state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (𝟙 (F.obj A.X) ⊗ F.ε ≫ F.map A.one) ≫ μ F A.X A.X ≫ F.map A.mul = (ρ_ (F.obj A.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | conv_lhs => rw [id_tensor_comp, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (𝟙 (F.obj A.X) ⊗ F.ε ≫ F.map A.one) ≫ μ F A.X A.X ≫ F.map A.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [id_tensor_comp, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (𝟙 (F.obj A.X) ⊗ F.ε ≫ F.map A.one) ≫ μ F A.X A.X ≫ F.map A.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [id_tensor_comp, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (𝟙 (F.obj A.X) ⊗ F.ε ≫ F.map A.one) ≫ μ F A.X A.X ≫ F.map A.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [id_tensor_comp, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ ((F.map (𝟙 A.X) ⊗ F.ε) ≫ (F.map (𝟙 A.X) ⊗ F.map A.one)) ≫ μ F A.X A.X ≫ F.map A.mul = (ρ_ (F.obj A.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 2 3 => rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (F.map (𝟙 A.X) ⊗ F.map A.one) ≫ μ F A.X A.X
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (F.map (𝟙 A.X) ⊗ F.map A.one) ≫ μ F A.X A.X
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (F.map (𝟙 A.X) ⊗ F.map A.one) ≫ μ F A.X A.X
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (F.map (𝟙 A.X) ⊗ F.ε) ≫ (μ F A.X (𝟙_ C) ≫ F.map (𝟙 A.X ⊗ A.one)) ≫ F.map A.mul = (ρ_ (F.obj A.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 3 4 => rw [← F.toFunctor.map_comp, A.mul_one] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| F.map (𝟙 A.X ⊗ A.one) ≫ F.map A.mul
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← F.toFunctor.map_comp, A.mul_one] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| F.map (𝟙 A.X ⊗ A.one) ≫ F.map A.mul
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← F.toFunctor.map_comp, A.mul_one] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| F.map (𝟙 A.X ⊗ A.one) ≫ F.map A.mul
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← F.toFunctor.map_comp, A.mul_one] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (F.map (𝟙 A.X) ⊗ F.ε) ≫ μ F A.X (𝟙_ C) ≫ F.map (ρ_ A.X).hom = (ρ_ (F.obj A.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (𝟙 (F.obj A.X) ⊗ F.ε) ≫ μ F A.X (𝟙_ C) ≫ F.map (ρ_ A.X).hom = (ρ_ (F.obj A.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.right_unitality] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (μ F A.X A.X ≫ F.map A.mul ⊗ 𝟙 (F.obj A.X)) ≫ μ F A.X A.X ≫ F.map A.mul =
(α_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom ≫
(𝟙 (F.obj ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | conv_lhs => rw [comp_tensor_id, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (μ F A.X A.X ≫ F.map A.mul ⊗ 𝟙 (F.obj A.X)) ≫ μ F A.X A.X ≫ F.map A.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [comp_tensor_id, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (μ F A.X A.X ≫ F.map A.mul ⊗ 𝟙 (F.obj A.X)) ≫ μ F A.X A.X ≫ F.map A.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [comp_tensor_id, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (μ F A.X A.X ≫ F.map A.mul ⊗ 𝟙 (F.obj A.X)) ≫ μ F A.X A.X ≫ F.map A.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [comp_tensor_id, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ ((μ F A.X A.X ⊗ F.map (𝟙 A.X)) ≫ (F.map A.mul ⊗ F.map (𝟙 A.X))) ≫ μ F A.X A.X ≫ F.map A.mul =
(α_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).ho... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 2 3 => rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (F.map A.mul ⊗ F.map (𝟙 A.X)) ≫ μ F A.X A.X
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (F.map A.mul ⊗ F.map (𝟙 A.X)) ≫ μ F A.X A.X
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (F.map A.mul ⊗ F.map (𝟙 A.X)) ≫ μ F A.X A.X
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (μ F A.X A.X ⊗ F.map (𝟙 A.X)) ≫ (μ F (A.X ⊗ A.X) A.X ≫ F.map (A.mul ⊗ 𝟙 A.X)) ≫ F.map A.mul =
