Éléments de géométrie algébrique

Grothendieck, Alexander; Dieudonné, Jean

doi:10.1007/bf02684747

Cited by 1,514 publications

(574 citation statements)

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“…In the terminology of Grothendieck [2] we conclude the fibrations are precisely the formally unramified morphisms.…”

Section: A'mentioning

confidence: 93%

Fibrations and Grothendieck topologies

Hiller

1976

Bull. Austral. Math. Soc.

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Given a site T , that is, a category equipped with a fixed Grothendieck topology, we provide a definition of fibration for morphisms of the presheaves on T . We verify that the notion is well-behaved with respect to composition, base change, and exponentiation, and is trivial on the topos of sheaves. We compare our definition to that of Kan fibration in the semisimplicial setting. Also we show how we can obtain a notion of fibration on our ground site T and investigate the resulting notion in certain ring-theoretic situations. . Introducti onLet 1 be a site; that is, a category equipped with a fixed Grothendieck topology. We have the adjoint pair S " * [T°, Se£6] sh where sh is the associated sheaf functor and S is the full topos of sheaves with respect to the topology. We define a notion of fibration for morphisms of presheaves that is well behaved with respect to composition, base change and exponentiation, and trivializes on the topos S . We investigate how our notion compares with that of Kan fibrations, when T = Old , the category of finite ordered sets equipped with an appropriate topology. We then observe we can pull our notion of fibration back to the ground site T and we investigate it in certain ring-theoretic situations.

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