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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associator_inv_naturality_middle (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d) :
f ◁ (η ▷ h) ≫ (α_ f g' h).inv = (α_ f g h).inv ≫ (f ◁ η) ▷ h | by simp | lemma | category_theory.bicategory.associator_inv_naturality_middle | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_assoc_symm (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d) :
f ◁ (η ▷ h) = (α_ f g h).inv ≫ (f ◁ η) ▷ h ≫ (α_ f g' h).hom | by simp | lemma | category_theory.bicategory.whisker_assoc_symm | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associator_naturality_right (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h') :
(f ≫ g) ◁ η ≫ (α_ f g h').hom = (α_ f g h).hom ≫ f ◁ (g ◁ η) | by simp | lemma | category_theory.bicategory.associator_naturality_right | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associator_inv_naturality_right (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h') :
f ◁ (g ◁ η) ≫ (α_ f g h').inv = (α_ f g h).inv ≫ (f ≫ g) ◁ η | by simp | lemma | category_theory.bicategory.associator_inv_naturality_right | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_whisker_left_symm (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h') :
f ◁ (g ◁ η) = (α_ f g h).inv ≫ (f ≫ g) ◁ η ≫ (α_ f g h').hom | by simp | lemma | category_theory.bicategory.comp_whisker_left_symm | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_unitor_naturality {f g : a ⟶ b} (η : f ⟶ g) :
𝟙 a ◁ η ≫ (λ_ g).hom = (λ_ f).hom ≫ η | by simp | lemma | category_theory.bicategory.left_unitor_naturality | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_unitor_inv_naturality {f g : a ⟶ b} (η : f ⟶ g) :
η ≫ (λ_ g).inv = (λ_ f).inv ≫ 𝟙 a ◁ η | by simp | lemma | category_theory.bicategory.left_unitor_inv_naturality | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_whisker_left_symm {f g : a ⟶ b} (η : f ⟶ g) :
η = (λ_ f).inv ≫ 𝟙 a ◁ η ≫ (λ_ g).hom | by simp | lemma | category_theory.bicategory.id_whisker_left_symm | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_unitor_naturality {f g : a ⟶ b} (η : f ⟶ g) :
η ▷ 𝟙 b ≫ (ρ_ g).hom = (ρ_ f).hom ≫ η | by simp | lemma | category_theory.bicategory.right_unitor_naturality | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_unitor_inv_naturality {f g : a ⟶ b} (η : f ⟶ g) :
η ≫ (ρ_ g).inv = (ρ_ f).inv ≫ η ▷ 𝟙 b | by simp | lemma | category_theory.bicategory.right_unitor_inv_naturality | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right_id_symm {f g : a ⟶ b} (η : f ⟶ g) :
η = (ρ_ f).inv ≫ η ▷ 𝟙 b ≫ (ρ_ g).hom | by simp | lemma | category_theory.bicategory.whisker_right_id_symm | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_left_iff {f g : a ⟶ b} (η θ : f ⟶ g) :
(𝟙 a ◁ η = 𝟙 a ◁ θ) ↔ (η = θ) | by simp | lemma | category_theory.bicategory.whisker_left_iff | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right_iff {f g : a ⟶ b} (η θ : f ⟶ g) :
(η ▷ 𝟙 b = θ ▷ 𝟙 b) ↔ (η = θ) | by simp | lemma | category_theory.bicategory.whisker_right_iff | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_unitor_whisker_right (f : a ⟶ b) (g : b ⟶ c) :
(λ_ f).hom ▷ g = (α_ (𝟙 a) f g).hom ≫ (λ_ (f ≫ g)).hom | by rw [←whisker_left_iff, whisker_left_comp, ←cancel_epi (α_ _ _ _).hom,
←cancel_epi ((α_ _ _ _).hom ▷ _), pentagon_assoc, triangle,
←associator_naturality_middle, ←comp_whisker_right_assoc, triangle,
associator_naturality_left]; apply_instance | lemma | category_theory.bicategory.left_unitor_whisker_right | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | We state it as a simp lemma, which is regarded as an involved version of
`id_whisker_right f g : 𝟙 f ▷ g = 𝟙 (f ≫ g)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_unitor_inv_whisker_right (f : a ⟶ b) (g : b ⟶ c) :
(λ_ f).inv ▷ g = (λ_ (f ≫ g)).inv ≫ (α_ (𝟙 a) f g).inv | eq_of_inv_eq_inv (by simp) | lemma | category_theory.bicategory.left_unitor_inv_whisker_right | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_left_right_unitor (f : a ⟶ b) (g : b ⟶ c) :
f ◁ (ρ_ g).hom = (α_ f g (𝟙 c)).inv ≫ (ρ_ (f ≫ g)).hom | by rw [←whisker_right_iff, comp_whisker_right, ←cancel_epi (α_ _ _ _).inv,
