22:51
Tags: #personal/idle-thoughts #higher-algebra/simplicial
Not sure how to get this to work yet, but here’s the condition for a functor to be a sheaf:
But we can write \(n{\hbox{-}}\)fold intersections as fiber products :
So the condition of \({\mathcal{F}}\) being a sheaf seems to look like letting \({\mathcal{U}}\rightrightarrows X\) be an open cover, setting \(M = {\textstyle\coprod}U_i\), then applying a bar construction \begin{align*} M: M{ {}^{ \scriptscriptstyle \underset{\scriptscriptstyle {X} }{\times} ^{1} } } \leftarrow M{ {}^{ \scriptscriptstyle \underset{\scriptscriptstyle {X} }{\times} ^{2} } } \leftarrow\cdots .\end{align*} Then apply \({\mathcal{F}}\), and look at some kind of image sequence? And ask for exactness for \(n\) many levels to get a sheaf, Unsorted/stacks MOC, etc:
The problem is that I don’t really know how to relate the bottom line (whose exactness is the usual condition for sheaves, stacks, etc) to the intermediate steps. This seems like it wants \({\mathcal{F}}({\textstyle\coprod}{-}) = \prod {\mathcal{F}}({-})\), so it commutes with (co?)limits, since probably contravariant functors send coproducts to products. Moreover the bar construction in the 2nd line might form a simplicial object? And the condition of satisfying descent is maybe related to either this being a simplicial object, or its image in the bottom line assembling to a simplicial object, since there are clear degeneracy maps and one would want sections in order to build face maps. Super vague, there are a lot of details missing here!!