Re: Categories and functors, query

[Michael Barr <barr@math.mcgill.ca>]

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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