* Re: quantum logic
@ 2003-10-22 18:07 Fred E.J. Linton
[not found] ` <20031022201258.GF22371@math-rs-n03.ucr.edu>
0 siblings, 1 reply; 13+ messages in thread
From: Fred E.J. Linton @ 2003-10-22 18:07 UTC (permalink / raw)
To: categories
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
>
>
>
>
^ permalink raw reply [flat|nested] 13+ messages in thread
* re: quantum logic
@ 2003-10-12 22:08 John Baez
2003-10-13 15:10 ` Michael Barr
2003-10-18 20:57 ` Michael Barr
0 siblings, 2 replies; 13+ messages in thread
From: John Baez @ 2003-10-12 22:08 UTC (permalink / raw)
To: categories
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.)
^ permalink raw reply [flat|nested] 13+ messages in thread
* re: quantum logic
2003-10-12 22:08 John Baez
@ 2003-10-13 15:10 ` Michael Barr
2003-10-18 20:57 ` Michael Barr
1 sibling, 0 replies; 13+ messages in thread
From: Michael Barr @ 2003-10-13 15:10 UTC (permalink / raw)
To: categories
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
^ permalink raw reply [flat|nested] 13+ messages in thread
* Re: quantum logic
2003-10-12 22:08 John Baez
2003-10-13 15:10 ` Michael Barr
@ 2003-10-18 20:57 ` Michael Barr
2003-10-20 19:51 ` Toby Bartels
1 sibling, 1 reply; 13+ messages in thread
From: Michael Barr @ 2003-10-18 20:57 UTC (permalink / raw)
To: categories
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.)
>
>
>
>
>
^ permalink raw reply [flat|nested] 13+ messages in thread
* Re: quantum logic
2003-10-18 20:57 ` Michael Barr
@ 2003-10-20 19:51 ` Toby Bartels
2003-10-22 16:01 ` Michael Barr
0 siblings, 1 reply; 13+ messages in thread
From: Toby Bartels @ 2003-10-20 19:51 UTC (permalink / raw)
To: categories
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
^ permalink raw reply [flat|nested] 13+ messages in thread
* Re: quantum logic
2003-10-20 19:51 ` Toby Bartels
@ 2003-10-22 16:01 ` Michael Barr
2003-10-22 20:14 ` Toby Bartels
0 siblings, 1 reply; 13+ messages in thread
From: Michael Barr @ 2003-10-22 16:01 UTC (permalink / raw)
To: categories
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
>
>
>
^ permalink raw reply [flat|nested] 13+ messages in thread
* quantum logic
@ 2003-10-12 0:57 John Baez
2003-10-12 18:31 ` Robert Seely
` (3 more replies)
0 siblings, 4 replies; 13+ messages in thread
From: John Baez @ 2003-10-12 0:57 UTC (permalink / raw)
To: categories
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
^ permalink raw reply [flat|nested] 13+ messages in thread
* Re: quantum logic
2003-10-12 0:57 John Baez
@ 2003-10-12 18:31 ` Robert Seely
2003-10-12 20:49 ` Michael Barr
` (2 subsequent siblings)
3 siblings, 0 replies; 13+ messages in thread
From: Robert Seely @ 2003-10-12 18:31 UTC (permalink / raw)
To: categories
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>
^ permalink raw reply [flat|nested] 13+ messages in thread
* Re: quantum logic
2003-10-12 0:57 John Baez
2003-10-12 18:31 ` Robert Seely
@ 2003-10-12 20:49 ` Michael Barr
2003-10-13 13:01 ` Pedro Resende
2003-10-13 13:21 ` Peter McBurney
3 siblings, 0 replies; 13+ messages in thread
From: Michael Barr @ 2003-10-12 20:49 UTC (permalink / raw)
To: categories
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
^ permalink raw reply [flat|nested] 13+ messages in thread
* Re: quantum logic
2003-10-12 0:57 John Baez
2003-10-12 18:31 ` Robert Seely
2003-10-12 20:49 ` Michael Barr
@ 2003-10-13 13:01 ` Pedro Resende
2003-10-13 13:21 ` Peter McBurney
3 siblings, 0 replies; 13+ messages in thread
From: Pedro Resende @ 2003-10-13 13:01 UTC (permalink / raw)
To: categories
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
>
>
>
>
^ permalink raw reply [flat|nested] 13+ messages in thread
* Re: quantum logic
2003-10-12 0:57 John Baez
` (2 preceding siblings ...)
2003-10-13 13:01 ` Pedro Resende
@ 2003-10-13 13:21 ` Peter McBurney
3 siblings, 0 replies; 13+ messages in thread
From: Peter McBurney @ 2003-10-13 13:21 UTC (permalink / raw)
To: categories
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
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2003-10-22 18:07 quantum logic Fred E.J. Linton
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2003-10-24 7:05 ` Fred E.J. Linton
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2003-10-12 22:08 John Baez
2003-10-13 15:10 ` Michael Barr
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2003-10-22 16:01 ` Michael Barr
2003-10-22 20:14 ` Toby Bartels
2003-10-12 0:57 John Baez
2003-10-12 18:31 ` Robert Seely
2003-10-12 20:49 ` Michael Barr
2003-10-13 13:01 ` Pedro Resende
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