Theorems · Definition · category theory

TopCat.Sheaf.restrictHomEquivHom

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {X : TopCat} →
      {ι : Type u_1} →
        {B : ι → TopologicalSpace.Opens ↑X} →
          (F : TopCat.Presheaf C X) →
            (F' : TopCat.Sheaf C X) →
              TopologicalSpace.Opens.IsBasis (Set.range B) →
                ((CategoryTheory.inducedFunctor B).op.comp F ⟶ (CategoryTheory.inducedFunctor B).op.comp F'.obj) ≃
                  (F ⟶ F'.obj)

If a family B of open sets forms a basis of the topology on X, and if F' is a sheaf on X, then a homomorphism between a presheaf F on X and F' is equivalent to a homomorphism between their restrictions to the indexing type ι of B, with the induced category structure on ι.

Defined in

Mathlib.Topology.Sheaves.SheafCondition.Sites

Cited by

4 results in Mathlib

Foundations

Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.Category