Re: Categories and functors, query

Re: Categories and functors, query

[Dana Scott]

On Sep 7, 2008, at 2:33 PM, R Brown wrote:

> There is another curiosity about the axioms for a category, namely the
> infuence of the known axioms for a groupoid (Brandt, 1926). Bill
> Cockcroft
> told me that these axioms had influenced E-M. These axioms were well
> used in
> the algebra group at Chicago.  However when I asked Sammy about this
> in 1985
> he firmly said `no, and was why the notion of groupoid did not
> appear as an
> example in the E-M paper'!
>
> Perhaps it was a case of forgetting the influence?

I certainly heard Saunders mention Brandt groupoids as examples.
(Not very good examples, since all maps are invertible.)  But, as
everyone knows, it is not the definition of a category that
is the key part, but seeing that functors and natural
transformations are interesting.

Re: Categories and functors, query

[R Brown]

Dear All,

I agree with Andre that part of the matter is sociological. It is also quite
fundamental, and is about the proper aims of mathematics. The need is for
discussion, rather than total agreement.

Miles Reid's infamous comment was "The study of category theory  for its own
sake (surely one of the most sterile of intellectual  pursuits) also
dates from this time; Grothendieck can't necessarily be  blamed
for this, [!!!] since his own use of categories was very successful in
solving problems. "  (My riposte in a paper was to suggest a game: `I can
think of a more intellectually sterile pursuit than you can!') This suggests
the view
that solving problems, presumably already formulated ones,  is the key part
of mathematics. (Miles did tell me he expected his student to use topoi or
whatever!)  A 1974 report on graduate mathematicians in employment suggested
they were good at solving problems but not so good at formulating them.

Grothendieck in one letter to me wrote on his aim for
`understanding'. (see my article on `Promoting Mathematics' on my
Popularisation web page) I believe many students come into mathematics
because they like finding out why things are true, they want to understand.
Loday told me he thought one of the strengths of French mathematics was to
try to realise this aim. By contrast, I  once
asked Frank Adams why he wrote that  a certain nonabelian cohomology was
trivial and he said `you just do a calculation' - Frank was a determined
problem solver!

So people have asked: "Where are the big theorems, the big problems,  in
category theory?" Are they there? Does it matter if they are not there?

Atiyah in his article on `20th century mathematics' (Bull LMS, 2001) talks
about the unity of mathematics, but the word `category' does not occur in
his
article. (Neither does groupoid.) He states a dichotomy between geometry
(good)and algebra (bad) but fails to recognise the combination given by,
say, Grothendieck's work, and also by higher categorical structures.
Indeed, underlying structures and processes may be of various types, all
very useful to know. I am *very* impressed by Henry Whitehead's finding so
many of these.

A word often omitted in mathematics teaching is `analogy'. Yet this is what
abstraction is about, and why it is so powerful. Category theory allows for
powerful analogies.

I am always puzzled, even horrified,  by mathematicians who use the word
`nonsense' to describe the work of others (as is all too common), yet often
themselves cannot well define professionalism in the subject. Indeed they
often cannot believe the direction others may take is chosen for good
professional reasons! They sometimes say `not mainstream'. Yet history shows
`the mainstream' shifts its course radically over the years. The lack is of
a consistent and well maintained mathematical criticism, recognising
historical trends and not just the `great man (or woman)', or famous
problem,  approach.

I believe we need to have prepared an answer to: What has category theory
done for mathematics? And indeed for evaluation of any subject areas. But a
good case is that category theory leads, or can lead,  and has led, to
clarity, to understanding and development of the rich variety of structures
there are and to be found.  However this does not rate for million $ prizes
(as it should, of course!).

When I see all the current fuss (rightly) about the LHC in Geneva, I do
wonder: who is going to speak up for mathematics, to attract students into
the subject, by getting over a message as to its value and achievements? and
also getting this message over to students studying the subject! (see
`Promoting Mathematics' and  Tim and my article on `the methodology of
mathematics')

Ronnie

www.bangor.ac.uk/r.brown/publar.html

----- Original Message -----
From: "jim stasheff" <jds@math.upenn.edu>
To: <categories@mta.ca>
Sent: Tuesday, September 09, 2008 11:22 PM
Subject: categories: Re: Categories and functors, query


