category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
Intuitively speaking, a braided monoidal category is a category with a tensor product and an isomorphism called the ‘braiding’ which lets us ‘switch’ two objects in a tensor product like $x \otimes y$.
A braided monoidal category, or braided tensor category, is a monoidal category $V$ equipped with a natural isomorphism
called the braiding, which must satisfy two axioms called hexagon identities encoding the compatibility of the braiding with the associator for the tensor product.
If the braiding squares to the identity, then the braided monoidal category is a symmetric monoidal category.
To see the hexagon identities, let us write $a_{x,y,z} : (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ for the components of the associator in $V$. Then the first hexagon identity says that for all $x,y,z \in Obj(V)$ the following diagram commutes:
The second hexagon identity says that for all $x,y,z \in Obj(V)$ the following diagram commutes:
Intuitively speaking, the first hexagon identity says we can braid $x \otimes y$ past $z$ all at once or in two steps. The second hexagon identity says we can braid $x$ past $y \otimes z$ ‘all at once’ or in two steps.
From these axioms, it follows that the braiding is compatible with the left and right unitors $l_x : I \otimes x \to x$ and $r_x : x \otimes I \to x$. That is to say, for all objects $x$ the diagram
commutes.
In terms of the language of k-tuply monoidal n-categories a braided monoidal category is a doubly monoidal 1-category .
Accordingly, by delooping twice, it may be identified with a tricategory with a single object and a single 1-morphism.
However, unlike the definition of a monoidal category as a bicategory with one object, this identification is not trivial; a doubly-degenerate tricategory is literally a category with two monoidal structures that interchange up to isomorphism. It requires the Eckmann-Hilton argument to deduce an equivalence with braided monoidal categories.
A commutative monoid is the same as a monoid in the category of monoids. Similarly, a braided monoidal category is equivalent to a monoidal-category object (that is, a pseudomonoid) in the monoidal 2-category of monoidal categories. This result goes back to the 1986 paper by Joyal and Street.
A braided monoidal category is equivalently a category that is equipped with the structure of an algebra over the little 2-cubes operad.
Details are in example 1.2.4 of
There is a strict 2-category BrMonCat with:
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 |
braided monoidal category, braided monoidal (∞,1)-category
symmetric monoidal category, symmetric monoidal (∞,1)-category
The original papers on braided monoidal categories are by Joyal and Street. The published version does not completely supersede the Macquarie Math Reports version, which has some nice extra results:
André Joyal and Ross Street, Braided monoidal categories, Macquarie Math Reports 860081 (1986).
André Joyal and Ross Street, Braided tensor categories , Adv. Math. 102 (1993), 20–78.
Around the same time the same definition was also proposed independently by Lawrence Breen in a letter to Pierre Deligne:
For a review of definitions of braided monoidal categories, braided monoidal functors and braided monoidal natural transformations, see:
For an elementary introduction to braided monoidal categories using string diagrams, see:
Eventually we should include all these diagrams here, along with the definition of braided monoidal functor and braided monoidal natural transformation! Can anyone help out?