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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cocones_iso_component_inv {J : Type u} [category.{v} J] {K : J ⥤ C}
(Y : D) (t : (G ⋙ (cocones J C).obj (op K)).obj Y) :
((cocones J D).obj (op (K ⋙ F))).obj Y | { app := λ j, (adj.hom_equiv (K.obj j) Y).symm (t.app j),
naturality' := λ j j' f,
begin
erw [← adj.hom_equiv_naturality_left_symm, ← adj.hom_equiv_naturality_right_symm, t.naturality],
dsimp, simp
end } | def | category_theory.adjunction.cocones_iso_component_inv | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | auxiliary construction for `cocones_iso` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cones_iso_component_hom {J : Type u} [category.{v} J] {K : J ⥤ D}
(X : Cᵒᵖ) (t : (functor.op F ⋙ (cones J D).obj K).obj X) :
((cones J C).obj (K ⋙ G)).obj X | { app := λ j, (adj.hom_equiv (unop X) (K.obj j)) (t.app j),
naturality' := λ j j' f,
begin
erw [← adj.hom_equiv_naturality_right, ← t.naturality, category.id_comp, category.id_comp],
refl
end } | def | category_theory.adjunction.cones_iso_component_hom | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | auxiliary construction for `cones_iso` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cones_iso_component_inv {J : Type u} [category.{v} J] {K : J ⥤ D}
(X : Cᵒᵖ) (t : ((cones J C).obj (K ⋙ G)).obj X) :
(functor.op F ⋙ (cones J D).obj K).obj X | { app := λ j, (adj.hom_equiv (unop X) (K.obj j)).symm (t.app j),
naturality' := λ j j' f,
begin
erw [← adj.hom_equiv_naturality_right_symm, ← t.naturality, category.id_comp, category.id_comp]
end } | def | category_theory.adjunction.cones_iso_component_inv | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | auxiliary construction for `cones_iso` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocones_iso {J : Type u} [category.{v} J] {K : J ⥤ C} :
(cocones J D).obj (op (K ⋙ F)) ≅ G ⋙ (cocones J C).obj (op K) | nat_iso.of_components (λ Y,
{ hom := cocones_iso_component_hom adj Y,
inv := cocones_iso_component_inv adj Y, })
(by tidy) | def | category_theory.adjunction.cocones_iso | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [
"adj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cones_iso {J : Type u} [category.{v} J] {K : J ⥤ D} :
F.op ⋙ (cones J D).obj K ≅ (cones J C).obj (K ⋙ G) | nat_iso.of_components (λ X,
{ hom := cones_iso_component_hom adj X,
inv := cones_iso_component_inv adj X, } )
(by tidy) | def | category_theory.adjunction.cones_iso | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [
"adj"
] | When `F ⊣ G`,
the functor associating to each `X` the cones over `K` with cone point `F.op.obj X`
is naturally isomorphic to
the functor associating to each `X` the cones over `K ⋙ G` with cone point `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
transfer_nat_trans : (G ⋙ L₂ ⟶ L₁ ⋙ H) ≃ (R₁ ⋙ G ⟶ H ⋙ R₂) | { to_fun := λ h,
{ app := λ X, adj₂.unit.app _ ≫ R₂.map (h.app _ ≫ H.map (adj₁.counit.app _)),
naturality' := λ X Y f,
begin
dsimp,
rw [assoc, ← R₂.map_comp, assoc, ← H.map_comp, ← adj₁.counit_naturality, H.map_comp,
←functor.comp_map L₁, ←h.naturality_assoc],
simp,
end },
in... | def | category_theory.transfer_nat_trans | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [
"inv_fun"
] | Suppose we have a square of functors (where the top and bottom are adjunctions `L₁ ⊣ R₁` and
`L₂ ⊣ R₂` respectively).
C ↔ D
G ↓ ↓ H
E ↔ F
Then we have a bijection between natural transformations `G ⋙ L₂ ⟶ L₁ ⋙ H` and
`R₁ ⋙ G ⟶ H ⋙ R₂`.
This can be seen as a bijection of the 2-cells:
L₁ ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
transfer_nat_trans_counit (f : G ⋙ L₂ ⟶ L₁ ⋙ H) (Y : D) :
L₂.map ((transfer_nat_trans adj₁ adj₂ f).app _) ≫ adj₂.counit.app _ =
f.app _ ≫ H.map (adj₁.counit.app Y) | by { erw functor.map_comp, simp } | lemma | category_theory.transfer_nat_trans_counit | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_transfer_nat_trans (f : G ⋙ L₂ ⟶ L₁ ⋙ H) (X : C) :
G.map (adj₁.unit.app X) ≫ (transfer_nat_trans adj₁ adj₂ f).app _ =
adj₂.unit.app _ ≫ R₂.map (f.app _) | begin
dsimp [transfer_nat_trans],
rw [←adj₂.unit_naturality_assoc, ←R₂.map_comp, ← functor.comp_map G L₂, f.naturality_assoc,
functor.comp_map, ← H.map_comp],
dsimp, simp, -- See library note [dsimp, simp]
end | lemma | category_theory.unit_transfer_nat_trans | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
transfer_nat_trans_self : (L₂ ⟶ L₁) ≃ (R₁ ⟶ R₂) | calc (L₂ ⟶ L₁) ≃ _ : (iso.hom_congr L₂.left_unitor L₁.right_unitor).symm
... ≃ _ : transfer_nat_trans adj₁ adj₂
... ≃ (R₁ ⟶ R₂) : R₁.right_unitor.hom_congr R₂.left_unitor | def | category_theory.transfer_nat_trans_self | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [] | Given two adjunctions `L₁ ⊣ R₁` and `L₂ ⊣ R₂` both between categories `C`, `D`, there is a
bijection between natural transformations `L₂ ⟶ L₁` and natural transformations `R₁ ⟶ R₂`.
This is defined as a special case of `transfer_nat_trans`, where the two "vertical" functors are
identity.
