statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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has_coe_to_prefunctor : has_coe (prelax_functor B C) (prefunctor B C) | ⟨to_prefunctor⟩ | instance | category_theory.prelax_functor.has_coe_to_prefunctor | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"prefunctor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prefunctor_eq_coe : F.to_prefunctor = F | rfl | lemma | category_theory.prelax_functor.to_prefunctor_eq_coe | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prefunctor_obj : (F : prefunctor B C).obj = F.obj | rfl | lemma | category_theory.prelax_functor.to_prefunctor_obj | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"prefunctor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prefunctor_map : @prefunctor.map B _ C _ F = @map _ _ _ _ _ _ F | rfl | lemma | category_theory.prelax_functor.to_prefunctor_map | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id (B : Type u₁) [quiver.{v₁+1} B] [∀ a b : B, quiver.{w₁+1} (a ⟶ b)] : prelax_functor B B | { map₂ := λ a b f g η, η, .. prefunctor.id B } | def | category_theory.prelax_functor.id | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"prefunctor.id"
] | The identity prelax functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp (F : prelax_functor B C) (G : prelax_functor C D) : prelax_functor B D | { map₂ := λ a b f g η, G.map₂ (F.map₂ η), .. (F : prefunctor B C).comp ↑G } | def | category_theory.prelax_functor.comp | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"prefunctor"
] | Composition of prelax functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
oplax_functor.map₂_associator_aux
(obj : B → C) (map : Π {X Y : B}, (X ⟶ Y) → (obj X ⟶ obj Y))
(map₂ : Π {a b : B} {f g : a ⟶ b}, (f ⟶ g) → (map f ⟶ map g))
(map_comp : Π {a b c : B} (f : a ⟶ b) (g : b ⟶ c), map (f ≫ g) ⟶ map f ≫ map g)
{a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : Prop | map₂ (α_ f g h).hom ≫ map_comp f (g ≫ h) ≫ map f ◁ map_comp g h =
map_comp (f ≫ g) h ≫ map_comp f g ▷ map h ≫ (α_ (map f) (map g) (map h)).hom | def | category_theory.oplax_functor.map₂_associator_aux | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
oplax_functor (B : Type u₁) [bicategory.{w₁ v₁} B] (C : Type u₂) [bicategory.{w₂ v₂} C]
extends prelax_functor B C | (map_id (a : B) : map (𝟙 a) ⟶ 𝟙 (obj a))
(map_comp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : map (f ≫ g) ⟶ map f ≫ map g)
(map_comp_naturality_left' : ∀ {a b c : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c),
map₂ (η ▷ g) ≫ map_comp f' g = map_comp f g ≫ map₂ η ▷ map g . obviously)
(map_comp_naturality_right' : ∀ {a b c :... | structure | category_theory.oplax_functor | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"map_comp",
"map_id"
] | An oplax functor `F` between bicategories `B` and `C` consists of a function between objects
`F.obj`, a function between 1-morphisms `F.map`, and a function between 2-morphisms `F.map₂`.
Unlike functors between categories, `F.map` do not need to strictly commute with the composition,
and do not need to strictly preser... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coe_to_prelax : has_coe (oplax_functor B C) (prelax_functor B C) | ⟨to_prelax_functor⟩ | instance | category_theory.oplax_functor.has_coe_to_prelax | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prelax_eq_coe : F.to_prelax_functor = F | rfl | lemma | category_theory.oplax_functor.to_prelax_eq_coe | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prelax_functor_obj : (F : prelax_functor B C).obj = F.obj | rfl | lemma | category_theory.oplax_functor.to_prelax_functor_obj | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prelax_functor_map : @prelax_functor.map B _ _ C _ _ F = @map _ _ _ _ F | rfl | lemma | category_theory.oplax_functor.to_prelax_functor_map | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prelax_functor_map₂ : @prelax_functor.map₂ B _ _ C _ _ F = @map₂ _ _ _ _ F | rfl | lemma | category_theory.oplax_functor.to_prelax_functor_map₂ | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_functor (a b : B) : (a ⟶ b) ⥤ (F.obj a ⟶ F.obj b) | { obj := λ f, F.map f,
map := λ f g η, F.map₂ η } | def | category_theory.oplax_functor.map_functor | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | Function between 1-morphisms as a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id (B : Type u₁) [bicategory.{w₁ v₁} B] : oplax_functor B B | { map_id := λ a, 𝟙 (𝟙 a),
map_comp := λ a b c f g, 𝟙 (f ≫ g),
.. prelax_functor.id B } | def | category_theory.oplax_functor.id | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"map_comp",
"map_id"
] | The identity oplax functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp (F : oplax_functor B C) (G : oplax_functor C D) : oplax_functor B D | { map_id := λ a,
(G.map_functor _ _).map (F.map_id a) ≫ G.map_id (F.obj a),
map_comp := λ a b c f g,
(G.map_functor _ _).map (F.map_comp f g) ≫ G.map_comp (F.map f) (F.map g),
map_comp_naturality_left' := λ a b c f f' η g, by
{ dsimp,
rw [←map₂_comp_assoc, map_comp_naturality_left, map₂_comp_assoc, ma... | def | category_theory.oplax_functor.comp | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"map_comp",
"map_id"
] | Composition of oplax functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_core (F : oplax_functor B C) | (map_id_iso (a : B) : F.map (𝟙 a) ≅ 𝟙 (F.obj a))
(map_comp_iso {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : F.map (f ≫ g) ≅ F.map f ≫ F.map g)
(map_id_iso_hom' : ∀ {a : B}, (map_id_iso a).hom = F.map_id a . obviously)
(map_comp_iso_hom' : ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c),
(map_comp_iso f g).hom = F.map_comp f g . obv... | structure | category_theory.oplax_functor.pseudo_core | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | A structure on an oplax functor that promotes an oplax functor to a pseudofunctor.
