Re: Categories and functors, query

[Walter Tholen <tholen@mathstat.yorku.ca>]

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.
>
>