A van Kampen theorem for toposes
by
Stephen Lack
University of Sydney
Coauthors: Marta Bunge

We set out to generalize the van Kampen theorems of topology to the case of toposes bounded over an elementary topos S (and so in particular to the case of Grothendieck toposes). To this end, we take an ``abstract van Kampen theorem'' of Brown and Janelidze, set in the context of extensive categories, and we formulate and prove a 2-dimensional version of the theorem. Both the statement and the proof involve the theory of descent.

We then apply the theorem to the 2-category Top_S of toposes bounded over S, after first proving that Top_S is an extensive 2-category. This can be used to obtain information about the fundamental groupoid (in S) of a locally connected and locally simply connected S-topos. The fundamental groupoid of such an S-topos E can be seen as the Galois groupoid, in the sense of Janelidze, of the adjunction between E and S for which the right adjoint is the inverse image of the structure map of E.

Date received: May 9, 2002