See `pseudofunctor.mk_of_oplax`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudofunctor.map₂_associator_aux
(obj : B → C) (map : Π {X Y : B}, (X ⟶ Y) → (obj X ⟶ obj Y))
(map₂ : Π {a b : B} {f g : a ⟶ b}, (f ⟶ g) → (map f ⟶ map g))
(map_comp : Π {a b c : B} (f : a ⟶ b) (g : b ⟶ c), map (f ≫ g) ≅ map f ≫ map g)
{a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : Prop | map₂ (α_ f g h).hom = (map_comp (f ≫ g) h).hom ≫ (map_comp f g).hom ▷ map h ≫
(α_ (map f) (map g) (map h)).hom ≫ map f ◁ (map_comp g h).inv ≫ (map_comp f (g ≫ h)).inv | def | category_theory.pseudofunctor.map₂_associator_aux | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudofunctor (B : Type u₁) [bicategory.{w₁ v₁} B] (C : Type u₂) [bicategory.{w₂ v₂} C]
extends prelax_functor B C | (map_id (a : B) : map (𝟙 a) ≅ 𝟙 (obj a))
(map_comp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : map (f ≫ g) ≅ map f ≫ map g)
(map₂_id' : ∀ {a b : B} (f : a ⟶ b), map₂ (𝟙 f) = 𝟙 (map f) . obviously)
(map₂_comp' : ∀ {a b : B} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h),
map₂ (η ≫ θ) = map₂ η ≫ map₂ θ . obviously)
(map₂_whiske... | structure | category_theory.pseudofunctor | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"map_comp",
"map_id"
] | A pseudofunctor `F` between bicategories `B` and `C` consists of a function between objects
`F.obj`, a function between 1-morphisms `F.map`, and a function between 2-morphisms `F.map₂`.
Unlike functors between categories, `F.map` do not need to strictly commute with the compositions,
and do not need to strictly preser... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coe_to_prelax_functor : has_coe (pseudofunctor B C) (prelax_functor B C) | ⟨to_prelax_functor⟩ | instance | category_theory.pseudofunctor.has_coe_to_prelax_functor | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_prelax_functor_eq_coe : F.to_prelax_functor = F | rfl | lemma | category_theory.pseudofunctor.to_prelax_functor_eq_coe | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_oplax : oplax_functor B C | { map_id := λ a, (F.map_id a).hom,
map_comp := λ a b c f g, (F.map_comp f g).hom,
.. (F : prelax_functor B C) } | def | category_theory.pseudofunctor.to_oplax | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"map_comp",
"map_id"
] | The oplax functor associated with a pseudofunctor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coe_to_oplax : has_coe (pseudofunctor B C) (oplax_functor B C) | ⟨to_oplax⟩ | instance | category_theory.pseudofunctor.has_coe_to_oplax | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_oplax_eq_coe : F.to_oplax = F | rfl | lemma | category_theory.pseudofunctor.to_oplax_eq_coe | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_oplax_obj : (F : oplax_functor B C).obj = F.obj | rfl | lemma | category_theory.pseudofunctor.to_oplax_obj | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_oplax_map : @oplax_functor.map B _ C _ F = @map _ _ _ _ F | rfl | lemma | category_theory.pseudofunctor.to_oplax_map | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_oplax_map₂ : @oplax_functor.map₂ B _ C _ F = @map₂ _ _ _ _ F | rfl | lemma | category_theory.pseudofunctor.to_oplax_map₂ | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_oplax_map_id (a : B) : (F : oplax_functor B C).map_id a = (F.map_id a).hom | rfl | lemma | category_theory.pseudofunctor.to_oplax_map_id | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_oplax_map_comp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) :
(F : oplax_functor B C).map_comp f g = (F.map_comp f g).hom | rfl | lemma | category_theory.pseudofunctor.to_oplax_map_comp | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_functor (a b : B) : (a ⟶ b) ⥤ (F.obj a ⟶ F.obj b) | (F : oplax_functor B C).map_functor a b | def | category_theory.pseudofunctor.map_functor | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [] | Function on 1-morphisms as a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id (B : Type u₁) [bicategory.{w₁ v₁} B] : pseudofunctor B B | { map_id := λ a, iso.refl (𝟙 a),
map_comp := λ a b c f g, iso.refl (f ≫ g),
.. prelax_functor.id B } | def | category_theory.pseudofunctor.id | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"map_comp",
"map_id"
] | The identity pseudofunctor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp (F : pseudofunctor B C) (G : pseudofunctor C D) : pseudofunctor B D | { map_id := λ a, (G.map_functor _ _).map_iso (F.map_id a) ≪≫ G.map_id (F.obj a),
map_comp := λ a b c f g,
(G.map_functor _ _).map_iso (F.map_comp f g) ≪≫ G.map_comp (F.map f) (F.map g),
