# Exercise 7.64#

Give an example of a space π, a sheaf $$π β \textbf{Shv(π)}$$, and two predicates $$p, q : π β Ξ©$$ for which $$p(π ) β’_{π :π} q(π )$$ holds. You do not have to be formal.

## Authorβs solution#

We need an example of a space $$π$$, a sheaf $$π β \textbf{Shv}(π)$$, and two predicates $$π, π : π β \Omega$$ for which $$π(π ) β’_{π :π} π(π )$$ holds. Take $$π$$ to be the one-point space, take $$π$$ to be the sheaf corresponding to the set $$π = β$$, let $$π(π )$$ be the predicate β24 β€ $$π$$ β€ 28β and let $$π(π )$$ be the predicate β$$π$$ is not prime.β Then $$π(π ) β’_{π :π} π(π )$$ holds.

As an informal example, take $$π$$ to be the surface of the earth, take $$π$$ to be the sheaf of vector fields as in Example 7.46 thought of in terms of wind-blowing. Let $$π$$ be the predicate βthe wind is blowing due east at somewhere between 2 and 5 kilometers per hourβ and let $$π$$ be the predicate βthe wind is blowing at somewhere between 1 and 5 kilometers per hour.β Then $$π(π ) β’_{π :π} π(π )$$ holds. This means that for any open set $$π$$, if the wind is blowing due east at somewhere between 2 and 5 kilometers per hour throughout $$π$$, then the wind is blowing at somewhere between 1 and 5 kilometers per hour throughout $$π$$ as well.

In the previous drawing we show one example section, representing βAliceβ and defined over the open set $$\{0,1\}$$. We also draw the βAliceβ sheaf more compactly as a gray box.
Weβ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 ββ¦ likes the weatherβ predicate $$p$$ as:
One can check that this is a natural transformation, and that $$S$$ is a sheaf. However, the authorβs language sometimes implies the people sheaf only includes sections representing people who exist throughout the whole of $$U$$. An attempt to βfixβ this issue:
Unfortunately, in this model there is no unique gluing of $$t_{0A}$$ and $$t_{1B}$$. See Exercise 7.62 for a discussion.