How does category theory help?
Charles Francis
>I have my ideas about what some of these concepts are like. I think
>n-categories are going to play a crucial role, because n-categories
>unify our ideas of space and time (i.e., manifolds and cobordisms
>between them) with our ideas of state and process (i.e., Hilbert
>spaces and operators between them). I tried my best to explain this
>in nontechnical terms here:
>
>http://math.ucr.edu/home/baez/planck/planck.html
I just don't see how category theory helps. If it were to describe a
morphism between a mathematical object and an ontology it could do some
good. But it is part of mathematics and really only describes morphisms
between mathematical objects. Try to open that up and get a general
definition then you will hit Russell's paradox, just as you do in set
theory. And then again if it only describes morphisms between
mathematical objects it is subsumed by set theory, and not the other way
around, so I do not see the grandiose claims that it lies at the
foundation of mathematics.
--
Regards
Charles Francis
cha...@clef.demon.co.uk
Chris Hillman
On Tue, 9 Jan 2001, Charles Francis wrote:
> I just don't see how category theory helps. If it were to describe a
> morphism between a mathematical object and an ontology it could do
> some good. But it is part of mathematics and really only describes
> morphisms between mathematical objects. Try to open that up and get a
> general definition then you will hit Russell's paradox, just as you do
> in set theory. And then again if it only describes morphisms between
> mathematical objects it is subsumed by set theory, and not the other
> way around, so I do not see the grandiose claims that it lies at the
> foundation of mathematics.
Well, these "grandiose claims" are in fact -very- well established and
moreover -very- "well known" (in the sense that every graduate student in
some specialities has been required to learn them, since the seventies---
in another sense, they are not at all well known to large groups of folk
who don't yet know why they want to learn this stuff, as I partially
explain below!). Consequently, while this is not particularly easy stuff
to learn, particularly if one has absolutely no background in logic/set
theory in general, there are -lots- of books and papers out there to help
beginning students:
@book{ls:cm,
author = {F. William Lawvere and Stephen H. Schanuel},
title = {Conceptual mathematics : a first introduction to categories},
publisher = {Cambridge University Press},
year = 1997}
(Very easy-going introduction to categorical thinking.)
@book{m:ecet,
author = {Colin McLarty},
title = {Elementary Categories, Elementary Toposes},
publisher = {Oxford Science Publications},
year = 1995}
(More demanding but covers more territory, up through one approach to the
relationship between a topos and logic and "set theory".)
@book{g:t,
author = {Robert Goldblatt},
title = {Topoi: the Categorial Analysis of Logic},
publisher = {North-Holland},
year = 1979}
(A bit muddled in places, but very detailed and readable, at about the
same level.)
@book{bw:ttt,
author = {Michael Barr and Charles Wells},
title = {Toposes, Theories, and Triples},
publisher = {Springer-Verlag},
year = 1985}
More about topoi and theories (in the sense of logic):
@book{b:tlst,
author = {J. L. Bell},
title = {Toposes and Local Set Theories},
publisher = {Clarendon Press},
year = 1988}
@book{mm:sgl,
author = {Saunders Mac Lane and Ieke Moerdijk},
title = {Sheaves in Geometry and Logic: a First Introduction to Topos
Theory},
publisher = {Springer-Verlag},
year = 1992}
You can also try these short lecture notes:
@unpublished{o:bct,
author = {Jaap van Oosten},
title = {Basic Category Theory},
series = {Basic Research in Computer Science Lecture Series},
note = {http://www.brics.aau.dk/BRICS/LS/95/1/},
month = {January},
year = 1995}
My own expository paper, "A Categorical Primer", currently under
revision/expansion, but available in truncated form on my web pages, is
much more concise but will attempt to cover correspondingly more ground
wrt logic and topos theory, bringing the story up to the stunning
unifications achieved by Grothendieck in the seventies.
(Caveat emptor!-- I am entirely self-taught in this area, this paper has
not been refereed, and indeed there is a metaexercise in the current
version: fix the numerous places where the compositions are written the
wrong way, or where I just plain had a brain blooper, or committed a more
serious conceptual error, and then please email me with your corrections!)
One way to understand where I am heading with the revised version is to
observe that the "local/global" and "internal/external" distinctions, as
well as the notion of "formal analogies", "abstract structure" and "levels
of structure", have been absolutely fundamental in math since the
nineteenth century. The notions of topoi, sites, sieves, and sheaves were
developed expressly to organize the manifold applications of these notions
in a simple and coherent way (no pun intended). Furthermore, it turns out
that logic, topology, algebraic geometry, and dynamics are very intimately
related in a deep way, as Grothendieck discovered in the seventies. His
ideas have been well known for decades to computer scientists, logicians,
algebraic geometers, number theorists, and some topologists, but
unfortunately have yet to "trickle down" ["well up"?] to dynamicists,
theoretical physicicsts, or to the "average mathematician". This is very
unfortunate, since these ideas are among the most useful/powerful ideas
developed in twentieth century mathematics, and IMO surely destined to
play an increasingly important role in the new century, in both math and
physics.
My growing recognition of the fundamental importance of his ideas have
provided, in fact, the major impetus to try to expand the original version
to cover the bare minimum of what I think every mathematician and
theoretical physicist should know about this stuff, and the belated
discovery that ideas I was kicking around as a graduate student (motivated
in part by the recognition that number theory and dynamical systems seem
poised for a unification, c.f. recent papers by Eskin, Kleinbock, and
Margulis, as well as the classic book by Furstenburg and the ergodic
theory proofs of van der Waerden's great theorem on arithmetic
progressions) had already been developed into a powerful theory by
Grothendieck in the early seventies.
Anyway, once you have a basic grounding, you can start looking for papers
in the math.LO section on LANL. As yet, LANL hasn't caught on in math
(except in algebraic geometry) as much as it should have done (I really
cannot understand why this is so), so you should also start looking
(helas!) at printed journals. However, there are already some very
interesting papers on logic, groupoids, n-categories, topoi, etc.,
available at LANL, as well some intriguing speculations on possible future
applications in physics:
@unpublished{ib:tt,
author = {C. J. Isham and J. Butterfield},
title = {Some possible roles for topos theory in quantum theory and
quantum gravity},
note = {gr-ac/9910005}
year = 1999}
@unpublished{bd:c,
author = {John C. Baez and James Dolan},
title = {Categorification},
note = {math.QA/9802029},
year = 1998}
@incollection{bd:fsfd,
author = {John C. Baez and James Dolan},
title ={From Finite Sets to {F}eynman Diagrams},
booktitle = {Mathematics Unlimited: 2000 and Beyond},
editor = {Bj\"orn Engquist and Wilfried Schmid},
publisher = {Springer-Verlag},
year = 2000}
See also:
@incollection{b:ncat,
author = {John C. Baez},
title = {An Introduction to n-Categories},
note = {q-alg/9705009},
booktitle = {7th Conference on Category Theory and Computer Science},
editor = {E. Moggi and G. Rosolini},
series = {Lecture Notes in Computer Science},
volume = 1290,
publisher = {Springer-Verlag},
pages = {1-33},
year = 1997}
@article{b:sfm,
author = {John C. Baez},
title = {Spin Foam Models},
note = {gr-qc/9709052},
journal = {Class.Quant.Grav.},
volume = 15,
year = 1998,
pages = {1827-1858}}
Also of interest are the ideas in the first few sections of this paper
http://xxx.lanl.gov/abs/hep-th/0011065
BTW, here is paper which I think offers a very nice introduction to the
"internal/external" distinction in logic and math:
@unpublished{cs:mp,
author = {A. Carbone and S. Semmes},
title = {Making proofs without {M}odus {P}onens: an introduction to the
combinatorics of cut elimination},
note = {math.LO/9607204},
year = 1996}
Here is a somewhat more speculative followup, also of interest in this
context:
@unpublished{cs:io,
author = {A. Carbone and S. Semmes},
title = {Looking from the inside and from the outside},
note = {math.LO/9607203},
year = 1996}
Chris Hillman
Home Page: http://www.math.washington.edu/~hillman/
Charles Francis
ngton.edu>, thus spake Chris Hillman <hil...@math.washington.edu>
>
>
>On Tue, 9 Jan 2001, Charles Francis wrote:
>
>> I just don't see how category theory helps. If it were to describe a
>> morphism between a mathematical object and an ontology it could do
>> some good. But it is part of mathematics and really only describes
>> morphisms between mathematical objects. Try to open that up and get a
>> general definition then you will hit Russell's paradox, just as you do
>> in set theory. And then again if it only describes morphisms between
>> mathematical objects it is subsumed by set theory, and not the other
>> way around, so I do not see the grandiose claims that it lies at the
>> foundation of mathematics.
>
>Well, these "grandiose claims" are in fact -very- well established and
>moreover -very- "well known" (in the sense that every graduate student in
>some specialities has been required to learn them, since the seventies---
That doesn't follow. I have spent many years being required to know
things without there being any reason why I was required to know them.
But here the idea that category theory is somehow more fundamental or
more encompassing than set theory and logic just seems plain wrong. Or
is that not the claim?
>onsequently, while this is not particularly easy stuff
>to learn, particularly if one has absolutely no background in logic/set
>theory in general, there are -lots- of books and papers out there to help
>beginning students:
I think I can claim background in set theory and logic. And I spent much
of my formative years pondering what mathematics actually is. So I find
it strange to be told that mathematics is something which I know it is
not. Especially as this something does not even seem to deal with the
real fundamental issues of understanding what mathematics is. The
logician Charles Dodgson (aka Lewis Carrol) seems to me to have had more
useful stuff to say on that than anyone else, and I cannot see how this
kind of thing even touches the issues he raised. Thanks for the
references by the way. I may get a book if someone can tell me what the
big idea actually is. So far I have read stuff by Baez, and also had an
introduction from Chris Isham & Jeremy Butterfield, and all I can see so
far is that there isn't a big idea. Chris notably made very few claims
for his speculations.
>@unpublished{o:bct,
>author = {Jaap van Oosten},
>title = {Basic Category Theory},
>series = {Basic Research in Computer Science Lecture Series},
>note = {http://www.brics.aau.dk/BRICS/LS/95/1/},
>month = {January},
>year = 1995}
I got nothing on this link.
>
>My own expository paper, "A Categorical Primer", currently under
>revision/expansion, but available in truncated form on my web pages, is
>much more concise but will attempt to cover correspondingly more ground
>wrt logic and topos theory, bringing the story up to the stunning
>unifications achieved by Grothendieck in the seventies.
>
This looks more useful.
>(Caveat emptor!-- I am entirely self-taught in this area, this paper has
>not been refereed, and indeed there is a metaexercise in the current
>version: fix the numerous places where the compositions are written the
>wrong way, or where I just plain had a brain blooper, or committed a more
>serious conceptual error, and then please email me with your corrections!)
OK - I hope I can make useful remarks. One thing I would like to pick up
on is that you say "..every science and engineering student studies
linear mappings between vector spaces and differentiable real valued
functions". The space of differentiable real valued functions *is* a
vector space, not a mapping of one.
> Furthermore, it turns out
>that logic, topology, algebraic geometry, and dynamics are very intimately
>related in a deep way, as Grothendieck discovered in the seventies.
This again illustrates my problem. We knew that all areas of mathematics
are intimately related long before the seventies. Although dynamics does
not fit very well here, as it is physics, and quite distinct from maths,
so this seems more like addled thinking.
<snip>
> I was kicking around as a graduate student (motivated
>in part by the recognition that number theory and dynamical systems seem
>poised for a unification,
-
In what sense? Either we have the sin of Pythogorean numerology (the
world actually *consists* of number) or we have something we already
knew.
>
>Also of interest are the ideas in the first few sections of this paper
>
> http://xxx.lanl.gov/abs/hep-th/0011065
I haven't yet been able to get to the stuff off lanl. My connection
doesn't always work. I'll try again tomorrow. But I did get the
following statement from the abstract of that one, which increases my
concern:
quote:
"What are strings made of? The possibility is discussed that strings are
purely mathematical objects, made of logical axioms"
And indeed from this it seems we have fallen back to Pythagorean number
worship. Substitute number for logical axiom and it is exactly the same.
Physical objects are not constructs of mind. These ideas were
discredited thousands of years ago, and should not now reawaken under a
new name.
As I read your paper Category Theory really does not seem to do more
than I had already understood. It is part of mathematics and really only
describes morphisms between mathematical objects. So it is subsumed by
set theory, and not the other way around.
By concentrating on the morphisms all we are really doing is showing
that mathematics reduces to the same logical relationships in different
cases. But that is already the function of algebra. I don't see why we
need category theory to explain a simple idea which has been an obvious
to competent mathematicians for two hundred or more years.
I'm afraid the most appropriate comment still seems to be the quote the
front of your paper:
"Category theory is bunk" (Marshall Cohen)
Well perhaps not bunk, but it can surely impress only those who
completely failed to understand the intentions of Boole.
Charles Francis
ngton.edu>, thus spake Chris Hillman <hil...@math.washington.edu>
>My own expository paper, "A Categorical Primer", currently under
>revision/expansion, but available in truncated form on my web pages, is
>much more concise but will attempt to cover correspondingly more ground
>wrt logic and topos theory, bringing the story up to the stunning
>unifications achieved by Grothendieck in the seventies.
>
>(Caveat emptor!-- I am entirely self-taught in this area, this paper has
>not been refereed, and indeed there is a metaexercise in the current
>version: fix the numerous places where the compositions are written the
>wrong way, or where I just plain had a brain blooper, or committed a more
>serious conceptual error, and then please email me with your corrections!)
Actually I would like to applaud what you do in your paper. It seems to
be just what I was looking for in terms of giving a handle on what
category theory is and what it is for. I think I had slightly the wrong
ideas about the claims of category theory, hence my concern. Your paper
makes things much clearer. It does have something to say about the
structure of mathematics. It is also the first time in very many years
that I have found an area of mathematics which I actually find
aesthetically appealing, so that feels quite strange in itself.
I have scarcely read much, but since you ask, may I make some minor
remarks. I think it is a mistake to use the term arrow rather than
morphism. For one thing I have just got used to morphism from other
sources, so the switch is confusing (unless arrow is widely used). For
another morphism has an obvious meaning from its greek root and the
different ...morphisms used in maths, while arrow could be anything (not
really important, but I would certainly find morphism more approachable.
Another thing I wondered about was the statement that "most mathematical
structures are comparatively specific and concrete". I would have said
that is not true. Typically algebraic structures are completely abstract
and can stand for anything. Yes, category theory introduces a new and
deeper abstraction, but even school level algebra is abstracted from
arithmetic, which itself is abstracted from reality; there is no such
thing as "five", only "five somethings".
Also I will still worry that maths becomes more and more abstract, and
goes of in directions of its own making without due concern about
applicability. I just found a quote from the Russian mathematician V. I.
Arnol'd in Michael Weiss' account of Lie groups, but it struck me as
particular pertinent:
It is almost impossible for me to read contemporary mathematicians who,
instead of saying Petya washed his hands," write simply:
There is a t1 < 0 such that the image of t1 under the natural map-
ping t1 -> Petya(t1) belongs to the set of dirty hands, and a t2,
t1 < t2 <= 0, such that the image of t2 under the above-mentioned
mapping belongs to the complement of the set defined in the pre-
ceding sentence."
John Forkosh
<snip>
: (More demanding but covers more territory, up through one approach to the
: relationship between a topos and logic and "set theory".)
: @book{bw:ttt,
: author = {Michael Barr and Charles Wells},
: title = {Toposes, Theories, and Triples},
: publisher = {Springer-Verlag},
: year = 1985}
: More about topoi and theories (in the sense of logic):
Although out of print, the full text of the above is available at
ftp://ftp.math.mcgill.ca/pub/barr/tttps.zip
And take a look at all the ttt* files in that directory for
pdf and dvi versions, and also for "cor"rections/errata to the
print version.
John (for...@panix.com)
Ralph E. Frost
news:rUkTDpCm...@clef.demon.co.uk...
> In article <Pine.OSF.4.21.010110...@goedel3.math.washi
> ngton.edu>, thus spake Chris Hillman <hil...@math.washington.edu>
> Another thing I wondered about was the statement that "most mathematical
> structures are comparatively specific and concrete". I would have said
> that is not true.
I'd agree, but a mathematician may be referring to "comparatively specific
and concrete _math structures_", there being not that many other kind of
artifacts present in their feid of vision. Maybe Chris can clarify?
>Typically algebraic structures are completely abstract
> and can stand for anything. Yes, category theory introduces a new and
> deeper abstraction, but even school level algebra is abstracted from
> arithmetic, which itself is abstracted from reality;
Don't you mean "associated with reality"?
>there is no such
> thing as "five", only "five somethings".
Huh? No such thing? What about 5 ? Plain old 5.
>
> Also I will still worry that maths becomes more and more abstract, and
> goes of in directions of its own making without due concern about
> applicability. I just found a quote from the Russian mathematician V. I.
> Arnol'd in Michael Weiss' account of Lie groups, but it struck me as
> particular pertinent:
>
> It is almost impossible for me to read contemporary mathematicians who,
> instead of saying Petya washed his hands," write simply:
> There is a t1 < 0 such that the image of t1 under the natural map-
> ping t1 -> Petya(t1) belongs to the set of dirty hands, and a t2,
> t1 < t2 <= 0, such that the image of t2 under the above-mentioned
> mapping belongs to the complement of the set defined in the pre-
> ceding sentence."
Folks resort to that sort of thing when they are really insecure about some
ontological issue.
Chris Hillman
> Actually I would like to applaud what you do in your paper. It seems
> to be just what I was looking for in terms of giving a handle on what
> category theory is and what it is for. I think I had slightly the
> wrong ideas about the claims of category theory, hence my concern.
> Your paper makes things much clearer. It does have something to say
> about the structure of mathematics. It is also the first time in very
> many years that I have found an area of mathematics which I actually
> find aesthetically appealing, so that feels quite strange in itself.
Hmm... well, glad you like it, but I certainly wish you had not sent your
previous message, since if you had not, I would not have sent my reply
(which however will probably be rejected by the moderators, since although
I tried hard, it kept coming out sounding sarcastic--- I also tried
replying by simply quoting a poem, but then I realized that too would be
unacceptable).
