Re: Fred

["René Guitart" <rene.guitart@orange.fr>]

Dear Emily,

many thanks for your message. It push me to precise some aspects of my link with the "op" things, with some words about your joint paper https://arxiv.org/abs/1208.4520. 

 	In fact in a talk at a the Louvain-la-Neuve's meeting in 2011, Category theory, algebra and geometry, 26-27 may 2011, I spoke on "Borromean Objects and Trijunctions". I do remember well that Eugenia was listening to this talk,
and so probably her attention was attracted on the notion of a trijunction (the notion you are explaining in your message). Some times later this was published in a paper "Trijunctions and Triadic Galois Connections" (Cahier Top. Géo. Diff. Cat, LIV-1 (2013), pp. 13-28) (accessible on my site : http://rene.guitart.pagesperso-orange.fr/publications.html). From the summary, we can learn why I did so : 
 	"In this paper we introduce the notion of a trijunction, which is related to a triadic Galois connection just as an adjunction is to a Galois connection. We construct the trifibered tripod associated to a trijunction, the
 	trijunction between toposes of presheaves associated to a discrete trifibration, and the generation of any trijunction by a bi-adjoint functor. While some examples are related to triadic Galois connections, to ternary relations, others are 	associated to some symmetric tensors, to toposes and algebraic universes". 
Now it is interesting to understand how this was achieved, in two steps: 
1 - Firstly I read a paper by Biedermann, on triadic Galois connections, related to ternary relations as Galois connections of Ore between ordered sets are related to binary relations. Immediately I try to extend that from order sets to categories.
2 - Fortunately in the same time I was conducted to read again carefully the famous paper by Kan on Adjoint functor. There I observed that in fact he he is mainly working with tensors and Gom, i.e. with bifunctors ;  and furthermore in the Mac Lane's book, the convenient lemma for parameterized adjunctions are reproduced. Then I notice that the perfect explicit ternary symmetry in Biedermann was in fact also implicit in Kan, but that only he "missed" to put the accent on it, by introducing the opposite of the opposite (E^op)^op in your message). So I did, and then I got the application to the descriptions of trifibrations and toposes, to the analysis clearly the system of functions or operations generated by a tripod.

So you can see that my motivations (to unified Kan and Biedermann at the level n = 3), in order to produce a functional analysis of tripod), seems rather different from yours (to enter in a game of general n-multiadjunctions).  Hence finally my question : to analyze the system of functions or operations generated by an n-pod, and to understand there the part play by the mysterious "op".

with my friendly greetings,

René.


Le 7 sept. 2017 à 19:03, Emily Riehl a écrit :

>> There is one other anecdote about UACT, nothing to do with Fred, that I
>> have always loved. In the course of MSRI director Bill Thurston's Galois Connections
>> opening remarks, he said words to the effect that the notion of the
>> opposite of a category made him nauseous. This was the only meeting I
>> have ever attended where fully half the attendees drew in enough breath
>> to drop the air pressure by an audible amount.
> 
> I’ll confess that the idea of an opposite category appearing as the codomain of a functor also makes me somewhat nauseated (the domain of course is no problem). 
> 
> But this said, in the interest of full disclosure, I should admit that in a joint paper with Cheng and Gurski someone — Eugenia, I believe? —  convinced us that the easiest way to think of a functor 
> 
> C x D —> E 
> 
> admitting right adjoints in both variables is as a functor 
> 
> C x D —> (E^op)^op
> 
> because in this way (writing E’ for E^op) the other two adjoints also have the form
> 
> D x E’ —> C^op
> 
> and 
> 
> E’ x C —> D^op.
> 
> Such two-variable adjunctions form the vertical binary morphisms in a “cyclic double multi category” of multivariable adjunctions and parametrized mates:
> 
> https://arxiv.org/abs/1208.4520
> 
> Regards,
> Emily
> 
> —
> Assistant Professor, Dept. of Mathematics
> Johns Hopkins University
> www.math.jhu.edu/~eriehl
> 
> 

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