Re: Challenge from Harvey Friedman

[categories <cat-dist@mta.ca>]

Date: Fri, 30 Jan 1998 10:40:55 -0500 (EST)
From: Michael Barr <barr@triples.math.mcgill.ca>



Well the first question has an easy answer.  It is just as possible to
talk about a category without knowing what a set is as it is to talk about
a set without knowing what a set is.  Of course, you cannot talk about
homsets.  It is interesting to read the very first Eilenberg-Mac Lane
paper, which did not talk about homsets.  A set is an undefined notion and
there is a relation, epsilon that may hold between one set and another,
subject to certain axioms, one version of which Friedman listed.  A
category consists of undefined things called arrows and three relations,
two functional and the third partially functional (actually better than
that, but leave that aside).  Friedman's axioms are not coherent, as has
been pointed out, while the categorical axioms are.  On the other hand,
one can state Friedman's axioms, in all their glorious incomprehensibility
(I think I could stare at the 8th one from now until the middle of next
year without understanding what it says, and the 6th, asserted to be the
axiom of infinity is not much clearer) in a couple hundred words, while it
is pretty much necessary to interrupt the topos axioms for some
definitions (at least monic and subobject) to do the topos axioms.  Thus
each one looks simpler to its devotees and there is really no point in
arguing about it.

Michael

On Thu, 29 Jan 1998, categories wrote:

> Date: Wed, 28 Jan 1998 22:15:27 +0000
> From: Carlos Simpson <carlos@picard.ups-tlse.fr>
> 
> Being a newcomer to the category list, I have a really naive and stupid question
> (concerning H. Friedman's challenge). Namely, I was always under the impression
> that you had to know what a set was before you could talk about what a
> category was (in particular a topos). Is it possible to talk about toposes
> without knowing what a set is?
> 
> This seems somewhat related to a question that has been bugging me for some
> time, namely how to talk about a ``category''
> which is enhanced over itself, but not necessarily having any functor to or
> from Sets. The very first part of the structure would be a class of objects
> O
> together with a function (x,y)\mapsto H(x,y) from O\times O to O, but I
> can't get beyond that.
> 
> ---Carlos Simpson
> 
> 
>