Still translating the introduction into French...

introduction.tex

@@ -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 $L(C)$.

mystyle.sty

@@ -3,6 +3,7 @@
 % Layout
 %\usepackage{times}
 \usepackage[utf8]{inputenc}
+\usepackage[francais,english]{babel}
 \usepackage[T1]{fontenc}
 
 %\usepackage{showlabels}

omegacat.tex

@@ -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"]\\