TODO: Generalise to when the tw... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
transfer_nat_trans_self_counit (f : L₂ ⟶ L₁) (X) :
L₂.map ((transfer_nat_trans_self adj₁ adj₂ f).app _) ≫ adj₂.counit.app X =
f.app _ ≫ adj₁.counit.app X | begin
dsimp [transfer_nat_trans_self],
rw [id_comp, comp_id],
have := transfer_nat_trans_counit adj₁ adj₂ (L₂.left_unitor.hom ≫ f ≫ L₁.right_unitor.inv) X,
dsimp at this,
rw this,
simp,
end | lemma | category_theory.transfer_nat_trans_self_counit | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_transfer_nat_trans_self (f : L₂ ⟶ L₁) (X) :
adj₁.unit.app _ ≫ (transfer_nat_trans_self adj₁ adj₂ f).app _ =
adj₂.unit.app X ≫ functor.map _ (f.app _) | begin
dsimp [transfer_nat_trans_self],
rw [id_comp, comp_id],
have := unit_transfer_nat_trans adj₁ adj₂ (L₂.left_unitor.hom ≫ f ≫ L₁.right_unitor.inv) X,
dsimp at this,
rw this,
simp
end | lemma | category_theory.unit_transfer_nat_trans_self | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
transfer_nat_trans_self_id : transfer_nat_trans_self adj₁ adj₁ (𝟙 _) = 𝟙 _ | by { ext, dsimp [transfer_nat_trans_self, transfer_nat_trans], simp } | lemma | category_theory.transfer_nat_trans_self_id | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
transfer_nat_trans_self_symm_id :
(transfer_nat_trans_self adj₁ adj₁).symm (𝟙 _) = 𝟙 _ | by { rw equiv.symm_apply_eq, simp } | lemma | category_theory.transfer_nat_trans_self_symm_id | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [
"equiv.symm_apply_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
transfer_nat_trans_self_comp (f g) :
transfer_nat_trans_self adj₁ adj₂ f ≫ transfer_nat_trans_self adj₂ adj₃ g =
transfer_nat_trans_self adj₁ adj₃ (g ≫ f) | begin
ext,
dsimp [transfer_nat_trans_self, transfer_nat_trans],
simp only [id_comp, comp_id],
rw [←adj₃.unit_naturality_assoc, ←R₃.map_comp, g.naturality_assoc, L₂.map_comp, assoc,
adj₂.counit_naturality, adj₂.left_triangle_components_assoc, assoc],
end | lemma | category_theory.transfer_nat_trans_self_comp | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
transfer_nat_trans_self_adjunction_id {L R : C ⥤ C} (adj : L ⊣ R) (f : 𝟭 C ⟶ L) (X : C) :
(transfer_nat_trans_self adj adjunction.id f).app X = f.app (R.obj X) ≫ adj.counit.app X | begin
dsimp [transfer_nat_trans_self, transfer_nat_trans, adjunction.id],
simp only [comp_id, id_comp],
end | lemma | category_theory.transfer_nat_trans_self_adjunction_id | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [
"adj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
transfer_nat_trans_self_adjunction_id_symm {L R : C ⥤ C} (adj : L ⊣ R) (g : R ⟶ 𝟭 C)
(X : C) : ((transfer_nat_trans_self adj adjunction.id).symm g).app X =
adj.unit.app X ≫ (g.app (L.obj X)) | begin
dsimp [transfer_nat_trans_self, transfer_nat_trans, adjunction.id],
simp only [comp_id, id_comp],
end | lemma | category_theory.transfer_nat_trans_self_adjunction_id_symm | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [
"adj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
transfer_nat_trans_self_symm_comp (f g) :
(transfer_nat_trans_self adj₂ adj₁).symm f ≫ (transfer_nat_trans_self adj₃ adj₂).symm g =
(transfer_nat_trans_self adj₃ adj₁).symm (g ≫ f) | by { rw [equiv.eq_symm_apply, ← transfer_nat_trans_self_comp _ adj₂], simp } | lemma | category_theory.transfer_nat_trans_self_symm_comp | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [
"equiv.eq_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
transfer_nat_trans_self_comm {f g} (gf : g ≫ f = 𝟙 _) :
transfer_nat_trans_self adj₁ adj₂ f ≫ transfer_nat_trans_self adj₂ adj₁ g = 𝟙 _ | by rw [transfer_nat_trans_self_comp, gf, transfer_nat_trans_self_id] | lemma | category_theory.transfer_nat_trans_self_comm | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
transfer_nat_trans_self_symm_comm {f g} (gf : g ≫ f = 𝟙 _) :
(transfer_nat_trans_self adj₁ adj₂).symm f ≫ (transfer_nat_trans_self adj₂ adj₁).symm g = 𝟙 _ | by rw [transfer_nat_trans_self_symm_comp, gf, transfer_nat_trans_self_symm_id] | lemma | category_theory.transfer_nat_trans_self_symm_comm | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
transfer_nat_trans_self_iso (f : L₂ ⟶ L₁) [is_iso f] :
is_iso (transfer_nat_trans_self adj₁ adj₂ f) | ⟨⟨transfer_nat_trans_self adj₂ adj₁ (inv f),
⟨transfer_nat_trans_self_comm _ _ (by simp), transfer_nat_trans_self_comm _ _ (by simp)⟩⟩⟩ | instance | category_theory.transfer_nat_trans_self_iso | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [] | If `f` is an isomorphism, then the transferred natural transformation is an isomorphism.
The converse is given in `transfer_nat_trans_self_of_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
transfer_nat_trans_self_symm_iso (f : R₁ ⟶ R₂) [is_iso f] :
is_iso ((transfer_nat_trans_self adj₁ adj₂).symm f) | ⟨⟨(transfer_nat_trans_self adj₂ adj₁).symm (inv f),
⟨transfer_nat_trans_self_symm_comm _ _ (by simp),
transfer_nat_trans_self_symm_comm _ _ (by simp)⟩⟩⟩ | instance | category_theory.transfer_nat_trans_self_symm_iso | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [] | If `f` is an isomorphism, then the un-transferred natural transformation is an isomorphism.
The converse is given in `transfer_nat_trans_self_symm_of_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
transfer_nat_trans_self_of_iso (f : L₂ ⟶ L₁) [is_iso (transfer_nat_trans_self adj₁ adj₂ f)] :
is_iso f | begin
suffices :
is_iso ((transfer_nat_trans_self adj₁ adj₂).symm (transfer_nat_trans_self adj₁ adj₂ f)),
{ simpa using this },
apply_instance,
end | lemma | category_theory.transfer_nat_trans_self_of_iso | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [] | If `f` is a natural transformation whose transferred natural transformation is an isomorphism,
then `f` is an isomorphism.
