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Ralph Sarkis (Feb 13 2024 at 16:21):I just learned from this set of notes that the fact that (co)limits can be computed by a (co)equalizer of (co)products was proven by Jean-Marie Maranda, hence that fact could be referred to as Maranda's theorem. I think it is the first time I even see this name, but the nlab says Maranda also "invented" enriched categories.
Why isn't Maranda better known? or more frequently credited for these two fundamental ideas?
Nathanael Arkor (Feb 13 2024 at 16:37):Unfortunately, I think this phenomenon is common more generally in the context of French (language) category theory from that time period. The same is true for ideas introduced by Bénabou, Burroni, Diers, Ehresmann, Guitart, to name just a few.
Nathanael Arkor (Feb 13 2024 at 16:38):Certainly the language barrier is partly an issue (and you see similar fundamental results overlooked in, say, the German categorical literature), but I think a bigger factor is political (but that would require the insight of someone who was around at that time).
Nathanael Arkor (Feb 13 2024 at 16:39):or more frequently credited for these two fundamental ideas?
I think most people don't care enough to find out who is responsible for which "well known" ideas in category theory.
Nathanael Arkor (Feb 13 2024 at 16:40):For instance, you see many instances in papers, books, and conversation of people attributing the concept of 2-category to Ehresmann, when the notion is in fact due to Bénabou and Maranda (along with the concept of enriched category).
Nathanael Arkor (Feb 13 2024 at 16:41):I, for one, would be very happy to see our community take more care in the attribution of ideas.
Ralph Sarkis (Feb 13 2024 at 16:42):re French CTists: But Maranda was in Montréal (right?) with Barr, Bunge, Joyal, Lambek, Makkai. Were they objectively significantly more important for CT?
Nathanael Arkor (Feb 13 2024 at 16:42):hence that fact could be referred to as Maranda's theorem
(But please let's not encourage this practice of naming theorems or definitions after people.)
Nathanael Arkor (Feb 13 2024 at 16:48):Ralph Sarkis said:
re French CTists: But Maranda was in Montréal (right?) with Barr, Bunge, Joyal, Lambek, Makkai. Were they objectively significantly more important for CT?
True, that wouldn't explain why Maranda's papers in English do not seem to be widely known.
Bryce Clarke (Feb 13 2024 at 17:00):Ralph Sarkis said:
I just learned from this set of notes that the fact that (co)limits can be computed by a (co)equalizer of (co)products was proven by Jean-Marie Maranda, hence that fact could be referred to as Maranda's theorem.
In the set of notes that you link, Maranda's Theorem (Theorem 3.2.2., p 34) is given by the equivalence of three statements. However, in Maranda's paper Some Remarks on Limits in Categories, I only see the equivalence of (1) and (2). In Categories Work, Chapter 5, Section 2 on Limits by Products and Equalizers, Mac Lane doesn't appear to mention Maranda at all, but does attribute the equivalence between statements (1) and (3) to [[Ernest Manes]] (see Exercise 1, p 114).
Bryce Clarke (Feb 13 2024 at 17:04):I wonder why Mac Lane did not acknowledge Maranda at all.
Bryce Clarke (Feb 13 2024 at 17:07):I am at least happy to learn from this thread that the construction of (co)limits by (co)equalisers and (co)products is due to Maranada, so thank you!
Nathanael Arkor (Feb 13 2024 at 17:13):Bryce Clarke said:
However, in Maranda's paper Some Remarks on Limits in Categories, I only see the equivalence of (1) and (2).
I think the notion of reflexive coequaliser may not have been introduced until Linton's paper Coequalizers in categories of algebras in 1969, so there would be no reason for Maranda to state it in this form. However, presumably it follows directly that one can require only reflexive coequalisers from inspecting Maranda's proof.
Bryce Clarke (Feb 13 2024 at 17:23):So I was curious as to why Mac Lane referred to Manes at all. The only paper of Manes cited in the bibliography of Categories Work is A triple theoretic construction of compact algebras from 1969.
Nathanael Arkor (Feb 13 2024 at 17:26):Maybe Mac Lane simply learned of the fact from Manes. In other places where he cites Manes, he explicitly refers to Manes' thesis, but does not do so here.
Bryce Clarke (Feb 13 2024 at 17:26):However, the 1967 PhD thesis of Manes A triple miscellany: Some aspects of the theory of algebra over a triple, Proposition 0.6.2 (p. 11) gives the statement: If K has coproducts and if every reflexive pair of morphisms has a coequaliser, then K has all colimits.
Bryce Clarke (Feb 13 2024 at 17:27):Unfortunately, I could only find a small preview of Manes' thesis on Proquest. Does anyone have the full pdf?
Bryce Clarke (Feb 13 2024 at 17:31):Manes does mention Maranda in the introduction of his thesis, but not in relation to this result. It would be great to see the full list of references of this thesis.
Nathanael Arkor (Feb 13 2024 at 17:39):Bryce Clarke said:
Unfortunately, I could only find a small preview of Manes' thesis on Proquest. Does anyone have the full pdf?
It's available here.
Morgan Rogers (he/him) (Feb 14 2024 at 09:49):It is nice to see a name attached to this considering that the result has been discussed recently around here.
fosco (Feb 14 2024 at 11:09):Nathanael Arkor said:
hence that fact could be referred to as Maranda's theorem
(But please let's not encourage this practice of naming theorems or definitions after people.)
I agree, and I think we should call this the "Arkor principle"