.. (F : prelax_functor B C).comp ↑G } | def | category_theory.pseudofunctor.comp | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"map_comp",
"map_id"
] | Composition of pseudofunctors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_of_oplax (F : oplax_functor B C) (F' : F.pseudo_core) : pseudofunctor B C | { map_id := F'.map_id_iso,
map_comp := λ _ _ _, F'.map_comp_iso,
map₂_whisker_left' := λ a b c f g h η, by
{ dsimp,
rw [F'.map_comp_iso_hom f g, ←F.map_comp_naturality_right_assoc,
←F'.map_comp_iso_hom f h, hom_inv_id, comp_id] },
map₂_whisker_right' := λ a b c f g η h, by
{ dsimp,
rw [F'.map_co... | def | category_theory.pseudofunctor.mk_of_oplax | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"map_comp",
"map_id"
] | Construct a pseudofunctor from an oplax functor whose `map_id` and `map_comp` are isomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_of_oplax' (F : oplax_functor B C)
[∀ a, is_iso (F.map_id a)] [∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), is_iso (F.map_comp f g)] :
pseudofunctor B C | { map_id := λ a, as_iso (F.map_id a),
map_comp := λ a b c f g, as_iso (F.map_comp f g),
map₂_whisker_left' := λ a b c f g h η, by
{ dsimp,
rw [←assoc, is_iso.eq_comp_inv, F.map_comp_naturality_right] },
map₂_whisker_right' := λ a b c f g η h, by
{ dsimp,
rw [←assoc, is_iso.eq_comp_inv, F.map_comp_natu... | def | category_theory.pseudofunctor.mk_of_oplax' | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"map_comp",
"map_id"
] | Construct a pseudofunctor from an oplax functor whose `map_id` and `map_comp` are isomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_left (η : F ⟶ G) {θ ι : G ⟶ H} (Γ : θ ⟶ ι) : η ≫ θ ⟶ η ≫ ι | { app := λ a, η.app a ◁ Γ.app a,
naturality' := λ a b f, by
{ dsimp, rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc], simp } } | def | category_theory.oplax_nat_trans.whisker_left | category_theory.bicategory | src/category_theory/bicategory/functor_bicategory.lean | [
"category_theory.bicategory.natural_transformation"
] | [] | Left whiskering of an oplax natural transformation and a modification. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_right {η θ : F ⟶ G} (Γ : η ⟶ θ) (ι : G ⟶ H) : η ≫ ι ⟶ θ ≫ ι | { app := λ a, Γ.app a ▷ ι.app a,
naturality' := λ a b f, by
{ dsimp, simp_rw [assoc, ←associator_inv_naturality_left, whisker_exchange_assoc], simp } } | def | category_theory.oplax_nat_trans.whisker_right | category_theory.bicategory | src/category_theory/bicategory/functor_bicategory.lean | [
"category_theory.bicategory.natural_transformation"
] | [] | Right whiskering of an oplax natural transformation and a modification. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associator (η : F ⟶ G) (θ : G ⟶ H) (ι : H ⟶ I) : (η ≫ θ) ≫ ι ≅ η ≫ (θ ≫ ι) | modification_iso.of_components (λ a, α_ (η.app a) (θ.app a) (ι.app a)) (by tidy) | def | category_theory.oplax_nat_trans.associator | category_theory.bicategory | src/category_theory/bicategory/functor_bicategory.lean | [
"category_theory.bicategory.natural_transformation"
] | [] | Associator for the vertical composition of oplax natural transformations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_unitor (η : F ⟶ G) : 𝟙 F ≫ η ≅ η | modification_iso.of_components (λ a, λ_ (η.app a)) (by tidy) | def | category_theory.oplax_nat_trans.left_unitor | category_theory.bicategory | src/category_theory/bicategory/functor_bicategory.lean | [
"category_theory.bicategory.natural_transformation"
] | [] | Left unitor for the vertical composition of oplax natural transformations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_unitor (η : F ⟶ G) : η ≫ 𝟙 G ≅ η | modification_iso.of_components (λ a, ρ_ (η.app a)) (by tidy) | def | category_theory.oplax_nat_trans.right_unitor | category_theory.bicategory | src/category_theory/bicategory/functor_bicategory.lean | [
"category_theory.bicategory.natural_transformation"
] | [] | Right unitor for the vertical composition of oplax natural transformations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
oplax_functor.bicategory : bicategory (oplax_functor B C) | { whisker_left := λ F G H η _ _ Γ, oplax_nat_trans.whisker_left η Γ,
whisker_right := λ F G H _ _ Γ η, oplax_nat_trans.whisker_right Γ η,
associator := λ F G H I, oplax_nat_trans.associator,
left_unitor := λ F G, oplax_nat_trans.left_unitor,
right_unitor := λ F G, oplax_nat_trans.right_unitor,
whisker_... | instance | category_theory.oplax_functor.bicategory | category_theory.bicategory | src/category_theory/bicategory/functor_bicategory.lean | [
"category_theory.bicategory.natural_transformation"