> Walter,
>
> I beg to differ only with
>
> In my experience, skepticism towards category theory is often rooted in
> the fear of the "illegitimately large" size, till today.
>
> In my experience, disdain for cat theory is due to papers with a very
> high density of unfamiliar names
> reminiscent of the minutia of PST and the (in) famous comment (by some
> one) about something like:
> hereditary hemi-demi-semigroups with chain condition
>
> jim
> Tholen wrote:
>> There is another aspect to the E-M achievement that I stressed in my
>> CT06 talk for the Eilenberg - Mac Lane Session at White Point. Given the
>> extent to which 20th-century mathematics was entrenched in set theory,
>> it was a tremendous psychological step to put structure on "classes" and
>> to dare regarding these (perceived) monsters as objects that one could
>> study just as one would study individual groups or topological spaces.
>> In my experience, skepticism towards category theory is often rooted in
>> the fear of the "illegitimately large" size, till today. By comparison,
>> Brandt groupoids lived in the cozy and familiar small world, and their
>> definition was arrived at without having to leave the universe. With the
>> definition of category (and functor and natural transformation)
>> Eilenberg and Moore had to do a lot more than just repeating at the
>> monoid level what Brandt did at the group level! In my view their big
>> psychological step here is comparable to Cantor's daring to think that
>> there could be different levels of infinity.
>>
>> Cheers,
>> Walter.
>

Re: Categories and functors, query

[Toby Bartels]

Dana Scott wrote in part:

>But, as
>everyone knows, it is not the definition of a category that
>is the key part, but seeing that functors and natural
>transformations are interesting.

Indeed, the notion of natural isomorphism (or canonical isomorphism)
should be available already to groupoid theorists before 1945.
To what extent did they know about functors and natural isomorphisms,
and to what extent did Saunders & Mac Lane have to tell them?
Or, pace Walter's remarks, did they know about the ~small~ ones
but not have the guts to apply them to large classes of strucures?

It's been said before that the real insight of category theory
--as something more general than groupoids, monoids, and posets--
is the notion of adjoint functors (including limits, etc).
I'm inclined to agree, so I'm interested in why and whether
groupoid theorists thought of (and applied) that which they ~did~ have.


--Toby

Re: Categories and functors, query

[jim stasheff]

Walter,

I beg to differ only with

In my experience, skepticism towards category theory is often rooted in
the fear of the "illegitimately large" size, till today.

In my experience, disdain for cat theory is due to papers with a very
high density of unfamiliar names
reminiscent of the minutia of PST and the (in) famous comment (by some
one) about something like:
hereditary hemi-demi-semigroups with chain condition

jim
 Tholen wrote:
> There is another aspect to the E-M achievement that I stressed in my
> CT06 talk for the Eilenberg - Mac Lane Session at White Point. Given the
> extent to which 20th-century mathematics was entrenched in set theory,
> it was a tremendous psychological step to put structure on "classes" and
> to dare regarding these (perceived) monsters as objects that one could
> study just as one would study individual groups or topological spaces.
> In my experience, skepticism towards category theory is often rooted in
> the fear of the "illegitimately large" size, till today. By comparison,
> Brandt groupoids lived in the cozy and familiar small world, and their
> definition was arrived at without having to leave the universe. With the
> definition of category (and functor and natural transformation)
> Eilenberg and Moore had to do a lot more than just repeating at the
> monoid level what Brandt did at the group level! In my view their big
> psychological step here is comparable to Cantor's daring to think that
> there could be different levels of infinity.
>
> Cheers,
> Walter.

Re: Categories and functors, query

[jim stasheff]

Michael

But they must have realized that the Hurevic map is a superior
example.  Still, Steenrod must have gotten the point immediately.

You lost me there.

jim

Re: Categories and functors, query

[Nikita Danilov]

Michael Barr writes:
 >
 > This reminds me of a speculation I have often had (although Saunders
 > denied and he knew Birkhoff pretty well).  In the 30s and 40s, the word
 > "homomorphism" was regularly used but always meant surjective.  By the
 > late 40s and 50s people were talking about "homomorphism into" meaning
not
 > necessarily surjective.  So groups had lattices of subgroups and lattices
 > of quotient groups and Birkhoff invented lattice theory at least partly
in
 > the hope that the structure of those two lattices would tell you a lot
 > about the structure of the group.  I don't think this actually happened
to
 > any great extent.  But I have wondered whether Birkhoff might instead
have

Noether's `set theoretic foundations of group theory', where group
axioms are based on a notion of coset decomposition rather than
multiplication, seems to be much earlier (20s) attempt to the same:

    http://www.math.jussieu.fr/~leila/grothendieckcircle/mclarty2.pdf

 >
 > Michael

Nikita.

Re: Categories and functors, query

[tholen]

Quoting Walter Tholen <tholen@mathstat.yorku.ca>:

> definition was arrived at without having to leave the universe. With the
> definition of category (and functor and natural transformation)
> Eilenberg and Moore had to do a lot more than just repeating at the
> monoid level what Brandt did at the group level! In my view their big

OOPS -- "Moore" should read "Mac Lane", of course. (Sorry, Saunders!) W.

