A van Kampen theorem for toposes
Marta Bunge and Stephen Lack
In this paper we introduce the notion of an extensive 2-category, to be thought of as "2-category of generalized spaces".
We consider and extensive 2-category K equipped with a binary-product-preserving pseudofunctor C : K_op → Cat, which we think of as specifying the "coverings" of the generalized spaces.
We prove, in this context, a van Kampen theorem which generalizes and refines one of Brown and Janelidze.
The local properties required in this theorem are stated in terms of morphism of effective descent for the pseudofunctor C.
We specialize the general van Kampen theorem to the 2-category Top_S of toposes bounded over an elementary topos S, and to its full sub-2-category LTop_S determined by the locally connected toposes, after showing both of these 2-categories to be extensive.
We then consider three particular notions of coverings on toposes corresponding respectively to local homeomorphisms, covering projections, and unramified morphisms;
in each case we deduce a suitable version of a van Kampen theorem in terms of coverings and, under further hypotheses, also one in terms of fundamental groupoids.
An application is also given to knot groupoids and branched coverings.
Along the way we are led to investigate locally constant objects in a topos bounded over an arbitrary base topos S and to establish some new facts about them.