←cancel_epi (f ◁ (α_ _ _ _).inv), pentagon_inv_assoc, triangle_assoc_comp_right,
←associator_inv_naturality_middle, ←whisker_left_comp_assoc, triangle_assoc_comp_right,
associator_inv_naturality_right]; apply_instance | lemma | category_theory.bicategory.whisker_left_right_unitor | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_left_right_unitor_inv (f : a ⟶ b) (g : b ⟶ c) :
f ◁ (ρ_ g).inv = (ρ_ (f ≫ g)).inv ≫ (α_ f g (𝟙 c)).hom | eq_of_inv_eq_inv (by simp) | lemma | category_theory.bicategory.whisker_left_right_unitor_inv | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_unitor_comp (f : a ⟶ b) (g : b ⟶ c) :
(λ_ (f ≫ g)).hom = (α_ (𝟙 a) f g).inv ≫ (λ_ f).hom ▷ g | by simp | lemma | category_theory.bicategory.left_unitor_comp | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_unitor_comp_inv (f : a ⟶ b) (g : b ⟶ c) :
(λ_ (f ≫ g)).inv = (λ_ f).inv ▷ g ≫ (α_ (𝟙 a) f g).hom | by simp | lemma | category_theory.bicategory.left_unitor_comp_inv | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_unitor_comp (f : a ⟶ b) (g : b ⟶ c) :
(ρ_ (f ≫ g)).hom = (α_ f g (𝟙 c)).hom ≫ f ◁ (ρ_ g).hom | by simp | lemma | category_theory.bicategory.right_unitor_comp | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_unitor_comp_inv (f : a ⟶ b) (g : b ⟶ c) :
(ρ_ (f ≫ g)).inv = f ◁ (ρ_ g).inv ≫ (α_ f g (𝟙 c)).inv | by simp | lemma | category_theory.bicategory.right_unitor_comp_inv | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unitors_equal : (λ_ (𝟙 a)).hom = (ρ_ (𝟙 a)).hom | by rw [←whisker_left_iff, ←cancel_epi (α_ _ _ _).hom, ←cancel_mono (ρ_ _).hom, triangle,
←right_unitor_comp, right_unitor_naturality]; apply_instance | lemma | category_theory.bicategory.unitors_equal | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unitors_inv_equal : (λ_ (𝟙 a)).inv = (ρ_ (𝟙 a)).inv | by simp [iso.inv_eq_inv] | lemma | category_theory.bicategory.unitors_inv_equal | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_path_aux {a : B} : ∀ {b : B}, path a b → hom a b | | _ nil := hom.id a
| _ (cons p f) := (inclusion_path_aux p).comp (hom.of f) | def | category_theory.free_bicategory.inclusion_path_aux | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [
"path"
] | Auxiliary definition for `inclusion_path`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion_path (a b : B) : discrete (path.{v+1} a b) ⥤ hom a b | discrete.functor inclusion_path_aux | def | category_theory.free_bicategory.inclusion_path | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [] | The discrete category on the paths includes into the category of 1-morphisms in the free
bicategory. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preinclusion (B : Type u) [quiver.{v+1} B] :
prelax_functor (locally_discrete (paths B)) (free_bicategory B) | { obj := id,
map := λ a b, (inclusion_path a b).obj,
map₂ := λ a b f g η, (inclusion_path a b).map η } | def | category_theory.free_bicategory.preinclusion | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [] | The inclusion from the locally discrete bicategory on the path category into the free bicategory
as a prelax functor. This will be promoted to a pseudofunctor after proving the coherence theorem.
See `inclusion`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preinclusion_obj (a : B) :
(preinclusion B).obj a = a | rfl | lemma | category_theory.free_bicategory.preinclusion_obj | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preinclusion_map₂ {a b : B} (f g : discrete (path.{v+1} a b)) (η : f ⟶ g) :
(preinclusion B).map₂ η = eq_to_hom (congr_arg _ (discrete.ext _ _ (discrete.eq_of_hom η))) | begin
rcases η with ⟨⟨⟩⟩,
cases discrete.ext _ _ η,
exact (inclusion_path a b).map_id _
end | lemma | category_theory.free_bicategory.preinclusion_map₂ | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_aux {a : B} : ∀ {b c : B}, path a b → hom b c → path a c | | _ _ p (hom.of f) := p.cons f
| _ _ p (hom.id b) := p
| _ _ p (hom.comp f g) := normalize_aux (normalize_aux p f) g | def | category_theory.free_bicategory.normalize_aux | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [
"path"
] | The normalization of the composition of `p : path a b` and `f : hom b c`.
