quantum logic

quantum logic

[John Baez]

Dear Categorists -

Do any of you know particularly insightful treatments of
quantum logic via category theory?  I'm more or less familiar
with quantum logic as the theory of the complete orthocomplemented
lattice of closed subspaces of a given Hilbert space.  But now I'm
interested in developing quantum logic starting as much as possible
from general properties of and structures on the category of
Hilbert spaces and bounded linear maps - for example, the fact
that it's an abelian category, and becomes a *-category and symmetric
monoidal category in a nice way (with Hilbert tensor product as the
monoidal structure).  And I'm interested in things like how the
2-dimensional Hilbert space acts a bit like a subobject classifier.

I don't mind sticking with finite-dimensional Hilbert spaces for now
to avoid certain subtleties.

On a related note: I've repeatedly heard people say something
like "the multiplicative fragment of linear logic is the internal
logic of (closed symmetric?) monoidal categories", but I've never heard
a precise result along these lines.  Has anyone worked out a sufficiently
general concept of "the internal logic of a category" or "the
internal logic of a certain 2-category of categories" so that one
could take something like a monoidal category, or a symmetric monoidal
category, or a closed symmetric monoidal category - or maybe the
2-category of all such - and extract by some systematic method the
corresponding "internal logic"?  I'm vaguely imagining some class
of generalizations of the Mitchell-Benabou language of a topos, or
something like that - but I'm really interested in the nonCartesian
case.

The reason I ask this is that it would be nice if you could
throw the (closed, symmetric, monoidal, *, etcetera...) category
of Hilbert spaces into some big machine and have "quantum logic"
pop out - and then throw in other similar categories, and have other
kinds of logic pop out.

Best,
jb

Re: quantum logic

[Robert Seely]

On Sat, 11 Oct 2003, John Baez wrote:

> On a related note: I've repeatedly heard people say something
> like "the multiplicative fragment of linear logic is the internal
> logic of (closed symmetric?) monoidal categories", but I've never heard
> a precise result along these lines.  Has anyone worked out a sufficiently
> general concept of "the internal logic of a category" or "the
> internal logic of a certain 2-category of categories" so that one
> could take something like a monoidal category, or a symmetric monoidal
> category, or a closed symmetric monoidal category - or maybe the
> 2-category of all such - and extract by some systematic method the
> corresponding "internal logic"?  I'm vaguely imagining some class
> of generalizations of the Mitchell-Benabou language of a topos, or
> something like that - but I'm really interested in the nonCartesian
> case.

Hi John -

You might want to take a look at the paper by Robin Cockett and me
"Proof theory for full intuitionistic linear logic, bilinear logic, and mix
categories " in TAC Vol 3 No 5.

  ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1997/n5/n5.ps

As for a general theory - there are plenty of examples, though I don't know
if anyone has really made a general theory of this notion of a categorical
doctrine, often referred to, and based on a paper of Kock and Reyes from the
70's.  But there are many examples (many in the papers  Robin and I have
written on linearly distributive categories and related structures - visit
my webpage if you're interested), which make clear how to go from the
internal logic of a category to the category and back.  I suggest also you
look at our "Introduction to  linear bicategories" (MSCS:10(2000)2 pp
165-203), also available on my webpage, for a higher dimensional approach.

 - all the best, Robert

-- 
<rags@math.mcgill.ca>
<www.math.mcgill.ca/rags>

Re: quantum logic

[Michael Barr]

I will let others answer about the connection between closed monoidal
categories and MLL, but I just wanted to say that I am not sure what you
mean by the category of Hilbert spaces.  If you want the inner product
preserved, then only isometric injections are permitted.  If you want just
bounded linear maps then you are not making any real use of the inner
product.  And the spaces are self-dual, so it is not a good model of
*-autonomy.  Perhaps of compact categories, I would have to think about
it.  But anyway, you have to say what category is meant.  Another
possibility is partial isometries (which can be thought of as total by
being zero on the subspace orthognal to the domain).  This is a lot like
sets and partial injections.

