nLab rigid monoidal category

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 $V$ in a monoidal category $M$ is an object $V^*$ equipped with unit $\eta : 1 \rightarrow V^* \otimes V$ and counit maps $\epsilon: V \otimes V^* \rightarrow 1$ 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 $F : C \rightarrow C$ has a right adjoint $F^* : C \rightarrow C$ if and only if $F^*$ is a right dual of $F$ in the monoidal category $End(C)$.

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 ${}^*\colon M \to M$. (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.

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

Free rigid monoidal categories

The inclusion of the 2-category of monoidal categories into the 2-category of rigid monoidal categories admits a left 2-adjoint functor? $L$.

Furthermore, the unit of the adjunction is a strong monoidal fully faithful functor, i.e., any monoidal category $C$ admits a fully faithful strong monoidal functor $C\to L(C)$, where $L(C)$ is a rigid monoidal category.

See Theorems 1 and 2 in Delpeuch.

References

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

• Antonin Delpeuch?, Autonomization of monoidal categories, arXiv, doi.

Related concepts

• star-autonomous category

• (∞,n)-category with duals

• natural language syntax