nLab free object

Definition

Let $U: C\to D$ be a forgetful functor and $x\in D$ an object of the category $D$.

A free $C$-object on $x$ with respect to $U$ is an object of $C$ that satisfies the universal property that $F(x)$ would have, if $F$ were a left adjoint to $U$ (the corresponding free functor) (the free construction on $x$).

If $U$ actually has a left adjoint, then $F(x)$ is a free $C$-object on $x$ for every $x$, and conversely if there exists a free $C$-object on every $x\in D$ then $U$ has a left adjoint. But individual free objects can exist without the whole left adjoint functor existing. In general, we have a “partially defined adjoint”, or $J$-relative adjoint where $J$ is the inclusion of a full subcategory (on those objects admitting free objects).

More precisely: a free $C$-object on $x$ consists of an object $y\in C$ together with a morphism $\eta_x \colon x\to U y$ in $D$ such that for any other $z\in C$ and morphism $f\colon x\to U z$ in $D$, there exists a unique $g\colon y\to z$ in $C$ with $U(g) \circ \eta_x = f$.

In other words, it is an initial object of the comma category $(x/U)$. A free $C$-object on $x$ is also sometimes called a universal arrow from $x$ to the functor $U$. It can also be identified with a semi-final lift of an empty $U$-structured sink.

Sometimes one says an object $c$ of $C$ is free (relative to a forgetful functor $U: C \to D$ which is often tacitly understood) if there is some object $x$ of $D$ and some arrow $x \to U c$ that is initial in $(x/U)$. For example, the Quillen-Suslin theorem says that finitely generated projective modules over polynomial algebras over a field are free; the tacit forgetful functor is from the category of modules over a polynomial algebra to $Set$. In this way, freeness is understood as a property of an object.

Similarly, a cofree object is given by a cofree functor.

Examples

• free abelian group

• free module

• distributivity pullback

For more examples see at free construction.

A general way to construct free objects 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).

Related concepts

• free functor, relative adjoint, universal property

• projective object, projective presentation, projective cover, projective resolution

  • projective module

• injective object, injective presentation, injective envelope, injective resolution

  • injective module

• free object, free resolution

  • free module

• flat object, flat resolution

  • flat module