2-commutative squares of functors

Similarly as CommSq.lean defines the notion of commutative squares, this file introduces the notion of 2-commutative squares of functors.

If T : C₁ ⥤ C₂, L : C₁ ⥤ C₃, R : C₂ ⥤ C₄, B : C₃ ⥤ C₄ are functors, then [CatCommSq T L R B] contains the datum of an isomorphism T ⋙ R ≅ L ⋙ B.

Future work: using this notion in the development of the localization of categories (e.g. localization of adjunctions).

class CategoryTheory.CatCommSq {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] (T : Functor C₁ C₂) (L : Functor C₁ C₃) (R : Functor C₂ C₄) (B : Functor C₃ C₄) : Type (max u_1 u_10)

CatCommSq T L R B expresses that there is a 2-commutative square of functors, where the functors T, L, R and B are respectively the left, top, right and bottom functors of the square.

• iso : T.comp R ≅ L.comp B

  Assuming [CatCommSq T L R B], iso T L R B is the isomorphism T ⋙ R ≅ L ⋙ B given by the 2-commutative square.

theorem CategoryTheory.CatCommSq.ext {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {inst✝ : Category.{u_7, u_1} C₁} {inst✝¹ : Category.{u_8, u_2} C₂} {inst✝² : Category.{u_9, u_3} C₃} {inst✝³ : Category.{u_10, u_4} C₄} {T : Functor C₁ C₂} {L : Functor C₁ C₃} {R : Functor C₂ C₄} {B : Functor C₃ C₄} {x y : CatCommSq T L R B} (iso : iso T L R B = iso T L R B) : x = y

theorem CategoryTheory.CatCommSq.ext_iff {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {inst✝ : Category.{u_7, u_1} C₁} {inst✝¹ : Category.{u_8, u_2} C₂} {inst✝² : Category.{u_9, u_3} C₃} {inst✝³ : Category.{u_10, u_4} C₄} {T : Functor C₁ C₂} {L : Functor C₁ C₃} {R : Functor C₂ C₄} {B : Functor C₃ C₄} {x y : CatCommSq T L R B} : x = y ↔ iso T L R B = iso T L R B

def CategoryTheory.CatCommSq.hComp {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₅ : Type u_5} {C₆ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₅] [Category.{u_12, u_6} C₆] (T₁ : Functor C₁ C₂) (T₂ : Functor C₂ C₃) (V₁ : Functor C₁ C₄) (V₂ : Functor C₂ C₅) (V₃ : Functor C₃ C₆) (B₁ : Functor C₄ C₅) (B₂ : Functor C₅ C₆) [CatCommSq T₁ V₁ V₂ B₁] [CatCommSq T₂ V₂ V₃ B₂] : CatCommSq (T₁.comp T₂) V₁ V₃ (B₁.comp B₂)

Horizontal composition of 2-commutative squares

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@[simp]

theorem CategoryTheory.CatCommSq.hComp_iso_inv_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₅ : Type u_5} {C₆ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₅] [Category.{u_12, u_6} C₆] (T₁ : Functor C₁ C₂) (T₂ : Functor C₂ C₃) (V₁ : Functor C₁ C₄) (V₂ : Functor C₂ C₅) (V₃ : Functor C₃ C₆) (B₁ : Functor C₄ C₅) (B₂ : Functor C₅ C₆) [CatCommSq T₁ V₁ V₂ B₁] [CatCommSq T₂ V₂ V₃ B₂] (X : C₁) : (iso (T₁.comp T₂) V₁ V₃ (B₁.comp B₂)).inv.app X = CategoryStruct.comp (B₂.map ((iso T₁ V₁ V₂ B₁).inv.app X)) ((iso T₂ V₂ V₃ B₂).inv.app (T₁.obj X))

