Subject: categories: Re: cracks and pots

From: dusko <[email-masked]>
Date: Tue, 28 Mar 2006 00:01:38 -0800

i think david yetter's analysis of the dichotomy "categories as  
foundations" vs "categories as algebra" was spot on ---  with respect  
to people and the community. indeed, one could split most of our  
papers into one category or the other.

but at the end of the day, i think, we'll all agree that the source  
of the unreasonable effectiveness of categorical algebra is its  
foundational content (although there is probably a lot of it that we  
dont understand yet); and the other way around. eg, if you look at  
grothendieck's work, he started working in algebra, and ended up  
developing foundational structures, because he needed them. and a lot  
on the "algebra" side now is built upon them. ok, then for a while it  
was thought that he exaggerated with foundations, and that a more  
direct approach "could have been in better taste" (to cite  
eilenberg). but maby the fermat theorem would have a more useful  
proof if it was developed in grothendieck style. and nowadays, there  
is a lot of foundational content in tannaka duality etc, in TQFT in  
general, but we only see hints of it at the moment (and i for one  
just see the reflections of these hints in other people's eyes).

i am of course saying things very clear and familiar to many people  
on this list, but maby they are worth saying nevertheless. it might  
be good if the links between "categories as algebras" and "categories  
as foundations" would not boil down just to the greatest of the  
category theorists, leaving the rest of us in two camps.

-- dusko

---

From: Alex Simpson <[email-masked]>
Date: Wed, 29 Mar 2006 13:57:05 +0100

Quoting dusko <[email-masked]>:

>  on the "algebra" side now is built upon them. ok, then for a while
> it  was thought that he exaggerated with foundations, and that a more
> general ...

TQFT!? It seems dusko has finally discovered the shift key
on his keyboard.

Alex

-- 
Alex Simpson, LFCS, School of Informatics, Univ. of Edinburgh, UK
Email: [email-masked]             Tel: +44 (0)131 650 5113
Web: http://homepages.inf.ed.ac.uk/als   Fax: +44 (0)131 667 7209