statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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hom_equiv_naturality_left (f : X' ⟶ X) (g : F.obj X ⟶ Y) :
(adj.hom_equiv X' Y) (F.map f ≫ g) = f ≫ (adj.hom_equiv X Y) g | by rw [← equiv.eq_symm_apply]; simp | lemma | category_theory.adjunction.core_hom_equiv.hom_equiv_naturality_left | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"equiv.eq_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_equiv_naturality_right_symm (f : X ⟶ G.obj Y) (g : Y ⟶ Y') :
(adj.hom_equiv X Y').symm (f ≫ G.map g) = (adj.hom_equiv X Y).symm f ≫ g | by rw [equiv.symm_apply_eq]; simp | lemma | category_theory.adjunction.core_hom_equiv.hom_equiv_naturality_right_symm | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"equiv.symm_apply_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
core_unit_counit (F : C ⥤ D) (G : D ⥤ C) | (unit : 𝟭 C ⟶ F.comp G)
(counit : G.comp F ⟶ 𝟭 D)
(left_triangle' : whisker_right unit F ≫ (functor.associator F G F).hom ≫ whisker_left F counit =
nat_trans.id (𝟭 C ⋙ F) . obviously)
(right_triangle' : whisker_left G unit ≫ (functor.associator G F G).inv ≫ whisker_right counit G =
nat_trans.id (G ⋙ 𝟭 C) . obvi... | structure | category_theory.adjunction.core_unit_counit | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | This is an auxiliary data structure useful for constructing adjunctions.
See `adjunction.mk_of_unit_counit`.
This structure won't typically be used anywhere else. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_of_hom_equiv (adj : core_hom_equiv F G) : F ⊣ G | { unit :=
{ app := λ X, (adj.hom_equiv X (F.obj X)) (𝟙 (F.obj X)),
naturality' :=
begin
intros,
erw [← adj.hom_equiv_naturality_left, ← adj.hom_equiv_naturality_right],
dsimp, simp -- See note [dsimp, simp].
end },
counit :=
{ app := λ Y, (adj.hom_equiv _ _).inv_fun (𝟙 (G.obj Y)),
... | def | category_theory.adjunction.mk_of_hom_equiv | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"adj",
"inv_fun"
] | Construct an adjunction between `F` and `G` out of a natural bijection between each
`F.obj X ⟶ Y` and `X ⟶ G.obj Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_of_unit_counit (adj : core_unit_counit F G) : F ⊣ G | { hom_equiv := λ X Y,
{ to_fun := λ f, adj.unit.app X ≫ G.map f,
inv_fun := λ g, F.map g ≫ adj.counit.app Y,
left_inv := λ f, begin
change F.map (_ ≫ _) ≫ _ = _,
rw [F.map_comp, assoc, ←functor.comp_map, adj.counit.naturality, ←assoc],
convert id_comp f,
have t := congr_arg (λ t : nat_... | def | category_theory.adjunction.mk_of_unit_counit | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"adj",
"inv_fun"
] | Construct an adjunction between functors `F` and `G` given a unit and counit for the adjunction
satisfying the triangle identities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : 𝟭 C ⊣ 𝟭 C | { hom_equiv := λ X Y, equiv.refl _,
unit := 𝟙 _,
counit := 𝟙 _ } | def | category_theory.adjunction.id | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"equiv.refl"
] | The adjunction between the identity functor on a category and itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_homset_left_of_nat_iso
{F F' : C ⥤ D} (iso : F ≅ F') {X : C} {Y : D} :
(F.obj X ⟶ Y) ≃ (F'.obj X ⟶ Y) | { to_fun := λ f, iso.inv.app _ ≫ f,
inv_fun := λ g, iso.hom.app _ ≫ g,
left_inv := λ f, by simp,
right_inv := λ g, by simp } | def | category_theory.adjunction.equiv_homset_left_of_nat_iso | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"inv_fun",
"iso"
] | If F and G are naturally isomorphic functors, establish an equivalence of hom-sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_homset_right_of_nat_iso
{G G' : D ⥤ C} (iso : G ≅ G') {X : C} {Y : D} :
(X ⟶ G.obj Y) ≃ (X ⟶ G'.obj Y) | { to_fun := λ f, f ≫ iso.hom.app _,
inv_fun := λ g, g ≫ iso.inv.app _,
left_inv := λ f, by simp,
right_inv := λ g, by simp } | def | category_theory.adjunction.equiv_homset_right_of_nat_iso | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"inv_fun",
"iso"
] | If G and H are naturally isomorphic functors, establish an equivalence of hom-sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_nat_iso_left
