Re: Categories of elements

[Jpdonaly@aol.com]

Dear Professor Lawvere,

Thanks for your clarifications and views in response to my latest note.
Coming from an applications-oriented environment, I do assume a set of
Zermelo-Fraenkel axioms with a universe of small sets (as prescribed in CWM) in order to
ensure access to a fully viable arithmetic of natural transformations. This
seems to allow for more than enough categories for my purposes, but it certainly
does give the category of small functions a prominence which can feel
artificially restrictive at times. Thus I would be especially attentive to any
comments which you might make specifically on the functorial isomorphism (I presume
to call it a "Lawvere isomorphism" )  which, in converting the Yoneda picture
(function-valued natural transformations) of categorical duality into the
Lawvere picture (cocompatible functors), represses the category of small functions
and, as I do realize, moves things into the context of the general existence
theory of adjunctions and Kan extensions, possibly providing a functorial
interpretation of your explanation of the origin of comma categories. By now this
isomorphism seems to me to be more of a perspicuous relabelling than a
redefiner of concepts, so that I have to plead innocent to your apparent conviction
that I agonize over the definition of elements. I am in full accord with the
doctrine of elements as you have described it, and the Lawvere isomorphism
actually relieves some conceptual agony in this regard by smoothly ensuring that, to
within a label, the elements of a function-valued functor constitute a
(limit) object which is in the functor's codomain category.  But I have to restate
my belief that the otherwise perfectly redeemable sentence, "An element of a
functor is an attaching functor into the category of elements of the functor,"
is unacceptably confusing due to the fact that the category of elements of a
functor does not in any sense consist of the elements of the functor (as you
would describe them). So I would rename it.

Pat Donaly