Questions on compact closed categories

Questions on compact closed categories

[Peterson Clayton]

Dear list members,

I am currently working in categorical logic with something that might be called a "compact closed deductive system", that is, a deductive system (in the sense of Lambek) defined as a compact closed category (i.e., a *-autonomous category where the tensor unit is a dualizing object).

I have two questions.

First, it appeared to me that we can show in a compact closed deductive system that every arrow is an isomorphism. Hence, if there is a deduction arrow from A to B, then A is isomorphic (logically equivalent) to B. Is this result accurate? Does this generalize to any compact closed category?

Secondly, I wonder what happens if we add an arbitrary arrow A --> B to the  category. Put differently, what happens if we add A --> B as an axiom to a  compact closed deductive system? Does this also yield an isomorphism between A and B (assuming that the first result is adequate)? Or is it possible to add some axioms that are not necessarily isomorphisms?

I hope my question is clear, and if not I would be happy to clarify myself,  so do not hesitate to contact me. 

Any lead will be appreciated.

Thanks in advance for those who will respond.

Yours,

Clayton Peterson
Ph. D. candidate
Université de Montréal
clayton.peterson@umontreal.ca


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Questions on compact closed categories

[Michael Barr]

I don't know about the second question, but first let point out that that
is not the definition of a compact closed category.  A compact closed
category requires that A -o B = A*\otimes B.  This is true in finite
dimensional vector spaces (and the tensor unit is the dualizing object).
But the tensor unit is the dualizing object in Chu(Vect,K) where Vect is
the category of vector space and K is the ground field.  The dualizing
object and tensor unit is (K,K) but it is not compact.  It is, of course,
*-autonomous.

Michael

On Tue, 19 Nov 2013, Peterson Clayton wrote:

> Dear list members,
>
> I am currently working in categorical logic with something that might be called a "compact closed deductive system", that is, a deductive system (in the sense of Lambek) defined as a compact closed category (i.e., a *-autonomous category where the tensor unit is a dualizing object).
>
> I have two questions.
>
> First, it appeared to me that we can show in a compact closed deductive system that every arrow is an isomorphism. Hence, if there is a deduction arrow from A to B, then A is isomorphic (logically equivalent) to B. Is this result accurate? Does this generalize to any compact closed category?
>
> Secondly, I wonder what happens if we add an arbitrary arrow A --> B to the  category. Put differently, what happens if we add A --> B as an axiom to a  compact closed deductive system? Does this also yield an isomorphism between A and B (assuming that the first result is adequate)? Or is it possible to add some axioms that are not necessarily isomorphisms?
>
> I hope my question is clear, and if not I would be happy to clarify myself,  so do not hesitate to contact me.
>
> Any lead will be appreciated.
>
> Thanks in advance for those who will respond.
>
> Yours,
>
> Clayton Peterson
> Ph. D. candidate
> Universit? de Montr?al
> clayton.peterson@umontreal.ca
>
-- 
         Every gun that is made, every warship launched, every rocket fired
signifies, in the final sense, a theft from those who hunger and are not
fed, those who are cold and are not clothed. -Dwight D. Eisenhower



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]