ON THE AXIOMS OF M, N-ADHESIVE CATEGORIES

Adhesive and quasiadhesive categories provide a general framework for the study of algebraic graph rewriting systems. In a quasiadhesive category any two regular subobjects have a join which is again a regular subobject. Vice versa, if regular monos are adhesive, then the existence of a regular join for any pair of regular subobjects entails quasiadhesivity. It is also known that (quasi)adhesive categories can be embedded in a Grothendieck topos via a functor preserving pullbacks and pushouts along (regular) monos. In this paper we extend these results to M, N-adhesive categories, a concept which generalizes the notion of (quasi)adhesivity. We introduce the notion of N-adhesive mor-phism, which allows us to express M, N-adhesivity as a condition on the subobjects’ posets. Moreover, N-adhesive morphisms allows us to show how an M, N-adhesive category can be embedded into a Grothendieck topos, preserving pullbacks and M, N-pushouts.