nLab
internally projective object

Idea

An object of a category is internally projective if it satisfies the internalization of the condition on a projective object.

Definition

An object $E$ of a topos $𝒯$ is called internally projective if the internal hom/exponential object functor

$$(-{)}^E:𝒯\to 𝒯$$(-)^E \colon \mathcal{T} \to \mathcal{T}

preserves epimorphisms.

Properties

If the terminal object in $𝒯$ is projective, then every internally projective object is projective. In the converse direction,

Proposition

If $𝒯$ has enough projectives and projectives are closed under binary products, then every projective object is internally projective. (In particular, if all objects of $𝒯$ are projective then all objects are internally projective.)

Proof

Let $P$ be a projective object. To show that $e^P:E^P\to B^P$ is epic whenever $e:E\to B$ is epic, choose an epi $\varphi :P\prime \to B^P$ where $P\prime$ is projective (using the assumption of enough projectives). Since $P\times P\prime$ is projective, there exists a lift through $e$ of the horizontal composite as shown:

$$\begin{array}{ccccc}& & & & E\\ & & & & {\downarrow }^e\\ P\prime \times P& \underset{\varphi \times 1}{\to }& B^P\times P& \underset{\mathrm{eval}}{\to }& B;\end{array}$$\array{ & & & & E \\ & & & & \downarrow ^\mathrlap{e} \\ P' \times P & \underset{\phi \times 1}{\to} & B^P \times P & \underset{eval}{\to} & B; }

this, by currying, provides a lift of $\varphi :P\prime \to B^P$ through $e^P$. Since $\varphi$ is epic, this immediately implies $e^P$ is epic, as desired.

Remark

Proposition 1 may fail without the assumption that projective objects are closed under binary products. An example is given here.

Remark

The internal axiom of choice (that is, the axiom of choice interpreted in the internal logic of the topos) is equivalent to the statement that every object is internally projective. This is strictly weaker than the “external” axiom of choice that every epimorphism in the topos is split.

Corollary

In a presheaf topos ${\mathrm{Set}}^{C^{\mathrm{op}}}$, if $C$ has binary products, then every projective presheaf is internally projective.

Proof

Representables, and arbitrary coproducts of representables, are projective, and every presheaf is covered by some coproduct of representables. This implies that projective presheaves are precisely retracts of coproducts of representables. Under the assumption that $C$ has binary products, coproducts of representables, and also their retracts, are also closed under binary products. Thus projective presheaves are closed under binary products. Now apply Proposition 1.

Related concepts

• projective object, projective presentation, projective cover, projective resolution

  • projective module

  • internally projective object

• injective object, injective presentation, injective envelope, injective resolution

  • injective module

• free object, free resolution

  • free module

• flat object, flat resolution

  • flat module