# Documentation

Mathlib.Topology.Sheaves.LocallySurjective

# Locally surjective maps of presheaves. #

Let X be a topological space, ℱ and 𝒢 presheaves on X, T : ℱ ⟶ 𝒢 a map.

In this file we formulate two notions for what it means for T to be locally surjective:

1. For each open set U, each section t : 𝒢(U) is in the image of T after passing to some open cover of U.

2. For each x : X, the map of stalks Tₓ : ℱₓ ⟶ 𝒢ₓ is surjective.

We prove that these are equivalent.

def TopCat.Presheaf.IsLocallySurjective {C : Type u} {X : TopCat} {ℱ : } {𝒢 : } (T : 𝒢) :

A map of presheaves T : ℱ ⟶ 𝒢 is locally surjective if for any open set U, section t over U, and x ∈ U, there exists an open set x ∈ V ⊆ U and a section s over V such that $T_*(s_V) = t|_V$.

See TopCat.Presheaf.isLocallySurjective_iff below.

Instances For
theorem TopCat.Presheaf.isLocallySurjective_iff {C : Type u} {X : TopCat} {ℱ : } {𝒢 : } (T : 𝒢) :
∀ (U : ) (t : ().obj (𝒢.obj ())) (x : X), x UV ι, (s, ↑(T.app ()) s = ) x V
theorem TopCat.Presheaf.locally_surjective_iff_surjective_on_stalks {C : Type u} {X : TopCat} {ℱ : } {𝒢 : } (T : 𝒢) :
∀ (x : X), Function.Surjective ↑(().map T)

An equivalent condition for a map of presheaves to be locally surjective is for all the induced maps on stalks to be surjective.