RE: For the record

RE: For the record

[Marta Bunge]

Dear Peter

I was also a participant at the Oberwolfach 1966 meeeting as you may
recall, and heard Lawvere give his lecture. Although I cannot find my own notes, those of Anders Kock bring back my recollection of it. The
following is relevant to your comments. I myself gave there a talk about an aspect of my thesis -- namely a characterization of functor categories ("diagrammatic categories") based on what I called "regular categories, a Set-version of your abelian categories, as well as on Bill's idea, expressed in his lecture, that, thinking of the category of  ordered sets, the analogue of Sets^{C} is 2^{\Lambda}. This analogue led to the idea of atoms in order to emphasize the analogous characterization of complete atomic Boolean algebras as fields of sets. 
For the record, my thesis (Marta C. Bunge Categories of Set-valued functors, University of Pennsylvania 1966) was unpublished as such and only parts of it in a different version appeared three years after completion (Marta C. Bunge "Relative Functor Categories and Categories of Algebras" J. Algebra 11-1 1969, 64-101), at the insistence of Saunders Mac Lane, who communicated my paper.  
About profunctors. My thesis as you may recall contains an entire chapter (III) on equivalences of diagrammatic categories and adjoint functors between them. Theorem 14.1 states that for M any complete category and B a small category Adj(Sets^B M) \equiv M^{B\op}. In particular Adj(Sets^B\, Sets^C) \equiv (Sets^C)^{B\op} \equiv Sets^{B\op x C}. Relevant to this is a previous paper by Michel Andre, Categories of functors and adjoint functors, Batelle Reports, Geneve 1964, which I quote both in my thesis and in the J. Algebra paper. 
It is furthermore remarked in my thesis that, when I is a discrete category, the statement Adj(\Sets^I \, Sets^I) \equiv Sets^{I x I) has an obvious interpretation analogous to that which exists between endomorphisms of a vector space and matrices. This remark was further expanded (section III.14) into a discussion of the analogue of matrix multiplication for "profunctors". Therefore, although I do not recollect Benabou's lecture or what followed it, Bill's lecture and my own thesis are, for me, enough indication of priority over Benabou's distributeurs. 
Thanks for giving me this opportunity to comment on a meeting that I had forgotten about. 
Best regards
Marta




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For the record

[Peter Freyd]

In Spring 1969 there was a conference in Rome. In the resulting volume
of Symposia Mathematica there appear papers by David Buchsbaum, Leon
Ehrenpreis, Peter Freyd, John Gray, Ioan.James, Barry Mitchell and
Graeme Segal. Talks also were given by Jean Benabou, Charles
Ehresmann, Bill Lawvere and Saunders Mac Lane. (Among others in
attendance -- I do not specifically recall if they gave talks -- were
Yitz Herstein and John Moore.)

Bill gave his first talk on elementary topoi. My talk was on the "more
general" adjoint functor theorem (but the paper I put in the
proceedings was on the concreteness of certain categories). Jean
Benabou talked about distributors. During his talk I decided to wait
until I could check with Bill before saying anything. In what I
expected to be an entirely private conversation I then brought to
Benabou's attention -- as gently as I have ever succeeded in being
with an adult -- that in July, 1966, he and I had heard Bill give a
talk in Oberwolfach in which Bill described a kind of "generalized
functor" he called a "bimodule." Within two seconds it ceased being a
private conversation. Nothing in my mathematical career had come close
to preparing me -- or anyone else I knew at the conference -- for the
scene that then occurred.

Under the circumstances I decided not to take the opportunity to point
out that at the lunch following Bill's 1966 talk many observed Benabou
pressing him for more details about these generalized functors called
bimodules; what good were they anyway; how would one use them.


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