(α_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).ho... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 3 4 => rw [← F.toFunctor.map_comp, A.mul_assoc] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| F.map (A.mul ⊗ 𝟙 A.X) ≫ F.map A.mul
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← F.toFunctor.map_comp, A.mul_assoc] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| F.map (A.mul ⊗ 𝟙 A.X) ≫ F.map A.mul
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← F.toFunctor.map_comp, A.mul_assoc] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| F.map (A.mul ⊗ 𝟙 A.X) ≫ F.map A.mul
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← F.toFunctor.map_comp, A.mul_assoc] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (μ F A.X A.X ⊗ F.map (𝟙 A.X)) ≫ μ F (A.X ⊗ A.X) A.X ≫ F.map ((α_ A.X A.X A.X).hom ≫ (𝟙 A.X ⊗ A.mul) ≫ A.mul) =
(α_ (F.obj A.X) (F.obj A.X... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | conv_lhs => rw [F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (μ F A.X A.X ⊗ F.map (𝟙 A.X)) ≫ μ F (A.X ⊗ A.X) A.X ≫ F.map ((α_ A.X A.X A.X).hom ≫ (𝟙 A.X ⊗ A.mul) ≫ A.mul) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (μ F A.X A.X ⊗ F.map (𝟙 A.X)) ≫ μ F (A.X ⊗ A.X) A.X ≫ F.map ((α_ A.X A.X A.X).hom ≫ (𝟙 A.X ⊗ A.mul) ≫ A.mul) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (μ F A.X A.X ⊗ F.map (𝟙 A.X)) ≫ μ F (A.X ⊗ A.X) A.X ≫ F.map ((α_ A.X A.X A.X).hom ≫ (𝟙 A.X ⊗ A.mul) ≫ A.mul) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (μ F A.X A.X ⊗ 𝟙 (F.obj A.X)) ≫ μ F (A.X ⊗ A.X) A.X ≫ F.map ((α_ A.X A.X A.X).hom ≫ (𝟙 A.X ⊗ A.mul) ≫ A.mul) =
(α_ (F.obj A.X) (F.obj A.X... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | conv_lhs => rw [F.toFunctor.map_comp, F.toFunctor.map_comp] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (μ F A.X A.X ⊗ 𝟙 (F.obj A.X)) ≫ μ F (A.X ⊗ A.X) A.X ≫ F.map ((α_ A.X A.X A.X).hom ≫ (𝟙 A.X ⊗ A.mul) ≫ A.mul) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.toFunctor.map_comp, F.toFunctor.map_comp] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (μ F A.X A.X ⊗ 𝟙 (F.obj A.X)) ≫ μ F (A.X ⊗ A.X) A.X ≫ F.map ((α_ A.X A.X A.X).hom ≫ (𝟙 A.X ⊗ A.mul) ≫ A.mul) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.toFunctor.map_comp, F.toFunctor.map_comp] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (μ F A.X A.X ⊗ 𝟙 (F.obj A.X)) ≫ μ F (A.X ⊗ A.X) A.X ≫ F.map ((α_ A.X A.X A.X).hom ≫ (𝟙 A.X ⊗ A.mul) ≫ A.mul) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.toFunctor.map_comp, F.toFunctor.map_comp] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (μ F A.X A.X ⊗ 𝟙 (F.obj A.X)) ≫
μ F (A.X ⊗ A.X) A.X ≫ F.map (α_ A.X A.X A.X).hom ≫ F.map (𝟙 A.X ⊗ A.mul) ≫ F.map A.mul =
(α_ (F.obj... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | conv_rhs => rw [id_tensor_comp, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (α_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom ≫ (𝟙 (F.obj A.X) ⊗ μ F A.X A.X ≫ F.map A.mul) ≫ μ F A.X A.X ≫ F.map A.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [id_tensor_comp, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (α_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom ≫ (𝟙 (F.obj A.X) ⊗ μ F A.X A.X ≫ F.map A.mul) ≫ μ F A.X A.X ≫ F.map A.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [id_tensor_comp, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (α_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom ≫ (𝟙 (F.obj A.X) ⊗ μ F A.X A.X ≫ F.map A.mul) ≫ μ F A.X A.X ≫ F.map A.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [id_tensor_comp, ← F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (μ F A.X A.X ⊗ 𝟙 (F.obj A.X)) ≫
μ F (A.X ⊗ A.X) A.X ≫ F.map (α_ A.X A.X A.X).hom ≫ F.map (𝟙 A.X ⊗ A.mul) ≫ F.map A.mul =
(α_ (F.obj... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_rhs 3 4 => rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (F.map (𝟙 A.X) ⊗ F.map A.mul) ≫ μ F A.X A.X
