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 (quasi)adhesive categories can be embedded in a Grothendieck topos via a functor preserving pullbacks and pushouts along (regular) monomorphisms. In this paper we extend these results to ℳ, 𝒩-adhesive categories, a concept recently introduced to generalize the notion of (quasi)adhesivity. We introduce the notion of 𝒩-adhesive morphism, which allows us to express ℳ, 𝒩-adhesivity as a condition on the subobjects's 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.