nLab monoidal bicategory

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):

• braided monoidal 2-categories
• sylleptic monoidal 2-categories
• symmetric monoidal 2-categories

There are also 2-categorical variants of other structures on monoidal 1-categories, such as:

• closed monoidal 2-category?
• balanced monoidal 2-category?
• compact closed 2-category

Examples

Example

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

• objects are $R$-algebras;
• morphisms are algebra bimodules;
• 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

• Richard Garner, Nick Gurski, The low-dimensional structures that tricategories form (arXiv:0711.1761)

This, and that the monoidal bicategory is even symmetric monoidal is given by the main theorem in

• Mike Shulman, Constructing symmetric monoidal bicategories (arXiv:1004.0993)

Example

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.

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:

• Inside a monoidal bicategory one can define a pseudomonoid, which when specialized to $Cat$ yields a monoidal category.
• Inside a braided monoidal bicategory one can define a braided pseudomonoid, which when specialized to $Cat$ yields a braided monoidal category.
• Inside a sylleptic monoidal bicategory one can define a symmetric pseudomonoid, which when specialized to $Cat$ yields a symmetric monoidal category.
• Inside a balanced monoidal bicategory one can define a balanced pseudomonoid, which when specialized to $Cat$ yields a balanced monoidal category.

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.

• Inside any compact closed bicategory one can define a closed pseudomonoid. A representable closed pseudomonoid in $Prof$ specializes to a closed monoidal category.
• Inside any compact closed bicategory one can define a star-autonomous pseudomonoid (a.k.a. a “Frobenius pseudomonoid”, since it also categorifies a Frobenius algebra). A representable star-autonomous pseudomonoid in $Prof$ specializes to a star-autonomous category.
• Inside any compact closed bicategory one can define a compact closed pseudomonoid. A representable compact closed pseudomonoid in $Prof$ specializes to a compact closed category.

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

• coherence theorem for monoidal bicategories
• coherence theorem for braided monoidal bicategories
• coherence theorem for symmetric monoidal bicategories

Related concepts

• braided monoidal 2-category

• sylleptic monoidal 2-category

• symmetric monoidal 2-category

• Picard 3-group

• compact closed 2-category

• monoidal category

• monoidal (2,1)-category

• monoidal double category

• monoidal (∞,1)-category

• monoidal (∞,n)-category

• module 2-category

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:

• Mikhail Kapranov, Vladimir Voevodsky, 2-Categories and Zamolodchikov tetrahedra equations, in: Proc. Symp. Pure Math. 56 Part 2 (1994), AMS, Providence, pp. 177–260 (doi:10.1090/pspum/056.2)

• Mikhail Kapranov, Vladimir Voevodsky, Braided monoidal 2-categories and Manin-Schechtman higher braid groups, Journal of Pure and Applied Algebra Volume 92, Issue 3, 25 March 1994, Pages 241-267 (doi:10.1016/0022-4049(94)90097-3)

Correction of the KV definition by adding one axiom:

• John C. Baez, Martin Neuchl, Higher-Dimensional Algebra I: Braided Monoidal 2-Categories, Adv. Math. 121 (1996), 196-244 (arXiv:q-alg/9511013)

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

• Brian Day, Ross Street, Monoidal Bicategories and Hopf Algebroids, Advances in Mathematics Volume 129, Issue 1, 15 July 1997, Pages 99-157 (doi:10.1006/aima.1997.1649)

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:

• Chris Schommer-Pries, Ph. D. thesis,

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

• Nick Gurski, Loop spaces, and coherence for monoidal and braided monoidal bicategories, Advances in Mathematics Volume 226, Issue 5, 20 March 2011, Pages 4225-4265 (arXiv:1102.0981, doi:10.1016/j.aim.2010.12.007)

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

• Nick Gurski, Angélica M. Osorno, Infinite loop spaces, and coherence for symmetric monoidal bicategories, Adv. Math. 246 (2013) 1-32 (arXiv:1210.1174)

Relation to differential linear logic and analytic functors:

• Marcelo Fiore, Nicola Gambino, Martin Hyland, Monoidal bicategories, differential linear logic, and analytic functors [arXiv:2405.05774]

See also:

• Brian Day and Ross Street, “Quantum categories, star autonomy, and quantum groupoids”, 2003 (arxiv:0301209)

• Ross Street, Frobenius monads and pseudomonoids (2004) [pdf]

  (on Frobenius monads)