Fiber functor

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 ${\displaystyle \pi _{1}(X,x)}$.

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.

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].