case a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Typ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (F.map (𝟙 A.X) ⊗ F.map A.mul) ≫ μ F A.X A.X
case a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Typ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (F.map (𝟙 A.X) ⊗ F.map A.mul) ≫ μ F A.X A.X
case a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Typ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.μ_natural] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (μ F A.X A.X ⊗ 𝟙 (F.obj A.X)) ≫
μ F (A.X ⊗ A.X) A.X ≫ F.map (α_ A.X A.X A.X).hom ≫ F.map (𝟙 A.X ⊗ A.mul) ≫ F.map A.mul =
(α_ (F.obj... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | conv_rhs => rw [F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (α_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom ≫
(F.map (𝟙 A.X) ⊗ μ F A.X A.X) ≫ (μ F A.X (A.X ⊗ A.X) ≫ F.map (𝟙 A.X ⊗ A.mul)) ≫ F.map A.mu... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (α_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom ≫
(F.map (𝟙 A.X) ⊗ μ F A.X A.X) ≫ (μ F A.X (A.X ⊗ A.X) ≫ F.map (𝟙 A.X ⊗ A.mul)) ≫ F.map A.mu... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (α_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom ≫
(F.map (𝟙 A.X) ⊗ μ F A.X A.X) ≫ (μ F A.X (A.X ⊗ A.X) ≫ F.map (𝟙 A.X ⊗ A.mul)) ≫ F.map A.mu... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [F.toFunctor.map_id] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (μ F A.X A.X ⊗ 𝟙 (F.obj A.X)) ≫
μ F (A.X ⊗ A.X) A.X ≫ F.map (α_ A.X A.X A.X).hom ≫ F.map (𝟙 A.X ⊗ A.mul) ≫ F.map A.mul =
(α_ (F.obj... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_rhs 1 3 => rw [← F.associativity] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (α_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom ≫ (𝟙 (F.obj A.X) ⊗ μ F A.X A.X) ≫ μ F A.X (A.X ⊗ A.X)
case a.a
C : Type u₁
inst✝³ : Cate... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← F.associativity] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (α_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom ≫ (𝟙 (F.obj A.X) ⊗ μ F A.X A.X) ≫ μ F A.X (A.X ⊗ A.X)
case a.a
C : Type u₁
inst✝³ : Cate... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← F.associativity] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
| (α_ (F.obj A.X) (F.obj A.X) (F.obj A.X)).hom ≫ (𝟙 (F.obj A.X) ⊗ μ F A.X A.X) ≫ μ F A.X (A.X ⊗ A.X)
case a.a
C : Type u₁
inst✝³ : Cate... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← F.associativity] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ (μ F A.X A.X ⊗ 𝟙 (F.obj A.X)) ≫
μ F (A.X ⊗ A.X) A.X ≫ F.map (α_ A.X A.X A.X).hom ≫ F.map (𝟙 A.X ⊗ A.mul) ≫ F.map A.mul =
(((μ F A.X... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp only [Category.assoc] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
X✝ Y✝ : Mon_ C
f : X✝ ⟶ Y✝
⊢ ((fun A => Mon_.mk (F.obj A.X) (F.ε ≫ F.map A.one) (μ F A.X A.X ≫ F.map A.mul)) X✝).one ≫ F.map f.hom =
((fun A => Mon_.mk ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | dsimp | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
X✝ Y✝ : Mon_ C
f : X✝ ⟶ Y✝
⊢ (F.ε ≫ F.map X✝.one) ≫ F.map f.hom = F.ε ≫ F.map Y✝.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [Category.assoc, ← F.toFunctor.map_comp, f.one_hom] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
X✝ Y✝ : Mon_ C
f : X✝ ⟶ Y✝
⊢ ((fun A => Mon_.mk (F.obj A.X) (F.ε ≫ F.map A.one) (μ F A.X A.X ≫ F.map A.mul)) X✝).mul ≫ F.map f.hom =
(F.map f.hom ⊗ F.ma... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | dsimp | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
X✝ Y✝ : Mon_ C
f : X✝ ⟶ Y✝
⊢ (μ F X✝.X X✝.X ≫ F.map X✝.mul) ≫ F.map f.hom = (F.map f.hom ⊗ F.map f.hom) ≫ μ F Y✝.X Y✝.X ≫ F.map Y✝.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [Category.assoc, F.μ_natural_assoc, ← F.toFunctor.map_comp, ← F.toFunctor.map_comp,
f.mul_hom] | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ { obj := fun A => Mon_.mk (F.obj A.X) (F.ε ≫ F.map A.one) (μ F A.X A.X ≫ F.map A.mul),
