Documentation

Mathlib.AlgebraicTopology.SimplicialSet.Monoidal

The monoidal category structure on simplicial sets #

This file defines an instance of chosen finite products for the category SSet. It follows from the fact the SSet if a category of functors to the category of types and that the category of types have chosen finite products. As a result, we obtain a monoidal category structure on SSet.

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@[simp]
theorem SSet.leftUnitor_hom_app_apply (K : SSet) {Δ : SimplexCategoryᵒᵖ} (x : (CategoryTheory.MonoidalCategoryStruct.tensorObj (CategoryTheory.MonoidalCategoryStruct.tensorUnit SSet) K).obj Δ) :
(CategoryTheory.MonoidalCategoryStruct.leftUnitor K).hom.app Δ x = x.2
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@[simp]
theorem SSet.leftUnitor_inv_app_apply (K : SSet) {Δ : SimplexCategoryᵒᵖ} (x : K.obj Δ) :
(CategoryTheory.MonoidalCategoryStruct.leftUnitor K).inv.app Δ x = (PUnit.unit, x)
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@[simp]
theorem SSet.rightUnitor_hom_app_apply (K : SSet) {Δ : SimplexCategoryᵒᵖ} (x : (CategoryTheory.MonoidalCategoryStruct.tensorObj K (CategoryTheory.MonoidalCategoryStruct.tensorUnit SSet)).obj Δ) :
(CategoryTheory.MonoidalCategoryStruct.rightUnitor K).hom.app Δ x = x.1
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@[simp]
theorem SSet.rightUnitor_inv_app_apply (K : SSet) {Δ : SimplexCategoryᵒᵖ} (x : K.obj Δ) :
(CategoryTheory.MonoidalCategoryStruct.rightUnitor K).inv.app Δ x = (x, PUnit.unit)
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@[simp]
theorem SSet.tensorHom_app_apply {K K' L L' : SSet} (f : K ⟶ K') (g : L ⟶ L') {Δ : SimplexCategoryᵒᵖ} (x : (CategoryTheory.MonoidalCategoryStruct.tensorObj K L).obj Δ) :
(CategoryTheory.MonoidalCategoryStruct.tensorHom f g).app Δ x = (f.app Δ x.1, g.app Δ x.2)
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@[simp]
theorem SSet.whiskerLeft_app_apply (K : SSet) {L L' : SSet} (g : L ⟶ L') {Δ : SimplexCategoryᵒᵖ} (x : (CategoryTheory.MonoidalCategoryStruct.tensorObj K L).obj Δ) :
(CategoryTheory.MonoidalCategoryStruct.whiskerLeft K g).app Δ x = (x.1, g.app Δ x.2)
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@[simp]
theorem SSet.whiskerRight_app_apply {K K' : SSet} (f : K ⟶ K') (L : SSet) {Δ : SimplexCategoryᵒᵖ} (x : (CategoryTheory.MonoidalCategoryStruct.tensorObj K L).obj Δ) :
(CategoryTheory.MonoidalCategoryStruct.whiskerRight f L).app Δ x = (f.app Δ x.1, x.2)
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@[simp]
theorem SSet.associator_hom_app_apply (K L M : SSet) {Δ : SimplexCategoryᵒᵖ} (x : (CategoryTheory.MonoidalCategoryStruct.tensorObj (CategoryTheory.MonoidalCategoryStruct.tensorObj K L) M).obj Δ) :
(CategoryTheory.MonoidalCategoryStruct.associator K L M).hom.app Δ x = (x.1.1, x.1.2, x.2)
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@[simp]
theorem SSet.associator_inv_app_apply (K L M : SSet) {Δ : SimplexCategoryᵒᵖ} (x : (CategoryTheory.MonoidalCategoryStruct.tensorObj K (CategoryTheory.MonoidalCategoryStruct.tensorObj L M)).obj Δ) :
(CategoryTheory.MonoidalCategoryStruct.associator K L M).inv.app Δ x = ((x.1, x.2.1), x.2.2)
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def SSet.unitHomEquiv (K : SSet) :
(CategoryTheory.MonoidalCategoryStruct.tensorUnit SSet ⟶ K) ≃ K.obj (Opposite.op (SimplexCategory.mk 0))

The bijection (𝟙_ SSet ⟶ K) ≃ K _⦋0⦌.

Instances For
    source
    def SSet.stdSimplex.isTerminalObj₀ :
    CategoryTheory.Limits.IsTerminal (stdSimplex.obj (SimplexCategory.mk 0))

    The object Δ[0] is terminal in SSet.

