UMP: Elementary vs. Higher
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> 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>, 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> 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> 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 ;)
That's amazing. I had never seen or known of Samuel's paper. Michael ________________________________ From: Posina Venkata Rayudu via Categories <categories-list@categories.org.au> Sent: Monday, October 19, 2026 1:18 PM To: Category Theory Mailing List <categories-list@categories.org.au> Cc: Posina Venkata Rayudu <posinavrayudu@gmail.com> Subject: [categories] UMP: Elementary vs. Higher 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://can01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1950-56-06%2FS0002-9904-1950-09427-0%2FS0002-9904-1950-09427-0.pdf&data=05%7C02%7Cbarr.michael%40mcgill.ca%7C22bbcf6313ca424c8f5108df2e71a0ab%7Ccd31967152e74a68afa9fcf8f89f09ea%7C0%7C0%7C639280737834436191%7CUnknown%7CTWFpbGZsb3d8eyJFbXB0eU1hcGkiOnRydWUsIlYiOiIwLjAuMDAwMCIsIlAiOiJXaW4zMiIsIkFOIjoiTWFpbCIsIldUIjoyfQ%3D%3D%7C0%7C%7C%7C&sdata=sbmNvfnrnoYMLoCmhMWCnunVdln023eHETa7eziZncI%3D&reserved=0<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://can01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1948-54-06%2FS0002-9904-1948-09052-8%2FS0002-9904-1948-09052-8.pdf&data=05%7C02%7Cbarr.michael%40mcgill.ca%7C22bbcf6313ca424c8f5108df2e71a0ab%7Ccd31967152e74a68afa9fcf8f89f09ea%7C0%7C0%7C639280737834452328%7CUnknown%7CTWFpbGZsb3d8eyJFbXB0eU1hcGkiOnRydWUsIlYiOiIwLjAuMDAwMCIsIlAiOiJXaW4zMiIsIkFOIjoiTWFpbCIsIldUIjoyfQ%3D%3D%7C0%7C%7C%7C&sdata=ooFbxbS6Xnu0F5m1CEcH0IBbohrv0rP2qQ9YoNJeWiI%3D&reserved=0<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://can01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.mat.uc.pt%2F~picado%2Flawvere%2Finterview.pdf&data=05%7C02%7Cbarr.michael%40mcgill.ca%7C22bbcf6313ca424c8f5108df2e71a0ab%7Ccd31967152e74a68afa9fcf8f89f09ea%7C0%7C0%7C639280737834462564%7CUnknown%7CTWFpbGZsb3d8eyJFbXB0eU1hcGkiOnRydWUsIlYiOiIwLjAuMDAwMCIsIlAiOiJXaW4zMiIsIkFOIjoiTWFpbCIsIldUIjoyfQ%3D%3D%7C0%7C%7C%7C&sdata=k8j9HPWW4umrCyZYA8WSKNB4zFWWcjzyOVCqzt5qnu8%3D&reserved=0<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 <https://can01.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.tac.mta.ca%2Ftac%2Freprints%2Farticles%2F5%2Ftr5.pdf&data=05%7C02%7Cbarr.michael%40mcgill.ca%7C22bbcf6313ca424c8f5108df2e71a0ab%7Ccd31967152e74a68afa9fcf8f89f09ea%7C0%7C0%7C639280737834472743%7CUnknown%7CTWFpbGZsb3d8eyJFbXB0eU1hcGkiOnRydWUsIlYiOiIwLjAuMDAwMCIsIlAiOiJXaW4zMiIsIkFOIjoiTWFpbCIsIldUIjoyfQ%3D%3D%7C0%7C%7C%7C&sdata=oyI%2BBANxW9ch59LzCqGEkdAvzLR8nNm13cUpd3xb6R8%3D&reserved=0<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
Dear All, Please accept my sincere apologies for my characterization of Americans (in contrasting with Europeans). I am truly thankful for the warm hospitality I enjoyed for a decade in San Diego and another in Boston (just as in the case with visiting my relatives, I had to disappear for, everytime I tried goodbye, it's invariably: stay some more time, which made me realize that what I hitherto thought of as Godavari culture has gone global :). I didn't intend: Americans are limited to salesmen; they are also, just to note one, saviors safeguarding math <https://doi.org/10.1126/science.ael8339>. Thank you Professor Emily Riehl! Thanking you, Yours respectfully, posina ---- On Wed, 21 Oct 2026 04:44:34 +0530 categories-list@categories.org.au wrote ---- That's amazing. I had never seen or known of Samuel's paper. Michael From: Posina Venkata Rayudu via Categories < mailto:categories-list@categories.org.au > Sent: Monday, October 19, 2026 1:18 PM To: Category Theory Mailing List < mailto:categories-list@categories.org.au > Cc: Posina Venkata Rayudu < mailto:posinavrayudu@gmail.com > Subject: [categories] UMP: Elementary vs. Higher 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/S0002-9... > 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/S0002-9... >, 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 > 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 > 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 -- mailto:categories-list@categories.org.au To unsubscribe send an email to mailto:categories-list-leave@categories.org.au _______________________________________________ Categories mailing list -- mailto:categories-list@categories.org.au To unsubscribe send an email to mailto:categories-list-leave@categories.org.au
participants (3)
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Michael Barr, Prof. -
posina venkata rayudu -
Posina Venkata Rayudu