Date: Tue, 14 May 1996 16:17:56 -0300 (ADT)
Subject: *-Autonomous categories ?

Date: Mon, 13 May 1996 12:50:28 +0100
From: Justin Pearson <justin@dcs.rhbnc.ac.uk>


Dear All,
Given a *-autonomous categories C, we have an isomorphism of Hom sets:
        Hom(A,B) iso HOM(B^\perp , A^\perp)

where B^\perp is the dual of B etc.

The question is, if you there is another isomorphism of Hom sets
        Hom(A,B) iso HOM(B,A)

Does this force the duality to become trivial, i.e. A^\perp iso A for
all A?

I suspect the answer is no, but something tells me it might be yes. But
my intuition and attempts to resolve the matter have, so far, failed me.

Regards
Justin Pearson
Computer Science
Royal Holloway
University of London
Egham
Surrey
TW20 0EX
U.K.
Tel: +44(0)1784 443912
Email: justin@dcs.rhbnc.ac.uk


Date: Wed, 15 May 1996 08:54:51 -0300 (ADT)
Subject: Re: *-Autonomous categories ?

Date: Wed, 15 May 96 09:55 BST
From: Dr. P.T. Johnstone <P.T.Johnstone@pmms.cam.ac.uk>

The answer is no. Take a closed symmetric monoidal category C which
happens to be self-dual (e.g. finite-dimensional vector spaces), and
perform (the trivial case of) the Chu construction on it -- i.e.
take C^op x C with the duality (A,B)^\perp = (B,A).

Peter Johnstone


Date: Wed, 15 May 1996 14:22:28 -0300 (ADT)
Subject: Re: *-Autonomous categories ?

Date: Wed, 15 May 1996 09:46:34 -0400
From: Michael Barr <barr@triples.math.mcgill.ca>

There answer is no.  One point to remember is that the category
could even be discrete, that is the only arrows are identities.
An easy example is this.  Let G be a group (and assume not every element
is of order dividing 2).  If you want a symmetric example, assume G is
commutative.  Now make a category whose objects are the elements of G
and arrows are only identities, so there is an arrow a --> b iff a = b
and then there is only one.  So Hom(a,b) = Hom(b,a).  The monoidal
structure is the group multiplication, a --o b = a\inv b (and
b o-- a = b a\inv) and a* = a\inv.

Here is a less trivial example.  Take a CMC with finite products
in which Hom(a,b) \iso Hom(b,a) (finite dimensional vector spaces,
say) and form Chu(C,1) (1 is terminal).  An object is a pair (a,a')
where a and a' are arbitrary objects of C and Hom((a,a'),(b,b'))=
Hom(a,b) x Hom(b',a') \iso Hom(b,a) x Hom(a',b') = Hom((b,b'),(a,a')),
while (a,a')* = (a',a).

Michael Barr