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 $\mathcal{M}, \mathcal{N}$-adhesive categories, a concept recently introduced to generalize the notion of (quasi)adhesivity. We introduce the notion of $\mathcal{N}$-adhesive morphism, which allows us to express $\mathcal{M}, \mathcal{N}$-adhesivity as a condition on the subobjects's posets. Moreover, $\mathcal{N}$-adhesive morphisms allows us to show how an $\mathcal{M},\mathcal{N}$-adhesive category can be embedded into a Grothendieck topos, preserving pullbacks and $\mathcal{M}, \mathcal{N}$-pushouts.

翻译：黏合范畴与拟黏合范畴为代数图重写系统的研究提供了通用框架。在拟黏合范畴中，任意两个正则子对象均存在并运算且结果仍为正则子对象。反之，若正则单态射具有黏合性，则任意正则子对象对的正则并存在性可推出拟黏合性。已知（拟）黏合范畴可通过保持（正则）单态射的拉回与推出之函子嵌入至格罗滕迪克拓扑斯。本文将这些结论推广至 $\mathcal{M}, \mathcal{N}$-黏合范畴——这一新近引入用于泛化（拟）黏合性概念的概念。我们引入 $\mathcal{N}$-黏合态射的概念，据此可将 $\mathcal{M}, \mathcal{N}$-黏合性表述为子对象偏序集上的条件。此外，利用 $\mathcal{N}$-黏合态射可证明 $\mathcal{M},\mathcal{N}$-黏合范畴可通过保持拉回与 $\mathcal{M}, \mathcal{N}$-推出的函子嵌入格罗滕迪克拓扑斯。