    Instances For
      source
      theorem SSet.stdSimplex.ext₀ {X : SSet} {f g : X ⟶ stdSimplex.obj (SimplexCategory.mk 0)} :
      f = g
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      theorem SSet.stdSimplex.ext₀_iff {X : SSet} {f g : X ⟶ stdSimplex.obj (SimplexCategory.mk 0)} :
      f = g ↔ True
      source
      instance SSet.instFiniteObjOppositeSimplexCategoryTensorObj (X Y : SSet) (n : SimplexCategoryᵒᵖ) [Finite (X.obj n)] [Finite (Y.obj n)] :
      Finite ((CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).obj n)
      source
      instance SSet.instFiniteTensorUnit :
      (CategoryTheory.MonoidalCategoryStruct.tensorUnit SSet).Finite
      source
      instance SSet.instHasDimensionLETensorUnitOfNatNat :
      (CategoryTheory.MonoidalCategoryStruct.tensorUnit SSet).HasDimensionLE 0
      source
      def SSet.Subcomplex.prod {X Y : SSet} (A : X.Subcomplex) (B : Y.Subcomplex) :
      (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).Subcomplex

      The external product of subcomplexes of simplicial sets.

      Instances For
        source
        @[simp]
        theorem SSet.Subcomplex.prod_obj {X Y : SSet} (A : X.Subcomplex) (B : Y.Subcomplex) (Δ : SimplexCategoryᵒᵖ) :
        (A.prod B).obj Δ = (A.obj Δ).prod (B.obj Δ)
        source
        theorem SSet.Subcomplex.prod_monotone {X Y : SSet} {A₁ A₂ : X.Subcomplex} (hX : A₁ ≤ A₂) {B₁ B₂ : Y.Subcomplex} (hY : B₁ ≤ B₂) :
        A₁.prod B₁ ≤ A₂.prod B₂
        source
        theorem SSet.Subcomplex.prod_le_top_prod {X Y : SSet} (A : X.Subcomplex) (B : Y.Subcomplex) :
        A.prod B ≤ ⊤.prod B
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        theorem SSet.Subcomplex.prod_le_prod_top {X Y : SSet} (A : X.Subcomplex) (B : Y.Subcomplex) :
        A.prod B ≤ A.prod ⊤
        source
        theorem SSet.Subcomplex.range_tensorHom {X₁ X₂ Y₁ Y₂ : SSet} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) :
        range (CategoryTheory.MonoidalCategoryStruct.tensorHom f₁ f₂) = (range f₁).prod (range f₂)
        source
        def SSet.Subcomplex.prodIso {X Y : SSet} (A : X.Subcomplex) (B : Y.Subcomplex) :
        (A.prod B).toSSet ≅ CategoryTheory.MonoidalCategoryStruct.tensorObj A.toSSet B.toSSet

        The isomorphism (A.prod B).toSSet ≅ A.toSSet ⊗ B.toSSet.

        Instances For
          source
          @[simp]
          theorem SSet.Subcomplex.prodIso_inv {X Y : SSet} (A : X.Subcomplex) (B : Y.Subcomplex) :
          (A.prodIso B).inv = lift (CategoryTheory.MonoidalCategoryStruct.tensorHom A.ι B.ι) ⋯
          source
          @[simp]
          theorem SSet.Subcomplex.prodIso_hom {X Y : SSet} (A : X.Subcomplex) (B : Y.Subcomplex) :
          (A.prodIso B).hom = CategoryTheory.CartesianMonoidalCategory.lift (lift (CategoryTheory.CategoryStruct.comp (A.prod B).ι (CategoryTheory.SemiCartesianMonoidalCategory.fst X Y)) ⋯) (lift (CategoryTheory.CategoryStruct.comp (A.prod B).ι (CategoryTheory.SemiCartesianMonoidalCategory.snd X Y)) ⋯)
          source
          noncomputable def SSet.ι₀ {X : SSet} :
          X ⟶ CategoryTheory.MonoidalCategoryStruct.tensorObj X (stdSimplex.obj (SimplexCategory.mk 1))

          The inclusion X ⟶ X ⊗ Δ[1] which is 0 on the second factor.

