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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 ${\eta }_x:x\to Uy$ in $D$ such that for any other $z\in C$ and morphism $f:x\to Uz$ in $D$, there exists a unique $g:y\to z$ 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.

Similarly, a cofree object (or fascist 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