category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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):
There are also 2-categorical variants of other structures on monoidal 1-categories, such as:
For $R$ a commutative ring, there is a symmetric monoidal bicategory $Alg(R)$ whose
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
For $V$ a cocomplete closed symmetric monoidal category, there is a symmetric (indeed compact closed) monoidal bicategory $V Prof$ whose objects are small $V$-enriched categories and whose morphisms are $V$-enriched profunctors.
Monoidal bicategories provide a fruitful context for examples of the microcosm principle; various kinds of monoidal category can naturally be generalized to corresponding kinds of pseudomonoids internal to corresponding kinds of monoidal bicategories. In some cases this is very immediate:
Of course, these also specialize to the appropriate kinds of monoidal categories when specialized to monoidal 2-categories of enriched categories, internal categories, etc.
But for fancier kinds of monoidal structure, we need instead to consider monoidal bicategories like Prof instead. An ordinary pseudomonoid in $Prof$ specializes to a promonoidal category, but ordinary monoidal categories can be regarded as particular “representable” promonoidal ones. In terms of the bicategory $Prof$, a Cauchy-complete monoidal category can be identified with a map pseudomonoid, i.e. a pseudomonoid whose structure 1-morphisms are maps (left adjoints). The non-Cauchy-complete case can be dealt with using a monoidal proarrow equipment instead. Of course, this also generalizes to bicategories of enriched profunctors, internal profunctors, etc.
Having passed from $Cat$ to $Prof$, we can now define more kinds of pseudomonoids. Note that $Prof$ has more structure than being merely (symmetric) monoidal: it is a compact closed bicategory, i.e. all objects have duals.
In fact compact closedness of the whole monoidal bicategory is not necessary to assume; it suffices to assume only that the pseudomonoid itself has a dual. (And in the star-autonomous case, this is actually a consequence of the definition.)
All the above definitions can be found in Day-Street 97, except for the star-autonomous case which is in Day-Street 03 and Street 04.
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,” (1997) 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).
Other references referred to above include: