Stalks

## Stalks

Let $X$ be a topological space, let $x\in X$ and let $ℱ:𝒯\to 𝒜$ be a functor.

Let ${𝒩}_{X}$ be the neighourhood filter at $x$.

The stalk of $𝒰$ at $x$ is $ℱ x = lim U ∈ 𝒩 x ℱ U (direct limit).$

Let $E = ∏ x ∈ X ℱ x$ with topology given by setting $W$ open if it satisfies the following constraint. $If U ∈ 𝒯 and s ∈ ℱ U then s ~ -1 W is open,$ where $s ~ : U ⟶ E x ⟼ s x is given by ℱ U ⟶ ℱ x s ⟼ s x$ is the natural homomorphism.

The sheafification of $ℱ$ is the sheaf $\stackrel{~}{ℱ}$ given by $ℱ ~ U = s ′ = s ′ x ∈ ∏ x ∈ U ℱ x | if x ∈ U then there exists V ∈ 𝒯 , and s ∈ ℱ V such that x ∈ V ⊆ U , and s ′ y y ∈ V = s y y ∈ V = s ′ : U → E | s ′ is continuous and p ∘ s ~ = id U$ where $p:E\to X$ sends ${ℱ}_{x}$ to $x$.

We have homomorphisms $ℱ U ↪ ℱ ~ U s ↦ s x x ∈ U .$