Editing Category of sets (section)

==Properties of the category of sets==
The axioms of a category are satisfied by '''Set''' because composition of functions is [[Associative property|associative]], and because every set ''X'' has an [[identity function]] id<sub>''X''</sub> : ''X'' → ''X'' which serves as identity element for function composition.

The [[epimorphism]]s in '''Set''' are the [[surjective]] maps, the [[monomorphism]]s are the [[injective]] maps, and the [[isomorphism (category theory)|isomorphism]]s are the [[bijective]] maps.

The [[empty set]] serves as the [[initial object]] in '''Set''' with [[empty function]]s as morphisms.  Every [[singleton (mathematics)|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 object]]s in '''Set'''. 

The category '''Set''' is [[complete category|complete and co-complete]]. The [[product (category theory)|product]] in this category is given by the [[cartesian product]] of sets. The [[coproduct (category theory)|coproduct]] is given by the [[disjoint union]]: given sets ''A''<sub>''i''</sub> where ''i'' ranges over some index set ''I'', we construct the coproduct as the union of ''A''<sub>''i''</sub>×{''i''} (the cartesian product with ''i'' serves to ensure that all the components stay disjoint).

'''Set''' is the prototype of a [[concrete category]]; 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 [[cartesian closed category|cartesian closed]] and [[Regular_category#Exact_%28effective%29_categories|exact in the sense of Barr]]).

'''Set''' is not [[abelian category|abelian]], [[additive category|additive]] nor [[preadditive category|preadditive]].

Every non-empty set is an [[injective object]] in '''Set'''. Every set is a [[projective module|projective object]] in '''Set''' (assuming the [[axiom of choice]]).

The [[Accessible category|finitely presentable objects]] in '''Set''' are the finite sets. Since every set is a [[direct limit]] of its finite subsets, the category '''Set''' is a [[Accessible category|locally finitely presentable category]].

If ''C'' is an arbitrary category, the [[Contravariant functor|contravariant functors]] from ''C'' to '''Set''' are often an important object of study. If ''A'' is an object of ''C'', then the functor from ''C'' to '''Set''' that sends ''X'' to Hom<sub>''C''</sub>(''X'',''A'') (the set of morphisms in ''C'' from ''X'' to ''A'') is an example of such a functor. If ''C'' is a [[Category_(mathematics)#Small_and_large_categories|small category]] (i.e. the collection of its objects forms a set), then the contravariant functors from ''C'' to '''Set''', together with natural transformations as morphisms, form a new category, a [[functor category]] known as the category of [[presheaves]] on ''C''.