---
date: 2021-09-19
tags: [ web/quick-notes ]
---


# 2021-09-19

## 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:

![](attachments/2021-09-19_22-52-18.png)

But we can write $n\dash$fold intersections as [fiber products](fiber%20products) :

![](attachments/2021-09-19_22-55-13.png)

So the condition of $\mcf$ being a sheaf seems to look like letting $\mcu \covers X$ be an open cover, setting $M = \disjoint U_i$, then applying a [bar construction](bar%20construction.md) 
\[
M: M\fiberpower{X}{1} \from M\fiberpower{X}{2} \from \cdots
.\]
Then apply $\mcf$, and look at some kind of image sequence? 
And ask for exactness for $n$ many levels to get a sheaf, [Unsorted/stacks MOC](Unsorted/stacks%20MOC.md), etc:


\begin{tikzcd}
	&& {M = M_0 \coloneqq \displaystyle\coprod_{i\in I} U_i} && {M_1 \coloneqq \displaystyle\coprod_{i,j\in I^{\times 2}} U_i\cap U_j} && {M_2 \coloneqq \displaystyle\coprod_{i,j, k\in I^{\times 3}} U_i\cap U_j \cap U_k} && \cdots \\
	&& {} \\
	&& M && {M^{\underset{X}\times 2} \coloneqq M\underset{X}\times M} && {M^{\underset{X}\times 3} \coloneqq M\underset{X}\times M \underset{X}\times M} && \cdots \\
	\\
	&& {\mathcal{F}(M)} && {\mathcal{F}(M^{\underset{X}\times 2})} && {\mathcal{F}(M^{\underset{X}\times 3})} && \cdots \\
	&&&&& {} \\
	{\mathcal{F}(\displaystyle\coprod_{i\in I}U_i)} && {\displaystyle\prod_{i\in I}\mathcal{F}(U_i)} && {\displaystyle\prod_{i, j \in I^{\times 2}}\mathcal{F}(U_i \cap U_j)} && {\displaystyle\prod_{i, j, k \in I^{\times 3}}\mathcal{F}(U_i \cap U_j \cap U_K)} && \cdots
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-5, to=1-7]
	\arrow[from=3-3, to=3-5]
	\arrow[from=3-5, to=3-7]
	\arrow["2", from=7-3, to=7-5]
	\arrow[from=1-7, to=1-9]
	\arrow[from=3-7, to=3-9]
	\arrow[Rightarrow, no head, from=1-3, to=3-3]
	\arrow[Rightarrow, no head, from=1-5, to=3-5]
	\arrow[Rightarrow, no head, from=1-7, to=3-7]
	\arrow["{\mathcal{F}}"{description}, from=3-3, to=5-3]
	\arrow["{\mathcal{F}}"{description}, from=3-5, to=5-5]
	\arrow["{\mathcal{F}}"{description}, from=3-7, to=5-7]
	\arrow[from=5-3, to=5-5]
	\arrow[from=5-5, to=5-7]
	\arrow[from=5-7, to=5-9]
	\arrow["1", from=7-1, to=7-3]
	\arrow["3", from=7-5, to=7-7]
	\arrow[from=7-7, to=7-9]
\end{tikzcd}


> [Link to Diagram](https://q.uiver.app/?q=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)

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 $\mcf(\disjoint \wait) = \prod \mcf(\wait)$, 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](Unsorted/descent.md) is maybe related to either this being a [simplicial object](simplicial%20object), 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!!