Twisted simplicial groups and twisted homology of categories

Abstract

Abstract. Let A be either a simplicial complex K or a small category 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 functionthere 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.