] | [] | A bicategory structure on the oplax functors between bicategories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_discrete (C : Type u) | C | def | category_theory.locally_discrete | category_theory.bicategory | src/category_theory/bicategory/locally_discrete.lean | [
"category_theory.discrete_category",
"category_theory.bicategory.functor",
"category_theory.bicategory.strict"
] | [] | A type synonym for promoting any type to a category,
with the only morphisms being equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_small_category (X Y : locally_discrete C) : small_category (X ⟶ Y) | category_theory.discrete_category (X ⟶ Y) | instance | category_theory.locally_discrete.hom_small_category | category_theory.bicategory | src/category_theory/bicategory/locally_discrete.lean | [
"category_theory.discrete_category",
"category_theory.bicategory.functor",
"category_theory.bicategory.strict"
] | [
"category_theory.discrete_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_hom {X Y : locally_discrete C} {f g : X ⟶ Y} (η : f ⟶ g) : f = g | begin
have : discrete.mk (f.as) = discrete.mk (g.as) := congr_arg discrete.mk (eq_of_hom η),
simpa using this
end | lemma | category_theory.locally_discrete.eq_of_hom | category_theory.bicategory | src/category_theory/bicategory/locally_discrete.lean | [
"category_theory.discrete_category",
"category_theory.bicategory.functor",
"category_theory.bicategory.strict"
] | [] | Extract the equation from a 2-morphism in a locally discrete 2-category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_discrete_bicategory : bicategory (locally_discrete C) | { whisker_left := λ X Y Z f g h η, eq_to_hom (congr_arg2 (≫) rfl (locally_discrete.eq_of_hom η)),
whisker_right := λ X Y Z f g η h, eq_to_hom (congr_arg2 (≫) (locally_discrete.eq_of_hom η) rfl),
associator := λ W X Y Z f g h, eq_to_iso $ by { unfold_projs, simp only [category.assoc] },
left_unitor := λ X Y f, e... | instance | category_theory.locally_discrete_bicategory | category_theory.bicategory | src/category_theory/bicategory/locally_discrete.lean | [
"category_theory.discrete_category",
"category_theory.bicategory.functor",
"category_theory.bicategory.strict"
] | [
"congr_arg2"
] | The locally discrete bicategory on a category is a bicategory in which the objects and the
1-morphisms are the same as those in the underlying category, and the 2-morphisms are the
equalities between 1-morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_discrete_bicategory.strict : strict (locally_discrete C) | { id_comp' := by { intros, ext1, unfold_projs, apply category.id_comp },
comp_id' := by { intros, ext1, unfold_projs, apply category.comp_id },
assoc' := by { intros, ext1, unfold_projs, apply category.assoc } } | instance | category_theory.locally_discrete_bicategory.strict | category_theory.bicategory | src/category_theory/bicategory/locally_discrete.lean | [
"category_theory.discrete_category",
"category_theory.bicategory.functor",
"category_theory.bicategory.strict"
] | [] | A locally discrete bicategory is strict. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.to_oplax_functor (F : I ⥤ B) : oplax_functor (locally_discrete I) B | { obj := F.obj,
map := λ X Y f, F.map f.as,
map₂ := λ i j f g η, eq_to_hom (congr_arg _ (eq_of_hom η)),
map_id := λ i, eq_to_hom (F.map_id i),
map_comp := λ i j k f g, eq_to_hom (F.map_comp f.as g.as) } | def | category_theory.functor.to_oplax_functor | category_theory.bicategory | src/category_theory/bicategory/locally_discrete.lean | [
"category_theory.discrete_category",
"category_theory.bicategory.functor",
"category_theory.bicategory.strict"
] | [
"map_comp",
"map_id"
] | If `B` is a strict bicategory and `I` is a (1-)category, any functor (of 1-categories) `I ⥤ B` can
be promoted to an oplax functor from `locally_discrete I` to `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
oplax_nat_trans (F G : oplax_functor B C) | (app (a : B) : F.obj a ⟶ G.obj a)
(naturality {a b : B} (f : a ⟶ b) : F.map f ≫ app b ⟶ app a ≫ G.map f)
(naturality_naturality' : ∀ {a b : B} {f g : a ⟶ b} (η : f ⟶ g),
F.map₂ η ▷ app b ≫ naturality g = naturality f ≫ app a ◁ G.map₂ η . obviously)
(naturality_id' : ∀ a : B,
naturality (𝟙 a) ≫ app a ◁ G.map_id a =... | structure | category_theory.oplax_nat_trans | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | If `η` is an oplax natural transformation between `F` and `G`, we have a 1-morphism
`η.app a : F.obj a ⟶ G.obj a` for each object `a : B`. We also have a 2-morphism
`η.naturality f : F.map f ≫ app b ⟶ app a ≫ G.map f` for each 1-morphism `f : a ⟶ b`.
