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

Abstract

Let M be a monoid, $C$ a category with pullbacks and X an object of $C$. 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 $C$. We then give a construction of such a reflection in terms of a colimit of a certain functor with values in $C$. We specify this construction to the case of categories admitting certain coproducts and coequalizers.