Axioms for Abstract Sets and Mappings

We have now seen most of the axioms we will require of the category S of abstract sets and mappings. As we progressed, some of the earlier axioms were included in later axioms. For example, the existence of the one-element set is part of the axiom that S has finite limits. Although we did not insist on it earlier, it is also the case that some of the axioms are more special than others. By this we mean that even though they hold in S they will not generally hold in categories of variable or cohesive sets. Thus, we are going to review the axioms here so that they can be considered all at once and grouped according to their generality.

The very first axiom, of course, is

AXIOM:S IS A CATEGORY

We have been emphasizing all along that the fundamental operation in a category, composition, is the basic tool for both describing and understanding all of the other properties of S.

The next group of three axioms is satisfied by any category of sets, variable or constant. In fact a category satisfying them is called a topos (in the elementary sense), and these categories have been studied intensively since 1969.