Dan Doel dan.doel at gmail.com
Wed Feb 23 01:55:44 CET 2011

```On Tuesday 22 February 2011 3:13:32 PM Vasili I. Galchin wrote:
>        What is the category that is used to interpret linear logic in
> a categorical logic sense?

This is rather a guess on my part, but I'd wager that symmetric monoidal
closed categories, or something close, would be to linear logic as Cartesian
closed categories are to intuitionistic logic. There's a tensor M (x) N, and a
unit (up to isomorphism) I of the tensor. And there's an adjunction:

M (x) N |- O  <=> M |- N -o O

suggestively named, hopefully. There's no diagonal A |- A (x) A like there is
for products, and I is not terminal, so no A |- I in general. Those two should
probably take care of the no-contraction, no-weakening rules. Symmetric
monoidal categories mean A (x) B ~= B (x) A, though, so you still get the
exchange rule.

Obviously a lot of connectives are missing above, but I don't know the
categorical analogues off the top of my head. Searching for 'closed monoidal'
or 'symmetric monoidal closed' along with linear logic may be fruitful,
though.

-- Dan

```