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TRIPLES, ALGEBRAS AND COHOMOLOGY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2003
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Topological and smooth stacks
"... Abstract. We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a “stack over a stack ” and the distinction between small stacks (which are algebraic objects) and large stacks (which are ..."
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Abstract. We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a “stack over a stack ” and the distinction between small stacks (which are algebraic objects) and large stacks (which are generalized spaces). 1.
Comparison of Abelian Categories Recollements
 DOCUMENTA MATH.
, 2004
"... We give a necessary and sufficient condition for a morphism between recollements of abelian categories to be an equivalence. ..."
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We give a necessary and sufficient condition for a morphism between recollements of abelian categories to be an equivalence.
Stacks for Everybody
"... Abstract. Let S be a category with a Grothendieck topology. A stack over S is a category fibered in groupoids over S, such that isomorphisms form a sheaf and every descent datum is effective. If S is the category of schemes with the étale topology, a stack is algebraic in the sense of DeligneMumfor ..."
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Abstract. Let S be a category with a Grothendieck topology. A stack over S is a category fibered in groupoids over S, such that isomorphisms form a sheaf and every descent datum is effective. If S is the category of schemes with the étale topology, a stack is algebraic in the sense of DeligneMumford (respectively Artin) if it has an étale (resp. smooth) presentation. I will try to explain the previous definitions so as to make them accessible to the widest possible audience. In order to do this, we will keep in mind one fixed example, that of vector bundles; if you know what pullback of vector bundles is in some geometric context (schemes, complex analytic spaces, but also varieties or manifolds) you should be able to follow this exposition. 1.
Localglobal principles for Galois cohomology. 2012 manuscript. Available at arXiv:math/1208.6359
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SUBMERSIONS AND EFFECTIVE DESCENT OF ÉTALE MORPHISMS
, 2007
"... Using the flatification by blowup result of Raynaud and Gruson, we obtain new results for submersive and subtrusive morphisms. We show that universally subtrusive morphisms, and in particular universally open morphisms, are morphisms of effective descent for the fibered category of étale morphism ..."
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Using the flatification by blowup result of Raynaud and Gruson, we obtain new results for submersive and subtrusive morphisms. We show that universally subtrusive morphisms, and in particular universally open morphisms, are morphisms of effective descent for the fibered category of étale morphisms. Our results extend and supplement previous treatments on submersive morphisms by Grothendieck, Picavet and Voevodsky. Applications include the universality of geometric quotients and the elimination of noetherian hypotheses in many instances.