{F G : C ⥤ D} {H : D ⥤ C} (adj : F ⊣ H) (iso : F ≅ G) :
G ⊣ H | adjunction.mk_of_hom_equiv
{ hom_equiv := λ X Y, (equiv_homset_left_of_nat_iso iso.symm).trans (adj.hom_equiv X Y) } | def | category_theory.adjunction.of_nat_iso_left | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"adj",
"iso"
] | Transport an adjunction along an natural isomorphism on the left. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_nat_iso_right
{F : C ⥤ D} {G H : D ⥤ C} (adj : F ⊣ G) (iso : G ≅ H) :
F ⊣ H | adjunction.mk_of_hom_equiv
{ hom_equiv := λ X Y, (adj.hom_equiv X Y).trans (equiv_homset_right_of_nat_iso iso) } | def | category_theory.adjunction.of_nat_iso_right | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"adj",
"iso"
] | Transport an adjunction along an natural isomorphism on the right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_adjoint_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [r : is_right_adjoint F] :
is_right_adjoint G | { left := r.left,
adj := of_nat_iso_right r.adj h } | def | category_theory.adjunction.right_adjoint_of_nat_iso | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"adj"
] | Transport being a right adjoint along a natural isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_adjoint_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [r : is_left_adjoint F] : is_left_adjoint G | { right := r.right,
adj := of_nat_iso_left r.adj h } | def | category_theory.adjunction.left_adjoint_of_nat_iso | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"adj"
] | Transport being a left adjoint along a natural isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp (adj₁ : F ⊣ G) (adj₂ : H ⊣ I) : F ⋙ H ⊣ I ⋙ G | { hom_equiv := λ X Z, equiv.trans (adj₂.hom_equiv _ _) (adj₁.hom_equiv _ _),
unit := adj₁.unit ≫
(whisker_left F $ whisker_right adj₂.unit G) ≫ (functor.associator _ _ _).inv,
counit := (functor.associator _ _ _).hom ≫
(whisker_left I $ whisker_right adj₁.counit H) ≫ adj₂.counit } | def | category_theory.adjunction.comp | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"equiv.trans"
] | Composition of adjunctions.
See <https://stacks.math.columbia.edu/tag/0DV0>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_adjoint_of_comp {E : Type u₃} [ℰ : category.{v₃} E] (F : C ⥤ D) (G : D ⥤ E)
[Fl : is_left_adjoint F] [Gl : is_left_adjoint G] : is_left_adjoint (F ⋙ G) | { right := Gl.right ⋙ Fl.right,
adj := Fl.adj.comp Gl.adj } | instance | category_theory.adjunction.left_adjoint_of_comp | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"adj"
] | If `F` and `G` are left adjoints then `F ⋙ G` is a left adjoint too. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_adjoint_of_comp {E : Type u₃} [ℰ : category.{v₃} E] {F : C ⥤ D} {G : D ⥤ E}
[Fr : is_right_adjoint F] [Gr : is_right_adjoint G] : is_right_adjoint (F ⋙ G) | { left := Gr.left ⋙ Fr.left,
adj := Gr.adj.comp Fr.adj } | instance | category_theory.adjunction.right_adjoint_of_comp | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"adj"
] | If `F` and `G` are right adjoints then `F ⋙ G` is a right adjoint too. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
he' {X Y Y'} (f g) : (e X Y').symm (f ≫ G.map g) = (e X Y).symm f ≫ g | by intros; rw [equiv.symm_apply_eq, he]; simp | lemma | category_theory.adjunction.he' | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"equiv.symm_apply_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_adjoint_of_equiv : C ⥤ D | { obj := F_obj,
map := λ X X' f, (e X (F_obj X')).symm (f ≫ e X' (F_obj X') (𝟙 _)),
map_comp' := λ X X' X'' f f', begin
rw [equiv.symm_apply_eq, he, equiv.apply_symm_apply],
conv { to_rhs, rw [assoc, ←he, id_comp, equiv.apply_symm_apply] },
simp
end } | def | category_theory.adjunction.left_adjoint_of_equiv | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"equiv.apply_symm_apply",
"equiv.symm_apply_eq"
] | Construct a left adjoint functor to `G`, given the functor's value on objects `F_obj` and
a bijection `e` between `F_obj X ⟶ Y` and `X ⟶ G.obj Y` satisfying a naturality law
`he : ∀ X Y Y' g h, e X Y' (h ≫ g) = e X Y h ≫ G.map g`.