map := fun {X Y} f => Mon_.Hom.mk (F.map f.hom... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | ext | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
A : Mon_ C
⊢ ({ obj := fun A => Mon_.mk (F.obj A.X) (F.ε ≫ F.map A.one) (μ F A.X A.X ≫ F.map A.mul),
map := fun {X Y} f => Mon_.Hom.mk (F... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
X✝ Y✝ Z✝ : Mon_ C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ { obj := fun A => Mon_.mk (F.obj A.X) (F.ε ≫ F.map A.one) (μ F A.X A.X ≫ F.map A.mul),
map := fun {X Y... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | ext | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F : LaxMonoidalFunctor C D
X✝ Y✝ Z✝ : Mon_ C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ ({ obj := fun A => Mon_.mk (F.obj A.X) (F.ε ≫ F.map A.one) (μ F A.X A.X ≫ F.map A.mul),
map :... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A :=
{ X := F.obj A.X
one := F.ε ≫ F.map A.one
mul := F.μ _ _ ≫ F.map A.m... | Mathlib.CategoryTheory.Monoidal.Mon_.202_0.NTUMzhXPwXsmsYt | /-- A lax monoidal functor takes monoid objects to monoid objects.
That is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.
-/
@[simps]
def mapMon (F : LaxMonoidalFunctor C D) : Mon_ C ⥤ Mon_ D where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
X✝ Y✝ : Mon_ C
f : X✝ ⟶ Y✝
x✝² x✝¹ : Discrete PUnit.{u + 1}
x✝ : x✝² ⟶ x✝¹
⊢ ((fun A =>
LaxMonoidalFunctor.mk
(CategoryTheory.Functor.mk { obj := fun x => A.X, map := fun {X Y} x => 𝟙 ((fun x => A.X) X) }) A.one
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | dsimp | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps]
def monToLaxMonoidal : Mon_ C ⥤ LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C where
obj A :=
{ obj := fun _ => A.X
map := fun _ => 𝟙 _
ε := A.one
μ := fun _ _ => A.mul
map_id := fun _ => rfl
map_comp := fun _ _ =... | Mathlib.CategoryTheory.Monoidal.Mon_.270_0.NTUMzhXPwXsmsYt | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps]
def monToLaxMonoidal : Mon_ C ⥤ LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
X✝ Y✝ : Mon_ C
f : X✝ ⟶ Y✝
x✝² x✝¹ : Discrete PUnit.{u + 1}
x✝ : x✝² ⟶ x✝¹
⊢ 𝟙 X✝.X ≫ f.hom = f.hom ≫ 𝟙 Y✝.X | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [Category.id_comp, Category.comp_id] | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps]
def monToLaxMonoidal : Mon_ C ⥤ LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C where
obj A :=
{ obj := fun _ => A.X
map := fun _ => 𝟙 _
ε := A.one
μ := fun _ _ => A.mul
map_id := fun _ => rfl
map_comp := fun _ _ =... | Mathlib.CategoryTheory.Monoidal.Mon_.270_0.NTUMzhXPwXsmsYt | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps]
def monToLaxMonoidal : Mon_ C ⥤ LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C where
obj A | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
F : LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C
x✝ : Discrete PUnit.{u + 1}
⊢ x✝ = (trivial (Discrete PUnit.{u + 1})).X | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | ext | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps!]
def unitIso :
𝟭 (LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C) ≅ laxMonoidalToMon C ⋙ monToLaxMonoidal C :=
NatIso.ofComponents
(fun F =>
MonoidalNatIso.ofComponents (fun _ => F.toFunctor.mapIso (eqToIso (by | Mathlib.CategoryTheory.Monoidal.Mon_.292_0.NTUMzhXPwXsmsYt | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps!]
def unitIso :
𝟭 (LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C) ≅ laxMonoidalToMon C ⋙ monToLaxMonoidal C | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
F : LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C
⊢ ∀ {X Y : Discrete PUnit.{u + 1}} (f : X ⟶ Y),
((𝟭 (LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C)).obj F).map f ≫
((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit.{u + 1})).... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | aesop_cat | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps!]