These 2-morphisms satisfies the naturality condition, and preserve th... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : oplax_nat_trans F F | { app := λ a, 𝟙 (F.obj a),
naturality := λ a b f, (ρ_ (F.map f)).hom ≫ (λ_ (F.map f)).inv } | def | category_theory.oplax_nat_trans.id | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | The identity oplax natural transformation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_left_naturality_naturality (f : a' ⟶ G.obj a) {g h : a ⟶ b} (β : g ⟶ h) :
f ◁ G.map₂ β ▷ θ.app b ≫ f ◁ θ.naturality h =
f ◁ θ.naturality g ≫ f ◁ θ.app a ◁ H.map₂ β | by simp_rw [←bicategory.whisker_left_comp, naturality_naturality] | lemma | category_theory.oplax_nat_trans.whisker_left_naturality_naturality | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right_naturality_naturality {f g : a ⟶ b} (β : f ⟶ g) (h : G.obj b ⟶ a') :
F.map₂ β ▷ η.app b ▷ h ≫ η.naturality g ▷ h =
η.naturality f ▷ h ≫ (α_ _ _ _).hom ≫ η.app a ◁ G.map₂ β ▷ h ≫ (α_ _ _ _).inv | by rw [←comp_whisker_right, naturality_naturality, comp_whisker_right, whisker_assoc] | lemma | category_theory.oplax_nat_trans.whisker_right_naturality_naturality | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_left_naturality_comp (f : a' ⟶ G.obj a) (g : a ⟶ b) (h : b ⟶ c) :
f ◁ θ.naturality (g ≫ h) ≫ f ◁ θ.app a ◁ H.map_comp g h =
f ◁ G.map_comp g h ▷ θ.app c ≫ f ◁ (α_ _ _ _).hom ≫
f ◁ G.map g ◁ θ.naturality h ≫ f ◁ (α_ _ _ _).inv ≫
f ◁ θ.naturality g ▷ H.map h ≫ f ◁ (α_ _ _ _).hom | by simp_rw [←bicategory.whisker_left_comp, naturality_comp] | lemma | category_theory.oplax_nat_trans.whisker_left_naturality_comp | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right_naturality_comp (f : a ⟶ b) (g : b ⟶ c) (h : G.obj c ⟶ a') :
η.naturality (f ≫ g) ▷ h ≫ (α_ _ _ _).hom ≫ η.app a ◁ G.map_comp f g ▷ h =
F.map_comp f g ▷ η.app c ▷ h ≫ (α_ _ _ _).hom ▷ h ≫ (α_ _ _ _).hom ≫
F.map f ◁ η.naturality g ▷ h ≫ (α_ _ _ _).inv ≫ (α_ _ _ _).inv ▷ h ≫
η.naturality... | by { rw [←associator_naturality_middle, ←comp_whisker_right_assoc, naturality_comp], simp } | lemma | category_theory.oplax_nat_trans.whisker_right_naturality_comp | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_left_naturality_id (f : a' ⟶ G.obj a) :
f ◁ θ.naturality (𝟙 a) ≫ f ◁ θ.app a ◁ H.map_id a =
f ◁ G.map_id a ▷ θ.app a ≫ f ◁ (λ_ (θ.app a)).hom ≫ f ◁ (ρ_ (θ.app a)).inv | by simp_rw [←bicategory.whisker_left_comp, naturality_id] | lemma | category_theory.oplax_nat_trans.whisker_left_naturality_id | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right_naturality_id (f : G.obj a ⟶ a') :
η.naturality (𝟙 a) ▷ f ≫ (α_ _ _ _).hom ≫ η.app a ◁ G.map_id a ▷ f =
F.map_id a ▷ η.app a ▷ f ≫ (λ_ (η.app a)).hom ▷ f ≫
(ρ_ (η.app a)).inv ▷ f ≫ (α_ _ _ _).hom | by { rw [←associator_naturality_middle, ←comp_whisker_right_assoc, naturality_id], simp } | lemma | category_theory.oplax_nat_trans.whisker_right_naturality_id | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vcomp (η : oplax_nat_trans F G) (θ : oplax_nat_trans G H) : oplax_nat_trans F H | { app := λ a, η.app a ≫ θ.app a,
naturality := λ a b f,
(α_ _ _ _).inv ≫ η.naturality f ▷ θ.app b ≫ (α_ _ _ _).hom ≫
η.app a ◁ θ.naturality f ≫ (α_ _ _ _).inv,
naturality_comp' := λ a b c f g, by
{ calc _ = _ ≫
F.map_comp f g ▷ η.app c ▷ θ.app c ≫ _ ≫
F.map f ◁ η.naturality g ▷ θ.app c ≫ _ ≫
... | def | category_theory.oplax_nat_trans.vcomp | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | Vertical composition of oplax natural transformations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
modification (η θ : F ⟶ G) | (app (a : B) : η.app a ⟶ θ.app a)
(naturality' : ∀ {a b : B} (f : a ⟶ b),
(F.map f ◁ app b) ≫ θ.naturality f = η.naturality f ≫ (app a ▷ G.map f) . obviously) | structure | category_theory.oplax_nat_trans.modification | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | A modification `Γ` between oplax natural transformations `η` and `θ` consists of a family of
2-morphisms `Γ.app a : η.app a ⟶ θ.app a`, which satisfies the equation
`(F.map f ◁ app b) ≫ θ.naturality f = η.naturality f ≫ (app a ▷ G.map f)`
for each 1-morphism `f : a ⟶ b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : modification η η | { app := λ a, 𝟙 (η.app a) } | def | category_theory.oplax_nat_trans.modification.id | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | The identity modification. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_left_naturality (f : a' ⟶ F.obj b) (g : b ⟶ c) :
f ◁ F.map g ◁ Γ.app c ≫ f ◁ θ.naturality g =
f ◁ η.naturality g ≫ f ◁ Γ.app b ▷ G.map g | by simp_rw [←bicategory.whisker_left_comp, naturality] | lemma | category_theory.oplax_nat_trans.modification.whisker_left_naturality | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right_naturality (f : a ⟶ b) (g : G.obj b ⟶ a') :
F.map f ◁ Γ.app b ▷ g ≫ (α_ _ _ _).inv ≫ θ.naturality f ▷ g =