> I have scarcely read much, but since you ask, may I make some minor
> remarks. I think it is a mistake to use the term arrow rather than
> morphism. For one thing I have just got used to morphism from other
> sources, so the switch is confusing (unless arrow is widely used).
It is, and repetition breeds familiarity. And once you get past some of
the basic stuff, you begin to realize that category theory is more closely
allied to good old graph theory than might at first appear. In using
"arrow", I am looking ahead to that discovery.
> For another morphism has an obvious meaning from its greek root and
> the different ...morphisms used in maths, while arrow could be
> anything (not really important, but I would certainly find morphism
> more approachable.
I agree that "morphism" will initially sound more friendly, but part of
what I try to do early on is to give some interesting examples of
-nonconcrete- categories where the connotations of "morphism" are
unhelpful.
> Another thing I wondered about was the statement that "most mathematical
> structures are comparatively specific and concrete". I would have said
> that is not true. Typically algebraic structures are completely abstract
> and can stand for anything. Yes, category theory introduces a new and
> deeper abstraction, but even school level algebra is abstracted from
> arithmetic, which itself is abstracted from reality; there is no such
> thing as "five", only "five somethings".
Well, you should like the paper by Baez & Dolan, "From Finite Sets to
Feynman Diagrams", then!
Also note that the notion of "models in a topos" is very well suited to
understanding the relationship between "the idea of a group" and a
specific group, for example. Furthermore, as you read further on, you
will come to see the algebraic categories I tick off in an early exercises
are indeed "concrete".
> Also I will still worry that maths becomes more and more abstract, and
> goes of in directions of its own making without due concern about
> applicability. I just found a quote from the Russian mathematician V. I.
> Arnol'd in Michael Weiss' account of Lie groups, but it struck me as
> particular pertinent:
>
> It is almost impossible for me to read contemporary mathematicians who,
> instead of saying Petya washed his hands," write simply:
> There is a t1 < 0 such that the image of t1 under the natural map-
> ping t1 -> Petya(t1) belongs to the set of dirty hands, and a t2,
> t1 < t2 <= 0, such that the image of t2 under the above-mentioned
> mapping belongs to the complement of the set defined in the pre-
> ceding sentence."
In case my previous message is rejected, I'll repeat something I said
there: the whole point of the two quotations which begin the paper is not
to make fun of Marshall Cohen, who is someone I like and respect (in fact
it was a throwaway remark of his in an algebraic topology class which
first put me onto what Louis Crane calls "categorification", or what I
might call for emphasis "categorification in good taste"), but that -both-
characterizations of "category theory" appear to me to have a grain of
truth. (No pun intended.)
The point of Marshall Cohen's barb is that papers in "category theory"
which are sufficiently dense or confusing to put off Arnold are probably
not "in good taste". There is a certain kind of specialization which
encourages inbreeding and that is never good mathematics, IMO. Indeed,
this kind of paper is in my view antithetical to the spirit in which
category theory was invented, which was to provide a common langauge for
all mathematicians to learn and understand and discuss certain
fundmamental constructions/notions which turned up all over the map in the
mathematics of the last century. The purpose of my paper is to promote
the use of this common "thought language" not only among mathematicians,
but among everyone who will use mathematics in the next century, including
physicists, biologists (OK, a tough sell, but hope springs eternal!), and
engineers (the kind who -engineer-, rather than the kind who post their
"theories of fundamental physics" to the InterNet, which needless to say
is the kind of engineer I lampooned in my recent sci.physics post--- since
the only engineers likely to read that book fall into the second class of
engineers, I think that was fair play). In my view, the notions Jacobson
had in mind are part of this common thought language.
Michael Weiss
: I think it is a mistake to use the term arrow rather than
: morphism.
My two cents: I also prefer morphism.
: (unless arrow is widely used).
No, morphism is far more common.
: Another thing I wondered about was the statement that "most mathematical
: structures are comparatively specific and concrete".
When I was in grad school, category theory was perceived by many as notching
up the abstraction level by a discontinuous jump, and accordingly both
vilified and glorified.
This strikes me as quite strange now, though at the time it seemed obvious.
OK, so the definition of a category does look a little different from (say)
the definition of a group --- the star player is a partially defined
operation --- but it's not really inherently more *abstract*.
Guess you had to be there.
: Also I will still worry that maths becomes more and more abstract, and
: goes of in directions of its own making without due concern about
: applicability.
Category theory arose directly out of applications (in topology), and has
flourished because it has found so many more, originally in math, but now
also in physics and comp-sci.
: I just found a quote from the Russian mathematician V. I.
: Arnol'd in Michael Weiss' account of Lie groups,
Thanks for reading!
: but it struck me as
: particular pertinent
[You can find the quote in http://math.ucr.edu/home/baez/lie/lie.html ]
I think Arnol'd was complaining about overly abstract *exposition*, not
abstraction per se. But you're right, the way I used the quote was
ambiguous.
I recently ran across a quote from Weyl, on Hilbert's style, that I like: he
says that Hilbert, unlike some authors, never assumed that the printer's
time and labor were far more valuable than the reader's.
As for the battle over abstraction, this has deep historical roots. Galileo
defends the use of geometry in his famous _Dialogues_.
In the intro to his book _Group Theory and Physics_, Sternberg quotes the
well-known physicist John Slater (inventor of the Slater determinant in
quantum chemistry):
It was at this point that Wigner, Hund, Heitler, and Weyl entered the
picture
with their "Gruppenpest": the pest of the group theory [actually, the
correct
translation is "the plague of group theory"] ... The authors of the
"Gruppenpest"
wrote papers which were incomprehensible to those like me who had not
studied
group theory... The practical consequences appeared to be negligible,
but everyone felt that to be in the mainstream one had to learn about it.
I had what I can only describe as a feeling of outrage at the turn
which the subject had taken...
As soon as this [Slater's] paper became known, it was obvious that a
great
many other physicists we are disgusted as I had been with the
group-theoretical
approach to the problem. As I heard later, there were remarks made
such as "Slater has slain the 'Gruppenpest'". I believe that no other
piece
of work I have done was so universally popular.
Sternberg's comment: "It is ... amazing to consider that [these remarks
were] published in 1975, after the major triumphs of group theory in
elementary particle physics."
John Baez
>The point of Marshall Cohen's barb is that papers in "category theory"
>which are sufficiently dense or confusing to put off Arnold are probably
>not "in good taste".
I would not want Arnol'd to be in charge of judging which papers
in category theory are in "good taste"!
Vladmir Igorevich Arnol'd is an excellent mathematical physicist,
but part of excellence is following ones own idiosyncratic taste
and using it to go places that no one has gone before. This does
not always lead to balanced judgement of other people following
other tastes. Arnol'd has a strong taste for visual/geometrical
as opposed to conceptual/algebraic reasoning, as anyone can tell
who has read his wonderful book:
Huygens and Barrow, Newton and Hooke: pioneers in mathematical analysis
and catastrophe theory from evolvents to quasicrystals, V. I. Arnol'd,
translated from the Russian by Eric J.F. Primrose. Basel, Boston:
Birkhauser Verlag, 1990.
Note how he prefers Newton to Leibniz because of Newton's more
geometrical reasoning! He is probably rather cold to Leibniz's
grand dreams regarding logic and the "
One can also see Arnol'd's taste for the geometrical in his work
on the "Kolmogorov-Arnol'd-Moser tori" and the "Arnol'd web". I
could explain these, but a picture is worth a thousand words in this
case! Look at the figures on this website:
Computation of the Arnol'd Web for the Hydrogen Atom in Crossed
Electric and Magnetic Fields, Jan von Milczewski, G. H. F. Diercksen
and T. Uzer, Phys. Rev. Lett., 2890 (1996).
http://www.physics.gatech.edu/people/faculty/uzer/arnold.html
Anyway, category theory tends to emphasize conceptual/algebraic
reasoning, so I can't imagine Arnol'd enjoying most of it. As with
*every* subject in math or physics, there are inbred papers which
are of little interest to anyone but specialists, and visionary
papers that reshape our whole thought-world, and many papers
that stand between these extremes. It's worth remembering that
it's rather hard to judge these papers until one has a good sense
of the subject. A paper can contain wonderful ideas but be written
in a way that doesn't make their wonderfulness evident.
Of all mathematical subjects, category theory does the best job
of cleaning your brain of its mental cobwebs and letting you see
mathematics as an undivided whole, simple and beautiful. It's very
abstract, but in a powerful way. It helps you understand many things
you thought you already understood - but didn't! For the raw amateur
out there who wants a taste of this experience, I highly recommend:
Conceptual mathematics: a first introduction to categories,
F. William Lawvere, Stephen H. Schanuel. Cambridge, New York:
Cambridge University Press, 1997.
It seems very easy and almost over-simple at first, but *do the
exercises*, or it may sneak up on you and become too difficult.
...................................................................
Caveat: while I say that category theory leans towards the
conceptual/algebraic, it's also true that this subject leans
more than any other towards proving things with the help of
diagrams - and it clarifies the relation between geometry and
logic! So even lovers of geometry should ultimately enjoy
category theory, once they get over the hump of abstraction.
Many never get over that hump. It helps to have a guru.
Chris Hillman
> Vladmir Igorevich Arnol'd is an excellent mathematical physicist, but
> part of excellence is following ones own idiosyncratic taste and using
> it to go places that no one has gone before. This does not always
> lead to balanced judgement of other people following other tastes.
> Arnol'd has a strong taste for visual/geometrical as opposed to
> conceptual/algebraic reasoning,
Agreed. The irony here is that -I- have a strong preference for
visual/geometrical reasoning, so I tend to agree with Arnol'd's judgement
of what is/isn't "interesting".
In any case, I (literally) see category theory :-/ although I guess I find
"easy algebraic reasoning" sufficiently easy :-/ to be able to prove the
kind of "trivial propositions" which crop up all of the place in CT (the
kind where you just unravel definitions to prove whatever is to be proven,
the same kind of reasoning which prevails in the first few weeks of course
in group theory or ring theory, as opposed to propositions which require a
"genuine idea" like the Sylow theorems [yes, I know that there are many
ways of proving these!--- but the point is, all the proofs require some
nontrivial insight]). I think what I'm trying to say is that I think
visually about all the definitions and statements of easy results, but
think algebraically when it comes to proving easy results, and that this
is probably the right way to proceed when first learning the subject.
> One can also see Arnol'd's taste for the geometrical in his work on
> the "Kolmogorov-Arnol'd-Moser tori" and the "Arnol'd web".
Wow, another irony: Arnol'd's claims in a little known announcement
@article{a:rqs,
author = {V. I. Arnol'd},
title = {Remarks on Quasicrystallic Symmetries},
journal = {Physica D},
volume = 33,
year = 1988,
pages = {21--25}}
to understand why these webs often bear a striking resemblance to
generalized Penrose tilings (aka "Sturmian tilings"). If anyone other
than VIA understands the basis for this claim, I'd like to know!
Yet another irony: Sturmian tilings exemplify one of the motivations for
"logical dynamics" (although I don't know if Grothendieck knew about them
when he was developing his ideas on this subject in the seventies, but
since the seminal work of Penrose and Conway on two-dimensional Penrose
tilings dates to the early seventies, and quickly interested people of the
caliber of Margulis and Bombieri, for example, he could well have done---
the two dimensional case is the lowest dimension in which a crucial
observation of Penrose holds true, although, re the above, dimension three
is the lowest in which Arnol'd webs arise).
David Combs
>the basic stuff, you begin to realize that category theory is more closely
>allied to good old graph theory than might at first appear. In using
>"arrow", I am looking ahead to that discovery.
>
>> For another morphism has an obvious meaning from its greek root and
>> the different ...morphisms used in maths, while arrow could be
>> anything (not really important, but I would certainly find morphism
>> more approachable.
>
>
>Well, you should like the paper by Baez & Dolan, "From Finite Sets to
>Feynman Diagrams", then!
>
Re arrow-diagrams and math (of physics, etc),
I just a day or two ago
received the most recent issue of "IEEE Computer Graphics and
Applications" (jan/feb 2001, vol 21, num 1), which has
its usual (or at least often) "Jim Blinn's Corner" article.
(For those of you who don't know who Jim Blinn is, he is
one of the master gurus of computer graphics, especially
of the (hairy) math behind getting clever things done.
He now has two collections of his articles, under the
name "Jim Blinn's Corner". Very apropos to physics
calculations.)
ANYWAY -- this issue's Blinn-article is "Polynomial
Discriminants -- Part 2: Tensor Diagrams". (Meaning
there was a part 1, not now in front of me as I type in).
Keeping things short (because all this math is *way* over
*my* head), the three references at the end of the
article are:
1: G.E. Stedman, Diagram Techniques in Group Theory,
Cambridge, 1990
(Blinn says this book gave him ideas he had never
even considered, from which he developed the
clever stuff in his article.)
2: Blinns 2nd collection of articles from IEEE CG&A,
"Jim Blinn's Corner: Dirty Pixels", Morgan Kaufmann, '98.
3: G. Kuperberg, "Involutory Hopf Algebras and 3-Manifold
Invariants", Intl J. Mathematics, vol 2, no. 1, '91, pp 41-66.
I give his refs so that some readers of this thread
can judge whether this IEEE article is worth tracking down.
(hope *someone* finds this interesting! :-)
David
John Baez
In article <945061$7kb$1...@news.state.mn.us>,
John Baez <ba...@galaxy.ucr.edu> wrote:
>Note how [Arnol'd] prefers Newton to Leibniz because of Newton's more
>geometrical reasoning! He is probably rather cold to Leibniz's
>grand dreams regarding logic and the "
Sorry, I meant to look up a phrase here, but I never got around
to it. I think the phrase I wanted was "characteristica universalis" -
the "universal language" that Leibniz dreamt of, in which all concepts
could be clearly and logically expressed. This dream led ultimately
to modern logic, which finds its clearest expression in the category-
theoretic approach pursued by Lawvere and others.
I've never heard Arnol'd talk about this stuff, but it's hard to imagine
him being fond of it - even though Lawvere got started on his thesis by
thinking about physics! He wanted to think of a physical "theory" as a
category, and a "model" of such a theory as a functor. But his work on
"algebraic theories" wound up being used more in logic and universal
algebra than in physics.
But history has many twists and turns. Much later, Graeme Segal
described "conformal field theories" using a variation of this same
idea. Atiyah's "topological quantum field theories" work the same
way too! This led to the current boom in applications of category
theory to physics.
John Baez
David Combs <dkc...@panix.com> wrote:
>Keeping things short (because all this math is *way* over
>*my* head), the three references at the end of the
>article are:
>
>1: G.E. Stedman, Diagram Techniques in Group Theory,
>Cambridge, 1990
This is one of the books I've been using in my quantum gravity
seminar. It's nice! Ultimately, when I turn the notes from my
seminar into a book, they should contain a lot of the information
in this book, but with more explicit emphasis on the category theory.
>2: Blinns 2nd collection of articles from IEEE CG&A,
> "Jim Blinn's Corner: Dirty Pixels", Morgan Kaufmann, '98.
I've never seen this.
>3: G. Kuperberg, "Involutory Hopf Algebras and 3-Manifold
> Invariants", Intl J. Mathematics, vol 2, no. 1, '91, pp 41-66.
This is one of an almost infinite set of papers which use
category theory, Hopf algebras and the like to rigorously construct
topological quantum field theories and invariants of manifolds.
The literature on this subject is full of pretty diagrams.
If anyone is interested in this particular paper, I can give them
references to some of the more recent literature on the same stuff.
David Hillman
John Baez wrote:
Yes, please!! I happened to be looking at this very paper today. Too
difficult (for me)! I want very pretty pictures showing just how the Hopf
algebra is associated with the 3-manifold. (Emphasis on invariants
preferred, since I am more liable to understand it. But if the paper
teaches me from scratch what a TQFT is, all the better. The key words are
"from scratch.")
John Baez
David Hillman <d...@olympus.net> wrote:
John Baez wrote:
>> >3: G. Kuperberg, "Involutory Hopf Algebras and 3-Manifold
>> > Invariants", Intl J. Mathematics, vol 2, no. 1, '91, pp 41-66.
>> The literature on this subject is full of pretty diagrams.
>> If anyone is interested in this particular paper, I can give them
>> references to some of the more recent literature on the same stuff.
>Yes, please!! I happened to be looking at this very paper today. Too
>difficult (for me)! I want very pretty pictures showing just how the Hopf
>algebra is associated with the 3-manifold. (Emphasis on invariants
>preferred [...]
I'd urge you to start by learning the *2-dimensional* version of the
theory, which is simpler. This involves plain old *algebras*, and the
pictures are all 2-dimensional, which makes it easy.
Then you can move on to the *3-dimensional* version, which involves
*Hopf algebras*.
I'll be covering the 2d version in the quantum gravity seminar very
soon - see Week 16 Track 1, when it comes out. But you should also
read this:
http://xxx.lanl.gov/abs/hep-th/9212154
Title: Lattice Topological Field Theory in Two Dimensions
Authors: M. Fukuma, S. Hosono, H. Kawai
Comments: 29 pages (Latex) + 19 figures (uuencoded through uufiles).
Journal-ref: Commun. Math. Phys. 161 (1994) 157-176
since these are the folks who did it first.
For the 3d theory, with lots of pretty pictures, try this:
http://xxx.lanl.gov/abs/hep-th/9305080
Title: Structure of Topological Lattice Field Theories in Three Dimensions
Author: Stephen-wei Chung, Masafumi Fukuma, Alfred Shapere
Comments: 63 pages, 46 figures
Subj-class: High Energy Physics - Theory; Quantum Algebra
Journal-ref: Int. J. Mod. Phys. A9 (1994) 1305-1360
If you start comparing what Kuperberg does to what these folks do,
you'll notice a bunch of differences - some subtle, others obvious.
At that point you may start wondering how the two approaches fit
together. And then I can give you some references that tackle that
question!
Both these papers focus on invariants of closed manifolds, rather
than TQFTs. In the quantum gravity seminar I'll explain how you
get TQFTs from these ideas.
George Hrabovsky
absence)...
"John Baez" <ba...@galaxy.ucr.edu> wrote in message
news:945061$7kb$1...@news.state.mn.us...
