Re: terminology

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

I have very much appreciated André's subtlety on this issue in conversation

2010/5/22 Joyal, André <joyal.andre@uqam.ca>:
> I wrote
> ------------------------------------------------------------
> One lies in the fact that equivalent categories are considered
> to be the "same",
> -------------------------------------------------------------
> I was careful not to write
> ------------------------------------------------------------
> One lies in the fact that equivalent categories are considered
> to be the same,
> -------------------------------------------------------------

John Baez has written carefully on this point too.

But not everyone is so careful and Ronnie has good reason to be
concerned about a tendency to sweep away distinctions that do need to
be made.

Isomorphic categories too must be distinguished from one another, some
times and for some purposes notably including all currently
articulated versions of categorical foundations.

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.

It is an interesting impulse in higher category theory to avoid
identity in favor of isomorphism on the level of objects, and to avoid
isomorphism in favor of equivalence on the level of categories.   But
so far as I know no one has yet articulated a way to avoid ever using
identity of objects and identity of categories.

Colin






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