Categories of elements

[Ross Street <street@ics.mq.edu.au>]

> "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