Dual to `right_adjoint_of_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjunction_of_equiv_left : left_adjoint_of_equiv e he ⊣ G | mk_of_hom_equiv
{ hom_equiv := e,
hom_equiv_naturality_left_symm' :=
begin
intros,
erw [← he' e he, ← equiv.apply_eq_iff_eq],
simp [(he _ _ _ _ _).symm]
end } | def | category_theory.adjunction.adjunction_of_equiv_left | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"equiv.apply_eq_iff_eq"
] | Show that the functor given by `left_adjoint_of_equiv` is indeed left adjoint to `G`. Dual
to `adjunction_of_equiv_right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
he' {X' X Y} (f g) : F.map f ≫ (e X Y).symm g = (e X' Y).symm (f ≫ g) | by intros; rw [equiv.eq_symm_apply, he]; simp | lemma | category_theory.adjunction.he' | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"equiv.eq_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_adjoint_of_equiv : D ⥤ C | { obj := G_obj,
map := λ Y Y' g, (e (G_obj Y) Y') ((e (G_obj Y) Y).symm (𝟙 _) ≫ g),
map_comp' := λ Y Y' Y'' g g', begin
rw [← equiv.eq_symm_apply, ← he' e he, equiv.symm_apply_apply],
conv { to_rhs, rw [← assoc, he' e he, comp_id, equiv.symm_apply_apply] },
simp
end } | def | category_theory.adjunction.right_adjoint_of_equiv | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"equiv.eq_symm_apply",
"equiv.symm_apply_apply"
] | Construct a right adjoint functor to `F`, given the functor's value on objects `G_obj` and
a bijection `e` between `F.obj X ⟶ Y` and `X ⟶ G_obj Y` satisfying a naturality law
`he : ∀ X Y Y' g h, e X' Y (F.map f ≫ g) = f ≫ e X Y g`.
Dual to `left_adjoint_of_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjunction_of_equiv_right : F ⊣ right_adjoint_of_equiv e he | mk_of_hom_equiv
{ hom_equiv := e,
hom_equiv_naturality_left_symm' := by intros; rw [equiv.symm_apply_eq, he]; simp,
hom_equiv_naturality_right' :=
begin
intros X Y Y' g h,
erw [←he, equiv.apply_eq_iff_eq, ←assoc, he' e he, comp_id, equiv.symm_apply_apply]
end } | def | category_theory.adjunction.adjunction_of_equiv_right | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"equiv.apply_eq_iff_eq",
"equiv.symm_apply_apply",
"equiv.symm_apply_eq"
] | Show that the functor given by `right_adjoint_of_equiv` is indeed right adjoint to `F`. Dual
to `adjunction_of_equiv_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_equivalence (adj : F ⊣ G) [∀ X, is_iso (adj.unit.app X)] [∀ Y, is_iso (adj.counit.app Y)] :
C ≌ D | { functor := F,
inverse := G,
unit_iso := nat_iso.of_components (λ X, as_iso (adj.unit.app X)) (by simp),
counit_iso := nat_iso.of_components (λ Y, as_iso (adj.counit.app Y)) (by simp) } | def | category_theory.adjunction.to_equivalence | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"adj"
] | If the unit and counit of a given adjunction are (pointwise) isomorphisms, then we can upgrade the
adjunction to an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_adjoint_to_is_equivalence [is_right_adjoint G]
[∀ X, is_iso ((adjunction.of_right_adjoint G).unit.app X)]
[∀ Y, is_iso ((adjunction.of_right_adjoint G).counit.app Y)] :
is_equivalence G | is_equivalence.of_equivalence_inverse (adjunction.of_right_adjoint G).to_equivalence | def | category_theory.adjunction.is_right_adjoint_to_is_equivalence | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | If the unit and counit for the adjunction corresponding to a right adjoint functor are (pointwise)
isomorphisms, then the functor is an equivalence of categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_adjunction (e : C ≌ D) : e.functor ⊣ e.inverse | mk_of_unit_counit ⟨e.unit, e.counit,
by { ext, dsimp, simp only [id_comp], exact e.functor_unit_comp _, },
by { ext, dsimp, simp only [id_comp], exact e.unit_inverse_comp _, }⟩ | def | category_theory.equivalence.to_adjunction | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | The adjunction given by an equivalence of categories. (To obtain the opposite adjunction,
simply use `e.symm.to_adjunction`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_equivalence_to_adjunction_unit {e : C ≌ D} :
e.functor.as_equivalence.to_adjunction.unit = e.unit | rfl | lemma | category_theory.equivalence.as_equivalence_to_adjunction_unit | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_equivalence_to_adjunction_counit {e : C ≌ D} :
e.functor.as_equivalence.to_adjunction.counit = e.counit | rfl | lemma | category_theory.equivalence.as_equivalence_to_adjunction_counit | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjunction (E : C ⥤ D) [is_equivalence E] : E ⊣ E.inv | (E.as_equivalence).to_adjunction | def | category_theory.functor.adjunction | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | An equivalence `E` is left adjoint to its inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_adjoint_of_equivalence {F : C ⥤ D} [is_equivalence F] : is_left_adjoint F | { right := _,
