nLab free functor




Informally, a free functor is a left adjoint to a forgetful functor – part of a free-forgetful adjunction. (This is informal because the concept of forgetful functor is informal; any functor might be viewed as forgetful, so any left adjoint might be viewed as free, while in practice only some are.)

Formally, with respect to a monad or algebraic theory or operad TT, and TAlg(C)T Alg(C) the corresponding category of algebras over a monad or algebras over an algebraic theory or algebras over an operad, respectively, in some category CC, the free TT-algebra functor is the left adjoint to the forgetful functor TAlg(C)CT Alg(C) \to C.

Such a functor may be thought of as sending any object of CC to the TT-algebra freely generated by it.

Free objects

In general, if U:CDU: C \to D is thought of as a forgetful functor and F:DCF: D \to C is its left adjoint, then F(x)F(x) is the free C-object on an object xx of DD.

More generally, even if the entire left adjoint FF doesn’t exist, a free object on xx can be defined using a universal property, as “what the value of F(x)F(x) would be if FF existed.” Conversely, if a free object on xx exists for all xDx\in D, then the left adjoint FF can be assembled from them.

Cofree functors

Dually, a cofree functor is a right adjoint to a forgetful functor.

For the classical functors which forget algebraic structure, cofree functors are less common than free functors.


Classically, examples of free constructions were characterized by a universal property. For example, in the case of the free group on a set XX the universal property states that any map XGX \to G as sets uniquely extends to a group homomorphism F(X)GF(X) \to G. When such a free construction can be realized as a left adjoint functor, this universal property is just a transliteration of the fact that the unit of the free-forgetful adjunction is an initial object in the comma category (XU)(X \downarrow U) (where UU is the forgetful functor out of the category of algebras, see e.g. the proof of Freyd’s general adjoint functor theorem.)

For free functors

Free constructions on categories:

One formal sort of free functor is the left adjoint CC TC\to C^T, where TT is a monad on the category CC and C TC^T is its Eilenberg-Moore category (the category of TT-algebras). This includes all of the examples above and many others.

A general way to construct free functors is with a transfinite construction of free algebras (in set-theoretic foundations), or with an inductive type or higher inductive type (in type-theoretic foundations).

For cofree functors

  • The cofree coalgebra on a vector space. More generally, if MM is an operad in a symmetric monoidal category VV, Prop(M)Prop(M) its associated PROP, and if CC is a monoidal VV-category, then an MM-coalgebra in CC may be identified with a monoidal VV-functor Prop(M) opCProp(M)^{op} \to C. Under suitable completeness assumptions on CC, the forgetful functor MM-Coalg CCCoalg_C \to C has a right adjoint, and this forgetful functor is comonadic.

  • If MM is a monoid, the forgetful functor Set MSetSet^M \to Set on (left) MM-sets has a right adjoint Xhom(M,X)X \mapsto \hom(M, X), where MM acts on functions f:MXf: M \to X according to the rule (mf)(m)=f(mm)(m f)(m') = f(m' m). This forgetful functor is comonadic. Much more generally, the right adjoint to the underlying functor Set CSet/C 0Set^C \to Set/C_0 (C 0C_0 the set of objects of a category CC) is comonadic. More generally still, if VV is complete and f:CDf: C \to D is a functor between small categories, the functor V f:V DV CV^f: V^D \to V^C has a right adjoint (although V fV^f will not normally be comonadic in this generality).

  • The forgetful functor CatSetCat \to Set, taking a small category to its set of objects, has a right adjoint KK for which KXK X is a category whose objects are elements of XX and where there is exactly one morphism xyx \to y for any x,yXx, y \in X. The category KXK X, which is a groupoid, is known as the chaotic category on XX, or the indiscrete category on XX.

  • When U:CSetU: C \to Set is topological concrete category over SetSet, as for example the forgetful functor U:TopSetU: Top \to Set, it frequently happens that UU possesses a right adjoint, assigning to a set an “indiscrete topology”.

  • The ring of Witt vectors is the co-free Lambda-ring.

  • A rich source of examples is coreflective subcategories, which are comonadic over the ambient category. For example, the category of compactly generated spaces is coreflective in the category of all spaces, TopTop.

Last revised on October 12, 2021 at 15:26:20. See the history of this page for a list of all contributions to it.