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.

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.

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.

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.

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.