Re: isomorphisms

[Colin McLarty <colin.mclarty@case.edu>]

2010/5/29 Eduardo J. Dubuc <edubuc@dm.uba.ar>

Expresses my main point when I quoted Grothendieck on equivalence and
isomorphism.

> isomorphisms appear not only in examples but are essential also in the
> theory.
> For example Grothendieck defines limits and colimits of categories (as
> universal pseudocones) in SGA4 by means of an isomorphism of categories.
> Same for toposes.


When AG says "none of the equivalences we meet in practice are
isomorphisms" he has in mind lots of examples that I will not even try
to survey.  (For a really simple one, the category of sheaves defined
as espaces etales on on a topological space versus the category of
sheaves defined as suitable functors on the site of open subsets.)

But when he defines functor categories, or derived categories, and a
lot of other things like that, he defines them up to unique
isomorphism over the data.  A topos E will often be defined only up to
equivalence.  But, given E, its derived category is defined up to
unique isomorphism and one constantly uses the fact that various
induced functors are isomorphisms.

AG's practice constantly distinguishes isomorphisms from equivalences,
and thus distinguishes identity of objects from isomorphism of them.

best, Colin


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