RE: Monads

RE: Monads

[Ross Street]

I hope I can add some jigsaw pieces towards the history of the term
"monad" in category theory without offending anyone.

1) It is clearly a fact that the term "monad" is used in Benabou's
paper SLNM 47 (1967). He recognized that it is a morphism of
bicategories from the terminal category 1.

2) I have a clear memory that Mac Lane told me (perhaps at Chicago
while I was a postdoc at Champaign-Urbana 1968-69) that Benabou
courteously asked him (possibly by airmail, maybe by phone call,
maybe at a conference) whether Mac Lane would mind whether he used
the term "bicategory" in the sense we now use it. Mac Lane had used
"bicategory" to mean a category with two distinguished classes of
morphisms: roughly speaking, what we now call a category with a
factorization system. Mac Lane told Benabou he did not mind. So
Benabou used it in SLNM 47.

3) Less clearly I remember Mac Lane said Benabou also suggested the
term "monad" for use in SLNM 47.

4) It is again my clear memory that, in his lecture marathon at the
Summer School on Category Theory at Bowdoin College (Maine,
mid-1969), Mac Lane expressed strong dislike for the term "triple"
but had not really settled on a term. Mac Lane actually used the term
"triad" in his lectures at Bowdoin.

5) At CT08, Lawvere told me Eilenberg suggested the term "monad".

Best wishes,
Ross

RE: Monads

[Patrik Eklund]

[Note from moderator: this thread has strayed; although this post is
allowed, comments closer to categories (not Kant's) are preferred.]

Reference to Leibniz is nice, and so is going back even more in history. 
Going forward into modern history leads to problems of who actually caused 
what. Probably because we then tend to mix history and politics.

Anyway, also having googled, I found this about Leibniz:

§. 1. Die Monaden (Das Worte Monade oder Monas) wovon wir allhier reden 
werden / sind nichts anders als einfache Substanzen / woraus die zusammen 
gesetzten Dinge oder composita bestehen. Unter dem Wort / einfach / 
verstehet man dasjenige / welches keine Teile hat.

"sind nichts anders als einfache Substanzen"
"is nothing but simple substances"
They are, but it is not a mathematical statement.

"woraus die zusammen gesetzten Dinge oder composita bestehen"
"using which you put them together or compose(!) them together"
Now he is cooking. Monad compositions are important. Leibniz and Beck 
working together, I like it. This is closer to mathematics.

"verstehet man dasjenige / welches keine Teile hat"
"is to be understood as something which doesn't have subparts"
I am sure there are non-trivial monads which are not composed (in Beck's 
sense) by other non-trivial monads. But more interestingely, composed 
monads are indeed monads, and even worse (from leibniz point of 
view) submonads do exist, like the filter monad being submonad to the 
ultrafilter monad (with the astonishing fact, yes, I know, I am repreating 
myself, that their respective algebras are Scott lattices and compact 
Hausdorff spaces).

So, basically I like Leibniz, even if he was wrong at this point. History 
is not easy. We say "Rome was destroyed" and we frequently say by the 
goths. Saying that leads us to ask "how could it be destroyed". Seldom do 
we hear "how could it stay alive so long".

Best,

Patrik

PS "Monas" seems mostly to be used for a sailing boat, the "Kiel", and 
"the Mona" is Louvre in Paris.

RE: Monads

[jim stasheff]

Ross Street wrote:
>
> I hope I can add some jigsaw pieces towards the history of the term
> "monad" in category theory without offending anyone.
>
>
<snip>...

> Best wishes,
> Ross
>
>

Some version of this and other responses whould be added to the Wiki
I'm not up to the job.

jim

Re: Monads

[Michael Barr]

Sorry if this offended you, but I heard from several places that you
claimed the invention of the term.  You will note that one other
respondent credited it to you, so there must have been a meme to that
effect.  If you never made that claim then I truly apologize.  Perhaps
people confused "monad" with "operad", which I do believe you invented.

Michael

Re: Monads

[jim stasheff]

Peter May wrote:
>  I never claimed to invent the term monad.  I did
> invent the term operad, as a portmanteau of operation and monad.
> Peter May
>
>
which I am happy to confirm
I think I was there at the time
or at least nearby

jim

Rainer,
    Is this covered in your memoir of those miravulaous years?

jim

RE: Monads

[Marta Bunge]

Dear all,Something to corroborate MacLane's abhorrence of the word "triple" is, in my view, his refusal to communicate my first paper (Marta Bunge, Relative Functor Categories and Categories of Algebras, J.of Algebra 11 (1969) 64-101) unless I changed the word "triple" in it for that of "monad". In order to show my independence (!), yet wishing to have the paper published, I changed "triple" back to "standard construction". This he accepted without objections. Nowadays I use monads like everybody else. I have no idea which, among the many possible reasons suggested in categories, was MacLane's reason for insisting on "monads", whether philosophical or mathematical. However, his acceptance of my use of "standard construction" suggests that his dislike of "triple" was stronger than his preference for "monad". Cordial regards,Marta Bunge

Monads

[Peter May]

Michael, where on earth did that piece of contemptible writing
come from.  I never claimed to invent the term monad.  I did
invent the term operad, as a portmanteau of operation and monad.
And I convinced MacLane to change from the silly term `triple'
to `monad' in Categories for the working mathematician.  He is
not here to corroborate, but look at his note on terminology,
page 138 of the second edition: ``The frequent but unfortunate
use of the word `triple' in this sense has achieved a maximum
of needless confusion, what with the conflict with ordered
triple, plus the use of associated terms such as ``triple
derived functor''for functors which are not three times
derived from anything in the world.  Hence the term monad.''

Michael, shame on you!

Peter May