Elementary theory of the category of sets


In mathematics, the elementary theory of the category of sets or ETCS is a set of axioms for set theory proposed by William Lawvere in 1964. Although it was originally stated in the language of category theory, as Tom Leinster pointed out, the axioms can be stated without references to category theory.
ETCS is a basic example of structural set theory, an approach to set theory that emphasizes sets as abstract structures.

Axioms

Informally, the axioms are as follows:
  1. Composition of functions is associative and has identities.
  2. There is a set with exactly one element.
  3. There is an empty set.
  4. A function is determined by its effect on elements.
  5. A Cartesian product exists for a pair of sets.
  6. Given sets and, there is a set of all functions from to.
  7. Given and an element, the pre-image is defined.
  8. The subsets of a set correspond to the functions.
  9. The natural numbers form a set.
  10. Every surjection has a right inverse.
The resulting theory is weaker than ZFC. If the axiom schema of replacement is added as another axiom, the resulting theory is equivalent to ZFC.