category with duals (list of them)
dualizable object (what they have)
A braided monoidal 2-category is a 2-category equipped with a tensor product 2-functor which satisfies the first in a hierarchy of conditions for being commutative up to equivalence: in the language of k-tuply monoidal n-categories a braided monoidal 2-category is a doubly monoidal 2-category.
braided monoidal 2-category, braided 3-group
Mikhail Kapranov and Vladimir Voevodsky, 2-Categories and Zamolodchikov tetrahedra equations, in Proc. Symp. Pure Math. 56 Part 2 (1994), AMS, Providence, pp. 177–260.
John Baez and Martin Neuchl, Higher-dimensional algebra I: braided monoidal 2-categories, Adv. Math. 121 (1996), 196-244. (pdf)
Sjoerd Crans, Generalized centers of braided and sylleptic monoidal 2-categories, Adv. Math. 136 (1998), 183-223. (pdf)