Re: Categories and functors, query

[Walter Tholen]

There is another aspect to the E-M achievement that I stressed in my
CT06 talk for the Eilenberg - Mac Lane Session at White Point. Given the
extent to which 20th-century mathematics was entrenched in set theory,
it was a tremendous psychological step to put structure on "classes" and
to dare regarding these (perceived) monsters as objects that one could
study just as one would study individual groups or topological spaces.
In my experience, skepticism towards category theory is often rooted in
the fear of the "illegitimately large" size, till today. By comparison,
Brandt groupoids lived in the cozy and familiar small world, and their
definition was arrived at without having to leave the universe. With the
definition of category (and functor and natural transformation)
Eilenberg and Moore had to do a lot more than just repeating at the
monoid level what Brandt did at the group level! In my view their big
psychological step here is comparable to Cantor's daring to think that
there could be different levels of infinity.

Cheers,
Walter.

Dana Scott wrote:

>
> On Sep 7, 2008, at 2:33 PM, R Brown wrote:
>
>> There is another curiosity about the axioms for a category, namely the
>> infuence of the known axioms for a groupoid (Brandt, 1926). Bill
>> Cockcroft
>> told me that these axioms had influenced E-M. These axioms were well
>> used in
>> the algebra group at Chicago.  However when I asked Sammy about this
>> in 1985
>> he firmly said `no, and was why the notion of groupoid did not
>> appear as an
>> example in the E-M paper'!
>>
>> Perhaps it was a case of forgetting the influence?
>
>
> I certainly heard Saunders mention Brandt groupoids as examples.
> (Not very good examples, since all maps are invertible.)  But, as
> everyone knows, it is not the definition of a category that
> is the key part, but seeing that functors and natural
> transformations are interesting.
>
>

Re: Categories and functors, query

[Michael Barr]

Interesting speculation, but how can we verify or refute it?  What I can
add is that when I sat in on Sammy's category theory course (called
homological algebra, but I am not sure Ext or Tor were ever mentioned), I
do not recall that he so much as mentioned groupoids.  I once mentioned to
Charles Ehresmann that he appeared to view categories as a generalization
of groupoids while Eilenberg and Mac Lane thought of them as a
generalization of posets.  Charles agreed.

This reminds me of a speculation I have often had (although Saunders
denied and he knew Birkhoff pretty well).  In the 30s and 40s, the word
"homomorphism" was regularly used but always meant surjective.  By the
late 40s and 50s people were talking about "homomorphism into" meaning not
necessarily surjective.  So groups had lattices of subgroups and lattices
of quotient groups and Birkhoff invented lattice theory at least partly in
the hope that the structure of those two lattices would tell you a lot
about the structure of the group.  I don't think this actually happened to
any great extent.  But I have wondered whether Birkhoff might instead have
invented categories had our more general notion of homomorphism been
rampant.  As I said Saunders didn't think so, but it still sounds
attractive to me.

One of the things that astonishes me about "General theory of natural
equivalences" is that they clearly knew about natural transformations in
general but chose to talk only about equivalences.  I once asked Sammy
about that and he more or less said something like one generalization at a
time.  But they must have realized that the Hurevic map is a superior
example.  Still, Steenrod must have gotten the point immediately.

Michael

On Sun, 7 Sep 2008, R Brown wrote:

> There is another curiosity about the axioms for a category, namely the
> infuence of the known axioms for a groupoid (Brandt, 1926). Bill Cockcroft
> told me that these axioms had influenced E-M. These axioms were well used in
> the algebra group at Chicago.  However when I asked Sammy about this in 1985
> he firmly said `no, and was why the notion of groupoid did not appear as an
> example in the E-M paper'!
>
> Perhaps it was a case of forgetting the influence?
>
> Ronnie
>
>
>
>
> ----- Original Message -----
> From: "Johannes Huebschmann" <huebschm@math.univ-lille1.fr>
> To: <categories@mta.ca>
> Sent: Saturday, September 06, 2008 11:48 AM
> Subject: categories: Categories and functors, query
>
>
>>  Dear All
>>
>>  I somewhat recall that, a while ago, we discussed the origins of
>>  the notions of category and functor. S. Mac Lane had once pointed
>>  out to me these origins but from my recollections we did not
>>  entirely reproduce them.
>>
>>  In his paper
>>
>>  Samuel Eilenberg and Categories, JPAA 168 (2002), 127-131
>>
>>  Saunders Mac Lane clearly pointed out the origins:
>>
>>  "Category" from Kant (which I had known all the time)
>>
>>  "Functor" from Carnap's book "Logical Syntax of Language" (which I
>>  had forgotten).
>>
>>
>>  Also I have a question, not directly related to the above issue:
>>
>>  I have seen, on some web page, a copy of
>>  the referee's report about the Eilenberg-Mac Lane paper
>>  where Eilenberg-Mac Lane spaces are introduced.
>>  I cannot find this web page (or the report)
>>  any more. Can anyone provide me with
>>  a hint where I can possibly find it?
>>
>>  Many thanks in advance
>>
>>  Johannes
>>
>>
>>
>>  HUEBSCHMANN Johannes
>>  Professeur de Mathematiques
>>  USTL, UFR de Mathematiques
>>  UMR 8524 Laboratoire Paul Painleve
>>  F-59 655 Villeneuve d'Ascq Cedex  France
>>  http://math.univ-lille1.fr/~huebschm
>>
>>  TEL. (33) 3 20 43 41 97
>>       (33) 3 20 43 42 33 (secretariat)
>>       (33) 3 20 43 48 50 (secretariat)
>>  Fax  (33) 3 20 43 43 02
>>
>>  e-mail Johannes.Huebschmann@math.univ-lille1.fr
>>
>>
>>
>
>
> --------------------------------------------------------------------------------
>
>
>
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> Checked by AVG - http://www.avg.com
> Version: 8.0.169 / Virus Database: 270.6.17/1657 - Release Date: 06/09/2008
> 20:07
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>
>

