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. On 13.07.19 11:45, Johannes Huebschmann wrote:

Dear All

The phrase

"the terminology of category theory came from Kant"

oversimplifies the situation.

Aristotle (Peri ton kategorion) discusses categories. Kant uses categories to mediate his thought (Kritik der Urteilskraft). Saunders Mac Lane's adviser in Goettingen was Paul Bernays. Bernays knew ancient Greek philosophy very well.

During my student's time at the ETH I still had occasion to talk to Paul Bernays (he then was in his 80s). He regularly attended the logic seminar and even contributed to the discussion.

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.

Also, in German, when you teach a course entitled "Kategorien" or "Kategorien und Funktoren", that synonymously means "Kategorientheorie". For example, D. Puppe taught such a course in the 1960s, and that was the origin of the Brinkmann-Puppe LNM.

Best regards

Johannes

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