Being Van Kampen is a universal property

Colimits that satisfy the Van Kampen condition have interesting exactness properties. We show that the elementary presentation of the Van Kampen condition is actually a characterisation of a universal property in the associated bicategory of spans. The main theorem states that Van Kampen cocones are precisely those diagrams in a category that induce bicolimit diagrams in its associated bicategory of spans, provided that the category has pullbacks and enough colimits. Introduction Exactness, or in other words, the relationship between limits and colimits in various categories of interest is a research topic with several applications in theoretical computer science, including the solution of recursive domain equations [35], semantics of concurrent programming languages [37] and the study of formal grammars and transformation systems [10]. Researchers have identified several classes of categories in which certain limits and colimits relate to each other in useful ways; extensive categories [7, 31] and adhesive categories [29] are two relatively recent examples. Going further back, research on toposes and quasitoposes involved elaborate study of their exactness properties [20, 38]. Extensive categories [7] have coproducts that are “well-behaved” with respect to pullbacks; more concretely, they are disjoint and stable under pullback. Extensivity has been used by mathematicians [6] and computer scientists [33] alike. In the presence of products, extensive categories are distributive [7] and thus can be used, for instance, to model circuits [36] or to give models of specifications [18]. Sets and topological spaces inhabit extensive categories while quasitoposes are not, in general, extensive [21]. Adhesive categories [28, 29] have pushouts along monos that are similarly “well-behaved” with respect to pullbacks—they are instances of Van Kampen squares. Adhesivity has been used as a categorical foundation for double-pushout graph transformation [28, 11] and has 1998 ACM Subject Classification: F.3.2, F.4.2.