# 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.