# 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.

## 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, weβll 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 βAliceβ and defined over the open set \(\{0,1\}\). We also draw the βAliceβ sheaf more compactly as a gray box.

Weβll say Alice only likes the weather at the first timestamp, and Bob only likes the weather at the second timestamp. We could visualize the ββ¦ likes the weatherβ predicate \(p\) as:

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

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