Hello, What is the category that is used to interpret linear logic in a categorical logic sense? Thank you, Vasili
Hi Vasili, not understanding clearly «in a categorical logic sense» -- but I can be sure you already checked out coherent spaces, which might be regarded as underlying Girard's original works in this sense?? I have a faint idea about improvements, but I don't have them present at the moment. Curiously -- is it allowed to ask about the motivation? Cheers, Nick On 02/22/2011 09:13 PM, Vasili I. Galchin wrote:
Hello,
What is the category that is used to interpret linear logic in a categorical logic sense?
Thank you,
Vasili
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On Tue, Feb 22, 2011 at 4:23 PM, Nick Rudnick
Hi Vasili,
not understanding clearly «in a categorical logic sense» -- but I can be sure you already checked out coherent spaces, which might be regarded as underlying Girard's original works in this sense?? I have a faint idea about improvements, but I don't have them present at the moment.
Curiously -- is it allowed to ask about the motivation?
Insofar as "I'm curious" is allowed as a legitimate response, yes. Luke
Cheers, Nick
On 02/22/2011 09:13 PM, Vasili I. Galchin wrote:
Hello,
What is the category that is used to interpret linear logic in a categorical logic sense?
Thank you,
Vasili
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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
I think http://www.cs.man.ac.uk/~schalk/notes/llmodel.pdf might be useful.
And John Baez and Matt Stay's math.ucr.edu/home/baez/rosetta.pdf (where I
found the citation for the first paper) has a fair amount about this sort of
question.
On Tue, Feb 22, 2011 at 7:55 PM, Dan Doel
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
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participants (5)
-
Dan Doel -
George Kulakowski -
Luke Palmer -
Nick Rudnick -
Vasili I. Galchin