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From rrosebru@mta.ca Sun Mar 2 19:42:10 2008 -0400
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Date: Sun, 2 Mar 2008 18:36:33 -0000 (GMT)
Subject: categories: Minimal abelian subcategory
From: "Tom Leinster"
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My colleague Walter Mazorchuk has the following question.
Being abelian is a *property* of a category, not extra structure. Given
an abelian category A, it therefore makes sense to define a subcategory o=
f
A to be an ABELIAN SUBCATEGORY if, considered as a category in its own
right, it is abelian. Note that a priori, the inclusion need not preserv=
e
sums, kernels etc.
Now let R be a ring and M an R-module. Is there a minimal abelian
subcategory of Mod-R containing M? If so, is there a canonical way to
describe it?
Any thoughts or pointers to the literature would be welcome. Feel free t=
o
assume hypotheses on R (it might be a finite-dimensional algebra etc), or
to answer the question for full subcategories only.
Thanks,
Tom
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From: Joshua P Nichols-Barrer
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Hi Tom,
Silly observation, but wouldn't the contractible category consisting only
of M and its identity morphism constitute an abelian subcategory by this
definition, albeit one that is trivial? It would seem that the question
for full subcategories is more interesting (and harder).
Best,
Josh
On Sun, 2 Mar 2008, Tom Leinster wrote:
> My colleague Walter Mazorchuk has the following question.
>
> Being abelian is a *property* of a category, not extra structure. Given
> an abelian category A, it therefore makes sense to define a subcategory of
> A to be an ABELIAN SUBCATEGORY if, considered as a category in its own
> right, it is abelian. Note that a priori, the inclusion need not preserve
> sums, kernels etc.
>
> Now let R be a ring and M an R-module. Is there a minimal abelian
> subcategory of Mod-R containing M? If so, is there a canonical way to
> describe it?
>
> Any thoughts or pointers to the literature would be welcome. Feel free to
> assume hypotheses on R (it might be a finite-dimensional algebra etc), or
> to answer the question for full subcategories only.
>
> Thanks,
> Tom
>
>
>
>
>
>
From rrosebru@mta.ca Sun Mar 2 21:02:11 2008 -0400
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Date: Mon, 3 Mar 2008 00:27:05 -0000 (GMT)
Subject: categories: Re: Minimal abelian subcategory
From: "Tom Leinster"
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A couple of people have pointed out to me - in private, I think - that th=
e
question has a trivial answer (namely, the subcategory consisting of just
M and its identity map). Sorry. I probably misinterpreted what Walter
said to me.
Tom
>> -----Original Message-----
>> From: cat-dist@mta.ca [mailto:cat-dist@mta.ca] On Behalf Of
>> Tom Leinster
>> Sent: Monday, March 03, 2008 5:37 AM
>> To: categories@mta.ca
>> Subject: categories: Minimal abelian subcategory
>>
>> My colleague Walter Mazorchuk has the following question.
>>
>> Being abelian is a *property* of a category, not extra
>> structure. Given an abelian category A, it therefore makes
>> sense to define a subcategory of A to be an ABELIAN
>> SUBCATEGORY if, considered as a category in its own right, it
>> is abelian. Note that a priori, the inclusion need not
>> preserve sums, kernels etc.
>>
>> Now let R be a ring and M an R-module. Is there a minimal
>> abelian subcategory of Mod-R containing M? If so, is there a
>> canonical way to describe it?
>>
>> Any thoughts or pointers to the literature would be welcome.
>> Feel free to assume hypotheses on R (it might be a
>> finite-dimensional algebra etc), or to answer the question
>> for full subcategories only.
>>
>> Thanks,
>> Tom
>>
>>
>>
>>
>>
>>
>
From rrosebru@mta.ca Sun Mar 2 21:02:11 2008 -0400
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From: Colin McLarty
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> Now let R be a ring and M an R-module. Is there a minimal abelian
> subcategory of Mod-R containing M? If so, is there a canonical way to
> describe it?
This question, as posed, is too easy: Just take M and its identity
arrow. It will be a zero-object in that subcategory. There may be a
better question here guiding Walter Mazorchuk's intuition, but it will
have to require something more than just containing the one object.
Colin
From rrosebru@mta.ca Mon Mar 3 10:42:32 2008 -0400
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From: peasthope@shaw.ca
Subject: categories: Re: A small cartesian closed concrete category
To: categories@mta.ca
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Folk,
At Thu, 14 Feb 2008 15:06:49 -0500 I wrote,
"Is there a cartesian closed concrete category which=20
is small enough to write out explicitly?"
=20
At Fri, 15 Feb 2008 08:47:57 +0000 Philip Wadler srote,
"... please summarize the replies ... and send ... to the ... list?
... interested to see if you receive a positive reply."
I've counted 16 respondents! The question is=20
answered well. With my limited knowledge, the=20
summary probably fails to credit some of the=20
responses adequately but this is not intentional.
Thanks to everyone who replied!
5 messages mentioned Hyting-algebras.
Never heard of them. Lawvere & Schanuel=20
do not mention them in the 1997 book. =20
Will store the terms for future reference.
Fred Linton wrote,
"... skeletal version of the full category
... having as only objects the ordinal numbers 0 and 1.
Here 0 x A =3D 0, 1 x A =3D A, 0^1 =3D 0, 0^0 =3D 1, 1^A =3D 1.
In other words, B x A =3D min(A, B), B^A =3D max(1-A, B)."
My product diagrams are at=20
http://carnot.yi.org/category01.jpg
.
Now I can try to illustrate the uniqueness=20
of map objects according to L&S, page page 314,=20
Exercise 1. Does this category have a name? =20
Is Boolean Category sensible?
Two messages mentioned lambda calculus.
Another topic for future reference.
Stephen Lack asked "How small is small?=20
How explicit is explicit?" Probably=20
several other readers wondered the same.
Fred's reply is small enough and explicit=20
enough to write out in detail.
One message addressed the term "concrete". =20
I referred to Concrete Categories in the=20
Wikipedia.
Matt Hellige mentioned categories a little=20
bigger than that described by Fred. =20
For instance, objects 0, 1, 2, 3.
Map A -> B exists iff A < B.
B x A =3D? min(A, B) =20
I should sketch the details of some of these=20
examples beyond the 0, 1 case above.
Andrej Bauer described Fred's category in the context=20
of Heyting algebra and noted a proof by=20
Peter Freyd.
Thorsten Altenkirch mentioned an equational=20
inconsistency which is beyond my present=20
grasp.
Apologies to anyone who's reply is not =20
addressed adequately. If someone requests,=20
I can revise the summary and resubmit it.
Thanks, ... Peter E.
Desktops.OpenDoc http://carnot.yi.org/
From rrosebru@mta.ca Mon Mar 3 20:30:05 2008 -0400
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From:
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Subject: categories: Re: A small cartesian closed concrete category
Date: Mon, 03 Mar 2008 16:30:55 -0500
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Peter Easthope points out that in
Lawvere & Schanuel there is no
mention of Arend Heyting. That is
unfortunate, especially since=20
pp 348-352 are devoted to
introducing Heyting's Algebras=20
and one of their possible
objective origins. The 2nd edition
should correct this omission.
Summarizing the 16 responses,
a common thought of many must=20
have been=20
"If small implies finite
then any example must be a poset
(category in which any two parallel
maps are equal) because of Freyd's
theorem. A CC poset is almost=20
by definition a Heying Algebra.
There are linearly ordered ones of=20
any size, but if the size is four or more,
there are also examples that are not=20
linearly ordered....
=20
On the other hand if infinite examples=20
are allowed, and posetal ones are not,
it is hard to think of a CCC smaller than
a skeletal category of all finite sets."
Bill
From rrosebru@mta.ca Mon Mar 3 20:30:05 2008 -0400
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Date: Mon, 03 Mar 2008 12:15:02 -0500 (EST)
From: Joshua P Nichols-Barrer
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Subject: categories: Re: Minimal abelian subcategory
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Hmm. I suppose that restricting to subcategories which respect the group
structure on the Hom-sets would be enough to render the problem harder
(the group structure of course can be recovered canonically from the
underlying category, so this merely refines the class of subcategories we
are considering). I would imagine this restriction would also have more
repercussions for algebra, anyway...
Josh
On Sun, 2 Mar 2008, Colin McLarty wrote:
>> Now let R be a ring and M an R-module. Is there a minimal abelian
>> subcategory of Mod-R containing M? If so, is there a canonical way to
>> describe it?
>
> This question, as posed, is too easy: Just take M and its identity
> arrow. It will be a zero-object in that subcategory. There may be a
> better question here guiding Walter Mazorchuk's intuition, but it will
> have to require something more than just containing the one object.
>
> Colin
>
>
>
From rrosebru@mta.ca Tue Mar 4 08:49:19 2008 -0400
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Date: Mon, 03 Mar 2008 17:59:40 -0800
From: Vaughan Pratt
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> 5 messages mentioned Hyting-algebras.
> Never heard of them.Lawvere & Schanuel
> do not mention them in the 1997 book.
> Will store the terms for future reference.
Nowadays when I hear "Never heard of x" my subconscious seems to turn it
into "never heard of Wikipedia." When five people tell you x is the
answer to your question, merely filing it "for future reference" misses
the point of the answer. (As one of the five, my examples consisted of
the finite nonempty chains and the finite Boolean algebras, which I
pointed out to Peter gave an example of every finite positive
cardinality, and two for the powers of two. My mistake was to lump
these examples together under the common rubric of "Heyting algebra,"
which appears to have made what was meant to be a simple answer
incomprehensible.)
As Bill points out, a Heyting algebra is almost the same thing as a CCC
in the case of categories that are posets. This is exactly the case
when there are finitely many objects (a case where Heyting algebras and
distributive lattices are "the same thing" in the sense that they have
the same underlying posets), and is close to true modulo existence of
joins in the infinite case. In particular a Heyting algebra needs the
empty join 0 in order to define negation as x->0, whence the negative
integers made a category with its standard ordering is cartesian closed
but is not a Heyting algebra for want of a least negative integer. More
generally Heyting algebras are required to have all finite joins, not a
requirement for posetal cartesian closed categories.
Vaughan Pratt
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Date: Tue, 4 Mar 2008 00:53:39 -0000 (GMT)
Subject: categories: Minimal abelian subcategory (corrected)
From: "Tom Leinster"
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Apologies for the previous trivial question. Here is the correct version=
.
(The mistake was omitting to say that the subcategory must contain all
endomorphisms of M.)
*
My colleague Walter Mazorchuk has the following question.
Being abelian is a *property* of a category, not extra structure. Given
an abelian category A, it therefore makes sense to define a subcategory o=
f
A to be an ABELIAN SUBCATEGORY if, considered as a category in its own
right, it is abelian. Note that a priori, the inclusion need not preserv=
e
sums, kernels etc.
Now let R be a ring and M an R-module. Is there a minimal abelian
subcategory of Mod-R containing M and all its endomorphisms? If so, is
there a canonical way to describe it?
Any thoughts or pointers to the literature would be welcome. Feel free t=
o
assume hypotheses on R (it might be a finite-dimensional algebra etc), or
to answer the question for full subcategories only.
Thanks,
Tom
From rrosebru@mta.ca Tue Mar 4 22:21:11 2008 -0400
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Date: Tue, 04 Mar 2008 16:20:14 +0100
From: Andrej Bauer
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Subject: categories: How to motivate a student of functional analysis
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This semester I am teaching rudimentary category theory at graduate
level. It is somewhat scary that I should be doing this, but other
faculty members do not seem to do much general category theory.
I have only few students (and they are very bright) but their areas of
research are quite diverse: discrete math/computer science, algebra,
algebraic topology, and functional analysis.
I can plenty motivate categories for discrete math and computer science,
with things like "initial algebras are inductive datatypes, final
coalgebras are coinductive (lazy) datatypes".
I also know enough general algebra to motivate algebraists with
tquestions like "What is an additive category with a single object?".
And we will study algebraic theories as well.
Algebraic topologists are self-motivated. Nevertheless, we'll do some
sheaves towards the end of the course.