def unitIso :
𝟭 (LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C) ≅ laxMonoidalToMon C ⋙ monToLaxMonoidal C :=
NatIso.ofComponents
(fun F =>
MonoidalNatIso.ofComponents (fun _ => F.toFunctor.mapIso (eqToIso (by ext))) (by | Mathlib.CategoryTheory.Monoidal.Mon_.292_0.NTUMzhXPwXsmsYt | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps!]
def unitIso :
𝟭 (LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C) ≅ laxMonoidalToMon C ⋙ monToLaxMonoidal C | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
F : LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C
⊢ ((𝟭 (LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C)).obj F).ε ≫
((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit.{u + 1})).X))) (𝟙_ (Discrete PUnit.{u + 1}))).hom =
((laxMono... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | aesop_cat | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps!]
def unitIso :
𝟭 (LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C) ≅ laxMonoidalToMon C ⋙ monToLaxMonoidal C :=
NatIso.ofComponents
(fun F =>
MonoidalNatIso.ofComponents (fun _ => F.toFunctor.mapIso (eqToIso (by ext))) (by aesop_cat)
... | Mathlib.CategoryTheory.Monoidal.Mon_.292_0.NTUMzhXPwXsmsYt | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps!]
def unitIso :
𝟭 (LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C) ≅ laxMonoidalToMon C ⋙ monToLaxMonoidal C | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
F : LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C
⊢ ∀ (X Y : Discrete PUnit.{u + 1}),
μ ((𝟭 (LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C)).obj F) X Y ≫
((fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete PUnit.{u + 1})).X))) (X ⊗ Y)... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | aesop_cat | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps!]
def unitIso :
𝟭 (LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C) ≅ laxMonoidalToMon C ⋙ monToLaxMonoidal C :=
NatIso.ofComponents
(fun F =>
MonoidalNatIso.ofComponents (fun _ => F.toFunctor.mapIso (eqToIso (by ext))) (by aesop_cat)
... | Mathlib.CategoryTheory.Monoidal.Mon_.292_0.NTUMzhXPwXsmsYt | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps!]
def unitIso :
𝟭 (LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C) ≅ laxMonoidalToMon C ⋙ monToLaxMonoidal C | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
⊢ ∀ {X Y : LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C} (f : X ⟶ Y),
(𝟭 (LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C)).map f ≫
((fun F =>
MonoidalNatIso.ofComponents (fun x => F.mapIso (eqToIso (_ : x = (trivial (Discrete... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | aesop_cat | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps!]
def unitIso :
𝟭 (LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C) ≅ laxMonoidalToMon C ⋙ monToLaxMonoidal C :=
NatIso.ofComponents
(fun F =>
MonoidalNatIso.ofComponents (fun _ => F.toFunctor.mapIso (eqToIso (by ext))) (by aesop_cat)
... | Mathlib.CategoryTheory.Monoidal.Mon_.292_0.NTUMzhXPwXsmsYt | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps!]
def unitIso :
𝟭 (LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C) ≅ laxMonoidalToMon C ⋙ monToLaxMonoidal C | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
⊢ ∀ {X Y : Mon_ C} (f : X ⟶ Y),
(monToLaxMonoidal C ⋙ laxMonoidalToMon C).map f ≫
((fun F =>
Iso.mk (Hom.mk (𝟙 ((monToLaxMonoidal C ⋙ laxMonoidalToMon C).obj F).X))
(Hom.mk (𝟙 ((𝟭 (Mon_ C)).obj F).X)))
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | aesop_cat | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps!]
def counitIso : monToLaxMonoidal C ⋙ laxMonoidalToMon C ≅ 𝟭 (Mon_ C) :=
NatIso.ofComponents
(fun F =>
{ hom := { hom := 𝟙 _ }
inv := { hom := 𝟙 _ } })
(by | Mathlib.CategoryTheory.Monoidal.Mon_.303_0.NTUMzhXPwXsmsYt | /-- Implementation of `Mon_.equivLaxMonoidalFunctorPUnit`. -/
@[simps!]
