Lax pullback complements in partial morphism categories

Abstract

The goal of this article is to characterize the lax pullback complement of a given partial morphism along a total morphism, in an M-partial  morphism category, where M is an exponentiable stable system. To achieve this, we first show that if a lax pullback complement of a  partial morphism exists in the partial morphism category, then a lax pullback complement of its total part exists in the base category. For  the converse, we consider two cases. In the first case the base category is assumed to be adhesive and the morphism along which we  take the lax pullback complement is admissible (i.e., its pushout along a monomorphism forms a pullback complement square), while in  the second case the base category is Mcohesive, however the morphism along which we take the lax pullback complement is an arbitrary  total morphism. Finally as a byproduct we provide the connection between exponentials in the partial morphism category and its base  category.