Date: Fri, 18 Oct 1996 11:02:48 -0300 (ADT)
Subject: *-autonomous 2-categories

Date: Fri, 18 Oct 1996 15:05:35 +0200
From: Gian Luca Cattani <luca@brics.dk>


I am a PhD student at the Computer Science Department of Aarhus University.

I would like to ask, whether anybody knows of extensions of the notion of
*-autonomous category to 2-categories, i.e., what does it mean for a
2-category to be *-autonomous.


Regards,

Luca Cattani

--
+-------------------------------------------------------------+
|  Gian Luca Cattani          (luca@brics.dk)                 |
+-------------------------------------------------------------+
| =========   BRICS:  Basic Research In Computer Science      |
| ==== ====   Computer Science Department                     |
| =========   Aarhus University                               |
| ==== ====   Ny Munkegade Bldg. 540                          |
| ==== ====   8000 Aarhus C, Denmark                          |
| ==== ====   Phone : (+45) 8942 3472   Fax : (+45) 8942 3255 |
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Date: Sun, 20 Oct 1996 19:46:07 -0300 (ADT)
Subject: Re: *-autonomous 2-categories

Date: Fri, 18 Oct 1996 11:01:43 -0700 (PDT)
From: john baez <baez@math.ucr.edu>

Luca Cattani writes:

> I am a PhD student at the Computer Science Department of Aarhus University.

> I would like to ask, whether anybody knows of extensions of the notion of
> *-autonomous category to 2-categories, i.e., what does it mean for a
> 2-category to be *-autonomous.

I'd like to know this too.  What Laurel Langford and I have are notions of:

2-category with duals,
monoidal 2-category with duals, and
braided monoidal 2-category with duals.

Here "with duals" refers to the fact that objects, morphisms,
and 2-morphisms all have "duals".  This may be stronger than
what you are thinking of, since one can consider situations
where only the j-morphisms for certain j have duals.  One may
be able to remove clauses and get a definition of *-autonomous
2-category.

If you like, I could send you our definition.  It is probably not
the ultimate definitive definition but it is good enough for our
present purposes.  It may help if I explain what our present purposes
are.  In the paper "Higher-dimensional algebra and topological quantum
field theory" (Jour. Math. Phys. 36 (1995), 6073-6105), James Dolan
and I proposed a general conjecture relating the topology of k-tangles
in n dimensions to n-category theory.  In a special case this says roughly
the following: isotopy classes of framed 2-dimensional surfaces embedded in
R^n are in 1-1 correspondence with certain 2-morphisms in a certain special
2-category.  In the case n = 2 this would be the "free 2-category with duals
on one object"; for n = 3 this would be the "free monoidal 2-category
with duals on one object", and for n = 4 this would be the "free
braided monoidal 2-category with duals on one object."

In "Higher-dimensional algebra I: braided monoidal 2-categories"
(Adv. Math. 121 (1996), 196-244) Martin Neuchl and I gave a precise
definition of braided monoidal 2-categories (which corrects the
definition of Kapranov and Voevodsky, and has subsequently been given
a more elegant formulation by Day and Street).  In "HDA IV: 2-tangles",
Laurel Langford and I prove the theorem relating surfaces in 4-dimensional
space to the free braided monoidal 2-category with duals on one object.
We are still writing this up.

A nice example of a braided monoidal 2-category with duals is
the 2-category of (finite-dimensional) 2-Hilbert spaces, discussed
in "HDA II: 2-Hilbert spaces".  This is available at

http://math.ucr.edu/home/baez/

I think this should be a *-autonomous 2-category by any
reasonable definition thereof.

Best,
John Baez


Date: Wed, 23 Oct 1996 11:37:37 -0300 (ADT)
Subject: Re: *-autonomous 2-categories

Date: Wed, 23 Oct 1996 11:15:10 +1000
From: Ross Street <street@mpce.mq.edu.au>

Response to Gian Luca Cattani <luca@brics.dk>

Perhaps I can add some remarks to John Baez's helpful reply.
The correct notion of monoidal bicategory seems to be tricategory
with one object (see Memoirs AMS #558 Sept 1995). There is a
coherence theorem (loc cit) which says each monoidal bicategory is
appropriately equivalent to a Gray monoid M (also called semistrict
monoidal 2-category); and these are easier to work with (although the
tensor product # : M x M --> M is only a special kind of pseudofunctor, not
a 2-functor in general). The paper by Brian Day and me that John Baez
mentions (called "Monoidal bicategories and Hopf algebroids" - submitted)
also defines closed Gray monoid. There are no surprises here: a right
hom in M is an object  [a,b]  together with a pseudonatural equivalence
of categories
                M(a#c,b) <--> M(c,[a,b]).
We write  [a,b]'  for a left hom: M(c#a,b) <--> M(c,[a,b]'). We
call  M  closed when each pair of objects has a left and right hom.
Of course, if  M  is braided then  [a,b]' = [a,b].

We do not discuss *-autonomous monoidal bicategories (although we
considered including *-autonomous objects in  M  since we have some
examples). What I suggest is that a monoidal bicategory is
*-autonomous when it is closed and has a dualising object  d.
By this last I mean that the canonical  a --> [[a,d]',d]  should
be an equivalence.

Hope this is of some use.

--Ross