A comparison theorem for simplicial resolutions

TL;DR

This paper investigates the properties of comonadic homology in semi-abelian categories, focusing on conditions that ensure the homology depends only on the induced class of projectives, extending known results from abelian categories.

Contribution

It provides a comparison theorem for simplicial resolutions that establishes when homology in semi-abelian categories is independent of the chosen comonad, generalizing classical results.

Findings

Homology depends only on the induced class of projectives under certain conditions

Extension of comonadic homology properties from abelian to semi-abelian categories

Conditions for independence of the homology theory from the choice of comonad

Abstract

It is well known that Barr and Beck's definition of comonadic homology makes sense also with a functor of coefficients taking values in a semi-abelian category instead of an abelian one. The question arises whether such a homology theory has the same convenient properties as in the abelian case. Here we focus on independence of the chosen comonad: conditions for homology to depend on the induced class of projectives only.