Re: The boringness of the dual of exponential

[Vaughan Pratt <pratt@cs.stanford.edu>]

On 11/9/2011 7:58 AM, RJ Wood wrote:
  > Your observation about lack of symmetry in \set is underscored by the
  > fact that the yoneda functor for \set has a left adjoint which has
  > a left adjoint which has a left adjoint which has a left adjoint but
  > but the co-yoneda functor for \set has a right adjoint that fails to
  > preserve even finite sums.
  > R_j

Richard modestly omitted that he and our esteemed moderator showed
(necessarily by classical reasoning) that what Richard just said
characterized \set up to equivalence.

On 11/10/2011 1:29 AM, Prof. Peter Johnstone wrote:
> One point that no-one has mentioned yet is that you can't have
> exponentiation and its dual in the same category, unless it is a
> preorder.

Peter modestly omitted that he is the go-to category theorist when it
comes to toposes.

He also omitted that he was confining himself to cartesian closed
categories when he mentioned his point, understandable given that every
topos is cartesian closed.

To expand a little on my (1965) classmate Ross Street's counterexample
of Set^op, Set x Set^op is yet another counterexample.  Here I've
one-upped Ross (I must be getting competitive in my dotage) by
contradicting Peter and giving a counterexample in which both
exponentiation *and* dual exponentiation are present simultaneously.

How did I know that?  Well, Set x Set^op is equivalent (in fact
isomorphic) to Chu(Set, 1).  For *any* set K, both exponentiation and
dual exponentiation are admissible in Chu(Set,K), product being of the
tensor kind in this case.

How did I know *that*?  Well, every Chu category is a *-autonomous
category in the sense of Barr 1979.  If you don't know why every
*-autonomous category contains both exponentiation and dual
exponentiation, then like Ebert and Siskel I'm not going to give away
the plot and you'll just have to fork out to see the movie, or steal it
if you're a nerd, or watch this space (someone is bound to be a spoiler).

Open question.  At this year's CT, conveniently held 3 km from my
sister's house so I could bike in, I talked about TAC's, or
topoalgebraic categories.  These are defined by picking two sets of
objects from an arbitrary category (TAC's for dummies), some details of
which may be found at http://boole.stanford.edu/pub/sortprop.pdf .  (At
question time PTJ insightfully observed that TACs would cause immense
confusion if I submitted my write-up to TAC.)

A Chu category is precisely a dense complete TAC for which J and L are
singleton monoids.  That is, one sort and one property, both rigid.

The open question:  Characterize those dense complete TAC's admitting
both exponentiation and dual exponentiation.  Chu categories do so, but
what about others?

Many thanks to Ross, Richard, and Jeff Eggers for their respective roles
in the representation of TAC objects (A,r,X) over a profunctor K as a
profunctor morphism r: AX --> K.  (Ross and I would call them bimodules,
much as Mike Barr calls monads triples.)

But toposes are fun too.  *-autonomous categories are to Democrats as
toposes are to Republicans.

Vaughan (donkey) Pratt


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