> Chris Hillman <hil...@math.washington.edu> wrote:
>
[Stuff Deleted]
>
> Vladmir Igorevich Arnol'd is an excellent mathematical physicist,
> but part of excellence is following ones own idiosyncratic taste
> and using it to go places that no one has gone before. This does
> not always lead to balanced judgement of other people following
> other tastes. Arnol'd has a strong taste for visual/geometrical
> as opposed to conceptual/algebraic reasoning, ...
I couldn't agree with you more. I have noticed three basic divisions in the
field of mathematical physics:
1) Algebraic thought built around notions of pure structure (i.e. Quantum
Logic, Quantum Groups, Symplectic Algebra, etc.).
2) Analytic thought built up around notions of functions and their
generalizations (i.e. Path Integrals, Functionals, PDEs, etc.).
3) Geometrical/Topological thought built up around notions of regions and
boundaries (i.e. Riemannian Manifolds, Vector Fields, Flows, etc.).
Of course this is a gross exaggeration, many of these topics are chock full
of elements of the other two.
>
> Of all mathematical subjects, category theory does the best job
> of cleaning your brain of its mental cobwebs and letting you see
> mathematics as an undivided whole, simple and beautiful. It's very
> abstract, but in a powerful way. It helps you understand many things
> you thought you already understood - but didn't! For the raw amateur
> out there who wants a taste of this experience, I highly recommend:
>
> Conceptual mathematics: a first introduction to categories,
> F. William Lawvere, Stephen H. Schanuel. Cambridge, New York:
> Cambridge University Press, 1997.
Another good book on category theory that encompasses the idea of the
overlap of subjects is:
Mathematical Physics
Robert Geroch. University of Chicago Press, Chicago and London:
University of Chicago Press, 1985.
I found this to be very useful and clear.
George
Urs Schreiber
lambda calculus, and physics. (This posting became somewhat longer than
expected, the question is found at the end.)
<Introduction>
In Penrose's book "The Emperors New Mind" I once read for the first time
about Church's "lambda calculus" (pp. 66-70) and found it very fascinating
due to its sheer fundamental and highly expressive character (the ultimate
book on lambda calculus is [1]). Even though lambda calculus has mainly (or
exclusively?) played a role in theoretical computer science and foundations
of mathematics, I always wondered if it were not be a perfect candidate for
a "primordial" formalism in physics. The (very vague) idea is this:
Say, in two hundred years or so, when the basics of quantum gravity and
grand unification have long moved from the research papers and settled in
the textbooks, one will probably teach and display "the final theory" in a
way current textbooks cover classical mechanics, where, starting from a few
assumptions, everything looks simple, (almost) obvious, and compelling,
since it has been worked out so thoroughly. For the final theory I suspect
that even the first assumption from which then everything is derived should
look simple, (almost) obvious, and compelling after being refined for
decades.
Perhaps Landau's grand-grand-child will begin a book on theoretical physics
with the words: "In order to describe an all-encompassing entity (the
universe) we will surely have to consider a formalism that expresses
/relations/ in their most pure form. There is apparently only one such
formalism, namely the celebrated mathematical framework called "A". This
"A" has elements called "B", "C" and "D", which can naturally be identified
with the physical notions of "x", "y", and "z", from which all of physics
is made of. By substituting "x", "y", "z" for "B", "C", "D" in the theorems
of "A", it will turn out that these are (equivalent to) the laws of
physics, which thus follow compellingly from the only assumption that
anything at all exists. ..."
Or so, if you know what I mean. ;)
This framework "A" is what I called the "primordial" formalism above ("the
dream that stuff is made of"). Probably category theory is a good candidate
for "A", but I always liked to think of lambda calculus as something like
"the god given way to properly describe 'existence' as such". This is
because lambda calculus really knows only one single notion, namely that of
"function of functions" (or "operator on operators"),
- which appears to be sufficiently circular to "bootstrap itself
into existence"
- which is in a way a notion of mere "structure" (because a lambda
function is defined by the pattern by which it applies its arguments on
each other, which are again lambda functions given by such patterns, and so
on)
- from which many (or all?) mathematical concepts can be
constructed: A lambda function can be both an "object" or a "morphism" (or
both), it can be a set of data (a tuple, a set) or an algorithm acting on
that data (or both), it can be a finite or infinite pattern, it can be used
to construct number theory and formal logic, and so on.
Now, there have been attempts to construct "models" of lambda calculus
within category theory. For example, in [2] concepts of lambda calculus like
"abstraction" and "application" are identified with certain morphisms
within Cartesian closed categories. However, this process looks awkward to
me, since it somehow destroys the pretty symmetry and elegance of the
concepts of lambda calculus. The central object that a model of lambda
calculus has to provide is a somewhat circular set \Lambda, whose elements
have an interpretation as functions with domain and range \Lambda itself
(e.g. [3]). *Naively*, the most natural way to model this within category
theory appears to me to identify lambda functions with objects (or
morphisms) of n-categories, for n -> infinity! I am aware (learned that
from this group and John Baez's web site) that defining n-categories for
arbitrary n is an unsolved problem, not to mention letting n go to
infinity. But an object of an (n=infinity)-category (supposed that it is
possible to give this idea a precise meaning) would be at the same time a
morphism of that category and vice versa. Thus, such objects could possibly
be identified with lambda functions.
<finally: the question>
Often things are simple if their order n is small, become immensely
complicated for arbitrary n but are manageable again for infinite n. Could
this be true for n-categories? Could an (n=infinite)-category be related to
lambda calculus? Could one or both of them be the "primordial" formalism?
I am of course not necessarily expecting any definite answers, but would
enjoy any comments on the above.
Regards,
Urs Schreiber
references:
-----------
[1]
@book{Barendregt:1981,
author = {Barendregt, H. P.},
title = {The Lambda Calculus},
year = {1981},
publisher = {North-Holland Publishing Company},
note = {All about lambda calculus.}
}
[2]
@techreport{Martini:19??,
author = {Martini, Alfio},
title = {Category Theory and the Simply-Typed $\lambda$-Calculus},
institution = {{T}echnische {U}niversit\"at {B}erlin
{F}achbereich 13},
year = {19??},
number = {96-7},
type = {{F}orschungsberichte des {Fa}chbereichs {I}nformatik},
note = {On how to provide a categorical model for the simply typed
$\lambda$-calculus.}
}
[3]
@book{Koymans:1980,
author = {Koymans, C. P. J.},
title = {Models of the lambda calculus},
year = {1980},
publisher = {Centrum voor Wiskunde en Informatica
Centre for Mathematics and Computer Science
Amsterdam, Netherlands},
volume = {9},
series = {CWI Tract}
}
Chris Hillman
> I would like to ask a question on the relation between category theory,
> lambda calculus, and physics.
You'd probably be interested in reading about topos theory and its
relation to the foundations of mathematics, formal logic, set theory, and
model theory. IIRC, lambda calculus appears as a detail :-/ in this
rather overwhelming circle of ideas... (Maybe someone else here has a
better memory than I?) Certainly, intuitionistic logic (Heyting algebras
rather than Boolean algebras, "forcing" (aka the "sheafification
functor"-- the latter was Paul Cohen's technique for proving the
independence of the continuum hypothesis from the usual axiomatization), a
much simpler set of axioms than Zermelo-Frankel for the foundations of
mathematics, a very simple expression of the role of the axiom of choice,
and a bunch of other neat stuff arises naturally in topos theory.
By the way, wrt to John Baez's mention the other day of simplices
appearing in topological field theory, it occurs to me that the category
Cho with objects = Choquet simplices and arrows = affine continuous maps
is probably a (Boolean) topos (I have almost verified this is some
detail.) I guess the point here might be that I have the impression that
many papers on quantum gravity often seem to emphasize algebra or category
theory a bit more than hard analysis, and Choquet simplices might offer an
elegant way to redress the balance.
What is a Choquet simplex? Well, it is the natural infinite
generalization of a simplex! It is a compact convex subset of a locally
convex topological vector space such as a Banach space. Such a Choquet
simplex X has a set of extreme points ext X which generalize the notion of
vertices, and the Choquet representation theorem generalizes the ergodic
decomposition. See
author = {Robert R. Phelps},
title = {Lectures on {C}hoquet's Theorem},
edition = {Second},
publisher = {Springer-Verlag},
year = 2001}
(due out later this month).
If so, we would have a very nice covariant functor from Chs (the category
of compact Hausdorff spaces) to Cho
X C(X) P(X)
f | ~~~> ^ f^* ~~~> | f_*
V | V
Y C(Y) P(Y)
where P(X) is the space of regular Borel probability measures on X. IOW,
every map f "pushes out" measures on X to measures on Y. Here, ext P(X)
are precisely the delta measures delta_x, x in X, and all the other ones
are convex combinations (integrals, not finite sums) of these delta
measures. Furthermore
X = cl con ext X
(X is the closed convex hull of the set of its own extreme points). And
you can define "faces" and so forth. If you are like me, you are thinking
"so what is the monad here?" :-/
Then, a topological dynamical system T: X-> X is a N-Chs object (Chs with
a monoid action by N) and induces an N-Cho object, where N-Cho is another
topos, but with a more interesting classifying object (see the paper by
Isham and Butterfield below)! Also, exp P(X,T) is precisely the set of
T-invariant regular Borel probability measures. See
author = {Peter Walters},
title = {Introduction to Ergodic Theory},
publisher = {Springer},
year = 1981}
In particular, if (X,T) is the full two-shift then P(X,T) is the Poulsen
simplex, the "largest" Choquet simplex in a several remarkable senses.
There are a number of surprising and remarkable theorems to the effect
that every Choquet simplex arises in this way from a -minimal-
homeomorphism T (one such that every orbit under T in X is -dense-).
I think this would be remarkable if true because so few really general
results concerning dynamical systems (in the sense of -arbitrary- group
actions G on spaces X) are known, and because Choquet simplices are
already known to be very interesting things indeed.
While I'm at it, let me throw out another idea: homotopy begins by
studying maps from small dimensional simplices Delta (let's even say one
dimensional for, uhm, -simplicity-!) to X, a topological space. Then one
can "multiply" paths to get the homotopy groupoid. This gives a covariant
functor with a simple neccessary criterion for when f:Delta -> X can lift
to f^~:Delta -> E:
E
| pi
V
f
Delta ---> X
So, has anyone thought about studying maps from X to simplices Delta,
giving a -contravariant functor-? I'm not quite sure what the algebraic
operations here might be, but for example one could "average" two maps
using the Choquet measure on P(X). This should in principle give a simple
neccessary criterion for when a map sigma: U -> E can be extended to
sigma^_: X -> E:
E
^
sigma |
i
U ---> X
(Motivation: cohomology only sees cocycles, so it clearly misses
"retractable obstructions" to extensions of local sections in sheaves.
For the same reason, cellular homology at least seems disappointing in
studying coverings by one dimensional branched manifolds.)
Anyway, anyone interested in learning a bit about topos theory, logic, and
the foundations of mathematics can try these resources (listed roughly in
order of increasing sophistication):
author = {F. William Lawvere and Stephen H. Schanuel},
title = {Conceptual mathematics : a first introduction to categories},
publisher = {Cambridge University Press},
year = 1997}
author = {Colin McLarty},
title = {Elementary Categories, Elementary Toposes},
publisher = {Oxford Science Publications},
year = 1995}
author = {Robert Goldblatt},
title = {Topoi: the Categorial Analysis of Logic},
publisher = {North-Holland},
year = 1979}
author = {Jaap van Oosten},
title = {Basic Category Theory},
series = {Basic Research in Computer Science Lecture Series},
note = {http://www.brics.aau.dk/BRICS/LS/95/1/},
month = {January},
year = 1995}
author = {Saunders Mac Lane},
title = {Categories for the Working Mathematician},
publisher = {Springer-Verlag},
year = 1971}
author = {Michael Barr and Charles Wells},
title = {Toposes, Theories, and Triples},
note = {http://www.cwru.edu/math/wells/pub/ttt.html},
publisher = {Springer-Verlag},
year = 1985}
author = {J. L. Bell},
title = {Toposes and Local Set Theories},
publisher = {Clarendon Press},
year = 1988}
author = {M. Makkai and G. Reyes},
title = {First Order Categorical Logic},
series = {Lecture Notes in Mathematics},
volume = 611,
publisher = {Springer-Verlag},
year = 1977}
author = {Saunders Mac Lane and Ieke Moerdijk},
title = {Sheaves in Geometry and Logic: a First Introduction to Topos
Theory},
publisher = {Springer-Verlag},
year = 1992}
and of course the speculative paper
author = {C. J. Isham and J. Butterfield},
title = {Some possible roles for topos theory in quantum theory and
quantum gravity},
note = {gr-qc/9910005},
year = 1999}
and these, which I haven't been able to study yet, so I don't know for
sure that they are related to topos theory, but they certainly seem on
casual inspection to use sheaf theory!:
http://xxx.lanl.gov/abs/gr-qc/0102108
http://xxx.lanl.gov/abs/gr-qc/0102097
http://xxx.lanl.gov/abs/gr-qc/9909056
(I can't remember which of the above listed books explains the topos
theory take on lambda calculus, but IIRC at least one of them does.)
Chris Hillman
Home Page: http://www.math.washington.edu/~hillman/
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
NOTE TO WOULD-BE CORRESPONDENTS: I have installed a mail filter which
deletes incoming messages not from the "*.edu" or "*.gov" domains, but
also deletes messages from some bad actors whose emails happen to be in
the "*.edu" domain and "passes" messages from a few friends with email
addresses in other domains.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Greg Kuperberg
>> > Invariants", Intl J. Mathematics, vol 2, no. 1, '91, pp 41-66.
>>
>> This is one of an almost infinite set of papers which use
>> category theory, Hopf algebras and the like to rigorously construct
>> topological quantum field theories and invariants of manifolds.
>> The literature on this subject is full of pretty diagrams.
>> If anyone is interested in this particular paper, I can give them
>> references to some of the more recent literature on the same stuff.
>
>Yes, please!! I happened to be looking at this very paper today. Too
>difficult (for me)! I want very pretty pictures showing just how the Hopf
>algebra is associated with the 3-manifold. (Emphasis on invariants
>preferred, since I am more liable to understand it. But if the paper
>teaches me from scratch what a TQFT is, all the better. The key words are
>"from scratch.")
If you are interested in this paper and its sequel, Non-involutory Hopf
algebras and 3-manifold invariants, arXiv:q-alg/9712047, I'd certainly
be happy to answer questions by e-mail, although I don't usually read
sci.physics.research.
Let me also point out that you don't need TQFTs or category theory or any
quasi-triangular or ribbon or modular what-have-you for my two papers,
although many of those concepts are implicitly there. All you need,
as input, is a finite-dimensional Hopf algebra H and a 3-manifold M. (M
must be framed if H is non-involutory.) The output is a number #(M,H),
which you can interpret as an invariant of the Hopf algebra or the
3-manifold as you prefer.
--
/\ Greg Kuperberg (UC Davis)
/ \
\ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/
\/ * All the math that's fit to e-print *
John Baez
>On 2 Mar 2001, Urs Schreiber wrote:
>> I would like to ask a question on the relation between category theory,
>> lambda calculus, and physics.
I didn't see the post where you actually asked the question, just
this reply by Hillman. So I can't answer your question. For all
I know, it's just one of those "I'd like to ask a question" sort of
questions. But I'll say some stuff anyway.
>You'd probably be interested in reading about topos theory and its
>relation to the foundations of mathematics, formal logic, set theory, and
>model theory. IIRC, lambda calculus appears as a detail :-/ in this
>rather overwhelming circle of ideas...
At some point Joachim Lambek realized that the lambda calculus was
basically just another way of talking about cartesian closed categories.
More precisely, if you ask in what sort of category you can do lambda
calculus, the answer is: a cartesian closed category.
He told me that he presented this result at a conference on lambda
calculus, naively hoping that henceforth the subject would be recognized
as a small branch of category theory and basically go away. As an older,
wiser man, he now realizes that not everyone doing lambda calculus *wants*
to see it subsumed by category theory!
By the way, Lambek also has applications of quaternions to physics
as a hobby, and he had a nice article about it in the Mathematical
Intelligencer. He was also the inventor of multicategories.
But what's a cartesian closed category???
A category is "cartesian" if it has finite products - or in other words,
binary products and a terminal object. A cartesian category is "closed"
if for each object A, the functor A x -- (that is, taking the product with
A) has a right adjoint. This adjoint is called HOM(A, --).
For example, the category of finite sets is Cartesian. The terminal
object is the one-element set, the product is the usual Cartesian one,
and HOM(A,B) is the finite set of functions from the finite set A to
the finite set B. The baloney about "right adjoint" means simply that
a function from A to HOM(B,C) is the same as a function from A x B to C.
Other examples include the category of sets, the category of abelian
groups, or the category of finite-dimensional vector spaces.
In all these contexts you can set up the lambda calculus.
Hans Aberg
>>> I would like to ask a question on the relation between category theory,
>>> lambda calculus, and physics.
>At some point Joachim Lambek realized that the lambda calculus was
>basically just another way of talking about cartesian closed categories.
>More precisely, if you ask in what sort of category you can do lambda
>calculus, the answer is: a cartesian closed category.
>
>He told me that he presented this result at a conference on lambda
>calculus, naively hoping that henceforth the subject would be recognized
>as a small branch of category theory and basically go away. As an older,
>wiser man, he now realizes that not everyone doing lambda calculus *wants*
>to see it subsumed by category theory!
Here is the context I came across the question:
Let I = identity, O the constant function O(x) := I, f * g function
composition written in reverse, (f * g)(x) := g(f(x)), x ^ f := f(x)
(evaluation), and
(f + g)(a) := f(a) * g(a)
Then, as Church's original lambda calculus does not include the constants,
{ I, +, *, ^ } is a primitive set for that theory (when those objects are
written as suitable lambda expressions), whereas { O, +, *, ^ } is a
primitive set for the lambda calculus that includes constants. See the
papers by Peter Hancock <p...@dcs.ed.ac.uk>.
I just briefly mention the Church functionals (n in Natural), defined by
\bar 0 := 0
\bar n := a |-> (b |-> a(...(a(b))...)), where `a' occurs n times
One can use the operators +, * and ^ above onto the Church functionals,
and the will behave as regular integer addition, multiplication, and
exponentiation. If S is the increment function S(x) := x + 1 (usual
integer addition, then one has
(\bar n)(S)(0) = n
Now, a category is roughly a mathematical object admitting { I, * }
(skipping some logical details). If one should be able to do lambda
calculus, one must also have { +, ^ } if constants are not required, and
also adding O if constants should be required.
