File: Mathlib/AlgebraicTopology/SimplicialCategory/Basic.lean

full_name: CategoryTheory.SimplicialCategory
code: /-- A simplicial category is a category `C` that is enriched over the
category of simplicial sets in such a way that morphisms in
`C` identify to the `0`-simplices of the enriched hom. -/
class SimplicialCategory extends EnrichedCategory SSet.{v} C where
  /-- morphisms identify to `0`-simplices of the enriched hom -/
  homEquiv (K L : C) : (K ⟶ L) ≃ (𝟙_ SSet.{v} ⟶ EnrichedCategory.Hom K L)
  homEquiv_id (K : C) : homEquiv K K (𝟙 K) = eId SSet K := by aesop_cat
  homEquiv_comp {K L M : C} (f : K ⟶ L) (g : L ⟶ M) :
    homEquiv K M (f ≫ g) = (λ_ _).inv ≫ (homEquiv K L f ⊗ homEquiv L M g) ≫
      eComp SSet K L M := by aesop_cat
goal: Type (max u (v + 1))

full_name: CategoryTheory.SimplicialCategory.homEquiv_id
code: /-- A simplicial category is a category `C` that is enriched over the
category of simplicial sets in such a way that morphisms in
`C` identify to the `0`-simplices of the enriched hom. -/
class SimplicialCategory extends EnrichedCategory SSet.{v} C where
  /-- morphisms identify to `0`-simplices of the enriched hom -/
  homEquiv (K L : C) : (K ⟶ L) ≃ (𝟙_ SSet.{v} ⟶ EnrichedCategory.Hom K L)
  homEquiv_id (K : C) : homEquiv K K (𝟙 K) = eId SSet K := by aesop_cat
  homEquiv_comp {K L M : C} (f : K ⟶ L) (g : L ⟶ M) :
    homEquiv K M (f ≫ g) = (λ_ _).inv ≫ (homEquiv K L f ⊗ homEquiv L M g) ≫
      eComp SSet K L M := by aesop_cat
goal: (CategoryTheory.SimplicialCategory.homEquiv K K) (𝟙 K) = CategoryTheory.eId SSet K

full_name: CategoryTheory.SimplicialCategory.homEquiv
code: /-- A simplicial category is a category `C` that is enriched over the
category of simplicial sets in such a way that morphisms in
`C` identify to the `0`-simplices of the enriched hom. -/
class SimplicialCategory extends EnrichedCategory SSet.{v} C where
  /-- morphisms identify to `0`-simplices of the enriched hom -/
  homEquiv (K L : C) : (K ⟶ L) ≃ (𝟙_ SSet.{v} ⟶ EnrichedCategory.Hom K L)
  homEquiv_id (K : C) : homEquiv K K (𝟙 K) = eId SSet K := by aesop_cat
  homEquiv_comp {K L M : C} (f : K ⟶ L) (g : L ⟶ M) :
    homEquiv K M (f ≫ g) = (λ_ _).inv ≫ (homEquiv K L f ⊗ homEquiv L M g) ≫
      eComp SSet K L M := by aesop_cat
goal: (K ⟶ L) ≃ (𝟙_ SSet ⟶ CategoryTheory.EnrichedCategory.Hom K L)

full_name: CategoryTheory.SimplicialCategory.homEquiv_comp
code: /-- A simplicial category is a category `C` that is enriched over the
category of simplicial sets in such a way that morphisms in
`C` identify to the `0`-simplices of the enriched hom. -/
class SimplicialCategory extends EnrichedCategory SSet.{v} C where
  /-- morphisms identify to `0`-simplices of the enriched hom -/
  homEquiv (K L : C) : (K ⟶ L) ≃ (𝟙_ SSet.{v} ⟶ EnrichedCategory.Hom K L)
  homEquiv_id (K : C) : homEquiv K K (𝟙 K) = eId SSet K := by aesop_cat
  homEquiv_comp {K L M : C} (f : K ⟶ L) (g : L ⟶ M) :
    homEquiv K M (f ≫ g) = (λ_ _).inv ≫ (homEquiv K L f ⊗ homEquiv L M g) ≫
      eComp SSet K L M := by aesop_cat
goal: (CategoryTheory.SimplicialCategory.homEquiv K M) (f ≫ g) =
  (λ_ (𝟙_ SSet)).inv ≫
    ((CategoryTheory.SimplicialCategory.homEquiv K L) f ⊗ (CategoryTheory.SimplicialCategory.homEquiv L M) g) ≫
      CategoryTheory.eComp SSet K L M

