Sets, category of

The category whose objects are all possible sets, and whose morphisms are all possible mappings of one set into another, composition of morphisms being defined as the usual composition of mappings. If category-theoretic concepts are interpreted within a fixed universe $U$, then the category of sets means the category whose objects are all sets belonging to $U$, with morphisms and composition as above. The category of sets may be denoted by $\mathfrak S$, ENS, Set or Me.

The empty set is an initial object (left zero) and any singleton is a terminal object (right zero) of the category of sets. Every non-empty set is a generator, and every set containing at least two elements is a cogenerator. Every monomorphism with non-empty domain is split (i.e. has a one-sided inverse); the assertion that every epimorphism is split is equivalent to the axiom of choice. The category of sets has a unique bicategory (factorization) structure.

The category of sets is locally small, complete, cocomplete, well-powered, and co-well-powered. In particular, the product of a family of sets (exists and) coincides with its Cartesian product, and the coproduct of a family of sets coincides with its disjoint union. The binary Cartesian product, the Hom-functor $\mathfrak S^*\times\mathfrak S\to\mathfrak S$ and a singleton set provide the category of sets with the structure of a Cartesian closed category. Furthermore, it is an (elementary) topos, with a two-element set as subobject classifier. Every locally small category can be regarded as a relative (enriched) category over the category of sets.

A category $\mathfrak K$ is equivalent to the category of sets if and only if: 1) it has a strict initial object; 2) the full subcategory of non-initial objects of $\mathfrak K$ has regular co-images and a unary generator; 3) each object $A$ has a square $A\times A$; and 4) each equivalence relation is the kernel pair of some morphism. Here an object $U$ is called unary if it has arbitrary copowers, and the only morphisms from $U$ to one of its copowers are the imbeddings of the summands (cf. Small object). For other characterizations of the category of sets, see [2], [3].

Categories equivalent to subcategories of the category of sets (equivalently, categories admitting a faithful functor into the category of sets) are called concrete. For necessary and sufficient conditions for a category to be concrete, see [1].

References

[1]	P. Freyd, "Concreteness" J. Pure Appl. Algebra , 3 (1973) pp. 171–191 MR0322006 Zbl 0277.18002
[2]	F.W. Lawvere, "An elementary theory of the category of sets" Proc. Nat. Acad. Sci. U.S.A. , 52 (1964) pp. 1506–1511 MR0396262 MR0172807 Zbl 0141.00603
[3]	L.A. Skornyakov, "A characterization of the category of polygons" Mat. Sb. , 80 (1969) pp. 492–502 (In Russian) Zbl 0203.31403

Comments

For a characterization of the category of sets amongs toposes, see [a1]. Cf. also Universe; Generator of a category; Faithful functor.

References

[a1]	M. Tierney, "Sheaf theory and the continuum hypothesis" F.W. Lawvere (ed.) , Toposes, algebraic geometry and logic (Dalhousic Univ., Jan. 1971) , Lect. notes in math. , 274 , Springer (1972) pp. 13–42 MR0373888 Zbl 0244.18005