adj := functor.adjunction F } | instance | category_theory.functor.left_adjoint_of_equivalence | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"adj"
] | If `F` is an equivalence, it's a left adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_adjoint_of_is_equivalence {F : C ⥤ D} [is_equivalence F] : right_adjoint F = inv F | rfl | lemma | category_theory.functor.right_adjoint_of_is_equivalence | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_adjoint_of_equivalence {F : C ⥤ D} [is_equivalence F] : is_right_adjoint F | { left := _,
adj := functor.adjunction F.inv } | instance | category_theory.functor.right_adjoint_of_equivalence | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"adj"
] | If `F` is an equivalence, it's a right adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_adjoint_of_is_equivalence {F : C ⥤ D} [is_equivalence F] : left_adjoint F = inv F | rfl | lemma | category_theory.functor.left_adjoint_of_is_equivalence | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_adjoint_of_structured_arrow_initials_aux (A : C) (B : D) :
((⊥_ (structured_arrow A G)).right ⟶ B) ≃ (A ⟶ G.obj B) | { to_fun := λ g, (⊥_ (structured_arrow A G)).hom ≫ G.map g,
inv_fun := λ f, comma_morphism.right (initial.to (structured_arrow.mk f)),
left_inv := λ g,
begin
let B' : structured_arrow A G :=
structured_arrow.mk ((⊥_ (structured_arrow A G)).hom ≫ G.map g),
let g' : ⊥_ (structured_arrow A G) ⟶ B' := s... | def | category_theory.left_adjoint_of_structured_arrow_initials_aux | category_theory.adjunction | src/category_theory/adjunction/comma.lean | [
"category_theory.adjunction.basic",
"category_theory.punit",
"category_theory.structured_arrow"
] | [
"inv_fun"
] | Implementation: If each structured arrow category on `G` has an initial object, an equivalence
which is helpful for constructing a left adjoint to `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_adjoint_of_structured_arrow_initials : C ⥤ D | adjunction.left_adjoint_of_equiv (left_adjoint_of_structured_arrow_initials_aux G) (λ _ _, by simp) | def | category_theory.left_adjoint_of_structured_arrow_initials | category_theory.adjunction | src/category_theory/adjunction/comma.lean | [
"category_theory.adjunction.basic",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | If each structured arrow category on `G` has an initial object, construct a left adjoint to `G`. It
is shown that it is a left adjoint in `adjunction_of_structured_arrow_initials`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjunction_of_structured_arrow_initials :
left_adjoint_of_structured_arrow_initials G ⊣ G | adjunction.adjunction_of_equiv_left _ _ | def | category_theory.adjunction_of_structured_arrow_initials | category_theory.adjunction | src/category_theory/adjunction/comma.lean | [
"category_theory.adjunction.basic",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | If each structured arrow category on `G` has an initial object, we have a constructed left adjoint
to `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_adjoint_of_structured_arrow_initials : is_right_adjoint G | { left := _, adj := adjunction_of_structured_arrow_initials G } | def | category_theory.is_right_adjoint_of_structured_arrow_initials | category_theory.adjunction | src/category_theory/adjunction/comma.lean | [
"category_theory.adjunction.basic",
"category_theory.punit",
"category_theory.structured_arrow"
] | [
"adj"
] | If each structured arrow category on `G` has an initial object, `G` is a right adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_adjoint_of_costructured_arrow_terminals_aux (B : D) (A : C) :
(G.obj B ⟶ A) ≃ (B ⟶ (⊤_ (costructured_arrow G A)).left) | { to_fun := λ g, comma_morphism.left (terminal.from (costructured_arrow.mk g)),
inv_fun := λ g, G.map g ≫ (⊤_ (costructured_arrow G A)).hom,
left_inv := by tidy,
right_inv := λ g,
begin
let B' : costructured_arrow G A :=
costructured_arrow.mk (G.map g ≫ (⊤_ (costructured_arrow G A)).hom),
let g' :... | def | category_theory.right_adjoint_of_costructured_arrow_terminals_aux | category_theory.adjunction | src/category_theory/adjunction/comma.lean | [
"category_theory.adjunction.basic",
"category_theory.punit",
"category_theory.structured_arrow"
] | [
"inv_fun"
] | Implementation: If each costructured arrow category on `G` has a terminal object, an equivalence
which is helpful for constructing a right adjoint to `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_adjoint_of_costructured_arrow_terminals : C ⥤ D | adjunction.right_adjoint_of_equiv (right_adjoint_of_costructured_arrow_terminals_aux G)
(λ B₁ B₂ A f g, by { rw ←equiv.eq_symm_apply, simp }) | def | category_theory.right_adjoint_of_costructured_arrow_terminals | category_theory.adjunction | src/category_theory/adjunction/comma.lean | [
"category_theory.adjunction.basic",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | If each costructured arrow category on `G` has a terminal object, construct a right adjoint to `G`.
