Twisted Simplicial Groups and Twisted Homology of Categories

TL;DR

This paper introduces a new framework for twisted simplicial groups and homology in categories and complexes, linking these structures to loop spaces and twisted smash products.

Contribution

It provides a canonical construction of twisted simplicial groups and homology, and characterizes their homotopy type as loop spaces over twisted smash products.

Findings

Constructed twisted simplicial groups and homology for categories and complexes.

Determined the homotopy type as loop spaces over twisted smash products.

Established conditions for commuting endomorphisms based on adjacency or arrows.

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.