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

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.