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Stream: learning: questions

Topic: ✔ epipresheaf, powerset, contractible cover

Naso (Sep 05 2022 at 13:47):

Let $(X,\mathcal{T})$ be a topological space and $O : \mathcal{T}^{op} \to \mathcal{Set}$ an epipresheaf (aka conjunctive presheaf).
Let $P : \mathcal{Set} \to \mathcal{Set}$ be the covariant powerset functor.
Let $\mathcal{U}=\{U_i\}_i$ be a cover of $X$ such that the Cech nerve $N\mathcal{U}$ is contractible.

Then is it true that $F := P \circ O$ is conjunctive w.r.t. $\mathcal{U}$, i.e., does every compatible family $\{s_i \in F U_i \}_i$ (with $s_i \mid_{U_i \cap U_j} = s_j \mid_{U_i \cap U_j}$ for all $i,j$) come from a (possibly not unique) $s \in F X$ with ${s \mid_{U_i}} = s_i$ for all $i$?

If it's not true I would be interested in if some extra assumptions on $X$, $O$ or $\mathcal{U}$ can make it true, e.g. finiteness.

EDIT: sorry about bumping this, I was trying to delete it. I think this is a counterexample for this too.

Notification Bot (Sep 21 2022 at 09:51):

naso has marked this topic as resolved.