Partial monoid actions on objects in categories with pullbacks and their globalizations
Mykola Khrypchenko, Francisco Klock
TL;DR
The paper develops a categorical framework for partial monoid actions on objects in categories with pullbacks, introducing partial action data, global actions, and the notion of restriction. It proves that, under mild cocompleteness or coproduct/coequalizer assumptions, a partial action admits a globalization realized as a reflection into the category of global actions, with explicit constructions via colimits or via coproducts and a coequalizer. The globalization concept is tied to pullback conditions when a reflection exists, providing a criterion parallel to dual globalization results in geopactions. The theory is specialized to sets, yielding an explicit enveloping space construction and re-deriving known results (e.g., Hollings) about globalization of partial monoid actions on sets. Overall, the work unifies and extends globalization results for partial actions in a broad categorical setting, offering practical tools to construct and recognize globalizations via colimits or 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 $α$ of $M$ on $X$ and study the globalization question for $α$. If $α$ 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.