`p` will eventually be taken to be `nil` and we then get the normalization
of `f` alone, but the auxiliary `p` is necessary for Lean to accept the definition of
`normalize_iso` and the `whisker_left` case of `normalize_aux_congr` and `normalize_n... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalize_iso {a : B} : ∀ {b c : B} (p : path a b) (f : hom b c),
(preinclusion B).map ⟨p⟩ ≫ f ≅ (preinclusion B).map ⟨normalize_aux p f⟩ | | _ _ p (hom.of f) := iso.refl _
| _ _ p (hom.id b) := ρ_ _
| _ _ p (hom.comp f g) := (α_ _ _ _).symm ≪≫
whisker_right_iso (normalize_iso p f) g ≪≫ normalize_iso (normalize_aux p f) g | def | category_theory.free_bicategory.normalize_iso | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [
"path"
] | A 2-isomorphism between a partially-normalized 1-morphism in the free bicategory to the
fully-normalized 1-morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalize_aux_congr {a b c : B} (p : path a b) {f g : hom b c} (η : f ⟶ g) :
normalize_aux p f = normalize_aux p g | begin
rcases η,
apply @congr_fun _ _ (λ p, normalize_aux p f),
clear p,
induction η,
case vcomp { apply eq.trans; assumption },
/- p ≠ nil required! See the docstring of `normalize_aux`. -/
case whisker_left : _ _ _ _ _ _ _ ih { funext, apply congr_fun ih },
case whisker_right : _ _ _ _ _ _ _ ih { fune... | lemma | category_theory.free_bicategory.normalize_aux_congr | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [
"congr_arg2",
"ih",
"path"
] | Given a 2-morphism between `f` and `g` in the free bicategory, we have the equality
`normalize_aux p f = normalize_aux p g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalize_naturality {a b c : B} (p : path a b) {f g : hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalize_iso p g).hom =
(normalize_iso p f).hom ≫
(preinclusion B).map₂ (eq_to_hom (discrete.ext _ _ (normalize_aux_congr p η))) | begin
rcases η, induction η,
case id : { simp },
case vcomp : _ _ _ _ _ _ _ ihf ihg
{ rw [mk_vcomp, bicategory.whisker_left_comp],
slice_lhs 2 3 { rw ihg },
slice_lhs 1 2 { rw ihf },
simp },
case whisker_left : _ _ _ _ _ _ _ ih
/- p ≠ nil required! See the docstring of `normalize_aux`. -/
{ ds... | lemma | category_theory.free_bicategory.normalize_naturality | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [
"ih",
"path"
] | The 2-isomorphism `normalize_iso p f` is natural in `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalize_aux_nil_comp {a b c : B} (f : hom a b) (g : hom b c) :
normalize_aux nil (f.comp g) = (normalize_aux nil f).comp (normalize_aux nil g) | begin
induction g generalizing a,
case id { refl },
case of { refl },
case comp : _ _ _ g _ ihf ihg { erw [ihg (f.comp g), ihf f, ihg g, comp_assoc] }
end | lemma | category_theory.free_bicategory.normalize_aux_nil_comp | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize (B : Type u) [quiver.{v+1} B] :
pseudofunctor (free_bicategory B) (locally_discrete (paths B)) | { obj := id,
map := λ a b f, ⟨normalize_aux nil f⟩,
map₂ := λ a b f g η, eq_to_hom $ discrete.ext _ _ $ normalize_aux_congr nil η,
map_id := λ a, eq_to_iso $ discrete.ext _ _ rfl,
map_comp := λ a b c f g, eq_to_iso $ discrete.ext _ _ $ normalize_aux_nil_comp f g } | def | category_theory.free_bicategory.normalize | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [
"map_comp",
"map_id",
"normalize"
] | The normalization pseudofunctor for the free bicategory on a quiver `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalize_unit_iso (a b : free_bicategory B) :
𝟭 (a ⟶ b) ≅ (normalize B).map_functor a b ⋙ inclusion_path a b | nat_iso.of_components (λ f, (λ_ f).symm ≪≫ normalize_iso nil f)
begin
intros f g η,
erw [left_unitor_inv_naturality_assoc, assoc],
congr' 1,
exact normalize_naturality nil η
end | def | category_theory.free_bicategory.normalize_unit_iso | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [
"normalize"
] | Auxiliary definition for `normalize_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalize_equiv (a b : B) : hom a b ≌ discrete (path.{v+1} a b) | equivalence.mk ((normalize _).map_functor a b) (inclusion_path a b)
(normalize_unit_iso a b)
(discrete.nat_iso (λ f, eq_to_iso (by { induction f; induction f; tidy }))) | def | category_theory.free_bicategory.normalize_equiv | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [
