Van Kampen Colimits and Path Uniqueness

TL;DR

This paper establishes an efficiently checkable condition for the Van Kampen property in presheaf topoi, linking it to a structural uniqueness feature of path-like structures, enhancing understanding of compositionality in fibred semantics.

Contribution

It provides a necessary and sufficient condition for the Van Kampen property in presheaf topoi based on path-like structure uniqueness, improving verification methods.

Findings

A new condition for Van Kampen property validity

Linking Van Kampen property to path-like structure uniqueness

Enhanced criteria for compositionality in fibred semantics

Abstract

Fibred semantics is the foundation of the model-instance pattern of software engineering. Software models can often be formalized as objects of presheaf topoi, i.e, categories of objects that can be represented as algebras as well as coalgebras, e.g., the category of directed graphs. Multimodeling requires to construct colimits of models, decomposition is given by pullback. Compositionality requires an exact interplay of these operations, i.e., diagrams must enjoy the Van Kampen property. However, checking the validity of the Van Kampen property algorithmically based on its definition is often impossible. In this paper we state a necessary and sufficient yet efficiently checkable condition for the Van Kampen property to hold in presheaf topoi. It is based on a uniqueness property of path-like structures within the defining congruence classes that make up the colimiting cocone of the models. We thus add to the statement "Being Van Kampen is a Universal Property" by Heindel and Soboci\'{n}ski the fact that the Van Kampen property reveals a presheaf-based structural uniqueness feature.