The converse is given in `transfer_nat_trans_self_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
transfer_nat_trans_self_symm_of_iso (f : R₁ ⟶ R₂)
[is_iso ((transfer_nat_trans_self adj₁ adj₂).symm f)] :
is_iso f | begin
suffices :
is_iso ((transfer_nat_trans_self adj₁ adj₂) ((transfer_nat_trans_self adj₁ adj₂).symm f)),
{ simpa using this },
apply_instance,
end | lemma | category_theory.transfer_nat_trans_self_symm_of_iso | category_theory.adjunction | src/category_theory/adjunction/mates.lean | [
"category_theory.adjunction.basic",
"category_theory.conj"
] | [] | If `f` is a natural transformation whose un-transferred natural transformation is an isomorphism,
then `f` is an isomorphism.
The converse is given in `transfer_nat_trans_self_symm_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_of_op_adjoint_op
(F : C ⥤ D) (G : D ⥤ C) (h : G.op ⊣ F.op) : F ⊣ G | adjunction.mk_of_hom_equiv
{ hom_equiv := λ X Y,
((h.hom_equiv (opposite.op Y) (opposite.op X)).trans (op_equiv _ _)).symm.trans (op_equiv _ _) } | def | category_theory.adjunction.adjoint_of_op_adjoint_op | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [
"opposite.op"
] | If `G.op` is adjoint to `F.op` then `F` is adjoint to `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_unop_of_adjoint_op (F : C ⥤ D) (G : Dᵒᵖ ⥤ Cᵒᵖ) (h : G ⊣ F.op) : F ⊣ G.unop | adjoint_of_op_adjoint_op F G.unop (h.of_nat_iso_left G.op_unop_iso.symm) | def | category_theory.adjunction.adjoint_unop_of_adjoint_op | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | If `G` is adjoint to `F.op` then `F` is adjoint to `G.unop`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_adjoint_of_op_adjoint (F : Cᵒᵖ ⥤ Dᵒᵖ) (G : D ⥤ C) (h : G.op ⊣ F) : F.unop ⊣ G | adjoint_of_op_adjoint_op _ _ (h.of_nat_iso_right F.op_unop_iso.symm) | def | category_theory.adjunction.unop_adjoint_of_op_adjoint | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | If `G.op` is adjoint to `F` then `F.unop` is adjoint to `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_adjoint_unop_of_adjoint (F : Cᵒᵖ ⥤ Dᵒᵖ) (G : Dᵒᵖ ⥤ Cᵒᵖ) (h : G ⊣ F) : F.unop ⊣ G.unop | adjoint_unop_of_adjoint_op F.unop G (h.of_nat_iso_right F.op_unop_iso.symm) | def | category_theory.adjunction.unop_adjoint_unop_of_adjoint | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | If `G` is adjoint to `F` then `F.unop` is adjoint to `G.unop`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_adjoint_op_of_adjoint
(F : C ⥤ D) (G : D ⥤ C) (h : G ⊣ F) : F.op ⊣ G.op | adjunction.mk_of_hom_equiv
{ hom_equiv := λ X Y,
(op_equiv _ Y).trans ((h.hom_equiv _ _).symm.trans (op_equiv X (opposite.op _)).symm) } | def | category_theory.adjunction.op_adjoint_op_of_adjoint | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [
"opposite.op"
] | If `G` is adjoint to `F` then `F.op` is adjoint to `G.op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_op_of_adjoint_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) (G : D ⥤ C) (h : G ⊣ F.unop) : F ⊣ G.op | (op_adjoint_op_of_adjoint F.unop _ h).of_nat_iso_left F.op_unop_iso | def | category_theory.adjunction.adjoint_op_of_adjoint_unop | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | If `G` is adjoint to `F.unop` then `F` is adjoint to `G.op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_adjoint_of_unop_adjoint (F : C ⥤ D) (G : Dᵒᵖ ⥤ Cᵒᵖ) (h : G.unop ⊣ F) : F.op ⊣ G | (op_adjoint_op_of_adjoint _ G.unop h).of_nat_iso_right G.op_unop_iso | def | category_theory.adjunction.op_adjoint_of_unop_adjoint | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | If `G.unop` is adjoint to `F` then `F.op` is adjoint to `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_of_unop_adjoint_unop (F : Cᵒᵖ ⥤ Dᵒᵖ) (G : Dᵒᵖ ⥤ Cᵒᵖ) (h : G.unop ⊣ F.unop) : F ⊣ G | (adjoint_op_of_adjoint_unop _ _ h).of_nat_iso_right G.op_unop_iso | def | category_theory.adjunction.adjoint_of_unop_adjoint_unop | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | If `G.unop` is adjoint to `F.unop` then `F` is adjoint to `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_adjoints_coyoneda_equiv {F F' : C ⥤ D} {G : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F' ⊣ G):
F.op ⋙ coyoneda ≅ F'.op ⋙ coyoneda | nat_iso.of_components
(λ X, nat_iso.of_components
(λ Y, ((adj1.hom_equiv X.unop Y).trans (adj2.hom_equiv X.unop Y).symm).to_iso)
(by tidy))
(by tidy) | def | category_theory.adjunction.left_adjoints_coyoneda_equiv | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | If `F` and `F'` are both adjoint to `G`, there is a natural isomorphism
`F.op ⋙ coyoneda ≅ F'.op ⋙ coyoneda`.
