# Gabriel-Morita theory for excisive model categories

## TL;DR

This paper extends Gabriel-Morita theory to excisive monoidal model categories, establishing conditions under which categories of monad algebras are Quillen equivalent to module categories, generalizing known results in stable homotopy theory.

## Contribution

It develops a Gabriel-Morita framework for strong monads in excisive pointed monoidal model categories, linking algebraic and module categories under new conditions.

## Key findings

- Category of T-algebras is Quillen equivalent to T(I)-modules.
- Recovers Schwede's theorem on connective stable homotopy as a special case.
- Provides criteria for equivalences in homotopical algebra contexts.

## Abstract

We develop a Gabriel-Morita theory for strong monads on pointed monoidal model categories. Assuming that the model category is excisive, i.e. the derived suspension functor is conservative, we show that if the monad T preserves cofibre sequences up to homotopy and has a weakly invertible strength, then the category of T-algebras is Quillen equivalent to the category of T(I)-modules where I is the monoidal unit. This recovers Schwede's theorem on connective stable homotopy over a pointed Lawvere theory as special case.