It is shown that it is a right adjoint in `adjunction_of_structured_arrow_initials`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjunction_of_costructured_arrow_terminals :
G ⊣ right_adjoint_of_costructured_arrow_terminals G | adjunction.adjunction_of_equiv_right _ _ | def | category_theory.adjunction_of_costructured_arrow_terminals | category_theory.adjunction | src/category_theory/adjunction/comma.lean | [
"category_theory.adjunction.basic",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | If each costructured arrow category on `G` has a terminal object, we have a constructed right
adjoint to `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_adjoint_of_costructured_arrow_terminals : is_left_adjoint G | { right := right_adjoint_of_costructured_arrow_terminals G,
adj := adjunction.adjunction_of_equiv_right _ _ } | def | category_theory.is_left_adjoint_of_costructured_arrow_terminals | category_theory.adjunction | src/category_theory/adjunction/comma.lean | [
"category_theory.adjunction.basic",
"category_theory.punit",
"category_theory.structured_arrow"
] | [
"adj"
] | If each costructured arrow category on `G` has an terminal object, `G` is a left adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_initial_of_left_adjoint (h : F ⊣ G) (A : C) :
is_initial (structured_arrow.mk (h.unit.app A) : structured_arrow A G) | { desc := λ B, structured_arrow.hom_mk ((h.hom_equiv _ _).symm B.X.hom) (by tidy),
uniq' := λ s m w,
begin
ext,
dsimp,
rw [equiv.eq_symm_apply, adjunction.hom_equiv_unit],
apply structured_arrow.w m,
end } | def | category_theory.mk_initial_of_left_adjoint | category_theory.adjunction | src/category_theory/adjunction/comma.lean | [
"category_theory.adjunction.basic",
"category_theory.punit",
"category_theory.structured_arrow"
] | [
"equiv.eq_symm_apply"
] | Given a left adjoint to `G`, we can construct an initial object in each structured arrow
category on `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_terminal_of_right_adjoint (h : F ⊣ G) (A : D) :
is_terminal (costructured_arrow.mk (h.counit.app A) : costructured_arrow F A) | { lift := λ B, costructured_arrow.hom_mk (h.hom_equiv _ _ B.X.hom) (by tidy),
uniq' := λ s m w,
begin
ext,
dsimp,
rw [h.eq_hom_equiv_apply, adjunction.hom_equiv_counit],
exact costructured_arrow.w m,
end } | def | category_theory.mk_terminal_of_right_adjoint | category_theory.adjunction | src/category_theory/adjunction/comma.lean | [
"category_theory.adjunction.basic",
"category_theory.punit",
"category_theory.structured_arrow"
] | [
"lift"
] | Given a right adjoint to `F`, we can construct a terminal object in each costructured arrow
category on `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_is_right_adjoint_iff_has_initial_structured_arrow {G : D ⥤ C} :
nonempty (is_right_adjoint G) ↔ ∀ A, has_initial (structured_arrow A G) | ⟨λ ⟨h⟩ A, by exactI (mk_initial_of_left_adjoint _ h.adj A).has_initial,
λ h, by exactI ⟨is_right_adjoint_of_structured_arrow_initials _⟩⟩ | lemma | category_theory.nonempty_is_right_adjoint_iff_has_initial_structured_arrow | category_theory.adjunction | src/category_theory/adjunction/comma.lean | [
"category_theory.adjunction.basic",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_is_left_adjoint_iff_has_terminal_costructured_arrow {F : C ⥤ D} :
nonempty (is_left_adjoint F) ↔ ∀ A, has_terminal (costructured_arrow F A) | ⟨λ ⟨h⟩ A, by exactI (mk_terminal_of_right_adjoint _ h.adj A).has_terminal,
λ h, by exactI ⟨is_left_adjoint_of_costructured_arrow_terminals _⟩⟩ | lemma | category_theory.nonempty_is_left_adjoint_iff_has_terminal_costructured_arrow | category_theory.adjunction | src/category_theory/adjunction/comma.lean | [
"category_theory.adjunction.basic",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
evaluation_left_adjoint (c : C) : D ⥤ C ⥤ D | { obj := λ d,
{ obj := λ t, ∐ (λ i : c ⟶ t, d),
map := λ u v f, sigma.desc $ λ g, sigma.ι (λ _, d) $ g ≫ f,
map_id' := begin
intros, ext ⟨j⟩, simp only [cofan.mk_ι_app, colimit.ι_desc, category.comp_id],
congr' 1, rw category.comp_id,
end,
map_comp' := begin
intros, ext, simp only [c... | def | category_theory.evaluation_left_adjoint | category_theory.adjunction | src/category_theory/adjunction/evaluation.lean | [
"category_theory.limits.shapes.products",
"category_theory.functor.epi_mono"
] | [] | The left adjoint of evaluation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
