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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