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Let $M$ be a monoid, $\mathscr{C}$ a category with pullbacks and $X$ anobject 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 theproblem 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 acolimit of a certain functor with values in $\mathscr{C}$. We specify thisconstruction to the case of categories admitting certain coproducts andcoequalizers.

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