[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. ...

Jerzy Karczmarczuk
/Caen, France/

L'absence de virus dans ce courrier électronique a été vérifiée par le logiciel antivirus Avast.

More information about the Haskell-Cafe mailing list