category of sets

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-complete. The product in this category is given by the cartesian product of sets. The coproduct is given by the disjoint union: 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 category; 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.

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