evaluation_adjunction_right (c : C) :
evaluation_left_adjoint D c ⊣ (evaluation _ _).obj c | adjunction.mk_of_hom_equiv
{ hom_equiv := λ d F,
{ to_fun := λ f, sigma.ι (λ _, d) (𝟙 _) ≫ f.app c,
inv_fun := λ f,
{ app := λ e, sigma.desc $ λ h, f ≫ F.map h,
naturality' := by { intros, ext, dsimp, simp } },
left_inv := begin
intros f,
ext x ⟨g⟩,
dsimp,
simp only [colimit... | def | category_theory.evaluation_adjunction_right | category_theory.adjunction | src/category_theory/adjunction/evaluation.lean | [
"category_theory.limits.shapes.products",
"category_theory.functor.epi_mono"
] | [
"inv_fun"
] | The adjunction showing that evaluation is a right adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
evaluation_is_right_adjoint (c : C) :
is_right_adjoint ((evaluation _ D).obj c) | ⟨_, evaluation_adjunction_right _ _⟩ | instance | category_theory.evaluation_is_right_adjoint | category_theory.adjunction | src/category_theory/adjunction/evaluation.lean | [
"category_theory.limits.shapes.products",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans.mono_iff_mono_app {F G : C ⥤ D} (η : F ⟶ G) :
mono η ↔ (∀ c, mono (η.app c)) | begin
split,
{ introsI h c,
exact (infer_instance : mono (((evaluation _ _).obj c).map η)) },
{ introsI _,
apply nat_trans.mono_of_mono_app }
end | lemma | category_theory.nat_trans.mono_iff_mono_app | category_theory.adjunction | src/category_theory/adjunction/evaluation.lean | [
"category_theory.limits.shapes.products",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
evaluation_right_adjoint (c : C) : D ⥤ C ⥤ D | { obj := λ d,
{ obj := λ t, ∏ (λ i : t ⟶ c, d),
map := λ u v f, pi.lift $ λ g, pi.π _ $ f ≫ g,
map_id' := begin
intros, ext ⟨j⟩, dsimp,
simp only [limit.lift_π, category.id_comp, fan.mk_π_app],
congr, simp,
end,
map_comp' := begin
intros, ext ⟨j⟩, dsimp,
simp only [limit.... | def | category_theory.evaluation_right_adjoint | category_theory.adjunction | src/category_theory/adjunction/evaluation.lean | [
"category_theory.limits.shapes.products",
"category_theory.functor.epi_mono"
] | [] | The right adjoint of evaluation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
evaluation_adjunction_left (c : C) :
(evaluation _ _).obj c ⊣ evaluation_right_adjoint D c | adjunction.mk_of_hom_equiv
{ hom_equiv := λ F d,
{ to_fun := λ f,
{ app := λ t, pi.lift $ λ g, F.map g ≫ f,
naturality' := by { intros, ext, dsimp, simp } },
inv_fun := λ f, f.app _ ≫ pi.π _ (𝟙 _),
left_inv := λ f, by { dsimp, simp },
right_inv := begin
intros f,
ext x ⟨g⟩,
ds... | def | category_theory.evaluation_adjunction_left | category_theory.adjunction | src/category_theory/adjunction/evaluation.lean | [
"category_theory.limits.shapes.products",
"category_theory.functor.epi_mono"
] | [
"inv_fun"
] | The adjunction showing that evaluation is a left adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
evaluation_is_left_adjoint (c : C) :
is_left_adjoint ((evaluation _ D).obj c) | ⟨_, evaluation_adjunction_left _ _⟩ | instance | category_theory.evaluation_is_left_adjoint | category_theory.adjunction | src/category_theory/adjunction/evaluation.lean | [
"category_theory.limits.shapes.products",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans.epi_iff_epi_app {F G : C ⥤ D} (η : F ⟶ G) :
epi η ↔ (∀ c, epi (η.app c)) | begin
split,
{ introsI h c,
exact (infer_instance : epi (((evaluation _ _).obj c).map η)) },
{ introsI,
apply nat_trans.epi_of_epi_app }
end | lemma | category_theory.nat_trans.epi_iff_epi_app | category_theory.adjunction | src/category_theory/adjunction/evaluation.lean | [
"category_theory.limits.shapes.products",
"category_theory.functor.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_is_iso_of_L_fully_faithful [full L] [faithful L] : is_iso (adjunction.unit h) | @nat_iso.is_iso_of_is_iso_app _ _ _ _ _ _ (adjunction.unit h) $ λ X,
@yoneda.is_iso _ _ _ _ ((adjunction.unit h).app X)
⟨⟨{ app := λ Y f, L.preimage ((h.hom_equiv (unop Y) (L.obj X)).symm f) },
⟨begin
ext x f, dsimp,
apply L.map_injective,
simp,
end, begin
ext x f, dsimp,
simp only [adjunction.h... | instance | category_theory.unit_is_iso_of_L_fully_faithful | category_theory.adjunction | src/category_theory/adjunction/fully_faithful.lean | [
"category_theory.adjunction.basic",
"category_theory.conj",
"category_theory.yoneda"
] | [] | If the left adjoint is fully faithful, then the unit is an isomorphism.
