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In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. It is the most basic and the most commonly used category in mathematics. The category is usually denoted simply as Set.

The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps.

The empty set serves as initial object in Set, while every singleton is a terminal object. There are thus no zero objects in Set.

The category Set is complete and co-completeIn category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. Accordingly, the dual notion of a col. The productIn category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces. Essentially, the product of a family of objects is the "most gen in this category is given by the cartesian productIn mathematics, the Cartesian product (or direct product X × Y of two sets X and Y is the set of all ordered pairs whose first component is a member of X and whose second component is a member of Y''. This concept is named after Rene Descartes. X × Y { x of sets. The coproduct is given by the disjoint unionIn set theory, a disjoint union or discriminated union is a set union in which each element of the resulting union is disjoint from each of the others; the intersection over a disjoint union is the empty set. The term is also used to refer to a modified u: given sets A_i where i ranges over some index set I, we construct the coproduct as the union of A_i×{i} (the cartesian product with i serves to insure that all the components stay disjoint).

Set is the prototype of a concrete categoryIn mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of function; other categories are concrete if they "resemble" Set in some well-defined way.

Every two-element set serves as a subobject classifier in Set. The power object of a set A is given by its power set, and the exponential object of the sets A and B is given by the set of all functions from A to B. Set is thus a topos (and in particular cartesian closed).

Set is not abelian, additive or preadditive; it doesn't even have zero morphisms.

Every not initial object in Set is injective and (assuming the axiom of choice) also projective.