## 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: K^{op} → 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 Top_{L} of toposes bounded over an elementary topos L, and to its full sub 2-category LTop_{L} 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 language | English |
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Pages (from-to) | 291-317 |

Number of pages | 27 |

Journal | Advances in Mathematics |

Volume | 179 |

Issue number | 2 |

DOIs | |

Publication status | Published - 10 Nov 2003 |

Externally published | Yes |

## Keywords

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