          Instances For
            source
            @[simp]
            theorem SSet.ι₀_comp {X Y : SSet} (f : X ⟶ Y) :
            CategoryTheory.CategoryStruct.comp ι₀ (CategoryTheory.MonoidalCategoryStruct.whiskerRight f (stdSimplex.obj (SimplexCategory.mk 1))) = CategoryTheory.CategoryStruct.comp f ι₀
            source
            @[simp]
            theorem SSet.ι₀_comp_assoc {X Y : SSet} (f : X ⟶ Y) {Z : SSet} (h : CategoryTheory.MonoidalCategoryStruct.tensorObj Y (stdSimplex.obj (SimplexCategory.mk 1)) ⟶ Z) :
            CategoryTheory.CategoryStruct.comp ι₀ (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight f (stdSimplex.obj (SimplexCategory.mk 1))) h) = CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp ι₀ h)
            source
            @[simp]
            theorem SSet.ι₀_fst (X : SSet) :
            CategoryTheory.CategoryStruct.comp ι₀ (CategoryTheory.SemiCartesianMonoidalCategory.fst X (stdSimplex.obj (SimplexCategory.mk 1))) = CategoryTheory.CategoryStruct.id X
            source
            @[simp]
            theorem SSet.ι₀_fst_assoc (X : SSet) {Z : SSet} (h : X ⟶ Z) :
            CategoryTheory.CategoryStruct.comp ι₀ (CategoryTheory.CategoryStruct.comp (CategoryTheory.SemiCartesianMonoidalCategory.fst X (stdSimplex.obj (SimplexCategory.mk 1))) h) = h
            source
            @[simp]
            theorem SSet.ι₀_snd (X : SSet) :
            CategoryTheory.CategoryStruct.comp ι₀ (CategoryTheory.SemiCartesianMonoidalCategory.snd X (stdSimplex.obj (SimplexCategory.mk 1))) = const (stdSimplex.obj₀Equiv.symm 0)
            source
            @[simp]
            theorem SSet.ι₀_snd_assoc (X : SSet) {Z : SSet} (h : stdSimplex.obj (SimplexCategory.mk 1) ⟶ Z) :
            CategoryTheory.CategoryStruct.comp ι₀ (CategoryTheory.CategoryStruct.comp (CategoryTheory.SemiCartesianMonoidalCategory.snd X (stdSimplex.obj (SimplexCategory.mk 1))) h) = CategoryTheory.CategoryStruct.comp (const (stdSimplex.obj₀Equiv.symm 0)) h
            source
            @[simp]
            theorem SSet.ι₀_app_fst {X : SSet} {m : SimplexCategoryᵒᵖ} (x : X.obj m) :
            (ι₀.app m x).1 = x
            source
            @[simp]
            theorem SSet.ι₀_app_snd_apply {X : SSet} {m : ℕ} (x : X.obj (Opposite.op (SimplexCategory.mk m))) (k : Fin (m + 1)) :
            (ι₀.app (Opposite.op (SimplexCategory.mk m)) x).2 k = 0
            source
            noncomputable def SSet.ι₁ {X : SSet} :
            X ⟶ CategoryTheory.MonoidalCategoryStruct.tensorObj X (stdSimplex.obj (SimplexCategory.mk 1))

            The inclusion X ⟶ X ⊗ Δ[1] which is 1 on the second factor.

