The index in $d$-exact categories

TL;DR

This paper extends the concept of the index in $d$-exact categories, showing it is additive up to an error term in categories with $d$-kernels, broadening its applicability in higher homological algebra.

Contribution

It introduces an index in $d$-exact categories with $d$-kernels, relaxing previous conditions and demonstrating its additivity up to an error term.

Findings

The index is well-defined in $d$-exact categories with $d$-kernels.

The index exhibits additivity on $d$-exact sequences up to an error term.

The framework generalizes previous notions of the index in higher homological algebra.

Abstract

Starting from its original definition in module categories with respect to projective modules, the index has played an important role in various aspects of homological algebra, categorification of cluster algebras and $K$-theory. In the last few years, the notion of index has been generalised to several different contexts in (higher) homological algebra, typically with respect to a (higher) cluster-tilting subcategory $\mathcal{X}$ of the relevant ambient category $\mathcal{C}$. The recent tools of extriangulated and higher-exangulated categories have permitted some conditions on the subcategory $\mathcal{X}$ to be relaxed. In this paper, we introduce the index with respect to a generating, contravariantly finite subcategory of a $d$-exact category that has $d$-kernels. We show that our index has the important property of being additive on $d$-exact sequences up to an error term.