Michael

Re: quantum logic

[Pedro Resende]

Hi John,

One of the ideas behind the theory of quantales is that the category of
"sheaves" on a given quantale should be a topos in *some* generalized
sense, whose subobject classifier would be a quantale (which is then
related to the multiplicative fragment of a noncommutative linear
logic). There are a few papers by various authors addressing sheaves on
quantales, however none getting near a satisfactory definition of
"quantum topos", but I don't think the field is exhausted. In
particular Chris Mulvey wrote a paper with his student Nawaz (you can
download it from Chris' web page), but restricted to idempotent
right-sided quantales, which form a rather limited class. Nevertheless
that paper gives you a category of sheaves which actually is a topos in
the classical sense, but equipped with additional structure that
provides the "quantum" part. I know that currently he has been working
with another student on an extension of this to a more general
situation encompassing all involutive quantales (= involutive monoids
in the monoidal category of sup-lattices), and last time I heard about
it the results looked promising.

The significance of this wrt Hilbert spaces is that once you consider
involutive quantales of the form "Max(A)" (ie, those consisting of all
the closed linear subspaces of a unital C*-algebra A), there is a
notion of "irreducible representation" of Max(A) that classifies up to
unitary equivalence the irreducible representations of A, and, to a
certain extent still in need of further clarification (very preliminary
material is in a paper of mine which is due to appear in the J. Algebra
and is downloadable from my web page - still a couple of typos and
minor bugs in the on-line version, I'm afraid), the category of
representations of A is approximated by the corresponding category of
quantale modules over Max(A). (Each representation of A on a Hilbert
space H induces in a natural way an action of Max(A) on the lattice of
closed linear subspaces of H.) By all of this I mean that ultimately
the category of sheaves on Max(A) should provide a logical handle on
the category of representations of A, and it seems reasonable to expect
that what you are saying about the category of Hilbert spaces and
bounded linear maps may relate to this general scheme.

Best,

Pedro.



On Sunday, October 12, 2003, at 01:57 AM, John Baez wrote:

> Dear Categorists -
>
> Do any of you know particularly insightful treatments of
> quantum logic via category theory?  I'm more or less familiar
> with quantum logic as the theory of the complete orthocomplemented
> lattice of closed subspaces of a given Hilbert space.  But now I'm
> interested in developing quantum logic starting as much as possible
> from general properties of and structures on the category of
> Hilbert spaces and bounded linear maps - for example, the fact
> that it's an abelian category, and becomes a *-category and symmetric
> monoidal category in a nice way (with Hilbert tensor product as the
> monoidal structure).  And I'm interested in things like how the
> 2-dimensional Hilbert space acts a bit like a subobject classifier.
>
> I don't mind sticking with finite-dimensional Hilbert spaces for now
> to avoid certain subtleties.
>
> On a related note: I've repeatedly heard people say something
> like "the multiplicative fragment of linear logic is the internal
> logic of (closed symmetric?) monoidal categories", but I've never heard
> a precise result along these lines.  Has anyone worked out a
> sufficiently
> general concept of "the internal logic of a category" or "the
> internal logic of a certain 2-category of categories" so that one
> could take something like a monoidal category, or a symmetric monoidal
> category, or a closed symmetric monoidal category - or maybe the
> 2-category of all such - and extract by some systematic method the
> corresponding "internal logic"?  I'm vaguely imagining some class
> of generalizations of the Mitchell-Benabou language of a topos, or
> something like that - but I'm really interested in the nonCartesian
> case.
>
> The reason I ask this is that it would be nice if you could
> throw the (closed, symmetric, monoidal, *, etcetera...) category
> of Hilbert spaces into some big machine and have "quantum logic"
> pop out - and then throw in other similar categories, and have other
> kinds of logic pop out.
>
> Best,
> jb
>
>
>
>

Re: quantum logic

[Peter McBurney]

John --

Although not a categorical treatment, a recent paper by Kurt Engesser
and Dov Gabbay in <Artificial Intelligence> discusses a connection
between Quantum Logic and Hilbert spaces.   (The reason the work
appeared in the leading AI journal is that there are applications to
nonmonotonic reasoning, which is a major area of research in AI.)

Citation details and abstract below.