full_name: CategoryTheory.SimplicialCategory.sHom
code: /-- Abbreviation for the enriched hom of a simplicial category. -/
abbrev sHom (K L : C) : SSet.{v} 
goal: SSet

full_name: CategoryTheory.SimplicialCategory.sHomComp
code: /-- Abbreviation for the enriched composition in a simplicial category. -/
abbrev sHomComp (K L M : C) : sHom K L ⊗ sHom L M ⟶ sHom K M 
goal: CategoryTheory.SimplicialCategory.sHom K L ⊗ CategoryTheory.SimplicialCategory.sHom L M ⟶
  CategoryTheory.SimplicialCategory.sHom K M

full_name: CategoryTheory.SimplicialCategory.homEquiv'
code: /-- The bijection `(K ⟶ L) ≃ sHom K L _[0]` for all objects `K` and `L`
in a simplicial category. -/
def homEquiv' (K L : C) : (K ⟶ L) ≃ sHom K L _[0] 
goal: (K ⟶ L) ≃ CategoryTheory.SimplicialCategory.sHom K L _[0]

full_name: CategoryTheory.SimplicialCategory.sHomWhiskerRight
code: /-- The morphism `sHom K' L ⟶ sHom K L` induced by a morphism `K ⟶ K'`. -/
noncomputable def sHomWhiskerRight {K K' : C} (f : K ⟶ K') (L : C) :
    sHom K' L ⟶ sHom K L 
goal: CategoryTheory.SimplicialCategory.sHom K' L ⟶ CategoryTheory.SimplicialCategory.sHom K L

full_name: CategoryTheory.SimplicialCategory.sHomWhiskerRight_id
code: @[simp]
lemma sHomWhiskerRight_id (K L : C) : sHomWhiskerRight (𝟙 K) L = 𝟙 _ 
goal: CategoryTheory.SimplicialCategory.sHomWhiskerRight (𝟙 K) L = 𝟙 (CategoryTheory.SimplicialCategory.sHom K L)

full_name: CategoryTheory.SimplicialCategory.sHomWhiskerRight_comp
code: @[simp, reassoc]
lemma sHomWhiskerRight_comp {K K' K'' : C} (f : K ⟶ K') (f' : K' ⟶ K'') (L : C) :
    sHomWhiskerRight (f ≫ f') L = sHomWhiskerRight f' L ≫ sHomWhiskerRight f L 
goal: CategoryTheory.SimplicialCategory.sHomWhiskerRight (f ≫ f') L =
  CategoryTheory.SimplicialCategory.sHomWhiskerRight f' L ≫ CategoryTheory.SimplicialCategory.sHomWhiskerRight f L

full_name: CategoryTheory.SimplicialCategory.sHomWhiskerLeft
code: /-- The morphism `sHom K L ⟶ sHom K L'` induced by a morphism `L ⟶ L'`. -/
noncomputable def sHomWhiskerLeft (K : C) {L L' : C} (g : L ⟶ L') :
    sHom K L ⟶ sHom K L' 
goal: CategoryTheory.SimplicialCategory.sHom K L ⟶ CategoryTheory.SimplicialCategory.sHom K L'

full_name: CategoryTheory.SimplicialCategory.sHomWhiskerLeft_id
code: @[simp]
lemma sHomWhiskerLeft_id (K L : C) : sHomWhiskerLeft K (𝟙 L) = 𝟙 _ 
goal: CategoryTheory.SimplicialCategory.sHomWhiskerLeft K (𝟙 L) = 𝟙 (CategoryTheory.SimplicialCategory.sHom K L)

