nLab
rigid monoidal category

Contents

Idea

A rigid (monoidal) category, also called an autonomous (monoidal) category is a kind of category with duals. Specifically, all of its objects are dualisable on both the left and the right.

Definition

A monoidal category is rigid if every object has duals on both sides. If only one type of dual exists, we speak of left rigid (or left autonomous) or right rigid categories.

Conventions differ regarding which type of duals are which. One convention is as follows: a right dual of an object VV in a monoidal category MM is an object V *V^* equipped with unit η:1V *V\eta : 1 \rightarrow V^* \otimes V and counit maps ϵ:VV *1\epsilon: V \otimes V^* \rightarrow 1 satisfying the triangle identities (the snake diagrams), while a left dual is the dual notion. This convention fits in with the standardized conventions regarding adjoint functors: an endofunctor F:CCF : C \rightarrow C has a right adjoint F *:CCF^* : C \rightarrow C if and only if F *F^* is a right dual of FF in the monoidal category End(C)End(C).

Remarks

Note that this definition only asserts the existence of the dual objects. It does not assert that specific duals have been chosen. However, the choice of duals is unique up to unique isomorphism, justifying reference to ‘the?’ dual of an object; in fact, this extends to a contravariant anafunctor *:MM{}^*\colon M \to M. (Using the axiom of choice to pick duals for every object at once, we can make this into a strict functor.)

Nor does this definition assert that the right dual of an object is isomorphic to its left dual: this need not be the case in general, though it is true in a braided monoidal category, and thus automatically also in a symmetric monoidal category. (A rigid monoidal category which is also symmetric is sometimes called compact closed or simply “compact”.)

In practice, algebraic geometers are the most frequent users of the term ‘rigid’, and they focus on the symmetric monoidal case, so they ignore the difference between right and left duals.

Properties

Tannaka duality

The statement of Tannaka duality for associative algebras says that rigid monoidal categories equipped with a fiber functor are categories of modules over a Hopf algebra.

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

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

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module

Revised on January 31, 2014 03:26:23 by Urs Schreiber (89.204.139.167)