2-category of internal categories

2-category of internal categories

[John Baez]

Dear Categorists -

Who first constructed internal categories, internal functors
and internal natural transformations in a given category, and
actually proved that these form a *2-category*?  I'm looking for
a reference.

Best,
jb

Re: 2-category of internal categories

[Max Kelly]

John Baez asks who constructed internal categories, functors, and natural
transformations, proving these to form a 2-category; and seeks a reference.
The answer is surely Ehresmann; and the precise reference must be in an
early part of his collected works, produced after his death, with a detailed
commentary, by his widow Andree Ehresmann-Bastiani. I have at least the
earlier volumes of these, which - being now a visitor here in Sydney, John -
you may certainly borrow. I'll have a bit of a look myself, if I have the time;
Ehresmann's language is at times far from what has now become the norm.
(Finding his categorical insights into differential geometry unappreciated by
his French colleagues, he cut himself off and set up an independent group
based in Paris VI (where he was) and in Amiens (where Bastiani was); with
their own journal, Cahiers de Topologie et Geometrie Differentialle. The
definitive rapprochement between this group and other category theorists
dates from 1973, when the first of several international conferences at Amiens
was arranged by Ehresmann and Bastiani.)

Probably there will be no need for me to take these volumes down tonight; for
Andree will doubtless see John's question when it dawns in Paris in an hour or
two, and will doubtless give us chapter and verse.

Max Kelly.