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Mathlib.Topology.Sheaves.PresheafOfFunctions

Presheaves of functions #

We construct some simple examples of presheaves of functions on a topological space.

def TopCat.presheafToTypes (X : TopCat) (T : XType v) :

The presheaf of dependently typed functions on X, with fibres given by a type family T. There is no requirement that the functions are continuous, here.

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    @[simp]
    theorem TopCat.presheafToTypes_obj (X : TopCat) {T : XType v} {U : (TopologicalSpace.Opens X)ᵒᵖ} :
    (TopCat.presheafToTypes X T).obj U = ((x : { x // x U.unop }) → T x)
    @[simp]
    theorem TopCat.presheafToTypes_map (X : TopCat) {T : XType v} {U : (TopologicalSpace.Opens X)ᵒᵖ} {V : (TopologicalSpace.Opens X)ᵒᵖ} {i : U V} {f : (TopCat.presheafToTypes X T).obj U} :
    (TopologicalSpace.Opens X)ᵒᵖ.map CategoryTheory.CategoryStruct.toQuiver (Type v) CategoryTheory.CategoryStruct.toQuiver (TopCat.presheafToTypes X T).toPrefunctor U V i f = fun x => f ((fun x => { val := x, property := (_ : x U.unop) }) x)

    The presheaf of functions on X with values in a type T. There is no requirement that the functions are continuous, here.

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      @[simp]
      theorem TopCat.presheafToType_obj (X : TopCat) {T : Type v} {U : (TopologicalSpace.Opens X)ᵒᵖ} :
      (TopCat.presheafToType X T).obj U = ({ x // x U.unop }T)
      @[simp]
      theorem TopCat.presheafToType_map (X : TopCat) {T : Type v} {U : (TopologicalSpace.Opens X)ᵒᵖ} {V : (TopologicalSpace.Opens X)ᵒᵖ} {i : U V} {f : (TopCat.presheafToType X T).obj U} :
      (TopologicalSpace.Opens X)ᵒᵖ.map CategoryTheory.CategoryStruct.toQuiver (Type v) CategoryTheory.CategoryStruct.toQuiver (TopCat.presheafToType X T).toPrefunctor U V i f = f fun x => { val := x, property := (_ : x U.unop) }

      The presheaf of continuous functions on X with values in fixed target topological space T.

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        The (bundled) commutative ring of continuous functions from a topological space to a topological commutative ring, with pointwise multiplication.

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          Pulling back functions into a topological ring along a continuous map is a ring homomorphism.

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            A homomorphism of topological rings can be postcomposed with functions from a source space X; this is a ring homomorphism (with respect to the pointwise ring operations on functions).

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              An upgraded version of the Yoneda embedding, observing that the continuous maps from X : TopCat to R : TopCommRingCat form a commutative ring, functorial in both X and R.

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                The presheaf (of commutative rings), consisting of functions on an open set U ⊆ X with values in some topological commutative ring T.

                For example, we could construct the presheaf of continuous complex valued functions of X as

                presheafToTopCommRing X (TopCommRing.of ℂ)
                

                (this requires import Topology.Instances.Complex).

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