Model category extensions of the Pirashvili-S{\l}omi\'{n}ska theorems

TL;DR

This paper introduces semi-stable model categories, generalizing classical equivalences, and establishes Morita equivalences among functor categories using conjugate pairs within a Quillen model framework.

Contribution

It extends the theory of model categories by defining semi-stability and constructs conjugate pairs to establish Quillen equivalences, broadening Morita theory applications.

Findings

Semi-stable model categories generalize product-coproduct equivalences.

Construction of conjugate pairs yields Quillen equivalences.

Framework extends Morita theory to new categorical contexts.

Abstract

We describe the class of semi-stable model categories, which generalize the equivalence of finite products and coproducts in abelian and stable model categories, and use this to establish Morita equivalences among categories of functors. We provide a construction of pairs of small categories--known as conjugate pairs--whose associated categories of diagrams are Quillen equivalent in the semi-stable setting. We frame our development in the context of Morita theory, following Slominska's work on similar questions for categories of functors enriched over and taking values in R-modules.