nLab
rigid monoidal category

Context

Monoidal categories

With symmetry

With duals for objects

  • (list of them)

  • (what they have)

  • , a.k.a.

  • , a.k.a.

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

  • (, , , )

Internal monoids

Examples

Theorems

In higher category theory

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 (this last fact can be considered an algebraic form of the “Whitney trick” for knots; see this MO discussion). Note that 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.

for over /

/
AAMod AMod_A
RR-Mod RMod_R-
= with -preserving
strict : with
with
(correct version) (without fiber functor)
with generalized
with
with
() with
with
() with and Schur smallness
form form

2-Tannaka duality for over

AAMod AMod_A
RR-Mod RMod_R-
(with some duality and strictness structure)

3-Tannaka duality for over

AAMod AMod_A
RR-Mod RMod_R-

References

  • N. Saavedra Rivano, “Catégories Tannakiennes.” Bulletin de la Société Mathématique de France 100 (1972): 417-430. EuDML

Last revised on February 1, 2016 at 08:53:23. See the history of this page for a list of all contributions to it.