ is free, then $C$ is also free. The proof of this
result is long and tedious, though conceptually not extremely hard, and first
appears in the paper \cite{guetta2020polygraphs}, which is dedicated to it.
......@@ -362,7 +362,8 @@ for the category of (strict) $\oo$\nbd{}categories.
For example, it is believed that there should be a folk model structure on the
category of weak $\oo$\nbd{}categories and that there should be a good notion
of free weak $\oo$\nbd{}category. In fact, this last notion should be defined
as weak $\oo$\nbd{}categories that are recursively obtained by freely
as weak $\oo$\nbd{}categories that are recursively obtained from the empty
$\oo$\nbd{}catégory by freely
adjoining cells, which is the formal analogue of the strict version but in the
weak context. The important point here is that a free strict
$\oo$\nbd{}category is \emph{never} free as a weak $\oo$\nbd{}category (except
......@@ -384,7 +385,7 @@ for the category of (strict) $\oo$\nbd{}categories.
of its polygraphic homology in the first place. From this observation, it is
tempting to make the following conjecture:
\begin{center}
The weak polygraphic homology of a (strict) $\oo$\nbd{}category coincides
The weak polygraphic homology of a strict $\oo$\nbd{}category coincides
with its singular homology.
\end{center}
In other words, we conjecture that the fact that polygraphic and singular
......@@ -402,8 +403,8 @@ for the category of (strict) $\oo$\nbd{}categories.
Grothendieck's conjecture (see \cite{grothendieck1983pursuing} and
\cite[Section 2]{maltsiniotis2010grothendieck}), the category of
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
$\oo$\nbd{}groupoids (see
\cite[Paragraph 2.2]{maltsiniotis2010grothendieck}) is a model for the homotopy
theory of spaces. In particular, every weak $\oo$\nbd{}groupoid has
homology groups and we can define the singular homology groups of a
weak $\oo$\nbd{}category $C$ as the homology groups of $L(C)$.
......
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......@@ -3,6 +3,7 @@
% Layout
%\usepackage{times}
\usepackage[utf8]{inputenc}
\usepackage[francais,english]{babel}
\usepackage[T1]{fontenc}
%\usepackage{showlabels}
......
......@@ -40,7 +40,7 @@
\[
\src_k(x)=\trgt_k(y).
\]
Note that the expression ``$x$ and $y$ are $k$\nbd{}composable'' is \emph{not} symmetric in $x$ and $y$ and we \emph{should} rather speak of a ``$k$\nbd{}composable pair $(x,y)$'', although we won't always do it. The set of pairs of $k$\nbd{}composable $n$\nbd{}cells is denoted by $X_n\underset{X_k}{\times}X_n$, and is characterized as the following fibred product
Note that the expression \og $x$ and $y$ are $k$\nbd{}composable\fg{} is \emph{not} symmetric in $x$ and $y$ and we \emph{should} rather speak of a ``$k$\nbd{}composable pair $(x,y)$'', although we won't always do it. The set of pairs of $k$\nbd{}composable $n$\nbd{}cells is denoted by $X_n\underset{X_k}{\times}X_n$, and is characterized as the following fibred product
\[
\begin{tikzcd}
X_n\underset{X_k}{\times}X_n \ar[r] \ar[dr,phantom,"\lrcorner", very near start] \ar[d] &X_n \ar[d,"\trgt_k"]\\
......
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