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Asphericity structures, smooth functors and fibrations. (Structures d’asphéricité, foncteurs lisses, et fibrations.) (French) Zbl 1093.18004
In his famous 1983’s manuscript “Pursuing Stacks”, A. Grothendieck introduces the notion of proper and smooth functors (defined, like for schemes, when some change of base morphisms are isos). Such functors are simply characterized and the theory only depends on a few formal properties (including Quillen’s Theorem A) of the class $$W_\infty$$ of weak equivalences in $${\mathcal C}at$$. This leads to define the notion of fundamental localizer $$W$$ (a class of functors satisfying suitable conditions yet well known in homotopy theory).
The present paper generalizes the theory of smooth functors in order to have a new characterization of fibred categories. Such are not exactly the $$W$$-smooth functors for a fundamental localizer $$W$$. But it is done by associating smooth functors to $$a$$, here defined, to get a minimal right asphericity structure for categories.

MSC:
 18D30 Fibered categories 18G55 Nonabelian homotopical algebra (MSC2010) 18A22 Special properties of functors (faithful, full, etc.) 14A99 Foundations of algebraic geometry 18F99 Categories in geometry and topology 14F20 Étale and other Grothendieck topologies and (co)homologies
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References:
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