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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:
[1] Artin, M.; Grothendieck, A.; Verdier, J.-L., Théorie des topos et cohomologie étale des schémas (SGA4), (19721973), Springer-Verlag
[2] Brown, K. S., Abstract homotopy and generalized sheaf cohomology, Transactions of the Amer. Math. Soc., 186, 419-458, (1973) · Zbl 0245.55007
[3] Cisinski, D.-C., Les préfaisceaux comme modèles des types d’homotopie, (2002)
[4] Cisinski, D.-C., Le localisateur fondamental minimal, Cahiers de topologie et géométrie différentielle catégoriques, 45-2, 109-140, (2004) · Zbl 1063.18013
[5] Grothendieck, A., Pursuing stacks, (1983)
[6] Grothendieck, A., Les dérivateurs, (1990)
[7] Heller, A., Homotopy theories, Memoirs of the Amer. Math. Soc., 71, 383, (1988) · Zbl 0643.55015
[8] Maltsiniotis, G., La théorie de l’homotopie de Grothendieck · Zbl 1104.18005
[9] Quillen, D., Higher algebraic K-theory : I, Algebraic K-theory I, 85-147, (1973), Springer-Verlag · Zbl 0292.18004
[10] Thomason, R. W.\(, \mathcal{C}\mathit{at}\) as a closed model category, Cahiers de topologie et géométrie différentielle catégoriques, XXI-3, 305-324, (1980) · Zbl 0473.18012
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