The invention of n-categories

The invention of n-categories

[Topos8]

Thanks to all who contributed information on history of the notion of a
strict n-category.

I'd like to summarize for the benefit of those interested what I have
learned.

The structure of CAT as a strict 2-category was of course implicit in
Eilenberg and MacLane's original 1945 definition of categories, functors
and natural transformations.

Around 1958 Godement observed the significance of what we now call the
exchange law relating horizontal and vertical compostion of natural
transformations.


In 1960 Charles Ehresmann codified our current notion of a strict 2-category
and this notion also appeared in the thesis of his student Jean Benabou
shortly thereafter.

In 1963 Ehresmann published a defintion of n-tuple categories which include
strict n-categories as a special case.  However, Ehresmann did not isolate
strict n-categories themselves as objects worthy of special attention.

At the 1965 La Jolla conference Eilenberg and Kelly presented their paper on
closed categories which defined strict n-categories recusively as categories
enriched in strict n-1 categories.

Around the same time the group surrounding Grothendieck at IHES had
informally developed a non-recursive definition of strict n-category
similar to what is used today.

Finally, at the 1969 Bowdin conference, category theorists recognized that
the recursive definition of Eilenberg-Kelly and the non recursive definition of
the IHES group were equivalent.

As far as I can tell the first significant technical development of the
theory of strict n-categories (n > 2 ) appeared in a 1981 Cahiers paper by
Ronnie Brown and PJ Higgins and in a 1984 Cahiers paper by Dominique
Bourn.

Carl Futia

Re: The invention of n-categories

[Vaughan Pratt]

>From: Topos8@aol.com (Carl Futia)
>Around the same time the group surrounding Grothendieck at IHES had
>informally developed a non-recursive definition of strict n-category
>similar to what is used today.

...prompting the question of whether x-categories for noninteger real x
have been looked at.  Presumably an early question here would be whether
the spaces of fractional dimension that have already been studied depend
on any geometric properties beyond those of n-categories.  In particular
what restrictions must one accept in order to have a coherent notion of
a 1.5-category?

Vaughan Pratt