A combinatorial construction of homology via ACGW categories

TL;DR

This paper introduces ACGW categories as a combinatorial framework for homology, demonstrating their utility in deriving classical homological algebra results like the Snake lemma and long exact sequences.

Contribution

It presents the first detailed exploration of homology theory within ACGW categories, highlighting their advantages over other structures due to their combinatorial nature.

Findings

ACGW categories facilitate homology theory development.

Classical results like the Snake lemma are derived within this framework.

The approach simplifies coherence issues in homological algebra.

Abstract

2-Segal spaces arise not only from $S_\dotp$-constructions associated to Waldhausen and (proto) exact categories, but also from $S_\dotp$-constructions associated to certain double-categorical structures. A major step in this direction is due to the work of Bergner--Osorno--Ozornova--Rovelli--Scheimbauer, who propose augmented stable double Segal objects as a natural input for an $S_\dotp$-construction. More recently, another such input has been put forth: ACGW categories. ACGW categories have the advantage that they are combinatorial in nature (as opposed to homotopical or algebraic), and thus have fewer difficult coherence issues to work with. The goal of this paper is to introduce the reader to the key ideas and techniques for working with ACGW categories. To do so, we focus on how homology theory generalizes to ACGW categories, particularly in the central example of finite sets. We show how the ACGW formalism can be used to produce various classical homological algebra results such as the Snake lemma and long exact sequences of relative pairs.