Projective object

An object $P$ is projective when every morphism $P\to B$ lifts through every epimorphism $A\twoheadrightarrow B$. Equivalently, the functor $\mathcal{C}(P,-)$ preserves epimorphisms.

In an abelian category with enough projectives, a projective object has a resolution concentrated in degree zero, so every first left derived functor vanishes on it. Conversely choose $0\to K\to P\to A\to 0$ with $P$ projective and use the additive functor $F=\mathcal{A}(-,K)$ with target the opposite of the category of abelian groups. This functor is right exact in that target. If $L_{1}F(A)=0$, the derived exact sequence makes $F(K)\to F(P)$ monic, meaning that $\mathrm{Hom}(P,K)\to \mathrm{Hom}(K,K)$ is surjective in abelian groups. Extending $1_K$ gives a retraction of $K\to P$, so the sequence splits and $A$ is a direct summand of the projective $P$.

For a locally small category $\mathcal{C}$, every representable functor $\mathcal{C}(A,-)$ is a projective object in a category in $[\mathcal{C},\mathbf{Set}]$. The Yoneda lemma identifies maps from this functor with elements at $A$, and the pointwise epimorphism in a functor category criterion lets such an element be lifted through any epimorphism.

A set-indexed coproduct in a category of projective objects in a category is projective. Restrict a map out of the coproduct to each summand, lift each restriction through the given epimorphism, and assemble the lifts by the coproduct universal property. Choosing an arbitrary family of lifts uses the usual axiom of choice. The empty coproduct, an initial object, is projective as well.

An object is indecomposable projective if every epic map from a coproduct to it has a component which is split epic. In a presheaf category, representables have this property by evaluation of the identity. Conversely the canonical representable cover shows such an object is a retract of a representable; if idempotents split in the indexing category, it is itself representable.

For a small category $\mathcal{C}$, an object $P$ of $[\mathcal{C},\mathbf{Set}]$ is an irreducible projective when $\mathrm{Nat}(P,-)$ preserves coproduct in a category objects and epimorphisms. Preservation of coproducts means every map from $P$ into a coproduct factors uniquely through one summand. Such objects are precisely retracts of representable functors; when the indexing category $\mathcal{C}$ is a Cauchy-complete category, they are representable. To see the retract, split the canonical epimorphism from a coproduct of representables using projectivity, and use coproduct preservation to factor the splitting through one summand.

For a small category $\mathcal{C}$ and $F:\mathcal{C}\to \mathbf{Set}$, the Yoneda lemma gives a canonical pointwise-surjective natural transformationEach representable is projective because evaluation preserves pointwise epimorphisms, and their coproduct is projective. Thus every set-valued functor on a small category is an epimorphic image of a projective object.