Fiber functor

Fiber functors in category theory, topology and algebraic geometry refer to several loosely related functors that generalise the functors taking a covering space ${\displaystyle \pi \colon X\rightarrow S}$ to the fiber ${\displaystyle \pi ^{-1}(s)}$ over a point ${\displaystyle s\in S}$.

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

${\displaystyle a^{*}:{\mathfrak {T}}(X)\leftrightarrows {\mathfrak {Set}}:a_{*}}$

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

From covering spaces

Consider the category of covering spaces over a topological space ${\displaystyle X}$, denoted ${\displaystyle {\mathfrak {Cov}}(X)}$. Then, from a point ${\displaystyle x\in X}$ there is a fiber functor^[3]

${\displaystyle {\text{Fib}}_{x}:{\mathfrak {Cov}}(X)\to {\mathfrak {Set}}}$

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

With étale topologies

There is an algebraic analogue of covering spaces coming from the étale topology on a connected scheme ${\displaystyle S}$. The underlying site consists of finite étale covers, which are finite^[4][5] flat surjective morphisms ${\displaystyle X\to S}$ such that the fiber over every geometric point ${\displaystyle s\in S}$ is the spectrum of a finite étale ${\displaystyle \kappa (s)}$-algebra. For a fixed geometric point ${\displaystyle {\overline {s}}:{\text{Spec}}(\Omega )\to S}$, consider the geometric fiber ${\displaystyle X\times _{S}{\text{Spec}}(\Omega )}$ and let ${\displaystyle {\text{Fib}}_{\overline {s}}(X)}$ be the underlying set of ${\displaystyle \Omega }$-points. Then,

${\displaystyle {\text{Fib}}_{\overline {s}}:{\mathfrak {Fet}}_{S}\to {\mathfrak {Sets}}}$

is a fiber functor where ${\displaystyle {\mathfrak {Fet}}_{S}}$ is the topos from the finite étale topology on ${\displaystyle S}$. In fact, it is a theorem of Grothendieck that the automorphisms of ${\displaystyle {\text{Fib}}_{\overline {s}}}$ form a profinite group, denoted ${\displaystyle \pi _{1}(S,{\overline {s}})}$, 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 ${\displaystyle H_{dR}}$ sends a motive ${\displaystyle M(X)}$ to its underlying de-Rham cohomology groups ${\displaystyle H_{dR}^{*}(X)}$.^[6]

See also

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

References

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 ${\displaystyle X\to S}$ is surjective, otherwise open subschemes of ${\displaystyle S}$ could be included.
6. Deligne; Milne. "Tannakian Categories" (PDF). p. 58.

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