Re: Fred

[Emily Riehl <eriehl@math.jhu.edu>]

> 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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