def counitIso : monToLaxMonoidal C ⋙ laxMonoidalToMon C ≅ 𝟭 (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
⊢ ((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp only [Category.assoc, Iso.cancel_iso_inv_left] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
| Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
⊢ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one) ≫ (α_ M.X N.X P.X).hom = M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 1 3 => rw [← Category.id_comp P.one, tensor_comp] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
| Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
| ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one) ≫ (α_ M.X N.X P.X).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp P.one, tensor_comp] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
| ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one) ≫ (α_ M.X N.X P.X).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp P.one, tensor_comp] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
| ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one) ≫ (α_ M.X N.X P.X).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp P.one, tensor_comp] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
⊢ (((λ_ (𝟙_ C)).inv ⊗ 𝟙 (𝟙_ C)) ≫ ((M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 2 3 => rw [associator_naturality] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
| ((M.one ⊗ N.one) ⊗ P.one) ≫ (α_ M.X N.X P.X).hom
case a C : Type u₁ inst✝¹ : Category.{v₁, u₁} C inst✝ : MonoidalCategory C M N P : Mon_ C | (λ_ (𝟙_ C)).inv ⊗ 𝟙 (𝟙_ C) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [associator_naturality] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
| ((M.one ⊗ N.one) ⊗ P.one) ≫ (α_ M.X N.X P.X).hom
case a C : Type u₁ inst✝¹ : Category.{v₁, u₁} C inst✝ : MonoidalCategory C M N P : Mon_ C | (λ_ (𝟙_ C)).inv ⊗ 𝟙 (𝟙_ C) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [associator_naturality] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
| ((M.one ⊗ N.one) ⊗ P.one) ≫ (α_ M.X N.X P.X).hom
case a C : Type u₁ inst✝¹ : Category.{v₁, u₁} C inst✝ : MonoidalCategory C M N P : Mon_ C | (λ_ (𝟙_ C)).inv ⊗ 𝟙 (𝟙_ C) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [associator_naturality] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
⊢ ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (𝟙_ C)) ≫ (α_ (𝟙_ C) (𝟙_ C) (𝟙_ C)).hom ≫ (M.one ⊗ N.one ⊗ P.one) =
M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_rhs 1 2 => rw [← Category.id_comp M.one, tensor_comp] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
| M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp M.one, tensor_comp] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
| M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp M.one, tensor_comp] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
| M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp M.one, tensor_comp] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
⊢ ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (𝟙_ C)) ≫ (α_ (𝟙_ C) (𝟙_ C) (𝟙_ C)).hom ≫ (M.one ⊗ N.one ⊗ P.one) =
(𝟙 (𝟙_ C) ⊗ (λ_ (𝟙_ C)).inv) ≫ (M.one ⊗ N.one ⊗ P.one) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 1 2 => rw [← leftUnitor_tensor_inv] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
| ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (𝟙_ C)) ≫ (α_ (𝟙_ C) (𝟙_ C) (𝟙_ C)).hom
case a C : Type u₁ inst✝¹ : Category.{v₁, u₁} C inst✝ : MonoidalCategory C M N P : Mon_ C | M.one ⊗ N.one ⊗ P.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← leftUnitor_tensor_inv] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
| ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (𝟙_ C)) ≫ (α_ (𝟙_ C) (𝟙_ C) (𝟙_ C)).hom
case a C : Type u₁ inst✝¹ : Category.{v₁, u₁} C inst✝ : MonoidalCategory C M N P : Mon_ C | M.one ⊗ N.one ⊗ P.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← leftUnitor_tensor_inv] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
| ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (𝟙_ C)) ≫ (α_ (𝟙_ C) (𝟙_ C) (𝟙_ C)).hom
case a C : Type u₁ inst✝¹ : Category.{v₁, u₁} C inst✝ : MonoidalCategory C M N P : Mon_ C | M.one ⊗ N.one ⊗ P.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← leftUnitor_tensor_inv] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
⊢ (λ_ (𝟙_ C ⊗ 𝟙_ C)).inv ≫ (M.one ⊗ N.one ⊗ P.one) = (𝟙 (𝟙_ C) ⊗ (λ_ (𝟙_ C)).inv) ≫ (M.one ⊗ N.one ⊗ P.one) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← cancel_epi (λ_ (𝟙_ C)).inv] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
⊢ (λ_ (𝟙_ C)).inv ≫ (λ_ (𝟙_ C ⊗ 𝟙_ C)).inv ≫ (M.one ⊗ N.one ⊗ P.one) =
(λ_ (𝟙_ C)).inv ≫ (𝟙 (𝟙_ C) ⊗ (λ_ (𝟙_ C)).inv) ≫ (M.one ⊗ N.one ⊗ P.one) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 1 2 => rw [leftUnitor_inv_naturality] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
| (λ_ (𝟙_ C)).inv ≫ (λ_ (𝟙_ C ⊗ 𝟙_ C)).inv
case a C : Type u₁ inst✝¹ : Category.{v₁, u₁} C inst✝ : MonoidalCategory C M N P : Mon_ C | M.one ⊗ N.one ⊗ P.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [leftUnitor_inv_naturality] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
| (λ_ (𝟙_ C)).inv ≫ (λ_ (𝟙_ C ⊗ 𝟙_ C)).inv
case a C : Type u₁ inst✝¹ : Category.{v₁, u₁} C inst✝ : MonoidalCategory C M N P : Mon_ C | M.one ⊗ N.one ⊗ P.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [leftUnitor_inv_naturality] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
| (λ_ (𝟙_ C)).inv ≫ (λ_ (𝟙_ C ⊗ 𝟙_ C)).inv
case a C : Type u₁ inst✝¹ : Category.{v₁, u₁} C inst✝ : MonoidalCategory C M N P : Mon_ C | M.one ⊗ N.one ⊗ P.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [leftUnitor_inv_naturality] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M N P : Mon_ C
⊢ ((λ_ (𝟙_ C)).inv ≫ (𝟙 (𝟙_ C) ⊗ (λ_ (𝟙_ C)).inv)) ≫ (M.one ⊗ N.one ⊗ P.one) =
(λ_ (𝟙_ C)).inv ≫ (𝟙 (𝟙_ C) ⊗ (λ_ (𝟙_ C)).inv) ≫ (M.one ⊗ N.one ⊗ P.one) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp only [Category.assoc] | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tens... | Mathlib.CategoryTheory.Monoidal.Mon_.380_0.NTUMzhXPwXsmsYt | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M : Mon_ C
⊢ ((λ_ (𝟙_ C)).inv ≫ (𝟙 (𝟙_ C) ⊗ M.one)) ≫ (λ_ M.X).hom = M.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 2 3 => rw [leftUnitor_naturality] | theorem one_leftUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (𝟙 (𝟙_ C) ⊗ M.one)) ≫ (λ_ M.X).hom = M.one := by
| Mathlib.CategoryTheory.Monoidal.Mon_.393_0.NTUMzhXPwXsmsYt | theorem one_leftUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (𝟙 (𝟙_ C) ⊗ M.one)) ≫ (λ_ M.X).hom = M.one | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M : Mon_ C
| (𝟙 (𝟙_ C) ⊗ M.one) ≫ (λ_ M.X).hom
case a C : Type u₁ inst✝¹ : Category.{v₁, u₁} C inst✝ : MonoidalCategory C M : Mon_ C | (λ_ (𝟙_ C)).inv | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [leftUnitor_naturality] | theorem one_leftUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (𝟙 (𝟙_ C) ⊗ M.one)) ≫ (λ_ M.X).hom = M.one := by
slice_lhs 2 3 => | Mathlib.CategoryTheory.Monoidal.Mon_.393_0.NTUMzhXPwXsmsYt | theorem one_leftUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (𝟙 (𝟙_ C) ⊗ M.one)) ≫ (λ_ M.X).hom = M.one | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M : Mon_ C
| (𝟙 (𝟙_ C) ⊗ M.one) ≫ (λ_ M.X).hom
case a C : Type u₁ inst✝¹ : Category.{v₁, u₁} C inst✝ : MonoidalCategory C M : Mon_ C | (λ_ (𝟙_ C)).inv | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [leftUnitor_naturality] | theorem one_leftUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (𝟙 (𝟙_ C) ⊗ M.one)) ≫ (λ_ M.X).hom = M.one := by
slice_lhs 2 3 => | Mathlib.CategoryTheory.Monoidal.Mon_.393_0.NTUMzhXPwXsmsYt | theorem one_leftUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (𝟙 (𝟙_ C) ⊗ M.one)) ≫ (λ_ M.X).hom = M.one | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M : Mon_ C
| (𝟙 (𝟙_ C) ⊗ M.one) ≫ (λ_ M.X).hom