See
* Lemma 4.5.13 from [Riehl][riehl2017]
* https://math.stackexchange.com/a/2727177
* https://stacks.math.columbia.edu/tag/07RB (we only prove the forward direction!) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
counit_is_iso_of_R_fully_faithful [full R] [faithful R] : is_iso (adjunction.counit h) | @nat_iso.is_iso_of_is_iso_app _ _ _ _ _ _ (adjunction.counit h) $ λ X,
@is_iso_of_op _ _ _ _ _ $
@coyoneda.is_iso _ _ _ _ ((adjunction.counit h).app X).op
⟨⟨{ app := λ Y f, R.preimage ((h.hom_equiv (R.obj X) Y) f) },
⟨begin
ext x f, dsimp,
apply R.map_injective,
simp,
end, begin
ext x f, dsimp,
... | instance | category_theory.counit_is_iso_of_R_fully_faithful | category_theory.adjunction | src/category_theory/adjunction/fully_faithful.lean | [
"category_theory.adjunction.basic",
"category_theory.conj",
"category_theory.yoneda"
] | [] | If the right adjoint is fully faithful, then the counit is an isomorphism.
See <https://stacks.math.columbia.edu/tag/07RB> (we only prove the forward direction!) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_map_unit {X : C} [is_iso (h.unit.app X)] :
inv (L.map (h.unit.app X)) = h.counit.app (L.obj X) | is_iso.inv_eq_of_hom_inv_id h.left_triangle_components | lemma | category_theory.inv_map_unit | category_theory.adjunction | src/category_theory/adjunction/fully_faithful.lean | [
"category_theory.adjunction.basic",
"category_theory.conj",
"category_theory.yoneda"
] | [] | If the unit of an adjunction is an isomorphism, then its inverse on the image of L is given
by L whiskered with the counit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_left_L_counit_iso_of_is_iso_unit [is_iso h.unit] :
L ⋙ R ⋙ L ≅ L | (L.associator R L).symm ≪≫ iso_whisker_right (as_iso h.unit).symm L ≪≫ functor.left_unitor _ | def | category_theory.whisker_left_L_counit_iso_of_is_iso_unit | category_theory.adjunction | src/category_theory/adjunction/fully_faithful.lean | [
"category_theory.adjunction.basic",
"category_theory.conj",
"category_theory.yoneda"
] | [] | If the unit is an isomorphism, bundle one has an isomorphism `L ⋙ R ⋙ L ≅ L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_counit_map {X : D} [is_iso (h.counit.app X)] :
inv (R.map (h.counit.app X)) = h.unit.app (R.obj X) | is_iso.inv_eq_of_inv_hom_id h.right_triangle_components | lemma | category_theory.inv_counit_map | category_theory.adjunction | src/category_theory/adjunction/fully_faithful.lean | [
"category_theory.adjunction.basic",
"category_theory.conj",
"category_theory.yoneda"
] | [] | If the counit of an adjunction is an isomorphism, then its inverse on the image of R is given
by R whiskered with the unit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_left_R_unit_iso_of_is_iso_counit [is_iso h.counit] :
(R ⋙ L ⋙ R) ≅ R | (R.associator L R).symm ≪≫ iso_whisker_right (as_iso h.counit) R ≪≫ functor.left_unitor _ | def | category_theory.whisker_left_R_unit_iso_of_is_iso_counit | category_theory.adjunction | src/category_theory/adjunction/fully_faithful.lean | [
"category_theory.adjunction.basic",
"category_theory.conj",
"category_theory.yoneda"
] | [] | If the counit of an is an isomorphism, one has an isomorphism `(R ⋙ L ⋙ R) ≅ R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
L_full_of_unit_is_iso [is_iso h.unit] : full L | { preimage := λ X Y f, (h.hom_equiv X (L.obj Y) f) ≫ inv (h.unit.app Y) } | def | category_theory.L_full_of_unit_is_iso | category_theory.adjunction | src/category_theory/adjunction/fully_faithful.lean | [
"category_theory.adjunction.basic",
"category_theory.conj",
"category_theory.yoneda"
] | [] | If the unit is an isomorphism, then the left adjoint is full | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
L_faithful_of_unit_is_iso [is_iso h.unit] : faithful L | { map_injective' := λ X Y f g H,
begin
rw ←(h.hom_equiv X (L.obj Y)).apply_eq_iff_eq at H,
simpa using H =≫ inv (h.unit.app Y),
end } | lemma | category_theory.L_faithful_of_unit_is_iso | category_theory.adjunction | src/category_theory/adjunction/fully_faithful.lean | [
