Partial pullback complement rewriting along admissible matches.

In this article, we first give two equivalent criteria for the existence of pullback complements along a universal morphism in a category in terms of the existence of pullbacks and co-universality of certain morphisms. Then we introduce the notion of admissible morphism and we characterize the existence of pullback complements in a partial morphism category along an admissible morphism in terms of the existence of pullback complements in its base category; when the base category is a certain admissible category. At the end we make a comparison between partial pullback complement rewriting and sesqui-pushout, double-pushout and single pushout rewritings, for an admissible match. [ABSTRACT FROM AUTHOR]