But how do I show the fun in categories to a student of functional
analysis? I would like to give him a class project that he will find
close to his interests. The course is covering (roughly) the following
material: basic category theory (limits, colimits, adjoints, we
mentioned additive and enriched categories), Lawvere's algebraic
categories, monads (up to stating Beck's theorem and working out some
examples), basics of presheaves and sheaves with a slant toward
topology. There must be some functional analysis in there.
I would very much appreciate some suggestions.
Best regards,
Andrej
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Date: Tue, 4 Mar 2008 23:49:39 +0000 (GMT)
From: "Prof. Peter Johnstone"
To: categories@mta.ca
Subject: categories: Re: Minimal abelian subcategory (corrected)
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On Tue, 4 Mar 2008, Tom Leinster wrote:
> Apologies for the previous trivial question. Here is the correct version.
>
There's still something odd about this question. Requiring the subcategory
to contain all endomorphisms of M of course requires it to contain
A(M,M) as a monoid. But if you don't require it to be closed under
biproducts in A, then presumably you don't require it to contain A(M,M)
as a ring. It therefore raises two questions of "pure algebra":
What conditions on a monoid (with 0) are needed to ensure that it occurs
as the multiplicative monoid of a ring?
Given that it does so occur, can there be several different additive
group structures making it into a ring?
I suspect that a fair amount must be known about these questions, but
the only result I know in this area is one which I quoted in "Stone
Spaces": for a ring of the form C(X), X a compact Hausdorff space,
the multiplicative monoid structure of C(X) is enough to determine
the topology of X (and hence the ring structure of C(X)) uniquely.
Peter Johnstone
>
> My colleague Walter Mazorchuk has the following question.
>
> Being abelian is a *property* of a category, not extra structure. Given
> an abelian category A, it therefore makes sense to define a subcategory of
> A to be an ABELIAN SUBCATEGORY if, considered as a category in its own
> right, it is abelian. Note that a priori, the inclusion need not preserve
> sums, kernels etc.
>
> Now let R be a ring and M an R-module. Is there a minimal abelian
> subcategory of Mod-R containing M and all its endomorphisms? If so, is
> there a canonical way to describe it?
>
> Any thoughts or pointers to the literature would be welcome. Feel free to
> assume hypotheses on R (it might be a finite-dimensional algebra etc), or
> to answer the question for full subcategories only.
>
> Thanks,
> Tom
>
>
>
>
>
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From: Paul Taylor
Subject: categories: Heyting algebras and Wikipedia
Date: Tue, 4 Mar 2008 14:17:18 +0000
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On the subject of Heyting algebras, usage seems to be ambiguous
as to whether they should have (and their morphisms preserve)
finite joins. I suggest that we should say "Heyting lattice"
if they should, and "Heyting semilattice" if not.
More generally, Vaughan said,
> Nowadays when I hear "Never heard of x" my subconscious seems
> to turn it into "never heard of Wikipedia."
I too turn to Wikipedia for information on most subjects. For
example its medical information is far superior to any other
lay source that I have seen. But I have two reservations:
Authority. Journalists like to take swipes at it on the grounds
that anyone can edit it, but in my opinion they over-estimate
the reliability of "authoritative" sources. A traditional paper
encyclopedia consults only a small number of experts on each topic,
so it's likely to be cliquey. On the other hand, there are
frequently stories in www.TheRegister.co.uk (online geek news)
about cliques taking over Wikipedia.
Closer to home, the coverage of mathematics is extremely poor in
comparison to other subjects. Usually, there is just the stark
classical undergraduate definition, with neither advanced
mainstream material nor any constructive critique. In my work
on ASD, particularly its application to real analysis, I have
wanted to refer to classical sources as a background, but on none
of the relevant topics have I considered the Wikipedia article
to be anywhere near satisfactory. All spaces are Hausdorff, and
Excluded Middle is a Fact. I have thought about rewriting the
articles on Dedekind cuts, locally compact spaces and some other
things, but am afraid that my contributions will just be "reverted".
Maybe if other categorists and constructivists joined in too,
I would feel in better company.
No, I don't want knock Wikipedia. It's a Good Thing, in principle.
And I would like to encourage others to improve the mathematical
coverage.
By the way, there's also PlanetMath.org, in which authors "own"
their articles, unless they have demonstrably abandoned them.
Since I'm here, I would like to point out that there are thoroughly
revised versions of
The Dedekind Reals in ASD (with Andrej Bauer)
and A Lambda Calculus for Real Analysis
on my web page at www.PaulTaylor.EU/ASD/analysis.php
The second of these contains a "need to know" introduction to the
Scott topology, proof theory and the lambda calculus, ie it is
written with the general mathematical audience in mind.
Paul Taylor
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Subject: categories: Re: Minimal abelian subcategory (corrected)
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As George Janelidze pointed out to me, there was an error in what I wrote
yesterday: the multiplicative monoid structure of C(X) determines X, and
hence the ring structure of C(X), up to isomorphism (this is a 1949
result of A.N. Milgram), but it doesn't determine the additive
structure uniquely, since one can take the standard addition and
"conjugate" it by a multiplicative automorphism of R, in the same way
that Steve points out for finite fields (e.g. one could define a new
addition by f +' g = (f^3 + g^3)^{1/3}).
Peter Johnstone
On Wed, 5 Mar 2008, Steve Vickers wrote:
> Dear Peter,
>
> A special case is that of groups (with 0 adjoined) and fields. But even that
> is hard. In Paul Cohn's book on Skew Fields (p.145 of 1995 edition) he says
> "The problem of characterizing the multiplicative group of a field has not
> yet been solved even in the commutative case, but there are some results on
> the subgroups of fields." (The main result he cites, one of Amitsur's, says
> that a finite group G can be embedded as a subgroup in the multiplicative
> group of a [skew]field if and only if G is (i) cyclic, or (ii) a certain kind
> of metacyclic group, or (iii) a certain from of soluble goup with a
> quaternion subgroup, or (iv) the binary icosahedral group SL_2(F_5) of order
> 120.)
>
> Even in finite fields F = Z/p, the additive structure is not determined on
> the nose by the multiplicative structure. If the group F* has a non-identity
> automorphism alpha then (extending alpha to take 0 to 0) a different addition
> for the same multiplication can be defined by x +' y = alpha^{-1}(alpha(x) +
> alpha(y)). This would be the original addition only of alpha preserves
> addition; but then since it preserves 0 and 1, and 1 is an additive
> generator, then it would have to be the identity. An example is F_5, where
> the multiplicative group is cyclic of order 4 and has a non-identity
> automorphism that swaps the two generators.
>
> There remains the deeper question of whether you can have non-isomorphic
> additive groups for the same multiplicative group.
>
> Regards,
>
> Steve.
>
> Prof. Peter Johnstone wrote:
>> On Tue, 4 Mar 2008, Tom Leinster wrote:
>>
>> > Apologies for the previous trivial question. Here is the correct
>> > version.
>> >
>> There's still something odd about this question. Requiring the subcategory
>> to contain all endomorphisms of M of course requires it to contain
>> A(M,M) as a monoid. But if you don't require it to be closed under
>> biproducts in A, then presumably you don't require it to contain A(M,M)
>> as a ring. It therefore raises two questions of "pure algebra":
>>
>> What conditions on a monoid (with 0) are needed to ensure that it occurs
>> as the multiplicative monoid of a ring?
>>
>> Given that it does so occur, can there be several different additive
>> group structures making it into a ring?
>>
>> I suspect that a fair amount must be known about these questions, but
>> the only result I know in this area is one which I quoted in "Stone
>> Spaces": for a ring of the form C(X), X a compact Hausdorff space,
>> the multiplicative monoid structure of C(X) is enough to determine
>> the topology of X (and hence the ring structure of C(X)) uniquely.
>>
>> Peter Johnstone
>
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Date: Wed, 05 Mar 2008 01:11:41 -0800
From: Vaughan Pratt
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Subject: categories: Re: Heyting algebras and Wikipedia
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Paul Taylor wrote:
> In my work
> on ASD, particularly its application to real analysis, I have
> wanted to refer to classical sources as a background, but on none
> of the relevant topics have I considered the Wikipedia article
> to be anywhere near satisfactory. All spaces are Hausdorff, and
> Excluded Middle is a Fact. I have thought about rewriting the
> articles on Dedekind cuts, locally compact spaces and some other
> things, but am afraid that my contributions will just be "reverted".
In my understanding of Heyting algebras/lattices/semilattices, excluded
middle fails for the algebras themselves but not for my understanding of
them, where the partial order x <= y in a Heyting algebra is either true
or false with no middle ground allowed.
I have had little luck absorbing the logic of Heyting algebras into my
own mathematical thinking. I furthermore worry that if ever I were to
succeed my insights might become even less penetrating than they already
are.
On a related note, a careful reading of Max Kelly's "Basic Concepts of
Enriched Category Theory" reveals that it is thoroughly grounded in Set,
as I pointed out in August 2006 in my initial Wikipedia article on Max.
I gave some thought to how one might eliminate Set from the treatment,
without much success, and concluded that Max's judgment there was spot on.
My feeling about these recommended Brouwerian modes of thoughts is that
they are something like locker room accounts of social and other
conquests: great stories about things that never actually happened, but
which with sufficient repetition convince one that they must surely have
occurred.
The self-evident is merely an hypothesis that is so convenient, and that
has been assumed for so long, that we can no longer imagine it false.
This is just as true for Excluded Middle itself as for its negation. I
happen to find Excluded Middle more convenient than its negation, but
that's just me and perhaps others have had the opposite experience.
Then there are those who accept neither Excluded Middle nor its
negation, which takes us into the Hall of Mirrors that I always find
myself in when I go down this particular rabbit-hole.
Vaughan Pratt
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Date: Wed, 05 Mar 2008 10:34:35 +0000
From: Steve Vickers
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Dear Peter,
A special case is that of groups (with 0 adjoined) and fields. But even
that is hard. In Paul Cohn's book on Skew Fields (p.145 of 1995 edition)
he says "The problem of characterizing the multiplicative group of a
field has not yet been solved even in the commutative case, but there
are some results on the subgroups of fields." (The main result he cites,
one of Amitsur's, says that a finite group G can be embedded as a
subgroup in the multiplicative group of a [skew]field if and only if G
is (i) cyclic, or (ii) a certain kind of metacyclic group, or (iii) a
certain from of soluble goup with a quaternion subgroup, or (iv) the
binary icosahedral group SL_2(F_5) of order 120.)
Even in finite fields F = Z/p, the additive structure is not determined
on the nose by the multiplicative structure. If the group F* has a
non-identity automorphism alpha then (extending alpha to take 0 to 0) a
different addition for the same multiplication can be defined by x +' y
= alpha^{-1}(alpha(x) + alpha(y)). This would be the original addition
only of alpha preserves addition; but then since it preserves 0 and 1,
and 1 is an additive generator, then it would have to be the identity.
An example is F_5, where the multiplicative group is cyclic of order 4
and has a non-identity automorphism that swaps the two generators.
There remains the deeper question of whether you can have non-isomorphic
additive groups for the same multiplicative group.
Regards,
Steve.
Prof. Peter Johnstone wrote:
> On Tue, 4 Mar 2008, Tom Leinster wrote:
>
>> Apologies for the previous trivial question. Here is the correct
>> version.
>>
> There's still something odd about this question. Requiring the subcategory
> to contain all endomorphisms of M of course requires it to contain
> A(M,M) as a monoid. But if you don't require it to be closed under
> biproducts in A, then presumably you don't require it to contain A(M,M)
> as a ring. It therefore raises two questions of "pure algebra":
>
> What conditions on a monoid (with 0) are needed to ensure that it occurs
> as the multiplicative monoid of a ring?
>
> Given that it does so occur, can there be several different additive
> group structures making it into a ring?
>
> I suspect that a fair amount must be known about these questions, but
> the only result I know in this area is one which I quoted in "Stone
> Spaces": for a ring of the form C(X), X a compact Hausdorff space,
> the multiplicative monoid structure of C(X) is enough to determine
> the topology of X (and hence the ring structure of C(X)) uniquely.
