nLab framed bicategory

Idea

For large classes of examples of bicategories the 1-morphisms naturally are of one of two different types:

1. special morphisms that may be taken to compose strictly among themselves;

2. more general morphisms that behave like bimodules.

The archetypical example is indeed the bicategory whose objects are algebras, and whose 1-morphisms are bimodules between these: every ordinary algebra homomorphism $f : A \to B$ induces an $A$-$B$-bimodule ${}_f B$ and this operation induces a 2-functor from the category of algebras and algebra homomorphisms into the bicategory of algebras and bimodules, which is the identity on objects.

For usefully working with bicategories of this kind, it is typically of crucial importance to remember this extra information. A framing on a bicategory is a way to encode this.

This is essentially the same as a proarrow equipment on a bicategory. See there for more.

Framed bicategories are also known as fibrant double categories, because they may be characterised by the property that $(L, R) : D_1 \to D_0 \times D_0$ is a fibration (Theorem 4.1 of Shulman ‘08). They are called gregarious double categories in DPP10.

Definition

This comes directly from Shulman ‘08:

Definition

A framed bicategory is a double category $\mathbb{D} = \mathbb{D}_0 \xleftarrow{L} \mathbb{D}_1 \xrightarrow{R} \mathbb{D}_0$ satisfying any of the following equivalent conditions:

1. $(L,R): \mathbb{D}_1 \to \mathbb{D}_0 \times \mathbb{D}_0$ is a fibration,
2. $(L,R): \mathbb{D}_1 \to \mathbb{D}_0 \times \mathbb{D}_0$ is an opfibration
3. $\mathbb{D}$ admits all companions and conjoints
4. $\mathbb{D}$ admits all restrictions
5. $\mathbb{D}$ admits all extensions

See also

• companion pair
• conjunction
• proarrow equipment

References

• Mike Shulman, Framed bicategories and monoidal fibrations, Theory and Applications of Categories 20 18 (2008) 650–738 [tac:2018, arXiv:0706.1286]

• R. Dawson, Robert Paré, Dorette Pronk, The span construction, Theory Appl. Categ. 24 (2010), No. 13, 302–377, TAC MR2720187

• Nicola Gambino, Joachim Kock, Polynomial functors and polynomial monads, (arXiv:0906.4931)

• Thomas Fiore, Nicola Gambino, Joachim Kock, Monads in double categories, (arXiv:1006.0797)

• Patrick Schultz, Regular and exact (virtual) double categories, (arXiv:1505.00712)

• Patrick Schultz, David Spivak, Christina Vasilakopoulou, Ryan Wisnesky, Algebraic Databases, (arXiv:1602.03501)

• Pierre-Evariste Dagand, Conor McBride, A Categorical Treatment of Ornaments, (arXiv:1212.3806)