In answer to John Baez

In answer to John Baez

[Andree Ehresmann]

In answer to John Baez

Charles Ehresmann introduced internal categories to unify several examples
of this notion that he had studied in earlier papers, namely topological
and differentiable categories and specially groupoids (i.e. internal to Top
and Diff) which he used extensively in his works on the foundations of
differential geometry in the early fifties, ordered categories of several
types (i.e. internal to sub-categories of the category of posets) also used
in these works, and double categories (of which the 2-categories are a
particular case) which he had introduced in his 1958 paper

         "Categorie des foncteurs types", Rev. Un. Mat. Argentina XX (1960).

He defines the notion of internal categories (which he called "categorie
structuree") and internal functors in the paper

         "Categories structurees", Ann. Ec. Normale Sup. 80 (1963),

and internal natural transformations in the sequel of this paper

"Categories structurees III: Quintettes et applications covariantes",
Cahiers Top. et GD V (1963)

where he constructs the double category of "quintettes structures" of which
the 2-category of internal categories is a sub-2-category.

However in these papers he defined only categories internal to a concrete
category, which explains the name "categorie structuree". Later on he
defined the general notion of an internal category, initially called
"categorie structuree generalisee", in

"Introduction to the theory of structured categories", Technical Report 10,
Un. of Kansas at Lawrence, 1966

where he introduced the theory of sketches and, in particular, the sketch
of a category. In this paper and in the paper

"Categories structurees generalisees", Cahiers de Top. et GD X-1 (1968)

he compares with the notion of a "C-category on (A,A0)" which Grothendieck
had defined in 1960-61 by the fact that the Hom(A,-) are equipped with a
natural strucutre of category.

All the papers of Charles are reprinted in "Charles Ehresmann: Oeuvres
completes et commentees", in the comments of which I give more historical
information on this subject..

                         With all my best wishes for 2003
                                         Andree C. Ehresmann