>
> Peter Johnstone
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From: "Katsov, Yefim"
To:
Date: Wed, 5 Mar 2008 10:25:36 -0500
Subject: categories: RE: How to motivate a student of functional analysis
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Dear Adrei,
May I suggest you to look at the monograph "Lectures and Exercises on Funct=
ional Analysis" by A. Ya. Helemskii published by AMS in 2006, where, I'm su=
re, you'll find a lot of good motivations for students interested in functi=
onal analysis to study category theory.
Good Luck and best regards,
Yefim
_______________________________________________________________________
Prof. Yefim Katsov
Department of Mathematics & CS
Hanover College
Hanover, IN 47243-0890, USA
telephones: office (812) 866-6119;
home (812) 866-4312;
fax (812) 866-7229
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From: Colin McLarty
To: categories@mta.ca
Date: Wed, 05 Mar 2008 09:24:02 -0500
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Vaughan Pratt
Wednesday, March 5, 2008 8:32 am
wrote, with much else:
> On a related note, a careful reading of Max Kelly's "Basic Concepts of
> Enriched Category Theory" reveals that it is thoroughly grounded in
> Set,as I pointed out in August 2006 in my initial Wikipedia article
> on Max.
> I gave some thought to how one might eliminate Set from the
> treatment,without much success, and concluded that Max's judgment
> there was spot on.
Without addressing this particular issue I want to say I appreciate the
phrase in the article: "the explicitly foundational role of the category
Set." I take it this is Vaughan's?
Various people including Sol Feferman promote the view that if you use
"sets" then you are admitting that you use ZF and not some categorical
foundations. Vaughan's phrase goes aptly against that: If you use
sets, then you use sets, but there is no reason it cannot be on
categorical foundations. He does not say it *is* on categorical
foundations, and that is fine in the context. He reminds people that it
*could* be.
best, Colin
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Date: Wed, 05 Mar 2008 16:10:43 +0100
From: Luigi Santocanale
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To: categories@mta.ca
Subject: categories: Postdoctoral research position in Theoretical Computer Science, Marseilles University
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=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D
POSTDOCTORAL RESEARCH POSITION IN THEORETICAL COMPUTER SCIENCE
Marseilles University - CNRS - ANR CHOCO
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D
The ANR project Curry-Howard for Concurrency (CHOCO) proposes a one year
postdoc research position in Marseilles in the field of theoretical=20
computer science, starting in September 2008 (or as soon as possible=20
thereafter).
The project CHOCO is focused on the applications of theoretical results=20
from mathematical logic and/or theoretical computer science to the=20
theory of concurrency.
Candidates should have their PhD and a good background in at least one=20
of the following themes:
- mathematical logic (lambda-calculus, complexity theory, linear logic),
- semantics of programming languages (theory of categories,=20
denotationnal and game semantics),
- models of concurrency (process calculi, bisimulation, event structures)=
.
The position will be taken in the logic group (LDP) of the Institut de
Math=E9matiques de Luminy (IML); strong interaction is expected with the
group MOVE of the Laboratoire d'Informatique Fondamentale (LIF) in=20
Marseilles, and the group Plume of the Laboratoire d'Informatique du=20
Parall=E9lisme in Lyon (LIP).
Application should be sent to:
postdoc-choco@choco.pps.jussieu.fr
before May 18th 2008 and should include (all documents in pdf):
- a CV (civil informations, universitary cursus, phd);
- a work programme (no more than one page);
- a publication list;
- contact information for 2 references.
Candidates will be notified by mid June.
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D
CHOCO: http://choco.pps.jussieu.fr/
IML : http://iml.univ-mrs.fr/
LDP : http://iml.univ-mrs.fr/ldp/
LIF : http://www.lif.univ-mrs.fr/
MOVE : http://www.lif.univ-mrs.fr/spip.php?article89
LIP : http://www.ens-lyon.fr/LIP/web/
Plume: http://www.ens-lyon.fr/LIP/PLUME/index.html.en
--=20
Luigi Santocanale
LIF/CMI Marseille T=E9l: 04 91 11 35 74
http://www.cmi.univ-mrs.fr/~lsantoca/ Fax: 04 91 11 36 02 =09
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Date: Wed, 5 Mar 2008 08:38:35 -0500 (EST)
From: Michael Barr
To: Categories list
Subject: categories: Re: How to motivate a student of functional analysis
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A student interested in functional analysis presumably knows some about
topological vector spaces in general and Mackey spaces in particular. He
might be interested in knowing that the full subcategory of Mackey spaces
has a *-autonomous structure. This means that if M and N are Mackey there
is a topology on the vector space of continuous linear maps M --> N that
makes it into a Mackey space, often denoted M -o N, and that if you let M*
= M -o C, then the canonical map M --> M** is an isomorphism. There is
also a tensor product @ and the usual isomorphism Hom(M@N,P) =
Hom(M,N-oP). See
M. Barr, On $*$-autonomous categories of topological vector spaces.
\cahiers {41} (2000), 243--254.
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From:
To: Categories list
Subject: categories: Re: How to motivate a student of functional analysis
Date: Wed, 05 Mar 2008 11:42:53 -0500
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Functional Analysis was one of the key origins
of categorical concepts and outlook, for example
that the functionals themselves should be collected=20
into a single object (Voltera-Hadamard) leads to
the Hom functor,etc. This was also one of the roads
followed by students in the 1950s, for example
from J. L. Kelley's "galactic" treatment of M. H. Stone's
functor C. However in North America (as distinct
from Europe) more recent functional analysists
have accepted categorical methods only=20
grudgingly, and hence piecemeal.
On the other hand, students who are not=20
specializing in analysis are often woefully ignorant
of the basics of functional analysis that are part=20
of what every mathematician should know. To combat=20
that ignorance in my Advanced Graduate Algebra=20
course I often devoted several weeks to topics
from functional analysis. It is a source of examples
both interesting and essential.
To begin to try to answer Andrej's question, I=20
rapidly recall some examples, and hope others will
also comment:
The double dual functor on Banch spaces is a
protype example of a composite of adjoints becoming
a monad. The EM algebras for this monoid were=20
computed by Fred Linton, in an exercise that should=20
be better known. It also illustrates the "descent"
principle that C. Houzel cited couple of months ago
(what I called semantics of structure of a given functor=20
in my thesis) : Objects constructed by a given functor
tend to have, by virtue of that, more structure than=20
originally contemplated in its codomain , hence
a lifted version of the functor comes closer to being
invertible.
As Peter Johnstone just recalled, if we consider
commutative monoids with zero and hom them into the
particular object of reals, the resulting set is "actually"
a compact space whose C-algebra reveals by=20
adjointness that the opposite of the spaces form a full=20
subcategory of the monoids with zero. Again a good=20
exercise, related to Kelley's "square root lemma".
Students might wonder why contiuous linear operators
are traditionally called "bounded" (when they are not even).
For many linear spaces (roughly those where sequentiality
suffices) , preserving sequential limits is equivalent to
preserving boundedness of sequences (for a linear map).
George Mackey started to functorize this crucial
observation before categories were fully explicit. Now
we can consider the category of all presheaves on the category
of all countable sets, define an "underlying" functor from
Banach spaces to it, and verify that it actually lands in
the subtopos of sheaves for the finite-disjoint-covering
topology. Indeed it not only gives abelian group objects in
the latter topos, but modules over R, the Dedekind reals
of the topos, and a FULL subcategory of those.
The above construction has an analogue using instead=20
Johnstone's coherent topos of sheaves on countable
compact spaces.
ETC
Bill
On Tue Mar 4 10:20 , Andrej Bauer sent:
>This semester I am teaching rudimentary category theory at graduate
>level. It is somewhat scary that I should be doing this, but other
>faculty members do not seem to do much general category theory.
>
...
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From:
To: categories@mta.ca
Subject: categories: Re: Minimal abelian subcategory (corrected)
Date: Wed, 05 Mar 2008 11:52:09 -0500
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Oops
Email is great. I cited Peter before
he could correct himself.=20=20
A possible remedy would be to
consider not just monoids with
zero, but those equipped with a=20
homomorphism from R, so that
points are retractions of that.
This cuts down on the automorphisms,
at least the naturally available ones.
Bill
On Wed Mar 5 6:22 , "Prof. Peter Johnstone" sent:
>As George Janelidze pointed out to me, there was an error in what I wrote
>yesterday: the multiplicative monoid structure of C(X) determines X, and
>hence the ring structure of C(X), up to isomorphism (this is a 1949
>result of A.N. Milgram), but it doesn't determine the additive
>structure uniquely, since one can take the standard addition and
>"conjugate" it by a multiplicative automorphism of R, in the same way
>that Steve points out for finite fields (e.g. one could define a new
>addition by f +' g =3D (f^3 + g^3)^{1/3}).
>
>Peter Johnstone
>
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Date: Wed, 5 Mar 2008 17:31:00 -0000 (GMT)
Subject: categories: Re: Minimal abelian subcategory (corrected)
From: "Tom Leinster"
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A message from Walter Mazorchuk:
Dear Colleagues,
thank you very much for your comments on my
abelian envelope question. Because of
my stereotype thinking I missed the point of the
determination of the additive structure by the
multiplicative one in the original formulation.
The stereotype is based on the
fact that I am a representation theorist and the
origin of the question is in module categories,
which are k-linear over some field k. So, the
subcategory I am looking for should be a k-linear
subcategory with the induced k-linear structure.
Best, Walter
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From: Thorsten Altenkirch
Subject: categories: Re: Heyting algebras and Wikipedia
Date: Wed, 5 Mar 2008 16:21:58 +0000
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Hi Vaughan,
On 5 Mar 2008, at 09:11, Vaughan Pratt wrote:
> My feeling about these recommended Brouwerian modes of thoughts is
> that
> they are something like locker room accounts of social and other
> conquests: great stories about things that never actually happened,
> but
> which with sufficient repetition convince one that they must surely
> have
> occurred.
>
> The self-evident is merely an hypothesis that is so convenient, and
> that
> has been assumed for so long, that we can no longer imagine it false.
> This is just as true for Excluded Middle itself as for its
> negation. I
> happen to find Excluded Middle more convenient than its negation, but
> that's just me and perhaps others have had the opposite experience.
Indeed, being a computer scientist the BHK interpretation (see
wikipedia) of logical connectives (which I can implement on a finite
machine) makes more sense to me than the idea of infinite truth tables.
You seem to think that the only alternative to excluded middle
(forall P:Prop. P \/ not P) is (exists P:Prop. not (P \/ not P))?
However, I'd say that "forall n:Nat. Halt n \/ not Halt n" is clearly
invalid in the BHK interpretation without claiming that there is a
particular statement which will never be decided, or a Turing
machine which can be never shown to be terminating or not, i.e. even
if we accepts Church's thesis, we arrive at "not (forall n:Nat. Halt
n \/ not Halt n)" but not "exists n:Nat.not (Halt n \/ not Halt n)".
To summarize, your reasoning seems to already presupposes that we
accept Excluded Middle.
>
> Then there are those who accept neither Excluded Middle nor its
> negation, which takes us into the Hall of Mirrors that I always find
> myself in when I go down this particular rabbit-hole.
>
Maybe this is related to my reply?
Thorsten
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From rrosebru@mta.ca Wed Mar 5 16:08:35 2008 -0400
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From: "Ronnie"
To: "Categories list"
Subject: categories: Re: categories and Wikipedia
Date: Wed, 5 Mar 2008 18:44:00 -0000
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I have made some minor contributions to wikipedia with information on for
example John Robinson, on groupoids, Grothendieck, and the van Kampen
theorem. The last three link to my web site and my counter (which registers
`came from') shows the utility of these links, among many others.
In the past a text had to assume or give an account of basic material. Why
give an account of say Yoneda when there is a reasonable one on wiki which
a reader can download?
So I would encourage category theorists to develop the accounts.
Ronnie
----- Original Message -----
From: "Paul Taylor"
To: "Categories list"
Sent: Tuesday, March 04, 2008 2:17 PM
Subject: categories: Heyting algebras and Wikipedia
> On the subject of Heyting algebras, usage seems to be ambiguous
> as to whether they should have (and their morphisms preserve)
> finite joins. I suggest that we should say "Heyting lattice"
> if they should, and "Heyting semilattice" if not.
