[Haskell-cafe] Monads vs. monoids
Jerzy Karczmarczuk
jerzy.karczmarczuk at unicaen.fr
Tue Jul 17 08:55:30 UTC 2018
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/
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