Dear Colleagues, May I only make a comment on what Mr. Posina Venkata Rayudu says about Mac Lane's "Duality of Groups"? We all are humans, not computers, and in some cases we might miss some important references (I am sure I did it more than once). But I think we should be more careful in saying this about others, especially about those who are not with us anymore, and those who made a great contribution to mathematics and greatly helped many others to become excellent mathematicians. I am glad that "...UMP definition of product clearly(?) beats MacLane by at least 2 years..." at least contains a question mark, because the answer to that question is NO. But it is so because "beats" is wrong, sorry! Let me explain, speaking, for simplicity, about the binary product AxB. Put simply, Samuel says that, for various types of structures, the product structure on the product of underlying sets of A and B can be characterized by saying that (the projections preserve structure and) given structure preserving maps f:C—>A and g:C—>B, the map h:C—>AxB defined by h(c)=(f(a),g(c)) is also structure preserving. Needless to say, this is fundamentally different from defining products by the universal property (not to mention anything else, this cannot be used to define products of sets). Well, one might argue that universal properties in the sense of Bourbaki were predecessors of what we call universal properties today, but one might equally argue that Bourbaki structures and their axiomatically defined morphisms were predecessors of categories, but predecessors themselves had predecessors... Or, one might reformulate what Bourbaki used in terms of suitable forgetful functors into the category of sets, but that is another story. Sincerely, George Janelidze ________________________________ From: Posina Venkata Rayudu via Categories <categories-list@categories.org.au> Sent: Monday, October 19, 2026 19:18 To: Category Theory Mailing List <categories-list@categories.org.au> Cc: Posina Venkata Rayudu <posinavrayudu@gmail.com> Subject: [categories] UMP: Elementary vs. Higher CAUTION: This email originated outside the UCT network. Do not click any links or open attachments unless you know and trust the source. Dear All, I hope all is well. (Reading Advisory: Boring ;) In the case of the elementary universal mapping property (UMP) definitions (e.g., product, sum), wherein commutativity i.e., the isomorphism between, say, (A + B) and (B + A), makes the ordering boring (both at the level of listing summands A, B and also at that of the sum). However, in stating the definition of exponentiation or functor category as a higher UMP: For any pair of objects A, B of a category C, there is an object B^A called the object (or category) of all maps (functors) from A to B. This always seemed little like withholding the truth, albeit for a femtosec, in the sense of: given objects A, B, there's also the object A^B of maps from B to A. Is this asymmetry: product / sum vs. exponential, where switching around the objects doesn't matter vs. does matter (turning the whole thing upside down ;) the content of the contrast: elementary vs. higher (UMP definitions)? On a related note, MacLane (1950) Duality for Groups <https://www.ams.org/journals/bull/1950-56-06/S0002-9904-1950-09427-0/S0002-9904-1950-09427-0.pdf<https://www.ams.org/journals/bull/1950-56-06/S0002-9904-1950-09427-0/S0002-9904-1950-09427-0.pdf>> construes products and sums as universal mapping properties, but, for whatever reasons, doesn't cite [Bourbaki] Samuel (1948) Universal Mappings paper published in the very same Bulletin of the AMS <https://www.ams.org/journals/bull/1948-54-06/S0002-9904-1948-09052-8/S0002-9904-1948-09052-8.pdf<https://www.ams.org/journals/bull/1948-54-06/S0002-9904-1948-09052-8/S0002-9904-1948-09052-8.pdf>>, which with its UMP definition of product clearly(?) beats MacLane by at least 2 years. Then again, it's MacLane who, in his own admission, got to pick and choose editors and reviewers of his papers (e.g., Mackey); I wish I could (one invariant of the reviews I receive is that I didn't cite this or that or dem ;) Moving along to higher UMPs, the definition of truth value object / subobject classifier had to wait for Professor F. William Lawvere (cf. Grothendieck's "Lawvere element" <https://www.mat.uc.pt/~picado/lawvere/interview.pdf<https://www.mat.uc.pt/~picado/lawvere/interview.pdf>> p. 7). In my limited reading, it seems to me that even the aforementioned higher UMP definition of exponentiation also appears, for the first time(?), in the thesis of Professor F. William Lawere (cf. Functorial Semantics of Algebraic Theories <http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf<http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf>> p. 29). I look forward to your clarifications and corrections! Thanking you, Yours respectfully, posina P.S. As much as I hate watching people fight, I can't help but share an anecdote: A Frenchman, upon returning from a visit to the then blazing USA, quipped: Americans confuse plumbing for culture ;) On a related note, going by the "Emily in Paris", France has a long way to go (plumbing ;) Along these lines, even at the Harvard Medical School, it is understood and admitted by none other than Americans, based on the accumulating evidence in the discipline of neuroscience, that European science is solid, while Americans are good salesmen ;) _______________________________________________ Categories mailing list -- categories-list@categories.org.au To unsubscribe send an email to categories-list-leave@categories.org.au Disclaimer - University of Cape Town This email is subject to UCT policies and email disclaimer published on our website at https://www.uct.ac.za/main/email-disclaimer or obtainable from +27 21 650 9111. 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