@[simp]

theorem CategoryTheory.CatCommSq.hComp_iso_hom_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₅ : Type u_5} {C₆ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₅] [Category.{u_12, u_6} C₆] (T₁ : Functor C₁ C₂) (T₂ : Functor C₂ C₃) (V₁ : Functor C₁ C₄) (V₂ : Functor C₂ C₅) (V₃ : Functor C₃ C₆) (B₁ : Functor C₄ C₅) (B₂ : Functor C₅ C₆) [CatCommSq T₁ V₁ V₂ B₁] [CatCommSq T₂ V₂ V₃ B₂] (X : C₁) : (iso (T₁.comp T₂) V₁ V₃ (B₁.comp B₂)).hom.app X = CategoryStruct.comp ((iso T₂ V₂ V₃ B₂).hom.app (T₁.obj X)) (B₂.map ((iso T₁ V₁ V₂ B₁).hom.app X))

def CategoryTheory.CatCommSq.vComp {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₅ : Type u_5} {C₆ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₅] [Category.{u_12, u_6} C₆] (L₁ : Functor C₁ C₂) (L₂ : Functor C₂ C₃) (H₁ : Functor C₁ C₄) (H₂ : Functor C₂ C₅) (H₃ : Functor C₃ C₆) (R₁ : Functor C₄ C₅) (R₂ : Functor C₅ C₆) [CatCommSq H₁ L₁ R₁ H₂] [CatCommSq H₂ L₂ R₂ H₃] : CatCommSq H₁ (L₁.comp L₂) (R₁.comp R₂) H₃

Vertical composition of 2-commutative squares

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theorem CategoryTheory.CatCommSq.vComp_iso_hom_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₅ : Type u_5} {C₆ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₅] [Category.{u_12, u_6} C₆] (L₁ : Functor C₁ C₂) (L₂ : Functor C₂ C₃) (H₁ : Functor C₁ C₄) (H₂ : Functor C₂ C₅) (H₃ : Functor C₃ C₆) (R₁ : Functor C₄ C₅) (R₂ : Functor C₅ C₆) [CatCommSq H₁ L₁ R₁ H₂] [CatCommSq H₂ L₂ R₂ H₃] (X : C₁) : (iso H₁ (L₁.comp L₂) (R₁.comp R₂) H₃).hom.app X = CategoryStruct.comp (R₂.map ((iso H₁ L₁ R₁ H₂).hom.app X)) ((iso H₂ L₂ R₂ H₃).hom.app (L₁.obj X))

@[simp]

theorem CategoryTheory.CatCommSq.vComp_iso_inv_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₅ : Type u_5} {C₆ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₅] [Category.{u_12, u_6} C₆] (L₁ : Functor C₁ C₂) (L₂ : Functor C₂ C₃) (H₁ : Functor C₁ C₄) (H₂ : Functor C₂ C₅) (H₃ : Functor C₃ C₆) (R₁ : Functor C₄ C₅) (R₂ : Functor C₅ C₆) [CatCommSq H₁ L₁ R₁ H₂] [CatCommSq H₂ L₂ R₂ H₃] (X : C₁) : (iso H₁ (L₁.comp L₂) (R₁.comp R₂) H₃).inv.app X = CategoryStruct.comp ((iso H₂ L₂ R₂ H₃).inv.app (L₁.obj X)) (R₂.map ((iso H₁ L₁ R₁ H₂).inv.app X))

def CategoryTheory.CatCommSq.hInv {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] (T : C₁ ≌ C₂) (L : Functor C₁ C₃) (R : Functor C₂ C₄) (B : C₃ ≌ C₄) : CatCommSq T.functor L R B.functor → CatCommSq T.inverse R L B.inverse

Horizontal inverse of a 2-commutative square

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@[simp]

theorem CategoryTheory.CatCommSq.hInv_iso_hom_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] (T : C₁ ≌ C₂) (L : Functor C₁ C₃) (R : Functor C₂ C₄) (B : C₃ ≌ C₄) (x✝ : CatCommSq T.functor L R B.functor) (X : C₂) : (iso T.inverse R L B.inverse).hom.app X = CategoryStruct.comp (B.unitIso.hom.app (L.obj (T.inverse.obj X))) (CategoryStruct.comp (B.inverse.map ((iso T.functor L R B.functor).inv.app (T.inverse.obj X))) (B.inverse.map (R.map (T.counitIso.hom.app X))))

