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Adjointness for 2-Categories

This entry is about the book

• John Gray,

  Formal category theory: adjointness for $2$-categories

  Lecture Notes in Mathematics, Vol. 391.

  Springer-Verlag, Berlin-New York, 1974. xii+282 pp.

  doi:10.1007/BFb0061280

on formal category theory, which is category theory formulated “formally” via the 2-category theory of the 2-category Cat of all categories.

This is one of the most influential and comprehensive historical books in low-dimensional higher category theory, following the spirit of:

• William Lawvere, The Category of Categories as a Foundation for Mathematics, pp.1-20 in Eilenberg, Harrison, MacLane, Röhrl (eds.), Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer Heidelberg 1966 (doi:10.1007/978-3-642-99902-4_1)

More recently, this approach is echoed in Riehl & Verity 13, where Cat is enhanced to the homotopy 2-category of (∞,1)-categories in order to provide 2-category theoretic foundations for (∞,1)-category theory. At least for presentable $\infty$-categories, this is also obtained as the 2-localization of the 2-category of combinatorial model categories and left Quillen functors at the Quillen equivalences: see at 2Ho(CombModCat)].

The book was supposed to be the first part of a four volume work, but unfortunately later volumes/chapters never appeared. It has some parts of 2- and 3-category theory; including the treatment of the famous Gray tensor product on 2-Cat. See also Gray-category.

Unfortunately, due to changes in terminology, the book may be difficult to read nowadays. Gray uses prefixes such as ‘quasi,’ ‘iso,’ and ‘weak’ to indicate various levels of weakness, but his choice of terminology is not entirely consistent, can be confusing, and is completely different from the standard modern terminology which uses ‘lax,’ ‘oplax,’ and ‘pseudo’ with (mostly) precise and consistent meanings.

The following is a list of some of the definitions given in the book, along with their modern names and links to nLab entries.

• Section 2

  • 2-category (see strict 2-category)
  • 2-functor/$Cat$-functor (i.e. strict 2-functor)
  • Cat-natural transformation (see enriched functor category)
  • Quasi-natural transformation (lax natural transformation)
  • Modification
  • 2-comma category?
  • Double category

• Section 3

  • Bicategory
  • Pseudo-functor (lax 2-functor), homomorphic pseudo-functor (pseudofunctor), strict pseudo-functor (= strict 2-functor – see enriched functor)

• Section 4

  • $Fun(A,B)$ and $Pseud(A,B)$ (see Gray tensor product)

• Section 6

  • Adjoint morphisms in a 2-category (adjunction)
  • Kan extension in a 2-category

• Section 7

  • Transcendental quasi-adjunction (lax 2-adjunction)

category: reference