(α_ _ _ _).inv ≫ η.naturality f ▷ g ≫ Γ.app a ▷ G.map f ▷ g | by simp_rw [associator_inv_naturality_middle_assoc, ←comp_whisker_right, naturality] | lemma | category_theory.oplax_nat_trans.modification.whisker_right_naturality | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vcomp (Γ : modification η θ) (Δ : modification θ ι) : modification η ι | { app := λ a, Γ.app a ≫ Δ.app a } | def | category_theory.oplax_nat_trans.modification.vcomp | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | Vertical composition of modifications. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category (F G : oplax_functor B C) : category (F ⟶ G) | { hom := modification,
id := modification.id,
comp := λ η θ ι, modification.vcomp } | instance | category_theory.oplax_nat_trans.category | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | Category structure on the oplax natural transformations between oplax_functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
modification_iso.of_components
(app : ∀ a, η.app a ≅ θ.app a)
(naturality : ∀ {a b} (f : a ⟶ b),
F.map f ◁ (app b).hom ≫ θ.naturality f = η.naturality f ≫ (app a).hom ▷ G.map f) :
η ≅ θ | { hom := { app := λ a, (app a).hom },
inv :=
{ app := λ a, (app a).inv,
naturality' := λ a b f, by simpa using
congr_arg (λ f, _ ◁ (app b).inv ≫ f ≫ (app a).inv ▷ _) (naturality f).symm } } | def | category_theory.oplax_nat_trans.modification_iso.of_components | category_theory.bicategory | src/category_theory/bicategory/natural_transformation.lean | [
"category_theory.bicategory.functor"
] | [] | Construct a modification isomorphism between oplax natural transformations
by giving object level isomorphisms, and checking naturality only in the forward direction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoidal_single_obj (C : Type*) [category C] [monoidal_category C] | punit | def | category_theory.monoidal_single_obj | category_theory.bicategory | src/category_theory/bicategory/single_obj.lean | [
"category_theory.bicategory.End",
"category_theory.monoidal.functor"
] | [] | Promote a monoidal category to a bicategory with a single object.
(The objects of the monoidal category become the 1-morphisms,
with composition given by tensor product,
and the morphisms of the monoidal category become the 2-morphisms.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star : monoidal_single_obj C | punit.star | def | category_theory.monoidal_single_obj.star | category_theory.bicategory | src/category_theory/bicategory/single_obj.lean | [
"category_theory.bicategory.End",
"category_theory.monoidal.functor"
] | [] | The unique object in the bicategory obtained by "promoting" a monoidal category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
End_monoidal_star_functor : monoidal_functor (End_monoidal (monoidal_single_obj.star C)) C | { obj := λ X, X,
map := λ X Y f, f,
ε := 𝟙 _,
μ := λ X Y, 𝟙 _,
μ_natural' := λ X Y X' Y' f g, begin
dsimp,
simp only [category.id_comp, category.comp_id],
-- Should we provide further simp lemmas so this goal becomes visible?
exact (tensor_id_comp_id_tensor _ _).symm,
end, } | def | category_theory.monoidal_single_obj.End_monoidal_star_functor | category_theory.bicategory | src/category_theory/bicategory/single_obj.lean | [
"category_theory.bicategory.End",
"category_theory.monoidal.functor"
] | [] | The monoidal functor from the endomorphisms of the single object
when we promote a monoidal category to a single object bicategory,
to the original monoidal category.
We subsequently show this is an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
End_monoidal_star_functor_is_equivalence :
is_equivalence (End_monoidal_star_functor C).to_functor | { inverse :=
{ obj := λ X, X,
map := λ X Y f, f, },
unit_iso := nat_iso.of_components (λ X, as_iso (𝟙 _)) (by tidy),
counit_iso := nat_iso.of_components (λ X, as_iso (𝟙 _)) (by tidy), } | def | category_theory.monoidal_single_obj.End_monoidal_star_functor_is_equivalence | category_theory.bicategory | src/category_theory/bicategory/single_obj.lean | [
"category_theory.bicategory.End",
"category_theory.monoidal.functor"
] | [] | The equivalence between the endomorphisms of the single object
when we promote a monoidal category to a single object bicategory,
and the original monoidal category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bicategory.strict : Prop | (id_comp' : ∀ {a b : B} (f : a ⟶ b), 𝟙 a ≫ f = f . obviously)
(comp_id' : ∀ {a b : B} (f : a ⟶ b), f ≫ 𝟙 b = f . obviously)
(assoc' : ∀ {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d),
(f ≫ g) ≫ h = f ≫ (g ≫ h) . obviously)
(left_unitor_eq_to_iso' : ∀ {a b : B} (f : a ⟶ b),
λ_ f = eq_to_iso (id_comp' f) . obvio... | class | category_theory.bicategory.strict | category_theory.bicategory | src/category_theory/bicategory/strict.lean | [
"category_theory.eq_to_hom",
"category_theory.bicategory.basic"
] | [] | A bicategory is called `strict` if the left unitors, the right unitors, and the associators are
isomorphisms given by equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_bicategory.category [bicategory.strict B] : category B | { id_comp' := λ a b, bicategory.strict.id_comp,
comp_id' := λ a b, bicategory.strict.comp_id,
