Locally surjective maps of presheaves. #
X be a topological space,
𝒢 presheaves on
T : ℱ ⟶ 𝒢 a map.
In this file we formulate two notions for what it means for
T to be locally surjective:
For each open set
U, each section
t : 𝒢(U)is in the image of
Tafter passing to some open cover of
x : X, the map of stalks
Tₓ : ℱₓ ⟶ 𝒢ₓis surjective.
We prove that these are equivalent.
A map of presheaves
T : ℱ ⟶ 𝒢 is locally surjective if for any open set
x ∈ U, there exists an open set
x ∈ V ⊆ U and a section
$T_*(s_V) = t|_V$.
An equivalent condition for a map of presheaves to be locally surjective is for all the induced maps on stalks to be surjective.