On Pushouts of Partial Maps

Abstract

The paper gives a sufficient condition for the existence of all pushouts in an arbitrary category of partial maps \(\mathbb{C}_{*\mathcal{M}}\) that is necessary whenever the category of total maps \(\mathbb{C} \subseteq \mathbb{C}_{*\mathcal{M}}\) has cocones of spans; the latter is the case in all slice categories of ℂ and thus the condition is necessary locally. The main theorem is that, given an admissible class of monos \(\mathcal{M}\) in a category ℂ that has cocones of spans, the category of partial maps \(\mathbb{C}_{*\mathcal{M}}\) has pushouts if and only if the category of total maps ℂ has hereditary pushouts and right adjoints to inverse image functors (where both properties are w.r.t. \(\mathcal{M}\)). This result clarifies previous work by Kennaway on graph rewriting in categories of partial maps that implicitly assumed existence of cocones of spans in the category of total maps.