"normalize"
] | Normalization as an equivalence of categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_thin {a b : free_bicategory B} : quiver.is_thin (a ⟶ b) | λ _ _, ⟨λ η θ, (normalize_equiv a b).functor.map_injective (subsingleton.elim _ _)⟩ | instance | category_theory.free_bicategory.locally_thin | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [
"quiver.is_thin"
] | The coherence theorem for bicategories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion_map_comp_aux {a b : B} : ∀ {c : B} (f : path a b) (g : path b c),
(preinclusion _).map (⟨f⟩ ≫ ⟨g⟩) ≅ (preinclusion _).map ⟨f⟩ ≫ (preinclusion _).map ⟨g⟩ | | _ f nil := (ρ_ ((preinclusion _).map ⟨f⟩)).symm
| _ f (cons g₁ g₂) := whisker_right_iso (inclusion_map_comp_aux f g₁) (hom.of g₂) ≪≫ α_ _ _ _ | def | category_theory.free_bicategory.inclusion_map_comp_aux | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [
"path"
] | Auxiliary definition for `inclusion`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion (B : Type u) [quiver.{v+1} B] :
pseudofunctor (locally_discrete (paths B)) (free_bicategory B) | { map_id := λ a, iso.refl (𝟙 a),
map_comp := λ a b c f g, inclusion_map_comp_aux f.as g.as,
-- All the conditions for 2-morphisms are trivial thanks to the coherence theorem!
.. preinclusion B } | def | category_theory.free_bicategory.inclusion | category_theory.bicategory | src/category_theory/bicategory/coherence.lean | [
"category_theory.path_category",
"category_theory.functor.fully_faithful",
"category_theory.bicategory.free",
"category_theory.bicategory.locally_discrete"
] | [
"map_comp",
"map_id"
] | The inclusion pseudofunctor from the locally discrete bicategory on the path category into the
free bicategory. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_hom {a b : B} (f : a ⟶ b) | (lift : of.obj a ⟶ of.obj b) | class | category_theory.bicategory.lift_hom | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [
"lift"
] | A typeclass carrying a choice of lift of a 1-morphism from `B` to `free_bicategory B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_hom_id : lift_hom (𝟙 a) | { lift := 𝟙 (of.obj a), } | instance | category_theory.bicategory.lift_hom_id | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom_comp (f : a ⟶ b) (g : b ⟶ c) [lift_hom f] [lift_hom g] : lift_hom (f ≫ g) | { lift := lift_hom.lift f ≫ lift_hom.lift g, } | instance | category_theory.bicategory.lift_hom_comp | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom_of (f : a ⟶ b) : lift_hom f | { lift := of.map f, } | instance | category_theory.bicategory.lift_hom_of | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom₂ {f g : a ⟶ b} [lift_hom f] [lift_hom g] (η : f ⟶ g) | (lift : lift_hom.lift f ⟶ lift_hom.lift g) | class | category_theory.bicategory.lift_hom₂ | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [
"lift"
] | A typeclass carrying a choice of lift of a 2-morphism from `B` to `free_bicategory B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_hom₂_id (f : a ⟶ b) [lift_hom f] : lift_hom₂ (𝟙 f) | { lift := 𝟙 _, } | instance | category_theory.bicategory.lift_hom₂_id | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom₂_left_unitor_hom (f : a ⟶ b) [lift_hom f] : lift_hom₂ (λ_ f).hom | { lift := (λ_ (lift_hom.lift f)).hom, } | instance | category_theory.bicategory.lift_hom₂_left_unitor_hom | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom₂_left_unitor_inv (f : a ⟶ b) [lift_hom f] : lift_hom₂ (λ_ f).inv | { lift := (λ_ (lift_hom.lift f)).inv, } | instance | category_theory.bicategory.lift_hom₂_left_unitor_inv | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom₂_right_unitor_hom (f : a ⟶ b) [lift_hom f] : lift_hom₂ (ρ_ f).hom | { lift := (ρ_ (lift_hom.lift f)).hom, } | instance | category_theory.bicategory.lift_hom₂_right_unitor_hom | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom₂_right_unitor_inv (f : a ⟶ b) [lift_hom f] : lift_hom₂ (ρ_ f).inv | { lift := (ρ_ (lift_hom.lift f)).inv, } | instance | category_theory.bicategory.lift_hom₂_right_unitor_inv | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom₂_associator_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d)
