Fiber functor explained

\pi\colon X → S to the fiber \pi ^{-1} (s) over a point s\in S .

Definition

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,

ak{Set} . If we have the topos of sheaves on a topological space X , denoted ak{T}(X) , then to give a point a in X is equivalent to defining adjoint functors

a ^{*: ak{T}(X)\leftrightarrows ak{Set}:a} _{*}

The functor a ^{*} sends a sheaf ak{F} on X to its fiber over the point a ; that is, its stalk.^[2]

From covering spaces

Consider the category of covering spaces over a topological space

X , denoted ak{Cov}(X) . Then, from a point x \in X there is a fiber functor^[3]

Fib _{x:} ak{Cov}(X) \to ak{Set}

sending a covering space \pi:Y \to X to the fiber \pi ^{-1} (x) . This functor has automorphisms coming from \pi _{1(X,x)} since the fundamental group acts on covering spaces on a topological space X . In particular, it acts on the set \pi ^{-1} (x) \subset Y . In fact, the only automorphisms of Fib _{x} come from \pi _{1(X,x)} .

With étale topologies

There is an algebraic analogue of covering spaces coming from the étale topology on a connected scheme

S . The underlying site consists of finite étale covers, which are finite^{[4] [5]} flat surjective morphisms X \to S such that the fiber over every geometric point s \in S is the spectrum of a finite étale \kappa(s) -algebra. For a fixed geometric point \overline{s}: Spec (\Omega) \to S , consider the geometric fiber X x _{SSpec(\Omega)} and let Fib _{\overline{s} }(X) be the underlying set of \Omega -points. Then,

Fib _{\overline{s} }: \mathfrak_S \to \mathfrak

is a fiber functor where ak{Fet} _{S} is the topos from the finite étale topology on S . In fact, it is a theorem of Grothendieck the automorphisms of Fib _{\overline{s} } form a profinite group, denoted \pi _{1(S,\overline{s})} , 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

Another class of fiber functors come from cohomological realizations of motives in algebraic geometry. For example, the De Rham cohomology functor

H _{dR} sends a motive M(X) to its underlying de-Rham cohomology groups H ^{*(X)} _{dR} .^[6]

See also

• Topos
• Étale topology
• Motive (algebraic geometry)
• Anabelian geometry

External links

• SGA 4 and SGA 4 IV
• Motivic Galois group - https://web.archive.org/web/20200408142431/https://www.him.uni-bonn.de/fileadmin/him/Lecture_Notes/motivic_Galois_group.pdf

Notes and References

1. Web site: SGA 4 Exp IV. Grothendieck. Alexander. 46–54. live. https://web.archive.org/web/20200501174937/http://www.normalesup.org/~forgogozo/SGA4/04/04.pdf. 2020-05-01.
2. Web site: A Mad Day's Work: From Grothendieck to Connes and Kontsevich – The Evolution of Concepts of Space and Symmetry. Cartier. Pierre. 400 (12 in pdf). live. https://web.archive.org/web/20200405212545/https://www.ams.org/journals/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf. 5 Apr 2020.
3. Web site: Heidelberg Lectures on Fundamental Groups. Szamuely. 2. live. https://web.archive.org/web/20200405211320/https://www.renyi.hu/~szamuely/heid.pdf. 5 Apr 2020.
4. Web site: Galois Groups and Fundamental Groups. 15–16. live. https://web.archive.org/web/20200406003039/https://math.berkeley.edu/~dcorwin/files/etale.pdf. 6 Apr 2020.
5. Which is required to ensure the étale map X \to S is surjective, otherwise open subschemes of S could be included.
6. Web site: Tannakian Categories. Deligne. Milne. 58.