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

2024-01-23

AI Summary

This paper addresses the limited expressiveness of classical adhesive category theory by introducing an axiomatic framework for $mathcal{M},mathcal{N}$-adhesive categories, featuring the novel notion of $mathcal{N}$-adhesive morphisms and characterizing adhesivity as a structural condition on the poset of subobjects. Methodologically, it integrates category theory, the theory of regular monomorphisms, and Grothendieck topos embedding techniques to establish a structural embedding theorem preserving pullbacks and $mathcal{M},mathcal{N}$-pushouts. The main contributions are: (1) a systematic generalization of core properties of adhesive and quasi-adhesive categories; (2) a rigorous proof that $mathcal{M},mathcal{N}$-adhesive categories admit full, faithful, and structure-preserving embeddings into Grothendieck topoi; and (3) a broader, verifiable semantic foundation for algebraic graph rewriting, enhancing both theoretical expressivity and practical applicability in rule-based system modeling.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

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'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.

Problem

• Adhesive Categories
• Grothendieck Topos
• Combination Rules

Innovation

• Adhesive Categories
• Grothendieck Topos
• Simplicity in Adhesiveness Description