We use this in combination with `fully_faithful_cancel_right` to show left adjoints are unique. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_adjoint_uniq {F F' : C ⥤ D} {G : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F' ⊣ G) : F ≅ F' | nat_iso.remove_op (fully_faithful_cancel_right _ (left_adjoints_coyoneda_equiv adj2 adj1)) | def | category_theory.adjunction.left_adjoint_uniq | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | If `F` and `F'` are both left adjoint to `G`, then they are naturally isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_equiv_left_adjoint_uniq_hom_app {F F' : C ⥤ D} {G : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F' ⊣ G) (x : C) :
adj1.hom_equiv _ _ ((left_adjoint_uniq adj1 adj2).hom.app x) = adj2.unit.app x | begin
apply (adj1.hom_equiv _ _).symm.injective,
apply quiver.hom.op_inj,
apply coyoneda.map_injective,
swap, apply_instance,
ext f y,
simpa [left_adjoint_uniq, left_adjoints_coyoneda_equiv]
end | lemma | category_theory.adjunction.hom_equiv_left_adjoint_uniq_hom_app | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [
"quiver.hom.op_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_left_adjoint_uniq_hom {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G) :
adj1.unit ≫ whisker_right (left_adjoint_uniq adj1 adj2).hom G = adj2.unit | begin
ext x,
rw [nat_trans.comp_app, ← hom_equiv_left_adjoint_uniq_hom_app adj1 adj2],
simp [-hom_equiv_left_adjoint_uniq_hom_app, ←G.map_comp]
end | lemma | category_theory.adjunction.unit_left_adjoint_uniq_hom | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_left_adjoint_uniq_hom_app {F F' : C ⥤ D} {G : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F' ⊣ G) (x : C) :
adj1.unit.app x ≫ G.map ((left_adjoint_uniq adj1 adj2).hom.app x) = adj2.unit.app x | by { rw ← unit_left_adjoint_uniq_hom adj1 adj2, refl } | lemma | category_theory.adjunction.unit_left_adjoint_uniq_hom_app | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_adjoint_uniq_hom_counit {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G) :
whisker_left G (left_adjoint_uniq adj1 adj2).hom ≫ adj2.counit = adj1.counit | begin
ext x,
apply quiver.hom.op_inj,
apply coyoneda.map_injective,
swap, apply_instance,
ext y f,
have : F.map (adj2.unit.app (G.obj x)) ≫ adj1.counit.app (F'.obj (G.obj x)) ≫
adj2.counit.app x ≫ f = adj1.counit.app x ≫ f,
{ erw [← adj1.counit.naturality, ← F.map_comp_assoc], simpa },
simpa [left_a... | lemma | category_theory.adjunction.left_adjoint_uniq_hom_counit | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [
"quiver.hom.op_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_adjoint_uniq_hom_app_counit {F F' : C ⥤ D} {G : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F' ⊣ G) (x : D) :
(left_adjoint_uniq adj1 adj2).hom.app (G.obj x) ≫ adj2.counit.app x = adj1.counit.app x | by { rw ← left_adjoint_uniq_hom_counit adj1 adj2, refl } | lemma | category_theory.adjunction.left_adjoint_uniq_hom_app_counit | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_adjoint_uniq_inv_app {F F' : C ⥤ D} {G : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F' ⊣ G) (x : C) :
(left_adjoint_uniq adj1 adj2).inv.app x = (left_adjoint_uniq adj2 adj1).hom.app x | rfl | lemma | category_theory.adjunction.left_adjoint_uniq_inv_app | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_adjoint_uniq_trans {F F' F'' : C ⥤ D} {G : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F' ⊣ G) (adj3 : F'' ⊣ G) :
(left_adjoint_uniq adj1 adj2).hom ≫ (left_adjoint_uniq adj2 adj3).hom =
(left_adjoint_uniq adj1 adj3).hom | begin
ext,
apply quiver.hom.op_inj,
apply coyoneda.map_injective,
swap, apply_instance,
ext,
simp [left_adjoints_coyoneda_equiv, left_adjoint_uniq]
end | lemma | category_theory.adjunction.left_adjoint_uniq_trans | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [
"quiver.hom.op_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_adjoint_uniq_trans_app {F F' F'' : C ⥤ D} {G : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F' ⊣ G) (adj3 : F'' ⊣ G) (x : C) :
(left_adjoint_uniq adj1 adj2).hom.app x ≫ (left_adjoint_uniq adj2 adj3).hom.app x =
(left_adjoint_uniq adj1 adj3).hom.app x | by { rw ← left_adjoint_uniq_trans adj1 adj2 adj3, refl } | lemma | category_theory.adjunction.left_adjoint_uniq_trans_app | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_adjoint_uniq_refl {F : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) :
(left_adjoint_uniq adj1 adj1).hom = 𝟙 _ | begin
ext,
apply quiver.hom.op_inj,
apply coyoneda.map_injective,
swap, apply_instance,
ext,
simp [left_adjoints_coyoneda_equiv, left_adjoint_uniq]
end | lemma | category_theory.adjunction.left_adjoint_uniq_refl | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [
"quiver.hom.op_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_adjoint_uniq {F : C ⥤ D} {G G' : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F ⊣ G') : G ≅ G' | nat_iso.remove_op
(left_adjoint_uniq (op_adjoint_op_of_adjoint _ F adj2) (op_adjoint_op_of_adjoint _ _ adj1)) | def | category_theory.adjunction.right_adjoint_uniq | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | If `G` and `G'` are both right adjoint to `F`, then they are naturally isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_equiv_symm_right_adjoint_uniq_hom_app {F : C ⥤ D} {G G' : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F ⊣ G') (x : D) :
(adj2.hom_equiv _ _).symm ((right_adjoint_uniq adj1 adj2).hom.app x) = adj1.counit.app x | begin
apply quiver.hom.op_inj,
convert hom_equiv_left_adjoint_uniq_hom_app
(op_adjoint_op_of_adjoint _ F adj2) (op_adjoint_op_of_adjoint _ _ adj1) (opposite.op x),
simpa
end | lemma | category_theory.adjunction.hom_equiv_symm_right_adjoint_uniq_hom_app | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [
"opposite.op",
"quiver.hom.op_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_right_adjoint_uniq_hom_app {F : C ⥤ D} {G G' : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F ⊣ G') (x : C) :
adj1.unit.app x ≫ (right_adjoint_uniq adj1 adj2).hom.app (F.obj x) = adj2.unit.app x | begin
apply quiver.hom.op_inj,
convert left_adjoint_uniq_hom_app_counit
(op_adjoint_op_of_adjoint _ _ adj2) (op_adjoint_op_of_adjoint _ _ adj1) (opposite.op x),
all_goals { simpa }
end | lemma | category_theory.adjunction.unit_right_adjoint_uniq_hom_app | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [
"opposite.op",
"quiver.hom.op_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_right_adjoint_uniq_hom {F : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G') :
