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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$ . less than the cardinality of $ U$ . In particular, $ U \notin 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 hear differs a bit from the treatments given there. We also refer to [13] as a good reference.


next up previous
Next: examples of categories. Up: Elementary category theory Previous: Elementary category theory
2007-12-11