category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
2-natural transformation?
A monoidal bicategory is a bicategory with a monoidal structure, which is up-to-equivalence in a suitable bicategorical sense. A concise definition is that a monoidal bicategory is a tricategory with one object. Just as every tricategory is equivalent to a Gray-category, every monoidal bicategory is equivalent to a Gray-monoid, i.e. a monoid in the monoidal category Gray.
Just as monoidal categories also come in braided and symmetric versions, monoidal bicategories have three extra levels of commutativity (see the periodic table and the stabilization hypothesis): braided, sylleptic, and symmetric.
For $R$ a commutative ring, there is a symmetric monoidal bicategory $Alg(R)$ whose
k-morphism are bimodule homomorphisms.
The monoidal product is given by tensor product over $R$.
By delooping this once, this gives an example of a tricategory with a single object.
The tricategory statement follows from theorem 21 of
This, and that the monoidal bicategory is even symmetric monoidal is given by the main theorem in
Slightly different definitions of these various structures can be found in the following sequence of papers:
Kapranov and Voevodsky, in “2-categories and Zamolodchikov tetrahedra equations” and “Braided monoidal 2-categories and Manin-Schechtman higher braid groups”, defined braided Gray monoids, i.e. Gray-monoids with a braiding.
Baez and Neuchl, in “Higher-Dimensional Algebra I. Braided monoidal 2-categories” corrected the KV definition by adding one axiom.
Day and Street, in “Monoidal bicategories and Hopf algebroids,” gave a definition of braided Gray-monoid equivalent to Baez-Neuchl, and also defined sylleptic and symmetric Gray-monoids and functors between them.
Crans, in “Generalized centers of braided and sylleptic monoidal 2-categories,” further modified the definition by adding an axiom relating to the tensor unit.
McCrudden?, in “Balanced coalgebroids,” defined braided and sylleptic monoidal bicategories (not Gray monoids).
Chris Schommer-Pries, in his Ph. D. thesis, gave the full definition of braided, sylleptic, and symmetric monoidal bicategories and also assembled them into a tricategory.
Nick Gurski, in “Loop spaces, and coherence for monoidal and braided monoidal bicategories” arXiv, proved a strictification theorem relating all these definitions, along with a coherence theorem for the braided case.
Nick Gurski and Angelica Osorno?, in “Infinite loop spaces, and coherence for symmetric monoidal bicategories” arXiv, proved a coherence and strictification theorem for the symmetric case (but the sylleptic case is perhaps still open).