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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