@[simp]

theorem CategoryTheory.CatCommSq.hInv_iso_inv_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] (T : C₁ ≌ C₂) (L : Functor C₁ C₃) (R : Functor C₂ C₄) (B : C₃ ≌ C₄) (x✝ : CatCommSq T.functor L R B.functor) (X : C₂) : (iso T.inverse R L B.inverse).inv.app X = CategoryStruct.comp (B.inverse.map (R.map (T.counitIso.inv.app X))) (CategoryStruct.comp (B.inverse.map ((iso T.functor L R B.functor).hom.app (T.inverse.obj X))) (B.unitIso.inv.app (L.obj (T.inverse.obj X))))

theorem CategoryTheory.CatCommSq.hInv_hInv {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} [Category.{u_8, u_1} C₁] [Category.{u_9, u_2} C₂] [Category.{u_10, u_3} C₃] [Category.{u_7, u_4} C₄] (T : C₁ ≌ C₂) (L : Functor C₁ C₃) (R : Functor C₂ C₄) (B : C₃ ≌ C₄) (h : CatCommSq T.functor L R B.functor) : hInv T.symm R L B.symm (hInv T L R B h) = h

def CategoryTheory.CatCommSq.hInvEquiv {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] (T : C₁ ≌ C₂) (L : Functor C₁ C₃) (R : Functor C₂ C₄) (B : C₃ ≌ C₄) : CatCommSq T.functor L R B.functor ≃ CatCommSq T.inverse R L B.inverse

In a square of categories, when the top and bottom functors are part of equivalence of categories, it is equivalent to show 2-commutativity for the functors of these equivalences or for their inverses.

Equations

• CategoryTheory.CatCommSq.hInvEquiv T L R B = { toFun := CategoryTheory.CatCommSq.hInv T L R B, invFun := CategoryTheory.CatCommSq.hInv T.symm R L B.symm, left_inv := ⋯, right_inv := ⋯ }

def CategoryTheory.CatCommSq.vInv {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] (T : Functor C₁ C₂) (L : C₁ ≌ C₃) (R : C₂ ≌ C₄) (B : Functor C₃ C₄) : CatCommSq T L.functor R.functor B → CatCommSq B L.inverse R.inverse T

Vertical inverse of a 2-commutative square

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• One or more equations did not get rendered due to their size.

@[simp]

theorem CategoryTheory.CatCommSq.vInv_iso_hom_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] (T : Functor C₁ C₂) (L : C₁ ≌ C₃) (R : C₂ ≌ C₄) (B : Functor C₃ C₄) (x✝ : CatCommSq T L.functor R.functor B) (X : C₃) : (iso B L.inverse R.inverse T).hom.app X = CategoryStruct.comp (R.inverse.map (B.map (L.counitIso.inv.app X))) (CategoryStruct.comp (R.inverse.map ((iso T L.functor R.functor B).inv.app (L.inverse.obj X))) (R.unitIso.inv.app (T.obj (L.inverse.obj X))))

@[simp]

theorem CategoryTheory.CatCommSq.vInv_iso_inv_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] (T : Functor C₁ C₂) (L : C₁ ≌ C₃) (R : C₂ ≌ C₄) (B : Functor C₃ C₄) (x✝ : CatCommSq T L.functor R.functor B) (X : C₃) : (iso B L.inverse R.inverse T).inv.app X = CategoryStruct.comp (R.unitIso.hom.app (T.obj (L.inverse.obj X))) (CategoryStruct.comp (R.inverse.map ((iso T L.functor R.functor B).hom.app (L.inverse.obj X))) (R.inverse.map (B.map (L.counitIso.hom.app X))))

theorem CategoryTheory.CatCommSq.vInv_vInv {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} [Category.{u_8, u_1} C₁] [Category.{u_9, u_2} C₂] [Category.{u_10, u_3} C₃] [Category.{u_7, u_4} C₄] (T : Functor C₁ C₂) (L : C₁ ≌ C₃) (R : C₂ ≌ C₄) (B : Functor C₃ C₄) (h : CatCommSq T L.functor R.functor B) : vInv B L.symm R.symm T (vInv T L R B h) = h

def CategoryTheory.CatCommSq.vInvEquiv {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] (T : Functor C₁ C₂) (L : C₁ ≌ C₃) (R : C₂ ≌ C₄) (B : Functor C₃ C₄) : CatCommSq T L.functor R.functor B ≃ CatCommSq B L.inverse R.inverse T

In a square of categories, when the left and right functors are part of equivalence of categories, it is equivalent to show 2-commutativity for the functors of these equivalences or for their inverses.

Equations

• CategoryTheory.CatCommSq.vInvEquiv T L R B = { toFun := CategoryTheory.CatCommSq.vInv T L R B, invFun := CategoryTheory.CatCommSq.vInv B L.symm R.symm T, left_inv := ⋯, right_inv := ⋯ }