The functor is the composite , and pullback preserves epis, so if preserves epis then so does .
Conversely, for any in , we have a pair of pullback squares:
The lower square and outer rectangle are easily seen to be pullbacks, hence so is the upper square. Since epimorphisms are closed under pullback, the other implication follows.
If is internally projective in , then is internally projective in .
equipped with the obvious projection to . Now we have a pullback square
in which the right-hand map is epi since is internally projective; hence so is the left-hand map.
is internally projective if and only if the statement ” is projective” is true in the stack semantics of .
By definition, truth of ” is projective” in the stack semantics means that for any and any epimorphism , there exists an epimorphism and a section of . (We have used the characterization that an object is projective just when every epimorphism with codomain is split.)
If is internally projective, then given an epimorphism as above, let , where is the projection. Since is internally projective in by Lemma 1, is an epimorphism. And is split by the counit of the adjunction .
Conversely, suppose the above condition holds, and let be an epimorphism in . Let , and let be the pullback of along the counit . Then is equivalently , where is the projection.
Morevoer, is epi, so by assumption there is an epi such that is split. Since pullback along epis reflects epis, it suffices to show that is split. However, we have a pullback square
so by the Beck-Chevalley condition, is equivalently . But is split, and all functors preserve split epis.
Note that Lemma 4.5.3(iii) of Sketches of an Elephant is the special case of the above stack-semantics version of internal projectivity when . This is insufficient for the implication (iii)(ii) of that lemma to hold, since if so, then every projective object would be internally projective, which as we show below is not the case.
If has enough projectives and projectives are closed under binary products, then every projective object is internally projective. (In particular, if all objects of are projective then all objects are internally projective.)
Let be a projective object. To show that is epic whenever is epic, choose an epi where is projective (using the assumption of enough projectives). Since is projective, there exists a lift through of the horizontal composite as shown:
this, by currying, provides a lift of through . Since is epic, this immediately implies is epic, as desired.
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.
In a presheaf topos , if has binary products, then every projective presheaf is internally projective.
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 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.
flat object, flat resolution