nLab
Adjointness for 2-Categories

This is one of the most influential and comprehensive historical books in “formal” low-dimensional higher category theory:

• John W. Gray, Formal category theory: adjointness for $2$-categories. Lecture Notes in Mathematics, Vol. 391. Springer-Verlag, Berlin-New York, 1974. xii+282 pp.

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, it is very difficult to read this book 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