Van Kampen squares and adhesive categories

Steve Lack – 8 October 2003

This talk was based on joint work with Pawel Sobocinski, and now forms part of a preprint entitled Adhesive categories, available as RS-03-31 in the BRICS Report Series. I defined {\em van Kampen squares} to be pushouts along monomorphisms which are "well-behaved" in the sense that coproducts in extensive categories are well-behaved. I defined a category to be {\em adhesive} if it has all pullbacks, pushouts along monomorphisms, and the latter are van Kampen squares. Every topos E is adhesive, but so are its slices e/E. Every adhesive category with a strict initial object is adhesive. I developed a little of the theory of adhesive categories, and explained the connection between van Kampen squares and the van Kampen theorems of topology; this involves the categorical van Kampen theorem of Brown and Janelidze [Van Kampen theorems for categories of covering morphisms in lextensive categories, JPAA 119:255-263, 1997.]