Subject: sets against categories

From: categories <[email-masked]>
Date: Fri, 15 Mar 1996 15:55:44 -0400 (AST)

Date: Fri, 15 Mar 1996 12:30:19 -0600
From: [email-masked]

The recent discussion of sets against categories on the internet appears to
miss the appropriate sources.

It is well known that it is easy to go from sets to categories, harder in
the reverse.  For this there is a very well-known equicoherence theorem,
which is presented in both the standard texts on topos
        Johnstone, Chapter 9, S 7
        Mac Lane / Moerdijk (Sheaves in Geometry and Logic, Chapter 6, S 10.)

I fondly imagine that the latter is a bit clearer.  Both sources will give
your the original literature--for example Mitchell JPAA 2(1972) p. 261 (I
suggested this question to Bill Mitchell when he was an instructor at
Chicago).

As far as I can make out; none of the many messages speaks to this fact. 
It is a reasonable question for Pratt to raise, though he should have known
that the Goldblatt book was hopeless from day one.

Of course most mathematicians find sets easier than cats--but they usually
can't recite ZFC axioms.  The fault may lie with Pat Suppes, who taught
sets in the Kindergarten.

Otherwise, the exchange convinces me tht e-mail is for the birds.  All
fluff with no professional substance

Saunders Mac Lane
Department of Mathematics
University of Chicago
[email-masked]

---

From: categories <[email-masked]>
Date: Sat, 16 Mar 1996 10:38:45 -0400 (AST)

Date: Fri, 15 Mar 1996 15:11:50 -0800
From: Vaughan Pratt <[email-masked]>

Reference *much* appreciated, which I am browsing now.  (Peter, will
Topos Theory be back in print soon?)

	It is well known that it is easy to go from sets to categories,
	harder in the reverse.

This is disturbing, since it is the opposite of the answer I came up
with myself a couple of days after asking my question.  (I am reading
"harder" as "necessarily harder" here, my apologies if this was not
your intended meaning.)

As often in these things, at least some of the problem lies with my
question, which was certainly fluffy when I asked it, not knowing then
quite what I really wanted to ask.  I think I can sharpen it now.

Is there a category equivalent to Set for which it is easy to recover
the membership relation from the category structure?

This is still not a mathematical question as it leaves "easy" open to
interpretation.  But I think the rest of it is unambiguous.

Now I thought I'd answered this sharper question in the affirmative,
with an interpretation of "easy" that surely *no* reasonable person
could complain about, namely a two line construction of the membership
relation.

Therefore if my answer was only "fluff", I have made an error
somewhere, either in my construction of this version of Set or in my
choice of problem.

With regard to the latter possibility, I freely admit to knowing less
than just about anyone on this list what it feels like to work in a
general topos, not enough to write even 6 pages about them let alone
600.  My experience of toposes is with one topos only, Set, which makes
me about the last person qualified to attempt a contribution to topos
theory.

However, that my question only concerns the one topos Set gives me hope
that, if there is indeed a problem with my construction, it is some
technical oversight that I need to repair if possible, and not
something to do with other toposes besides Set.

In passing, let me again draw people's attention to the fact that I
described not just the category Set but its (cartesian) closed
structure as well, including complete verification of the coherence
conditions.  (Not that this was particularly difficult in this case. :)

Without giving the full closed structure I do not understand how one
can claim to have fully specified which topos one is speaking of.  Does
the topos literature attend adequately to this detail?  (It may well, I
just don't know where to find it.)

I interpret the reference to Isbell at the end of VII.1 of CTWM as
(inter alia) a warning that one cannot take the closed structure for
granted merely because it is cartesian closed.  If this misinterprets
the situation for the cartesian closed case, and coherence is in fact
routine there, then my apologies for the misunderstanding.

Vaughan Pratt