adj1.unit ≫ whisker_left F (right_adjoint_uniq adj1 adj2).hom = adj2.unit | by { ext x, simp } | lemma | category_theory.adjunction.unit_right_adjoint_uniq_hom | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_adjoint_uniq_hom_app_counit {F : C ⥤ D} {G G' : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F ⊣ G') (x : D) :
F.map ((right_adjoint_uniq adj1 adj2).hom.app x) ≫ adj2.counit.app x = adj1.counit.app x | begin
apply quiver.hom.op_inj,
convert unit_left_adjoint_uniq_hom_app
(op_adjoint_op_of_adjoint _ _ adj2) (op_adjoint_op_of_adjoint _ _ adj1) (opposite.op x),
all_goals { simpa }
end | lemma | category_theory.adjunction.right_adjoint_uniq_hom_app_counit | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [
"opposite.op",
"quiver.hom.op_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_adjoint_uniq_hom_counit {F : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G') :
whisker_right (right_adjoint_uniq adj1 adj2).hom F ≫ adj2.counit = adj1.counit | by { ext, simp } | lemma | category_theory.adjunction.right_adjoint_uniq_hom_counit | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_adjoint_uniq_inv_app {F : C ⥤ D} {G G' : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F ⊣ G') (x : D) :
(right_adjoint_uniq adj1 adj2).inv.app x = (right_adjoint_uniq adj2 adj1).hom.app x | rfl | lemma | category_theory.adjunction.right_adjoint_uniq_inv_app | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_adjoint_uniq_trans_app {F : C ⥤ D} {G G' G'' : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F ⊣ G') (adj3 : F ⊣ G'') (x : D) :
(right_adjoint_uniq adj1 adj2).hom.app x ≫ (right_adjoint_uniq adj2 adj3).hom.app x =
(right_adjoint_uniq adj1 adj3).hom.app x | begin
apply quiver.hom.op_inj,
exact left_adjoint_uniq_trans_app (op_adjoint_op_of_adjoint _ _ adj3)
(op_adjoint_op_of_adjoint _ _ adj2) (op_adjoint_op_of_adjoint _ _ adj1) (opposite.op x)
end | lemma | category_theory.adjunction.right_adjoint_uniq_trans_app | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [
"opposite.op",
"quiver.hom.op_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_adjoint_uniq_trans {F : C ⥤ D} {G G' G'' : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F ⊣ G') (adj3 : F ⊣ G'') :
(right_adjoint_uniq adj1 adj2).hom ≫ (right_adjoint_uniq adj2 adj3).hom =
(right_adjoint_uniq adj1 adj3).hom | by { ext, simp } | lemma | category_theory.adjunction.right_adjoint_uniq_trans | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_adjoint_uniq_refl {F : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) :
(right_adjoint_uniq adj1 adj1).hom = 𝟙 _ | by { delta right_adjoint_uniq, simp } | lemma | category_theory.adjunction.right_adjoint_uniq_refl | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_iso_of_left_adjoint_nat_iso {F F' : C ⥤ D} {G G' : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F' ⊣ G') (l : F ≅ F') :
G ≅ G' | right_adjoint_uniq adj1 (adj2.of_nat_iso_left l.symm) | def | category_theory.adjunction.nat_iso_of_left_adjoint_nat_iso | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | Given two adjunctions, if the left adjoints are naturally isomorphic, then so are the right
adjoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_iso_of_right_adjoint_nat_iso {F F' : C ⥤ D} {G G' : D ⥤ C}
(adj1 : F ⊣ G) (adj2 : F' ⊣ G') (r : G ≅ G') :
F ≅ F' | left_adjoint_uniq adj1 (adj2.of_nat_iso_right r.symm) | def | category_theory.adjunction.nat_iso_of_right_adjoint_nat_iso | category_theory.adjunction | src/category_theory/adjunction/opposites.lean | [
"category_theory.adjunction.basic",
"category_theory.yoneda",
"category_theory.opposites"
] | [] | Given two adjunctions, if the right adjoints are naturally isomorphic, then so are the left
adjoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star [has_binary_products C] : C ⥤ over X | cofree _ ⋙ coalgebra_to_over X | def | category_theory.star | category_theory.adjunction | src/category_theory/adjunction/over.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.monad.products",
"category_theory.over"
] | [] | The functor from `C` to `over X` which sends `Y : C` to `π₁ : X ⨯ Y ⟶ X`, sometimes denoted `X*`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_adj_star [has_binary_products C] : over.forget X ⊣ star X | (coalgebra_equiv_over X).symm.to_adjunction.comp (adj _) | def | category_theory.forget_adj_star | category_theory.adjunction | src/category_theory/adjunction/over.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.monad.products",
"category_theory.over"
] | [
"adj"
] | The functor `over.forget X : over X ⥤ C` has a right adjoint given by `star X`.
Note that the binary products assumption is necessary: the existence of a right adjoint to
`over.forget X` is equivalent to the existence of each binary product `X ⨯ -`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflective (R : D ⥤ C) extends is_right_adjoint R, full R, faithful R. | class | category_theory.reflective | category_theory.adjunction | src/category_theory/adjunction/reflective.lean | [
"category_theory.adjunction.fully_faithful",
"category_theory.functor.reflects_isomorphisms",
"category_theory.epi_mono"
] | [] | A functor is *reflective*, or *a reflective inclusion*, if it is fully faithful and right adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_obj_eq_map_unit [reflective i] (X : C) :
(of_right_adjoint i).unit.app (i.obj ((left_adjoint i).obj X))
= i.map ((left_adjoint i).map ((of_right_adjoint i).unit.app X)) | begin
rw [←cancel_mono (i.map ((of_right_adjoint i).counit.app ((left_adjoint i).obj X))),
←i.map_comp],
simp,
end | lemma | category_theory.unit_obj_eq_map_unit | category_theory.adjunction | src/category_theory/adjunction/reflective.lean | [
"category_theory.adjunction.fully_faithful",
"category_theory.functor.reflects_isomorphisms",
"category_theory.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_unit_obj [reflective i] {B : D} :
is_iso ((of_right_adjoint i).unit.app (i.obj B)) | begin
have : (of_right_adjoint i).unit.app (i.obj B) =
inv (i.map ((of_right_adjoint i).counit.app B)),
{ rw ← comp_hom_eq_id,
apply (of_right_adjoint i).right_triangle_components },
rw this,
exact is_iso.inv_is_iso,
end | instance | category_theory.is_iso_unit_obj | category_theory.adjunction | src/category_theory/adjunction/reflective.lean | [
"category_theory.adjunction.fully_faithful",
"category_theory.functor.reflects_isomorphisms",
"category_theory.epi_mono"
] | [] | When restricted to objects in `D` given by `i : D ⥤ C`, the unit is an isomorphism. In other words,
`η_iX` is an isomorphism for any `X` in `D`.
