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Fiber functor

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Fiber functors in category theory, topology and algebraic geometry refer to several loosely related functors that generalise the functors taking a covering space to the fiber over a point .

Definition

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A fiber functor (or fibre functor) is a loose concept which has multiple definitions depending on the formalism considered. One of the main initial motivations for fiber functors comes from Topos theory.[1] Recall a topos is the category of sheaves over a site. If a site is just a single object, as with a point, then the topos of the point is equivalent to the category of sets, . If we have the topos of sheaves on a topological space , denoted , then to give a point in is equivalent to defining adjoint functors

The functor sends a sheaf on to its fiber over the point ; that is, its stalk.[2]

From covering spaces

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Consider the category of covering spaces over a topological space , denoted . Then, from a point there is a fiber functor[3]

sending a covering space to the fiber . This functor has automorphisms coming from since the fundamental group acts on covering spaces on a topological space . In particular, it acts on the set . In fact, the only automorphisms of come from .

With étale topologies

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There is an algebraic analogue of covering spaces coming from the étale topology on a connected scheme . The underlying site consists of finite étale covers, which are finite[4][5] flat surjective morphisms such that the fiber over every geometric point is the spectrum of a finite étale -algebra. For a fixed geometric point , consider the geometric fiber and let be the underlying set of -points. Then,

is a fiber functor where is the topos from the finite étale topology on . In fact, it is a theorem of Grothendieck the automorphisms of form a profinite group, denoted , and induce a continuous group action on these finite fiber sets, giving an equivalence between covers and the finite sets with such actions.

From Tannakian categories

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Another class of fiber functors come from cohomological realizations of motives in algebraic geometry. For example, the De Rham cohomology functor sends a motive to its underlying de-Rham cohomology groups .[6]

See also

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References

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  1. ^ Grothendieck, Alexander. "SGA 4 Exp IV" (PDF). pp. 46–54. Archived (PDF) from the original on 2020-05-01.
  2. ^ Cartier, Pierre. "A Mad Day's Work: From Grothendieck to Connes and Kontsevich – The Evolution of Concepts of Space and Symmetry" (PDF). p. 400 (12 in pdf). Archived (PDF) from the original on 5 Apr 2020.
  3. ^ Szamuely. "Heidelberg Lectures on Fundamental Groups" (PDF). p. 2. Archived (PDF) from the original on 5 Apr 2020.
  4. ^ "Galois Groups and Fundamental Groups" (PDF). pp. 15–16. Archived (PDF) from the original on 6 Apr 2020.
  5. ^ Which is required to ensure the étale map is surjective, otherwise open subschemes of could be included.
  6. ^ Deligne; Milne. "Tannakian Categories" (PDF). p. 58.
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