Different definitions of projective objects

Solution 1:

For clarity, let us reserve the phrase projective object for an object satisfying (i) and use choice object for an object satisfying (ii). (One formulation of the axiom of choice is to say that every object in $\textbf{Set}$ is a choice object.)

Proposition. If $\mathcal{C}$ is a category such that

• all pullbacks exist, and
• epimorphisms are preserved by pullbacks,

then choice objects in $\mathcal{C}$ are projective objects.

Proof. Let $C$ be a choice object in $\mathcal{C}$, let $f : A \twoheadrightarrow B$ be an epimorphism, and let $h : C \to B$ be any morphism. Let $r : A \times_B C \to C$ be the pullback of $f$ along $h$. By hypothesis, $r$ is an epimorphism, so there is a morphism $s : C \to A \times_B C$ with $r \circ s = \textrm{id}$. Let $\ell : A \times_B C \to A$ be the pullback of $h$ along $f$, and set $g = \ell \circ s$. By construction, $$f \circ g = f \circ \ell \circ s = h \circ r \circ s = h$$ so we have the required lifting of $h$ along $f$. Thus $C$ is projective. $\quad \blacksquare$

This is a much milder assumption than assuming $\mathcal{C}$ has enough projectives. For example, every topos and every abelian category satisfies the hypotheses of the proposition. If $\mathcal{C}$ is a regular category then the same proof shows that every choice object is regular-projective. (A regular-projective object is an object with the lifting property for regular epimorphisms.) In some cases every epimorphism is regular: for example, the category of groups is a regular category in which every epimorphism is regular.