>
> More generally, Vaughan said,
> > Nowadays when I hear "Never heard of x" my subconscious seems
> > to turn it into "never heard of Wikipedia."
>
...
From rrosebru@mta.ca Wed Mar 5 16:11:04 2008 -0400
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Date: Wed, 5 Mar 2008 14:30:43 -0500 (EST)
From: Jeff Egger
Subject: categories: Re: How to motivate a student of functional analysis
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Dear Andrej,
Conventional functional analysis is largely concerned with Banach=20
spaces, and there's certainly alot that can be said about the=20
category of Banach spaces and linear contractions (i.e., continuous=20
linear transformations with norm less-than-or-equal-to 1). =20
For example, it is symmetric monoidal closed (with internal hom,=20
the space of _all_ continuous linear transformations!) and locally=20
countably presentable. In fact, it is a countable-ary quasi-variety,=20
and the full subcategory of its finite-dimensional objects is a good=20
example of a *-autonomous category that is not compact closed. =20
[Although it is not hard to prove the latter directly, it can also=20
be seen as an interesting application of Robin Houston's theorem=20
that products and coproducts can not differ in a compact closed=20
category.] =20
In fact, I think Ban is a fine example which can teach any student=20
of category theory a number of salutary lessons:
1. in category theory, the meaning of isomorphism is fixed---so if=20
you have a pre-existing class of isomorphisms in mind (in this case,=20
the isometric (norm-preserving) isomorphisms), then you must take=20
care in choosing an appropriate class of morphisms;
2a. there's more to defining internal homs than just slapping an =20
extra structure on the external homs;=20
2b. forgetful functors don't have to be "the obvious thing";
3. you can't always have your cake and eat it too!---the whole=20
category can not hope to be self-dual, precisely because it is=20
locally presentable (and not a poset).
Of course there are also (unital) C*-algebras, and I can make an=20
interesting point about them too---sometimes one needs to consider=20
maps between C*-algebras which are not *-homomorphisms: for example,
there are "completely positive maps" and "completely bounded maps".
Now, as important as the b.o./f.f. factorisation may be in general,=20
it seems fishy to speak of a category whose objects are C*-algebras
but whose morphisms preserve only part of the C*-algebraic structure;
and so it was that analysts were led to develop the notions of=20
"operator space" and "operator system" which provide the correct=20
level of structure to define c.b. maps and c.p. maps, respectively. =20
In fact, these are quite interesting categories in their own right:
operator spaces are said to model "non-commutative functional analysis"
---but I only have a tenuous grasp of what that is supposed to mean!
I meant to discuss quantale theory and Banach sheaves too, but I've
run out of time---perhaps someone else will pick up the thread.
Cheers,
Jeff.
--- Andrej Bauer wrote:
> This semester I am teaching rudimentary category theory at graduate
> level. It is somewhat scary that I should be doing this, but other
> faculty members do not seem to do much general category theory.
>=20
> I have only few students (and they are very bright) but their areas of
> research are quite diverse: discrete math/computer science, algebra,
> algebraic topology, and functional analysis.
>=20
> I can plenty motivate categories for discrete math and computer science=
,
> with things like "initial algebras are inductive datatypes, final
> coalgebras are coinductive (lazy) datatypes".
>=20
> I also know enough general algebra to motivate algebraists with
> tquestions like "What is an additive category with a single object?".
> And we will study algebraic theories as well.
>=20
> Algebraic topologists are self-motivated. Nevertheless, we'll do some
> sheaves towards the end of the course.
>=20
> But how do I show the fun in categories to a student of functional
> analysis? I would like to give him a class project that he will find
> close to his interests. The course is covering (roughly) the following
> material: basic category theory (limits, colimits, adjoints, we
> mentioned additive and enriched categories), Lawvere's algebraic
> categories, monads (up to stating Beck's theorem and working out some
> examples), basics of presheaves and sheaves with a slant toward
> topology. There must be some functional analysis in there.
>=20
> I would very much appreciate some suggestions.
>=20
> Best regards,
>=20
> Andrej
>=20
>=20
>=20
Looking for the perfect gift? Give the gift of Flickr!=20
http://www.flickr.com/gift/
From rrosebru@mta.ca Wed Mar 5 16:12:36 2008 -0400
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Date: Wed, 05 Mar 2008 10:13:55 -0800
From: PETER EASTHOPE
Subject: categories: Re^3: A small cartesian closed concrete category
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Vaughan P.,
vp> ... "never heard of Wikipedia." When five people
tell you x is the answer to your question, ...
I appreciate your exasperation. Am afraid
that almost everyone who replied to my question
is severely over-estimating my state of
comprehension. My background is primarily in
engineering and physics, whereas most of you
teach at the honours undergraduate and graduate
levels.
At present I am trying to understand the concept of
map object and the exercises on pp. 314 and 315 of L&S.
Long ago, a professional mathematician, as you all
are, advised: When stuck, find examples until you
come to an understanding. Presently I am looking
for examples illustrating Exercises 1-6. Fred's
reply is a good start. Longer chains suggested by
Matt H. will be interesting, if not necessary.
Skipping ahead 34 pages, I see that map objects
are an ingredient of a topos. A scan of
http://en.wikipedia.org/wiki/Heyting_algebra
reports that "map object" is not in the page.
Perhaps it should be. Neverthless, working
through the book systematically seems more
promising than reading about toposes and Heyting
algebras before understanding map objects.
fwl> ... Heyting's Algebras and one of their
possible objective origins. The 2nd edition
should correct this omission.
I don't want to be presumptuous, but if some
of the tiny categories mentioned by Fred and
Matt can also fit into the second edition, that
would certainly interest me. Without this
text, my endeavour to learn category th. would
be quite a battle. Thanks!
I should have explained at the beginning,
the intention in seeking the examples. Sorry
for the aggravation.
Regards, ... Peter E.
http://carnot.yi.org/
From rrosebru@mta.ca Thu Mar 6 09:42:38 2008 -0400
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Date: Wed, 05 Mar 2008 20:22:37 -0500
From: "Fred E.J. Linton"
To: Categories list
Subject: categories: Re: How to motivate a student of functional analysis
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Following Jeff Egger, who wrote, in part, "Ban is a fine example which
can teach any student of category theory a number of salutary lessons,"
but asking forgiveness for tooting my own horn, I'd like to point out =
another one of those lessons -- my old characterization of Banach =
conjugate spaces as the algebras over the double-dualization monad =
on {Ban}. Neat mix of Beck Theorem, functional analysis, and more, on
pp. 227-240 of: =
Proc. Conf. Integration, Topology, and Geometry in Linear Spaces,
in: Contemporary Mathematics, Volume 2, AMS, Providence, 1980.
Might even serve as one student's "individual reading report" project.
There's also my even older squib on "Functorial Measure Theory," in pp.
36-49 of: Proc. Conf. Functional Analysis, UC Irvine, 3/28-4/1, 1966,
Thompson Book Co., Wash., DC, & Academic Press, London, 1967. This one
breathes life into the slogan, "Measures are adjoint to functions."
Cheers,
-- Fred
From rrosebru@mta.ca Thu Mar 6 09:42:39 2008 -0400
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From: Robert L Knighten
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Most of the material connecting analysis and category theory seems to be
written by specialists in category theory who have observed some of the ways
that insights from category theory can be brought to bear (for example look on
Math Sci Net at the many papers (co)authored by Joan Wick Pelletier.) But
here's an example which is a book on functional analysis that has a strong use
of categories:
@book {MR0296671,
AUTHOR = {Semadeni, Zbigniew},
TITLE = {Banach spaces of continuous functions. {V}ol. {I}},
NOTE = {Monografie Matematyczne, Tom 55},
PUBLISHER = {PWN---Polish Scientific Publishers},
ADDRESS = {Warsaw},
YEAR = {1971},
PAGES = {584 pp. (errata insert)},
MRCLASS = {46E15 (46M99)},
MRNUMBER = {MR0296671 (45 \#5730)},
MRREVIEWER = {H. E. Lacey},
}
-- Bob
--
Robert L. Knighten
RLK@knighten.org
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Date: Wed, 5 Mar 2008 17:53:54 -0500 (EST)
From: Michael Barr
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Subject: categories: graphics and dvi
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If you read the TAC instructions for authors, you will find that we
discourage the use of the graphics package unless absolutely necessary
because, for the time being, dvi is still our basic archive format and
graphics specials were not rendered properly by dvi viewers. I recently
discovered that at least one dvi viewer does indeed render graphics
specials. Namely the yap viewer that comes with miktex, a
windows-specific implementation of tex enters something called dvips mode
(meaning, I imagine, an on-the fly conversion to ps and then rendering
that). To be more precise, miktex2.5 asks if you want dvips mode (why
wouldn't you?) and miktex2.7 enters it automatically. (I don't know about
2.6).
As far as I know no Unix (or Linux) viewer does this. Neither pdflatex
nor dvipdfm renders graphics specials correctly. The only way I have been
able to get correct pdf files is to first use dvips to make a ps file and
then use ghostscript to convert to pdf.
Still, one can hope that these problems will disappear in some future
implementations and we may withdraw our objections to the use of the
graphics package. At that point, our reluctance to use Paul Taylor's
diagrams will also disappear. For the time being, however, we will
continue to recommend the use of xy-pic and, in particular, the diagxy
front end.
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Date: Thu, 06 Mar 2008 11:15:57 +0000
To: categories@mta.ca
Subject: categories: CiE 2008 - accepted papers, informal presentations, participation
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****************************************************************
Computability in Europe 2008: Logic and Theory of Algorithms
University of Athens, June 15-20 2008
http://www.cs.swan.ac.uk/cie08/
CONTENTS:
1) List of accepted papers
2) Call for informal presentations
3) Call for participation
1) Accepted papers
The list of accepted papers can be found at
http://www.cs.swan.ac.uk/cie08/give-page.php?18
2) Informal Presentations
There is a remarkable difference in conference style between
computer science and mathematics conferences. Mathematics
conferences allow for informal presentations that are prepared
very shortly before the conference and inform the participants
about current research and work in progress. The format of
computer science conferences with pre-produced proceedings
volumes is not able to accommodate this form of scientific
communication.
Continuing the tradition of past CiE conferences, also this
year's CiE conference endeavours to get the best of both worlds.
In addition to the formal presentations based on our LNCS
proceedings volume, we invite researchers to present informal
presentations. For this, please send us a brief description of
your talk (between one paragraph and half a page) before
30 April 2008.
Please submit your abstract via our Submission Form, now online
at:
http://www.cs.swansea.ac.uk/cie08/abstract-submission.php
You will be notified whether your informal presentation has been
accepted before 15 May 2008.
Let us remind you that there will be three post-conference
publications of CiE 2008, see
http://www.cs.swansea.ac.uk/cie08/publications.php
All speakers, including the speakers of informal presentations,
are eligible to be invited to submit a full journal version of
their talk to one of the post-conference publications.
3) Registration for CiE 2008 is now open:
http://www.cs.swan.ac.uk/cie08/registration.php
The early registration deadline is
4 May 2008.
You can also use the registration process to book accommodation.
Please note that the current prices as listed on our website
http://www.cs.swan.ac.uk/cie08/accommodation.php
are only guaranteed until 31 March 2008.
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Subject: categories: CiE 2008 - grants
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****************************************************************
Computability in Europe 2008: Logic and Theory of Algorithms
University of Athens, June 15-20 2008
http://www.cs.swan.ac.uk/cie08/
Call for Grant Applications
Deadline: 15 APRIL, 2008
A number of grants are available for attenting CiE 2008. They
are intended for students, post-docs and persons with limited
means. Also, student members of the ASL may apply for travel
funds. For more details see our website
http://www.cs.swansea.ac.uk/cie08/grants.php
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for categories-list@mta.ca; Thu, 06 Mar 2008 09:35:51 -0400
Date: Wed, 05 Mar 2008 21:15:46 -0800
From: Vaughan Pratt
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Colin is exactly right on all points.
I tend to look at sets from the perspective neither of a set theorist
nor a category theorist but a combinatorialist. As long as people agree
on the cardinalities of the homsets between sets, particularly the
finite ones, I figure they must be talking about essentially the same
objects. Infinite domains are problematic for everyone, infinite
codomains much less so (we understand the homset N^2 much better than 2^N).
