Eilenberg-Watts theorems

The classical Eilenberg-Watts theorem gives a complete description of colimit preserving additive functors between categories of modules (over unital rings) as precisely the functors that act by tensoring with a bimodule. A similar theorem can be formulated in more generality [1], which, for instance, allows one to set up a correspondence between colimit preserving functors between varieties of algebras and bimodules (in a certain sense) between algebraic theories.

We will give an overview of some standard category-theoretic considerations from which such theorems arise. Finally, we will take a look at how the existing descriptions of functors between the various categories of semigroup actions fit into this picture.

[1] L. Poinsot, H.E. Porst, Internal coalgebras in cocomplete categories: generalizing the Eilenberg-Watts theorem. J. Algebra Appl. 20 (2021), no. 9, Paper No. 2150165, 35 pp. arXiv: https://arxiv.org/abs/2003.08113