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)
    • Influential book on low-dimensional (2- and 3-dimensional) higher category theory
    • nLab warns that terminology is outdated, giving a partial translation into modern terminology (see also Math.SE )

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 )
  • 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.