Twisted Simplicial Groups and Twisted Homology of Categories

J. Y. Li; V. V. Vershinin; J. Wu

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arXiv · 1509.06424

Twisted Simplicial Groups and Twisted Homology of Categories

J. Y. Li·V. V. Vershinin·J. Wu

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Abstract

Let $A$ be either a simplicial complex $K$ or a small category $\mathcal C$ with $V(A)$ as its set of vertices or objects. We define a twisted structure on $A$ with coefficients in a simplicial group $G$ as a function $$ \delta\colon V(A)\longrightarrow \operatorname{End}(G), \quad v\mapsto \delta_v $$ such that $\delta_v\circ \delta_w=\delta_w\circ \delta_v$ if there exists an edge in $A$ joining $v$ with $w$ or an arrow either from $v$ to $w$ or from $w$ to $v$. We give a canonical construction of twisted simplicial group as well as twisted homology for $A$ with a given twisted structure. Also we determine the homotopy type of of this simplicial group as the loop space over certain twisted smash product.

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BibTeX RIS

J. Y. Li, V. V. Vershinin, J. Wu. 2015-09-21. Twisted Simplicial Groups and Twisted Homology of Categories. https://arxiv.org/abs/1509.06424

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