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# Homology, Homotopy and Applications

## Volume 19 (2017)

### Number 2

### Twisted simplicial groups and twisted homology of categories

Pages: 111 – 130

DOI: https://dx.doi.org/10.4310/HHA.2017.v19.n2.a7

#### Authors

#### 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\[\delta \colon V(A) \longrightarrow \mathrm{End}(G), \quad v\mapsto \delta_v,\]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 groups as well as twisted homology for $A$ with a given twisted structure. Also we determine the homotopy type of this simplicial group as the loop space over certain twisted smash product.

#### Keywords

homology, simplicial group, category

#### 2010 Mathematics Subject Classification

Primary 55U10. Secondary 18G30.

Received 29 August 2016

Published 18 October 2017