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.
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. 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, formula_1. If we have the topos of sheaves on a topological space formula_2, denoted formula_3, then to give a point formula_4 in formula_2 is equivalent to defining adjoint functorsformula_6The functor formula_7 sends a sheaf formula_8 on formula_2 to its fiber over the point formula_4; that is, its stalk.
From covering spaces.
Consider the category of covering spaces over a topological space formula_2, denoted formula_12. Then, from a point formula_13 there is a fiber functorformula_14sending a covering space formula_15 to the fiber formula_16. This functor has automorphisms coming from formula_17 since the fundamental group acts on covering spaces on a topological space formula_2. In particular, it acts on the set formula_19. In fact, the only automorphisms of formula_20 come from formula_17.
With etale topologies.
There is algebraic analogue of covering spaces coming from the Étale topology on a connected scheme formula_22. The underlying site consists of finite etale covers, which are finite flat surjective morphisms formula_23 such that the fiber over every geometryic point formula_24 is the spectrum of a finite etale formula_25-algebra. For a fixed geometric point formula_26, consider the geometric fiber formula_27 and let formula_28 be the underlying set of formula_29-points. Then,formula_30is a fiber functor where formula_31 is the topos from the finite etale topology on formula_22. In fact, it is a theorem of Grothendieck the automorphisms of formula_33 form a Profinite group, denoted formula_34, 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 formula_35 sends a motive formula_36 to its underlying de-Rham cohomology groups formula_37.