[lift_hom f] [lift_hom g] [lift_hom h] :
lift_hom₂ (α_ f g h).hom | { lift := (α_ (lift_hom.lift f) (lift_hom.lift g) (lift_hom.lift h)).hom, } | instance | category_theory.bicategory.lift_hom₂_associator_hom | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom₂_associator_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d)
[lift_hom f] [lift_hom g] [lift_hom h] :
lift_hom₂ (α_ f g h).inv | { lift := (α_ (lift_hom.lift f) (lift_hom.lift g) (lift_hom.lift h)).inv, } | instance | category_theory.bicategory.lift_hom₂_associator_inv | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom₂_comp {f g h : a ⟶ b} [lift_hom f] [lift_hom g] [lift_hom h]
(η : f ⟶ g) (θ : g ⟶ h) [lift_hom₂ η] [lift_hom₂ θ] : lift_hom₂ (η ≫ θ) | { lift := lift_hom₂.lift η ≫ lift_hom₂.lift θ } | instance | category_theory.bicategory.lift_hom₂_comp | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom₂_whisker_left (f : a ⟶ b) [lift_hom f] {g h : b ⟶ c} (η : g ⟶ h)
[lift_hom g] [lift_hom h] [lift_hom₂ η] : lift_hom₂ (f ◁ η) | { lift := lift_hom.lift f ◁ lift_hom₂.lift η } | instance | category_theory.bicategory.lift_hom₂_whisker_left | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom₂_whisker_right {f g : a ⟶ b} (η : f ⟶ g) [lift_hom f] [lift_hom g] [lift_hom₂ η]
{h : b ⟶ c} [lift_hom h] : lift_hom₂ (η ▷ h) | { lift := lift_hom₂.lift η ▷ lift_hom.lift h } | instance | category_theory.bicategory.lift_hom₂_whisker_right | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bicategorical_coherence (f g : a ⟶ b) [lift_hom f] [lift_hom g] | (hom [] : f ⟶ g)
[is_iso : is_iso hom . tactic.apply_instance] | class | category_theory.bicategory.bicategorical_coherence | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | A typeclass carrying a choice of bicategorical structural isomorphism between two objects.
Used by the `⊗≫` bicategorical composition operator, and the `coherence` tactic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl (f : a ⟶ b) [lift_hom f] : bicategorical_coherence f f | ⟨𝟙 _⟩ | instance | category_theory.bicategory.bicategorical_coherence.refl | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_left
(f : a ⟶ b) (g h : b ⟶ c) [lift_hom f][lift_hom g] [lift_hom h] [bicategorical_coherence g h] :
bicategorical_coherence (f ≫ g) (f ≫ h) | ⟨f ◁ bicategorical_coherence.hom g h⟩ | instance | category_theory.bicategory.bicategorical_coherence.whisker_left | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right
(f g : a ⟶ b) (h : b ⟶ c) [lift_hom f] [lift_hom g] [lift_hom h] [bicategorical_coherence f g] :
bicategorical_coherence (f ≫ h) (g ≫ h) | ⟨bicategorical_coherence.hom f g ▷ h⟩ | instance | category_theory.bicategory.bicategorical_coherence.whisker_right | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tensor_right (f : a ⟶ b) (g : b ⟶ b) [lift_hom f] [lift_hom g]
[bicategorical_coherence (𝟙 b) g] :
bicategorical_coherence f (f ≫ g) | ⟨(ρ_ f).inv ≫ (f ◁ bicategorical_coherence.hom (𝟙 b) g)⟩ | instance | category_theory.bicategory.bicategorical_coherence.tensor_right | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tensor_right' (f : a ⟶ b) (g : b ⟶ b) [lift_hom f] [lift_hom g]
[bicategorical_coherence g (𝟙 b)] :
bicategorical_coherence (f ≫ g) f | ⟨(f ◁ bicategorical_coherence.hom g (𝟙 b)) ≫ (ρ_ f).hom⟩ | instance | category_theory.bicategory.bicategorical_coherence.tensor_right' | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left (f g : a ⟶ b) [lift_hom f] [lift_hom g] [bicategorical_coherence f g] :
bicategorical_coherence (𝟙 a ≫ f) g | ⟨(λ_ f).hom ≫ bicategorical_coherence.hom f g⟩ | instance | category_theory.bicategory.bicategorical_coherence.left | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left' (f g : a ⟶ b) [lift_hom f] [lift_hom g] [bicategorical_coherence f g] :
bicategorical_coherence f (𝟙 a ≫ g) | ⟨bicategorical_coherence.hom f g ≫ (λ_ g).inv⟩ | instance | category_theory.bicategory.bicategorical_coherence.left' | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right (f g : a ⟶ b) [lift_hom f] [lift_hom g] [bicategorical_coherence f g] :
bicategorical_coherence (f ≫ 𝟙 b) g | ⟨(ρ_ f).hom ≫ bicategorical_coherence.hom f g⟩ | instance | category_theory.bicategory.bicategorical_coherence.right | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right' (f g : a ⟶ b) [lift_hom f] [lift_hom g] [bicategorical_coherence f g] :
