On Thu, 10 Nov 2011, Vaughan Pratt wrote:
... 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).
I hadn't intended to say this, but since Vaughan brought up *-autonomous cats, here goes. In Cockett-Seely "Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories" (TAC 1997) we showed that bilinear logic, formulated with both exponentials (ie suitable left adjoints to tensoring with an object) and dual exponentials (ie suitable right adjoints to co-tensoring ("par'ing") with an object), are just *-autonomous categories. So not only do *-autonomous cats have these two types of "internal homs" (4 operators in all, in the non-symmetric case), but if (eg) a linearly distributive category has them all, then it must be *-autonomous. (In the paper the result is a bit "finer", since we consider two variants of bilinear logic, the Lambek-style one as above, and what we call "Grishin categories", BILL and GILL in the paper. Both amount to different presentations of *-autonomous cats.) So, in the non-Cartesian context, suitable duals to exponentials are anything but boring ... -= rags =- -- <rags@math.mcgill.ca> <www.math.mcgill.ca/rags> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]