Partial monoid actions on objects in categories with pullbacks and their globalizations

TL;DR

This paper introduces partial monoid actions on objects in categories with pullbacks, explores conditions for their globalization, and provides a construction for the reflection into global actions using colimits.

Contribution

It formalizes partial monoid actions in categories with pullbacks and offers a colimit-based construction for their globalizations, extending previous work on globalizing partial actions.

Findings

Reduction of globalization verification to a pullback condition

Construction of globalizations via colimits in suitable categories

Application to categories with coproducts and coequalizers

Abstract

Let $M$ be a monoid, $\mathscr{C}$ a category with pullbacks and $X$ an object of $\mathscr{C}$. We introduce the notion of a partial action $\alpha$ of $M$ on $X$ and study the globalization question for $\alpha$. If $\alpha$ admits a reflection in the subcategory of global actions, then we reduce the problem to the verification that a certain diagram is a pullback in $\mathscr{C}$. We then give a construction of such a reflection in terms of a colimit of a certain functor with values in $\mathscr{C}$. We specify this construction to the case of categories admitting certain coproducts and coequalizers.