The remark in my post about the self-evident being merely a convenient
long-held hypothesis (which I put on my "sayings" website
http://boole.stanford.edu/dotsigs.html less than a month ago) applies in
spades to membership as characteristic of sets, the premise for ZF.
Those who identify acceptance of the category Set with acceptance of ZF
have not only not accepted but not even grasped that the wholesale
replacement of the binary relation of membership by the (partial) binary
operation of composition, with a set of axioms radically different from
those of ZF, is a foundational move. ZF is so deeply ingrained in their
thought processes that they have no idea how to think about mathematical
structures without falling back on its axioms. Borrowing from Hilbert,
they are unable to replace "set," "function," and "composite" by
"table," "chair," and "beermug."
If you find it hard to imagine how anyone could find it hard to imagine
mathematics without ZF, just read Steve Simpson on 2/25/98 (almost
exactly a decade ago) at
http://cs.nyu.edu/pipermail/fom/1998-February/001228.html
The bit "I totally repudiate every syllable of every word of every
subclaim of every claim that McLarty has ever made about what he is
pleased to call `categorical foundations'" made abundantly clear back
then that Steve could not begin to concieve of replacing membership by
composition as the basis for an alternative foundation of mathematics.
While I can't speak for Steve today, this remains a stumbling block for
those raised to believe that rigorous mathematics would not be possible
in a world where propositions such as "for all x and y there exists z
such that x is a subset of z and y is a member of z" did not hold. How
could x U {y} fail to exist and the walls of mathematics not come
tumbling down?
Vaughan
Colin McLarty wrote:
> Vaughan Pratt
> Wednesday, March 5, 2008 8:32 am
>
> wrote, with much else:
>
>> On a related note, a careful reading of Max Kelly's "Basic Concepts of
>> Enriched Category Theory" reveals that it is thoroughly grounded in
>> Set,as I pointed out in August 2006 in my initial Wikipedia article
>> on Max.
>> I gave some thought to how one might eliminate Set from the
>> treatment,without much success, and concluded that Max's judgment
>> there was spot on.
>
> Without addressing this particular issue I want to say I appreciate the
> phrase in the article: "the explicitly foundational role of the category
> Set." I take it this is Vaughan's?
>
> Various people including Sol Feferman promote the view that if you use
> "sets" then you are admitting that you use ZF and not some categorical
> foundations. Vaughan's phrase goes aptly against that: If you use
> sets, then you use sets, but there is no reason it cannot be on
> categorical foundations. He does not say it *is* on categorical
> foundations, and that is fine in the context. He reminds people that it
> *could* be.
>
> best, Colin
>
>
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Date: Thu, 06 Mar 2008 11:35:59 +0100
From: Luigi Santocanale
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To: categories@mta.ca
Subject: categories: Call for participation: workshop on MODAL FIXPOINT LOGICS, Amsterdam, March 25-27 2008
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[Apologies for multiple copies]
Call for participation.
Workshop on
MODAL FIXPOINT LOGICS
Amsterdam, March 25-27 2008
http://staff.science.uva.nl/~yde/mfl
Registration deadline: March 19, 2008
Modal fixpoint logics constitute a research field of considerable
interest, not only because of their many applications, but also
because of their rich logical/mathematical theory. Systems such as
LTL, PDL, CTL, and the modal mu-calculus, originate from computer
science, and are for instance applied in the theory of program
specification and verification. The richness of their theory stems
from the deep connections with various fields in logic, mathematics,
and theoretical computer science, such as lattices and universal
(co-)algebra, modal logic, automata, and game theory.
Large areas of the theory of modal fixpoint logics, in particular the
connection with the theory of automata and games, have been intensively
investigated and are by now are well understood. Nevertheless, there
are still many aspects that are less explored. This applies in particular
to the model theory, intended as the study of a logic as a function of
classes of models, the proof theory, the algebraic logic, duality theory
in the spirit of Stone/Priestly duality, and the relation to the theory
of ordered sets as grounding the concept of "least fixpoint".
The aim of the workshop is to bring together researchers from various
backgrounds, in particular, computer scientists and pure logicians,
who share an interest in the area.
The workshop program is available from the web site=20
http://staff.science.uva.nl/~yde/mfl.
Invited speakers:
Marcello Bonsangue, Leiden
Johan van Benthem, Amsterdam
Dietmar Berwanger, Aachen
Giovanna D'Agostino, Udine
Dexter Kozen, Cornell
Giacomo Lenzi, Pisa
Damian Niwinski, Warszawa
Colin Stirling, Edinburgh
Thomas Studer, Bern
Albert Visser, Utrecht
Igor Walukiewicz, Bordeaux
Thomas Wilke, Kiel
Organizers:
Luigi Santocanale, Marseille
Yde Venema, Amsterdam
--=20
Luigi Santocanale
LIF/CMI Marseille T=E9l: 04 91 11 35 74
http://www.cmi.univ-mrs.fr/~lsantoca/ Fax: 04 91 11 36 02 =09
From rrosebru@mta.ca Thu Mar 6 09:42:41 2008 -0400
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for categories-list@mta.ca; Thu, 06 Mar 2008 09:28:58 -0400
Date: Wed, 05 Mar 2008 20:40:13 +0000
From: Tim Porter
To: Categories list
Subject: categories: Re: How to motivate a student of functional analysis
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Jeff did not mention the excellent very categorical lectures given on=20=20=
=0A=
operator spaces by Matthias Neufang at the=0A=
FIELDS INSTITUTE=0A=
Summer School in Operator Algebras=0A=
held last summer at the University of Ottawa and that we both=20=20=0A=
attended. I do not know if any version of Matthias' notes is=20=20=0A=
available. The theme of tensor products was important. Not only did=20=20=
=0A=
his lectures provide good motivation for studying the subject from a=20=20=
=0A=
categorical viewpoint. He did not do the category theory of operator=20=20=
=0A=
spaces but rather was explicitly conscious of the categorical content=20=20=
=0A=
of what he was saying. His notes may be of some interest to others so=20=
=20=0A=
let us hope he will put some of the material on the web.=0A=
=0A=
=0A=
Tim=0A=
=0A=
=0A=
=0A=
=0A=
Quoting Jeff Egger :=0A=
=0A=
> Dear Andrej,=0A=
>=0A=
> Conventional functional analysis is largely concerned with Banach=0A=
> spaces, and there's certainly alot that can be said about the=0A=
> category of Banach spaces and linear contractions (i.e., continuous=0A=
> linear transformations with norm less-than-or-equal-to 1).=0A=
>=0A=
From rrosebru@mta.ca Thu Mar 6 14:03:16 2008 -0400
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From: "Ronnie"
To: "Categories list"
Subject: categories: Re: How to motivate a student of functional analysis
Date: Thu, 6 Mar 2008 15:15:11 -0000
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You could also look at
MR1471480 (98i:58015)
Kriegl, Andreas; Michor, Peter W.
The convenient setting of global analysis. (English summary)
Mathematical Surveys and Monographs, 53. American Mathematical Society,
Providence, RI, 1997. x+618 pp. ISBN: 0-8218-0780-3
for which an e-version has been downloadable. However as the review says:
"the exposition is based on functional analysis rather than on category
theory; this fact will, undoubtedly, allow the subject to reach a wider
audience. "
Ronnie
----- Original Message -----
From: "Robert L Knighten"
To: "Categories list"
Sent: Thursday, March 06, 2008 2:37 AM
Subject: categories: How to motivate a student of functional analysis
> Most of the material connecting analysis and category theory seems to be
> written by specialists in category theory who have observed some of the
> ways
> that insights from category theory can be brought to bear (for example
> look on
> Math Sci Net at the many papers (co)authored by Joan Wick Pelletier.) But
> here's an example which is a book on functional analysis that has a strong
> use
> of categories:
>
>
> @book {MR0296671,
> AUTHOR = {Semadeni, Zbigniew},
> TITLE = {Banach spaces of continuous functions. {V}ol. {I}},
> NOTE = {Monografie Matematyczne, Tom 55},
> PUBLISHER = {PWN---Polish Scientific Publishers},
> ADDRESS = {Warsaw},
> YEAR = {1971},
> PAGES = {584 pp. (errata insert)},
> MRCLASS = {46E15 (46M99)},
> MRNUMBER = {MR0296671 (45 \#5730)},
> MRREVIEWER = {H. E. Lacey},
> }
>
> -- Bob
>
> --
> Robert L. Knighten
> RLK@knighten.org
>
>
>
>
> --
> No virus found in this incoming message.
> Checked by AVG Free Edition.
> Version: 7.5.516 / Virus Database: 269.21.4/1312 - Release Date:
> 04/03/2008 21:46
>
From rrosebru@mta.ca Thu Mar 6 14:03:16 2008 -0400
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Date: Thu, 6 Mar 2008 12:35:44 -0500 (EST)
From: Jeff Egger
Subject: categories: Re: How to motivate a student of functional analysis
To: Categories list
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Hi Tim,
> Jeff did not mention the excellent very categorical lectures given on =20
> operator spaces by Matthias Neufang at the
> FIELDS INSTITUTE
> Summer School in Operator Algebras
> held last summer at the University of Ottawa and that we both =20
> attended.=20
There are, of course, many people I could have credited and cited in=20
my previous posting, but only the newest readers of this list will be
unaware of the perils with which such an attempt is fraught. [For=20
instance, I first read of the local presentability of Ban in Adamek=20
and Rosicky's book, but I would not like to hazard a guess as to the=20
origin of this result.]
But you are right: I should have made an exception in Matthias' case.
I should also credit Vladimir Pestov, a topologist who knows enough
category theory to wonder whether operator spaces might be internal=20
Banach spaces in some Grothendieck topos (but perhaps not enough to=20
realise that this might take a student more than one term to prove),
for having introduced me to Matthias several years ago. =20
While at Dalhousie, I gave a talk on Pestov's conjecture; but when=20
I started writing up my notes, I was distracted by an unrelated=20
observation about the category of operator spaces which ultimately=20
led to my ill-fated C*-algebra paper. I still haven't gotten back=20
to the original project. =20
> I do not know if any version of Matthias' notes is available.=20
Nor do I, but I am sure he would rather point people towards the=20
pre-Wikipedia-era "online dictionary" of operator space theory to=20
which he contributed:=20
[German] http://www.math.uni-sb.de/ag/wittstock/projekt99.html
[English] http://www.math.uni-sb.de/ag/wittstock/projekt2001.html
These notes are quite good in the sense that, to use Tim's words,=20
they are
> explicitly conscious of the categorical content =20
In particular, it is quite gratifying to see a theorem such as "the=20
forgetful functor from operator spaces to Banach space admits both=20
a left and a right adjoint" stated (more or less) ungrudgingly. =20
Cheers,
Jeff.
P.S. I should say that I don't think that operator spaces would be a=20
suitable topic for an introductory CT course to a general audience;=20
they are rather intricate. But it might be possible to craft an=20
interesting set of exercises for the functional analysis contingent=20
of such a course around operator space theory. =20
From rrosebru@mta.ca Thu Mar 6 14:03:17 2008 -0400
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From: Colin McLarty
To: categories@mta.ca
Date: Thu, 06 Mar 2008 09:10:52 -0500
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Subject: categories: How to motivate me to become a student of functional analysis
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Robert L Knighten
Thursday, March 6, 2008 8:48 am
mentioned Semadeni _Banach spaces of continuous functions_ as using a
categorical perspective. Maybe that is the book I need.
I want to understand Grothendieck's functional analysis in more detail
than just to say he used categorical definitions of different tensor
products to explain Fredholm kernels. For a start, I know nothing about
Fredholm kernels except what is on Wikipedia. Grothendieck's own
writings on it are long and start with many definitions so that it is
hard for me to see the point -- he even says in Recoltes et Semailles
that he never really *felt* the point but did it as an assignment. So
that work shows nothing like the very clear motivation he gives for
schemes and etale cohomology in SGA.
What is a good introduction to his contributions in functional analysis?
best, Colin
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From: Pedro Resende
Subject: categories: Re: How to motivate a student of functional analysis
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Every student who learned the basics of operator algebras knows the
Gelfand-Naimark representation theorem, usually stated non-
categorically as "every commutative unital C*-algebra is isomorphic to
the algebra of continuous functions on a compact Hausdorff space".
