nLab fiber functor

Idea

A forgetful functor from a category of actions/representations to the underlying sets/spaces is often called a fiber functor, notably in the context of Tannaka duality and Galois theory.

The archetypical example which gives rise to the term is the following. If one has the category $Et(X)$ of covering spaces of a (nice enough) topological space $X$, then after picking any point $x \in X$ the operation of forming the fibre over that point gives a functor $fib_x \colon Et(X)\to Set$ to the category Set of sets. The natural automorphisms of this functor form the (algebraic) fundamental group, $\pi_1(X)$. The main theorem of the Galois-Poincaré theory of covering spaces can be viewed as stating that this sets up an equivalence of categories between that category of covering spaces and the category of $\pi_1$-sets. This equivalence is compatible with the chosen fibre functor and the further forgetful functor from $\pi_1-Sets$ to $Sets$. Extracting from this situation, that forgetful functor is thought of as being a fibre functor as well. Any category of G-sets, for $G$ a group, gives a monoidal category, and the forgetful functor is a monoidal functor; of course, the category of G-sets corresponds to the category of permutation representations of $G$, and generalising this basic example leads to the following idea.

The forgetful strict monoidal functor from a monoidal category to some standard monoidal category, usually the category Vect of vector spaces over a field is called the fiber functor in some contexts, especially in Tannaka reconstruction in which the symmetry object is reconstructed from the (object of) endomorphisms of the fiber functor. In mixed Tannaka duality, a single fiber functor does not suffice for reconstruction, but rather a family of fiber functors to different bases.

Historically, the notion was used extensively, starting in the 1960s by Grothendieck and his collaborators. The terminology is from the Grothendieck Galois theory: namely Grothendieck reconstructs the (profinite) fundamental group in algebraic geometry from a fiber functor: the fundamental group acts on a covering by deck transformations and by monodromy transformation for bundles over the covering, algebraic analogues of such a picture can thus be used to define a fundamental group, not by using some idea of loops which are often hard to define in abstract setups, but by a form of Tannakian reconstruction. Grothendieck also introduced the idea of using many “base points” that is, many fibre functors, thus giving an abstract analogue of the fundamental groupoid of a space.

Warning

Please do not confuse the terminology with the case of a functor which is a Grothendieck fibration (i.e. a fibered category); nor with a fiber (“preimage” of a sort) of a functor. These are related ideas but are best kept separate.

Properties

Tannaka duality

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebra	category of modules
$A$	$Mod_A$
$R$-algebra	$Mod_R$-2-module
sesquialgebra	2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebra	strict 2-ring: monoidal category with fiber functor
Hopf algebra	rigid monoidal category with fiber functor
hopfish algebra (correct version)	rigid monoidal category (without fiber functor)
weak Hopf algebra	fusion category with generalized fiber functor
quasitriangular bialgebra	braided monoidal category with fiber functor
triangular bialgebra	symmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)	rigid braided monoidal category with fiber functor
triangular Hopf algebra	rigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)	rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld double	form Drinfeld center
trialgebra	Hopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category	2-category of module categories
$A$	$Mod_A$
$R$-2-algebra	$Mod_R$-3-module
Hopf monoidal category	monoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category	3-category of module 2-categories
$A$	$Mod_A$
$R$-3-algebra	$Mod_R$-4-module

See also

• Tannakian category

References

• P. Cartier, A mad day’s work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry, Bull. Amer. Math. Soc. 38 (2001), 389-408, pdf

• P. Deligne, Catégories Tannakiennes

For fusion 2-categories:

• Thibault Decoppet, Matthew Yu. Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories. (2023) [arXiv:2306.08117]