# mathlib3documentation

topology.sheaves.locally_surjective

# Locally surjective maps of presheaves. #

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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 Top.presheaf.is_locally_surjective {C : Type u} {X : Top} {ℱ 𝒢 : X} (T : ℱ 𝒢) :
Prop

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 is_locally_surjective_iff below.

Equations
theorem Top.presheaf.is_locally_surjective_iff {C : Type u} {X : Top} {ℱ 𝒢 : X} (T : ℱ 𝒢) :
(U : (t : (𝒢.obj (opposite.op U))) (x : X), x U ( (V : (ι : V U), ( (s : (ℱ.obj (opposite.op V))), (T.app (opposite.op V)) s = x V)
theorem Top.presheaf.locally_surjective_iff_surjective_on_stalks {C : Type u} {X : Top} {ℱ 𝒢 : X} (T : ℱ 𝒢) :
(x : X),

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