full_name: CategoryTheory.SimplicialCategory.sHomWhiskerLeft_comp
code: @[simp, reassoc]
lemma sHomWhiskerLeft_comp (K : C) {L L' L'' : C} (g : L ⟶ L') (g' : L' ⟶ L'') :
    sHomWhiskerLeft K (g ≫ g') = sHomWhiskerLeft K g ≫ sHomWhiskerLeft K g' 
goal: CategoryTheory.SimplicialCategory.sHomWhiskerLeft K (g ≫ g') =
  CategoryTheory.SimplicialCategory.sHomWhiskerLeft K g ≫ CategoryTheory.SimplicialCategory.sHomWhiskerLeft K g'

full_name: CategoryTheory.SimplicialCategory.sHom_whisker_exchange
code: @[reassoc]
lemma sHom_whisker_exchange {K K' L L' : C} (f : K ⟶ K') (g : L ⟶ L') :
    sHomWhiskerLeft K' g ≫ sHomWhiskerRight f L' =
      sHomWhiskerRight f L ≫ sHomWhiskerLeft K g 
goal: CategoryTheory.SimplicialCategory.sHomWhiskerLeft K' g ≫ CategoryTheory.SimplicialCategory.sHomWhiskerRight f L' =
  CategoryTheory.SimplicialCategory.sHomWhiskerRight f L ≫ CategoryTheory.SimplicialCategory.sHomWhiskerLeft K g

full_name: CategoryTheory.SimplicialCategory.sHomFunctor
code: /-- The bifunctor `Cᵒᵖ ⥤ C ⥤ SSet.{v}` which sends `K : Cᵒᵖ` and `L : C` to `sHom K.unop L`. -/
@[simps]
noncomputable def sHomFunctor : Cᵒᵖ ⥤ C ⥤ SSet.{v} 
goal: Cᵒᵖ ⥤ C ⥤ SSet

full_name: CategoryTheory.SimplicialCategory.sHomFunctor_map_app
code: /-- The bifunctor `Cᵒᵖ ⥤ C ⥤ SSet.{v}` which sends `K : Cᵒᵖ` and `L : C` to `sHom K.unop L`. -/
@[simps]
noncomputable def sHomFunctor : Cᵒᵖ ⥤ C ⥤ SSet.{v} 
goal: ∀ {X Y : Cᵒᵖ} (φ : X ⟶ Y) (L : C),
  ((CategoryTheory.SimplicialCategory.sHomFunctor C).map φ).app L =
    CategoryTheory.SimplicialCategory.sHomWhiskerRight φ.unop L

full_name: CategoryTheory.SimplicialCategory.sHomFunctor_obj_map
code: /-- The bifunctor `Cᵒᵖ ⥤ C ⥤ SSet.{v}` which sends `K : Cᵒᵖ` and `L : C` to `sHom K.unop L`. -/
@[simps]
noncomputable def sHomFunctor : Cᵒᵖ ⥤ C ⥤ SSet.{v} 
goal: ∀ {X Y : C} (φ : X ⟶ Y),
  ((CategoryTheory.SimplicialCategory.sHomFunctor C).obj K).map φ =
    CategoryTheory.SimplicialCategory.sHomWhiskerLeft (Opposite.unop K) φ

full_name: CategoryTheory.SimplicialCategory.sHomFunctor_obj_obj
code: /-- The bifunctor `Cᵒᵖ ⥤ C ⥤ SSet.{v}` which sends `K : Cᵒᵖ` and `L : C` to `sHom K.unop L`. -/
@[simps]
noncomputable def sHomFunctor : Cᵒᵖ ⥤ C ⥤ SSet.{v} 
goal: ((CategoryTheory.SimplicialCategory.sHomFunctor C).obj K).obj L =
  CategoryTheory.SimplicialCategory.sHom (Opposite.unop K) L