I came across this in building what computer scientist call "functional
covers", which are used in implementing functional languages, or languages
built upon the lambda calculus. Forget about the lambda calculus for a
moment. Then a functional cover is merely a data structure of something
that should be executed later on. So say that one does not want to execute
a functional application f(x) immediately; then one can replace it by the
object x ^ f, that is, a data structure holding references to x and f, and
knowing about the ^ evaluations that should be taken place later on.
So in this way, one will implement the lambda calculus, even though it may
not appear to be so explicitly. -- I should add that it is not practically
a good idea to try implement the lambda calculus via the operators { O, +,
*, ^ }, because the translation between lambda expressions and those
primitives will be very complicated.
But this gives hints to as why the lambda calculus may show up in other
contexts: If for some reason one needs to abstract, and make sure that
certain operations such as { O, +, *, ^ } are also available as objects,
then that will then include the lambda calculus.
Hans Aberg * Anti-spam: remove "remove." from email address.
* Email: Hans Aberg <remove...@member.ams.org>
* Home Page: <http://www.matematik.su.se/~haberg/>
* AMS member listing: <http://www.ams.org/cml/>
Chris Hillman
> At some point Joachim Lambek realized that the lambda calculus was
> basically just another way of talking about cartesian closed categories.
> More precisely, if you ask in what sort of category you can do lambda
> calculus, the answer is: a cartesian closed category.
Oh wow, that is too nice not to include in the next edition of my
"Categorical Primer". Do you have a citation?
By the way, several days I tried to post a correction to my incautious
suggestion that Choquet simplices (which are fundamental in functional
analysis and ergodic theory) might form a topos--- the trouble is that
Cartesian products of simplices obviously aren't simplices, in general!---
but that post hasn't yet appeared, so I'll try again in this remark. As
far as I can see (still haven't taken time to check very carefully, since
one would need to decide what topology to use in certain spaces and things
like that), the considerably larger category of continuous affine maps
between compact convex sets might form a topos, and that would be
interesting-- it would say mathematics can be completely "affinized", I
think.
Chris Hillman
Home Page: http://www.math.washington.edu/~hillman/
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Urs Schreiber
[snipped lots of information and references on topos theory and the
like]
> and of course the speculative paper
>
> author = {C. J. Isham and J. Butterfield},
> title = {Some possible roles for topos theory in quantum theory and
> quantum gravity},
> note = {gr-qc/9910005},
> year = 1999}
This, I have read by now. (It is concerned with proposing a certain
topos for interpretation of QM which induces a "quantum logic" with
"new truth values" such that, roughly,
"the new truth-value of a proposition is given by the set of its
consequences that are true in the old sense"
(p. 20), where "old" refers to boolean logic.)
Since the paper is concerned with "interpretation" of QM, it made me
think about the following:
The authors discuss the fact that common boolean logic is ill suited to
reason about QM, but alternative systems of logic (such as arise in
topoi different from "Set") which, basically, know more truth values
than TRUE and FALSE, might be fine. The authors see a relation to the
different interpretations of quantum mechanics (p. 19).
It seems to me that everything that can be expressed in an alternative
system of logic (such as quantum logic) can always be expressed in
boolean logic, too, (and vice versa), possibly less elegantly. To be
specific, consider the example on p. 10 of the above paper:
The (topos) category of endomaps induces a multi valued logic, where the
question "Is x in X element of the sub-endomap Z of X?" has a truth
value in the set of natural numbers including "infinity". This truth
value n is the minimum number of times that we have to apply the
function alpha associated to the endomap X to x so that alpha^n(x) is in
Z. So besides TRUE (x is in Z, n=0) and FALSE (there is no n such that
alpha^n(x) is in Z, n = infinity) there are a countable number of
"degrees" of truth.
Now, at least if the endomaps are finite sets, the situation can clearly
be described iteratively by two-valued boolean logic by associtaing a
tuple of boolean truth values to each endomap truth value. For example:
Is n infinite? No. Is n greater than 0? Yes. Greater than 1? No. Thus we
have truth value "1" in endomap logic and truth values FALSE-TRUE-FALSE
in boolean logic. I hope this does not appear to be too silly, I just
want to express the idea that one can express multi-valued propositions
by boolean propositions. (I believe this is the very reason that it is
possible to write a paper about alternative logic: We describe a new
logic in terms of the conventional boolean one. It is hard to imagine a
paper where the reader is supposed to assign truth value from
multi-valued logic to each sentence. Like: "The following definition
makes sense only when you think about it n>5 times." ;-)
Is it possible that boolean and quantum logic are related in a similar
fashion as "realistic" QM models are related to the standard QM
interpretation? Does the following (informal) diagram commute in any
sense:
boolean logic -------------> quantum logic
| | |
| | |
| | |
| | |
| | |
v v v
"realistic" QM models --------> Hilbert space formulation
(e.g Bohm, Nelson) of QM
?
> (I can't remember which of the above listed books explains the topos
> theory take on lambda calculus, but IIRC at least one of them does.)
So existence of the sought for element is guaranteed, finding it is left
as an exercise ;-) I am not sure if I have the time to work through all
this, but: Since lambda calculus can be implemented in Cartesian Closed
Categories, are these topoi? If not, is there a generalization of a CCC
so that it becomes a topos? What would be its internal logic?
Thanks for your help.
Urs Schreiber
P.S.
I am aware that there was a thread about quantum logic on s.p.r
recently.
--
eMail: Urs.Sc...@uni-essen.de
Justin Pearson
> But what's a cartesian closed category???
>
> A category is "cartesian" if it has finite products - or in other words,
> binary products and a terminal object.
One thing that got me early on is the following misanalogue:
i) Well known theorem (nice to prove as a beginning exercise in
category theory) all finite limits can be constructed only using
finite products and equalisers.
ii) Cartesian category has all finite products BUT NOT ALL FINITE
LIMITS, it just happens most examples that people give (set, a
topos) have all finite limits.
I found the whole Cartesian terminology choice a strange one; after
all, Descartes solved equations as well.
> A cartesian category is "closed"
> if for each object A, the functor A x -- (that is, taking the product with
> A) has a right adjoint. This adjoint is called HOM(A, --).
> For example, the category of finite sets is Cartesian. The terminal
> object is the one-element set, the product is the usual Cartesian one,
> and HOM(A,B) is the finite set of functions from the finite set A to
> the finite set B. The baloney about "right adjoint" means simply that
> a function from A to HOM(B,C) is the same as a function from A x B to C.
The baloney is a good way of understanding adjoints. But I don't think
I've ever eaten baloney.
> Other examples include the category of sets, the category of abelian
> groups, or the category of finite-dimensional vector spaces.
>
> In all these contexts you can set up the lambda calculus.
Typed lambda calculus.
Regards
Justin
--
Justin Pearson - Uppsala Sweden http://www.docs.uu.se/~justin
John Baez
>On 7 Mar 2001, John Baez wrote:
>> At some point Joachim Lambek realized that the lambda calculus was
>> basically just another way of talking about cartesian closed categories.
>> More precisely, if you ask in what sort of category you can do lambda
>> calculus, the answer is: a cartesian closed category.
>Oh wow, that is too nice not to include in the next edition of my
>"Categorical Primer". Do you have a citation?
No, alas - you should probably ask him. You can find his
email address here:
http://www.math.mcgill.ca/retiredmembers.html
And if you include something about this, please get the statement
straight - as someone pointed out, I was being sort of sloppy in
my statement above.
Speaking of sloppy: some of the categories I said were
Cartesian closed, weren't. The infamous James Dolan pointed
this out by email. For example, Vect has finite products
(often called "direct sums"), but it doesn't have an internal
hom that's adjoint to this sort of product - we can't invent a
vector space HOM(V,W) such that a linear map from V x W to Z
is the same as a linear map from V to HOM(W,Z). There is
of course a famous vector space HOM(V,W): the space of all
linear maps from V to W. But this is adjoint to the tensor
product of vector spaces, not the sort of product we're talking
about here. So Vect is a closed monoidal category w.r.t. tensor
product, and a cartesian category, but not cartesian closed. I
guess I made the same dumb mistake for the category of abelian
groups.
By the way, someone recently sent me some email wondering if you
really were a computer program, as you claimed on your webpage.
He said he'd emailed you but couldn't tell from your replies if
you were or not.
Hans Aberg
<Pine.OSF.4.21.010307...@goedel3.math.washington.edu>,
Chris Hillman <hil...@math.washington.edu> wrote:
>...I tried to post a correction to my incautious
>suggestion that Choquet simplices (which are fundamental in functional
>analysis and ergodic theory) might form a topos--- the trouble is that
>Cartesian products of simplices obviously aren't simplices, in general!---
Simplicial categories can be described using monads, and the latter can be
used in describing (co-)homology. Have you tried that one? -- See for
example, MacLane's book "Cat's for Math's".
Urs Schreiber
>
> On 7 Mar 2001, John Baez wrote:
>
> > At some point Joachim Lambek realized that the lambda calculus was
> > basically just another way of talking about cartesian closed categories.
> > More precisely, if you ask in what sort of category you can do lambda
> > calculus, the answer is: a cartesian closed category.
>
> Oh wow, that is too nice not to include in the next edition of my
> "Categorical Primer". Do you have a citation?
The following two references that I had given in my original posting
discuss cartesian closed categories as one model (among others) of
lambda calculus. [3] contains lots of relevant citations.
WRT my original question it should be remarked (as Justin Pearson also
pointed out elsewhere on this thread) that ccc only models *typed*
lambda calculus, which is really less interesting than the general
untyped lambda calculus that I was referring to in the context of
"primordial" formalisms. Types are introduced in typed lambda calculus
in order to avoid the "circularity" of the general calculus and to make
it more suitable for applications (mainly in computer science). But
exactly this circularity (identification of functions and arguments) is
what makes the general lambda calculus interesting as a formalism from a
more philosophical point of view. Since a similar identification should
be exhibited by objects and morphism of "omega-categories" (if there are
such) I had been asking if there is a possible relation.
For example, the general lambda calculus allows Goedel numbering and
contains an operater that takes a lambda representaion of a Goedel
number as argument and returns the lambda operator corresponding to that
number.
Urs Schreiber
[2]
@techreport{Martini:19??,
author = {Martini, Alfio},
title = {Category Theory and the Simply-Typed
$\lambda$-Calculus},
institution = {{T}echnische {U}niversit\"at {B}erlin
{F}achbereich 13},
year = {19??},
number = {96-7},
type = {{F}orschungsberichte des {Fa}chbereichs {I}nformatik},
note = {On how to provide a categorical model for the simply
typed
$\lambda$-calculus.}
}
[3]
@book{Koymans:1980,
author = {Koymans, C. P. J.},
title = {Models of the lambda calculus},
year = {1980},
publisher = {Centrum voor Wiskunde en Informatica
Centre for Mathematics and Computer Science
Amsterdam, Netherlands},
volume = {9},
series = {CWI Tract}
}
--
eMail: Urs.Sc...@uni-essen.de
url: http://www.uni-essen.de/~hnr00s
Urs Schreiber
> At some point Joachim Lambek realized that the lambda calculus was
> basically just another way of talking about cartesian closed categories.
> More precisely, if you ask in what sort of category you can do lambda
BTW, there is a neat "diagrammatic" exposition of lambda calculus at
http://www.uq.net.au/~zzdkeena/Lambda/
called:
>>>
To Dissect a Mockingbird:
A Graphical Notation for the Lambda Calculus with Animated Reduction
Abstract
The lambda calculus, and the closely related theory of combinators, are
important in the foundations of mathematics, logic and
computer science. This paper provides an informal and entertaining
introduction by means of an animated graphical notation.
<<<
It (naturally) represents lambda functions ("combinators") as machines
that have an input and an output
|
|
|
__/\__
| |
f = | f |
|_ __|
\/
|
|
|
but because of THE COOL FACT of lambda calulus, namely that the
application operator op() that applies a function on its argument
op()(f,x) = f(x)
is nothing but the *identity* operator
op() = \lambda f x . f x
(i.e. that functions and arguments are identified)
the only diagrammatic operations are "multiplexing" an input
|
|
|
/\
/ \
/ \
| |
| |
and "feeding one line into the other"
| | | |
| | | |
_|____|_ \ |
| | \ |
| op() | = \_/\
|_______| \/
| |
| |
| |
so that
f(x) =
f x f x
| | | |
| | | |
_|____|_ \ |
| | \ |
| op() | = \_/\
|_______| \/
| |
| |
| |
x
|
|
|
__/\__
| |
= | f |
|_ __|
\/
|
|
|
|
|
|
__/\__
| |
= | f(x)|
|_ __|
\/
|
|
|
For example, the diagrams of some commom combinators are
I = \lambda x . x =
|
|
|
__/|\__
| | |
| | |
|_ | _|
\|/
|
|
|
K = \lambda x y . y =
|
|
|
__/ \__
| __ |
| |I| |
| \/ |
|_ | __|
\|/
|
|
|
=
|
|
|
______/\_______
| |
| |
| |
| | |
| | |
| | |
| __/|\__ |
| | | | |
| | | | |
| |_ | _| |
| \|/ |
| | |
| | |
|_____ | ______|
\|/
|
|
and last not least, the first really interesting one:
S = \lambda x y z . x(z)(y(z)) =
| | |
| | |
| | |
| ____|_____/|
| | | |
| | | |
|__/\ |______/\
\/ \/
| |
\ |
\ |
\________/\
\/
|
|
|
To see lambda reduction at work in this framework please refer to the
above mentioned URL, I will not do that in ASCII. It's pretty cool how
one ends up with "movies" of functions that feed on each other, mutate,
pop out of each other, etc.
The diagrammatic reasoning here is somewhat different from that
presented in John Baez' quantum gravity seminar, but it is also somewhat
similar. In fact, it makes me wonder even more if there is a connection
to omega-categories: The above does not really work in typed lambda
calculus (there one would have to unnaturally decide and keep track of
which diagrams were allowed to be connected to each other) so it does
not work in cartesian closed categories.
But the central diagram
| |
| |
\ |
\ |
\_/\
\/
|
|
|
which in essence converts an object to a morphism (a line to a vertex)
should also play a role in omega-categories, shouldn't it?
Urs Schreiber
--
eMail: Urs.Sc...@uni-essen.de
Hans Aberg
>>> I would like to ask a question on the relation between category theory,
>>> lambda calculus, and physics.
>At some point Joachim Lambek realized that the lambda calculus was
>basically just another way of talking about cartesian closed categories.
>More precisely, if you ask in what sort of category you can do lambda
>calculus, the answer is: a cartesian closed category.
>
>He told me that he presented this result at a conference on lambda
>calculus, naively hoping that henceforth the subject would be recognized
>as a small branch of category theory and basically go away. As an older,
>wiser man, he now realizes that not everyone doing lambda calculus *wants*
>to see it subsumed by category theory!
In a functional (that is, using lambda calculus) computer programming
language like Haskell http://haskell.org/, categories show up as an
interpretation of the type theory.
However, such functional languages usually use "currying" (after the
mathematician Haskell Curry), that is, one uses the natural equivalence
Hom(A x B, C) = Hom(A, Hom(B, C))
f |-> (a |-> (b |-> f(a, b))
((a, b) |-> g(a)(b)) <-| g
valid in a Cartesian closed category. That is, functions f with two
arguments are always expressed as a functional f(a)(b) and not as f(a, b).
This practise possibly has to do with the kinds of type theories one is
using typically some variations of Hindley-Milner type theory.
But category theory and lambda calculus do show up together, even though
this is not category in the strict mathematical sense, but a
interpretation of it, implemented in a computer language.
One is also using type theory with quantifiers (all x): P(x), exists: P(x).
Chris Hillman
> In article
> <Pine.OSF.4.21.010307...@goedel3.math.washington.edu>,
> Chris Hillman <hil...@math.washington.edu> wrote:
>
>> the trouble is that Cartesian products of simplices obviously aren't
>> simplices, in general!
>
> Simplicial categories can be described using monads, and the latter
> can be used in describing (co-)homology. Have you tried that one? --
> See for example, MacLane's book "Cat's for Math's".
Aha!--- I have been trying to understand that section in that book for
months (on and off, mostly off, not continuously). I don't know why I
find it so difficult to follow. If you are anyone can summarize the gist,
that might help me get over whatever mental block is holding me up.
Chris Hillman
Home Page: http://www.math.washington.edu/~hillman/
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Hans Aberg
<Urs.Sc...@uni-essen.de> wrote:
> I would like to ask a question on the relation between category theory,
>lambda calculus, and physics. (This posting became somewhat longer than
>expected, the question is found at the end.)
...
>Now, there have been attempts to construct "models" of lambda calculus
>within category theory. For example, in [2] concepts of lambda calculus like
>"abstraction" and "application" are identified with certain morphisms
>within Cartesian closed categories. However, this process looks awkward to
>me, since it somehow destroys the pretty symmetry and elegance of the
>concepts of lambda calculus. The central object that a model of lambda
>calculus has to provide is a somewhat circular set \Lambda, whose elements
>have an interpretation as functions with domain and range \Lambda itself
>(e.g. [3]). *Naively*, the most natural way to model this within category
>theory appears to me to identify lambda functions with objects (or
>morphisms) of n-categories, for n -> infinity! I am aware (learned that
>from this group and John Baez's web site) that defining n-categories for
>arbitrary n is an unsolved problem, not to mention letting n go to
>infinity. But an object of an (n=infinity)-category (supposed that it is
>possible to give this idea a precise meaning) would be at the same time a
>morphism of that category and vice versa. Thus, such objects could possibly
>be identified with lambda functions.
Here is suggestion on how n-categories may enter the picture:
In another post in this thread, I described how the operators
{ O, +, *, ^ } are a primitive set for (i.e. generates) the
lambda calculus that includes the constants.
It is easy to show that these operations have the usual expected
distributivity laws on the RHS (right hand side), and further,
{ O, + } and { I, * } are monoids.
