Topos Theory (Part 3)

Can We Fix The Air? Topos Theory (Part 4)

Topos Theory (Part 3)

Last time I described two viewpoints on sheaves. In the first, a sheaf on a topological space is a special sort of presheaf

$F \colon \mathcal{O}(X)^{\mathrm{op}} \to \mathsf{Set}$

Namely, it’s one obeying the ‘sheaf condition’.

I explained this condition in Part 1, but here’s a slicker way to say it. Suppose $U \subseteq X$ is an open set covered by a collection of open sets $U_i \subseteq U.$ Then we get this diagram:

$\displaystyle{ FU \rightarrow \prod_i FU_i \rightrightarrows \prod_{i,j} F(U_i \cap U_j) }$

The first arrow comes from restricting elements of to the smaller sets U_i. The other two arrows come from this: we can either restrict from FU_i to $F(U_i \cap U_j),$ or restrict from...