Morita theory in enriched context

Abstract

We develop a homotopy theoretical version of classical Morita theory using the notion of a strong monad. It was Anders Kock who proved that a monad T in a monoidal category E is strong if and only if T is enriched in E. We prove that this correspondence between strength and enrichment follows from a 2-isomorphism of 2-categories. Under certain conditions on T, we prove that the category of T-algebras is Quillen equivalent to the category of modules over the endomorphism monoid of the T-algebra T(I) freely generated by the unit I of E. In the special case where E is the category of Gamma-spaces equipped with Bousfield-Friedlander's stable model structure and T is the strong monad associated to a well-pointed Gamma-theory, we recover a theorem of Stefan Schwede, as an instance of a general homotopical Morita theorem.