Re: Sets vs. categories as a foundation

 : In article <[EMAIL PROTECTED]>,
 : Jesse F. Hughes <[EMAIL PROTECTED]> wrote:
 : >Perhaps.  I came into this thread secondhand.  Why don't you tell us
 : >what you mean when you write, "all categories are in a sense
 : >imitations of the category of sets, the objects being imitations of
 : >sets and the morphisms being imitations of functions."  It's very
 : >plausible that I don't know what the heck you mean.

[EMAIL PROTECTED] writes:
 : Seems pretty clear to me.

That's your problem.

 :  The fact that set theory and category theory
 : can be used as foundations for mathematics is irrelevant to james dolan's
 : point,

No, it isn't.
Suppose you started out as a committed categorist,
believing that using categories as a foundation was good
and using sets was bad.  Then, suppose a fellow categorist
had reminded you, "all categories are in a sense imitations
of the category of sets."  That INVITES you to react, "Hmmm --
WHY is the category of SETS so SPECIAL -- WHAT is it ABOUT
THAT category that makes it beat this relationship to ALL
other categories?  Why is it so central?  Could it be -- gasp --
adequately FOUNDATIONAL??"

 : and is being dragged in uninvited.

No, really, it isn't.  The person alleging an unusually
important central basic role for sets here IS JAMES DOLAN, not
anybody else.

 : Suppose one had said instead,
 : "All homology and cohomology theories are in a sense imitations of
 : simplicial (co)homology."  Presumably nobody is tempted to use simplicial
 : homology as a foundation for mathematics, and hence nobody is tempted
 : into failing to understand this statement.

The analogy fails.  I don't personally know WHETHER 
other homology and cohomology theories "imitate"
simplicial cohomology.  But the question of whether other
categories "imitate" "the" category of sets is a lot more
attackable, even on the basis of very limited general concerns
like the ones I have been raising.

 : The fact that there are other ways of thinking about categories 

Other THAN WHAT??
Nobody ever suggested that any particular way was more or less
relevant to the question of whether other categories "imitate"
the category of sets!  If the imitation is real then one would
certainly expect it to remain equally real under ALL such ways!