Fiber functor

In category theory, a branch of mathematics, a fiber functor is a faithful k-linear tensor functor from a tensor category to the category of finite-dimensional k-vector spaces.^[1]

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.^[2] 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.^[3]

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^[4]

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

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 etale topologies[]

There is algebraic analogue of covering spaces coming from the Étale topology on a connected scheme ${\displaystyle S}$. The underlying site consists of finite etale covers, which are finite^[5][6] flat surjective morphisms ${\displaystyle X\to S}$ such that the fiber over every geometryic point ${\displaystyle s\in S}$ is the spectrum of a finite etale ${\displaystyle \kappa (s)}$-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 etale topology on ${\displaystyle S}$. In fact, it is a theorem of Grothendieck 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 ${\displaystyle H_{dR}^{*}(X)}$.^[7]

See also[]

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

References[]

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