-- Peter


==================================================================

Artificial Intelligence
Volume 136, Issue 1 , March 2002 , Pages 61-100

"Quantum logic, Hilbert space, revision theory"

Kurt Engesser and Dov M. Gabbay

a Birkenweg 3, 78573 Wurmlingen, Germany
b Department of Computer Science, King's College London, Strand, London
WC2R 2LS, UK


Abstract

Our starting point is the observation that with a given Hilbert space H
we may, in a way to be made precise, associate a class of non-monotonic
consequence relations in such a way that there exists a one-to-one
correspondence between the rays of H and these consequence relations.
The projectors in Hilbert space may then be viewed as a sort of revision
operators. The lattice of closed subspaces appears as a natural
generalisation of the concept of a Lindenbaum algebra in classical
logic. The logics presentable by Hilbert spaces are investigated and
characterised. Moreover, the individual consequence relations are
studied. A key concept in this context is that of a consequence relation
having a pointer to itself. It is proved that such consequence relations
have certain remarkable properties in that they reflect their metatheory
at the object level to a surprising extent. The tools used in the
investigation stem from two different areas of research, namely from the
disciplines of non-monotonic logic on the one hand and from Hilbert
space theory on the other. There exist surprising connections between
these two fields of research the investigation of which constitutes the
purpose of this paper.

Author Keywords: Quantum logic; Hilbert space; Revision theory;
Consequence relation; Non-monotonic logic

====================================================================

re: quantum logic

[John Baez]

Michael Barr wrote:

> I will let others answer about the connection between closed monoidal
> categories and MLL, but I just wanted to say that I am not sure what you
> mean by the category of Hilbert spaces. If you want the inner product
> preserved, then only isometric injections are permitted.  If you want just
> bounded linear maps then you are not making any real use of the inner
> product.

Right.  I wanted to leave things flexible so different readers could
interpret my question in different ways, but I also tried to hint
that I think it's crucial to work with the *-category Hilb whose objects
are Hilbert spaces, whose morphisms are bounded linear maps, and whose
*-structure sends the bounded linear map f: H -> H' to its Hilbert
space adjoint f*: H' -> H.  This *-structure can be used to define
concepts crucial for quantum mechanics, like "self-adjoint" and
"unitary" operators, as well as "isometric injections".  Isometric
injections are a nice way to study subobjects in Hilb, but they're
not good enough for doing full-fledged quantum mechanics, nor is
ignoring the inner product altogether.

Category theorists are often a bit uncomfortable with *-categories
because they prefer "adjoints" that are defined using other structure
rather than put in by brute force.  However, I'm convinced that we
can only understand how quantum field theory exploits the analogy
between differential topology and Hilbert space theory if we think
about *-categories.  For example, a topological quantum field theory
is a symmetric monoidal functor from some *-category of cobordisms
to the *-category Hilb - but the most physically realistic TQFTs are
the "unitary" ones, which preserve the *-structure.

I've talked about this *-stuff and the nascent concept of "n-categories
with duals" in my papers on 2-Hilbert spaces

http://math.ucr.edu/home/baez/2hilb.ps

and 2-tangles

http://math.ucr.edu/home/baez/hda4.ps

and now I want to say a bit about how it impinges on quantum
logic - but to avoid reinventing the wheel, I'd like to hear
anything vaguely relevant anyone knows about approaching quantum
logic with an eye on category theory.

(I know a bit about quantales, but maybe there's other stuff
I've never heard of.)

re: quantum logic

[Michael Barr]

I think Rick Blute (+ collaborators) has done some things with this. It is
not clear whether you want a self-duality or a *-autonomous category.  If
you stick to finite dimensional Hilbert spaces, the situation seems
simple.  If V and W are inner product spaces, then for f, g: V --> W, let
f.g = \sum f(v_i).g(v_i) the sum taken over an orthonormal basis.  I
believe this is invariant to an orthonormal base change and it is
obviously positive definite.  For infinite dimensional spaces, you would
have to stick to f for which \sum f(v_i)^2 < oo.  But this isn't a
category.  It is closed under composition (I think) but certainly lacks
identities.  This gives rise to something called a nuclear category.  The
category has all maps and there is sub-non-category of nuclear maps.  This
all goes back (needless to say) to Grothendieck.

If by *-category you just mean self dual, well then Hilbert spaces
certainly are that.  Self dual categories are a dime a dozen.  Just take C
x C^op.  The amazing thing is that if C is closed, C x C^op is
*-autonomous, (assuming C has binary cartesian products).

