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arXiv · 2401.12638

On The Axioms Of $\mathcal{M},\mathcal{N}$-Adhesive Categories

Davide Castelnovo·Marino Miculan
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Abstract

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' 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.

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Davide Castelnovo, Marino Miculan. 2024-01-23. On The Axioms Of $\mathcal{M},\mathcal{N}$-Adhesive Categories. https://doi.org/10.46298/lmcs-21(1%3A22)2025

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