Re: we do meet isomorphisms of categories

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

Yes, we do meet isomorphisms of categories; my favourite algebraic
example is the isomorphism between (Boolean algebras) and (Boolean
rings), and another good one is the isomorphism between (finite
T_0-spaces) and (finite partial orders). But there's a sense in which
these isomorphisms are "accidental", arising from the fact that both
categories are based on the same category of sets, and in practice
(so far as I know) one never makes use of the fact that they are
isomorphisms rather than mere equivalences.

An even better example occurs in realizability. Some years ago on this
list I queried the need for the condition "Sxy is defined for all x
and y" in the definition of a partial combinatory algebra, and John
Longley came up with a beautiful proof that, given a "weak pca" A
which fails to satisfy this condition, there is a pca A' which does
satisfy it, such that the category of A-valued assemblies is *identical*
(not just equivalent, or even isomorphic) to the category of A'-valued
assemblies. (Details can be found in Jaap van Oosten's book.) The
accident arises in this case from the fact that A' happens to have the
same underlying set as A. But, once again, I don't know of any use for
the fact that the correspondence between the categories of assemblies is
anything more than an equivalence.

Peter Johnstone
---------------------------
On Mon, 24 May 2010, Marco Grandis wrote:

> We do meet isomorphisms of categories. Only, they are so obvious that
> sometimes we do not see them.
>
> For instance:
>
> The category of abelian groups is (canonically) isomorphic to the
> category
> of Z-modules.
>
> Groups are often defined as semigroups satisfying two conditions; but
> they
> can also be defined as sets with a zeroary operation, a unary
> operation and
> a binary operation satisfying certain axioms. Again, we have two
> isomorphic
> categories. An unbiased definition would give a third isomorphic
> category
> (and one can form infinitely many intermediate cases between the second
> and the third, likely of little interest). Algebras for the free
> group monad are
> directly linked with the unbiased version, yet not the same.
>
> Lattices (with 0 and 1) can be defined as ordered sets satisfying
> some conditions;
> or as sets with two binary operations satisfying other conditions;
> then one can
> add two zeroary operations;...
>
> Best regards
>
> Marco Grandis

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