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 $\pi\colon X\rightarrow S$ 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.{{Cite web|title=SGA 4 Exp IV|url=http://www.normalesup.org/~forgogozo/SGA4/04/04.pdf|last=Grothendieck|first=Alexander|date=|website=|pages=46–54|url-status=live|archive-url=https://web.archive.org/web/20200501174937/http://www.normalesup.org/~forgogozo/SGA4/04/04.pdf|archive-date=2020-05-01|access-date=}} 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, $\mathfrak{Set}$ . If we have the topos of sheaves on a topological space $X$ , denoted $\mathfrak{T}(X)$ , then to give a point $a$ in $X$ is equivalent to defining adjoint functors

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

The functor

a^*

sends a sheaf

\mathfrak{F}

X

to its fiber over the point

a

; that is, its stalk.{{Cite web|url=https://www.ams.org/journals/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf|title=A Mad Day's Work: From Grothendieck to Connes and Kontsevich – The Evolution of Concepts of Space and Symmetry|last=Cartier|first=Pierre|date=|website=|page=400 (12 in pdf)|url-status=live|archive-url=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|archive-date=5 Apr 2020|access-date=}}

= From covering spaces =

Consider the category of covering spaces over a topological space $X$ , denoted $\mathfrak{Cov}(X)$ . Then, from a point $x \in X$ there is a fiber functor{{Cite web|url=https://www.renyi.hu/~szamuely/heid.pdf|title=Heidelberg Lectures on Fundamental Groups|last=Szamuely|date=|website=|page=2|url-status=live|archive-url=https://web.archive.org/web/20200405211320/https://www.renyi.hu/~szamuely/heid.pdf|archive-date=5 Apr 2020|access-date=}}

$\text{Fib}_x: \mathfrak{Cov}(X) \to \mathfrak{Set}$

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

\text{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{{Cite web|url=https://math.berkeley.edu/~dcorwin/files/etale.pdf|title=Galois Groups and Fundamental Groups|last=|first=|date=|website=|pages=15–16|url-status=live|archive-url=https://web.archive.org/web/20200406003039/https://math.berkeley.edu/~dcorwin/files/etale.pdf|archive-date=6 Apr 2020|access-date=}}Which is required to ensure the étale map $X \to S$ is surjective, otherwise open subschemes of $S$ could be included. 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 $\overline{s}:\text{Spec}(\Omega) \to S$ , consider the geometric fiber $X\times_S\text{Spec}(\Omega)$ and let $\text{Fib}_{\overline{s}}(X)$ be the underlying set of $\Omega$ -points. Then,

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

is a fiber functor where

\mathfrak{Fet}_S

is the topos from the finite étale topology on

S

. In fact, it is a theorem of Grothendieck that the automorphisms of

\text{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 $H_{dR}^*(X)$ .{{Cite web|url=https://www.jmilne.org/math/xnotes/tc.pdf|title=Tannakian Categories|last1=Deligne|last2=Milne|date=|website=|page=58|archive-url=|archive-date=|access-date=}}

References

External links

[http://fabrice.orgogozo.perso.math.cnrs.fr/SGA4/index.html SGA 4] and [https://web.archive.org/web/20200501174937/http://www.normalesup.org/~forgogozo/SGA4/04/04.pdf 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

Category:Category theory

Category:Monoidal categories