This talk was based on joint work with Marta Bunge, and now forms part of a paper entitled Van Kampen theorem for toposes. to appear in Advances in Mathematics. The talk builds on my talk of the previous week Extensivity for 2-categories.
Brown and Janelidze (Van Kampen theorems for categories of covering morphisms in lextensive categories, J. Pure Appl. Alg. 119:255-263, 1997) proved an abstract van Kampen theorem in the context of an extensive category K equipped with a suitable class F of morphisms of K to be thought of as ``coverings'. In this talk I described a generalization of the theorem of Brown and Janelidze. The extensive category K was replaced by an extensive 2-category K, and the class F was replaced by a pseudofunctor A:Kop→CAT which preserves binary products. Our abstract van Kampen theorem then stated that A sends certain pullback diagrams in K to pullbacks in CAT; the hypotheses on the pullback diagrams were expressed in terms of descent theory.
In our main example, K is the 2-category TopS of toposes bounded over a base topos S, and A is the pseudofunctor sending a topos E to its subcategory of locally constant objects.
Some applications involving branched coverings and knot groups were given by Marta in the following talk, Locally path simply connected toposes and their fundamental groupoids.