Categories of elements

Categories of elements

[Ross Street]

> "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).

I'm not sure where the above quote is taken from but I agree it is confusing.

Here is my argument in favour of the traditional name.

As Bill says, an element of an object F in a category is generally
any morphism A --> F  into  F.  It just happens that in many
categories  F  is determined by elements with a restricted class of
domains  A.  In Set, we can restrict  A  to be terminal.  In a
presheaf category, we can restrict  A  to be representable.  The
objects of the category  elF  of elements of  F  are (up to
isomorphism) elements  A --> F  with  A  representable.  It is also
conventional to name categories after their objects (although the
Ehresmann convention of naming them after their morphisms is more
precise). Hence  elF  is the category of elements of  F.


Regards,
Ross

Re: Categories of elements

[Jpdonaly]

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