Re: Fred

[Patrik Eklund <peklund@cs.umu.se>]

Dear Emily and Ren??,

Under this list, and e.g. in connection with G??del's Incompleteness
Theorem (which I still call the Incompleteness Paradox), I have thrown
out the idea that logic may and perhaps even should be "lative", in the
sense that

- first we fix sorts and operators, i.e., the signature
- then we build terms based on that signature, but terms cannot change
anything about the underlying signature, so the door is closed, so as to
say
- sentences then build upon terms, and similarly cannot manipulate or
change whatever is in the term set

And so on, so an entailment, building e.g. upon sentences, even if being
"true" cannot be seen as a sentence, i.e., we should allow ourselves to
view entailments also as sentences and throw them back into the bag of
sentences. This is what G??del is doing, and this is widely accepted. I
don't accept, but this is now not my message here.

---

Triadic relations are potentially "illative" in this respect. Port-Royal
is dyadically lative, I would say, and maybe Peirce "triadically
lative", but a bit les so. Both are not formal enough, but my question
is how trijunctions or multivariable adjunctions think in theses
respects?

Best,

Patrik

PS "Lative logic", yet to be better defined, is an extension of Goguen's
Institutions and Meseguer's Entailment Systems.
http://umu.diva-portal.org/smash/get/diva2:619702/FULLTEXT01.pdf



On 2017-09-27 12:10, Ren?? Guitart wrote:
> 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. G??o. 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,
>
> Ren??.
>
>
> Le 7 sept. 2017 ?? 19:03, Emily Riehl a ??crit :
>
>>> 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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