Commit db4f274a authored by Leonard Guetta's avatar Leonard Guetta
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parent c8b9fbf5
......@@ -91,35 +91,35 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equiv
\end{proof}
\section{Abelianization}
\begin{paragr}
Let $X$ be an $\oo$-category. For every $n\geq 0$, we define $\lambda_n(X)$ as the abelian group obtained by quotienting out $\mathbb{Z}X_n$, the free abelian group on $X_n$, by the congruence generated by the relations
Let $C$ be an $\oo$-category. For every $n\geq 0$, we define $\lambda_n(C)$ as the abelian group obtained by quotienting out $\mathbb{Z}C_n$, the free abelian group on $C_n$, by the congruence generated by the relations
\[
x \comp_k y \sim x+y
\]
for all $x,y \in X_n$ that are $k$-composable for some $k<n$. For $n=0$, we have $\lambda_0(X)=\mathbb{Z}X_0$. For $n>0$, consider the linear map
for all $x,y \in C_n$ that are $k$-composable for some $k<n$. For $n=0$, we have $\lambda_0(C)=\mathbb{Z}C_0$. For $n>0$, consider the linear map
\begin{align*}
\mathbb{Z}X_n &\to \mathbb{Z}X_{n-1}\\
x \in X_n &\mapsto t(x)-s(x).
\mathbb{Z}C_n &\to \mathbb{Z}C_{n-1}\\
x \in C_n &\mapsto t(x)-s(x).
\end{align*}
The axioms of $\oo$-categories implies that it induces a map
\[
\partial : \lambda_{n}(X) \to \lambda_{n-1}(X)
\partial : \lambda_{n}(C) \to \lambda_{n-1}(C)
\]
which furtheremore satisfy the equation $\partial \circ \partial = 0$. Thus, we have defined a chain complex $\lambda(X)$:
which furthermore satisfies the equation $\partial \circ \partial = 0$. Thus, we have defined a chain complex $\lambda(C)$:
\[
\lambda_0(X) \overset{\partial}{\longleftarrow} \lambda_1(X) \overset{\partial}{\longleftarrow} \lambda_2(X) \overset{\partial}{\longleftarrow} \cdots
\lambda_0(C) \overset{\partial}{\longleftarrow} \lambda_1(C) \overset{\partial}{\longleftarrow} \lambda_2(C) \overset{\partial}{\longleftarrow} \cdots
\]
Now let $f : X \to Y$ be an $\oo$-functor. The map
Now let $f : C \to D$ be an $\oo$-functor. The map
\begin{align*}
\mathbb{Z}X_n &\to \mathbb{Z}Y_{n}\\
x \in X_n &\mapsto f(x)
\mathbb{Z}C_n &\to \mathbb{Z}D_{n}\\
x \in C_n &\mapsto f(x)
\end{align*}
induces a map
\[
\lambda_n(f) : \lambda_n(X) \to \lambda_n(Y).
\lambda_n(f) : \lambda_n(C) \to \lambda_n(D).
\]
Since $f$ commutes with source and target, we obtain a morphism of chain complexes
\[
\lambda(f) : \lambda(X) \to \lambda(Y).
\lambda(f) : \lambda(C) \to \lambda(D).
\]
Altogether, this defines a functor
\[
......@@ -127,6 +127,13 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equiv
\]
which we call the \emph{abelianization functor}.
\end{paragr}
\begin{lemma}\label{lemma:adjlambda}
The functor $\lambda$ is a left ajdoint.
\end{lemma}
\begin{proof}
The category of chain complexes is equivalent to the category $\omega\Cat(\Ab)$ of $\omega$-categories internal to abelian groups (see \cite[Theorem 3.3]{bourn1990another}) and with this identification, the functor $\lambda : \omega\Cat \to \omega\Cat(\Ab)$ is nothing but the left adjoint of the canonical forgetful functor $\omega\Cat(\Ab) \to \omega\Cat$.
\end{proof}
As we shall now see, when $C$ is a \emph{free} $\oo$\nbd-category, the chain complex $\lambda(C)$
\begin{paragr}
Let $X$ be an $\oo$-category, $n \in \mathbb{N}$ and $E \subseteq X_n$ a subset of the $n$-cells. We obtain a map $\mathbb{Z}E \to \lambda_n(X)$ defined as the composition
\[
......@@ -177,12 +184,6 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equiv
&\simeq \Hom_{\Ab}(\mathbb{Z}\Sigma_n,G).
\end{align*}
\end{proof}
\begin{lemma}\label{lemma:adjlambda}
The functor $\lambda$ is a left ajdoint.
\end{lemma}
\begin{proof}
The category of chain complexes is equivalent to the category $\omega\Cat(\Ab)$ of $\omega$-categories internal to abelian groups (see \cite[Theorem 3.3]{bourn1990another}) and with this identification, the functor $\lambda : \omega\Cat \to \omega\Cat(\Ab)$ is nothing but the left adjoint of the canonical forgetful functor $\omega\Cat(\Ab) \to \omega\Cat$.
\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 $\lambda(v)$.
......
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