Commit 0455d9f4 authored by Leonard Guetta's avatar Leonard Guetta
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I need to take up from Section 4.3

parent 1a3115ea
......@@ -195,7 +195,7 @@ As we shall now see, when $C$ is a \emph{free} $\oo$\nbd-category the chain comp
\Hom_{n\Cat}(C,B^nG)\simeq \Hom_{\Ab}(\lambda_n(C),G).\qedhere
\]
\end{proof}
\begin{paragr}
\begin{paragr}\label{paragr:abelpolmap}
Let $C$ be an $\oo$-category, $n \in \mathbb{N}$ and $E \subseteq C_n$ a subset of the $n$-cells. We obtain a map $\mathbb{Z}E \to \lambda_n(C)$ defined as the composition
\[
\mathbb{Z}E \to \mathbb{Z}C_n \to \lambda_n(C),
......@@ -278,30 +278,47 @@ As we shall now see, when $C$ is a \emph{free} $\oo$\nbd-category the chain comp
\]
for every $x \in \Sigma_n$.
With this identification, if $C'$ is another free $\oo$\nbd-category and if $F : C \to C'$ is an $\oo$\nbd-functor (not necessarily rigid), then the linear map $\lambda_n(F) : \lambda_n(C) \to \lambda_{n}(C')$ reads
With this identification, if $C'$ is another free $\oo$\nbd-category and if $F : C \to C'$ is an $\oo$\nbd-functor (not necessarily rigid), then the map $\lambda_n(F) : \lambda_n(C) \to \lambda_{n}(C')$ reads
\[
\lambda_n(F)(x)=w_n(F(x))
\]
for every $x \in \Sigma_n$.
\end{proposition}
\begin{proof}
\end{proof}
For $n \geq 0$, write $\phi_n : \mathbb{Z}\Sigma_n \to \lambda_n(C)$ for the map defined in \ref{paragr:abelpolmap} (which we know is an isomorphism from Lemma \ref{lemma:abelpol}).
The map $w_n: C_n \to \mathbb{Z}\Sigma_n$ induces a map $\mathbb{Z}C_n \to \mathbb{Z}\Sigma$ by linearity, which in turn induces a map $\lambda_n(C) \to \mathbb{Z}\Sigma_n$ (because $w_n(x \comp_k y) = w_n(x)+w_n(y)$ for every pair $(x,y)$ of $k$\nbd-composable $n$\nbd-cells). Write $\psi_n$ for this last map. It is immediate to check that the composition
\[
\mathbb{Z}\Sigma_n \overset{\phi_n}{\longrightarrow} \lambda_n(C) \overset{\psi_n}{\longrightarrow} \mathbb{Z}\Sigma_n
\]
gives the identity on $\mathbb{Z}\Sigma_n$. Hence, $\psi_n$ is the inverse of $\phi_n$.
Now, for $n>0$, notice that the map $\partial : \mathbb{Z}\Sigma_n \to \mathbb{Z}\Sigma_{n-1}$ given in the statement of the proposition is nothing but the composition
\[
\mathbb{Z}\Sigma_n \overset{\phi_n}{\longrightarrow} \lambda_n(C) \overset{\partial}{\longrightarrow} \lambda_{n-1}(C) \overset{\psi_n}{\longrightarrow} \mathbb{Z}\Sigma_{n-1}.
\]
The first part of the proposition follows then from Lemma \ref{lemma:abelpol}.
As for the second part, it suffices to notice that if we identify $\lambda_n(C)$ with $\mathbb{Z}\Sigma_n$ via $\phi_n$ for every free $\oo$\nbd-category $C$, then map $\mathbb{Z}\Sigma_n \to \mathbb{Z}\Sigma'_n$ (where $\Sigma'_n$ is the $n$-basis of $C'$) induced by $F$ is given by the composition
\[
\mathbb{Z}\Sigma_n \overset{\phi_n}{\longrightarrow} \lambda_n(C) \overset{\lambda_n(F)}{\longrightarrow} \lambda_n(C') \overset{\psi_n}{\longrightarrow} \mathbb{Z}\Sigma'_n.\qedhere
\]
\end{proof}
\section{Polygraphic homology}
\begin{lemma}\label{lemma:abeloplax}
Let $u, v : X \to Y$ be two $\oo$-functors. If there is an oplax transformation $\alpha : u \Rightarrow v$, then there is a homotopy of chain complexes from $\lambda(u)$ to $\lambda(v)$.
Let $u, v : C \to D$ be two $\oo$-functors. If there is an oplax transformation $\alpha : u \Rightarrow v$, then there is a homotopy of chain complexes from $\lambda(u)$ to $\lambda(v)$.
\end{lemma}
\begin{proof}
For any $n$-cell $x$ of $X$ (resp.\ $Y$), let us use the notation $[x]$ for the image of $x$ in $\lambda_n(X)$ (resp. $\lambda_n(Y)$).
For any $n$-cell $x$ of $C$ (resp.\ $D$), let us use the notation $[x]$ for the image of $x$ in $\lambda_n(C)$ (resp.\ $\lambda_n(D)$).
Let $h_n$ be the map
\[
\begin{aligned}
h_n : \lambda_n(X) &\to \lambda_{n+1}(Y)\\
h_n : \lambda_n(C) &\to \lambda_{n+1}(D)\\
[x] & \mapsto [\alpha_x].
\end{aligned}
\]
The formulas for oplax transformations from Paragraph \ref{paragr:formulasoplax} imply that $h_n$ is linear and that for every $n$-cell $x$ of $X$,
The formulas for oplax transformations from Paragraph \ref{paragr:formulasoplax} imply that $h_n$ is linear and that for every $n$-cell $x$ of $C$,
\[
\partial (h_n(x)) + h_{n-1}(\partial(x)) = [v(x)] - [u(x)].
\]
......@@ -326,7 +343,7 @@ As we shall now see, when $C$ is a \emph{free} $\oo$\nbd-category the chain comp
\begin{definition}\label{de:polhom}
The \emph{polygraphic homology functor}
\[
\sH^{\pol} : \ho(\oo\Cat^{\Th}) \to \ho(\Ch)
\sH^{\pol} : \ho(\oo\Cat^{\folk}) \to \ho(\Ch)
\]
is the left derived functor of $\lambda : \oo\Cat \to \Ch$ with respect to the folk model structure.
\end{definition}
......
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