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 $\mathcal{C}$ is a morphism $f : A \to B$ with the following extension property:

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

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

  then there exists a (not necessarily unique) morphism $\bar{a} : B' \to A$ in $\mathcal{C}$ such that $a = \bar{a} \circ f'$. (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 $\kappa' \le \kappa$, then any $\kappa$-pure morphism is also $\kappa'$-pure.

Proposition

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

This is LPAC, Prop. 2.29.

Proposition

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

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

This is LPAC, Prop. 2.30.

Proposition

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

This is LPAC, Prop. 2.31.

Applications

Theorem

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

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

This is LPAC, Cor. 2.36.

References

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