On Colimits in Categories of Relations

We study (finite) coproducts and colimits of -chains in Rel(C), the 2-category of relations over a given category C. The former exist and are the same as in C provided that C is extensive. The latter do not exist for example in Rel(Set). However, the canonical construction of those colimits in the category of sets can be generalized to Rel(Set). The canonical cocone is shown to satisfy a 2-categorical universal property, namely that of an lax adjoint cooplimit. Sufficient conditions for any base category C to admit the construction are given.A necessary and sufficient condition for the construction to yield colimits of -chains in the category of maps of Rel(C) is also given.