Re: Categories and functors, query

[R Brown]

There is another curiosity about the axioms for a category, namely the
infuence of the known axioms for a groupoid (Brandt, 1926). Bill Cockcroft
told me that these axioms had influenced E-M. These axioms were well used in
the algebra group at Chicago.  However when I asked Sammy about this in 1985
he firmly said `no, and was why the notion of groupoid did not appear as an
example in the E-M paper'!

Perhaps it was a case of forgetting the influence?

Ronnie




----- Original Message -----
From: "Johannes Huebschmann" <huebschm@math.univ-lille1.fr>
To: <categories@mta.ca>
Sent: Saturday, September 06, 2008 11:48 AM
Subject: categories: Categories and functors, query


> Dear All
>
> I somewhat recall that, a while ago, we discussed the origins of
> the notions of category and functor. S. Mac Lane had once pointed
> out to me these origins but from my recollections we did not
> entirely reproduce them.
>
> In his paper
>
> Samuel Eilenberg and Categories, JPAA 168 (2002), 127-131
>
> Saunders Mac Lane clearly pointed out the origins:
>
> "Category" from Kant (which I had known all the time)
>
> "Functor" from Carnap's book "Logical Syntax of Language" (which I
> had forgotten).
>
>
> Also I have a question, not directly related to the above issue:
>
> I have seen, on some web page, a copy of
> the referee's report about the Eilenberg-Mac Lane paper
> where Eilenberg-Mac Lane spaces are introduced.
> I cannot find this web page (or the report)
> any more. Can anyone provide me with
> a hint where I can possibly find it?
>
> Many thanks in advance
>
> Johannes
>
>
>
> HUEBSCHMANN Johannes
> Professeur de Mathematiques
> USTL, UFR de Mathematiques
> UMR 8524 Laboratoire Paul Painleve
> F-59 655 Villeneuve d'Ascq Cedex  France
> http://math.univ-lille1.fr/~huebschm
>
> TEL. (33) 3 20 43 41 97
>      (33) 3 20 43 42 33 (secretariat)
>      (33) 3 20 43 48 50 (secretariat)
> Fax  (33) 3 20 43 43 02
>
> e-mail Johannes.Huebschmann@math.univ-lille1.fr
>
>
>


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20:07

Categories and functors, query

[Johannes Huebschmann]

Dear All

I somewhat recall that, a while ago, we discussed the origins of
the notions of category and functor. S. Mac Lane had once pointed
out to me these origins but from my recollections we did not
entirely reproduce them.

In his paper

Samuel Eilenberg and Categories, JPAA 168 (2002), 127-131

Saunders Mac Lane clearly pointed out the origins:

"Category" from Kant (which I had known all the time)

"Functor" from Carnap's book "Logical Syntax of Language" (which I
had forgotten).


Also I have a question, not directly related to the above issue:

I have seen, on some web page, a copy of
the referee's report about the Eilenberg-Mac Lane paper
where Eilenberg-Mac Lane spaces are introduced.
I cannot find this web page (or the report)
any more. Can anyone provide me with
a hint where I can possibly find it?

Many thanks in advance

Johannes



HUEBSCHMANN Johannes
Professeur de Mathematiques
USTL, UFR de Mathematiques
UMR 8524 Laboratoire Paul Painleve
F-59 655 Villeneuve d'Ascq Cedex  France
http://math.univ-lille1.fr/~huebschm

TEL. (33) 3 20 43 41 97
      (33) 3 20 43 42 33 (secretariat)
      (33) 3 20 43 48 50 (secretariat)
Fax  (33) 3 20 43 43 02

e-mail Johannes.Huebschmann@math.univ-lille1.fr