Re: terminology in definitions of limits

[Richard Garner <rhgg2@hermes.cam.ac.uk>]

I have always used the phrase "test object" in a slightly
different sense. Namely, to refer to a tractably small
collection of objects that one may use, not only to detect,
but also to calculate some right adjoint. Thus in Set, one
may take the terminal object; in Set/X, the elements 1-->X;
in Cat, the ordinals 1, 2 and 3; in presheaf categories, the
representables; and so on. The best case is that these test
objects are colimit dense, since then your calculations
always yield a right adjoint as soon as the functor you start
with preserves colimits.

Richard