Tensor Enriched Categorical Generalization of the Eilenberg-Watts Theorem

TL;DR

This paper establishes an equivalence between categories of commutative monoids and cocontinuous lax monoidal enriched functors in a Bénabou cosmos, inspired by algebraic geometry formalisms.

Contribution

It introduces a tensor enriched categorical generalization of the Eilenberg-Watts theorem within a Bénabou cosmos framework.

Findings

Proves an equivalence of categories involving commutative monoids and enriched functors.

Extends classical results to a tensor enriched setting inspired by algebraic geometry.

Provides a new categorical perspective relevant for six-functor formalisms.

Abstract

Let $b$, $b'$ be commutative monoids in a B\'{e}nabou cosmos. Motivated by six-functor formalisms in algebraic geometry, we prove that the category of commutative monoids over $b\otimes b'$ is equivalent to the category of cocontinuous lax monoidal enriched functors between the monoidal enriched categories of right modules over $b$, $b'$.