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Mathlib.AlgebraicTopology.TopologicalSimplex

Topological simplices #

We define the natural functor from SimplexCategory to TopCat sending [n] to the topological n-simplex. This is used to define TopCat.toSSet in AlgebraicTopology.SingularSet.

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instance SimplexCategory.instFintypeObjSimplexCategoryToQuiverToCategoryStructSmallCategoryTypeToQuiverToCategoryStructTypesToPrefunctorForget (x : SimplexCategory) :
Fintype (CategoryTheory.ConcreteCategory.forget.toPrefunctor.obj x)
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def SimplexCategory.toTopObj (x : SimplexCategory) :
Set ((CategoryTheory.forget SimplexCategory).toPrefunctor.obj x → NNReal)

The topological simplex associated to x : SimplexCategory. This is the object part of the functor SimplexCategory.toTop.

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    instance SimplexCategory.instCoeFunElemForAllObjSimplexCategoryToQuiverToCategoryStructSmallCategoryTypeToQuiverToCategoryStructTypesToPrefunctorForgetInstConcreteCategorySimplexCategorySmallCategoryNNRealToTopObj (x : SimplexCategory) :
    CoeFun ↑(SimplexCategory.toTopObj x) fun (x_1 : ↑(SimplexCategory.toTopObj x)) => (CategoryTheory.forget SimplexCategory).toPrefunctor.obj x → NNReal
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    theorem SimplexCategory.toTopObj.ext {x : SimplexCategory} (f : ↑(SimplexCategory.toTopObj x)) (g : ↑(SimplexCategory.toTopObj x)) :
    ↑f = ↑g → f = g
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    def SimplexCategory.toTopMap {x : SimplexCategory} {y : SimplexCategory} (f : x ⟶ y) (g : ↑(SimplexCategory.toTopObj x)) :
    ↑(SimplexCategory.toTopObj y)

    A morphism in SimplexCategory induces a map on the associated topological spaces.

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      @[simp]
      theorem SimplexCategory.coe_toTopMap {x : SimplexCategory} {y : SimplexCategory} (f : x ⟶ y) (g : ↑(SimplexCategory.toTopObj x)) (i : (CategoryTheory.forget SimplexCategory).toPrefunctor.obj y) :
      ↑(SimplexCategory.toTopMap f g) i = Finset.sum (Finset.filter (fun (x_1 : (CategoryTheory.forget SimplexCategory).toPrefunctor.obj x) => f x_1 = i) Finset.univ) fun (j : (CategoryTheory.forget SimplexCategory).toPrefunctor.obj x) => ↑g j
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      theorem SimplexCategory.continuous_toTopMap {x : SimplexCategory} {y : SimplexCategory} (f : x ⟶ y) :
      Continuous (SimplexCategory.toTopMap f)
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      @[simp]
      theorem SimplexCategory.toTop_obj (x : SimplexCategory) :
      SimplexCategory.toTop.toPrefunctor.obj x = TopCat.of ↑(SimplexCategory.toTopObj x)
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      @[simp]
      theorem SimplexCategory.toTop_map_apply :
      ∀ {X Y : SimplexCategory} (f : X ⟶ Y) (g : ↑(SimplexCategory.toTopObj X)), (SimplexCategory.toTop.toPrefunctor.map f) g = SimplexCategory.toTopMap f g
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      def SimplexCategory.toTop :
      CategoryTheory.Functor SimplexCategory TopCat

      The functor associating the topological n-simplex to [n] : SimplexCategory.

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