# 2021-09-19

## 22:51

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!!

#web/quick-notes #personal/idle-thoughts #higher-algebra/simplicial