| Summary: | 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.
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