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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeMar 5th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThe 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?

   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?