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


J. Y. Li (Department of Mathematics, Shijiazhuang Tiedao University, Shijiazhuang, China)

V. V. Vershinin (Département des Sciences Mathématiques, Université de Montpellier, France; and Sobolev Institute of Mathematics, Novosibirsk, Russia)

J. Wu (Department of Mathematics, National University of Singapore)


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.


homology, simplicial group, category

2010 Mathematics Subject Classification

Primary 55U10. Secondary 18G30.

Received 29 August 2016

Published 18 October 2017