nLab free functor

Idea

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

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

Free objects

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

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

Examples

Classically, examples of free constructions were characterized by a universal property. For example, in the case of the free group on a set $X$ the universal property states that any map $X \to G$ as sets uniquely extends to a group homomorphism $F(X) \to G$. 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 $(X \downarrow U)$ (where $U$ 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

• the free monoid functor Set $\to$ Mon;

• the free module functor Set $\to$ $K$ Mod for a rig $K$;

• the free group functor Set $\to$ Grp;

• the group completion functor Mon $\to$ Grp

  in the abelian case in particular:

  the Grothendieck group completion functor CMon $\to$ Ab

• the free abelian group functor Set $\to$ Ab;

• the abelianization functor Grp $\to$ Ab;

• the free category functor $Grph_{d,r} \to$ Cat;

• the free operad functor;

• the unitisation functor Rng $\to$ Ring.

Free constructions on categories:

• free cocompletion, free coproduct completion, idempotent completion, Cauchy completion

One formal sort of free functor is the left adjoint $C\to C^T$, where $T$ is a monad on the category $C$ and $C^T$ is its Eilenberg-Moore category (the category of $T$-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 $M$ is an operad in a symmetric monoidal category $V$, $Prop(M)$ its associated PROP, and if $C$ is a monoidal $V$-category, then an $M$-coalgebra in $C$ may be identified with a monoidal $V$-functor $Prop(M)^{op} \to C$. Under suitable completeness assumptions on $C$, the forgetful functor $M$-$Coalg_C \to C$ has a right adjoint, and this forgetful functor is comonadic.

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

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

• When $U: C \to Set$ is topological concrete category over $Set$, as for example the forgetful functor $U: Top \to Set$, it frequently happens that $U$ 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, $Top$.

Related concepts

• free object, universal property