FOM: poll

[Vaughan Pratt]

From: kanovei at wminf2.math.uni-wuppertal.de (Kanovei)
>*We* do not need a poll to understand the difference 

Kanovei is quite right.  When you're polling your mathematical colleagues
about the number of groups of order four, don't poll FOMers, they know
the difference.

>Everyone knows that (1), even being algebraically isomorphic, can bear
>some extra structure (say topological) which is the mathematical reason
>to distinguish them from each other despite the group isomorphism.

Ok, so the scenario you envisage would go something like:

Q: How many groups of order four are there?
A: Two.
Q: How many topological groups of order four are there?
A: A proper class.

You seem to be arguing that you can put group structure on an "anonymous"
set (one defined only up to isomorphism) but not topological structure.
My understanding was that group isomorphisms, homeomorphisms, and group
homeomorphisms are all instances of isomorphisms, meaning opposite pairs
of maps whose composition in either order is the identity map.

>It remains to wonder is the whole "system of foundations", so to speak, of
>category theories, concentrated on topics of equal mathematical *depth*.

Fortunately not all refutations of categorical foundations are of equal
depth.  (I'm painfully aware that my contribution to this exchange is
in the shallow end of the pool.)

Vaughan Pratt