symmetric monoidal (∞,1)-category of spectra
abstract duality: opposite category,
The archetypical example which gives rise to the term is the following. If one has the category of covering spaces of a (nice enough) topological space , then after picking any point the operation of forming the fibre over that point gives a functor to the category Set of sets. The natural automorphisms of this functor form the (algebraic) fundamental group, . 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 -sets. This equivalence is compatible with the chosen fibre functor and the further forgetful functor from to . Extracting from this situation, that forgetful functor is thought of as being a fibre functor as well. Any category of G-sets, for 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 , 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.
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.
|monoid/associative algebra||category of modules|
|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|
|form Drinfeld double||form Drinfeld center|
|trialgebra||Hopf monoidal category|
|monoidal category||2-category of module categories|
|Hopf monoidal category||monoidal 2-category (with some duality and strictness structure)|
|monoidal 2-category||3-category of module 2-categories|