Non-abelian and $\varepsilon$-curved homological algebra with arrow categories

TL;DR

This paper extends Grandis's non-abelian homological algebra to arrow categories, enabling homology theory in broader contexts including approximate chain complexes in normed spaces, with applications to group representations.

Contribution

It proves that arrow categories of categories with null morphisms and existing (co)kernels are homological, broadening the scope of Grandis's framework to include $ extit{ extvarepsilon}$-curved homological algebra.

Findings

Arrow categories of certain categories are homological.

Homology can be defined via morphisms in these categories.

Facilitates application of homological methods to approximate structures.

Abstract

Grandis's non-abelian homological algebra generalizes standard homological algebra in abelian categories to \textit{homological categories}, which are a broader class of categories including for example the category of lattices and Galois connections. Here, we prove that if $\mathsf{C}$ is any category with an ideal of null morphisms with respect to which (co)kernels exist, then the arrow category of $\mathsf{C}$ is a homological category. This broadens the applicability of Grandis's framework substantially. In particular, one can form the homology of chain complexes in $\mathsf{C}$ by taking the homology objects to be morphisms of $\mathsf{C}$, which one may think of as maps from an object of cycles to an object of chains modulo boundaries. One situation to which Grandis's original framework does not apply is \textit{$\varepsilon$-curved homological algebra}. This refers to chain complexes of normed spaces whose differential squares to zero only approximately, in the sense that $\|d^2\| \leq \varepsilon$ for some $\varepsilon > 0$. This is relevant for example in the theory of approximate representations of groups, where Kazhdan has successfully employed $\varepsilon$-curved homological techniques in an ad-hoc manner. We develop some basics of $\varepsilon$-curved homological algebra and note that our result on arrow categories facilitates the application of Grandis's theory.