Exercise 7.53

For part 1. we need to check that Ω is functorial i.e. that it preserves identities and composition. It preserves identities because the identity on every open set \(U\) in 𝓒 (the trivial inclusion ⊆) is mapped to the identity on \(Ω(U)\) in Set. This latter identity is \(Ω(U) → Ω(U)\) or \(U' ↦ U' ∩ U\) or \(U' ↦ U'\) (every subset of the open set gets mapped to itself).

To preserve composition we must have that if \(f⨟g = h\) then \(Ω(f)⨟Ω(g) = Ω(h)\). Using syntax from Sheaf (mathematics), and including a reversal because presheafs are contravariant, if W ⊆ V ⊆ U then \(res_{V,U}⨟res_{W,V} = res_{W,U}\). The first two restriction maps are:

\[\begin{split} \begin{align} U' & ↦ U' ∩ V \\ V' & ↦ V' ∩ W \end{align} \end{split}\]

Replacing the dummy variables with \(S\) (for set) to make for easier reading:

\[\begin{split} \begin{align} S & ↦ S ∩ V \\ S & ↦ S ∩ W \end{align} \end{split}\]

Composing these and using the fact that W ⊆ V so that V ∩ W = W:

\[\begin{split} \begin{align} S & ↦ (S ∩ V) ∩ W \\ S & ↦ S ∩ (V ∩ W) \\ S & ↦ S ∩ W \end{align} \end{split}\]

For part 2., it’s true that all you need to do to check that something like Ω is a presheaf is to check that it is a functor, because a functor and a presheaf are essentially synonymous.