From: Ross Street <street@ics.mq.edu.au> Subject: categories: RE:
Monads Date: Sat, 4 Apr 2009 14:31:39 +1100 To: categories
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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