Exercise 7.64

Give an example of a space 𝑋, a sheaf \(𝑆 ∈ \textbf{Shv(𝑋)}\), and two predicates \(p, q : 𝑆 → Ω\) for which \(p(𝑠) ⊢_{𝑠:𝑆} q(𝑠)\) holds. You do not have to be formal.

Author’s solution

We need an example of a space \(𝑋\), a sheaf \(𝑆 ∈ \textbf{Shv}(𝑋)\), and two predicates \(𝑝, 𝑞 : 𝑆 → \Omega\) for which \(𝑝(𝑠) ⊢_{𝑠:𝑆} 𝑞(𝑠)\) holds. Take \(𝑋\) to be the one-point space, take \(𝑆\) to be the sheaf corresponding to the set \(𝑆 = ℕ\), let \(𝑝(𝑠)\) be the predicate “24 ≤ \(𝑠\) ≤ 28” and let \(𝑞(𝑠)\) be the predicate “\(𝑠\) is not prime.” Then \(𝑝(𝑠) ⊢_{𝑠:𝑆} 𝑞(𝑠)\) holds.

As an informal example, take \(𝑋\) to be the surface of the earth, take \(𝑆\) to be the sheaf of vector fields as in Example 7.46 thought of in terms of wind-blowing. Let \(𝑝\) be the predicate “the wind is blowing due east at somewhere between 2 and 5 kilometers per hour” and let \(𝑞\) be the predicate “the wind is blowing at somewhere between 1 and 5 kilometers per hour.” Then \(𝑝(𝑠) ⊢_{𝑠:𝑆} 𝑞(𝑠)\) holds. This means that for any open set \(𝑈\), if the wind is blowing due east at somewhere between 2 and 5 kilometers per hour throughout \(𝑈\), then the wind is blowing at somewhere between 1 and 5 kilometers per hour throughout \(𝑈\) as well.

Alternative answer

As before, we must set up a bit of a story to give a concrete example of a topological predicate. Continuing with previous examples, we’ll assume that time is discrete and our universe has only two timestamps. Bob only exists at the second timestamp, and Alice (c.f. Alice and Bob) exists at both timestamps. One possible visualization of these people in a single drawing is:

In the previous drawing we show one example section, representing “Alice” and defined over the open set \(\{0,1\}\). We also draw the “Alice” sheaf more compactly as a gray box.

We’ll say Alice only likes the weather at the first timestamp, and Bob only likes the weather at the second timestamp. We could visualize the “… likes the weather” predicate \(p\) as:

One can check that this is a natural transformation, and that \(S\) is a sheaf. However, the author’s language sometimes implies the people sheaf only includes sections representing people who exist throughout the whole of \(U\). An attempt to “fix” this issue:

Unfortunately, in this model there is no unique gluing of \(t_{0A}\) and \(t_{1B}\). See Exercise 7.62 for a discussion.