Subject: categories: Re: cracks and pots

From: "George Janelidze" <[email-masked]>
Date: Fri, 17 Mar 2006 11:36:46 +0200

I join Bob in saying that I fully agree with Marta, and I fully agree with
Bob's second sentence. However, I have a problem with "look the gift horse
in the mouth", since the horses we get are so often headless...

I would also like to make just one comment to Paul's message (although I
disagree with most of it; sorry!). Paul says:

"Which generation was it that alienated other mathematicians by making
outrageous claims about the foundations of mathematics that it never
backed up with theorems?   Which generation actually got its hands dirty
and proved the theorems that relate category theory to other foundational
disciplines?"

Well, our colleagues active in the 1960s and 70s invented elementary
toposes, for example, and proved many theorems about them. Those theorems
did not convince set-theorists to forget sets, but are they convinced now?
On the other hand those theorems were very beautiful, along with many others
from several areas of category theory; I would describe 1960s and 70s as
Golden Age of category theory. I am not saying of course that nothing
important was discovered after 70s, but I see problems, and growing chaos,
often created by ambitiously presented pseudo-relations with "other
foundational disciplines".

Moreover, talking about "relations": According to the classical work of
Sammy and Saunders, the first "relation" was with algebraic topology. As we
all know, there are various (co)homology/homotopy functors from topological
spaces to groups, or to more complicated algebraic (or coalgebraic, Hopf,
etc.) structures. There are also simplicial sets and other combinatorial
intermediate players, and the relationship between geometric and
combinatorial objects goes back to Euler (if not to Plato...). As we know
from 1960s, the universal property of Yoneda embedding yields various
adjoint functors, including those between simplicial sets and topological
spaces - and this is why combinatorial objects are there! And what do recent
algebraic topology text books do instead of explaining this? They are still
talking about gluing cells instead. I think if we really care about
relations between category theory and "other foundational disciplines", we
should begin by explaining that category theory is not just a language
allowing one to call homology a functor, but that category theory has
beautiful constructions and results (some already from 1940s and 50s!)
making enormous simplifications/applications/illuminations in neighbour
areas of pure mathematics, such as abstract algebra, geometry, and logic.

George Janelidze

----- Original Message -----
From: "RFC Walters" <[email-masked]>
To: <[email-masked]>
Sent: Wednesday, March 15, 2006 3:35 PM
Subject: categories: Re: cracks and pots

> I also would like to support the remarks of Marta with which I am in
> full agreement.
> The category theory community seems happy to accept uncritically, and
> give centre-stage to, any interest shown by an external field. In this
> context one should certainly look the gift horse in the mouth.
>
> Bob Walters