Extensivity for 2-categories
Steve Lack (7/11/01)
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.
A category E with finite coproducts is said to be extensive if for any two objects a and b of E, the functor E/a x E/b --> E/(a+b) given by coproduct is an equivalence of categories. For an introduction to extensive categories see my joint paper An introduction to distributive and extensive categories with Carboni and Walters (published in JPAA, 1993).
In this talk I defined a notion of extensive 2-category and developed the basic theory of these: all the main results about extensive categories carry over when suitably adapted to the 2-dimensional case. I also showed that the 2-category Top of elementary toposes is extensive, as is the 2-category TopS of toposes bounded over an arbitrary elementary topos S. The proof involves first showing that the opposite of the 2-category Ext1 of extensive categories with terminal object is extensive.
Other examples include the 2-categories Cat of categories, Gpd of groupoids, and any extensive category, seen as a 2-category with no non-identity 2-cells.
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Steve Lack Last modified: Thu Aug 14 09:57:11 EST 2003