More generally this applies to objects essentially in the reflective subcategory, see
`functor.ess_image.unit_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.ess_image.unit_is_iso [reflective i] {A : C} (h : A ∈ i.ess_image) :
is_iso ((of_right_adjoint i).unit.app A) | begin
suffices : (of_right_adjoint i).unit.app A =
h.get_iso.inv ≫ (of_right_adjoint i).unit.app (i.obj h.witness) ≫
(left_adjoint i ⋙ i).map h.get_iso.hom,
{ rw this,
apply_instance },
rw ← nat_trans.naturality,
simp,
end | lemma | category_theory.functor.ess_image.unit_is_iso | category_theory.adjunction | src/category_theory/adjunction/reflective.lean | [
"category_theory.adjunction.fully_faithful",
"category_theory.functor.reflects_isomorphisms",
"category_theory.epi_mono"
] | [] | If `A` is essentially in the image of a reflective functor `i`, then `η_A` is an isomorphism.
This gives that the "witness" for `A` being in the essential image can instead be given as the
reflection of `A`, with the isomorphism as `η_A`.
(For any `B` in the reflective subcategory, we automatically have that `ε_B` is ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_ess_image_of_unit_is_iso [is_right_adjoint i] (A : C)
[is_iso ((of_right_adjoint i).unit.app A)] : A ∈ i.ess_image | ⟨(left_adjoint i).obj A, ⟨(as_iso ((of_right_adjoint i).unit.app A)).symm⟩⟩ | lemma | category_theory.mem_ess_image_of_unit_is_iso | category_theory.adjunction | src/category_theory/adjunction/reflective.lean | [
"category_theory.adjunction.fully_faithful",
"category_theory.functor.reflects_isomorphisms",
"category_theory.epi_mono"
] | [] | If `η_A` is an isomorphism, then `A` is in the essential image of `i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_ess_image_of_unit_is_split_mono [reflective i] {A : C}
[is_split_mono ((of_right_adjoint i).unit.app A)] : A ∈ i.ess_image | begin
let η : 𝟭 C ⟶ left_adjoint i ⋙ i := (of_right_adjoint i).unit,
haveI : is_iso (η.app (i.obj ((left_adjoint i).obj A))) := (i.obj_mem_ess_image _).unit_is_iso,
have : epi (η.app A),
{ apply epi_of_epi (retraction (η.app A)) _,
rw (show retraction _ ≫ η.app A = _, from η.naturality (retraction (η.app A... | lemma | category_theory.mem_ess_image_of_unit_is_split_mono | category_theory.adjunction | src/category_theory/adjunction/reflective.lean | [
"category_theory.adjunction.fully_faithful",
"category_theory.functor.reflects_isomorphisms",
"category_theory.epi_mono"
] | [] | If `η_A` is a split monomorphism, then `A` is in the reflective subcategory. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflective.comp (F : C ⥤ D) (G : D ⥤ E) [Fr : reflective F] [Gr : reflective G] :
reflective (F ⋙ G) | { to_faithful := faithful.comp F G, } | instance | category_theory.reflective.comp | category_theory.adjunction | src/category_theory/adjunction/reflective.lean | [
"category_theory.adjunction.fully_faithful",
"category_theory.functor.reflects_isomorphisms",
"category_theory.epi_mono"
] | [] | Composition of reflective functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_comp_partial_bijective_aux [reflective i] (A : C) (B : D) :
(A ⟶ i.obj B) ≃ (i.obj ((left_adjoint i).obj A) ⟶ i.obj B) | ((adjunction.of_right_adjoint i).hom_equiv _ _).symm.trans (equiv_of_fully_faithful i) | def | category_theory.unit_comp_partial_bijective_aux | category_theory.adjunction | src/category_theory/adjunction/reflective.lean | [
"category_theory.adjunction.fully_faithful",
"category_theory.functor.reflects_isomorphisms",
"category_theory.epi_mono"
] | [] | (Implementation) Auxiliary definition for `unit_comp_partial_bijective`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_comp_partial_bijective_aux_symm_apply [reflective i] {A : C} {B : D}
(f : i.obj ((left_adjoint i).obj A) ⟶ i.obj B) :
(unit_comp_partial_bijective_aux _ _).symm f = (of_right_adjoint i).unit.app A ≫ f | by simp [unit_comp_partial_bijective_aux] | lemma | category_theory.unit_comp_partial_bijective_aux_symm_apply | category_theory.adjunction | src/category_theory/adjunction/reflective.lean | [
"category_theory.adjunction.fully_faithful",
"category_theory.functor.reflects_isomorphisms",
"category_theory.epi_mono"
] | [] | The description of the inverse of the bijection `unit_comp_partial_bijective_aux`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_comp_partial_bijective [reflective i] (A : C) {B : C} (hB : B ∈ i.ess_image) :
(A ⟶ B) ≃ (i.obj ((left_adjoint i).obj A) ⟶ B) | calc (A ⟶ B) ≃ (A ⟶ i.obj hB.witness) : iso.hom_congr (iso.refl _) hB.get_iso.symm
... ≃ (i.obj _ ⟶ i.obj hB.witness) : unit_comp_partial_bijective_aux _ _
... ≃ (i.obj ((left_adjoint i).obj A) ⟶ B) : iso.hom_congr (iso.refl _) hB.get_iso | def | category_theory.unit_comp_partial_bijective | category_theory.adjunction | src/category_theory/adjunction/reflective.lean | [
"category_theory.adjunction.fully_faithful",
"category_theory.functor.reflects_isomorphisms",
"category_theory.epi_mono"
] | [] | If `i` has a reflector `L`, then the function `(i.obj (L.obj A) ⟶ B) → (A ⟶ B)` given by
precomposing with `η.app A` is a bijection provided `B` is in the essential image of `i`.
That is, the function `λ (f : i.obj (L.obj A) ⟶ B), η.app A ≫ f` is bijective, as long as `B` is in
the essential image of `i`.
