Michael Barr | 1 Apr 20:13 2009

Re: Where does the term monad come from?

I have told this story many times, but I guess one more can't hurt.  Of
course, it was originally used by Leibniz to describe the set of
infintesimals surrounding an ordinary point.

In the summer (or maybe late spring, the Oberwohlfach records will show
this) of 1966, there was a category meeting there.  It was, as far as I
know, the third meeting ever devoted to categories.  The first was the
first Midwest Category meeting, an invitation affair that consisted of
five people from Urbana (Jon Beck, John Gray, Alex Heller, Max Kelly, and
me), John Isbell and Fred Linton visiting Chicago that year, and a couple
people from U. Chicago, Mac Lane who was the host and arranged to pay our
expenses, Dick Swan, and maybe a couple others.  The second was in La
Jolla and this was the third.  The attendance consisted of practically
everyone in the world who had any interest in categories, with the notable
exception of Charles Ehresmann.

What, with one exception, most categorists call monads had by that time
been called "Standard constructions", "fundamental constructions" (in a
little-known paper by Jean-Marie Maranda pointed out to me by Peter
Huber), and, of course, "Triples".  The latter was created by
Eilenberg-Moore and I once asked Sammy (who was known to agonize over good
terminology--e.g. "Exact") why.  He answered that the concept seemed to be
of little importance, so he and John Moore spent no time on it!  So much
for the predictive ability of a great mathematician.

At any rate, the big unanswered question of the meeting, where the
importance of the concept was becoming clear (Jon and I had proved our
Acyclic models theorem, for example, and the fact of the triplebleness of
compact Hausdorff  spaces over sets, along with many mor familiar
examples), the search was on for a better name.  We tried many ideas (mine
was "Standard Natural Algebraic Functor with Unit" (try the acronym).  One
day at lunch or dinner I happened to be sitting next to Jean Benabou and
he turned to me and said something like "How about `monad'?"  I thought
about and said it sounded pretty good to me.  (Yes, I did.)  So Jean
proposed it to the general audience and there was general agreement.  It
suggested "monoid" of course and it is a monoid in a functor category.
The one dissenter was Jon Beck, who had invested as much into studying
them as anyone.  His argument was that while "triples" was not a good
name, "monad" wasn't either and we shouldn't change the name from a poor
one to a mediocre one, but instead continue to search for a better one.

Out of solidarity with Jon (we collaborated on several papers), I
continued to use "triple".  SLN 80 was (and is) known as the "Zurich
Triples Book".  By 1980, Jon was no longer doing serious mathematics and I
was ready to change.  Except that the book title "Toposes, Triples and
Theories" was too attactive to let go of.  Try "Toposes, Monads and

Incidentally, Peter May also claims to have invented the term.  Treat that
claim with the contempt it deserves.  The most charitable explanation I
have is that he heard it from Mac Lane, forgot that he had and then came
up with it later.

On Wed, 1 Apr 2009, Thorsten Altenkirch wrote:

> A question just came up at the Midland Graduate School (actually in
> the functional programming lecture):
> Where does the word monad come from?
> I know that a monad is a monoid in the category of endofunctors, but
> what is the logic monoid => monad?
> Btw, I frequently encounter monads in a categories of functors which
> are not endofunctors. An example are finite dimensional vectorspaces
> which can be constructed via a monoid in the category of functors
> FinSet -> Set, here I is the embedding and (x) can be constructed from
> the left kan extension and composition.
> The unit is given by the Kronecker delta and join can be constructed
> from Matrix multiplication. Should one call these beasts monads as
> well? Is there a good reference for this type of construction?
> Cheers,
> Thorsten