Michael

Re: quantum logic

[Michael Barr]

After giving the matter some thought, I finally decided that the
category of Hilbert spaces should have as its morphisms norm-reducing
linear maps.  At the very least that will ensure that an isomorphism is
an isometry.  And yes the category of finite dimensional Hilbert spaces
is *-autonomous.  The internal hom of two is the space of all linear
maps and the inner product <f,g> = \sum f(u_i)g(u_i), taken over an
orthonormal basis of the domain.  This can be shown to be invariant
under orthogonal change of  basis.  The norm of the identity on an
n-dimensional space is sqrt(n).  The dual of a space is itself, of
course with the duality being adjunction (or transpose).  Then the
tensor product H # G = (H --o G^*)^*.

Here is another approach to the same structure.  Consider a pair
(V,\phi) where V is a finite dimensional space and \phi is an
isomorphism of V with its dual space.  You have to add positive
definiteness and symmetry, but that is no problem.  Maps again are norm
reducing.  Now we can define (V,\phi) # (W,\psi) = (V # W,\phi # \psi),
(V,\phi)^* = (V,\phi^{-1}), and (V,\phi) --o (W,\psi) = (V # W,\phi^{-1}
# \psi).  The resultant category is exactly the same as before.

BTW, it is easy to see that the transpose of a norm-reducing map is norm
reducing.


On Sun, 12 Oct 2003, John Baez wrote:

> Michael Barr wrote:
>
> > I will let others answer about the connection between closed monoidal
> > categories and MLL, but I just wanted to say that I am not sure what you
> > mean by the category of Hilbert spaces. If you want the inner product
> > preserved, then only isometric injections are permitted.  If you want just
> > bounded linear maps then you are not making any real use of the inner
> > product.
>
> Right.  I wanted to leave things flexible so different readers could
> interpret my question in different ways, but I also tried to hint
> that I think it's crucial to work with the *-category Hilb whose objects
> are Hilbert spaces, whose morphisms are bounded linear maps, and whose
> *-structure sends the bounded linear map f: H -> H' to its Hilbert
> space adjoint f*: H' -> H.  This *-structure can be used to define
> concepts crucial for quantum mechanics, like "self-adjoint" and
> "unitary" operators, as well as "isometric injections".  Isometric
> injections are a nice way to study subobjects in Hilb, but they're
> not good enough for doing full-fledged quantum mechanics, nor is
> ignoring the inner product altogether.
>
> Category theorists are often a bit uncomfortable with *-categories
> because they prefer "adjoints" that are defined using other structure
> rather than put in by brute force.  However, I'm convinced that we
> can only understand how quantum field theory exploits the analogy
> between differential topology and Hilbert space theory if we think
> about *-categories.  For example, a topological quantum field theory
> is a symmetric monoidal functor from some *-category of cobordisms
> to the *-category Hilb - but the most physically realistic TQFTs are
> the "unitary" ones, which preserve the *-structure.
>
> I've talked about this *-stuff and the nascent concept of "n-categories
> with duals" in my papers on 2-Hilbert spaces
>
> http://math.ucr.edu/home/baez/2hilb.ps
>
> and 2-tangles
>
> http://math.ucr.edu/home/baez/hda4.ps
>
> and now I want to say a bit about how it impinges on quantum
> logic - but to avoid reinventing the wheel, I'd like to hear
> anything vaguely relevant anyone knows about approaching quantum
> logic with an eye on category theory.
>
> (I know a bit about quantales, but maybe there's other stuff
> I've never heard of.)
>
>
>
>
>

Re: quantum logic

[Toby Bartels]

Michael Barr wrote in part:

>After giving the matter some thought, I finally decided that the
>category of Hilbert spaces should have as its morphisms norm-reducing
>linear maps.  At the very least that will ensure that an isomorphism is
>an isometry.

True, but are you begging the question by trying to ensure that?
After all, an invertible bounded linear map is enough to deduce
that Hilbert spaces are isomorphic (even in the sense of isometric),
so why not count those maps as isomorphisms themselves?

This matter is much bigger than Hilbert spaces, of course;
moving to Banach spaces (a closed category even for arbitrary dimension),
we can even see how, /as/ a closed category, it doesn't really matter!
The question is, what is the forgetful functor from Ban to Set?
Do we take the set of all vectors? or do we take the closed unit ball?
The former corresponds to allowing all bounded linear maps as morphisms,
while the latter corresponds to requiring norm-reducing linear maps.
But in the closed category Ban, the Banach space of morphisms
is, whatever your conventions, the space of all bounded linear maps.
Still, this can be consistent with either choice of hom-SET,
since the closed unit ball in the Banach space of bounded linear maps
is none other than your preferred hom-set of norm-reducing maps.

