### movie time

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 $k0$, consider the linear map for all $x,y \in C_n$ that are $k$-composable for some $k0$, 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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