category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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.
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 $End(C)$.
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.
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 |