we do meet isomorphisms of categories

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

On 23 May 2010, at 17:39, Colin McLarty wrote:

> Grothendieck gave it a fine nuance in Tohoku (p. 125) saying "Aucune
> des equivalences de categories qu'on rencontre en pratique n'est un
> isomorphisme (none of the equivalences one meets in practice are
> isomorphisms)."  He stressed that we must distinguish isomorphisms
> from equivalences.  Throughout that and later works he *constructs* a
> great many categories up to isomorphism, and not just up to
> equivalence.  We do not meet these isomorphisms, we construct them --
> and it is quite important that once constructed they are not merely
> equivalences.

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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