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
reply to r.brown@bangor.ac.uk 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 There was an unpublished manuscript by O. Wyler (1972) referred to in 34, which my memory suggests did define n-fold categories.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 ----- Original Message ----- From: <Topos8@aol.com> To: <categories@mta.ca> Sent: Monday, February 23, 2004 4:01 PM Subject: categories: Who invented n-categories?
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
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
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Andree Ehresmann -
Ronald Brown -
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