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 (or fascist object) is given by a cofree functor.
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).
projective object, projective presentation, projective cover, projective resolution
injective object, injective presentation, injective envelope, injective resolution
free object, free resolution
flat object, flat resolution
Last revised on April 15, 2016 at 11:53:16. See the history of this page for a list of all contributions to it.