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 $x \otimes y$. Thus the tensor product is “commutative” in a sense, but not as coherently commutative as in a symmetric monoidal category.
A braided monoidal category is a special case of the notion of braided pseudomonoid in a braided monoidal 2-category.
A braided monoidal category, or (“braided tensor category”, but see there), is a monoidal category $\mathcal{C}$ equipped with a natural isomorphism
called the braiding, such that the following two kinds of diagrams commute for all objects involved (called the hexagon identities encoding the compatibility of the braiding with the associator for the tensor product):
and
where $a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the components of the associator of $\mathcal{C}^\otimes$.
If the braiding in def. 1 “squares” to the identity in that $B_{y,x} \circ B_{x,y} = id_{x \otimes y}$, then the braided monoidal category is called a symmetric monoidal category.
Intuitively speaking, the first hexagon identity in def. 1 says we may braid $x$ past $y \otimes z$ ‘all at once’ or in two steps. The second hexagon identity says that we may braid $x \otimes y$ past $z$ all at once or in two steps.
From these axioms in def. 1, 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. (There is also a notion of braided pseudomonoid that specializes directly in Cat to braided monoidal categories.)
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:
Any symmetric monoidal category is a braided monoidal category.
The monoidal category of graded modules over a commutative ring (with the usual tensor product of graded modules) can be made into a braided monoidal category with the braiding $V \otimes W \to W \otimes V,\ x \otimes y \mapsto y \otimes x$. The braiding $x \otimes y \mapsto (-1)^{|x| |y|} y \otimes x$ (where $|x|$ and $|y|$ denote the degrees) is also commonly used.
More generally, for any invertible element $u$ of the base ring, there is the braiding $x \otimes y \mapsto u^{|x| |y|} y \otimes x$, and these braidings are the only possible. The resulting braided monoidal category is symmetric if and only if $u^2 = 1$.
The maps $x \otimes y \mapsto u y \otimes x$ define a braiding on the monoidal category of (ungraded) modules only if $u = 1$.
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:
Textbook accounts include
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? * John Baez and Mike Stay, Physics, topology, logic and computation: a Rosetta Stone, to appear in New Structures in Physics, ed. Bob Coecke.
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?