assoc' := λ a b c d, bicategory.strict.assoc } | instance | category_theory.strict_bicategory.category | category_theory.bicategory | src/category_theory/bicategory/strict.lean | [
"category_theory.eq_to_hom",
"category_theory.bicategory.basic"
] | [] | Category structure on a strict bicategory | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_left_eq_to_hom {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g = h) :
f ◁ eq_to_hom η = eq_to_hom (congr_arg2 (≫) rfl η) | by { cases η, simp only [whisker_left_id, eq_to_hom_refl] } | lemma | category_theory.bicategory.whisker_left_eq_to_hom | category_theory.bicategory | src/category_theory/bicategory/strict.lean | [
"category_theory.eq_to_hom",
"category_theory.bicategory.basic"
] | [
"congr_arg2"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_hom_whisker_right {a b c : B} {f g : a ⟶ b} (η : f = g) (h : b ⟶ c) :
eq_to_hom η ▷ h = eq_to_hom (congr_arg2 (≫) η rfl) | by { cases η, simp only [id_whisker_right, eq_to_hom_refl] } | lemma | category_theory.bicategory.eq_to_hom_whisker_right | category_theory.bicategory | src/category_theory/bicategory/strict.lean | [
"category_theory.eq_to_hom",
"category_theory.bicategory.basic"
] | [
"congr_arg2"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
category_struct (obj : Type u)
extends quiver.{v+1} obj : Type (max u (v+1)) | (id : Π X : obj, hom X X)
(comp : Π {X Y Z : obj}, (X ⟶ Y) → (Y ⟶ Z) → (X ⟶ Z)) | class | category_theory.category_struct | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | A preliminary structure on the way to defining a category,
containing the data, but none of the axioms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category (obj : Type u)
extends category_struct.{v} obj : Type (max u (v+1)) | (id_comp' : ∀ {X Y : obj} (f : hom X Y), 𝟙 X ≫ f = f . obviously)
(comp_id' : ∀ {X Y : obj} (f : hom X Y), f ≫ 𝟙 Y = f . obviously)
(assoc' : ∀ {W X Y Z : obj} (f : hom W X) (g : hom X Y) (h : hom Y Z),
(f ≫ g) ≫ h = f ≫ (g ≫ h) . obviously) | class | category_theory.category | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | The typeclass `category C` describes morphisms associated to objects of type `C`.
The universe levels of the objects and morphisms are unconstrained, and will often need to be
specified explicitly, as `category.{v} C`. (See also `large_category` and `small_category`.)
See <https://stacks.math.columbia.edu/tag/0014>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
large_category (C : Type (u+1)) : Type (u+1) | category.{u} C | abbreviation | category_theory.large_category | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | A `large_category` has objects in one universe level higher than the universe level of
the morphisms. It is useful for examples such as the category of types, or the category
of groups, etc. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
small_category (C : Type u) : Type (u+1) | category.{u} C | abbreviation | category_theory.small_category | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | A `small_category` has objects and morphisms in the same universe level. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_whisker {f g : X ⟶ Y} (w : f = g) (h : Y ⟶ Z) : f ≫ h = g ≫ h | by rw w | lemma | category_theory.eq_whisker | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | postcompose an equation between morphisms by another morphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_eq (f : X ⟶ Y) {g h : Y ⟶ Z} (w : g = h) : f ≫ g = f ≫ h | by rw w | lemma | category_theory.whisker_eq | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | precompose an equation between morphisms by another morphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_comp_left_eq {f g : X ⟶ Y} (w : ∀ {Z : C} (h : Y ⟶ Z), f ≫ h = g ≫ h) : f = g | by { convert w (𝟙 Y), tidy } | lemma | category_theory.eq_of_comp_left_eq | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_comp_right_eq {f g : Y ⟶ Z} (w : ∀ {X : C} (h : X ⟶ Y), h ≫ f = h ≫ g) : f = g | by { convert w (𝟙 Y), tidy } | lemma | category_theory.eq_of_comp_right_eq | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_comp_left_eq' (f g : X ⟶ Y)
(w : (λ {Z : C} (h : Y ⟶ Z), f ≫ h) = (λ {Z : C} (h : Y ⟶ Z), g ≫ h)) : f = g | eq_of_comp_left_eq (λ Z h, by convert congr_fun (congr_fun w Z) h) | lemma | category_theory.eq_of_comp_left_eq' | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_comp_right_eq' (f g : Y ⟶ Z)
(w : (λ {X : C} (h : X ⟶ Y), h ≫ f) = (λ {X : C} (h : X ⟶ Y), h ≫ g)) : f = g | eq_of_comp_right_eq (λ X h, by convert congr_fun (congr_fun w X) h) | lemma | category_theory.eq_of_comp_right_eq' | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_of_comp_left_id (f : X ⟶ X) (w : ∀ {Y : C} (g : X ⟶ Y), f ≫ g = g) : f = 𝟙 X | by { convert w (𝟙 X), tidy } | lemma | category_theory.id_of_comp_left_id | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_of_comp_right_id (f : X ⟶ X) (w : ∀ {Y : C} (g : Y ⟶ X), g ≫ f = g) : f = 𝟙 X | by { convert w (𝟙 X), tidy } | lemma | category_theory.id_of_comp_right_id | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_ite {P : Prop} [decidable P]
