mathlib3 documentation

topology.sheaves.locally_surjective

Locally surjective maps of presheaves. #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

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.

source
def Top.presheaf.is_locally_surjective {C : Type u} [category_theory.category C] [category_theory.concrete_category C] {X : Top} {ℱ 𝒢 : Top.presheaf C 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
source
theorem Top.presheaf.is_locally_surjective_iff {C : Type u} [category_theory.category C] [category_theory.concrete_category C] {X : Top} {ℱ 𝒢 : Top.presheaf C X} (T : ℱ ⟶ 𝒢) :
Top.presheaf.is_locally_surjective T ↔ ∀ (U : topological_space.opens ↥X) (t : ↥(𝒢.obj (opposite.op U))) (x : ↥X), x ∈ U → (∃ (V : topological_space.opens ↥X) (ι : V ⟶ U), (∃ (s : ↥(ℱ.obj (opposite.op V))), ⇑(T.app (opposite.op V)) s = Top.presheaf.restrict t ι) ∧ x ∈ V)
source
theorem Top.presheaf.locally_surjective_iff_surjective_on_stalks {C : Type u} [category_theory.category C] [category_theory.concrete_category C] {X : Top} {ℱ 𝒢 : Top.presheaf C X} [category_theory.limits.has_colimits C] [category_theory.limits.preserves_filtered_colimits (category_theory.forget C)] (T : ℱ ⟶ 𝒢) :
Top.presheaf.is_locally_surjective T ↔ ∀ (x : ↥X), function.surjective ⇑((Top.presheaf.stalk_functor C x).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.