case a C : Type u₁ inst✝¹ : Category.{v₁, u₁} C inst✝ : MonoidalCategory C M : Mon_ C | (λ_ (𝟙_ C)).inv | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [leftUnitor_naturality] | theorem one_leftUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (𝟙 (𝟙_ C) ⊗ M.one)) ≫ (λ_ M.X).hom = M.one := by
slice_lhs 2 3 => | Mathlib.CategoryTheory.Monoidal.Mon_.393_0.NTUMzhXPwXsmsYt | theorem one_leftUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (𝟙 (𝟙_ C) ⊗ M.one)) ≫ (λ_ M.X).hom = M.one | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M : Mon_ C
⊢ (λ_ (𝟙_ C)).inv ≫ (λ_ (𝟙_ C)).hom ≫ M.one = M.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp | theorem one_leftUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (𝟙 (𝟙_ C) ⊗ M.one)) ≫ (λ_ M.X).hom = M.one := by
slice_lhs 2 3 => rw [leftUnitor_naturality]
| Mathlib.CategoryTheory.Monoidal.Mon_.393_0.NTUMzhXPwXsmsYt | theorem one_leftUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (𝟙 (𝟙_ C) ⊗ M.one)) ≫ (λ_ M.X).hom = M.one | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M : Mon_ C
⊢ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ 𝟙 (𝟙_ C))) ≫ (ρ_ M.X).hom = M.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 2 3 => rw [rightUnitor_naturality, ← unitors_equal] | theorem one_rightUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ 𝟙 (𝟙_ C))) ≫ (ρ_ M.X).hom = M.one := by
| Mathlib.CategoryTheory.Monoidal.Mon_.399_0.NTUMzhXPwXsmsYt | theorem one_rightUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ 𝟙 (𝟙_ C))) ≫ (ρ_ M.X).hom = M.one | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M : Mon_ C
| (M.one ⊗ 𝟙 (𝟙_ C)) ≫ (ρ_ M.X).hom
case a C : Type u₁ inst✝¹ : Category.{v₁, u₁} C inst✝ : MonoidalCategory C M : Mon_ C | (λ_ (𝟙_ C)).inv | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [rightUnitor_naturality, ← unitors_equal] | theorem one_rightUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ 𝟙 (𝟙_ C))) ≫ (ρ_ M.X).hom = M.one := by
slice_lhs 2 3 => | Mathlib.CategoryTheory.Monoidal.Mon_.399_0.NTUMzhXPwXsmsYt | theorem one_rightUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ 𝟙 (𝟙_ C))) ≫ (ρ_ M.X).hom = M.one | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M : Mon_ C
| (M.one ⊗ 𝟙 (𝟙_ C)) ≫ (ρ_ M.X).hom
case a C : Type u₁ inst✝¹ : Category.{v₁, u₁} C inst✝ : MonoidalCategory C M : Mon_ C | (λ_ (𝟙_ C)).inv | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [rightUnitor_naturality, ← unitors_equal] | theorem one_rightUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ 𝟙 (𝟙_ C))) ≫ (ρ_ M.X).hom = M.one := by
slice_lhs 2 3 => | Mathlib.CategoryTheory.Monoidal.Mon_.399_0.NTUMzhXPwXsmsYt | theorem one_rightUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ 𝟙 (𝟙_ C))) ≫ (ρ_ M.X).hom = M.one | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M : Mon_ C
| (M.one ⊗ 𝟙 (𝟙_ C)) ≫ (ρ_ M.X).hom
case a C : Type u₁ inst✝¹ : Category.{v₁, u₁} C inst✝ : MonoidalCategory C M : Mon_ C | (λ_ (𝟙_ C)).inv | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [rightUnitor_naturality, ← unitors_equal] | theorem one_rightUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ 𝟙 (𝟙_ C))) ≫ (ρ_ M.X).hom = M.one := by
slice_lhs 2 3 => | Mathlib.CategoryTheory.Monoidal.Mon_.399_0.NTUMzhXPwXsmsYt | theorem one_rightUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ 𝟙 (𝟙_ C))) ≫ (ρ_ M.X).hom = M.one | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
M : Mon_ C
⊢ (λ_ (𝟙_ C)).inv ≫ (λ_ (𝟙_ C)).hom ≫ M.one = M.one | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp | theorem one_rightUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ 𝟙 (𝟙_ C))) ≫ (ρ_ M.X).hom = M.one := by
slice_lhs 2 3 => rw [rightUnitor_naturality, ← unitors_equal]
| Mathlib.CategoryTheory.Monoidal.Mon_.399_0.NTUMzhXPwXsmsYt | theorem one_rightUnitor {M : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ 𝟙 (𝟙_ C))) ≫ (ρ_ M.X).hom = M.one | Mathlib_CategoryTheory_Monoidal_Mon_ |
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