Projective

Let be a class of Epimorphisms in a Locally Small Category An Object in is projective if Hom-Functor preserves Epimorphisms in i.e. is surjective for all in . i.e. if given

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\begin{tikzcd}
& P 
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Q \arrow[r,two heads,"g"] & R
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where is an Epimorphism in then there exists with

We say that is projective if are all Epimorphisms in

Lemma

For any Small Category , the functors are pointwise projective in

Proof

Let and some Functors. Suppose is a Natural Transformation and that is surjective. Let be a Natural Transformation. By Yoneda Lemma, corresponds to some (i.e. ) Then there is some such that But then there is some Natural Transformation such that (again by Yoneda Lemma) We conclude that

But is also a Natural Transformation so we can write

and is bijective so

Proposition

Let be a Small Category. For any Functor there is a pointwise Epimorphism in with being pointwise projective.

Proof

Given , let be the disjoint union

where the union is taken over and . Now is a Coproduct A Coproduct of -projective objects is -projective so is pointwise projective. The morphism whose th component

is pointwise Epimorphism, since is in the image of .