But for the LHS rules, one has to put a functor requirement:
c functor (i.e., c(a*b) = c(a)*c(b), c(I) = I) =>
(a*b)^c = a^c * b^c
(a+b)*c = a*c + b*c
I^c = I
So it means that if one wants to have such relations, the category must
have objects that are categories that are categories. The thing perhaps
then iterates, so that one ends up with a category containing all n
categories.
Urs Schreiber
> >> I would like to ask a question on the relation between category theory,
> >> lambda calculus, and physics.
> I didn't see the post where you actually asked the question, just
> this reply by Hillman. So I can't answer your question.
I knew about lambda calculus in Cartesian Closed Categories. Part of my
question was if there is a connection between lambda calculus and n-categories
for "n to infinity" (is this officially called "omega-categories"?). Since in
lambda calculus a function is both "object" and "morphism", in a sense, and the
same should be true for objects and morphisms of omega-categories, I wondered
if lambda calculus might be more naturally related to "omega-categories" than
to other models (like CCC or Scott's models).
Urs Schreiber
Hans Aberg
<Pine.OSF.4.21.010309...@goedel2.math.washington.edu>,
Chris Hillman <hil...@math.washington.edu> wrote:
>> Simplicial categories can be described using monads, and the latter
>> can be used in describing (co-)homology. Have you tried that one? --
>> See for example, MacLane's book "Cat's for Math's".
>Aha!--- I have been trying to understand that section in that book for
>months (on and off, mostly off, not continuously). I don't know why I
>find it so difficult to follow. If you are anyone can summarize the gist,
>that might help me get over whatever mental block is holding me up.
According loc.cit., ch. 7.6, any comonad induces a simplicial object. If
the category of the comonad is Abelian, one can compute the usual
cohomology. There is an example given how to get the standard group
cohomology this way.
It does not say anything if say homology can be gotten from monads.
Archisman Rudra
> Simplicial categories can be described using monads, and the latter can be
> used in describing (co-)homology. Have you tried that one? -- See for
> example, MacLane's book "Cat's for Math's".
Are these the same monads that show up in haskell io, for example? I
know, this might be slightly off topic, but it sort of leads on
directly from the ideas of categories etc., and for some people might
give a more concrete idea of what categories are.
thanks,
archi
Hans Aberg
<ar...@mosaic.mrl.nyu.edu> wrote:
>> Simplicial categories can be described using monads, and the latter can be
>> used in describing (co-)homology. Have you tried that one? -- See for
>> example, MacLane's book "Cat's for Math's".
>
>Are these the same monads that show up in haskell io, for example? I
>know, this might be slightly off topic, but it sort of leads on
>directly from the ideas of categories etc., and for some people might
>give a more concrete idea of what categories are.
The monads used in Haskell are taken straightly out of math, so in that
sense they are the same, but on the other hand, it is only a computer
language interpretation of it, so viewed as mathematical objects, they are
not the same.
One interprets the Haskell types as categorical objects, and the arrows of
this type theory as categorical homomorphism. But there is no way to
ensure that the monad relations are true: Computers do not have a
mechanism to ensure such relations are true via proofs.
Also (relating to some other stuff in this thread), the Hindley-Milner
computer theory of types is somewhat fishy, because a "type" is a
relation, not an object in the sense of category theory. It happens
because one wants to describe polymorphic types, or formulas that behave
the same when applied many different objects. Objects such as Integer,
Float, etc. are called "constant types".
Compare this with mathematical category theory: In category theory one
could use objects to describe to constant types and ranges of such.
So the effect of the use of a computer language type theory is that it
becomes more _special_. After all, one idea with the Hindley-Milner type
theory is that it is possible to infer the type of a formula.
Summing it up, when using category theory in math and physics, one
certainly get a inspiration and a feel fro it from the corresponding
objects in computer science. But those objects are only as close as the
current implementation admits, and one should not jump to any conclusions
from the computer science results and assume one will be able to have the
same in math or physics: One better check those results carefully in a
mathematically rigorous manner.
This also applies not only to the use of the code in a language like
Haskell, but also to the papers written in computer science: It appears at
least to me to be quite common that the kind of interpretations that one
has to make when implementing say monads also slips into the
interpretation made in computer science papers. So it can be quite
difficult to say what the exact mathematical formulation of a result in a
computer science papers should be.
So the suggestion is, for math and physics, always stick to the rigorous
mathematical formulations, and use the computer science equivalents as an
input.
On the other hand, I know mathematical which use Haskell to implement
things like uncountable ordinals. Again, one then gets a formula which is
similar to that the math formula, and which can be manipulated in similar
ways, but the exact relation to math is unclear. But it is possible to
extract mathematically useful results this way, using say Haskell as a
toll in the process.
Doug Merritt
> The above does not really work in typed lambda
> calculus (there one would have to unnaturally decide and keep track of
> which diagrams were allowed to be connected to each other)
This is untrue in the more interesting case of typed lambda calculii
in which all functions are total (polymorphic), which can be arranged
as trivially as by having the algebra of parameter types force a
function to return "bottom" whenever it is passed parameters of a type
that it did not specifically allow, even in the complex cases of parameter
types specified by regular expressions (or more powerful grammars) over
types.
This is more than a trivial formal variation on a theme; the (long out
of print) book "Anatomy of Lisp" accentuated this formal approach as
a means of solving a variety of issues that can arise in real world
use of languages like Lisp. (The book is about half math and half C.S.,
and is an unusual gem with deep insight.)
The basic notion of combining total functions with added element "bottom"
or the equivalent has been used very heavily in both the theoretical
math world and also in C.S. in design of higher order languages, but that's
the only reference I know from memory.
Doug
--
Professional Wild-Eyed Visionary Member, Crusaders for a Better Tomorrow
John Baez
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>Since in
>lambda calculus a function is both "object" and "morphism", in a sense, and
>the same should be true for objects and morphisms of omega-categories, I
>wondered if lambda calculus might be more naturally related to
"omega-categories" than to other models (like CCC or Scott's models).
Well, there ought to be *some* interesting relation between the
lambda calculus and omega-categories based on the idea you mention.
And in fact, Andre Joyal once suggested to James Dolan and I that
there should be some way to develop the theory of omega-categories
based on the theory of CCC's, using this idea. But nothing seems
to have come of Joyal's suggestion. There were some problems with
it, which nobody seems to have solved.
In another article, Urs Schreiber wrote:
> The above does not really work in typed lambda
> calculus (there one would have to unnaturally decide and keep track of
> which diagrams were allowed to be connected to each other)
I like those diagrams a lot! Just one comment, though:
There's nothing unnatural about imposing restrictions on which
diagrams can be hooked up to which. For example, in the theory
of spin networks, the spins must match before we can hook them
up. This is just a reflection of the fact that in a category,
we can only compose morphisms if the source of one matches the
target of the other. This is a simple example of "type-matching".
Computer scientists also like using diagrams of this sort.
Remember what the Wiz said on the first day of class:
"Let's start with diagrams of operators on vector spaces," said the Wizard.
"By "vector space", I mean a finite dimensional complex vector space.
Suppose we have a linear operator T from the space V to the space W.
Now, we could write this as T: V -> W, but I'm going to do it differently.".
And, waving his wand:
|
|
V v
|
|
/ \
| T |
\_/
|
|
W v
|
|
"You can think of this as a bit of pipe with a machine in it.
An element v of V falls through the pipe labeled "V".
Then it enters the machine labeled "T" and comes out
as an element of W falling through the pipe labeled "W".
Now, one thing we can do with linear operators is to compose them.
If T: V -> W and S: U -> V, then TS: U -> W is the composition.
We can draw TS in either of 2 ways. We can separate it as:
|
|
U v
|
|
/ \
| S |
\_/
|
|
V v
|
|
/ \
| T |
\_/
|
|
W v
|
|
or put it all together into one machine as
|
|
U v
|
|
/ \
| TS|
\_/
|
|
W v
|
|
Dr Tim Robinson
>However, such functional languages usually use "currying" (after the
>mathematician Haskell Curry), that is, one uses the natural equivalence
> Hom(A x B, C) = Hom(A, Hom(B, C))
> f |-> (a |-> (b |-> f(a, b))
> ((a, b) |-> g(a)(b)) <-| g
>valid in a Cartesian closed category. That is, functions f with two
>arguments are always expressed as a functional f(a)(b) and not as f(a, b).
>
>This practise possibly has to do with the kinds of type theories one is
>using typically some variations of Hindley-Milner type theory.
Try doing Scott-Strachey semantics with uncurried functions!
------------------------------------------------------------
When deja.com got taken over I switched to
http://MailAndNews.com
Doug Merritt
> I always wondered if [lambda calculus] were not be a perfect candidate for
> a "primordial" formalism in physics. [...]
> Say, in two hundred years or so [...]
> For the final theory I suspect
> that even the first assumption from which then everything is derived should
> look simple, (almost) obvious, and compelling after being refined for
> decades.
On the one hand, lambda calculus is, in fact, very cool, and this has
inspired a vast number of devotees and enthusiasts.
On the other hand, this is also true of other powerful yet simple systems.
Consider Euclid's geometry; consider the Peano axioms.
Consider Whitehead & Russell's Principia Mathematica, which started with
very simple postulates, and with which they hoped to re-derive all of
mathematics.
Skipping the foundational crisis that crippled their masterpiece in
a formal sense, for the purpose of the current discussion, it's even
more important that it took them, what was it, 1000 pages to prove
the corollary that "therefore 1 + 1 = 2" ?!?! I understand why they
took that approach, but IMHO, that's too much work to accomplish too
little (unless you're just out to prove a point, like they were), and
that therefore, it'd be much better to start with a more powerful
underlying system.
That's a particularly famous example, but there are thousands of other
examples that show that too simple of a starting system will typically
lead to complex proof or computational techniques. Utter simplicity does
not breed simplicity, in this sense.
I believe that what usually works is to pick, not the simplest possible
starting point, but rather to start with a system that (say) maximizes the
product of Simplicity X Powerfulness.
I've only studied "kindergarten" level category theory, but I believe
that many of its fans in this newsgroup might say that it yields a
much higher degree of powerful expressiveness in exchange for only
a small loss of simplicity, and therefore they might have reason to
think that it, rather than lambda calculus, is more likely to withstand
the passage of time, in the sense that you're talking about.
I will go out on a limb and say that the actual basis, two hundred years
hence, hasn't even been invented yet, but that it will likely be
(A) simple, but not minimally simple, and (B) surprisingly powerful
in subtle ways, without needing to throw out the goal of relative simplicity
in order to accomplish that.
It's like what's-his-name, who said, "I apologize that I didn't have enough
time to write a shorter letter." It's extremely difficult to be both
expressive and terse and the same time, and there is no reason to think
that we are anywhere close to the ultimate in these regards.
Chris Hillman
http://xxx.lanl.gov/abs/gr-qc/9910005):
> It seems to me that everything that can be expressed in an alternative
> system of logic (such as quantum logic) can always be expressed in
> boolean logic, too, (and vice versa), possibly less elegantly.
Hmm... there is a -huge- variety of "alternative logics" (algebraic
structures) out there, so I suspect this statement is probably wrong
unless by "Boolean logic" you mean not the elementary Boolean
propositional calculus but something much stronger, including quantifiers
and such like.
A safer statement would be this: any topos which can serve as the
foundation of mathematics can... serve as the foundation of mathematics
:-/ The simplest of these is probably the topos of sets, and for many
purposes it is therefore still the best choice for laying the foundations.
However, for various special purposes there are other topoi which are
clearly superior. Most people would probably prefer that there be only
one possible way of laying the foundations of mathematics, but that simply
isn't true, and I think its important to recognize this and learn to make
a virtue of what might at first appear as a source of confusion and
controversy. "It's a feature, not a bug".
> To be specific, consider the example on p. 10 of the above paper:
>
> The (topos) category of endomaps induces a multi valued logic, where the
> question "Is x in X element of the sub-endomap Z of X?" has a truth
> value in the set of natural numbers including "infinity". This truth
> value n is the minimum number of times that we have to apply the
> function alpha associated to the endomap X to x so that alpha^n(x) is in
> Z. So besides TRUE (x is in Z, n=0) and FALSE (there is no n such that
> alpha^n(x) is in Z, n = infinity) there are a countable number of
> "degrees" of truth.
>
> Now, at least if the endomaps are finite sets,
^
defined on
Of course, in dynamical systems, we almost -never- consider that
situation-- it's much too easy!
Hmm... now someone will probably mention a dynamical system on a finite
set which I have forgotten about which is hard to analyze. Come to think
of it, I suppose many cellular automata would qualify. Still, one is
generally interested in cellular automata on finite lattices only for
purposes of taking a "thermodynamic limit", if for no other reason than as
is we all know, infinite structures often admit simple answers to simple
questions, whereas this is frequently -not- true for finite structures,
where "arbitrary coincidences" of finite numbers usually prevent
"telescoping" or other simplifying phenomena. Hmm... it seems I wound up
saying that we avoid studying finite dynamical systems because they are,
being "anarchistic", too -hard-! Oh well, whatever...
> the situation can clearly be described iteratively by two-valued
> boolean logic by associtaing a tuple of boolean truth values to each
> endomap truth value. For example: Is n infinite? No. Is n greater
> than 0? Yes. Greater than 1? No. Thus we have truth value "1" in
> endomap logic and truth values FALSE-TRUE-FALSE in boolean logic. I
> hope this does not appear to be too silly, I just want to express the
> idea that one can express multi-valued propositions by boolean
> propositions.
Agreed, although to sharpen a point you have referred to yourself: this
particular construction is rather arbitrary. The constructions of topos
theory on the other hand guarantee all kinds of very powerful and
desirable properties. Topos theory is a machine, if you like, which tells
you right off not only what the right question is, but what kind of
answers that question can have. That saves a lot of time and energy.
> (I believe this is the very reason that it is possible to write a
> paper about alternative logic: We describe a new logic in terms of the
> conventional boolean one.
I wish we had a real logician hanging around (or do we already?), because
I am rapidly getting out of my depth (although I am trying to learn this
stuff properly). That said, I think this description is seriously
oversimplified. Rather, logicians seem to talk about modeling one topos
in another (say, the topos of sets). This "modeling" might involve--- if I
understand things correctly--- some rather arbitrary, unnatural, and even
ugly constructions, but it does allow you to construction the new
foundations from the old--- and vice versa!
> It is hard to imagine a paper where the reader is supposed to assign
> truth value from multi-valued logic to each sentence. Like: "The
> following definition makes sense only when you think about it n>5
> times." ;-)
I think you might not have yet appreciated the point of the example
offered by Isham and Butterfield. (By the way, for a much more extensive
discussion of the topos of M-sets, where M is a monoid--- in the IB
example the monoid is N, the natural numbers--- see the book by Goldblatt
which I cited in one of my earlier posts. Goldblatt's discussion may help
to understand the rationale for pursuing a categorical understanding of
logic and foundations.) When doing elementary propositional logic -in the
context of endomaps-, the topos of endomaps automatically gives the right
logic to use. In that context, it is very natural. In others, it might
rightly seem like overkill.
For example: G-sets (for a fixed group G) form a topos sufficiently rich
that we could in principle start all over and rewrite all our math books
using G-sets as the foundation instead of sets. That wouldn't really make
much sense because for many purposes, the action would be extra baggage.
And if I wanted to use G = Z/(3Z), someone else would no doubt prefer G =
SU(2), and so forth! But for dynamical systems, especially when one is
interested in exploring deep connections between logical, arithmetical,
combinatorial, topological, ergodic theoretic, and dynamical properties,
using the right propositional logic (and indeed, the right predicate
logic, with logical quantifiers included) is very advantageous. Here, a
particular choice of G would amount to focusing attention on a large class
of dynamical systems. For example, Z^2 actions already give a very rich
and so far very little understood class of dynamical systems. Another
obvious choice would be some nonabelian but "amenable" group.
(In principle one could hope to do ergodic theory for actions by any
group, say the free group Z*Z. Unfortunately, the Birkhoff and von
Neumann ergodic theorems, which were classically stated for Z actions [or
N actions], can be generalized in a useful way only in the case of
amenable groups. [So far as we now know.] So, one of the things you
probably have to give up in studying actions by nonamenable groups are the
types of questions one asks and tries to answer in ergodic theory.
Fortunately, there are many other important types of questions one can
still ask, and in any case, the class of amenable groups is large enough
so that ergodic theorists have not fallen into a funk about the the future
of their field--- to the contrary, exploration of actions by groups other
than Z and N is still in its infancy, but various preliminary results are
very encouraging. However, even for Z^2-actions, which is just about the
simplest case after Z, you do run right away into the fact that elementary
problems which are solvable by -effective- algorithms for Z actions are no
longer solvable by -any- algorithm. As I have already said in another
post, this "bug" is one of the motivations for pursuing Grothendieck's way
of studying dynamical systems, in which it becomes a "feature".)
> > (I can't remember which of the above listed books explains the topos
> > theory take on lambda calculus, but IIRC at least one of them does.)
I know that wasn't optimally helpful, but for once I have a good excuse!
Our Math Research Library was still closed due to earthquake damage (10:55
AM, Feb. 28, magnitude 6.8, epicenter 35 mi. from Seattle) when I wrote
that post. No kidding--- some of the bolts holding the shelves to the
walls and ceiling partially ripped out, and although none of the stacks
actually collapsed, some very nearly did, and books and journals certainly
went flying:
http://www.lib.washington.edu/Math/earthquake22001.html
In one of the pictures, if you look hard, you can see one of the places
where a bolt partially ripped out of the ceiling. By the way, in the
Engineering Library, the stacks -did- collapse. Some CS and math books I
use are (or rather, were) actually shelved on the fourth floor of that
library, which will be closed for several months while they try to repair
the damage:
http://www.lib.washington.edu/engineering/images/quake/4floor_wavex.jpg
(The QA stacks are at the far end in this picture, if I am not mistaken,
and the information theory books, in the mid ground.)
The irony is that two years ago the Industrial Engineering Program even
took the trouble to sacrifice a steer to Neptune, the ingrate! OK,
-officially- it was called an "outdoor barbecue", but everyone knew what
it was -really- all about. This year they had some bad weather and some
financial problems and couldn't afford even a small sheep, and look what
happened as a result!
