nLab
pure subobject

Contents

Idea

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

Definition

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

  • Given any morphisms f:ABf' : A' \to B', a:AAa : A' \to A, b:BBb : B' \to B in 𝒞\mathcal{C}, if both AA' and BB' are κ\kappa-compact and fa=bff \circ a = b \circ f',

    A a A f f B b B, \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 a¯:BA\bar{a} : B' \to A in 𝒞\mathcal{C} such that a=a¯fa = \bar{a} \circ f'. (We do not assert any compatibility with bb, however.)

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

Examples

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

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

  • If AA is an injective module and BB is any module containing AA as a submodule, then the inclusion ABA \hookrightarrow B is κ\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 κ\kappa-pure subgroup, since it is a filtered colimit of direct summands. (See below.)

Properties

Lemma

In any category:

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

  • If gfg \circ f is a κ\kappa-pure morphism, then so is ff.

  • If κκ\kappa' \le \kappa, then any κ\kappa-pure morphism is also κ\kappa'-pure.

Proposition

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

This is LPAC, Prop. 2.29.

Proposition

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

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

This is LPAC, Prop. 2.30.

Proposition

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

This is LPAC, Prop. 2.31.

Applications

Theorem

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

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

This is LPAC, Cor. 2.36.

References

Revised on November 27, 2013 02:15:34 by Zhen Lin (131.111.24.211)