Jim Dolan (IIRC) suggested that Ban is more fundamentally a closed category
than a category in the first place.

We can do this on a more elementary level with metric spaces;
is the hom-set the set of all Lipschitz continuous functions,
or is it only the set of distance-reducing functions?
But unlike with Banach (or Hilbert) spaces, this makes a difference
even to the classification of metric spaces into isomorphism classes.
The question becomes, is an isomorphism of metric spaces
merely a relabelling of points keeping all distances the same,
or does it also allow for a recalibration of ones ruler?
Which is the correct interpretation may depend on the application,
and how absolute -- rather than measured in some unit -- the distances are.
(One can even recalibrate more generously to allow as morphisms
all uniformly continuous maps, or even all continuous maps.
Thus classically one speaks of variously "equivalent" metric spaces,
such as "uniformly equivalent" or "topologically equivalent".)
To get closed categories here, one must restrict to bounded metric spaces;
the analysis is a little more fun than for Banach spaces,
especially with the degeneracy surrounding the initial and terminal spaces.


-- Toby

Re: quantum logic

[Michael Barr]

I will stick to my perception that if you dealing with Hilbert or Banach
spaces isomorphisms should be just that.  It makes no difference to the
*-autonomous structure anyway.

For Banach spaces, if you take as underlying functor the closed unit ball,
it has an adjoint.  It is not tripleable, however, but C^*-algebras are
(with the unit ball underlying functor).

On Mon, 20 Oct 2003, Toby Bartels wrote:

> Michael Barr wrote in part:
>
> >After giving the matter some thought, I finally decided that the
> >category of Hilbert spaces should have as its morphisms norm-reducing
> >linear maps.  At the very least that will ensure that an isomorphism is
> >an isometry.
>
> True, but are you begging the question by trying to ensure that?
> After all, an invertible bounded linear map is enough to deduce
> that Hilbert spaces are isomorphic (even in the sense of isometric),
> so why not count those maps as isomorphisms themselves?
>
> This matter is much bigger than Hilbert spaces, of course;
> moving to Banach spaces (a closed category even for arbitrary dimension),
> we can even see how, /as/ a closed category, it doesn't really matter!
> The question is, what is the forgetful functor from Ban to Set?
> Do we take the set of all vectors? or do we take the closed unit ball?
> The former corresponds to allowing all bounded linear maps as morphisms,
> while the latter corresponds to requiring norm-reducing linear maps.
> But in the closed category Ban, the Banach space of morphisms
> is, whatever your conventions, the space of all bounded linear maps.
> Still, this can be consistent with either choice of hom-SET,
> since the closed unit ball in the Banach space of bounded linear maps
> is none other than your preferred hom-set of norm-reducing maps.
>
> Jim Dolan (IIRC) suggested that Ban is more fundamentally a closed category
> than a category in the first place.
>
> We can do this on a more elementary level with metric spaces;
> is the hom-set the set of all Lipschitz continuous functions,
> or is it only the set of distance-reducing functions?
> But unlike with Banach (or Hilbert) spaces, this makes a difference
> even to the classification of metric spaces into isomorphism classes.
> The question becomes, is an isomorphism of metric spaces
> merely a relabelling of points keeping all distances the same,
> or does it also allow for a recalibration of ones ruler?
> Which is the correct interpretation may depend on the application,
> and how absolute -- rather than measured in some unit -- the distances are.
> (One can even recalibrate more generously to allow as morphisms
> all uniformly continuous maps, or even all continuous maps.
> Thus classically one speaks of variously "equivalent" metric spaces,
> such as "uniformly equivalent" or "topologically equivalent".)
> To get closed categories here, one must restrict to bounded metric spaces;
> the analysis is a little more fun than for Banach spaces,
> especially with the degeneracy surrounding the initial and terminal spaces.
>
>
> -- Toby
>
>
>

Re: quantum logic

[Toby Bartels]

Michael Barr wrote:

>For Banach spaces, if you take as underlying functor the closed unit ball,
>it has an adjoint.  It is not tripleable, however, but C^*-algebras are
>(with the unit ball underlying functor).

OK, that's a good point.  I agree (with the L-1 norm on the free space).


