PPS. I see now that the 1958 paper of Rosen is indeed listed as a reference in Kurosh et al. 1960 "Foundations of the theory of categories" (and, of course, Rosen too uses "the theory of categories" in the title). Curiously, however, it seems there is no quotation of Rosen in the body of the paper and it's not clear to me that it was Rosen who convinced them this is a separate research area. Throughout, they refer to "the theory of categories" as an already established field, impossible to summarize in one article. Furthermore, the introduction ends with one an expression of?? "gratitude to the participants in the seminar on the theory of categories, which worked under the direction of one of the authors in Moscow University". Its participants apparently produced quite a few papers listed by MacLane. Not sure when it started, but I doubt if there is any example of an earlier seminar devoted to (foundations of) category theory. As I said earlier and as mentioned by MacLane, in 1960 Kurosh also lectured about the subject in Prague.

PS. Actually, I stand corrected. I found out now that MacLane's "Concepts and Categories in Perspective"

http://www.ams.org/samplings/math-history/hmath1-maclane25.pdf

discusses in some detail when exactly he started to see "category theory" as a viable field for further research in its own right, rather than just "a handy language" or "a conceptual view of parts of mathematics".

He claims that for him, 1963 was the critical year. Details to be found in Section 17 of the paper linked above. He says, for example, that being invited as the 1963 AMS Colloquium Lecturer, he chose to give four lectures on category theory (more specifically, he spoke about "Categorical Algebra"), whereas a year or two later he would have spoken about homological algebra or algebraic topology.

Notably, one can also find there a list of early papers on category theory, which illustrates how the field suddenly exploded in early 1960's.?? I can't find Robert Rosen being mentioned, so it might be that MacLane missed the first occurrence of the phrase "category theory" in print. But he mentions, for example, a 1960 paper by a mathematician from the Kurosh school (M. S. Calenko) whose title is "On the foundations of the theory of categories". Another very early paper with "theory of categories"?? or "category theory" in the title is a 1962 one by V??ra Trnkov??.

Actually, MacLane seems to omit an even earlier reference (quoted, for example, by Eckmann and Hilton 1962, who also speak occasionally about "theory of categories"):

Kurosh, A. G., A. Kh. Livshits and E. G. Shul'geifer: Foundations of the theory of categories. Uspekhi Matem. Nauk XV,6, 1 (1960).

Is this the first paper which openly mentions category theory in its title then? It seems that researchers influenced by Kurosh (who also lectured in Prague in 1960)?? were the earliest ones who had the courage to write papers explicitly on "the theory of categories".?? I guess, e.g., Ji???? Ad??mek?? or Ale?? Pultr could have much more to say here.

On 14.07.19 17:58, Tadeusz Litak wrote:

...

...

As for the terminology "functor" I vaguely remember this derives from Carnap but I may be wrong and perhaps my memory fails. Perhaps someone knows better. This is indeed commonly accepted original inspiration; I've never heard about Eilenberg or Mac???Lane protesting against this explanation (maybe somebody knows better?). See for example:

https://plato.stanford.edu/entries/category-theory/#2

which seems a better summary of historical origins than anything you might find on wikipedia.

...

The central notion at the time, as their title indicates, was that of natural transformation. In order to give a general definition of the latter, they defined functor, borrowing the term from Carnap, and in order to define functor, they borrowed the word ???category??? from the philosophy of Aristotle, Kant, and C. S. Peirce, but redefining it mathematically.

As regards Mathematical Reviews, while this is an interesting observation, I am not sure if it settles definitively the question?? who and when started to see category theory as a field in its own right. There is no guarantee that the authors of these reviews paid particular attention to terminological issues, or that they were sympathetic enough to the goals of papers under discussion. In particular, that they shared the views of reviewed authors that mathematics needs a new subdiscipline.

I haven't checked if the phrase "category theory" was ever used in the 1950's books of Eilenberg & Steenrod or Cartan & Eilenberg. But Kan's 1958 paper on adjoint functors contains, by a quick count, 120 occurrences of the term "category" on merely 36 pages and Grothendieck's 1957 T??hoku paper---over 200 (it is almost four times as large that of Kan though). It'd seem that if you talk about a mathematical entity so much, you are developing its theory.

In fact, the 1945 paper itself does seem to suggest quite openly that a grand unifying foundational theory is the ultimate goal. It is enough to read the final paragraphs of its intro:

...

In a metamathematical sense our theory provides general concepts applicable to all branches of abstract mathematics, and so contributes to the current trend towards uniform treatment of different mathematical disciplines. In particular, it provides opportunities for the comparison of constructions and of the isomorphisms occurring in different branches of mathematics; in this way it may occasionally suggest new results by analogy. (...) This may be regarded as a continuation of the Klein Erlanger Programm, in the sense that a geometrical space with its group of transformations is generalized to a category with its algebra of mappings.

Best, t.

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