Asking such students to check that this is part of a dual equivalence
of categories is probably a good idea, and based on this one can do
exercises about particular algebras they know - for instance to
compute presentations by generators and relations of C(T^n), the
algebra of continuous functions on the n-dimensional torus, which by
the duality is instantly reduced to finding a presentation of C(S^1),
etc. Also, some students might be willing to work out how the
existence of presentations of C*-algebras by generators and relations
relates to the fact that the category of C*-algebras is algebraic over
Sets.
On Mar 4, 2008, at 3:20 PM, Andrej Bauer wrote:
> This semester I am teaching rudimentary category theory at graduate
> level. It is somewhat scary that I should be doing this, but other
> faculty members do not seem to do much general category theory.
>
> I have only few students (and they are very bright) but their areas of
> research are quite diverse: discrete math/computer science, algebra,
> algebraic topology, and functional analysis.
>
> I can plenty motivate categories for discrete math and computer
> science,
> with things like "initial algebras are inductive datatypes, final
> coalgebras are coinductive (lazy) datatypes".
>
> I also know enough general algebra to motivate algebraists with
> tquestions like "What is an additive category with a single object?".
> And we will study algebraic theories as well.
>
> Algebraic topologists are self-motivated. Nevertheless, we'll do some
> sheaves towards the end of the course.
>
> But how do I show the fun in categories to a student of functional
> analysis? I would like to give him a class project that he will find
> close to his interests. The course is covering (roughly) the following
> material: basic category theory (limits, colimits, adjoints, we
> mentioned additive and enriched categories), Lawvere's algebraic
> categories, monads (up to stating Beck's theorem and working out some
> examples), basics of presheaves and sheaves with a slant toward
> topology. There must be some functional analysis in there.
>
> I would very much appreciate some suggestions.
>
> Best regards,
>
> Andrej
>
>
From rrosebru@mta.ca Thu Mar 6 14:07:08 2008 -0400
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From: Dan Christensen
To: Categories list
Subject: categories: Re: graphics and dvi
Date: Thu, 06 Mar 2008 12:41:30 -0500
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[Note from moderator: Discussion of TeX-nicalities is admittedly
off-topic, but the last paragraph below is of wide interest.]
Michael Barr writes:
> If you read the TAC instructions for authors, you will find that we
> discourage the use of the graphics package unless absolutely necessary
> because, for the time being, dvi is still our basic archive format and
> graphics specials were not rendered properly by dvi viewers.
I'm curious what you mean by this. One of my duties as Managing Editor
of Homology, Homotopy and Applications is to oversee the copyediting and
typesetting, and I produce the final versions of all files in dvi, ps
and pdf format on my linux machine. We receive files using quite a
variety of graphics packages, including the "graphics" package, and
rarely have any problems with the resulting dvi files. (I test using
xdvi.) The only problem I can recall is with figures that are rotated
90 degrees, and for such papers we simply don't make the dvi file
publicly available.
> Neither pdflatex nor dvipdfm renders graphics specials correctly.
The pdf files for HHA are almost always produced using dvipdfm, and
again this works quite reliably in my experience. In some cases, we use
pdflatex, and again I have had no trouble with it. dvipdfm works even
in the one or two cases where xdvi didn't display a file correctly,
such as with rotated figures.
And while for most files I regard dvi as the primary processed archive,
I also use the snapshot package to save most of the .sty files each
article includes, so that if necessary in the future the articles can be
reprocessed to produce some hypothetical new format that contains
information not in the dvi file. (E.g. to add hyperlinks to all
articles.)
While I am writing, I'd like to encourage readers of this list to submit
articles to HHA and to ask their libraries to subscribe if they don't
already. The price is quite reasonable, and we regularly publish
articles with a categorical bent. All articles are available online at
http://intlpress.com/HHA
You can receive announcements of new articles by writing directly to me
and asking to be put on the announcements mailing list.
Best wishes,
Dan
From rrosebru@mta.ca Thu Mar 6 22:43:59 2008 -0400
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for categories-list@mta.ca; Thu, 06 Mar 2008 22:36:01 -0400
Date: Thu, 6 Mar 2008 16:01:33 +0000 (GMT)
From: Paul B Levy
To: categories@mta.ca
Subject: categories: Re: Re: Heyting algebras and Wikipedia
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> While I can't speak for Steve today, this remains a stumbling block for
> those raised to believe that rigorous mathematics would not be possible
> in a world where propositions such as "for all x and y there exists z
> such that x is a subset of z and y is a member of z" did not hold. How
> could x U {y} fail to exist and the walls of mathematics not come
> tumbling down?
Maybe not the walls of mathematics, but what about theorems like "every
polynomial functor on Set has a unique initial algebra whose structure map
is an identity"? I think theorems like this are worth retaining (and
antifoundation makes even more of them).
I'd also like to suggest that "foundations" is being used in two very
different senses. In FoM, it's about quantifying the philosophical risks
involved in particular formal systems and proofs, i.e. issues such as
relative consistency, omega-consistency, etc. For this purpose the
primacy of membership vs composition is quite immaterial. One could, I
suppose, make a formal theory based on composition equal in strength (in
whatever sense) to ZF.
Category theory on the other hand is about fundamental algebraic
structures. I don't think it makes sense to ask "is category theory
omega-consistent?" as one can for ZF (not that anyone knows the answer).
Paul
>
> Vaughan
>
> Colin McLarty wrote:
>> Vaughan Pratt
>> Wednesday, March 5, 2008 8:32 am
>>
>> wrote, with much else:
>>
>>> On a related note, a careful reading of Max Kelly's "Basic Concepts of
>>> Enriched Category Theory" reveals that it is thoroughly grounded in
>>> Set,as I pointed out in August 2006 in my initial Wikipedia article
>>> on Max.
>>> I gave some thought to how one might eliminate Set from the
>>> treatment,without much success, and concluded that Max's judgment
>>> there was spot on.
>>
>> Without addressing this particular issue I want to say I appreciate the
>> phrase in the article: "the explicitly foundational role of the category
>> Set." I take it this is Vaughan's?
>>
>> Various people including Sol Feferman promote the view that if you use
>> "sets" then you are admitting that you use ZF and not some categorical
>> foundations. Vaughan's phrase goes aptly against that: If you use
>> sets, then you use sets, but there is no reason it cannot be on
>> categorical foundations. He does not say it *is* on categorical
>> foundations, and that is fine in the context. He reminds people that it
>> *could* be.
>>
>> best, Colin
>>
>>
>
>
>
--
Paul Blain Levy email: pbl@cs.bham.ac.uk
School of Computer Science, University of Birmingham
Birmingham B15 2TT, U.K. tel: +44 121-414-4792
http://www.cs.bham.ac.uk/~pbl
From rrosebru@mta.ca Thu Mar 6 22:43:59 2008 -0400
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From:
To: "Categories list"
Subject: categories: Re: How to motivate a student of functional analysis
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Ronnie points out the very excellent 1997
book on smooth analysis by Kriegl & Michor.
In fact, not the reviewer but the authors themselves=20
originally stated the principle of=20
functional analysis "rather than" category theory.
It is rather strange since much of the=20
material in the book was arrived at
by very categorical means. For example,
results published in Kriegl's joint work
with Alfred Frolicher are basic. My dismay
is reflected in my RCMP paper on Volterra,
where I praise the book for its powerful=20
combination of
functional analysis "and" category theory.
In a related expositional choice the book
claims to be about topological vector spaces,
but the definition of morphism used betrays the fact
that the weaker structures of bounded=20
sequences and of C-infinity paths are the
actual underpinning.
It would be instructive to know whether this
strategy actually widened the audience in the=20
past 10 years.
Bill
On Thu Mar 6 10:15 , "Ronnie" sent:
>You could also look at
>MR1471480 (98i:58015)
>Kriegl, Andreas; Michor, Peter W.
>The convenient setting of global analysis. (English summary)
>Mathematical Surveys and Monographs, 53. American Mathematical Society,
>Providence, RI, 1997. x+618 pp. ISBN: 0-8218-0780-3
>
>for which an e-version has been downloadable. However as the review says:
>"the exposition is based on functional analysis rather than on category
>theory; this fact will, undoubtedly, allow the subject to reach a wider
>audience. "
>
>
>
>
>Ronnie
>
>
>----- Original Message -----
>From: "Robert L Knighten" RLK@knighten.org>
>To: "Categories list" categories@mta.ca>
>Sent: Thursday, March 06, 2008 2:37 AM
>Subject: categories: How to motivate a student of functional analysis
>
>
>> Most of the material connecting analysis and category theory seems to be
>> written by specialists in category theory who have observed some of the
>> ways
>> that insights from category theory can be brought to bear (for example
>> look on
>> Math Sci Net at the many papers (co)authored by Joan Wick Pelletier.) B=
ut
>> here's an example which is a book on functional analysis that has a stro=
ng
>> use
>> of categories:
>>
>>
>> @book {MR0296671,
>> AUTHOR =3D {Semadeni, Zbigniew},
>> TITLE =3D {Banach spaces of continuous functions. {V}ol. {I}},
>> NOTE =3D {Monografie Matematyczne, Tom 55},
>> PUBLISHER =3D {PWN---Polish Scientific Publishers},
>> ADDRESS =3D {Warsaw},
>> YEAR =3D {1971},
>> PAGES =3D {584 pp. (errata insert)},
>> MRCLASS =3D {46E15 (46M99)},
>> MRNUMBER =3D {MR0296671 (45 \#5730)},
>> MRREVIEWER =3D {H. E. Lacey},
>> }
>>
>> -- Bob
>>
>> --
>> Robert L. Knighten
>> RLK@knighten.org
>>
>>
>>
>>
>> --
>> No virus found in this incoming message.
>> Checked by AVG Free Edition.
>> Version: 7.5.516 / Virus Database: 269.21.4/1312 - Release Date:
>> 04/03/2008 21:46
>>
>
>
>
>
From rrosebru@mta.ca Fri Mar 7 15:28:07 2008 -0400
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for categories-list@mta.ca; Fri, 07 Mar 2008 15:20:36 -0400
Date: Thu, 06 Mar 2008 23:59:05 -0500
From: "Fred E.J. Linton"
To: Categories list
Subject: categories: Re: Heyting algebras and Wikipedia
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Vaughan has written, in part, that he would =
> happen to find Excluded Middle more convenient than its negation, but
> that's just me and perhaps others have had the opposite experience.
Me too, for the most part(*), but as regards
=
> Then there are those who accept neither Excluded Middle nor its
> negation, which takes us into the Hall of Mirrors that I always find
> myself in when I go down this particular rabbit-hole.
I find that it's NOT the case that I "accept neither" -- rather,
it's that I sometimes prefer neither to accept it, nor to reject it,
but to remain uncommitted.
Noncommittally yours,
-- Fred
(*) I'm reminded of the legendary airline passenger who, faced with
the stewardess's classic offer of "coffee, tea, or me," countered with: =
"Any chance of some tonic water instead, please?" -- F.
From rrosebru@mta.ca Fri Mar 7 15:28:07 2008 -0400
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for categories-list@mta.ca; Fri, 07 Mar 2008 15:18:06 -0400
From: Bas Spitters
Subject: categories: Re: How to motivate a student of functional analysis
Date: Fri, 7 Mar 2008 09:57:09 +0100
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Dear Andrej,
Two examples that have not been mentioned before:
* The use of the Giry monad in stochastic processes. This should motivate CS
students as well as (functional) analysists. You can also use co-algebras
here.
* Gelfand's theorem: commutative C*-algebras are precisely the complex numbers
in the topos of sheaves over its spectrum. This will also teach them that the
axiom of choice is almost never needed in functional analysis and that there
are good reasons to avoid it: E.g. continuous fields of C*-algebras.
This is the fundamental work by Banaschewski and Mulvey.
Bas
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From: "Ronnie"
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Subject: categories: Re: How to motivate a student of functional analysis
Date: Fri, 7 Mar 2008 10:31:18 -0000
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Bill is quite right on what the author's say. I'd also be glad of any of
Bill's comments on the `Historical remarks on the development of smooth
calculus', pp, 79-83, which seem very carefully put.
