Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
For large classes of examples of bicategories the 1-morphisms naturally are of one of two different types:
special morphisms that may be taken to compose strictly among themselves;
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 induces an --bimodule 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 fibrant double category is a way to achieve this.
A fibrant double category is essentially the same as a proarrow equipment on a bicategory. See there for more.
Fibrant double categories have also been called framed bicategories (Shulman β08) and gregarious double categories (DPP10.
This comes directly from Shulman β08:
A double category is fibrant if it satisfies any of the following equivalent conditions:
Mike Shulman, Framed bicategories and monoidal fibrations, Theory and Applications of Categories 20 18 (2008) 650β738 [tac:2018, arXiv:0706.1286]
R. Dawson, Robert ParΓ©, Dorette Pronk, The span construction, Theory Appl. Categ. 24 (2010), No. 13, 302β377, 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)
Last revised on October 10, 2024 at 16:19:52. See the history of this page for a list of all contributions to it.