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

Davide Castelnovo, Marino Miculan

TL;DR

This work extends the theory of adhesive and quasiadhesive categories to the broader framework of $\mathcal{M}, \mathcal{N}$-adhesive categories. It introduces $\mathcal{N}$-adhesive morphisms, establishes how adhesive-like behavior can be read from subobject posets, and proves that, under suitable hypotheses, $\mathcal{M}, \mathcal{N}$-adhesive categories embed into Grothendieck toposes while preserving pullbacks and $\mathcal{M}, \mathcal{N}$-pushouts. The authors develop a consistent ecosystem of VK squares, pushouts, pullbacks, and unions to generalize classic results and provide embedding theorems akin to the adhesive/quasiadhesive case, with illustrated examples in graph-theoretic categories. Overall, the paper broadens the algebraic graph rewriting framework by permitting more flexible mono-class interactions and showing how topos-embedding techniques extend to this richer setting. The results have potential impact on categorical rewriting formalisms and their semantic guarantees, enabling new application domains and more robust compositional reasoning.

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.

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

TL;DR

This work extends the theory of adhesive and quasiadhesive categories to the broader framework of -adhesive categories. It introduces -adhesive morphisms, establishes how adhesive-like behavior can be read from subobject posets, and proves that, under suitable hypotheses, -adhesive categories embed into Grothendieck toposes while preserving pullbacks and -pushouts. The authors develop a consistent ecosystem of VK squares, pushouts, pullbacks, and unions to generalize classic results and provide embedding theorems akin to the adhesive/quasiadhesive case, with illustrated examples in graph-theoretic categories. Overall, the paper broadens the algebraic graph rewriting framework by permitting more flexible mono-class interactions and showing how topos-embedding techniques extend to this richer setting. The results have potential impact on categorical rewriting formalisms and their semantic guarantees, enabling new application domains and more robust compositional reasoning.

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 -adhesive categories, a concept recently introduced to generalize the notion of (quasi)adhesivity. We introduce the notion of -adhesive morphism, which allows us to express -adhesivity as a condition on the subobjects' posets. Moreover, -adhesive morphisms allows us to show how an -adhesive category can be embedded into a Grothendieck topos, preserving pullbacks and -pushouts.
Paper Structure (12 sections, 44 theorems, 25 equations)

This paper contains 12 sections, 44 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Synopsis
  3. $\mathcal{M}, \mathcal{N}$-adhesive categories
  4. The Van Kampen condition
  5. Definition of $\mathcal{M}, \mathcal{N}$-adhesivity
  6. Relation with $\mathcal{M}$-adhesivity
  7. $\mathcal{N}$-(pre)adhesive morphisms
  8. From $\mathcal{M}, \mathcal{N}$-unions to $\mathcal{M}, \mathcal{N}$-adhesivity
  9. Application to toposes
  10. From $\mathcal{M},\mathcal{N}$-adhesivity to $\mathcal{M},\mathcal{N}$-unions
  11. An embedding theorem for $\mathcal{M}, \mathcal{N}$-adhesive categories
  12. Conclusions and further work

Key Result

Lemma 2.3

Let $\textbf{X}$ be a category, and consider the following diagram in which the right square is a pullback. \xymatrix{X \ar[d]_{a} \ar[r]^{f}& \ar[r]^{g} Y \ar[d]^{b}& Z \ar[d]^{c}\\ A \ar[r]_{h}& B \ar[r]_{k}& C}Then the whole rectangle is a pullback if and only if the left square is one.

Theorems & Definitions (83)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Lemma 2.7
  • Definition 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 73 more