FOM: Foundations of category theory

[Till Mossakowski]

I have forwarded Steve Simpson's mails from Wed, 19 May 1999 17:25:14
and Thu May 20 02:19:16 1999 to Horst Herrlich, since they directly 
address the books written by him (togehter with G. Strecker 
and J. Adamek). His reply is appended below. The exercise and theorem
numbers refer to Adamek-Herrlich-Strecker: "Abstract and concrete
categories", Wiley New York 1990.

Till Mossakowski

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...
thanks for your interesting notes on the discussion going on right now
concerning the foundations of category theory. I would love to join in,
however I have to leave in a few days for a one month vacation. So I
must
restrict myself to the following very sketchy remarks:
1) Steve asked for the meaning of "extensions" in our book. We had in
mind
such things as those described briefly in the exercises 21G, 21H, 21I
and 21J.
2) I agree with almost everything Steve writes. The exception is his
idea
that (if I understand him right) categories could be treated just like
groups etc and that for this reason ZFC would suffice as foundation. I
don't agree here since I consider the distinction between "large" and
"small" in category theory important and unavoidable, and this
distinction
is not available in ZFC alone. Such important concepts as "has products"
(10.29(1)) and "completeness" (12.2(2)) - and many other considerations
(as
one example see my paper "On the failure of Birkhoff's Theorem for
locally
small based equational categories of algebras" in Cahiers
Topol.Geom.Diff.Categ.34(1993) 185-192) - require this distinction. If
it
would be dropped the corresponding concepts would be far too restrictive
to
be interesting as theorem 10.32 demonstrates clearly. See the remark
10.33.
This is all for the moment.
Please keep me informed. Once I am back I could join in again.
Good luck with your further discussions, yours Horst