With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
The structure of a braided monoidal category on top of the underlying monoidal category with tensor product “$\otimes$” is a natural isomorphism of the form
called the braiding.
A braided monoidal category is called symmetric if and only if $B_{x,y}$ and $B_{y,x}$ are inverse morphisms to each other (while they are isomorphisms in any case).
In Vect (or generally Mod), the braiding maps elements $a\otimes b$ of a tensor product of vector spaces (of modules) $X \otimes Y$ to $b \otimes a$.
The braiding for the tensor product of chain complexes and that of super vector spaces is as in Exp. up to multiplication by sign $a \otimes b \mapsto (-1)^{deg(a) deg(b)} (b \otimes a)$ (see at signs in supergeometry).
See the references at braided monoidal category
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