bicategorical_coherence f (g ≫ 𝟙 b) | ⟨bicategorical_coherence.hom f g ≫ (ρ_ g).inv⟩ | instance | category_theory.bicategory.bicategorical_coherence.right' | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
assoc (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : a ⟶ d)
[lift_hom f] [lift_hom g] [lift_hom h] [lift_hom i] [bicategorical_coherence (f ≫ (g ≫ h)) i] :
bicategorical_coherence ((f ≫ g) ≫ h) i | ⟨(α_ f g h).hom ≫ bicategorical_coherence.hom (f ≫ (g ≫ h)) i⟩ | instance | category_theory.bicategory.bicategorical_coherence.assoc | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
assoc' (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : a ⟶ d)
[lift_hom f] [lift_hom g] [lift_hom h] [lift_hom i] [bicategorical_coherence i (f ≫ (g ≫ h))] :
bicategorical_coherence i ((f ≫ g) ≫ h) | ⟨bicategorical_coherence.hom i (f ≫ (g ≫ h)) ≫ (α_ f g h).inv⟩ | instance | category_theory.bicategory.bicategorical_coherence.assoc' | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bicategorical_iso (f g : a ⟶ b) [lift_hom f] [lift_hom g] [bicategorical_coherence f g] :
f ≅ g | as_iso (bicategorical_coherence.hom f g) | def | category_theory.bicategory.bicategorical_iso | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | Construct an isomorphism between two objects in a bicategorical category
out of unitors and associators. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bicategorical_comp {f g h i : a ⟶ b} [lift_hom g] [lift_hom h]
[bicategorical_coherence g h] (η : f ⟶ g) (θ : h ⟶ i) : f ⟶ i | η ≫ bicategorical_coherence.hom g h ≫ θ | def | category_theory.bicategory.bicategorical_comp | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | Compose two morphisms in a bicategorical category,
inserting unitors and associators between as necessary. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bicategorical_iso_comp {f g h i : a ⟶ b} [lift_hom g] [lift_hom h]
[bicategorical_coherence g h] (η : f ≅ g) (θ : h ≅ i) : f ≅ i | η ≪≫ as_iso (bicategorical_coherence.hom g h) ≪≫ θ | def | category_theory.bicategory.bicategorical_iso_comp | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | Compose two isomorphisms in a bicategorical category,
inserting unitors and associators between as necessary. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bicategorical_comp_refl {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) :
η ⊗≫ θ = η ≫ θ | by { dsimp [bicategorical_comp], simp, } | lemma | category_theory.bicategory.bicategorical_comp_refl | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bicategorical_coherence : tactic unit | focus1 $
do
o ← get_options, set_options $ o.set_nat `class.instance_max_depth 128,
try `[dsimp],
`(%%lhs = %%rhs) ← target,
to_expr ``((free_bicategory.lift (prefunctor.id _)).map₂ (lift_hom₂.lift %%lhs) =
(free_bicategory.lift (prefunctor.id _)).map₂ (lift_hom₂.lift %%rhs))
>>= tactic.change,
congr | def | tactic.bicategorical_coherence | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | Coherence tactic for bicategories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_simps : tactic unit | `[simp only [
category_theory.category.assoc,
category_theory.bicategory.comp_whisker_left,
category_theory.bicategory.id_whisker_left,
category_theory.bicategory.whisker_right_comp,
category_theory.bicategory.whisker_right_id,
category_theory.bicategory.whisker_left_comp,
category_theory.bi... | def | tactic.bicategory.whisker_simps | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | Simp lemmas for rewriting a 2-morphism into a normal form. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
assoc_lift_hom₂ {f g h i : a ⟶ b} [lift_hom f] [lift_hom g] [lift_hom h]
(η : f ⟶ g) (θ : g ⟶ h) (ι : h ⟶ i) [lift_hom₂ η] [lift_hom₂ θ] :
η ≫ (θ ≫ ι) = (η ≫ θ) ≫ ι | (category.assoc _ _ _).symm | lemma | tactic.bicategory.coherence.assoc_lift_hom₂ | category_theory.bicategory | src/category_theory/bicategory/coherence_tactic.lean | [
"category_theory.bicategory.coherence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
