Thread by @HigherGeometer

I have a stupid question: I'm pretty sure I remember having seen a construction that takes a bounded hypercover of a topological space X (I even only care about some cosk₁(V ⇉ U) → X and produces an open cover of X refining the hypercover. Or am I being dense?

I might just #trymathslive with this one.

Tomorrow. Unless someone beats me to it.

OK, so here goes. I had an idea last night and tried not to think about it.

I have a bounded hypercover cosk₁(V ⇉ U) → X of a space X, where U = ∐_α U(α) for an open cover {U(α)}_α of X, and V = ∐_{α1,α2,β} V(α1,α2,β), for {V(α1,α2,β)}_β an open cover of U(α1)∩U(α2).

1/

What I want is an open cover {W(γ)}_γ of X refining {U(α)}_α with the property that each W(γ1)∩W(γ2) ⊂ U(α1)∩U(α2), is contained in V(α1,α2,β) for some β.

Since {V(α1,α2,β)}_{α1,α2,β} is an open cover of X I could try that (what else?)

2/

Then V(α1,α2,β) ⊂ U(α1), for instance. Now

V(α11,α21,β1)∩V(α12,α22,β2) ⊂ U(α11)∩U(α12)

and I want V(α11,α21,β1)∩V(α12,α22,β2) ⊂ V(α11,α12,β3) for some β3.

3/

There's no a priori reason why such a V(α11,α12,β3) should exist.

However, I *am* allowed to refine my hypercover first, namely replace V by V' with more and finer open sets, perhaps duplicating some sets so as to be attached to different U(α)s.

4/

So for instance, I can arrange it so that the cover {V(α1,α2,β)}_{α1,α2,β} is closed under intersections. Moreover, I can make sure that for fixed α1,α2, if
U(α1)∩U(α2)∩V(α3,α4,β) is non-empty, then there is a β' with

V(α1,α2,β') = U(α1)∩U(α2)∩V(α3,α4,β).

5/

I said "...the cover {V(α1,α2,β)}_{α1,α2,β} is closed under intersections" but really what I should have said is what this is supposed to be covering. It should be

"...the cover {V(α1,α2,β)}_β *of U(α1)∩U(α2)* is closed under intersections"

which is stronger, and is doable

6/

Reminder (to me): the goal is, for all α11, α21, β1, α12, α22, β2,

V(α11,α21,β1)∩V(α12,α22,β2) ⊂ V(α11,α12,β3) for some β3

and notice the particular combination of indices on the RHS (wlog LHS≠∅).

But note

LHS ⊂ U(α11)∩U(α21)∩U(α12)∩U(α22) ⊂U(α11)∩U(α12)

7/

I should also note I can also refine U so that it is also a cover closed under intersections. I'm not sure I need to use this, yet.

8/

But since LHS≠∅ above, this means

U(α11)∩U(α12)∩V(α1i,α2i,βi)≠∅ for i=1,2

so there are β'1,β'2 with

V(α11,α12,β'i) = U(α11)∩U(α12)∩V(α1i,α2i,βi)

for some V(α11,α12,β'1), V(α11,α12,β'2) in the open cover of U(α11)∩U(α12).

9/

So then

V(α11,α21,β1)∩V(α12,α22,β2)
=
U(α11)∩U(α12)∩V(α11,α21,β1)∩U(α11)∩U(α12)∩V(α12,α22,β2)
= V(α11,α12,β'1)∩V(α11,α12,β'2)

but I assumed the cover {V(α11,α12,β)}_β is closed under ∩, so there is β'3 with

V(α11,α12,β'1)∩V(α11,α12,β'2)=V(α11,α12,β'3)

10/

...now we have β'3 with

V(α11,α21,β1)∩V(α12,α22,β2) = V(α11,α12,β'3)

which implies what I needed!

To recap: given cosk₁(V ⇉ U) → X, I want a cover W → X so that cosk₀(W) → X refines it: so take W to be a certain refinement of the cover V → X.

#trymathslive

11/11

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