category with duals (list of them)
dualizable object (what they have)
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 .
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 for the components of the associator in . Then the first hexagon identity says that for all the following diagram commutes:
The second hexagon identity says that for all the following diagram commutes:
Intuitively speaking, the first hexagon identity says we can braid past all at once or in two steps. The second hexagon identity says we can braid past ‘all at once’ or in two steps.
From these axioms, it follows that the braiding is compatible with the left and right unitors and . That is to say, for all objects the diagram
In terms of the language of k-tuply monoidal n-categories a braided monoidal category is a doubly monoidal 1-category .
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.
Details are in example 1.2.4 of
There is a strict 2-category BrMonCat with:
|monoid/associative algebra||category of modules|
|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|
|form Drinfeld double||form Drinfeld center|
|trialgebra||Hopf monoidal category|
|monoidal category||2-category of module categories|
|Hopf monoidal category||monoidal 2-category (with some duality and strictness structure)|
|monoidal 2-category||3-category of module 2-categories|
braided monoidal category, braided monoidal (∞,1)-category
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?