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Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat

The homotopy category of a simplicial set #

The homotopy category of a simplicial set is defined as a quotient of the free category on its underlying reflexive quiver (equivalently its one truncation). The quotient imposes an additional hom relation on this free category, asserting that f ≫ g = h whenever f, g, and h are respectively the 2nd, 0th, and 1st faces of a 2-simplex.

In fact, the associated functor

SSet.hoFunctor : SSet.{u} ⥤ Cat.{u, u} := SSet.truncation 2 ⋙ SSet.hoFunctor₂

is defined by first restricting from simplicial sets to 2-truncated simplicial sets (throwing away the data that is not used for the construction of the homotopy category) and then composing with an analogously defined SSet.hoFunctor₂ : SSet.Truncated.{u} 2 ⥤ Cat.{u,u} implemented relative to the syntax of the 2-truncated simplex category.

In the file Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.lean we show the functor SSet.hoFunctor to be left adjoint to the nerve by providing an analogous decomposition of the nerve functor, made by possible by the fact that nerves of categories are 2-coskeletal, and then composing a pair of adjunctions, which factor through the category of 2-truncated simplicial sets.

A 2-truncated simplicial set S has an underlying refl quiver with S _⦋0⦌₂ as its underlying type.

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    structure SSet.OneTruncation₂.Hom {S : Truncated 2} (X Y : OneTruncation₂ S) :
    Type u_1

    The hom-types of the refl quiver underlying a simplicial set S are types of edges in S _⦋1⦌₂ together with source and target equalities.

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      theorem SSet.OneTruncation₂.Hom.ext_iff {S : Truncated 2} {X Y : OneTruncation₂ S} {x y : X.Hom Y} :
      x = y x.edge = y.edge
      theorem SSet.OneTruncation₂.Hom.ext {S : Truncated 2} {X Y : OneTruncation₂ S} {x y : X.Hom Y} (edge : x.edge = y.edge) :
      x = y

      A 2-truncated simplicial set S has an underlying refl quiver SSet.OneTruncation₂ S.

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      The functor that carries a 2-truncated simplicial set to its underlying refl quiver.

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        @[simp]
        theorem SSet.oneTruncation₂_map_map_edge {S T : Truncated 2} (F : S T) {X✝ Y✝ : (CategoryTheory.ReflQuiv.of (OneTruncation₂ S))} (f : X✝ Y✝) :
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        theorem SSet.oneTruncation₂_map_obj {S T : Truncated 2} (F : S T) (a✝ : S.obj (Opposite.op { obj := SimplexCategory.mk 0, property := OneTruncation₂._proof_1 })) :
        theorem SSet.OneTruncation₂.hom_ext {S : Truncated 2} {x y : OneTruncation₂ S} {f g : x y} :
        f.edge = g.edgef = g
        theorem SSet.OneTruncation₂.hom_ext_iff {S : Truncated 2} {x y : OneTruncation₂ S} {f g : x y} :
        f = g f.edge = g.edge
        @[simp]
        theorem SSet.OneTruncation₂.homOfEq_edge {X : Truncated 2} {x₁ y₁ x₂ y₂ : OneTruncation₂ X} (f : x₁ y₁) (hx : x₁ = x₂) (hy : y₁ = y₂) :

        An equivalence between the type of objects underlying a category and the type of 0-simplices in the 2-truncated nerve.

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          A hom equivalence over the function OneTruncation₂.nerveEquiv.

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            def SSet.Truncated.ι0₂ :
            { obj := SimplexCategory.mk 0, property := } { obj := SimplexCategory.mk 2, property := }

            The map that picks up the initial vertex of a 2-simplex, as a morphism in the 2-truncated simplex category.

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              def SSet.Truncated.ι1₂ :
              { obj := SimplexCategory.mk 0, property := } { obj := SimplexCategory.mk 2, property := }

              The map that picks up the middle vertex of a 2-simplex, as a morphism in the 2-truncated simplex category.

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                def SSet.Truncated.ι2₂ :
                { obj := SimplexCategory.mk 0, property := } { obj := SimplexCategory.mk 2, property := }

                The map that picks up the final vertex of a 2-simplex, as a morphism in the 2-truncated simplex category.

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                  def SSet.Truncated.ev0₂ {V : Truncated 2} (φ : V.obj (Opposite.op { obj := SimplexCategory.mk 2, property := })) :

                  The initial vertex of a 2-simplex in a 2-truncated simplicial set.

