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Groupoids in categories with pretopology

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Ralf Meyer and Chenchang Zhu

We survey the general theory of groupoids, groupoid actions,
groupoid principal bundles, and various kinds of morphisms between
groupoids in the framework of categories with
pretopology. The categories of topological spaces and finite or
infinite dimensional manifolds are examples of such categories. We
study extra assumptions on pretopologies that are needed for this
theory. We check these extra assumptions in several categories with
pretopologies.

Functors between groupoids may be localised at equivalences in two
ways. One uses spans of functors, the other bibundles (commuting
actions) of groupoids. We show that both approaches give equivalent
bicategories. Another type of groupoid morphism, called an actor,
is closely related to functors between the categories of groupoid
actions. We also generalise actors using bibundles, and show that
this gives another bicategory of groupoids.

Keywords:
Grothendieck topology; cover; groupoid; groupoid action;
groupoid sheaf; principal bundle; Hilsum--Skandalis morphism;
anafunctor; bicategory; comorphism; infinite dimensional groupoid

2010 MSC:
51H25

*Theory and Applications of Categories,*
Vol. 30, 2015,
No. 55, pp 1906-1998.

Published 2015-12-31.

http://www.tac.mta.ca/tac/volumes/30/55/30-55.pdf

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