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For a presheaf $F$ on a category equipped with a pretopology, one has the sheafification $F^{\sharp}$ of $F$.

I know well the plus-construction of sheafification, which is presented in Artin's paper "Grothendieck Topologies", for example.


I have heard that there is another construction of sheafification by using hypercovering, but I can not find good explanation of it.

So, I want some examples to feel the essence of the construction.

In particular, I am interested in the following situation:

Let $C$ be a category with a pretopology $T$.

For an object $X \in C$, denote by $F_{X}$ the presheaf on $C$ represented by $X$ i.e. $F_{X}=Hom_{C}(-,X)$.

Now, let $X_{\bullet} \to X$ be a $T$-hypercover of $X \in C$.

Then, how can the sheafification of $F_{X}$ be written by using $F_{X_{\bullet}}$ ?

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You have to look at all hypercovers, not just one of them. Related question: – Mike Shulman Jan 13 '13 at 18:11

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