{X Y Z : C} (f : X ⟶ Y) (g g' : (Y ⟶ Z)) :
(f ≫ if P then g else g') = (if P then f ≫ g else f ≫ g') | by { split_ifs; refl } | lemma | category_theory.comp_ite | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ite_comp {P : Prop} [decidable P]
{X Y Z : C} (f f' : (X ⟶ Y)) (g : Y ⟶ Z) :
(if P then f else f') ≫ g = (if P then f ≫ g else f' ≫ g) | by { split_ifs; refl } | lemma | category_theory.ite_comp | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_dite {P : Prop} [decidable P]
{X Y Z : C} (f : X ⟶ Y) (g : P → (Y ⟶ Z)) (g' : ¬P → (Y ⟶ Z)) :
(f ≫ if h : P then g h else g' h) = (if h : P then f ≫ g h else f ≫ g' h) | by { split_ifs; refl } | lemma | category_theory.comp_dite | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dite_comp {P : Prop} [decidable P]
{X Y Z : C} (f : P → (X ⟶ Y)) (f' : ¬P → (X ⟶ Y)) (g : Y ⟶ Z) :
(if h : P then f h else f' h) ≫ g = (if h : P then f h ≫ g else f' h ≫ g) | by { split_ifs; refl } | lemma | category_theory.dite_comp | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi (f : X ⟶ Y) : Prop | (left_cancellation : Π {Z : C} (g h : Y ⟶ Z) (w : f ≫ g = f ≫ h), g = h) | class | category_theory.epi | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | A morphism `f` is an epimorphism if it can be "cancelled" when precomposed:
`f ≫ g = f ≫ h` implies `g = h`.
See <https://stacks.math.columbia.edu/tag/003B>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono (f : X ⟶ Y) : Prop | (right_cancellation : Π {Z : C} (g h : Z ⟶ X) (w : g ≫ f = h ≫ f), g = h) | class | category_theory.mono | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | A morphism `f` is a monomorphism if it can be "cancelled" when postcomposed:
`g ≫ f = h ≫ f` implies `g = h`.
See <https://stacks.math.columbia.edu/tag/003B>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cancel_epi (f : X ⟶ Y) [epi f] {g h : Y ⟶ Z} : (f ≫ g = f ≫ h) ↔ g = h | ⟨λ p, epi.left_cancellation g h p, congr_arg _⟩ | lemma | category_theory.cancel_epi | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_mono (f : X ⟶ Y) [mono f] {g h : Z ⟶ X} : (g ≫ f = h ≫ f) ↔ g = h | ⟨λ p, mono.right_cancellation g h p, congr_arg _⟩ | lemma | category_theory.cancel_mono | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_epi_id (f : X ⟶ Y) [epi f] {h : Y ⟶ Y} : (f ≫ h = f) ↔ h = 𝟙 Y | by { convert cancel_epi f, simp, } | lemma | category_theory.cancel_epi_id | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_mono_id (f : X ⟶ Y) [mono f] {g : X ⟶ X} : (g ≫ f = f) ↔ g = 𝟙 X | by { convert cancel_mono f, simp, } | lemma | category_theory.cancel_mono_id | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_comp {X Y Z : C} (f : X ⟶ Y) [epi f] (g : Y ⟶ Z) [epi g] : epi (f ≫ g) | begin
split, intros Z a b w,
apply (cancel_epi g).1,
apply (cancel_epi f).1,
simpa using w,
end | lemma | category_theory.epi_comp | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_comp {X Y Z : C} (f : X ⟶ Y) [mono f] (g : Y ⟶ Z) [mono g] : mono (f ≫ g) | begin
split, intros Z a b w,
apply (cancel_mono f).1,
apply (cancel_mono g).1,
simpa using w,
end | lemma | category_theory.mono_comp | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_of_mono {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [mono (f ≫ g)] : mono f | begin
split, intros Z a b w,
replace w := congr_arg (λ k, k ≫ g) w,
dsimp at w,
rw [category.assoc, category.assoc] at w,
exact (cancel_mono _).1 w,
end | lemma | category_theory.mono_of_mono | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_of_mono_fac {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} {h : X ⟶ Z} [mono h] (w : f ≫ g = h) :
mono f | by { substI h, exact mono_of_mono f g, } | lemma | category_theory.mono_of_mono_fac | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_of_epi {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [epi (f ≫ g)] : epi g | begin
split, intros Z a b w,
replace w := congr_arg (λ k, f ≫ k) w,
dsimp at w,
rw [←category.assoc, ←category.assoc] at w,
exact (cancel_epi _).1 w,
end | lemma | category_theory.epi_of_epi | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_of_epi_fac {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} {h : X ⟶ Z} [epi h] (w : f ≫ g = h) :
epi g | by substI h; exact epi_of_epi f g | lemma | category_theory.epi_of_epi_fac | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift_category : category.{v} (ulift.{u'} C) | { hom := λ X Y, (X.down ⟶ Y.down),
id := λ X, 𝟙 X.down,
comp := λ _ _ _ f g, f ≫ g } | instance | category_theory.ulift_category | category_theory.category | src/category_theory/category/basic.lean | [
"combinatorics.quiver.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Bipointed : Type.{u + 1} | (X : Type.{u})
(to_prod : X × X) | structure | Bipointed | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [] | The category of bipointed types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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