Re: Who invented n-categories?

[Andree Ehresmann <Andree.Ehresmann@u-picardie.fr>]

In answer to Carl Futia

The first given example of a strict 2-category is the example of a
2-category of natural transformations. It has been given by Charles
Ehresmann in his paper "Foncteurs types" of 1960 (reprinted in "Charles
Ehresmann: Oeuvres completes et commentees" Part IV-1, page 103). He does
not give the name 2-category but he explicits the "permutability" of the
two laws of which Godement had given some particular cases in his book on
sheaf theory in 1958.

It is this example as well as the double category of squares of a category
(which Charles called 'quatuors' and defined about the same time) that
suggested the definition of double categories.
I don't know who introduced the name 2-category nor when, but I remembers
that Benabou used it around 1962-63.

The general definition of an n-fold category is given by Charles in his
paper "Categories structurees" in 1963 (reprinted in the "Oeuvres" Part
III-1, p. 68), as an example of the general notion of an internal category
in a concrete category (which he then called a structured category).
The particular case of (strict) n-categories is not specified there. We
used it in the last series of papers I wrote with Charles on n-fold
categories in 1978 (reprinted in "Oeuvres", Part IV-2, p. 681, but it was
well-known by this time.

                         Andree C. Ehresmann