Re: "First" use of 'Category theory' to describe our field

[David Roberts <droberts.65537@gmail.com>]

Dear George,

thanks for supplying that quote about Eilenberg. After emails with
Peter May I had tracked down a secondary source that did not cite that
Mac Lane article you give, so it's good to know the provenance of the
claim.

As far as people studying categories acknowledging the field by name
goes, Peter Hilton in the intro to the Battelle conference proceedings
in 1968 (titled Category theory, Homology theory and their
applications, LNM 86, 92, and 99), writes thus in vol 1:

"The object of this conference was to bring together research workers
in the fields of category theory and homology theory and those who
applied the results of these theories to their own mathematical
disciplines within algebra or topology. Thus this was not, and was not
intended to be, a tightly specialized conference in categorical
algebra (by comparison with the Midwest Seminars), the expectation of
its organizers being that the roles of category theory and homology
theory within mathematics would emerge the more clearly from the
conference and that the interplays of these theories with other parts
of mathematics would be highlighted."

Interestingly, Mac Lane contributed an article titled "Possible
programs for categorists" (note: "categorists", not category
theorists), in which he writes

"Category theory today is both a specialty and a generality.
Specialities are the many particular fields in which current
Mathematical knowledge and folklore develops; a new specialty arises
in a field when the knowledge in that field and its prospects of
further development demand full time workers. In the last six or eight
years, category theory has become a flourishing specialty."

So we might have a lower bound of 1960-62 according to Mac Lane's
written estimate as to when category theory started to 'became a
flourishing speciality' (around the time of Freyd's thesis, it seems).

Going back a few years in the publication record, the 1965 La Jolla
conference was published as "Proceedings of the Conference on
Categorical Algebra" (https://doi.org/10.1007/978-3-642-99902-4), so
perhaps it was a lot more focussed in nature. However, the
introduction states

"The editors hope to have achieved a representative, if incomplete,
cover­ age of the present activities in Categorical Algebra within the
United States by bringing together this group of mathematicians and by
solici­ting the articles contained in this volume. They also hope that
these Proceedings indicate the trend of research in Categorical
Algebra in this country."

So it looks like 'categorical algebra' was at least a working phrase
(modulo having to satisfy the United States Air Force Office of
Scientific Research, which I read elsewhere was not pleased with
having funded such abstract work, and promised to never fund such a
conference again)

In between these two there is the "Seminar on Triples and Categorical
Homology Theory" (LNM 80):

"The papers in this volume were presented to the seminar on category
theory held during the academic year 1966-67 at the Forschungsinstitut
für Mathematik of the Eidgenossische Technische Hochschule, Zürich."

Someone pointed out off-list the reference to which Colin McLarty alluded:

Rosen, Robert. 1958. “The Representation of Biological Systems from
the Standpoint of  the Theory of Categories.” Bulletin of Mathematical
Biophysics 20 (4): 317–42.

in which he talks of "the theory of categories and functors" in his
abstract, and closes with

"The application of category theory to more general kinds of systems
becomes correspondingly more complicated, but at the very least, we
hope to have indicated in the foregoing that the notion of systems
introduced here can be put on a rigorous basis and that the results
obtained by using those notions can be formally justified."

Thanks to all who replied here and elsewhere.

Best regards, and apologies for so many bit of historical trivia,

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com
On Mon, 15 Jul 2019 at 00:19, George Janelidze
<george.janelidze@uct.ac.za> wrote:
>
> Dear Colleagues,
>
> I would like to add three remarks to this discussion:
>
> 1. In his paper "Samuel Eilenberg and Categories" (Journal of Pure and
> Applied Algebra 168 (2002) 127–131), Saunders Mac Lane, talking about [S.
> Eilenberg and S. Mac Lane, General theory of natural equivalences,
> Transactions of the American Mathematical Society 58, 2 (1945) 231-294]
> says:
>
>      "...At the time, Sammy stated firmly that this would be the only paper
> needed for category theory. Probably what he had in mind was that the trio
> of notions - category, functor, and natural transformation - was enough to
> make good applications possible; in particular it was enough to formulate
> the axiomatic treatment of homology theory carried out in the famous
> Eilenberg--Steenrod text “Foundations of Algebraic Topology”.
>      This initial paper on category theory was certainly a “far out”
> endeavor; it might not have seen the light of day! Also the terminology was
> largely purloined: “category” from Kant, “natural” from vector spaces and
> “functor” from Carnap. (It was used in a different sense in Carnap’s
> influential book “Logical Syntax of Language”; I had reviewed the English
> translation of the book (in the Bulletin, AMS) and had spotted some errors;
> since Carnap never acknowledged my finding, I did not mind using his
> terminology.)
>      Sammy’s initial idea that one paper would be enough turned out to be
> wildly wrong. Other basic examples such as adjoint functors were developed;
> at Columbia University Sammy subsequently inspired and guided a remarkable
> group of young mathematicians who took up category theory: John Gray, Daniel
> Kan, Bill Lawvere, Mike Barr, Jon Beck, Alex Heller, Peter Freyd, and many
> others. Sammy and I were very fortunate in our students and associates..."
>
> 2. We celebrated 50th Anniversary of Category Theory in 1995 twice: in
> Halifax (Canada) and then in Cambridge (UK). In particular, the webpage
> https://www.mta.ca/~cat-dist/ct95.html says:
>
> "...Fifty years after the paper which founded Category Theory and
> twenty-five years after the discovery of Elementary Topos Theory, the
> Category Theory community met in Halifax..."
>
> 3. Yes, the title "General theory of natural equivalences" has no categories
> in it, and one might have different opinions on "which paper has the most
> important contribution in transforming 'language' into 'theory'" (what about
> [S. Mac Lane, Duality for groups, Bulletin of the American Mathematical
> Society 56 (1950) 485-516]?). But I think the citations above clearly
> suggest to say that Category Theory was 'officially' born in 1945, and let
> us hope to celebrate its 100th Anniversary in 2045!
>
> Of course all this means no disrespect for great contributions of
> non-North-American authors mentioned (or not mentioned) in various messages
> on this topic.
>
> Best regards,
> George
>
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