It might interest people to give what seems the origin of the word
`convenient category'. In my 1963 paper `Ten topologies for X x Y' (a title
frivolously influenced by `Seven brides for seven brothers') I wrote in the
Introduction:
`It may be that the category of Hausdorff k-spaces is adequate and
convenient for all purposes of topology'.
`Convenient' here meant cartesian closed. One of the above ten topologies
gives monoidal closed on all Hausdorff spaces. Some later writers removed
the Hausdorff restrictions (I tried, but I was at that time not too good on
final topologies). The current acount in `Topology and Groupoids' was
influenced by Eldon Dyer.
But for analysis the Kriegl-Michor comments show how the emphasis moved from
k-spaces to ideas from Frohlicher, Bill and others.
Ronnie
----- Original Message -----
From:
To: "Categories list"
Sent: Thursday, March 06, 2008 8:30 PM
Subject: categories: Re: How to motivate a student of functional analysis
Ronnie points out the very excellent 1997
book on smooth analysis by Kriegl & Michor.
In fact, not the reviewer but the authors themselves
originally stated the principle of
functional analysis "rather than" category theory.
It is rather strange since much of the
material in the book was arrived at
by very categorical means. For example,
results published in Kriegl's joint work
with Alfred Frolicher are basic. My dismay
is reflected in my RCMP paper on Volterra,
where I praise the book for its powerful
combination of
functional analysis "and" category theory.
In a related expositional choice the book
claims to be about topological vector spaces,
but the definition of morphism used betrays the fact
that the weaker structures of bounded
sequences and of C-infinity paths are the
actual underpinning.
It would be instructive to know whether this
strategy actually widened the audience in the
past 10 years.
Bill
On Thu Mar 6 10:15 , "Ronnie" sent:
>You could also look at
>MR1471480 (98i:58015)
>Kriegl, Andreas; Michor, Peter W.
>The convenient setting of global analysis. (English summary)
>Mathematical Surveys and Monographs, 53. American Mathematical Society,
>Providence, RI, 1997. x+618 pp. ISBN: 0-8218-0780-3
>
>for which an e-version has been downloadable. However as the review says:
>"the exposition is based on functional analysis rather than on category
>theory; this fact will, undoubtedly, allow the subject to reach a wider
>audience. "
>
>
>
>
>Ronnie
>
>
>----- Original Message -----
>From: "Robert L Knighten" RLK@knighten.org>
>To: "Categories list" categories@mta.ca>
>Sent: Thursday, March 06, 2008 2:37 AM
>Subject: categories: How to motivate a student of functional analysis
>
>
>> Most of the material connecting analysis and category theory seems to be
>> written by specialists in category theory who have observed some of the
>> ways
>> that insights from category theory can be brought to bear (for example
>> look on
>> Math Sci Net at the many papers (co)authored by Joan Wick Pelletier.)
>> But
>> here's an example which is a book on functional analysis that has a
>> strong
>> use
>> of categories:
>>
>>
>> @book {MR0296671,
>> AUTHOR = {Semadeni, Zbigniew},
>> TITLE = {Banach spaces of continuous functions. {V}ol. {I}},
>> NOTE = {Monografie Matematyczne, Tom 55},
>> PUBLISHER = {PWN---Polish Scientific Publishers},
>> ADDRESS = {Warsaw},
>> YEAR = {1971},
>> PAGES = {584 pp. (errata insert)},
>> MRCLASS = {46E15 (46M99)},
>> MRNUMBER = {MR0296671 (45 \#5730)},
>> MRREVIEWER = {H. E. Lacey},
>> }
>>
>> -- Bob
>>
>> --
>> Robert L. Knighten
>> RLK@knighten.org
>>
>>
>>
>>
>> --
>> No virus found in this incoming message.
>> Checked by AVG Free Edition.
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>
>
>
>
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From rrosebru@mta.ca Fri Mar 7 15:28:08 2008 -0400
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From: Thorsten Altenkirch
Subject: categories: Re: Heyting algebras and Wikipedia
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Hi Paul,
>> While I can't speak for Steve today, this remains a stumbling
>> block for
>> those raised to believe that rigorous mathematics would not be
>> possible
>> in a world where propositions such as "for all x and y there exists z
>> such that x is a subset of z and y is a member of z" did not
>> hold. How
>> could x U {y} fail to exist and the walls of mathematics not come
>> tumbling down?
>
> Maybe not the walls of mathematics, but what about theorems like
> "every
> polynomial functor on Set has a unique initial algebra whose
> structure map
> is an identity"? I think theorems like this are worth retaining (and
> antifoundation makes even more of them).
If we leave out "the structure map is the identity", I have no
problem. The 2nd part seems to be rather cosmetical anyway, but in a
bad sense of hiding the structure. Yes, I know you save some ink...
>
> I'd also like to suggest that "foundations" is being used in two very
> different senses. In FoM, it's about quantifying the philosophical
> risks
> involved in particular formal systems and proofs, i.e. issues such as
> relative consistency, omega-consistency, etc. For this purpose the
> primacy of membership vs composition is quite immaterial. One could, I
> suppose, make a formal theory based on composition equal in
> strength (in
> whatever sense) to ZF.
>
Exactly, the discussion is similar to the question whether the carta
of human rights should be written in English or French. I don't care
whether foundations are expressed in the language of predicate logic,
category theory or type theory as long as they make sense (to me).
Having said this I prefer the latter two, which work very well
together, but this again has to do with beauty as opposed to cosmetics.
> Category theory on the other hand is about fundamental algebraic
> structures. I don't think it makes sense to ask "is category theory
> omega-consistent?" as one can for ZF (not that anyone knows the
> answer).
Precisely!
Cheers,
Thorsten
>
> Paul
>
>
>
>>
>> Vaughan
>>
>> Colin McLarty wrote:
>>> Vaughan Pratt
>>> Wednesday, March 5, 2008 8:32 am
>>>
>>> wrote, with much else:
>>>
>>>> On a related note, a careful reading of Max Kelly's "Basic
>>>> Concepts of
>>>> Enriched Category Theory" reveals that it is thoroughly grounded in
>>>> Set,as I pointed out in August 2006 in my initial Wikipedia article
>>>> on Max.
>>>> I gave some thought to how one might eliminate Set from the
>>>> treatment,without much success, and concluded that Max's judgment
>>>> there was spot on.
>>>
>>> Without addressing this particular issue I want to say I
>>> appreciate the
>>> phrase in the article: "the explicitly foundational role of the
>>> category
>>> Set." I take it this is Vaughan's?
>>>
>>> Various people including Sol Feferman promote the view that if
>>> you use
>>> "sets" then you are admitting that you use ZF and not some
>>> categorical
>>> foundations. Vaughan's phrase goes aptly against that: If you use
>>> sets, then you use sets, but there is no reason it cannot be on
>>> categorical foundations. He does not say it *is* on categorical
>>> foundations, and that is fine in the context. He reminds people
>>> that it
>>> *could* be.
>>>
>>> best, Colin
>>>
>>>
>>
>>
>>
>
> --
> Paul Blain Levy email: pbl@cs.bham.ac.uk
> School of Computer Science, University of Birmingham
> Birmingham B15 2TT, U.K. tel: +44 121-414-4792
> http://www.cs.bham.ac.uk/~pbl
>
>
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From rrosebru@mta.ca Fri Mar 7 15:28:08 2008 -0400
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Date: Thu, 6 Mar 2008 22:18:05 -0600
From: "Michael Shulman"
Subject: categories: Re: replacing set theory
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On Thu, Mar 6, 2008 at 10:01 AM, Paul B Levy wrote:
> Maybe not the walls of mathematics, but what about theorems like "every
> polynomial functor on Set has a unique initial algebra whose structure map
> is an identity"? I think theorems like this are worth retaining (and
> antifoundation makes even more of them).
I'm not familiar with that particular result, but I know other
categorical proofs which use set-theoretic ideas like transfinite
induction, and so cannot be detached from ZF in an obvious way. On
the other hand, there is nothing intrinsically "membership-based" in
transfinite induction. The problem seems to be the lack of a
categorical analogue of ZF's axiom of replacement, since the sets in
V_{\omega+\omega} already form a well-pointed elementary topos with a
NNO. I find this especially mysterious because on the surface,
replacement merely replaces a set by an isomorphic one (or at most a
quotient)!
One categorical analogue of replacement comes from categories of classes in
algebraic set theory. That is, we move from a categorical analogue of ZF
to an analogue of Godel-Bernays set theory. But it seems natural to wonder
whether there could be a categorical analogue of replacement expressible
solely as a property of the category Set, without reference to how it sits
in a category of classes. Has anyone studied this question?
Mike
From rrosebru@mta.ca Fri Mar 7 15:28:08 2008 -0400
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Date: Thu, 6 Mar 2008 23:04:19 -0800
From: Toby Bartels
To: categories@mta.ca
Subject: categories: Categorial foundations (Was: Heyting algebras and Wikipedia)
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Paul B Levy wrote in part:
>what about theorems like "every
>polynomial functor on Set has a unique initial algebra whose structure map
>is an identity"? I think theorems like this are worth retaining (and
>antifoundation makes even more of them).
This is so fundamental that I'm inclined to make it an axiom.
(Well, we can leave uniquiness --up to isomorphism, you mean--
and the invertibility of the structure map for theorems.)
This is essentially an axiom of the Calculus of Inductive Constructions,
which (like most modern type theory) is easily put in categorial language.
>I'd also like to suggest that "foundations" is being used in two very
>different senses. In FoM, it's about quantifying the philosophical risks
>involved in particular formal systems and proofs, i.e. issues such as
>relative consistency, omega-consistency, etc. For this purpose the
>primacy of membership vs composition is quite immaterial.
All the same, I find these matters much easier to understand
when I think about them in terms of categories of sets,
rather than in terms of (models of a) membership-based set theory.
I would be able to read FoM if it weren't so hostile to this
(although I'll follow Vaughn in noting that I haven't looked lately,
so I can't speak for what it's like now).
>Category theory on the other hand is about fundamental algebraic
>structures. I don't think it makes sense to ask "is category theory
>omega-consistent?" as one can for ZF (not that anyone knows the answer).
No, but one can ask of a topos with a natural-numbers object N
(and satisfying other properties that match various axioms of ZF),
given a morphism X -> N whose pullbacks 0, 1, 2, ...: 1 -> N
are all occupied (so each pullback has a morphism from 1),
whether the negation of X over N (the internal hom [0, X]
taken in the slice category over N) can also be occupied.
--Toby
From rrosebru@mta.ca Fri Mar 7 17:09:43 2008 -0400
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Date: Fri, 7 Mar 2008 11:07:59 -0800
From: Toby Bartels
To: categories@mta.ca
Subject: categories: Re: Categorial foundations
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I wrote in part:
>[...] ask of a topos with [...]
>whether the negation of X over N (the internal hom [0, X]
>taken in the slice category over N) can also be occupied.
I have one and a half things backwards here.
First of all, of course negation is [X, 0] rather than [0, X].
(But in exponential notation, it is 0^X; that is my excuse.)
Also, my placement of "can" implies that the relevant question
is whether there ~exists~ a topos E (with given properties)
and there exists an object X in E (with the properties that I described);
rather, the question is whether for ~every~ E (with given properties)
there exists an object X in E (with the properties that I described).
Iff so, then the properties required of E are omega-inconsistent.
(Iff E must be a terminal category, then they are simply inconsistent.
Thus omega-inconsistency is weaker than inconsistency,
and omega-consistency is stronger than mere consistency.)
--Toby
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Date: Fri, 7 Mar 2008 11:37:15 -0800
From: Toby Bartels
To: Categories list
Subject: categories: Re: How to motivate a student of functional analysis
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Bill Lawvere wrote in part:
>Students might wonder why contiuous linear operators
>are traditionally called "bounded" (when they are not even).
Then Jeff Egger wrote in part:
>there's certainly alot that can be said about the
>category of Banach spaces and linear contractions
and:
>forgetful functors don't have to be "the obvious thing";
Indeed, the "obvious" forgetful functor from Ban to Set (or Top or Met)
takes a Banach space to its space of all points,
while the "good" one takes the space to its unit ball.
