Talk:Category theory

Why is this article separate from Category (mathematics)?

It seems to me that either they should be merged, or that at the very least, there should be hatnotes for disambiguation and context. It is too far out of my competence for me to do it, but something either is wrong, or incomplete.JonRichfield (talk) 04:33, 3 September 2020 (UTC)

My understanding is that Category (mathematics) deals with the specific object called a category, while category theory deals with categories, morphisms, natural transformations, and related subjects. Adding things like natural transformations into the category page would be flat out wrong, and I think there is plenty of evidence that such a summary page as this is notable and related. For a similar thing, I would compare set (mathematics) and set theory. _{Integral Python} ^{click here to argue with me} 15:45, 3 September 2020 (UTC)

Labeled directed graph?

If someone has the time, please remove the reference to "labeled directed graph". For once, it is wrong, a graph has a set of nodes, a category has a class. Also, the word graph never appears again in the whole article. And what even is the labelling supposed to represent? --345Kai (talk) 16:51, 12 April 2022 (UTC)

I agree that the the reference to graphs is definitively wrong and must be removed. The article has many other issues and needs to be completely rewritten. Just now, I have not the time for fixing them. D.Lazard (talk) 20:00, 12 April 2022 (UTC)

I have rewritten the lead completely and removed the confusing essay-like sections. D.Lazard (talk) 14:58, 13 April 2022 (UTC)

For once?

Definitively?

פשוט pashute ♫ (talk) 20:32, 7 November 2022 (UTC)

Why were the letters hom() chosen

Is it short for home? If so how does that relate to morphs? If not, why were those letters chosen? פשוט pashute ♫ (talk) 20:33, 7 November 2022 (UTC)

This the abbreviation of "homomorphism". D.Lazard (talk) 20:38, 7 November 2022 (UTC)

reference to Ulam

I've been trying to understand better what connection if any Stanisław Ulam had with category theory. Currently, this article states that "Stanislaw Ulam, and some writing on his behalf, have claimed that related ideas were current in the late 1930s in Poland", but without giving any sources. It seems that this information was originally added to the article by User:Charles_Matthews over twenty years ago, in an edit on 21 November 2003. Charles, if you see this message (and if you remember!), could you please clarify what were your sources?

Actually, I only noticed the reference to Ulam in this article after reading a passage from his autobiography "Adventures of a Mathematician", 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.

I wonder if this is the source that Matthews had in mind? However, it's not clear what exactly Ulam was referring to in this quote. I asked about it on the categories mailing list, and Zoran Škoda suggested that he might have meant "category" in the sense of the Baire category theorem, which is related to the Kuratowski–Ulam theorem. So it might just be a terminological confusion. In any case, it would be good if the article clarified the source for this claim of a claim, at the very least someone should add a "citation needed". Noamz (talk) 19:10, 11 June 2024 (UTC)

@Noamz: I think my source was some edition of the "Scottish Book". As I once had a volume of Ulam's papers, I can't be sure. This quote about Ulam and Banach and "analogies between analogies" makes the whole thing plausible.[1] Charles Matthews (talk) 19:18, 11 June 2024 (UTC)

Hmm, if it was the Scottish Book, then I suspect it might have been a reference to "category" in the other sense, not of category theory but of first category / second category. For example I just found this quote from "A Personal History of the Scottish Book" by Marc Kac, where it's clear from context that he is talking about "category" in that latter sense:

Now, one final observation in connection with other people’s involvement in the Scottish Book Conference, namely with Professor Zygmund’s, who referred in his talk to one of the greatest Polish discoveries, the category method. As a matter of fact, this discovery is so well known that one does not even recognize what a remarkable discovery it was. It was remarkable because it showed that sometimes it is easier to prove that most objects have a certain property than to exhibit a particular example.

Noamz (talk) 19:45, 11 June 2024 (UTC)

And if there is no verified source, I would suggest to simply remove this claim of a claim by Ulam, as it seems to me very likely to be the result of a terminology clash. Noamz (talk) 07:38, 12 June 2024 (UTC)

but as a counterargument that maybe Ulam really did mean "category theory" in the sense we use it today, I found the following quote from Kuratowski's report on Ulam's thesis in Polish statisticians: Biographical notes (p.145):

The work is a study of the "product" operation, which has been little researched until now, but is playing an increasingly important role in mathematics. The author analyses this concept in relation to problems in set theory, group theory, topology, metric space geometry, combinatorics and measure theory in connection with probability. Since the author has demonstrated complete mastery of the subject and due knowledge of the relevant literature, that moreover the work contains a number of original results, and finally that the author presents many interesting problems in this work, I consider this thesis to be perfect.

So perhaps the thesis formulated something like the definition of product (category theory)? Without a copy of the original source it's all quite mysterious! Noamz (talk) 09:00, 12 June 2024 (UTC)

Lack of examples

When we are presented for the first time with group theory, it appears abstract, arcane, and difficult to understand what it is useful for, until we are shown that Z, or Q, or permutations, or symmetries are groups with the appropriate operations. Similarly, when exposing category theory, one should give examples right after each notion introduced. If the article followed this principle, it would make category theory much easier to understand. 2A01:CB1C:8040:4C00:8552:2D17:F906:92BF (talk) 02:15, 30 July 2024 (UTC)