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                    def SSet.Truncated.ev1₂ {V : Truncated 2} (φ : V.obj (Opposite.op { obj := SimplexCategory.mk 2, property := })) :

                    The middle vertex of a 2-simplex in a 2-truncated simplicial set.

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                      def SSet.Truncated.ev2₂ {V : Truncated 2} (φ : V.obj (Opposite.op { obj := SimplexCategory.mk 2, property := })) :

                      The final vertex of a 2-simplex in a 2-truncated simplicial set.

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                        def SSet.Truncated.δ0₂ :
                        { obj := SimplexCategory.mk 1, property := } { obj := SimplexCategory.mk 2, property := }

                        The 0th face of a 2-simplex, as a morphism in the 2-truncated simplex category.

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                          def SSet.Truncated.δ1₂ :
                          { obj := SimplexCategory.mk 1, property := } { obj := SimplexCategory.mk 2, property := }

                          The 1st face of a 2-simplex, as a morphism in the 2-truncated simplex category.

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                            def SSet.Truncated.δ2₂ :
                            { obj := SimplexCategory.mk 1, property := } { obj := SimplexCategory.mk 2, property := }

                            The 2nd face of a 2-simplex, as a morphism in the 2-truncated simplex category.

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                              def SSet.Truncated.ev12₂ {V : Truncated 2} (φ : V.obj (Opposite.op { obj := SimplexCategory.mk 2, property := })) :

                              The arrow in the ReflQuiver OneTruncation₂ V of a 2-truncated simplicial set arising from the 0th face of a 2-simplex.

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                                def SSet.Truncated.ev02₂ {V : Truncated 2} (φ : V.obj (Opposite.op { obj := SimplexCategory.mk 2, property := })) :

                                The arrow in the ReflQuiver OneTruncation₂ V of a 2-truncated simplicial set arising from the 1st face of a 2-simplex.

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                                  def SSet.Truncated.ev01₂ {V : Truncated 2} (φ : V.obj (Opposite.op { obj := SimplexCategory.mk 2, property := })) :

                                  The arrow in the ReflQuiver OneTruncation₂ V of a 2-truncated simplicial set arising from the 2nd face of a 2-simplex.

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                                    The 2-simplices in a 2-truncated simplicial set V generate a hom relation on the free category on the underlying refl quiver of V.

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                                      theorem SSet.Truncated.HoRel₂.mk' {V : Truncated 2} (φ : V.obj (Opposite.op { obj := SimplexCategory.mk 2, property := })) {X₀ X₁ X₂ : OneTruncation₂ V} (f₀₁ : X₀ X₁) (f₁₂ : X₁ X₂) (f₀₂ : X₀ X₂) (h₀₁ : f₀₁.edge = V.map (SimplexCategory.Truncated.δ₂ 2 ).op φ) (h₁₂ : f₁₂.edge = V.map (SimplexCategory.Truncated.δ₂ 0 ).op φ) (h₀₂ : f₀₂.edge = V.map (SimplexCategory.Truncated.δ₂ 1 ).op φ) :

                                      A 2-simplex whose faces are identified with certain arrows in OneTruncation₂ V defines a term of type HoRel₂ between those arrows.

                                      The type underlying the homotopy category of a 2-truncated simplicial set V.

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                                        A canonical functor from the free category on the refl quiver underlying a 2-truncated simplicial set V to its homotopy category.

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                                          By Quotient.lift_unique' (not Quotient.lift) we have that quotientFunctor V is an epimorphism.

                                          A map of 2-truncated simplicial sets induces a functor between homotopy categories.

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                                            The functor that takes a 2-truncated simplicial set to its homotopy category.

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                                              The functor that takes a simplicial set to its homotopy category by passing through the 2-truncation.

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                                                Since ⦋0⦌ : SimplexCategory is terminal, Δ[0] has a unique point and thus OneTruncation₂ ((truncation 2).obj Δ[0]) has a unique inhabitant.

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                                                Since ⦋0⦌ : SimplexCategory is terminal, Δ[0] has a unique edge and thus the homs of OneTruncation₂ ((truncation 2).obj Δ[0]) have unique inhabitants.

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                                                The homotopy category functor preserves generic terminal objects.

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