-- Toby

Re: quantum logic

[Fred E.J. Linton]

I'll  address two of these questions.  The first:

> The question is, what is the forgetful functor from Ban to Set?
> Do we take the set of all vectors? or do we take the closed unit ball?
> The former corresponds to allowing all bounded linear maps as morphisms,
> while the latter corresponds to requiring norm-reducing linear maps.

Actually, when the "underlying-set functor" for Banach spaces
is taken to be the unit disk functor, and the morphisms are
taken as the norm-decreasing maps, the situation is really great,
because the norm-decreasing maps DO constitute the unit disk
of the Banach space of bounded linear transformations, as you know.
And products and coproducts are as Banach spacists like to see them
(the familiar L-infinity style "full direct product" and and L-1 
style "weak direct product", respectively).

When the underlying-set functor is taken to be ALL the vectors 
of the Banach space, on the other hand, products and coproducts 
misbehave quite badly. 

As for the question,

> After all, an invertible bounded linear map is enough to deduce
> that Hilbert spaces are isomorphic (even in the sense of isometric),
> so why not count those maps as isomorphisms themselves?

I'd answer by saying that unless the invertible bounded linear map
in the question IS an isometry I'd never dare call it one.

-- Fred (usually <FLinton@Wesleyan.edu>)

Toby Bartels <toby@math.ucr.edu> wrote:

> Michael Barr wrote in part:
> 
> >After giving the matter some thought, I finally decided that the
> >category of Hilbert spaces should have as its morphisms norm-reducing
> >linear maps.  At the very least that will ensure that an isomorphism is
> >an isometry.
> 
> True, but are you begging the question by trying to ensure that?
> After all, an invertible bounded linear map is enough to deduce
> that Hilbert spaces are isomorphic (even in the sense of isometric),
> so why not count those maps as isomorphisms themselves?
> 
> This matter is much bigger than Hilbert spaces, of course;
> moving to Banach spaces (a closed category even for arbitrary dimension),
> we can even see how, /as/ a closed category, it doesn't really matter!
> The question is, what is the forgetful functor from Ban to Set?
> Do we take the set of all vectors? or do we take the closed unit ball?
> The former corresponds to allowing all bounded linear maps as morphisms,
> while the latter corresponds to requiring norm-reducing linear maps.
> But in the closed category Ban, the Banach space of morphisms
> is, whatever your conventions, the space of all bounded linear maps.
> Still, this can be consistent with either choice of hom-SET,
> since the closed unit ball in the Banach space of bounded linear maps
> is none other than your preferred hom-set of norm-reducing maps.
> 
> Jim Dolan (IIRC) suggested that Ban is more fundamentally a closed category
> than a category in the first place.
> 
> We can do this on a more elementary level with metric spaces;
> is the hom-set the set of all Lipschitz continuous functions,
> or is it only the set of distance-reducing functions?
> But unlike with Banach (or Hilbert) spaces, this makes a difference
> even to the classification of metric spaces into isomorphism classes.
> The question becomes, is an isomorphism of metric spaces
> merely a relabelling of points keeping all distances the same,
> or does it also allow for a recalibration of ones ruler?
> Which is the correct interpretation may depend on the application,
> and how absolute -- rather than measured in some unit -- the distances are.
> (One can even recalibrate more generously to allow as morphisms
> all uniformly continuous maps, or even all continuous maps.
> Thus classically one speaks of variously "equivalent" metric spaces,
> such as "uniformly equivalent" or "topologically equivalent".)
> To get closed categories here, one must restrict to bounded metric spaces;
> the analysis is a little more fun than for Banach spaces,
> especially with the degeneracy surrounding the initial and terminal spaces.
> 
> 
> -- Toby
> 
> 
> 
> 

Re: quantum logic

[Fred E.J. Linton]

Toby Bartels wrote:

 > For finitary products/coproducts, the L-p style norm will work for any p,
 > which is no surprise since the results are isomorphic (in either
category).
 > In fact, we get a biproduct diagram that works regardless of norm
 > (so long as the projections and injections are norm-reducing).

Sorry, in the maps-norm-decreasing, disk-as-underlying-set category,
even  RxR  gives you isometrically different Banach spaces for different
values of  p .(*)  Only the L-1 style norm gives you a coproduct,
only the L-infinity style norm gives you a product (using the
usual "as-vector-space" injections and projections); the other
choices of  p  give you god-only-knows-what.