Anyway, if you mix these, then a linear transformation is bounded
iff it is bounded as a function from the unit ball to the space of all points.
Similarly for compact linear transformations (the image is compact).
That may not be the origin of these terms, but it's how I understand them.
More related to category theory itself:
Jeff also wrote:
>in category theory, the meaning of isomorphism is fixed
I'd say that the meaning of a term like "Banach space"
necessarily includes the idea of what an isomorphism of such is.
If different definitions define equivalent groupoids
(or equivalent omega-groupoids in the most general case,
as with different definitions of n-category, for example),
then we can consider them equivalent defintions.
So to define the essence of what Banach spaces are,
one must specify (up to equivalence) the groupoid Ban_0
of Banach spaces and linear isometries between them.
That said, there is some sense in the category Ban_b
of Banach spaces and bounded linear transformations between them,
but it is only a secondary notion compared to Ban_0.
To be useful at all, it needs some extra structure,
such as (at least) the dagger operator (giving duals of morphisms);
then the actual isomorphisms of Banach spaces (those in Ban_0)
are only the ~unitary~ (dual = inverse) isomorphisms in Ban_b.
(In contrast, the category Ban as Jeff defined it
needs no extra structure to be a sensible concept,
since all of its isomorphisms are in Ban_0 already.)
This dagger operator is used, for example, to make Hilb_b
(the full subcategory of Ban_b whose objects are Hilbert spaces)
into a 2-Hilbert space (from John Baez's HDA4),
which is useful if you want examples of 2-Hilbert spaces;
but the ~essence~ of what Hilbert spaces are
is given by the groupoid Hilb_0 of linear isometries.
So here is another lesson of category theory,
to be taken together with Jeff's lesson last quoted above:
Sometimes different notions of morphism are useful for different purposes.
--Toby
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From: Steve Awodey
Subject: categories: Re: replacing set theory
Date: Fri, 7 Mar 2008 21:52:39 +0100
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On Mar 7, 2008, at 5:18 AM, Michael Shulman wrote:
>
>
> One categorical analogue of replacement comes from categories of
> classes in
> algebraic set theory. That is, we move from a categorical analogue
> of ZF
> to an analogue of Godel-Bernays set theory. But it seems natural
> to wonder
> whether there could be a categorical analogue of replacement
> expressible
> solely as a property of the category Set, without reference to how
> it sits
> in a category of classes. Has anyone studied this question?
yes: Carsten Butz, Thomas Streicher, Alex Simpson and I did.
See the first two items under 2007 on the AST site:
http://www.phil.cmu.edu/projects/ast/
The short answer is, it depends on how "Sets" sits in the category of
classes.
In fact, *any* topos can occur as a category of "Sets" satisfying
replacement in a suitable category of classes constructed from the
topos.
Steve Awodey
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Date: Fri, 7 Mar 2008 14:39:24 -0800
From: Toby Bartels
To: categories@mta.ca
Subject: categories: Re: Categorial foundations
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I wrote in part:
>given a morphism X -> N whose pullbacks 0, 1, 2, ...: 1 -> N
>are all occupied [...]
Another typo; the word "along" is missing; it should be
>given a morphism X -> N whose pullbacks along 0, 1, 2, ...: 1 -> N
>are all occupied [...]
So:
Given a collection of conditions on a locally cartesian-closed category E
with an initial object 0, a final object 1, and a natural-numbers object N
(various refinements should be possible for more general theories),
these conditions are _omega-inconsistent_ if in every such E
there exists an object X and a morphism p: X -> N such that:
* defining the numerals [i]: 1 -> N using the stucture maps of N
(so [0]: 1 -> N, [1]: 1 -> N -> N, [2]: 1 -> N -> N -> N, etc)
and letting X_i be the pullback of p: X -> N and [i]: 1 -> N,
each X_i has a morphism a_i: 1 -> X_i;
* letting [X,0]_N be the internal hom from X to 0 in the slice category E/N,
there is (in E itself) a morphism b: 1 -> [X,0]_N.
I've read this 5 times, in different orders, so there should be no mistakes.
I apologise for any confusion from my abbreviations and corrections.
--Toby
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Date: Fri, 7 Mar 2008 17:51:50 -0600
From: "Michael Shulman"
Subject: categories: Re: replacing set theory
To: categories@mta.ca
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On Fri, Mar 7, 2008 at 2:52 PM, Steve Awodey wrote:
> http://www.phil.cmu.edu/projects/ast/
>
> The short answer is, it depends on how "Sets" sits in the category
> of classes.
> In fact, *any* topos can occur as a category of "Sets"
> satisfying replacement in a suitable category of classes constructed
> from the topos.
Very interesting! But I don't think that is the answer to the
question I intended to ask, although perhaps I phrased the question
poorly. As far as I can tell, you give a way of interpreting
replacement/collection in such a way that it is satisfied in all
toposes, by "constructivizing" the existential quantifier. But as you
say, "In consequence, the standard arguments using Replacement that
take one outside of V_\lambda(A) for \lambda non-inaccessible, are not
reproducible." What I would really like to know is, can one formulate
an elementary property of a topos which *does* allow one to reproduce
the standard arguments of Replacement?
Here's another way to phrase the same (or a similar) question.
Suppose I meet a mathematician who thinks categorically enough to
dislike the membership-based nature of ZF(C), but doesn't want to give
up any of its consequences. In particular, he wants to be able to use
transfinite induction beyond \omega+\omega. For instance, he wants
Borel determinacy to be true, which is provable in ZFC but not in
Zermelo set theory (ZFC minus Replacement). Is there a categorical
foundation I can tell him to use? That is, is there an elementary
categorical theory which is as strong as ZF(C)?
Mike
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Paul Levy wrote:
> Maybe not the walls of mathematics, but what about theorems like "every
> polynomial functor on Set has a unique initial algebra whose
> structure map is an identity"? I think theorems like this are worth retaining (and
> antifoundation makes even more of them).
I'm not sure what "retaining" means here. Does the category Set, even
as a cartesian category, have *any* properties that are open to debate?
(Other than by intuitionists, who seem to thrive on quicksand.) For
Set as a cartesian closed category I can see room for debate about the
number of nonisomorphic sets that can appear along a chain of monics
from X to 2^X when X is infinite (defined say as admitting endomonics
that are not automorphisms), but relatively little in practical
mathematics seems to hinge on the outcome.
Correct me if I'm wrong, but my impression is that for any given
language in which to express properties of Set, whether that of
categories, cartesian categories, cartesian closed categories, or
toposes, the properties of Set, understood classically and up to
equivalence, are essentially fixed modulo largely irrelevant minutiae
such as the above.
Your example is a perfectly identifiable property of any category with
polynomial functors (suitably defined) such as Set. If the polynomial
functors are those generated from the identity functor by binary product
and coproduct, e.g. X, X+X, X^2, X+X^2, etc. then it holds of Set
because then the initial algebra is always the empty set (but you
probably had the empty product 1 in mind as well). If Set didn't have
that property it wouldn't be Set, just as Z wouldn't be Z if integer
addition wasn't commutative.
The property P = "for all objects x and y there exists a set z for which
x is a subset of z and y is a member of z" holds of all models of ZF.
It cannot be said to hold of the category Set however, not because we
can't prove it, i.e. can't imagine how assuming it false could do any
harm, but because we can't define it, i.e. can't imagine how it could be
either true or false. What does it even mean when applied to Set as a
category, or as a cartesian closed category, or even as a topos? More
structure than that has to be added to Set to make P meaningful.
The same goes for AFA. Chapters 1-6 of Aczel are developed starting
within the ZF framework. Categories enter at Chapter 7, but Set is
already fully encumbered at that point with all the machinery necessary
to interpret all sentences of the language of ZF, where P is true. In
that sense even FA (the Foundation Axiom) creates properties of Set that
are not meaningful for Set as a mere topos.
AFA as a weakening of FA means that generically there are fewer
properties than with FA, not more. Fixing a particular model of AFA
creates properties specific to the model, which may or may not
contradict FA. (Every model of ZF is automatically a model of ZF-FA+AFA.)
Vaughan
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From: Thomas Streicher
Subject: categories: Re: replacing set theory
To: categories@mta.ca
Date: Sat, 8 Mar 2008 03:42:09 +0100 (CET)
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Augmenting Steve Awodey's reply to M. Shulman I want to mention
a further possibility which is more in the spirit of type / category
theory, namely that of universes in toposes as described in my
article with the same title (available under
(www.mathematik.tu-darmstadt.de/~streicher/NOTES/UniTop.ps.gz).
It is essentially a catgorical variant of Martin-Loef's notion
of universe albeit an impredicative one. It was used a lot
in categorical semantics of type theory (starting ~1985)
but certainly part of the categorical folklore. The first
written account I know of is Jean B'enabou's "Probl`emes
dans le topos" from 1973. His main example that time was
decidable K-finite objects in a topos with nno.
A universe in a topos EE is a pullback stable class SS
of morphism admitting a generic element in SS, i.e. a
map E -> U in SS from which all other maps in SS can be obtained
via pullback. Replacement is modelled by the requirement that
SS be closed under composition. Of course, one usually requires
more further closure properties (as in type theory).
In the above mentioned paper I have shown that all Grothendieck
and realizability toposes admit such universes (exploiting
Grothendieck universes on the meta-level).
As far as I can see universes serve well the purpose of replacement
in mathematics, namely defining families of types by recursion.
They achieve this goal in a more direct way than replacement does.
The reason why they are presumably weaker than the setting Steve
mentioned is that one needs a type-theoretic collection axiom
(as in Joyal and Moerdijk's "Algebraic Set Theory") besides W-types
for constructing set theoretic universes from type theoretic ones.
I don't know why universes have hardly been considered in topos theory.
(One notable exception being B'enabou's calibrations giving notions of
size when considering locally small fibrations.) I think they are most
useful and actually indispensible for doing category theory in the
internal language of a topos.
Algebraic Set Theory is an instance of universes, namely universes
within categories modelling first order logic.
Thomas
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From: Steve Awodey
Subject: categories: Re: replacing set theory
Date: Sat, 8 Mar 2008 01:09:25 +0100
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On Mar 8, 2008, at 12:51 AM, Michael Shulman wrote:
> On Fri, Mar 7, 2008 at 2:52 PM, Steve Awodey wrote:
>> http://www.phil.cmu.edu/projects/ast/
>>
>> The short answer is, it depends on how "Sets" sits in the category
>> of classes.
>> In fact, *any* topos can occur as a category of "Sets"
>> satisfying replacement in a suitable category of classes constructed
>> from the topos.
>
> Very interesting! But I don't think that is the answer to the
> question I intended to ask, although perhaps I phrased the question
> poorly. As far as I can tell, you give a way of interpreting
> replacement/collection in such a way that it is satisfied in all
> toposes, by "constructivizing" the existential quantifier.
no, the existential quantifier has its standard (categorical)
interpretation (direct image), not the "constructive" one from type
theory. We do not reinterpret replacement/collection either -- they
have their usual interpretation.
What is a bit delicate is the background category of classes in which
the (set-theoretically) unbounded quantifiers are interpreted.
> But as you
> say, "In consequence, the standard arguments using Replacement that
> take one outside of V_\lambda(A) for \lambda non-inaccessible, are not
> reproducible." What I would really like to know is, can one formulate
> an elementary property of a topos which *does* allow one to reproduce
> the standard arguments of Replacement?
>
> Here's another way to phrase the same (or a similar) question.
> Suppose I meet a mathematician who thinks categorically enough to
> dislike the membership-based nature of ZF(C), but doesn't want to give
> up any of its consequences. In particular, he wants to be able to use
> transfinite induction beyond \omega+\omega. For instance, he wants
> Borel determinacy to be true, which is provable in ZFC but not in
> Zermelo set theory (ZFC minus Replacement). Is there a categorical
> foundation I can tell him to use? That is, is there an elementary
> categorical theory which is as strong as ZF(C)?
>
AST?
Steve
> Mike
>
From rrosebru@mta.ca Sat Mar 8 19:27:10 2008 -0400
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