universe

In order to deal with some set theoretical difficulties, we assume the existence of sufficiently many universes.

DEFINITION 5.2   A universe $U$ is a nonempty set satisfying the following axioms:

1. If $x\in U$ and $y\in x$ , then $y\in U$ .
2. If $x,y\in U$ , then $\{x,y\}\in U$ .
3. If $x\in U$ , then the power set $2^x\in U$ .
4. If $\{x_i \vert i\in I\}$ is a family of elements of $U$ indexed by an element $I\in U$ , then $\cup_{i\in I} x_i\in U$ .

LEMMA 5.3   Let $U$ be an universe. Then the following statements hold.

1. If $x\in U$ , then $\{x\}\in U$ .
2. If $x$ is a subset of $y\in U$ , then $x\in U$ .
3. If $x,y\in U$ , then the ordered pair $(x,y) = \{\{x,y\},x\}$ is in $U$ .
4. If $x,y\in U$ , then $x\cup y$ and $x\times y$ are in $U$ .
5. If $\{x_i \vert i\in I\}$ is a family of elements of $U$ indexed by an element $I\in U$ , then we have $\prod_{i\in I} x_i \in U$ .

In this text we always assume the following.

For any set $S$ , there always exists a universe $U$ such that $S\in U$ .

The assumption above is related to a ``hard part'' of set theory. So we refrain ourselves from arguing the ``validity" of it.

DEFINITION 5.4   Let $U$ be a universe.

1. A set $S$ is said to be $U$ -small if it is an element of $U$ .
2. A category $\mathcal{C}$ is said to be $U$ -small if
   1. $\operatorname{Ob}(\mathcal{C})$ is a $U$ -small set.
   2. For any $X,Y \in \operatorname{Ob}(\mathcal{C})$ , $\operatorname{Hom}(X,Y)$ is $U$ -small.

Note: The treatment in this subsection owes very much on those of wikipedia:

http://en.wikipedia.org/wiki/Small_set_(category_theory)

and planetmath.org:

http://planetmath.org/encyclopedia/Small.html

but the treatment here differs a bit from the treatments given there. We also refer to [13] as a good reference.