Editing Free object (section)

==Free functor<!--'Free functor' and 'Cofree functor' redirect here-->==
The most general setting for a free object is in [[category theory]], where one defines a [[functor]], the '''free functor'''<!--boldface per WP:R#PLA-->, that is the [[left adjoint]] to the [[forgetful functor]].

Consider a category '''C''' of [[algebraic structure]]s; the objects can be thought of as sets plus operations, obeying some laws. This category has a functor, <math>U:\mathbf{C}\to\mathbf{Set}</math>, the [[forgetful functor]], which maps objects and morphisms in '''C''' to '''Set''', the [[category of sets]]. The forgetful functor is very simple: it just ignores all of the operations.

The free functor ''F'', when it exists, is the left adjoint to ''U''. That is, <math>F:\mathbf{Set}\to\mathbf{C}</math> takes sets ''X'' in '''Set''' to their corresponding free objects ''F''(''X'') in the category '''C'''.  The set ''X'' can be thought of as the set of "generators" of the free object ''F''(''X'').

For the free functor to be a left adjoint, one must also have a '''Set'''-morphism <math>\eta_X:X\to U(F(X))\,\!</math>. More explicitly, ''F'' is, up to isomorphisms in '''C''', characterized by the following [[universal property]]:
:Whenever {{Math|''B''}} is an algebra in {{Math|'''C'''}}, and <math>g\colon X\to U(B)</math> is a function (a morphism in the category of sets), then there is a unique {{Math|'''C'''}}-morphism <math>f\colon F(X)\to B</math> such that <math>g=U(f)\circ \eta_X</math>.

Concretely, this sends a set into the free object on that set; it is the "inclusion of a basis". Abusing notation, <math>X \to F(X)</math> (this abuses notation because ''X'' is a set, while ''F''(''X'') is an algebra; correctly, it is <math>X \to U(F(X))</math>).

The [[natural transformation]] <math>\eta:\operatorname{id}_\mathbf{Set}\to UF</math> is called the [[unit (category theory)|unit]]; together with the [[counit]] <math>\varepsilon:FU\to \operatorname {id}_\mathbf{C}</math>, one may construct a [[T-algebra]], and so a [[monad (category theory)|monad]].

The '''cofree functor'''<!--boldface per WP:R#PLA--> is the [[right adjoint]] to the forgetful functor.

===Existence===
There are general existence theorems that apply; the most basic of them guarantees that 
:Whenever '''C''' is a [[variety (universal algebra)|variety]], then for every set ''X'' there is a free object ''F''(''X'') in '''C'''.

Here, a variety is a synonym for a [[finitary algebraic category]], thus implying that the set of relations are [[finitary relation|finitary]], and ''algebraic'' because it is [[monad (category theory)|monadic]] over '''Set'''.

===General case===
Other types of forgetfulness also give rise to objects quite like free objects, in that they are left adjoint to a forgetful functor, not necessarily to sets.

For example, the [[tensor algebra]] construction on a [[vector space]] is the left adjoint to the functor on [[associative algebra]]s that ignores the algebra structure. It is therefore often also called a [[free algebra]]. Likewise the [[symmetric algebra]] and [[exterior algebra]] are free symmetric and anti-symmetric algebras on a vector space.