Editing Free object (section)

==Definition==

Free objects are the direct generalization to [[Category (mathematics)|categories]] of the notion of [[Basis (linear algebra)|basis]] in a vector space. A linear function {{math|''u'' : ''E''<sub>1</sub> → ''E''<sub>2</sub>}} between vector spaces is entirely determined by its values on a basis of the vector space {{math|''E''<sub>1</sub>.}} The following definition translates this to any category.

A [[concrete category]] is a category that is equipped with a [[faithful functor]] to '''Set''', the [[category of sets]]. Let {{math|'''C'''}} be a concrete category with a faithful functor {{math|''U'' : '''C''' → '''Set'''}}. Let {{math|''X''}} be a set (that is, an object in '''Set'''), which will be the ''basis'' of the free object to be defined. A '''free object''' on {{mvar|X}} is a pair consisting of an object <math>A</math> in {{math|'''C'''}} and an injection <math>i:X\to U(A)</math> (called the ''[[Inclusion map|canonical injection]]''), that satisfies the following [[universal property]]:
:For any object {{math|''B''}} in {{math|'''C'''}} and any map between sets <math>g:X\to U(B)</math>, there exists a unique morphism <math>f:A\to B</math> in {{math|'''C'''}} such that <math>g=U(f)\circ i</math>. That is, the following [[Commutative diagram|diagram]] commutes:

{{Dark mode invert|[[File:Free-object-universal-property-tweak-color.svg|center|Commutative diagram]]}}

If free objects exist in {{math|'''C'''}}, [[Universal property#Relation to adjoint functors|the universal property implies]] every map between two sets induces a unique morphism between the free objects built on them, and this defines a functor <math>F:\mathbf{Set}\to \mathbf C</math>. It follows that, if free objects exist in {{math|'''C'''}}, the functor {{mvar|F}}, called the '''free functor''' is a [[left adjoint]] to the faithful functor {{mvar|U}}; that is, there is a bijection
:<math>\operatorname{Hom}_\mathbf{Set}(X, U(B))\cong \operatorname{Hom}_\mathbf{C}(F(X), B).</math>