End_monoidal (X : C) | X ⟶ X | def | category_theory.End_monoidal | category_theory.bicategory | src/category_theory/bicategory/End.lean | [
"category_theory.bicategory.basic",
"category_theory.monoidal.category"
] | [] | The endomorphisms of an object in a bicategory can be considered as a monoidal category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free_bicategory (B : Type u) | B | def | category_theory.free_bicategory | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | Free bicategory over a quiver. Its objects are the same as those in the underlying quiver. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom : B → B → Type (max u v)
| of {a b : B} (f : a ⟶ b) : hom a b
| id (a : B) : hom a a
| comp {a b c : B} (f : hom a b) (g : hom b c) : hom a c | inductive | category_theory.free_bicategory.hom | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | 1-morphisms in the free bicategory. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom₂ : Π {a b : B}, hom a b → hom a b → Type (max u v)
| id {a b} (f : hom a b) : hom₂ f f
| vcomp {a b} {f g h : hom a b} (η : hom₂ f g) (θ : hom₂ g h) : hom₂ f h
| whisker_left {a b c} (f : hom a b) {g h : hom b c} (η : hom₂ g h) : hom₂ (f.comp g) (f.comp h)
-- `η` cannot be earlier than `h` since it is a recursive a... | inductive | category_theory.free_bicategory.hom₂ | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | Representatives of 2-morphisms in the free bicategory. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel : Π {a b : B} {f g : hom a b}, hom₂ f g → hom₂ f g → Prop
| vcomp_right {a b} {f g h : hom a b} (η : hom₂ f g) (θ₁ θ₂ : hom₂ g h) :
rel θ₁ θ₂ → rel (η ≫ θ₁) (η ≫ θ₂)
| vcomp_left {a b} {f g h : hom a b} (η₁ η₂ : hom₂ f g) (θ : hom₂ g h) :
rel η₁ η₂ → rel (η₁ ≫ θ) (η₂ ≫ θ)
| id_comp {a b} {f g : hom a b} (η ... | inductive | category_theory.free_bicategory.rel | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [
"rel"
] | Relations between 2-morphisms in the free bicategory. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_category (a b : B) : category (hom a b) | { hom := λ f g, quot (@rel _ _ _ _ f g),
id := λ f, quot.mk rel (hom₂.id f),
comp := λ f g h, quot.map₂ hom₂.vcomp rel.vcomp_right rel.vcomp_left,
id_comp' := by { rintros f g ⟨η⟩, exact quot.sound (rel.id_comp η) },
comp_id' := by { rintros f g ⟨η⟩, exact quot.sound (rel.comp_id η) },
ass... | instance | category_theory.free_bicategory.hom_category | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [
"quot.map₂",
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bicategory : bicategory (free_bicategory B) | { hom := λ a b : B, hom a b,
id := hom.id,
comp := λ a b c, hom.comp,
hom_category := free_bicategory.hom_category,
whisker_left := λ a b c f g h η,
quot.map (hom₂.whisker_left f) (rel.whisker_left f g h) η,
whisker_left_id' := λ a b c f g, quot.sound (rel.whisker_left_id f g),
whisker_left_comp' ... | instance | category_theory.free_bicategory.bicategory | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [
"quot.map",
"rel"
] | Bicategory structure on the free bicategory. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_vcomp {f g h : a ⟶ b} (η : hom₂ f g) (θ : hom₂ g h) :
quot.mk rel (η.vcomp θ) = (quot.mk rel η ≫ quot.mk rel θ : f ⟶ h) | rfl | lemma | category_theory.free_bicategory.mk_vcomp | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_whisker_left (f : a ⟶ b) {g h : b ⟶ c} (η : hom₂ g h) :
quot.mk rel (hom₂.whisker_left f η) = (f ◁ quot.mk rel η : f ≫ g ⟶ f ≫ h) | rfl | lemma | category_theory.free_bicategory.mk_whisker_left | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_whisker_right {f g : a ⟶ b} (η : hom₂ f g) (h : b ⟶ c) :