By the way, the Physics-Astronomy Library was undamaged, although some
5000 items were redistributed in a rather nice demonstration of the
principle that Nature generally adores anything that increases entropy and
makes a mess:
http://www.lib.washington.edu/physics/quake.html
Anyway, the bolts in the Math Research Library were replaced last week,
and now the stacks are open again. Alas, that doesn't completely solve my
problem because people often have checked out books that this humble
software package would like to reconsult. Since no-one else in the Math.
Department will admit to an interest in category theory or logic :-/ I
assume I am being pestered by a poltergeist (one with -library
privileges-, yuk).
I can't help you with quantum logic either because I know very little
about that, in fact I know very little about quantum theory itself.
(Every now and again I try to change that, but haven't found a book I
really like yet--- what I need is "Quantum Theory for Functional Analysts
and Operator Theorists" or something like that.)
> Since lambda calculus can be implemented in Cartesian Closed
> Categories, are these topoi?
It seems that the correct statement is that -typed- lambda calculus (less
powerful) can be implemented in any cartesian closed category. Every
topos is a cartesian closed category, but the converse is definitely
-false-.
> If not, is there a generalization of a CCC so that it becomes a topos?
A topos -is- a generalization of a cartesian closed category!
Maybe you meant to ask: given a CCC, is there some procedure for
identifying a subcategory which forms a topos? If so, good question, but
I don't know any answers--- maybe someone else does.
> What would be its internal logic?
The internal logic of the topos is defined by constructions described in
detail (here and there) in the books I already cited. I also intend to
sketch the key ideas in the next edition of my "Categorical Primer".
Chris "I Try, Sometimes, Sort Of, I Guess" Hillman
Urs Schreiber
> Here is suggestion on how n-categories may enter the picture:
>
> In another post in this thread, I described how the operators
> { O, +, *, ^ } are a primitive set for (i.e. generates) the
> lambda calculus that includes the constants.
Could you please give me the representation of
K = \lambda a b . a
S = \lambda a b c . a c ( b c )
by means of {O, +, *, ^} ? I fiddled around a bit, but with no success
so far.
> It is easy to show that these operations have the usual expected
> distributivity laws on the RHS (right hand side),
Are you referring here to the fact that with
+ = A = \lambda f g x y . f x ( g x y )
* = M = \lambda f g x . f ( g x )
^ = P = \lambda f g . f g
we automatically have
1) M (A x y) z = A (M x z) (M y z)
2) P (A x y) z = M (P x z) (P y z)
for all x,y,z, but not necessarily
M z (A x y) = A (M z x) (M z y)
(except for special x,y,z, like, for example, the Church numerals)?
and further,
> { O, + } and { I, * } are monoids.
>
> But for the LHS rules, one has to put a functor requirement:
> c functor (i.e., c(a*b) = c(a)*c(b), c(I) = I) =>
a) > (a*b)^c = a^c * b^c
b) > (a+b)*c = a*c + b*c
c) > I^c = I
I do not quite see what you are saying here. a) does hold for a,b,c
numerals (or more generally for c=numeral and a*b = b*a), but we would
not want it to hold for general a,b,c. Or? On the other hand b) is 1)
above and true for all a,b,c. Or is your c above not supposed to be a
lambda function?
> So it means that if one wants to have such relations, the category must
> have objects that are categories that are categories. The thing perhaps
> then iterates, so that one ends up with a category containing all n
> categories.
I had been thinking about something like this:
Let L1 be the category whose set of objects is the set of lambda
functions and with a morphism F_A from A to B iff there is a lambda
function F such that F(A)=B. (Whith some notion of equality for lambda
functions, say \beta\eta-convertibility.)
Let L1* be the category with single object \Lambda (being the set of
lambda functions) and the set of morphisms being the set of functions
\Lambda ---> \Lambda that are continuous in the tree topology.
We now have a functor Abstr from L1 to L1* that carries all objects of L
to the single object \Lambda of L1* and maps morphisms F_A, F_B, ... of
L1 to the morphism (x |-> F(x)) of L1*.
Next, a 2-category L2 can be defined whose objects are the morphisms of
L1*. Clearly this set of objects is isomorphic to the set of objetcs of
L1 so that we can define morphisms in L2 as induced by the morphisms of
L1. By this construction L1 and L2 are "the same" and the above process
can be repeated:
Abstr "n+1" Abstr "n+1"
L1 ----> L1* ------> L2 --...--> Ln ----> Ln* ------> L(n+1)
| ^| ^ ^
| isom. || isom. | |
-------------------- --------------------------------- |
| |
|________________________________________________________|
isom.
Thus here it seems natural to consider L(\infty) with functors
Abstr
L(\infty) ------> L(\infty)
"n+1"
L(\infty) ------> L(\infty) .
Morphisms of L(\infty) are converted to objects of L(\infty) by "n+1"
and vice versa.
John Baez
>On 8 Mar 2001, Hans Aberg wrote:
>> Simplicial categories can be described using monads, and the latter
>> can be used in describing (co-)homology. Have you tried that one? --
>> See for example, MacLane's book "Cat's for Math's".
>Aha!--- I have been trying to understand that section in that book for
>months (on and off, mostly off, not continuously). I don't know why I
>find it so difficult to follow. If you are anyone can summarize the gist,
>that might help me get over whatever mental block is holding me up.
I summarized it here:
http://math.ucr.edu/home/baez/week118.html
More precisely, I described here how monads give rise to simplicial
objects. This was part of a series of This Week's Finds about basic
concepts in algebraic topology. In other issues of that series, I
described how simplicial objects give rise to (co)chain complexes.
If you combine these two ideas, you can work out the (co)homology
of pretty much any algebraic structure which can be described using
a monad. And that covers a lot of territory! Examples include the
(co)homology of groups, the (co)homology of rings, the (co)homology
of algebras, the (co)homology of Lie algebras, ... and so on.
In fact, this framework gives a unified explanation of all the
different flavors of (co)homology found in Cartan and Eilenberg's
book "Homological Algebra"!
So it's definitely worth understanding.
Here's a quick sketch of what you need to go through to understand
this stuff:
0) Learn to love Delta, the category of simplices. There are many
descriptions of it. The most basic is perhaps "the category of
finite totally ordered sets and order-preserving functions". But
make sure you see what this has to do with simplices as described
geometrically!
1) A "simplicial object" in a category C is a functor F: Delta^{op} -> C.
Learn to love simplicial objects. Learn to visualize them in all
your favorite categories. For example:
2) A simplicial object in the category of abelian groups is nothing
other than a chain complex! So we can take its homology... and
this explains why homology and simplices are so intimately related.
3) Now learn what a monoidal category is. (Technically, a "strict"
monoidal category - a category with tensor products and unit object
that obey the associative law and left/right unit laws "strictly",
i.e. as equations.)
4) Now learn what a monoid object in a monoidal category is: an object
x equipped with a product m: x tensor x -> x and unit i: 1 -> x
obeying the associative law and left/right unit laws.
5) Now learn that Delta is a monoidal category containing a monoid
object - in fact the *free* monoidal category on a monoid object!
In other words, whenever you have a monoidal category C with a monoid
object, you get a unique monoidal functor F: Delta -> C carrying
the monoid object in Delta to the monoid object in C. This is
a profound feature of Delta and the real reason simplices and
(co)homology show up in so many unexpected places in math.
6) Now put 1) and 5) together and realize that given a monoidal category
C containing a monoid object, you instantly get a simplicial object in
C^{op}. If you can find a functor from C^{op} to the category of
abelian groups, you then get a simplicial object in the category of
abelian groups, hence by 2) a chain complex! Then you can take its
homology....
7) Now realize that monads are nothing but monoid objects in
endofunctor categories.
8) Now put items 6) and 7) together and start computing the homology
of all the chain complexes you get from monads this way. Reinvent
everything in Cartan and Eilenberg... or at least see how all the
stuff in there is a special case of this idea. Generalize the idea
and apply it to physics, e.g. the cohomology of operads used in
string theory. Run wild!
Alas, this is a dried-up and desiccated sketch of a very beautiful
circle of ideas, which should really be explained in a long book full
of examples, pictures and philosophical digressions. But I don't
have time to write that now....
John Baez
Chris Hillman <hil...@math.washington.edu> wrote:
>Every topos is a cartesian closed category,
>but the converse is definitely -false-.
and then:
>A topos -is- a generalization of a cartesian closed category!
The first statement here is true, so the second one is false unless
a special case can count as a "generalization".
A topos is a special sort of cartesian closed category: one with
equalizers (and thus all finite limits) and also a subobject classifier.
......................................................................
Puzzle 14 - Where was the Republic of Rough and Ready, and why did it
secede from the United States of America?
For the answer see http://math.ucr.edu/home/baez/puzzles/14.html
David Hillman
"Theorem: A 2d topological lattice field theory is a semisimple algebra."
This seems to me to be an overly strong statement. For one thing (as was
mentioned in the seminar), the g_{i,j} was assumed to be invertible; this
doesn't follow from the topology. Also, it is not obvious to me that this
result doesn't follow only because of the particular setup chosen.
To illustrate my problem, let me give another setup for 2D TLFT. This setup
has two nice features:
1) it only involves one type of tensor, instead of two
2) in this setup, orientation reversal and Poincare duality are very simple
transformations
I assume an oriented combinatorial manifold made of glued-together polygons
(no restriction on number of sides of a polygon or degree of a vertex, except
that these are finite). Instead of coloring edges, color the manifold like
this: on each polygon, put a color next to (but not on) each vertex. So each
n-sided polygon has n colors, which go around the face in a circle in the
direction of its orientation. Similarly, each degree-n vertex has n colors
(one for each of the n adjacent faces), and these similarly go around the
vertex in a circle in the direction of its orientation.
Now, represent an edge by a (2,2) tensor T_{a,c}^{b,d}, like this:
\ /
\ a b /
? -------------- ?
/ d c \
/ \
Let the faces be oriented counterclockwise and the vertices clockwise. (Where
the two question marks are, there can be many edges hitting that vertex.) Each
edge has four colors associated with it. Here the colors are a b in the
direction of the orientation of the face on one side of the edge, and c d on
the other side of the edge. Also, we have a d in the direction of orientation
along one vertex, and c b on the other vertex. So, to understand the tensor,
picture the upper two indices written directly above the lower two, and read
straight up to follow colors along an edge in the direction of orientation,
and diagonally up to follow colors along a vertex in the direction of
orientation. Any scalar made out of T's corresponds to a compact 2D manifold.
There are three symmetries. One is a local symmetry: there is no way to
distinguish the two sides of an edge (rotate the diagram 180 degrees and we
get the same situation). So:
T_{a,c}^{b,d} = T_{c,a}^{d,b}
There are two global symmetries:
1) orientation reversal: flip all T's upside down
2) Poincare duality: switch the order of the upper indices of each T
Now for the topology moves. It turns out that these are all generated by:
1a) T_{a,c}^{b,d} T_{b,f}^{a,e} = T_{b,f}^{d,e}
1b) T_{a,c}^{d,b} T_{b,f}^{e,a} = T_{b,f}^{e,d}
2) T_{a,c}^{b,d} T_{d,f}^{e,g} = T_{a,d}^{b,g} T_{f,c}^{d,e}
Move 1a is the move that says the two edges of a two-sided face can be
collapsed to a single edge. Move 1b is the Poincare dual of this move: it says
that the two edges adjoining a degree-two vertex can be collapsed to a single
edge. (Orientation reversal leaves this move invariant.)
Move 2 is similar to the "crossing symmetry" move described in Week 16 Track 1
of the seminar. If this move is named m, then m^3 is the identity, and the
Poincare dual of m equals the orientation reversal of m equals m^2.
Okay: here would seem to be another formulation of 2D TLFTs (and because of
its simplicity and its nice behavior vis-a-vis symmetries, it seems at least
as natural as the original formulation). How do these two setups relate to one
another? What does the invertibility of g in the original setup mean in terms
of this setup? How are semisimple algebras related to these T's? What other
formulations of 2D TLFTs are possible? Can we write down a reasonable
definition of the class of all setups, and describe precisely how they are
related to one another?
John Baez
David Hillman <d...@olympus.net> wrote:
>from the Quantum Gravity Seminar, Week 16, Track 1:
>"Theorem: A 2d topological lattice field theory is a semisimple algebra."
>
>This seems to me to be an overly strong statement. For one thing (as was
>mentioned in the seminar), the g_{i,j} was assumed to be invertible; this
>doesn't follow from the topology. Also, it is not obvious to me that this
>result doesn't follow only because of the particular setup chosen.
Of course, the theorem is correct given the definition of 2d TLFT that
we're using here. But you're right that one can imagine other definitions
of "TLFT" which might give other theorems. In the thesis of my student
Sharon Newman-Gomez, she considers a generalization of the definition
given here. The main differences are:
1) one allows labellings of vertices, not just edges
2) one does not require nondegeneracy of the g_{ij}.
Dropping condition 1) means that the necessary algebraic data is
a kind of *category* with extra structure instead of just a *set*
with extra structure. Dropping 2) means that one is not forcing
semisimplicity on the situation. Nonetheless the basic ideas are
all very similar.
>To illustrate my problem, let me give another setup for 2D TLFT.
>[....]
>Okay: here would seem to be another formulation of 2D TLFTs (and because of
>its simplicity and its nice behavior vis-a-vis symmetries, it seems at least
>as natural as the original formulation).
The main way in which it's less nice is that, at least superficially,
it does not correspond to a well-understood sort of algebraic gadget.
Part of the game here is getting correspondences between "nice topology"
and "nice algebra". However, I haven't really *tried* to relate your
setup to known algebra - so it's possible that with a little work
it would turn out to a disguised version of some more familiar kind of
gadget!
>How do these two setups relate to one
>another? What does the invertibility of g in the original setup mean in terms
>of this setup? How are semisimple algebras related to these T's?
Well, you've outlined a nice project here.... and I hope you figure out
the answers! One step is to figure out the TQFTs that your new sort
of TLFT gives. (I'm pretty sure they will indeed give TQFTs.) Since
2d TQFTs are just commutative Frobenius algebras, we get a functor
from the category of your new algebraic gadgets to the category of
these well-known gadgets, which should be a step towards understanding
them.
However, this functor will almost surely lose a lot of information, so
it would better to figure out how to get your TLFTs from the traditional
ones, and/or vice versa. To do this, one should either:
1) think about different ways of formulating setup in the special case
of a manifold chopped up into triangles,
or maybe
2) think about how the usual setup extends to manifolds chopped up into
arbitrary polygons.
Part 2) has already been done and is also part of Sharon's thesis;
it's not terribly hard because any polygon can be chopped up into
triangles.
>What other
>formulations of 2D TLFTs are possible? Can we write down a reasonable
>definition of the class of all setups, and describe precisely how they are
>related to one another?
This is more ambitious. In the 2d case the results are not likely to
be very interesting to the mathematical physics community as a whole,
because 2d topology and field theory (at least of this sort) just isn't
that interesting. However, a similar program in higher dimensions might
be very interesting, and the 2d case would make a good warmup.
Hans Aberg
>>>..I described here how monads give rise to simplicial>objects. This
was part of a series of This Week's Finds about basic
concepts in algebraic topology. In other issues of that series, I
described how simplicial objects give rise to (co)chain complexes.
If you combine these two ideas, you can work out the (co)homology
of pretty much any algebraic structure which can be described using
a monad. And that covers a lot of territory! Examples include the
(co)homology of groups, the (co)homology of rings, the (co)homology
of algebras, the (co)homology of Lie algebras, ... and so on.
>>>0) Learn to love Delta, the category of simplices. There are many
descriptions of it. The most basic is perhaps "the category of
finite totally ordered sets and order-preserving functions". But
make sure you see what this has to do with simplices as described
geometrically!
>>>5) Now learn that Delta is a monoidal category containing a monoid
> object - in fact the *free* monoidal category on a monoid object!
> In other words, whenever you have a monoidal category C with a monoid
> object, you get a unique monoidal functor F: Delta -> C carrying
> the monoid object in Delta to the monoid object in C. This is
> a profound feature of Delta and the real reason simplices and
> (co)homology show up in so many unexpected places in math.
It seems that a monoid in C gives rise to monoidal functor Delta -> C,
which produces a simplicial object, from which the homology can be
computed if C is Abelian.
In the same manner, Delta^op contains the free comonoid on a single
element, so a comonoid in C gives rise to monoidal functor Delta^op -> C,
which produces a co-simplicial object (also called a "simplicial object"
by abuse of language), from which the cohomology can be computed if C is
Abelian.
Is this right?
Ralph Hartley
> functions and with a morphism F_A from A to B iff there is a lambda
> function F such that F(A)=B. (Whith some notion of equality for lambda
> functions, say \beta\eta-convertibility.)
But there is a lambda function with F(A)=B for ANY A and B, namely
lambda x.B, the constant fuction that ignores its argument and returns
B. That definition has a single morphism between each pair of
objects, so it isn't a real interesting category.
More interesting would be if there were a morphism from A to B for
EACH F such that F(A)=B. The obvious composition of morphisms would be
composition of functions, but that doesn't work because composition of
lambda functions is not associative.
I suppose you could get a category by moding out by (ab)c ~ a(bc) or
something like that. I don't know what, if anything, would be left of
the rich structure of lambda.
Ralph Hartley
Chris Hillman
> Chris Hillman <hil...@math.washington.edu> wrote:
>
> >A topos -is- a generalization of a cartesian closed category!
>
> The first statement here is true, so the second one is false unless
> a special case can count as a "generalization".
(LOL)
You are correct, of course--- that was a brain spasm on my part. Thanks
for catching this.
> A topos is a special sort of cartesian closed category: one with
> equalizers (and thus all finite limits) and also a subobject classifier.
That is what I was trying to say.
Chris Hillman
Urs Schreiber
>
> > Let L1 be the category whose set of objects is the set of lambda
> > functions and with a morphism F_A from A to B iff there is a lambda
> > function F such that F(A)=B. (Whith some notion of equality for lambda
> > functions, say \beta\eta-convertibility.)
>
> But there is a lambda function with F(A)=B for ANY A and B, namely
> lambda x.B, the constant fuction that ignores its argument and returns
> B. That definition has a single morphism between each pair of
> objects, so it isn't a real interesting category.
Instead, it has *at least* this one morphism between each pair of
objects.
> More interesting would be if there were a morphism from A to B for
> EACH F such that F(A)=B.
