Colimits of monoids

TL;DR

This paper proves that under certain conditions, the category of monoids in a cocomplete monoidal category is itself cocomplete, and applies this to a lifting theorem for adjunctions involving monoidal right adjoints.

Contribution

It establishes conditions under which the category of monoids inherits cocompleteness and introduces a lifting theorem for adjunctions with monoidal right adjoints.

Findings

Category of monoids is cocomplete under specified conditions

Lifting theorem for adjunctions with monoidal right adjoints

Applicability to categories with regular factorizations

Abstract

If $\mathcal{C}$ is a cocomplete monoidal category in which tensoring from both sides preserves coequalizers, then the category of monoids over $\mathcal{C}$ is cocomplete. The same holds if $\mathcal{C}$ has regular factorizations and tensoring only preserves regular epimorphisms. As an application a lifting theorem for an adjunction with a monoidal right adjoint to an adjunction between the respective categories of monoids is proved.