Category of sets

In the mathematical field of , the category of sets, denoted as Set, is the whose are sets. The arrows or morphisms between sets A and B are the total functions from A to B, and the composition of morphisms is the composition of functions. Many other categories (such as the , with group homomorphisms as arrows) add structure to the objects of the category of sets and/or restrict the arrows to functions of a particular kind. The axioms of a category are satisfied by Set because composition of functions is associative, and because every set X has an identity function idX : X → X which serves as identity element for function composition. The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the s are the bijective maps. The empty set serves as the initial object in Set with empty functions as morphisms. Every singleton is a terminal object, with the functions mapping all elements of the source sets to the single target element as morphisms. There are thus no zero objects in Set. The category Set is . The in this category is given by the cartesian product of sets. The is given by the disjoint union: given sets Ai where i ranges over some index set I, we construct the coproduct as the union of Ai×{i} (the cartesian product with i serves to ensure that all the components stay disjoint). Set is the prototype of a ; other categories are concrete if they are "built on" 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 and ). Set is not , nor . Every non-empty set is an injective object in Set. Every set is a projective object in Set (assuming the axiom of choice). The in Set are the finite sets. Since every set is a direct limit of its finite subsets, the category Set is a . If C is an arbitrary category, the contravariant functors from C to Set are often an important object of study.