Bicategories
In a bicategory, there are morphisms between morphisms, called 2-morphisms. In contrast to a 2-cell in a double category, the domain and codomain of a 2-morphism are required to have equal domains and equal codomains.
Bicategories belong to a hierarchy of higher categories. According to the periodic table , a bicategory with one object is a monoidal category.
Warning: Traditionally, a “2-category” is strict and a “bicategory” is a kind of weak 2-category, where the composition and unit laws hold only up to coherent natural isomorphism. However, some authors, including those of the nLab , take a weak-by-default attitude, speaking of “strict 2-categories” and “general 2-categories.” This page is about both 2-categories and bicategories.
Literature
Book treatments
- Johnson & Yau, 2020: 2-dimensional categories (arxiv, nCat Cafe )
- Borceux, 1994: Handbook of categorical algebra, Vol. 1, Ch. 7: Bicategories and distributors
- Gray, 1974: Formal category theory: adjointness for 2-categories (doi)
Notes and surveys
- Lack, 2010: A 2-categories companion (doi, arxiv)
- Lengthy informal guide to 2-categories and bicategories
- Chapter in: Baez & May, 2010: Towards higher categories
- Leinster, 2008: Basic bicategories (arxiv)
- Power, 1998: 2-categories (pdf)
- Nice motivation from PLT point of view, but generally quite terse
- Material on pasting diagrams elaborated in: Power, 1990: A 2-categorical pasting theorem (doi, pdf)
- Kelly & Street, 1974: Review of the elements of 2-categories (doi, nCat Cafe )
- Sec. 1 introduces 2-categories as a special case of double categories
- Sec. 3 discusses doctrines
Limits and colimits
- Borceux, 1994: Handbook of categorical algebra, Vol. 1
- Sec. 7.4: 2-limits and bilimits
- Sec. 7.6: Lax limits and pseudolimits
- Kelly, 1989: Elementary observations on 2-categorical limits (doi, nCat Cafe )
- Sec. 4 contains a helpful list of notable specific 2-limits, such as inserters , equifiers , inverters , and comma objects
- Lack & Shulman, 2012: Enhanced 2-categories and limits for lax morphisms (doi,
arxiv, nCat Cafe )
- Limits in 2-categories whose objects are categories with extra structure
- Clingman & Moser, 2020: 2-limits and 2-terminal objects are too different (arxiv)
- Brandenberg, 2020: Bicategorical colimits of tensor categories (arxiv)
Monoidal bicategories
Monoidal bicategories are bicategories with a monoidal product, the analogue of monoidal categories in one higher dimension. Just as a monoidal category is a bicategory with one objet, a monoidal bicategory is a tricategory with one object. Cartesian bicategories (monoidal bicategories with certain extra structure) are the foundation of Carboni and Walters’ axiomatization of relations as bicategories of relations.
- Baez & Neuchl, 1996: Higher-dimensional algebra I: Braided monoidal 2-categories (doi, arxiv, pdf)
- Gurski & Orsono, 2013: Infinite loop spaces, and coherence for symmetric monoidal bicategories (doi, arxiv, nCat Cafe )
- Stay, 2016: Compact closed bicategories (pdf, arxiv, nCat Cafe )
- Ahmadi, 2020: Monoidal 2-categories: A review (arxiv)
- Abstract: “We review the complete definition of monoidal 2-categories and recover Kapranov and Voevodsky’s definition from the algebraic definition of weak 3-category(or tricategory).”
See also Shulman’s work on framed bicategories, aka fibrant double categories, and on constructing monoidal bicategories from monoidal double categories.