View source for Sets, category of

{{TEX|done}}{{MSC|18B05}}

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 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(2)|bicategory]] (factorization) structure.

The category of sets is locally small, complete, cocomplete, [[Well-powered category|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|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|Small object]]). For other characterizations of the category of sets, see [[#References|[2]]], [[#References|[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 [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P. Freyd, "Concreteness" ''J. Pure Appl. Algebra'' , '''3''' (1973) pp. 171–191 {{MR|0322006}} {{ZBL|0277.18002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> F.W. Lawvere, "An elementary theory of the category of sets" ''Proc. Nat. Acad. Sci. U.S.A.'' , '''52''' (1964) pp. 1506–1511 {{MR|0396262}} {{MR|0172807}} {{ZBL|0141.00603}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.A. Skornyakov, "A characterization of the category of polygons" ''Mat. Sb.'' , '''80''' (1969) pp. 492–502 (In Russian) {{MR|}} {{ZBL|0203.31403}} </TD></TR></table>

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> 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 {{MR|0373888}} {{ZBL|0244.18005}} </TD></TR></table>