van Kampen theorems for toposes

Marta Bunge*, Stephen Lack

*Corresponding author for this work

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

In this paper we introduce the notion of an extensive 2-category, to be thought of as a "2-category of generalized spaces". We consider an extensive 2-category K equipped with a binary-product-preserving pseudofunctor C: Kop → CAT, which we think of as specifying the "coverings" of our 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 morphisms of effective descent for the pseudofunctor C. We specialize the general van Kampen theorem to the 2-category TopL of toposes bounded over an elementary topos L, and to its full sub 2-category LTopL 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 L and to establish some new facts about them.

Original languageEnglish
Pages (from-to)291-317
Number of pages27
JournalAdvances in Mathematics
Volume179
Issue number2
DOIs
Publication statusPublished - 10 Nov 2003
Externally publishedYes

Keywords

  • 2-categories
  • Branched coverings
  • Descent
  • Extensive categories
  • Fundamental groupoids
  • Intensive and extensive quantities
  • Locally constant objects
  • Toposes
  • Unramified maps
  • van Kampen theorems

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