(joint work with Steve Lack)
We shall begin by briefly recalling the definition of adhesive categories and mention some of their applications to computer science. Adhesive categories can be described, roughly, as categories which have well-behaved pushouts along monos, analogous to the way that extensive categories have well-behaved coproducts.
It turns out that adhesive categories can be seen as special cases of a more general theory of quasiadhesive categories, where only pushouts along regular monos are required to be well-behaved. We shall explore some of the basic theory of quasiadhesive categories and mention several relevant examples, which include the category of binary relations (or directed graphs with at most one edge from one vertex to any other) and the category of algebraic specifications. We shall conclude by noticing that many interesting examples of quasiadhesive categories can be described uniformly using some results of Carboni and Johnstone's work on Artin glueing.