Le 17/07/2018 à 09:30, Joachim Durchholz a écrit :
being a monoid in a category does not make it a monoid directly.
??? Could you please explain what do you mean by this?
There's also a final argument: If monad and monoid are really the same, why do mathematicians still keep the separate terminology?
I am sure that you see yourself that this is a non-argument. Mathematics is a human activity, not a formal, distilled language. On math.stackexchange.com there is a discussion about "monoid" term history. One user says: Oxford English Dictionary traces monoid in this sense back to Chevalley's Fundamental Concept of Algebra published in 1956. Arthur Mattuck's review of the book in 1957 suggests that this use may be new... Others trace the term to MacLane, or to something which appeared in 1954. So, it was a term which lived separately from monads. == Mathematicians don't quarrel often on terminological issues, unless they have nothing more interesting to do. In the Barr & Wells book monads figure once, just to tell the readers that the term "have also been used in place of “triple”" (Even without the attribution: "Kleisli"...). (And they mention "triads", "fundamental constructions", etc.). In abstract algebra some people say "magma", others: "grupoid" , and --- There is also the inverse phenomenon, the existence of distinct enities with the same name. In Differential geometry, the "pullback" is used differently than in Categories. --- the literature will warn you that grupoid in Category Theory means something different. (Former: a structure with a single binary op; here: a group with partial function replacing the binop). Wikipedia will tell you: "In non-standard analysis, a monad (also called halo) is the set of points infinitesimally close to a given point." Anybody here heard about this?... etc. etc. ... Thanks. Jerzy Karczmarczuk /Caen, France/ --- L'absence de virus dans ce courrier électronique a été vérifiée par le logiciel antivirus Avast. https://www.avast.com/antivirus