Fiber functors in category theory, topology and algebraic geometry refer to several loosely related functors that generalise the functors taking a covering space
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,
If we have the topos of sheaves on a topological space
, then to give a point
is equivalent to defining adjoint functors
Consider the category of covering spaces over a topological space
{\displaystyle {\text{Fib}}_{x}:{\mathfrak {Cov}}(X)\to {\mathfrak {Set}}}
sending a covering space
This functor has automorphisms coming from
since the fundamental group acts on covering spaces on a topological space
In particular, it acts on the set
There is an algebraic analogue of covering spaces coming from the étale topology on a connected scheme
The underlying site consists of finite étale covers, which are finite[4][5] flat surjective morphisms
such that the fiber over every geometric point
is the spectrum of a finite étale
For a fixed geometric point
, consider the geometric fiber
be the underlying set of
{\displaystyle {\text{Fib}}_{\overline {s}}:{\mathfrak {Fet}}_{S}\to {\mathfrak {Sets}}}
is the topos from the finite étale topology on
In fact, it is a theorem of Grothendieck that the automorphisms of
form a profinite group, denoted
, 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
sends a motive
to its underlying de-Rham cohomology groups