Let $C$ be category, let $PSh(C) = [C^{op},$ *Set*$]$ be the category of presheaves on $C$, the Karoubi envelope of $C$, denoted $\overline{C}$, is defined as the full subcategory of $PSh(C)$ which are retracts of objects in $C$ under the Yoneda embedding.
I'm reading this paper on Karoubi envelopes ("Cauchy completion in category theory" by Borceux and Dejean), on page 3 (theorem 1) they say:
"(4) The category $PSh(\overline{C})$ of presheaves on $\overline{C}$ is equivalent to the category $PSh(C)$ of presheaves on $C$"
I'm interested to see how they proved this equivalence so I've been trying to follow the whole proof to get to where they prove it but frankly I'm stuck and a lot of what they're saying is going over my head, like for example this:
"If every idempotent of $C$ splits, every retract of a representable functor $C(-,c)$ induces an idempotent on $C(-,c)$, thus an idempotent $e$ on $C$, $e$ splits in $C$, which produces a retraction of $c$ and thus a retraction of $C(-,c)$, which..."
That's where I first got lost, every retract of a representable functor induces an idempotent on itself thus an idempotent $e$ on $C$ that splits?? Don't follow at all, I tried to skip to the next paragraph (where they prove the equivalence), it says:
"To prove (4), it suffices to show that every presheaf $F$ on $C$ can be uniquely (up to an isomorphism) extended in a presheaf $\overline{F}$ on $\overline{C}$."
("extended IN a presheaf $\overline{F}$"? I guess they mean "extended TO a presheaf $\overline{F}$" right?), then they write:
"From the uniqueness of the splitting in $\overline{C}$ of an idempotent $e \in C$ and the Cauchy completeness of the category of sets (Proposition 1), $\overline{F}$ has to map the splitting of $e$ on the splitting of $F e$."
(Don't they mean $\overline{F}$ has to map the splitting of $e$ TO the splitting of $F(e)$? What do they mean by ON?)
The proof is longer but I just about stopped there, couldn't follow the rest of the proof, can anyone provide some clarification on this?
Please be nice to me :)