This is exactly what I have in mind. My wording "with a morphism F_A
from A to B iff there is a lambda function F such that F(A)=B" implies
that whenever you find any lambda function F with F(A)=B then there is a
corresponding morphism F_A from A to B, doesn't it? Another function G
with G(A)=B will correspond to *another* morphism G_A from A to B.
> The obvious composition of morphisms would be
> composition of functions, but that doesn't work because composition of
> lambda functions is not associative.
This is not correct. After all, lambda functions are *functions*
(mappings) and as such certainly have associative composition:
(f o ( g o h))(x) = f(g(h(x))) = ((f o g) o h)(x)
> I suppose you could get a category by moding out by (ab)c ~ a(bc) or
> something like that. I don't know what, if anything, would be left of
> the rich structure of lambda.
Ok, you seem to be talking about lambda calculus not being associative
when regarded as an *algebra*. But in this context the symbol "fg" does
not correspond to *composition* of f and g but to *application* f(g),
which is not associative.
However, in the category L1 that I defined above, composition of
morphisms is really composition of functions (or rather: of functional
covers), not application of functions:
F_A G_B
A ---------> B -----------> C
| ^ <=> G(F(A)) = (G o F)(A)
|____________________________|
(G o F)_A = G_B o F_A
All morphisms have this suffix _A or _B because they do not correspond
to lambda functions per se (in L1!) but to applications of them to A
resp. B. So a morphism in L1 is really rather a "functional cover" , a
combination of a function and an argument.
My motivation to define it this way is the following: I want to have
lambda functions as objects and as morphisms. But to have both at the
same time is not possible in an ordinary category. Therefore I invent
the category L1 in which the *objects* are lambda functions. But then
the only reasonable morphisms are the above "functional covers" and not
the functions themselves. Therefore I make up a functor Abstr that
"abstracts the functions from the functional covers" in that it sends
all the functional-cover-morphism in L1 to their respective
function-morphism in L1*
L1: L1*:
Abstr
F_A ----------------> F
^
Abstr |
F_B -------------------|
...
Abstr
G_A -----------------> G
...
Now, in L1* each *morphism* corresponds to a lambda function, but there
is only one single object (the set of lambda functions) that these
morphism act upon. But in the 2-category L2 of L1* morphism become
objects, so that here we have again lambda functions as objects, just
like in L1. By introducing the proper 2-morphisms (which are now
functional covers whose argument part is a function) there is a trivial
isomorphism between L1 and L2. This isomorphism relates objects in L1 to
objects in L2. But since the latter are really morphism, this is an
isomorphism identifying objects with morphisms.
Hans Aberg
>theory in physics" to "category theory in computer science". - jb]
Not really: I don't think that the current use of category theory in
computer science will be of much direct use for category theory in physics
or math, which I pointed out in another post, even though it might serve
as an input in view of that there are computer programs available which
allows one to experiment with features similar to that in use in physics
and math theories.
So whatever inputs one gets from computer science must be reworked in a
mathematically rigorous way before it can be useful in say physical
theories.
Those inputs can though become useful, if given the proper rigorous examination.
One should also note that the lack of rigorous handling in computer
science and computer programs of mathematical objects (such as categories)
is also due to the fact that the computers are not yet powerful enough to
do it fully rigorously: It is just typical of computer implementations
that they strip out whatever is time consuming to compute.
But this also means that the computers will probably improve a great deal
in this respect in the near future (say the next decade), as they now
start become truly powerful.
It also means that the interaction between math-physics-computers will
probably also increase a great deal also in the theoretical domains, where
there currently is not so much interaction.
Actually, I can give a simple philosophical example on how math, physics
and computers hang together, which I find rather intriguing:
In a uniprocessor computer, a logical expression such as x | y (x or y) is
not symmetrical (in general, x | y /= y | x), because one of x or y must
be computed before the other.
But in math, x | y = y | x; why is that so? -- Probably because
historically it is an empiric somehow extracted from physical experience.
But now put x and y in different threads in the computer, so that they are
computer in parallel; thus there is no causality that one of x and y must
come before the other. Then x | y = y | x; however, the reason one is not
implementing this way in computers is that it is slow.
But it means that when computers become more powerful (in this case able
to compute in parallel), they will be in the need of a logics which
approaches that of math, which in its turn somehow is extracted from
physical experiences.
So things do hang together, even though it is not easy to immediately see how.
Ralph Hartley
GOOD physical theory must be describable in the lambda calculus. Here
good is not exactly the same thing as "true". The Church/Turing thesis
says that lambda definability is the same as effective computability.
That means that to the extent that a theory can not be so described,
it can't be used to make any predictions either.
So the TRUE theory (whatever that means) might not fit within the
lambda calculus, but such a theory would be of no value to us. It
could not be used to predict experiments, to design bridges, or
anything.
Penrose denies the Church/Turing thesis. I think I understand his
arguments on this point, they are largely philosophical/mathematical.
They have no merit whatsoever.
The fact that Penrose is completely wrong on a subject I know well,
makes me suspicious of what he says on other topics I don't have
independent knowledge of. I just don't give him the benefit of the
doubt. How is Penrose viewed in the physics community?
Ralph Hartley
Ralph Hartley
>
> Ralph Hartley wrote:
> > More interesting would be if there were a morphism from A to B for
> > EACH F such that F(A)=B.
>
> This is exactly what I have in mind. My wording "with a morphism F_A
> from A to B iff there is a lambda function F such that F(A)=B" implies
> that whenever you find any lambda function F with F(A)=B then there is a
> corresponding morphism F_A from A to B, doesn't it? Another function G
> with G(A)=B will correspond to *another* morphism G_A from A to B.
I see how it could be read to imply that. I mistook "there is" as a
existential quantifier.
> > The obvious composition of morphisms would be
> > composition of functions, but that doesn't work because composition of
> > lambda functions is not associative.
>
> This is not correct.
...
> Ok, you seem to be talking about lambda calculus not being associative
> when regarded as an *algebra*. But in this context the symbol "fg" does
> not correspond to *composition* of f and g but to *application* f(g),
> which is not associative.
OK so FoG <--> \x.F(Gx) which does make Fo(GoH) = (FoG)oH (leaving out
the arguments for clarity). Define all the identities by I_A <--> \x.x
and we do get a category. ("\" == mangled ascii for "lambda")
What you loose is the relationship between f, g, and f(g). You need
that if you want to preserve the full structure of lambda calculus. I
assume that you are hoping to get it back from n-categories as
n->infinity. I haven't figured out yet if you do.
> All morphisms have this suffix _A or _B because they do not correspond
> to lambda functions per se (in L1!) but to applications of them to A
> resp. B. So a morphism in L1 is really rather a "functional cover" , a
> combination of a function and an argument.
Because for any lambda function F, there is more than one morphism.
There is one morphism F_A for each object A (also a lambda function),
and it is in Hom(A,F(A)).
> My motivation to define it this way is the following: I want to have
> lambda functions as objects and as morphisms. But to have both at the
> same time is not possible in an ordinary category. Therefore I invent
> the category L1 in which the *objects* are lambda functions. But then
> the only reasonable morphisms are the above "functional covers" and not
> the functions themselves. Therefore I make up a functor Abstr that
> "abstracts the functions from the functional covers" in that it sends
> all the functional-cover-morphism in L1 to their respective
> function-morphism in L1*
>
> L1: L1*:
>
> Abstr
> F_A ----------------> F
> ^
> Abstr |
> F_B -------------------|
>
So L1* = L1 mod (A=B) gotten by identifying all the objects in L1.
> Now, in L1* each *morphism* corresponds to a lambda function, but there
> is only one single object (the set of lambda functions) that these
> morphism act upon. But in the 2-category L2 of L1* morphism become
> objects, so that here we have again lambda functions as objects, just
> like in L1. By introducing the proper 2-morphisms (which are now
> functional covers whose argument part is a function)
I will assume for now that the 2-morphisms satisfy the requirements
for a 2-category.
> there is a trivial
> isomorphism between L1 and L2. This isomorphism relates objects in L1 to
> objects in L2. But since the latter are really morphism, this is an
> isomorphism identifying objects with morphisms.
So you hope that gets you a well defined n-category for every n. It
isn't clear to me if you have the full structure lambda, or not.
Doesn't that require something like morphisms between different levels
(e.g. between objects an morphisms)? You have an isomorphism between
objects and morphisms, but I thought the (a) point of category theory
was that being isomorphic was kept distinct from being equal.
I'm a little unclear about how you get from typed lambda calculus to
untyped. Maybe I need to go back to your posts again, but this post is
long enough. If I misinterpreted you again at the start, writing more
now could be a waste of time.
Ralph Hartley
John Baez
Hans Aberg <remove...@matematik.su.se> wrote:
>John Baez wrote:
>>5) Now learn that Delta is a monoidal category containing a monoid
>> object - in fact the *free* monoidal category on a monoid object!
>> In other words, whenever you have a monoidal category C with a monoid
>> object, you get a unique monoidal functor F: Delta -> C carrying
>> the monoid object in Delta to the monoid object in C. This is
>> a profound feature of Delta and the real reason simplices and
>> (co)homology show up in so many unexpected places in math.
>It seems that a monoid in C gives rise to monoidal functor Delta -> C,
>which produces a simplicial object, from which the homology can be
>computed if C is Abelian.
We need to be a bit careful: a simplicial object in C is a functor
F: Delta^{op} -> C, so a functor Delta -> C is a simplicial object
in C^{op}. So we really want C^{op} to be abelian here.
However, unless I'm hallucinating, C is abelian iff C is abelian,
since the definition of "abelian" is self-dual. Is this right???
And by the way, it's only hardcore algebraists who call a functor
F: Delta^{op} -> C of "simplicial object". Topologists would call
it an "augmented simplicial object", since the object 0 in Delta
corresponds to the "-1-simplex", which is a bit strange from the
viewpoint of topology.
However, I side with the hardcore algebraists in this dispute.
>In the same manner, Delta^op contains the free comonoid on a single
>element,
I'd prefer to say "Delta^{op} is the free monoidal category on a
comonoid object".
>so a comonoid in C gives rise to monoidal functor Delta^op -> C,
>which produces a co-simplicial object [...]
It actually produces a simplicial object in C, at least by the
hardcore algebraist's definition.
And this is the way the game is usually played: working with
comonoids rather than monoids! If you know about "comonads"
you'll find this perfectly pleasant.
In this game C is often not abelian. However, we often have a
functor F: C -> A where A *is* abelian; then we can push our
simplicial object in C forwards to get a simplicial object in A,
and take the homology of *that*.
John Baez
Ralph Hartley <har...@aic.nrl.navy.mil> wrote:
>How is Penrose viewed in the physics community?
He is greatly respected for his work on general relativity and
quasicrystals (Penrose tiles), but regarded with great suspicion
when it comes to his thoughts on quantum mechanics and consciousness,
as expounded in his popular books. There's a curiously large gap
between the two perceptions.
Urs Schreiber
> Hmm, I knew I don't know lambda calculus, but I thought I at least
> understoodd the basic idea. Now I'm entirely confused... How can it be
> non-associative???
Composition of lambda functions is certainly associative. Lambda
calculus "not being associative" refers to the non-associativity of the
"product operation"
A . B = AB = A(B) = A evaluated at B
of the underlying "applicative structure" {\Lambda, "."} (combinatory
algebra).
Here, of course,
(A . B) . C = (AB)C = A(B)(C) =/= A(B(C)) = A(BC) = A . (B . C)
--
eMail: Urs.Sc...@uni-essen.de
Hans Aberg
>>More interesting would be if there were a morphism from A to B for
>>composition of functions, but that doesn't work because composition of
>>lambda functions is not associative.
>
>understoodd the basic idea. Now I'm entirely confused... How can it be
>non-associative???
This has already been corrected in another post: Lambda functions
compositions are associative. The poster above simply has been confused by
the fact that in the style of notation common in lambda calculus, f(x) and
f o g looks the same: One simply has to think a bit more of what the
formulas actually mean.
This is one reason I translated those formulas into something that one is
used to in standard math.
Hans Aberg
<Urs.Sc...@uni-essen.de> wrote:
> I would like to ask a question on the relation between category theory,
>lambda calculus, and physics. (This posting became somewhat longer than
>expected, the question is found at the end.)
There are known relations between category thoery (extensions) & lambda
calculus.
There is a message at the Haskell mailing list <http://haskell.org/>
1999/06/07, which I take the liberty to quote:
From: Christoph Lueth <c...@Informatik.Uni-Bremen.DE>
Date: 07 Jun 1999 13:11:50 +0200
Hans Aberg <hab...@matematik.su.se> writes:
> Exactly how is this connection between the lambda calculus and
> category theory described? -- That is, one would expect to know that
> if one has a category of some sort, it is equivalent to the lambda
> calculus, or something like that.
There is a very beautiful connection here, which is explained at
length e.g. in [1]. In short, every simply typed lambda-calculus gives
rise to a a cartesian closed category (ccc) (by taking its term
model). On the other hand, every ccc gives us a simply typed
lambda-calculus, by considering its so-called internal language, which
has its objects as types, and its morphisms as terms: a morphism
f:A--> B is a a term of B in the context of a variable (or a tuple of
variables) of type A. These two mappings are adjoint (the internal
language construction is the right adjoint).
To get the untyped lambda-calculus, you move from ccc's to so-called
c-monoids, but I've forgotten the exact definition of that; it's like
a ccc, but without a terminal object I think to eliminate trivial
(one-point) models.
--Christoph.
[1]
@Book{LambekScott,
author = "J.~Lambek and P.~J.~Scott",
title = "Introduction to Higher Order Categorical Logic",
publisher = "Cambridge University Press",
year = 1986,
volume = 7,
series = "Cambridge studies in advanced mathematics"
Ralph Hartley
> This is exactly what I have in mind. My wording "with a morphism F_A
> from A to B iff there is a lambda function F such that F(A)=B" implies
> that whenever you find any lambda function F with F(A)=B then there is a
> corresponding morphism F_A from A to B, doesn't it? Another function G
> with G(A)=B will correspond to *another* morphism G_A from A to B.
I also notice that for any object A, there are always at least 2
morphisms from A to A:
I_A <-> (\x.x)(A)
and
(KA)_A <-> (KA)(A) = ((\x.(\y.x))A)(A) = (\x.A)(A)
I think I can even prove that there are infinitely many morphisms
between any pair of objects.
If I understand the definitions correctly, that implies there are
neither initial nor final objects in the category. So it isn't a
Cartesian category.
I don't mean to imply that you said it was (you didn't), I just find
it interesting, considering the connections (which I don't yet
understand very well) that people keep making between closed Cartesian
categories and lambda calculus.
Ralph Hartley
Daryl McCullough
I don't want to pile on Penrose, but in his popular book
_The Emperor's New Mind_ he mentions Sakharov's ideas about
gravity, only to say that they can't possibly be correct.
Since this is a claim about General Relativity, which is
Penrose' area of competence, I take it more seriously.
Sakharov's idea is described in Misner, Thorne, and Wheeler's
_Gravitation_. In one sentence, his idea is that
"...gravity could be understood as an induced effect brought about
by changes in the quantum fluctuation energy of the vacuum due
to the presence of matter"
This quote was found in a paper by Puthoff, available online at
http://www.ldolphin.org/energetic.html. Look there for a reference
to Sakharov's original 1968 paper.
Do most physicists agree with Penrose' negative assessment of
Sakharov's idea? It seemed to me that the problem with Sakharov's
idea wasn't that it was wrong, but that it didn't really predict
anything new.
--
Daryl McCullough
CoGenTex, Inc.
Ithaca, NY
Urs Schreiber
> Actually, I can give a simple philosophical example on how math, physics
> and computers hang together, which I find rather intriguing:
>
> In a uniprocessor computer, a logical expression such as x | y (x or y) is
> not symmetrical (in general, x | y /= y | x), because one of x or y must
> be computed before the other.
>
> But in math, x | y = y | x; why is that so? -- Probably because
> historically it is an empiric somehow extracted from physical experience.
and concluded
> So things do hang together, even though it is not easy to immediately see
> how.
This observations has a close relationship to the ideas expressed in
Louis Kauffman, "Noncommutativity and Discrete Physics", q-alg/9709012
(and references therein)
which is essentially concerned with the point that operations "with side
effect" (that the above quotation gives an example of) give rise to
certain noncommutativity relations which are foreign to classical
physics but are exactly what one encounters in quantum physics. Thus the
above mentioned incongruency between "the way computers work" and "the
way physics works" might proof to be a classical illusion and instead
"the way computers work" may be closely related to "the way quantum
physics works".
The above example of the noncommutativity of the arguments in a
computer-evaluated OR expression hinges on the temporal assymetry that
one argument is evaluated *after* the other. This notion of ordering
operators not only algebraically by their applicative order but also
"temporally" (i.e. by some external parameter t) is formalized in the
above mentioned paper:
The author considers a ring of elements, that are indexed by a discrete
parameter t, and introduces an additional element, J, which behaves like
the identity algebraically but furthermore produces the *side effect* of
incrementing t by one unit. Applying this operator J formalizes the
notion of applying an operator now or "later". E.g., for some element
X_t of the ring we have ("A B" means composition, resp. ring
multiplication of A and B):
X_t J = X_{t+1}
Read: First J increases time by 1, then X is applied *at that new time*.
The derivative of an operator, its discrete rate of change, can thus be
discribed by
DX := X_{t+1} - X_t = X J - J X = [X,J]
(The appearance of J in the second term assures that DX has the
consistent side effect of increasing t by one unit.)
The author can show that from this rather simple construction a wealth
of formulas and relationships can be derived that, at least, very much
look like (quantum) physics. For example, the equation DX = [X,J]
immediately suggest to identify J with the Hamiltonian.
Of course, in a usual computer program a function can possibly have much
more severe side effects than just increasing the value of a single
global variable.
Regarding lambda calculus:
--------------------------
I once wondered if and how lambda calculus could get in direct contact
with physics. For that matter I tried to find a lambda representation of
the commutator of two operators. Most lambda operators do not commute
under composition
M a b := a o b := \lambda x . a ( b x ) ,
in general
a o b \neq b o a ,
but the problem is how to represent the "difference" of A o B and B o A,
since lambda operators do not form a vector space. (This immediately
leads to the idea that one could construct the free vector space over
the set of lambda functions. Has this been examined by anyone?) On the
other hand, the addition operator of Church numerals is a natural
addition with respect to the composition of functions, so that one could
perhaps define the commutator [a,b] of a and b (if it exists) by
a o b = (b o a) + [a,b]
where "+" is the Church addition
A a b = a + b = \lambda x y . a x (b x y) .