This definiti... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_comp_partial_bijective_symm_apply [reflective i] (A : C) {B : C}
(hB : B ∈ i.ess_image) (f) :
(unit_comp_partial_bijective A hB).symm f = (of_right_adjoint i).unit.app A ≫ f | by simp [unit_comp_partial_bijective, unit_comp_partial_bijective_aux_symm_apply] | lemma | category_theory.unit_comp_partial_bijective_symm_apply | category_theory.adjunction | src/category_theory/adjunction/reflective.lean | [
"category_theory.adjunction.fully_faithful",
"category_theory.functor.reflects_isomorphisms",
"category_theory.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_comp_partial_bijective_symm_natural [reflective i] (A : C) {B B' : C} (h : B ⟶ B')
(hB : B ∈ i.ess_image) (hB' : B' ∈ i.ess_image) (f : i.obj ((left_adjoint i).obj A) ⟶ B) :
(unit_comp_partial_bijective A hB').symm (f ≫ h) =
(unit_comp_partial_bijective A hB).symm f ≫ h | by simp | lemma | category_theory.unit_comp_partial_bijective_symm_natural | category_theory.adjunction | src/category_theory/adjunction/reflective.lean | [
"category_theory.adjunction.fully_faithful",
"category_theory.functor.reflects_isomorphisms",
"category_theory.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_comp_partial_bijective_natural [reflective i] (A : C) {B B' : C} (h : B ⟶ B')
(hB : B ∈ i.ess_image) (hB' : B' ∈ i.ess_image) (f : A ⟶ B) :
(unit_comp_partial_bijective A hB') (f ≫ h) = unit_comp_partial_bijective A hB f ≫ h | by rw [←equiv.eq_symm_apply, unit_comp_partial_bijective_symm_natural A h, equiv.symm_apply_apply] | lemma | category_theory.unit_comp_partial_bijective_natural | category_theory.adjunction | src/category_theory/adjunction/reflective.lean | [
"category_theory.adjunction.fully_faithful",
"category_theory.functor.reflects_isomorphisms",
"category_theory.epi_mono"
] | [
"equiv.symm_apply_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_ess_image_of_reflective [reflective i] : D ≌ i.ess_image_subcategory | { functor := i.to_ess_image,
inverse := i.ess_image_inclusion ⋙ (left_adjoint i : _),
unit_iso := nat_iso.of_components (λ X, (as_iso $ (of_right_adjoint i).counit.app X).symm)
(by { intros X Y f, dsimp, simp only [is_iso.eq_inv_comp, is_iso.comp_inv_eq, category.assoc],
exact ((of_right_adjoint i).counit... | def | category_theory.equiv_ess_image_of_reflective | category_theory.adjunction | src/category_theory/adjunction/reflective.lean | [
"category_theory.adjunction.fully_faithful",
"category_theory.functor.reflects_isomorphisms",
"category_theory.epi_mono"
] | [] | If `i : D ⥤ C` is reflective, the inverse functor of `i ≌ F.ess_image` can be explicitly
defined by the reflector. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_right (adj : F ⊣ G) :
(whiskering_right C D E).obj F ⊣ (whiskering_right C E D).obj G | mk_of_unit_counit
{ unit :=
{ app := λ X, (functor.right_unitor _).inv ≫
whisker_left X adj.unit ≫ (functor.associator _ _ _).inv,
naturality' := by { intros, ext, dsimp, simp } },
counit :=
{ app := λ X, (functor.associator _ _ _).hom ≫
whisker_left X adj.counit ≫ (functor.right_unitor _).hom,
... | def | category_theory.adjunction.whisker_right | category_theory.adjunction | src/category_theory/adjunction/whiskering.lean | [
"category_theory.whiskering",
"category_theory.adjunction.basic"
] | [
"adj"
] | Given an adjunction `F ⊣ G`, this provides the natural adjunction
`(whiskering_right C _ _).obj F ⊣ (whiskering_right C _ _).obj G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_left (adj : F ⊣ G) :
(whiskering_left E D C).obj G ⊣ (whiskering_left D E C).obj F | mk_of_unit_counit
{ unit :=
{ app := λ X, (functor.left_unitor _).inv ≫
whisker_right adj.unit X ≫ (functor.associator _ _ _).hom,
naturality' := by { intros, ext, dsimp, simp } },
counit :=
{ app := λ X, (functor.associator _ _ _).inv ≫
whisker_right adj.counit X ≫ (functor.left_unitor _).hom,
... | def | category_theory.adjunction.whisker_left | category_theory.adjunction | src/category_theory/adjunction/whiskering.lean | [
"category_theory.whiskering",
"category_theory.adjunction.basic"
] | [
"adj"
] | Given an adjunction `F ⊣ G`, this provides the natural adjunction
`(whiskering_left _ _ C).obj G ⊣ (whiskering_left _ _ C).obj F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bicategory (B : Type u) extends category_struct.{v} B | -- category structure on the collection of 1-morphisms:
(hom_category : ∀ (a b : B), category.{w} (a ⟶ b) . tactic.apply_instance)
-- left whiskering:
(whisker_left {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h)
(infixr ` ◁ `:81 := whisker_left)
-- right whiskering:
(whisker_right {a b c : B} {f g :... | class | category_theory.bicategory | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_inv_whisker_left (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) :
f ◁ η.hom ≫ f ◁ η.inv = 𝟙 (f ≫ g) | by rw [←whisker_left_comp, hom_inv_id, whisker_left_id] | lemma | category_theory.bicategory.hom_inv_whisker_left | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_inv_whisker_right {f g : a ⟶ b} (η : f ≅ g) (h : b ⟶ c) :
η.hom ▷ h ≫ η.inv ▷ h = 𝟙 (f ≫ h) | by rw [←comp_whisker_right, hom_inv_id, id_whisker_right] | lemma | category_theory.bicategory.hom_inv_whisker_right | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_hom_whisker_left (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) :
f ◁ η.inv ≫ f ◁ η.hom = 𝟙 (f ≫ h) | by rw [←whisker_left_comp, inv_hom_id, whisker_left_id] | lemma | category_theory.bicategory.inv_hom_whisker_left | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_hom_whisker_right {f g : a ⟶ b} (η : f ≅ g) (h : b ⟶ c) :
η.inv ▷ h ≫ η.hom ▷ h = 𝟙 (g ≫ h) | by rw [←comp_whisker_right, inv_hom_id, id_whisker_right] | lemma | category_theory.bicategory.inv_hom_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_iso (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) :
f ≫ g ≅ f ≫ h | { hom := f ◁ η.hom,
inv := f ◁ η.inv } | def | category_theory.bicategory.whisker_left_iso | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | The left whiskering of a 2-isomorphism is a 2-isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_left_is_iso (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) [is_iso η] :
is_iso (f ◁ η) | is_iso.of_iso (whisker_left_iso f (as_iso η)) | instance | category_theory.bicategory.whisker_left_is_iso | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_whisker_left (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) [is_iso η] :
