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Zbl 0871.18003
Pronk, Dorette A.
Étendues and stacks as bicategories of fractions. (English)
[J] Compos. Math. 102, No.3, 243-303 (1996). ISSN 0010-437X; ISSN 1570-5846

The main purpose of this paper is to give the construction of a bicategory of fractions, as a generalization of the Gabriel-Zisman notion of a category of fractions. This construction leads to the following theorem: There is a canonical equivalence of bicategories $(T_1$-Étendues)$\cong (T_1$-Étale Groupoids) $[W^{-1}]$. Here $(T_1$-Étendues) is the 2-category of toposes which roughly speaking locally look like a $T_1$-space, $(T_1$-Étale Groupoids) is the 2-category of étale groupoids in the category of $T_1$-spaces; and $W$ denotes the class of weak equivalences of groupoids (the equivalence in the theorem above is an equivalence of bicategories because in general the category of fractions of a 2-category will turn out to be a bicategory and is called a bicategory of fractions).\par Subsequently, the bicategory of fractions construction is applied to give some results about differentiable stacks and about étale groupoids in schemes and algebraic stacks (the differentiable stacks are defined over the category of differentiable manifolds). It is shown that étendues and stacks (among others) arise as bicategories of fractions from appropriate categories of groupoids.
[I.Tofan (Iaşi)]

MSC 2000:: *18D05 2-categories and generalizations
18B25 Topoi
18B40 Generalizations of groups viewed as categories
14A20 Generalizations (algebraic spaces, etc.)

Keywords: $T\sb 1$-spaces; bicategory of fractions; 2-category of toposes; étale groupoids; weak equivalences; equivalence of bicategories; differentiable stacks; schemes; algebraic stacks; étendues

Cited in: Zbl 1175.18008

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