nLab
pure subobject

Idea

A pure subobject is a monomorphism $A\rightarrowtail B$ – hence a subobject $A$ of some object $B$ in some category – which is a pure morphism: such that any sufficiently small system of equations involving constants in $A$ that admits a solution in $B$ also admits a solution in $A$. This generalises the classical notions of ‘pure group’ and ‘pure submodule’.

Definition

Let $\kappa$ be a regular cardinal. A $\kappa$-pure morphism in a category $𝒞$ is a morphism $f:A\to B$ with the following extension property:

• Given any morphisms $f\prime :A\prime \to B\prime$, $a:A\prime \to A$, $b:B\prime \to B$ in $𝒞$, if both $A\prime$ and $B\prime$ are $\kappa$-compact and $f\circ a=b\circ f\prime$,

  $$\begin{array}{ccc}A\prime & \stackrel{a}{\to }& A\\ {\downarrow }^{f\prime }& & {\downarrow }^f\\ B\prime & \stackrel{b}{\to }& B\end{array},$$\array{ A' &\stackrel{a}{\to}& A \\ \downarrow^{\mathrlap{f'}} && \downarrow^{\mathrlap{f}} \\ B' &\stackrel{b}{\to}& B } \,,

  then there exists a (not necessarily unique) morphism $\overline{a}:B\prime \to A$ in $𝒞$ such that $a=\overline{a}\circ f\prime$. (We do not assert any compatibility with $b$, however.)

A $\kappa$-pure subobject is a $\kappa$-pure monomorphism.

Examples

• A retract is a $\kappa$-pure subobject in any category, for any $\kappa$.

• Conversely, any $\kappa$-pure subobject in Set is a retract.

• If $A$ is an injective module and $B$ is any module containing $A$ as a submodule, then the inclusion $A\hookrightarrow B$ is $\kappa$-pure. (This can be checked directly without recourse to the fact that any injective submodule is a retract!)

• The torsion subgroup of any abelian group is a $\kappa$-pure subgroup, since it is a filtered colimit of direct summands. (See below.)

Properties

Lemma

In any category:

• The class of $\kappa$-pure morphisms is closed under composition.

• If $g\circ f$ is a $\kappa$-pure morphism, then so is $f$.

• If $\kappa \prime \ge \kappa$, then any $\kappa$-pure morphism is also $\kappa \prime$-pure.

Proposition

In a $\kappa$-accessible category, any $\kappa$-pure morphism is necessarily monic.

This is LPAC, Prop. 2.29.

Proposition

If $𝒞$ is a $\kappa$-accessible category, then $\kappa$-pure subobjects in $𝒞$ are closed under $\kappa$-filtered colimits in the arrow category $\mathrm{Arr}(𝒞)$.

If $𝒞$ is a $\kappa$-accessible category with pushouts, then any $\kappa$-pure subobject in $𝒞$ is a $\kappa$-filtered colimit in $\mathrm{Arr}(𝒞)$ of retracts in $𝒞$.

This is LPAC, Prop. 2.30.

Proposition

In a $\kappa$-accessible category, every $\kappa$-pure morphism is a regular monomorphism.

This is LPAC, Prop. 2.31.

Applications

Theorem

Let $𝒞$ be a $\kappa$-accessible category, and let $𝒟$ be a full subcategory of $𝒞$ that is closed under $\kappa$-filtered colimits for some regular cardinal $\kappa$. Then, $𝒟$ is a $\mu$-accessible category for some regular cardinal $\mu$ sharply larger than $\kappa$ if and only if $𝒟$ is closed under $\kappa$-pure subobjects in $𝒞$.

In particular, a category $𝒟$ is accessible if and only if there is a fully faithful functor $R:𝒟\to {\mathrm{Set}}^{𝒜}$ where $𝒜$ is small, $R$ creates colimits for all $\kappa$-filtered diagrams, and $𝒟$ is closed under $\kappa$-pure subobjects in ${\mathrm{Set}}^{𝒜}$.

This is LPAC, Cor. 2.36.

References

• LPAC Jiri Adamek, Jiri Rosicky, Locally presentable and accessible categories