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Mathlib.Topology.Sheaves.SheafCondition.Sites

Coverings and sieves; from sheaves on sites and sheaves on spaces #

In this file, we connect coverings in a topological space to sieves in the associated Grothendieck topology, in preparation of connecting the sheaf condition on sites to the various sheaf conditions on spaces.

We also specialize results about sheaves on sites to sheaves on spaces; we show that the inclusion functor from a topological basis to TopologicalSpace.Opens is cover dense, that open maps induce cover preserving functors, and that open embeddings induce continuous functors.

Given a presieve R on U, we obtain a covering family of open sets in X, by taking as index type the type of dependent pairs (V, f), where f : V ⟶ U is in R.

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    If R is a presieve in the grothendieck topology on Opens X, the covering family associated to R really is covering, i.e. the union of all open sets equals U.

    Given a family of opens U : ι → Opens X and any open Y : Opens X, we obtain a presieve on Y by declaring that a morphism f : V ⟶ Y is a member of the presieve if and only if there exists an index i : ι such that V = U i.

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      Take Y to be iSup U and obtain a presieve over iSup U.

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        @[simp]

        Given a presieve R on Y, if we take its associated family of opens via coveringOfPresieve (which may not cover Y if R is not covering), and take the presieve on Y associated to the family of opens via presieveOfCoveringAux, then we get back the original presieve R.

        An index i : ι can be turned into a dependent pair (V, f), where V is an open set and f : V ⟶ iSup U is a member of presieveOfCovering U f.

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          By using the axiom of choice, a dependent pair (V, f) where f : V ⟶ iSup U is a member of presieveOfCovering U f can be turned into an index i : ι, such that V = U i.

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            theorem OpenEmbedding.functor_isContinuous {X : TopCat} {Y : TopCat} {f : X Y} (h : OpenEmbedding f) :
            .functor.IsContinuous (Opens.grothendieckTopology X) (Opens.grothendieckTopology Y)
            theorem TopCat.Presheaf.isSheaf_of_openEmbedding {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} {Y : TopCat} {f : X Y} {F : TopCat.Presheaf C Y} (h : OpenEmbedding f) (hF : F.IsSheaf) :
            TopCat.Presheaf.IsSheaf (.functor.op.comp F)

            The empty component of a sheaf is terminal.

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              A variant of isTerminalOfEmpty that is easier to apply.

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              • F.isTerminalOfEqEmpty h = .mpr F.isTerminalOfEmpty
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                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.

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                  @[simp]
                  theorem TopCat.Sheaf.extend_hom_app {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} {ι : Type u_1} {B : ιTopologicalSpace.Opens X} (F : TopCat.Presheaf C X) (F' : TopCat.Sheaf C X) (h : TopologicalSpace.Opens.IsBasis (Set.range B)) (α : (CategoryTheory.inducedFunctor B).op.comp F (CategoryTheory.inducedFunctor B).op.comp F'.val) (i : ι) :
                  ((TopCat.Sheaf.restrictHomEquivHom F F' h) α).app (Opposite.op (B i)) = α.app (Opposite.op i)
                  theorem TopCat.Sheaf.hom_ext {C : Type u} [CategoryTheory.Category.{v, u} C] {X : TopCat} {ι : Type u_1} {B : ιTopologicalSpace.Opens X} (F : TopCat.Presheaf C X) (F' : TopCat.Sheaf C X) (h : TopologicalSpace.Opens.IsBasis (Set.range B)) {α : F F'.val} {β : F F'.val} (he : ∀ (i : ι), α.app (Opposite.op (B i)) = β.app (Opposite.op (B i))) :
                  α = β