1) Two days ago by chance I come across an article of Bill Thurston: https://arxiv.org/pdf/math/9404236.pdf and seeing his name mentioned in this thread it occurs to me that everybody in this list should read it. In my opinion it is an extraordinary document about mathematics, mathematical activity and mathematicians. 2) Respect to to subject of this thread, the formal opposite of a category, denoted "op", is simply a notation very useful to work with functors which are contravariant in some variables, either with the "op" in the domain or the codomain of the functor arrow. Notations are important, and the "op" notation is essential in the language of categories and functors. 3) Finally, concerning Fred Linton, his death sadness me, he did important work in the early days of category theory, but more important, he was one of us, it was always a pleasure to encounter him, an he was a good guy. all the best e.d. On 07/09/17 14:03, Emily Riehl wrote:
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 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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