FOM: Nice Idea! Thanks to Kanovei

[Till Mossakowski]

Kanovei wrote (Mon, 12 Apr 99 08:51:24):

>I think the "experts" call those universes and 
>corresponding categories "small" (univ. and cat.), 
>that is, I have definitely seen this term in a book, 
>as opposed to "large" ones, by full Grothendieck. 

I think you cannot mix up small categories and small universes.
The terminology used in the book by Herrlich and Strecker,
generalized to an arbitrary universe, is the following:

Given a univsere U, a subset of U is called an U-class,
and a member of U is called an U-set (or just U-small).
A U-category is a U-class of objects together with some
structure, while a U-small category has an U-set of objects.

Many familiar U-categories have (some) U-small limits,
but U-categories having all U-class-indexed limits are degenerated.
U-small categories are of not so much interest, because
U-small categories having all U-small limits are again degenerated,
and the only natural limitation on the size of the
limits to take is a universe U' smaller than U - but
then you can directly work with U'-categories.

When working with just a single universe U, "U-small"
is abbreviated by "small".

The question of smallness of a universe is orthogonal
to this.
It makes much more sense to consider the theory
of U-categories for a small universe U, than the
theory of U-small categories for a universe U.

Till Mossakowski