nLab
monoidal bicategory

Contents

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

2-Category theory

Contents

Idea

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:

Examples

Example

For RR a commutative ring, there is a symmetric monoidal bicategory Alg(R)Alg(R) whose

The monoidal product is given by tensor product over RR.

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

Example

For VV a cocomplete closed symmetric monoidal category, there is a symmetric (indeed compact closed) monoidal bicategory VProfV Prof whose objects are small VV-enriched categories and whose morphisms are VV-enriched profunctors.

Internalization

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 ProfProf specializes to a promonoidal category, but ordinary monoidal categories can be regarded as particular “representable” promonoidal ones. In terms of the bicategory ProfProf, 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 CatCat to ProfProf, we can now define more kinds of pseudomonoids. Note that ProfProf 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.

Coherence theorems

References

Slightly different definitions of these various structures can be found in the following sequence of papers:

Definition of braided Gray monoids, i.e. Gray-category-monoids with a braiding:

Correction of the KV definition by adding one axiom:

A definition of braided Gray-monoid equivalent to Baez-Neuchl, also defining sylleptic and symmetric Gray-monoids:

Further modification of the definition by adding an axiom relating to the tensor unit:

  • Sjoerd E. Crans, Generalized Centers of Braided and Sylleptic Monoidal 2-Categories, Advances in Mathematics, Volume 136, Issue 2, 25 June 1998, Pages 183-223 (doi:10.1006/aima.1998.1720)

  • Paddy McCrudden, in “Balanced coalgebroids,” defined braided and sylleptic monoidal bicategories (not Gray monoids).

A full definition of braided, sylleptic, and symmetric monoidal bicategories and assembing into a tricategory:

A strictification theorem relating all these definitions, along with a coherence theorem for the braided case:

A coherence and strictification theorem for the symmetric case (but the sylleptic case is perhaps still open):

See also:

Last revised on July 21, 2021 at 13:25:07. See the history of this page for a list of all contributions to it.