we do meet isomorphisms of categories

[Marco Grandis <grandis@dima.unige.it>]

I should have mentioned another quite elementary example, that is
perhaps more intriguing.

Let us write Top for (topological spaces defined by open sets) and
Top' for
the isomorphic category (topological spaces defined by closed sets).

Let (X, L) be a set X equipped with a complete sublattice L of its
lattice of parts.
Viewing it as on object of Top or Top' will interchange an Alexandrov
topology
for X with the opposite one, generally different.

This says that - formally - we cannot think of these two isomorphic
categories as being the same thing. Even if, of course, we do think
that way, informally and in practice.

I am not entirely convinced by a comment of Peter:
"in practice (so far as I know) one never makes use of the fact that
they are
isomorphisms rather than mere equivalences".

I am happy with the fact that, going from Top to Top' and back, we
get the same
space on the nose; this spares a lot of complications.

Best regards

Marco Grandis


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