On the other hand, there is a set of lambda functions (the "hereditary
permutations", HPs) that are invertible with respect to the above
multiplication and addition operations. For this restricted set the
equation
[a,b] = (a o b) - (b o a)
is defineable by
a - b = \lambda x . (a x) o INV(b x)
where INV is the operator that inverts an element of the hereditary
permutations. (Such an operator is always defineable for a subset HPn of
the hereditary permutations that contains all elements X with X^n = Id
by
INV x = x^(n-1) ).
The question here is if the HPs are rich enough to model anything
physical.
another idea (discrete time, cellular automata, lambda automata):
-----------------------------------------------------------------
On page 7 of the above mentioned paper the author expresses the point
that a discrete calculus as presented in his paper fits the model of the
universe as a cellular automaton, where each step of time is a discrete
step in the calculation of the automaton.
Since a cellular automaton is effectively computable, it is lambda
representable. However, a usual implementation will have a huge and very
awkward lambda formula. (As always: It is nice to reason about lambda
calculus, but one does not want to write programs in it). Therefore I
came to wonder if there might be some merit in studying "lambda
automata" instead of the cellular ones, by which I mean the following:
The fascination of cellular automata is due to the wealth of structure
that arises from simple rules applied iteratively. Translating "wealth
of structure by simple iterative rules" to lambda calculus makes me
think about simple lambda functions *without* normal form (i.e. those
that are never done evaluating themselves). By applying a reduction
scheme to them, some will produce an ever more complex formula, or "Bohm
tree", that will iteratively grow and mutate. This might even produce
"game of life"-like visual effects when displayed appropriately, e.g.
like on
http://www.uq.net.au/~zzdkeena/Lambda/ . A step of time may naturally
and consistently be identified with a reduction of all "innermost"
redexes.
Consider for example the "lambda automaton"
A = \lambda s a b . s s (s a (s b)) (s b (s a))
with the "initial value"
A A A A .
The (admittedly very vague) question behind all this is: Could this have
a physical meaning? The reasoning being: It seems possible that discrete
models and concepts of computation (like space-time-graphs, cellular
automata) play an important role in fundamental physics. Since lambda
calculus is, in a way, the ultimate "concept of computation", a lambda
formula being a "Feynman graph of computation", so to say, could it
reveal a more closer look at the inner workings of anything?
Hans Aberg
>We need to be a bit careful: a simplicial object in C is a functor
>F: Delta^{op} -> C, so a functor Delta -> C is a simplicial object
>in C^{op}. So we really want C^{op} to be abelian here.
>
>However, unless I'm hallucinating, C is abelian iff C is abelian,
>since the definition of "abelian" is self-dual. Is this right???
Yes, so it seems. For every diagram chasing theorem, there is a dual
theorem, I recall.
>And by the way, it's only hardcore algebraists who call a functor
>F: Delta^{op} -> C of "simplicial object". Topologists would call
>it an "augmented simplicial object", since the object 0 in Delta
>corresponds to the "-1-simplex", which is a bit strange from the
>viewpoint of topology.
Are you sure your dimension haven't slipped one step? -- The way it looks
in MacLane, "Cat's for Math's", ch 7.6, "Monads & Homology", he gets the
usual unreduced cohomology out of a comonad. In topology, H^0 is free
Abelian with basis the components of the topological space. It the space
ic conncected, one can reduce the (co)homology, by looking at the
(co)kernel of the map to(form?) a point.
>And this is the way the game is usually played: working with
>comonoids rather than monoids! If you know about "comonads"
>you'll find this perfectly pleasant.
The comonoids give rise to a cohomology, that is clear. But were you able
to figure out how the homology is produced (if it is via monads or
whatever)?
John Baez
Daryl McCullough <da...@cogentex.com> wrote:
>I don't want to pile on Penrose, but in his popular book
>_The Emperor's New Mind_ he mentions Sakharov's ideas about
>gravity, only to say that they can't possibly be correct.
Did he say why? It's hard to evaluate his claim without
hearing his evidence. It's true that a simple-minded application
of Sakharov's idea gives a ridiculously large figure for the
cosmological constant. On the other hand, this problem plagues
most other theories of quantum gravity, too. So it's hard for
me to see why Sakharov's ideas "can't possibly be correct", except
insofar as *all* existing ideas on quantum gravity can't possibly
be correct - i.e., they can't be the last word on the matter.
>Do most physicists agree with Penrose' negative assessment of
>Sakharov's idea?
It's hard to know exactly what to do with Sakharov's idea, so
it hasn't become the basis for a program to quantize gravity.
But I don't see people running around saying it "can't possibly
be correct". And in fact, Ted Jacobson has used a very similar
idea to explain why the area-entropy law
S = (1/4) (A/L^2) (L = Planck length)
should work with the same constant no matter how what species of
elementary particles there are. (Having lots of particles around
affects S, but it also affects the gravitational constant and thus
L!)
John Baez
Hans Aberg <remove...@matematik.su.se> wrote:
>In article <99958v$ach$1...@glue.ucr.edu>, ba...@galaxy.ucr.edu (John Baez) wrote:
>>We need to be a bit careful: a simplicial object in C is a functor
>>F: Delta^{op} -> C, so a functor Delta -> C is a simplicial object
>>in C^{op}. So we really want C^{op} to be abelian here.
>>
>>However, unless I'm hallucinating, C is abelian iff C is abelian,
>>since the definition of "abelian" is self-dual. Is this right???
>Yes, so it seems.
Heh. I meant to write "C^{op} is abelian iff C is abelian", and
apparently that's what you read, but it's not what I actually wrote. :-)
>For every diagram chasing theorem, there is a dual theorem, I recall.
Okay, but what got me scared (and caused my hesitancy) was the
sudden realization that I don't know what C^{op} is like for all
the abelian categories C that I know. I thought there was some
theorem saying any abelian category can be thought of as a subcategory
of the category of R-modules of some ring, or something like that.
What does this theorem say about C^{op} for the abelian categories
C that I know and love?
If C is the category of finite-dimensional real or complex vector
spaces, C^{op} is equivalent to C. But what about the category of
abelian groups, or more generally R-modules for some ring R? I can't
see why these would be equivalent to their opposites....
>>And by the way, it's only hardcore algebraists who call a functor
>>F: Delta^{op} -> C of "simplicial object". Topologists would call
>>it an "augmented simplicial object", since the object 0 in Delta
>>corresponds to the "-1-simplex", which is a bit strange from the
>>viewpoint of topology.
>Are you sure your dimension haven't slipped one step? -- The way it looks
>in MacLane, "Cat's for Math's", ch 7.6, "Monads & Homology", he gets the
>usual unreduced cohomology out of a comonad. In topology, H^0 is free
>Abelian with basis the components of the topological space. It the space
>ic conncected, one can reduce the (co)homology, by looking at the
>(co)kernel of the map to(form?) a point.
He probably pulls a slight trick somewhere, restricting from the
"algebraist's Delta" to the "topologist's Delta" (which he calls
something else, like Delta_0). The topologist's Delta is a
subcategory of algebraist's, so any simplicial object of the
algebraist's sort effortlessly gives one of the topologist's sort.
>>And this is the way the game is usually played: working with
>>comonoids rather than monoids! If you know about "comonads"
>>you'll find this perfectly pleasant.
>The comonoids give rise to a cohomology, that is clear. But were you able
>to figure out how the homology is produced (if it is via monads or
>whatever)?
Wait a minute. A comonoid object in a category C gives a simplicial
object in C. If C is abelian, this gives a chain complex in C.
We can then take the homology of this chain complex. We can also
hom this chain complex into an object of C, getting a cochain complex
in AbGp, and take the cohomology of that. (This is called cohomology
"with coefficients in" the given object of C.)
That's how I usually think about it. But now you're persuading me that
if C is abelian, so is C^{op}. If so, a monoid object in C gives
a comonoid object in C^{op} and thus a simplicial object in C^{op}...
so we can take the homology or cohomology of this, as well!
Alternatively, a monoid object in C gives a cosimplicial object in C.
And you seem to be claiming that this gives a cochain complex in C!
Sure, why not? If so, we can take the cohomology of this cochain
complex, or else hom it into an object in C and take the homology
of the resulting chain complex!
And unless I'm confused, the last two paragraphs are just different
ways of talking about the same thing.
But I'm sure most people must think I'm talking gobbledygook, and
this is drifting away from physics, I'll stop here.
Hans Aberg
>To get back to physics, I think it could be reasonably argued that a
>GOOD physical theory must be describable in the lambda calculus.
I doubt it myself: We know that QM does not allow full predictions via
observations, in view of the Heisenberg uncertainty principle: There is no
way to get full information about the quantum fields.
>The Church/Turing thesis
>says that lambda definability is the same as effective computability.
I am not so sure about this particular interpretation.
From the point of view of standard math, constructive math is a subset
with special properties.
>That means that to the extent that a theory can not be so described,
>it can't be used to make any predictions either.
That is not true: It means that in standard math, there is no guarantee of
finding a proof of statement. Which means that one has to make a separate
investigation on that particular question, instead of relying on a general
mechanism, as in constructive math.
From the practical point of view, there is little difference, as one
anyhow must make sure to make a theory useful, and it does make little
difference exactly how that is being done.
Instead, the hunch I have is that if one somehow is enriching category
theory somehow, perhaps via n-categories and the relations I posted, then
that will contain lambda theory. If such a theory now is used in physics,
that will give hints of what role (the mathematical version of) lambda
theory will play in physics.
Hans Aberg
>> Though forgetting to cite himself, it was Hans Aberg who wrote:
>>> John Baez wrote:
>>>However, unless I'm hallucinating, C is Abelian iff C is Abelian,
>>>since the definition of "Abelian" is self-dual. Is this right???
>>Yes, so it seems.
>Heh. I meant to write "C^{op} is Abelian iff C is Abelian", and
>apparently that's what you read, but it's not what I actually wrote. :-)
Yes, I was blind at to your intent as well. :-) I guess this has something
to do with all those mathematical errors that are correctable... :-)
>Okay, but what got me scared (and caused my hesitancy) was the
>sudden realization that I don't know what C^{op} is like for all
>the Abelian categories C that I know. I thought there was some
>theorem saying any Abelian category can be thought of as a subcategory
>of the category of R-modules of some ring, or something like that.
>What does this theorem say about C^{op} for the Abelian categories
>C that I know and love?
Such theorems depends on the axiom of choice, which does not give any clue
how to do things in explicit computation.
For example every vectorspace has a basis. But which basis has an infinite
product \prod_I R?
In fact, this is the reason some math fellows became constructivists,
because of the moral seeming moral dilemma.
On the other hand, in QM, one regularly works with infinite dimensional
vectorspaces for which one does not know how to extract an explicit basis:
QM solutions just extract some small explicit subspaces.
>>The comonoids give rise to a cohomology, that is clear. But were you able
>>to figure out how the homology is produced (if it is via monads or
>>whatever)?
>Wait a minute. A comonoid object in a category C gives a simplicial
>object in C. If C is Abelian, this gives a chain complex in C.
>We can then take the homology of this chain complex. We can also
>hom this chain complex into an object of C, getting a cochain complex
>in AbGp, and take the cohomology of that. (This is called cohomology
>"with coefficients in" the given object of C.)
>That's how I usually think about it. But now you're persuading me that
>if C is Abelian, so is C^{op}. If so, a monoid object in C gives
>a comonoid object in C^{op} and thus a simplicial object in C^{op}...
>so we can take the homology or cohomology of this, as well!
>Alternatively, a monoid object in C gives a cosimplicial object in C.
>And you seem to be claiming that this gives a cochain complex in C!
>Sure, why not? If so, we can take the cohomology of this cochain
>complex, or else hom it into an object in C and take the homology
>of the resulting chain complex!
Strictly speaking, there is only one homology operator H, which can be
applied to a pair (A, delta), where A is an Abelian object, and delta: A
-> A is a homomorphism satisfying delta^2 = 0.
A can have a grading, and delta can be graded with respect to that. The
degree of delta is -1, one gets "cohomology", and if the degree is 1, one
gets "homology".
>And unless I'm confused, the last two paragraphs are just different
>ways of talking about the same thing.
So that was my impression. But I do not know, and thought it would be known.
>But I'm sure most people must think I'm talking gobbledygook, and
>this is drifting away from physics, I'll stop here.
Perhaps. But cohomology shows up as constructions to compute the solutions
of differential equations, as via Sobolev spaces, which produce the
Hilbert spaces that show up in QM. So the distance is not as far as it may
appear.
Dan Christensen
> Heh. I meant to write "C^{op} is abelian iff C is abelian"
This is true. The axioms are self-dual. Moreover, the homology
of a chain complex in C is the same as the cohomology of the
corresponding cochain complex in C^{op}.
> Okay, but what got me scared (and caused my hesitancy) was the
> sudden realization that I don't know what C^{op} is like for all
> the abelian categories C that I know.
It's usually not much like categories we are familiar with.
For example, filtered colimits are typically not exact.
> I thought there was some
> theorem saying any abelian category can be thought of as a subcategory
> of the category of R-modules of some ring, or something like that.
> What does this theorem say about C^{op} for the abelian categories
> C that I know and love?
I think there is more than one such theorem, but one I can quote
precisely states that for any *small* abelian category C there is a
ring R and a full and faithful exact functor from C to R-modules.
("Small" means that C has only a set of objects, as opposed to a
proper class. The existence of such a functor means that C can be
regarded as a full subcategory of R-modules, with exactness in the
subcategory agreeing with exactness in the larger category.) Since
this only works for small categories, you can't apply it to the
opposite category of R-modules. What it does let you do is take
any small diagram you are considering, and view it as being inside
a category of R-modules. But, when you consider a different diagram,
you may have to pass to a different category of R-modules.
--
Dan Christensen
jdc+...@uwo.ca
Hans Aberg
> states that for any *small* abelian category C there is a
>ring R and a full and faithful exact functor from C to R-modules.
>("Small" means that C has only a set of objects, as opposed to a
>proper class.
Some call all statements "sets", that is, a set is just an expression of
the form { x | P(x) }, where P is any logical statement. Then a small set
is a set which can be constructed from the set theory axioms.(axiom of
choice etc.).
> The existence of such a functor means that C can be
>regarded as a full subcategory of R-modules, with exactness in the
>subcategory agreeing with exactness in the larger category.) Since
>this only works for small categories, you can't apply it to the
>opposite category of R-modules.
I don't the restriction to a small category is any essential limitation
here, because the essential mathematics usually belongs in a subcategory
which is small.
For example, just when taking the ordinary (co)homology of an algebraic
object, one gets into a problems because it is really only defined up to
an isomorphism, which is problematic if the Abelian category isn't small.
But then one can reason that the important objects one ever will study
belong to a small category. Then there is not any problem.
>From the technical point of view, one can get around it by introducing
derived categories. This produces a cleaner theory. But I figure that in
explicit computations, one will get back to the usual (co)homology
constructions.
But really no one working with (co)homology and homology in order to
produce explicit results (like a working theoretical physicist and most
mathematicians) should have to worry much about this problem.
> What it does let you do is take
>any small diagram you are considering, and view it as being inside
>a category of R-modules. But, when you consider a different diagram,
>you may have to pass to a different category of R-modules.
So this should not be any bother, as the interesting diagrams are likely
to be small (or having small equivalents).
I think the problem is really that the axioms of set theory, especially
the axiom of choice, is such that one cannot make the constructions
explicit.
David Treumann
>
> If C is the category of finite-dimensional real or complex vector
> spaces, C^{op} is equivalent to C. But what about the category of
> abelian groups, or more generally R-modules for some ring R? I can't
> see why these would be equivalent to their opposites....
You can look at Ab(A,S^1) as a closed subset of the topological product of
A copies of S^1. Ab(A,S^1) is thus a compact abelian group. (That is, an
abelian group object in the category of compact spaces). The functor
Ab(-,S^1) is an equivalence of categories between Ab^op and the category of
compact abelian groups (I think it would be cute to call this category ab).
This is called 'Pontrjagin duality'.
As far as R-modules go, if M is an R-Module, UM the underlying group, we
have an adjunction Ab(UM,S^1) ~ R-Mod(M,S^1 (x) R). So R-Mod^op is
equivalent the category of compact R-Modules.
I think that the ring in the Mitchell embedding theorem is the ring of
endomorphisms of the injective cogenerator. For R-Mod^op, this would be the
projective generator of R-Mod, which is R, right? But some of the other
posts in this thread make me think this is wrong.
If C is a category with finite limits, what are some sufficient conditions
for the category of abelian group objects in C to be an abelian category?
It would be nice if those conditions were satisfied by sets and compact
spaces.
David
My e-mail address does not have a q in it.
John Baez
David Treumann <dt...@q.yahoo.com> wrote:
>John Baez wondered:
>> If C is the category of finite-dimensional real or complex vector
>> spaces, C^{op} is equivalent to C. But what about the category of
>> abelian groups, or more generally R-modules for some ring R? I can't
>> see why these would be equivalent to their opposites....
>You can look at Ab(A,S^1) as a closed subset of the topological product of
>A copies of S^1. Ab(A,S^1) is thus a compact abelian group. (That is, an
>abelian group object in the category of compact spaces). The functor
>Ab(-,S^1) is an equivalence of categories between Ab^op and the category of
>compact abelian groups (I think it would be cute to call this category ab).
>This is called 'Pontrjagin duality'.
That's cool. I should have realized that.... I supposedly know
about Pontrjagin duality.
>As far as R-modules go, if M is an R-Module, UM the underlying group, we
>have an adjunction Ab(UM,S^1) ~ R-Mod(M,S^1 (x) R). So R-Mod^op is
>equivalent the category of compact R-Modules.
Zounds!
>If C is a category with finite limits, what are some sufficient conditions
>for the category of abelian group objects in C to be an abelian category?
>It would be nice if those conditions were satisfied by sets and compact
>spaces.
Bill Rowan asked this question on the category theory mailing list.
For his question and some erudite replies, see:
http://merwan.sunyerie.edu/~alsani/cat-dist2html9900/msg00298.html