Commit 83986a13 by Leonard Guetta

### Still translating the introduction into French...

parent 8eb2617c
 ... @@ -200,7 +200,7 @@ for the category of (strict) $\oo$\nbd{}categories. ... @@ -200,7 +200,7 @@ for the category of (strict) $\oo$\nbd{}categories. \end{tikzcd} \end{tikzcd} \] \] This triangle is \emph{not} commutative (even up to isomorphism), since this 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 $\oo$\nbd{}category. However, since both functors $\LL \lambda^{\folk}$ and $\LL \lambda^{\Th}$ are left derived functors of the same functor $\lambda$, $\LL \lambda^{\Th}$ are left derived functors of the same functor $\lambda$, the existence of a natural transformation $\pi : \LL \lambda^{\Th} \circ \J the existence of a natural transformation$\pi : \LL \lambda^{\Th} \circ \J ... @@ -295,7 +295,7 @@ for the category of (strict) $\oo$\nbd{}categories. ... @@ -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''. 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 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 $\oo$\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 result is long and tedious, though conceptually not extremely hard, and first appears in the paper \cite{guetta2020polygraphs}, which is dedicated to it. appears in the paper \cite{guetta2020polygraphs}, which is dedicated to it. ... @@ -362,7 +362,8 @@ for the category of (strict) $\oo$\nbd{}categories. ... @@ -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 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 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 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 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 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 $\oo$\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. ... @@ -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 of its polygraphic homology in the first place. From this observation, it is tempting to make the following conjecture: tempting to make the following conjecture: \begin{center} \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. with its singular homology. \end{center} \end{center} In other words, we conjecture that the fact that polygraphic and singular In other words, we conjecture that the fact that polygraphic and singular ... @@ -402,8 +403,8 @@ for the category of (strict) $\oo$\nbd{}categories. ... @@ -402,8 +403,8 @@ for the category of (strict) $\oo$\nbd{}categories. Grothendieck's conjecture (see \cite{grothendieck1983pursuing} and Grothendieck's conjecture (see \cite{grothendieck1983pursuing} and \cite[Section 2]{maltsiniotis2010grothendieck}), the category of \cite[Section 2]{maltsiniotis2010grothendieck}), the category of weak $\oo$\nbd{}groupoids equipped with the weak equivalences of weak weak $\oo$\nbd{}groupoids equipped with the weak equivalences of weak $\oo$\nbd{}groupoids (see Paragraph 2.2 of $\oo$\nbd{}groupoids (see \cite{maltsiniotis2010grothendieck}) is a model for the homotopy \cite[Paragraph 2.2]{maltsiniotis2010grothendieck}) is a model for the homotopy theory of spaces. In particular, every weak $\oo$\nbd{}groupoid has theory of spaces. In particular, every weak $\oo$\nbd{}groupoid has homology groups and we can define the singular homology groups of a homology groups and we can define the singular homology groups of a weak $\oo$\nbd{}category $C$ as the homology groups of $L(C)$. weak $\oo$\nbd{}category $C$ as the homology groups of $L(C)$. ... ...
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 ... @@ -3,6 +3,7 @@ ... @@ -3,6 +3,7 @@ % Layout % Layout %\usepackage{times} %\usepackage{times} \usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc} \usepackage[francais,english]{babel} \usepackage[T1]{fontenc} \usepackage[T1]{fontenc} %\usepackage{showlabels} %\usepackage{showlabels} ... ...
 ... @@ -40,7 +40,7 @@ ... @@ -40,7 +40,7 @@ $\[ \src_k(x)=\trgt_k(y). \src_k(x)=\trgt_k(y).$ \] 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} \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"]\\ 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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