Strong homotopy types of acyclic categories and $\Delta$-complexes

TL;DR

This paper extends homotopy theories to finite acyclic categories and Δ-complexes, emphasizing the role of morphism and simplex universality, and explores compatibility with classifying space and face poset functors.

Contribution

It introduces a new homotopy framework for acyclic categories and Δ-complexes, highlighting the importance of universality in morphisms and simplices.

Findings

Homotopy theories are extended to acyclic categories and Δ-complexes.

Classifying space and face poset functors are compatible with these homotopy theories.

Universality of morphisms and simplices is central to the new framework.

Abstract

We extend the homotopy theories based on point reduction for finite spaces and simplicial complexes to finite acyclic categories and $\Delta$-complexes, respectively. The functors of classifying spaces and face posets are compatible with these homotopy theories. In contrast with the classical settings of finite spaces and simplicial complexes, the universality of morphisms and simplices plays a central role in this paper.