Being Van Kampen in Presheaf Topoi is a Uniqueness Property

Fibred semantics is the foundation of the model-instance pattern of software engineering. Software models can often be formalized as objects of presheaf topoi, e.g. the category of directed graphs. Multimodeling requires to construct colimits of diagrams of single models and their instances, while decomposition of instances of the multimodel is given by pullback. Compositionality requires an exact interplay of these operations, i.e., the 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 easily checkable condition for the Van Kampen property to hold for diagrams 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 Sobocinski presented at CALCO 2009 the fact that the Van Kampen property reveals a set-based structural uniqueness feature