Internal categories, anafunctors and localisations

Details Internal categories, anafunctors and localisations Internal categories, anafunctors and localisations

2011-01-12

arxiv.org/abs/1101.2363v1

Mathematics

Category Theory

33 pages

Scientific paper

We show that given a category S (not necessarily finitely complete!) and a 2-category Cat'(S)\subset Cat(S) of internal categories in S, closed under some natural operations, the bicategorical localisation Cat'(S)[W_E^{-1}] exists for W_E a class of weak equivalences analogous to fully faithful and essentially surjective functors. Secondly we show that this localisation is given by a bicategory with anafunctors as 1-arrows when S is a site, and give conditions when various such bicategories of anafunctors are equivalent. While the connections to stacks are not pursued here, this work provides a precise setting to the claim that stacks on arbitrary sites are internal groupoids up to essential equivalence. Finally, we make some conjectural comments about localisation of bicategories qua (\infty,2)-categories.

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