# Sets, category of

(Redirected from Category of sets)
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, $\mathsf{Set}$ or Me.
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 , .