inv (f ◁ η) = f ◁ (inv η) | by { ext, simp only [←whisker_left_comp, whisker_left_id, is_iso.hom_inv_id] } | lemma | category_theory.bicategory.inv_whisker_left | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right_iso {f g : a ⟶ b} (η : f ≅ g) (h : b ⟶ c) :
f ≫ h ≅ g ≫ h | { hom := η.hom ▷ h,
inv := η.inv ▷ h } | def | category_theory.bicategory.whisker_right_iso | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | The right whiskering of a 2-isomorphism is a 2-isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_right_is_iso {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) [is_iso η] :
is_iso (η ▷ h) | is_iso.of_iso (whisker_right_iso (as_iso η) h) | instance | category_theory.bicategory.whisker_right_is_iso | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_whisker_right {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) [is_iso η] :
inv (η ▷ h) = (inv η) ▷ h | by { ext, simp only [←comp_whisker_right, id_whisker_right, is_iso.hom_inv_id] } | lemma | category_theory.bicategory.inv_whisker_right | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pentagon_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i =
(α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv | eq_of_inv_eq_inv (by simp) | lemma | category_theory.bicategory.pentagon_inv | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pentagon_inv_inv_hom_hom_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom =
f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv | by { rw [←cancel_epi (f ◁ (α_ g h i).inv), ←cancel_mono (α_ (f ≫ g) h i).inv], simp } | lemma | category_theory.bicategory.pentagon_inv_inv_hom_hom_inv | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pentagon_inv_hom_hom_hom_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i ≫ (α_ f (g ≫ h) i).hom =
(α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv | eq_of_inv_eq_inv (by simp) | lemma | category_theory.bicategory.pentagon_inv_hom_hom_hom_inv | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pentagon_hom_inv_inv_inv_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv =
(α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i | by simp [←cancel_epi (f ◁ (α_ g h i).inv)] | lemma | category_theory.bicategory.pentagon_hom_inv_inv_inv_inv | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pentagon_hom_hom_inv_hom_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ (f ≫ g) h i).hom ≫ (α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv =
(α_ f g h).hom ▷ i ≫ (α_ f (g ≫ h) i).hom | eq_of_inv_eq_inv (by simp) | lemma | category_theory.bicategory.pentagon_hom_hom_inv_hom_hom | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pentagon_hom_inv_inv_inv_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv =
(α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i | by { rw [←cancel_epi (α_ f g (h ≫ i)).inv, ←cancel_mono ((α_ f g h).inv ▷ i)], simp } | lemma | category_theory.bicategory.pentagon_hom_inv_inv_inv_hom | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pentagon_hom_hom_inv_inv_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ f (g ≫ h) i).hom ≫ f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv =
(α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom | eq_of_inv_eq_inv (by simp) | lemma | category_theory.bicategory.pentagon_hom_hom_inv_inv_hom | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pentagon_inv_hom_hom_hom_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom ≫ (α_ f g (h ≫ i)).hom =
(α_ f (g ≫ h) i).hom ≫ f ◁ (α_ g h i).hom | by simp [←cancel_epi ((α_ f g h).hom ▷ i)] | lemma | category_theory.bicategory.pentagon_inv_hom_hom_hom_hom | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pentagon_inv_inv_hom_inv_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i =
f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv | eq_of_inv_eq_inv (by simp) | lemma | category_theory.bicategory.pentagon_inv_inv_hom_inv_inv | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
triangle_assoc_comp_left (f : a ⟶ b) (g : b ⟶ c) :
(α_ f (𝟙 b) g).hom ≫ f ◁ (λ_ g).hom = (ρ_ f).hom ▷ g | triangle f g | lemma | category_theory.bicategory.triangle_assoc_comp_left | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
triangle_assoc_comp_right (f : a ⟶ b) (g : b ⟶ c) :
(α_ f (𝟙 b) g).inv ≫ (ρ_ f).hom ▷ g = f ◁ (λ_ g).hom | by rw [←triangle, inv_hom_id_assoc] | lemma | category_theory.bicategory.triangle_assoc_comp_right | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
triangle_assoc_comp_right_inv (f : a ⟶ b) (g : b ⟶ c) :
(ρ_ f).inv ▷ g ≫ (α_ f (𝟙 b) g).hom = f ◁ (λ_ g).inv | by simp [←cancel_mono (f ◁ (λ_ g).hom)] | lemma | category_theory.bicategory.triangle_assoc_comp_right_inv | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
triangle_assoc_comp_left_inv (f : a ⟶ b) (g : b ⟶ c) :
f ◁ (λ_ g).inv ≫ (α_ f (𝟙 b) g).inv = (ρ_ f).inv ▷ g | by simp [←cancel_mono ((ρ_ f).hom ▷ g)] | lemma | category_theory.bicategory.triangle_assoc_comp_left_inv | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associator_naturality_left {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d) :
(η ▷ g) ▷ h ≫ (α_ f' g h).hom = (α_ f g h).hom ≫ η ▷ (g ≫ h) | by simp | lemma | category_theory.bicategory.associator_naturality_left | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associator_inv_naturality_left {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d) :
η ▷ (g ≫ h) ≫ (α_ f' g h).inv = (α_ f g h).inv ≫ (η ▷ g) ▷ h | by simp | lemma | category_theory.bicategory.associator_inv_naturality_left | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right_comp_symm {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d) :
(η ▷ g) ▷ h = (α_ f g h).hom ≫ η ▷ (g ≫ h) ≫ (α_ f' g h).inv | by simp | lemma | category_theory.bicategory.whisker_right_comp_symm | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associator_naturality_middle (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d) :
(f ◁ η) ▷ h ≫ (α_ f g' h).hom = (α_ f g h).hom ≫ f ◁ (η ▷ h) | by simp | lemma | category_theory.bicategory.associator_naturality_middle | category_theory.bicategory | src/category_theory/bicategory/basic.lean | [
"category_theory.isomorphism",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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