Equivalences between localisations of categories provided by replacements

TL;DR

This paper characterizes functors between categories that induce equivalences on localizations, especially when inverses are given by replacements like projective resolutions or cofibrant replacements.

Contribution

It provides a new criterion for when functors induce equivalences on localizations, focusing on the role of replacements such as projective and cofibrant replacements.

Findings

Characterizes functors inducing equivalences on localizations

Identifies conditions involving replacements like projective resolutions

Clarifies when inverses are induced by specific replacements

Abstract

We give a characterisation of functors whose induced functor on the level of localisations is an equivalence and where the isomorphism inverse is induced by some kind of replacements such as projective resolutions or cofibrant replacements.