Subject: categories: Who invented n-categories?

From: [email-masked]
Date: Mon, 23 Feb 2004 11:01:19 EST

Can anyone offer a reference to the first published work which defined a
notion of strict n-category equivalent to that used today?

I know that Ehresmann invented n-tuple ( or n-fold ) categories which
contain strict n-categories as special cases.  If this is the first
implicit defintion of strict n-category does anyone know who was the first
to isolate our current notion of strict n-category as a particularly
interesting special case of an n-tuple category?

Carl Futia

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From: "Ronald Brown" <[email-masked]>
Date: Wed, 25 Feb 2004 09:57:21 -0000

reply to [email-masked]

There is the following paper

34.  (with P.J. HIGGINS), ``The equivalence of $\infty$-groupoids
and crossed  complexes'', {\em Cah. Top. G\'eom. Diff.} 22 (1981)
371-386.

which first defines  an n-fold category, specialises to an n-category,  and
relates that to a notion of globular set in (2.2), (2.3),  (without using
the term globular, which I think came from Pursuing Stacks, 1983).

Published at the same time was

33.  (with P.J. HIGGINS), ``The equivalence of $\omega$-groupoids
and cubical  $T$-complexes'', {\em Cah. Top. G\'eom. Diff.} 22
(1981) 349-370

which deals with the cubical, groupoid,  case (essential for the topological
applications) and of which some announcement was made in

22.  (with P.J. HIGGINS), ``Sur les complexes crois\'es,
$\omega$-groupo\"{\i}des et  T-complexes'', {\em C.R. Acad. Sci.
Paris S\'er. A.} 285 (1977) 997-999.

Were there earlier definitions? There was an unpublished manuscript by O.
Wyler (1972) referred to in 34, which my memory suggests did define n-fold
categories.

Ronnie Brown
http://www.bangor.ac.uk/~mas010

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From: Andree Ehresmann <[email-masked]>
Date: Thu, 26 Feb 2004 19:52:29 +0100

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