As regards another point, I think this is dead wrong:

 > The L-oo style full direct product and the L-1 style weak direct product
 > work as (respectively) product and coproduct using /either/ hom-set
 > (and hence using either corresponding choice of underlying-set functor).
 > This is because |f| <= sup_i |f_i| holds (for both product and coproduct,
 > albeit by a different calculation for L-oo product than for L-1
coproduct).

Here's why: a bounded linear transformation to the L-oo style product
of a bunch of real lines, say (in the real Banach space case) arises
from a BOUNDED family of bounded linear functionals.  An UNbounded
family of bounded linear functionals WILL give you a continuous linear
transformation, of course, but NOT to the L-oo style product of  R 's
-- it will be taking values in the topological-vector-space product
of those  R 's.

Same problem, in reverse, for the L-1 style weak direct product as
coproduct: the bounded maps from, say, l_1(aleph-0) to a Banach space  B
correspond, after composing with the injections, to BOUNDED families
of maps  R --> B  (i.e., bounded families of vectors in  B ).
But ARBITRARY families of maps  R --> B  should have a common
extension to a continuous map from the coproduct of those  R 's.
So their L-1 style weak direct product (which is what l_1(aleph-0) is)
won't be the coproduct in the continuous-linear-transformation category.

Eilenberg, may he rest in peace, once summed up the dilemma:
are you talking about Banach spaces? or about Banachable spaces?
(Banachable spaces are topological vector spaces, complete in
their (uniform) topology, whose topology can come from a norm.)
In the latter case, continuous linear transformations
are all there is.  And if you want invertible bounded
linear transformations to be isomorphisms, Banachable
spaces is all you can be capturing.  But products, as
topological vector spaces, of too many Banachable spaces
are no longer Banachable; and coproducts ... are no longer
even uniformly complete.  So if you want to talk about
Banach spaces, with the expected L-1 style weak products
as coproducts and the expected L-oo style products as products,
then you are obviously focussed on the norms, and you've
got to be focussed on maps that don't increase the norms,
for otherwise you're only focussing on the Banachable aspect
of the topological vector spaces underlying your Banach spaces.

As to other remarks:

> If I were talking with John Baez, and he had just said
> that he was accepting all bounded linear maps as morphism,
> then I /would/ dare call an invertible bounded linear map an isomorphism,
> because it would in fact /be/ an isomorphism in that category.

And I'd understand he was interested only in Banachable TVSes, and
not actually in Banach spaces.

> (But in a general context, I would call /only/ isometries isomorphisms,
> because otherwise people might get confused about what I meant!)

This would tell me you're interested not merely in Banachable
spaces, but in actual Banach spaces.

> I say this just to remind us that we're discussing which category is /best/,
> not which category is /correct/.

Both categories (Banach spaces -- with norm-non-increasing maps,
and Banachable spaces, with continuous linear transformations)
are useful categories.  But even their objects are different, not
just the maps allowed between two particular Banach spaces.

> (There's the additional matter that the "unit disk functor"
> isn't a functor at all if all bounded linear maps are morphisms,
> but the correspondence is stronger than that.)

That's because Banachable spaces have no unit disks -- it's
not that "the 'unit disk functor' isn't a functor at all,"
it's that there isn't even a CANDIDATE for object-function
of a putative unit disk functor!

Nonetheless, both categories -- Ban , and  Banachable -- though
far from equivalent, have their uses.

And  Ban , though not monadic over  Sets  via its unit disk functor, as
Mike Barr has correctly pointed out, IS a full reflective subcategory
of the category of algebras over the monad for that unit-disk functor;
in that regard, it somewhat resembles the category of torsion-free
abelian groups (likewise not monadic, yet fully reflective in the
category of algebras for its underlying set functor, viz., in Ab.Gps).

Hope these comments help.

(*)PS: by a fluke, l_1(n) and l_oo(n)
can be made isometric, for n=2:

send (1, 0) in l_1(2) to (1, 1) in l_oo(2), and
send (0, 1) in l_1(2) to (-1, 1) in l_oo(2), and
extend by linearity.

This is the linear map  R^2 --> R^2  that rotates by 45 degrees
and then multiplies by  square.root(2) , and it carries
the l_1 unit diamond onto the l_oo unit square.

I don't think anything like this can work for exhibiting
isometries between l_1(2) and l_p(2) for any other p,
and I don't think anything like this can work for l_1(n) and l_oo(n)
for any n > 2.

But enough for now.

-- F.