            Instances For
              source
              @[simp]
              theorem SSet.ι₁_fst (X : SSet) :
              CategoryTheory.CategoryStruct.comp ι₁ (CategoryTheory.SemiCartesianMonoidalCategory.fst X (stdSimplex.obj (SimplexCategory.mk 1))) = CategoryTheory.CategoryStruct.id X
              source
              @[simp]
              theorem SSet.ι₁_fst_assoc (X : SSet) {Z : SSet} (h : X ⟶ Z) :
              CategoryTheory.CategoryStruct.comp ι₁ (CategoryTheory.CategoryStruct.comp (CategoryTheory.SemiCartesianMonoidalCategory.fst X (stdSimplex.obj (SimplexCategory.mk 1))) h) = h
              source
              @[simp]
              theorem SSet.ι₁_snd (X : SSet) :
              CategoryTheory.CategoryStruct.comp ι₁ (CategoryTheory.SemiCartesianMonoidalCategory.snd X (stdSimplex.obj (SimplexCategory.mk 1))) = const (stdSimplex.obj₀Equiv.symm 1)
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              @[simp]
              theorem SSet.ι₁_snd_assoc (X : SSet) {Z : SSet} (h : stdSimplex.obj (SimplexCategory.mk 1) ⟶ Z) :
              CategoryTheory.CategoryStruct.comp ι₁ (CategoryTheory.CategoryStruct.comp (CategoryTheory.SemiCartesianMonoidalCategory.snd X (stdSimplex.obj (SimplexCategory.mk 1))) h) = CategoryTheory.CategoryStruct.comp (const (stdSimplex.obj₀Equiv.symm 1)) h
              source
              @[simp]
              theorem SSet.ι₁_comp {X Y : SSet} (f : X ⟶ Y) :
              CategoryTheory.CategoryStruct.comp ι₁ (CategoryTheory.MonoidalCategoryStruct.whiskerRight f (stdSimplex.obj (SimplexCategory.mk 1))) = CategoryTheory.CategoryStruct.comp f ι₁
              source
              @[simp]
              theorem SSet.ι₁_comp_assoc {X Y : SSet} (f : X ⟶ Y) {Z : SSet} (h : CategoryTheory.MonoidalCategoryStruct.tensorObj Y (stdSimplex.obj (SimplexCategory.mk 1)) ⟶ Z) :
              CategoryTheory.CategoryStruct.comp ι₁ (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight f (stdSimplex.obj (SimplexCategory.mk 1))) h) = CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp ι₁ h)
              source
              @[simp]
              theorem SSet.ι₁_app_fst {X : SSet} {m : SimplexCategoryᵒᵖ} (x : X.obj m) :
              (ι₁.app m x).1 = x
              source
              @[simp]
              theorem SSet.ι₁_app_snd_apply {X : SSet} {m : ℕ} (x : X.obj (Opposite.op (SimplexCategory.mk m))) (k : Fin (m + 1)) :
              (ι₁.app (Opposite.op (SimplexCategory.mk m)) x).2 k = 1
              source
              @[simp]
              theorem SSet.prod_map_fst (X Y : SSet) {m n : SimplexCategoryᵒᵖ} (f : m ⟶ n) (z : (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).obj m) :
              ((CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).map f z).1 = X.map f z.1
              source
              @[simp]
              theorem SSet.prod_map_snd (X Y : SSet) {m n : SimplexCategoryᵒᵖ} (f : m ⟶ n) (z : (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).obj m) :
              ((CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).map f z).2 = Y.map f z.2
              source
              @[simp]
              theorem SSet.prod_δ_fst (X Y : SSet) {n : ℕ} (i : Fin (n + 2)) (z : (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).obj (Opposite.op (SimplexCategory.mk (n + 1)))) :
              (CategoryTheory.SimplicialObject.δ (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y) i z).1 = CategoryTheory.SimplicialObject.δ X i z.1
              source
              @[simp]
              theorem SSet.prod_δ_snd (X Y : SSet) {n : ℕ} (i : Fin (n + 2)) (z : (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).obj (Opposite.op (SimplexCategory.mk (n + 1)))) :
              (CategoryTheory.SimplicialObject.δ (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y) i z).2 = CategoryTheory.SimplicialObject.δ Y i z.2
              source
              @[simp]
              theorem SSet.prod_σ_fst (X Y : SSet) {n : ℕ} (i : Fin (n + 1)) (z : (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).obj (Opposite.op (SimplexCategory.mk n))) :
              (CategoryTheory.SimplicialObject.σ (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y) i z).1 = CategoryTheory.SimplicialObject.σ X i z.1
              source
              @[simp]
              theorem SSet.prod_σ_snd (X Y : SSet) {n : ℕ} (i : Fin (n + 1)) (z : (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).obj (Opposite.op (SimplexCategory.mk n))) :
              (CategoryTheory.SimplicialObject.σ (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y) i z).2 = CategoryTheory.SimplicialObject.σ Y i z.2
              source
              def SSet.Subcomplex.unionProd {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
              (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).Subcomplex

              Given S ≤ X and T ≤ Y, this is the subcomplex of X ⊗ Y given by (X ⊗ T) ⊔ (S ⊗ Y).

              Instances For
                source
                theorem SSet.Subcomplex.mem_unionProd_iff {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) {n : SimplexCategoryᵒᵖ} (x : (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).obj n) :
                x ∈ (S.unionProd T).obj n ↔ x.2 ∈ T.obj n ∨ x.1 ∈ S.obj n
                source
                theorem SSet.Subcomplex.top_prod_le_unionProd {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                ⊤.prod T ≤ S.unionProd T
                source
                theorem SSet.Subcomplex.prod_top_le_unionProd {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                S.prod ⊤ ≤ S.unionProd T
                source
                theorem SSet.Subcomplex.prod_le_unionProd {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                S.prod T ≤ S.unionProd T
                source
                noncomputable def SSet.Subcomplex.unionProd.ι₁ {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                CategoryTheory.MonoidalCategoryStruct.tensorObj X T.toSSet ⟶ (S.unionProd T).toSSet

                The inclusion X ⊗ T ⟶ S.unionProd T as simplicial sets.

