(The following query by Noam Zeilberger has recently appeared on the Categories mailing list; I am taking the liberty of asking it here.)

In Ulam's autobiography Adventures of a Mathematician, there appears the following passage where he talks about his Master's thesis at the Lwów Polytechnic Institute:

I worked for a week on the thesis, then wrote it up in one night, from about ten in the evening until four in the morning, on my father's long sheets of legal paper. I still have the original manuscript. (It is unpublished to this day.) The paper contains general ideas on the operations of products of sets, and some of it outlines what is now called Category Theory.

After quoting this passage, Noam writes, "According to the information I found online, the Stanislaw M. Ulam Papers are stored in the American Philosophical Society library in Philadelphia, and include copies of both his Master's thesis in Polish titled "O operacje produkto" as well as a 1973 English translation "On the operation of product". But I was not able to find digital copies of these."

One respondent said (in essence) that it would not be surprising if Ulam had used specific examples of functors and natural transformations but without developing any general theory of them. Another suggested that "category theory" could very well refer to a theory about first or second category in the sense of Baire, not the Eilenberg-Mac Lane notion. It also occurs to me as I write this that in topology there is also the concept of Lyusternik–Schnirelmann category.

Does anyone have any idea what Ulam meant?


1 Answer 1


Not discovered, but Ulam might have anticipated some aspects of category theory and possibly influenced Eilenberg. Kuratowski's comment on Ulam's 1932 Master's thesis reads:"The work is a study or the 'product' operation... in relation to problems in set theory, group theory, topology, metric space geometry, combinatorics and measure theory in connection with probability" (quoted from Krzysko's obituary). Ulam himself writes about "treating very abstractly the idea of a general theory of many variables" there.

Product measures were introduced by Lomnicki and Ulam in Sur la théorie de la mesure dans les espaces combinatoires et son application au calcul des probabilités (1934). Some of what Ulam was up to can be gleaned from von Neumann's 1936 letter to him (quoted from Mauldin):

"I agree wholeheartedly with your plans to write an up-to-date presentation of measure-theory. Caratheodory's exposition, which is perhaps the relatively best one existing, is hopelessly obsolete. A thoroughly modern one as much combinatorial and as little topological as possible, making extensive use of finite and infinite direct products, and — above all — interpreting measure much more as probability and much less as volume, would really be a very good thing. At least I often felt how badly such a thing is lacking in the present literature... I [am] looking forward, too, with great interest for your mscr. on the general product operation."

This does not sound related to Baire's category, and the latter topic would not be called "category theory" in 1976, when Ulam wrote, either. Ulam's claim is generally taken as Polish school exploring some ideas of category theory in the 1930s. Eilenberg was Polish and got his PhD from the University of Warsaw in 1936. One of his thesis advisors was Kuratowski, same as Ulam's.

Of course, some ideas do not amount to discovering category theory, which is more than abstract treatment of products. MacLane in The development and prospects for category theory is even more categorical (pardon the pun):

"Stan Ulam, who in his autobiography, claimed to have invented, but not published, ideas on categories in Poland in the early 30's. I sharply doubt this claim; I knew Ulam well in the late 1930's, and could not then discern any interest on his part in such conceptual developments."

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    $\begingroup$ does not sound related to Baire's category --- The excerpt from Ulam's autobiography is clearly (to me, at least) not about Baire category. Also, "Oxtoby" in the index (chosen because of the Oxtoby-Ulam theorem in ergodic theory, which is a Baire category "almost all" result), leads to this on pp. 84−85: Right off, we started to discuss problems concerning the idea of "category" of sets. "Category" is a notion in a way parallel to but less quantitative than the measure of sets—that is, length, area, volume, and their generalizations. $\endgroup$ Commented Jun 13 at 10:35
  • $\begingroup$ r.e. the OP's excerpt: I initially considered but then dismissed that it could have referred to notions involving "operations of sets" as mentioned in this MSE answer (and Kantorovitch/Livenson papers I & II, and subsequent work by several Russian mathematicians). $\endgroup$ Commented Jun 13 at 11:10
  • $\begingroup$ Thanks, Conifold. The title of my post was a little bit clickbait-y, because I strongly incline to Mac Lane's side: this was almost certainly a piece of bravado on Ulam's part (as well as some regrettable "flag-planting"). It's one thing to discover a universal property or two; it's another to pin down what a natural transformation is, by having to conceptually backtrack from there to functor, and from functor to category, and to realize that "category" could mean far more than "collection of structured sets, and functions preserving that structure". The Ulam advocates are pushing it. $\endgroup$ Commented Jun 13 at 12:19
  • $\begingroup$ First of all, we should define what we mean by category theory. In the narrow sense, IMHO, it says that in your profs, you are allowed to use function composition, and all your proofs will be manipulations of paths in directed graphs. $\endgroup$
    – bandi
    Commented 2 days ago
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    $\begingroup$ Thanks @Conifold for the pointers, and thanks Todd for posting here. For the record, my original Q re: Ulam was "Without wanting to read too much into his offhand remark, I'm curious about what he was talking about". It is interesting to see Mac Lane's negative reaction, and understandable given Ulam seemed to suggest having written up the "outlines of category theory" in a night. OTOH, Ulam's master's thesis is an actual document, which is why I wondered what he really meant by the cryptic remark, beyond bravado. From the Kuratowski quote, it sounds like it did have some conceptual content. $\endgroup$ Commented 2 days ago

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