Theorems · Definition · category theory

StalkSkyscraperPresheafAdjunctionAuxs.toSkyscraperPresheaf

{X : TopCat} →
  (p₀ : ↑X) →
    [inst : (U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] →
      {C : Type v} →
        [inst_1 : CategoryTheory.Category.{u, v} C] →
          [inst_2 : CategoryTheory.Limits.HasTerminal C] →
            [inst_3 : CategoryTheory.Limits.HasColimits C] →
              {𝓕 : TopCat.Presheaf C X} → {c : C} → (𝓕.stalk p₀ ⟶ c) → (𝓕 ⟶ skyscraperPresheaf p₀ c)

If f : 𝓕.stalk p₀ ⟶ c, then a natural transformation 𝓕 ⟶ skyscraperPresheaf p₀ c can be defined by: 𝓕.germ p₀ ≫ f : 𝓕(U) ⟶ c if p₀ ∈ U and the unique morphism to a terminal object if p₀ ∉ U.

Defined in

Mathlib.Topology.Sheaves.Skyscraper

Cited by

4 results in Mathlib

Foundations

Depth 41 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DecidableCategoryTheory.CategoryCategoryTheory.Limits.HasTerminalCategoryTheory.Limits.HasColimits