                Instances For
                  source
                  noncomputable def SSet.Subcomplex.unionProd.ι₂ {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                  CategoryTheory.MonoidalCategoryStruct.tensorObj S.toSSet Y ⟶ (S.unionProd T).toSSet

                  The inclusion S ⊗ Y ⟶ S.unionProd T as simplicial sets

                  Instances For
                    source
                    @[simp]
                    theorem SSet.Subcomplex.unionProd.ι₁_ι {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                    CategoryTheory.CategoryStruct.comp (ι₁ S T) (S.unionProd T).ι = CategoryTheory.MonoidalCategoryStruct.whiskerLeft X T.ι
                    source
                    @[simp]
                    theorem SSet.Subcomplex.unionProd.ι₁_ι_assoc {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) {Z : SSet} (h : CategoryTheory.MonoidalCategoryStruct.tensorObj X Y ⟶ Z) :
                    CategoryTheory.CategoryStruct.comp (ι₁ S T) (CategoryTheory.CategoryStruct.comp (S.unionProd T).ι h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X T.ι) h
                    source
                    @[simp]
                    theorem SSet.Subcomplex.unionProd.ι₂_ι {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                    CategoryTheory.CategoryStruct.comp (ι₂ S T) (S.unionProd T).ι = CategoryTheory.MonoidalCategoryStruct.whiskerRight S.ι Y
                    source
                    @[simp]
                    theorem SSet.Subcomplex.unionProd.ι₂_ι_assoc {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) {Z : SSet} (h : CategoryTheory.MonoidalCategoryStruct.tensorObj X Y ⟶ Z) :
                    CategoryTheory.CategoryStruct.comp (ι₂ S T) (CategoryTheory.CategoryStruct.comp (S.unionProd T).ι h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight S.ι Y) h
                    source
                    theorem SSet.Subcomplex.unionProd.bicartSq {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                    (S.prod T).BicartSq (⊤.prod T) (S.prod ⊤) (S.unionProd T)
                    source
                    theorem SSet.Subcomplex.unionProd.isPushout {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                    CategoryTheory.IsPushout (CategoryTheory.MonoidalCategoryStruct.whiskerRight S.ι T.toSSet) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft S.toSSet T.ι) (ι₁ S T) (ι₂ S T)
                    source
                    @[simp]
                    theorem SSet.Subcomplex.unionProd.preimage_β_hom {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                    (S.unionProd T).preimage (β_ Y X).hom = T.unionProd S
                    source
                    @[simp]
                    theorem SSet.Subcomplex.unionProd.preimage_β_inv {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                    (S.unionProd T).preimage (β_ X Y).inv = T.unionProd S
                    source
                    @[simp]
                    theorem SSet.Subcomplex.unionProd.image_β_hom {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                    (S.unionProd T).image (β_ X Y).hom = T.unionProd S
                    source
                    @[simp]
                    theorem SSet.Subcomplex.unionProd.image_β_inv {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                    (S.unionProd T).image (β_ Y X).inv = T.unionProd S
                    source
                    noncomputable def SSet.Subcomplex.unionProd.symmIso {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                    (S.unionProd T).toSSet ≅ (T.unionProd S).toSSet

                    The isomorphism unionProd S T ≅ unionProd T S as simplicial sets.

                    Instances For
                      source
                      @[simp]
                      theorem SSet.Subcomplex.unionProd.symmIso_hom {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                      (symmIso S T).hom = lift (CategoryTheory.CategoryStruct.comp (S.unionProd T).ι (β_ X Y).hom) ⋯
                      source
                      @[simp]
                      theorem SSet.Subcomplex.unionProd.symmIso_inv {X Y : SSet} (S : X.Subcomplex) (T : Y.Subcomplex) :
                      (symmIso S T).inv = lift (CategoryTheory.CategoryStruct.comp (T.unionProd S).ι (β_ Y X).hom) ⋯
                      source
                      @[implicit_reducible]
                      instance SSet.Truncated.instMonoidalTruncation (n : ℕ) :
                      (truncation n).Monoidal
                      source
                      @[simp]
                      theorem SSet.Truncated.tensor_map_apply_fst {n : ℕ} {X Y : Truncated n} {d e : (SimplexCategory.Truncated n)ᵒᵖ} (f : d ⟶ e) (x : (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).obj d) :
                      ((CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).map f x).1 = X.map f x.1
                      source
                      @[simp]
                      theorem SSet.Truncated.tensor_map_apply_snd {n : ℕ} {X Y : Truncated n} {d e : (SimplexCategory.Truncated n)ᵒᵖ} (f : d ⟶ e) (x : (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).obj d) :
                      ((CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).map f x).2 = Y.map f x.2