quot.mk rel (hom₂.whisker_right h η) = (quot.mk rel η ▷ h : f ≫ h ⟶ g ≫ h) | rfl | lemma | category_theory.free_bicategory.mk_whisker_right | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_def : hom.id a = 𝟙 a | rfl | lemma | category_theory.free_bicategory.id_def | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_def : hom.comp f g = f ≫ g | rfl | lemma | category_theory.free_bicategory.comp_def | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_id : quot.mk _ (hom₂.id f) = 𝟙 f | rfl | lemma | category_theory.free_bicategory.mk_id | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_associator_hom : quot.mk _ (hom₂.associator f g h) = (α_ f g h).hom | rfl | lemma | category_theory.free_bicategory.mk_associator_hom | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_associator_inv : quot.mk _ (hom₂.associator_inv f g h) = (α_ f g h).inv | rfl | lemma | category_theory.free_bicategory.mk_associator_inv | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_left_unitor_hom : quot.mk _ (hom₂.left_unitor f) = (λ_ f).hom | rfl | lemma | category_theory.free_bicategory.mk_left_unitor_hom | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_left_unitor_inv : quot.mk _ (hom₂.left_unitor_inv f) = (λ_ f).inv | rfl | lemma | category_theory.free_bicategory.mk_left_unitor_inv | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_right_unitor_hom : quot.mk _ (hom₂.right_unitor f) = (ρ_ f).hom | rfl | lemma | category_theory.free_bicategory.mk_right_unitor_hom | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_right_unitor_inv : quot.mk _ (hom₂.right_unitor_inv f) = (ρ_ f).inv | rfl | lemma | category_theory.free_bicategory.mk_right_unitor_inv | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of : prefunctor B (free_bicategory B) | { obj := id,
map := λ a b, hom.of } | def | category_theory.free_bicategory.of | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [
"prefunctor"
] | Canonical prefunctor from `B` to `free_bicategory B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_hom : ∀ {a b : B}, hom a b → (F.obj a ⟶ F.obj b) | | _ _ (hom.of f) := F.map f
| _ _ (hom.id a) := 𝟙 (F.obj a)
| _ _ (hom.comp f g) := lift_hom f ≫ lift_hom g | def | category_theory.free_bicategory.lift_hom | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | Auxiliary definition for `lift`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_hom_id (a : free_bicategory B) : lift_hom F (𝟙 a) = 𝟙 (F.obj a) | rfl | lemma | category_theory.free_bicategory.lift_hom_id | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom_comp {a b c : free_bicategory B} (f : a ⟶ b) (g : b ⟶ c) :
lift_hom F (f ≫ g) = lift_hom F f ≫ lift_hom F g | rfl | lemma | category_theory.free_bicategory.lift_hom_comp | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom₂ : ∀ {a b : B} {f g : hom a b}, hom₂ f g → (lift_hom F f ⟶ lift_hom F g) | | _ _ _ _ (hom₂.id _) := 𝟙 _
| _ _ _ _ (hom₂.associator _ _ _) := (α_ _ _ _).hom
| _ _ _ _ (hom₂.associator_inv _ _ _) := (α_ _ _ _).inv
| _ _ _ _ (hom₂.left_unitor _) := (λ_ _).hom
| _ _ _ _ (hom₂.left_unitor_inv _) := (λ_ _).inv
| _ _ _ _ (hom₂.right_unitor _) := (ρ_ _... | def | category_theory.free_bicategory.lift_hom₂ | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [] | Auxiliary definition for `lift`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_hom₂_congr {a b : B} {f g : hom a b} {η θ : hom₂ f g} (H : rel η θ) :
lift_hom₂ F η = lift_hom₂ F θ | by induction H; tidy | lemma | category_theory.free_bicategory.lift_hom₂_congr | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift : pseudofunctor (free_bicategory B) C | { obj := F.obj,
map := λ a b, lift_hom F,
map₂ := λ a b f g, quot.lift (lift_hom₂ F) (λ η θ H, lift_hom₂_congr F H),
map_id := λ a, iso.refl _,
map_comp := λ a b c f g, iso.refl _ } | def | category_theory.free_bicategory.lift | category_theory.bicategory | src/category_theory/bicategory/free.lean | [
"category_theory.bicategory.functor"
] | [
"lift",
"map_comp",
"map_id"
] | A prefunctor from a quiver `B` to a bicategory `C` can be lifted to a pseudofunctor from
`free_bicategory B` to `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prelax_functor
(B : Type u₁) [quiver.{v₁+1} B] [∀ a b : B, quiver.{w₁+1} (a ⟶ b)]
(C : Type u₂) [quiver.{v₂+1} C] [∀ a b : C, quiver.{w₂+1} (a ⟶ b)] extends prefunctor B C | (map₂ {a b : B} {f g : a ⟶ b} : (f ⟶ g) → (map f ⟶ map g)) | structure | category_theory.prelax_functor | category_theory.bicategory | src/category_theory/bicategory/functor.lean | [
"category_theory.bicategory.basic"
] | [
"prefunctor"
] | A prelax functor between bicategories consists of functions between objects,
1-morphisms, and 2-morphisms. This structure will be extended to define `oplax_functor`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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