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The sheafification of a (1-)presheaf on a site is classically constructed in a two-step process $X^{++}$, where $X^+$ consists of matching families in $X$, is always separated, and is a sheaf if $X$ is separated. But the sheafification can also be constructed in a single step by looking at matching families over hypercovers. However, the only published reference I can find which mentions this latter fact is Higher Topos Theory (section 6.5.3), and it doesn’t really give a proof. Does anyone know of a reference on “good old” 1-sheaves which discusses sheafification via hypercovers?
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