Colimits of Monads

TL;DR

This paper investigates the existence of colimits of monads in various categories, establishing conditions under which coequalizers, cointersections, and coproducts exist, and providing explicit formulas for coproducts of separated monads.

Contribution

It extends the theory of colimits of monads to many-sorted sets and other categories, introducing the concept of separated monads and providing concrete formulas for their coproducts.

Findings

Category of monads over many-sorted sets has coequalizers and strong cointersections.

Colimits of monads exist under preservation of monomorphisms and large joint pre-fixpoints.

Explicit formulas for coproducts of separated monads are provided.

Abstract

The category of all monads over many-sorted sets (and over other "set-like" categories) is proved to have coequalizers and strong cointersections. And a general diagram has a colimit whenever all the monads involved preserve monomorphisms and have arbitrarily large joint pre-fixpoints. In contrast, coequalizers fail to exist e.g. for monads over the (presheaf) category of graphs. For more general categories we extend the results on coproducts of monads from [2]. We call a monad separated if, when restricted to monomorphisms, its unit has a complement. We prove that every collection of separated monads with arbitrarily large joint pre-fixpoints has a coproduct. And a concrete formula for these coproducts is presented.