CATEGORY OF SETS

In mathematics, the 'category of sets', denoted as 'Set', 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.
Because of Russell's paradox, which shows assuming the existence of the set of all sets leads to a contradiction, the object class of 'Set' is a proper class, and thus the category is large.
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, they are usually noted respectively ''0'' and ''1''. 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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References

References

★ Categories for the Working Mathematician, , Saunders, Mac Lane, Springer, 1998, ISBN 0-387-98403-8 (Volume 5 in the series Graduate Texts in Mathematics)