"category_theory.adjunction.basic",
"category_theory.conj",
"category_theory.yoneda"
] | [] | If the unit is an isomorphism, then the left adjoint is faithful | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
R_full_of_counit_is_iso [is_iso h.counit] : full R | { preimage := λ X Y f, inv (h.counit.app X) ≫ (h.hom_equiv (R.obj X) Y).symm f } | def | category_theory.R_full_of_counit_is_iso | category_theory.adjunction | src/category_theory/adjunction/fully_faithful.lean | [
"category_theory.adjunction.basic",
"category_theory.conj",
"category_theory.yoneda"
] | [] | If the counit is an isomorphism, then the right adjoint is full | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
R_faithful_of_counit_is_iso [is_iso h.counit] : faithful R | { map_injective' := λ X Y f g H,
begin
rw ←(h.hom_equiv (R.obj X) Y).symm.apply_eq_iff_eq at H,
simpa using inv (h.counit.app X) ≫= H,
end } | lemma | category_theory.R_faithful_of_counit_is_iso | category_theory.adjunction | src/category_theory/adjunction/fully_faithful.lean | [
"category_theory.adjunction.basic",
"category_theory.conj",
"category_theory.yoneda"
] | [] | If the counit is an isomorphism, then the right adjoint is faithful | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_left_counit_iso_of_L_fully_faithful
[full L] [faithful L] : is_iso (whisker_left L h.counit) | begin
have := h.left_triangle,
rw ←is_iso.eq_inv_comp at this,
rw this,
apply_instance
end | instance | category_theory.whisker_left_counit_iso_of_L_fully_faithful | category_theory.adjunction | src/category_theory/adjunction/fully_faithful.lean | [
"category_theory.adjunction.basic",
"category_theory.conj",
"category_theory.yoneda"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right_counit_iso_of_L_fully_faithful
[full L] [faithful L] : is_iso (whisker_right h.counit R) | begin
have := h.right_triangle,
rw ←is_iso.eq_inv_comp at this,
rw this,
apply_instance
end | instance | category_theory.whisker_right_counit_iso_of_L_fully_faithful | category_theory.adjunction | src/category_theory/adjunction/fully_faithful.lean | [
"category_theory.adjunction.basic",
"category_theory.conj",
"category_theory.yoneda"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_left_unit_iso_of_R_fully_faithful
[full R] [faithful R] : is_iso (whisker_left R h.unit) | begin
have := h.right_triangle,
rw ←is_iso.eq_comp_inv at this,
rw this,
apply_instance
end | instance | category_theory.whisker_left_unit_iso_of_R_fully_faithful | category_theory.adjunction | src/category_theory/adjunction/fully_faithful.lean | [
"category_theory.adjunction.basic",
"category_theory.conj",
"category_theory.yoneda"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker_right_unit_iso_of_R_fully_faithful
[full R] [faithful R] : is_iso (whisker_right h.unit L) | begin
have := h.left_triangle,
rw ←is_iso.eq_comp_inv at this,
rw this,
apply_instance
end | instance | category_theory.whisker_right_unit_iso_of_R_fully_faithful | category_theory.adjunction | src/category_theory/adjunction/fully_faithful.lean | [
"category_theory.adjunction.basic",
"category_theory.conj",
"category_theory.yoneda"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjunction.restrict_fully_faithful (iC : C ⥤ C') (iD : D ⥤ D') {L' : C' ⥤ D'} {R' : D' ⥤ C'}
(adj : L' ⊣ R') {L : C ⥤ D} {R : D ⥤ C} (comm1 : iC ⋙ L' ≅ L ⋙ iD) (comm2 : iD ⋙ R' ≅ R ⋙ iC)
[full iC] [faithful iC] [full iD] [faithful iD] :
L ⊣ R | adjunction.mk_of_hom_equiv
{ hom_equiv := λ X Y,
calc (L.obj X ⟶ Y) ≃ (iD.obj (L.obj X) ⟶ iD.obj Y) : equiv_of_fully_faithful iD
... ≃ (L'.obj (iC.obj X) ⟶ iD.obj Y) : iso.hom_congr (comm1.symm.app X) (iso.refl _)
... ≃ (iC.obj X ⟶ R'.obj (iD.obj Y)) : adj.hom_equiv _ _
... ≃ (iC.obj X ⟶ iC.obj (... | def | category_theory.adjunction.restrict_fully_faithful | category_theory.adjunction | src/category_theory/adjunction/fully_faithful.lean | [
"category_theory.adjunction.basic",
"category_theory.conj",
"category_theory.yoneda"
] | [
"adj"
] | If `C` is a full subcategory of `C'` and `D` is a full subcategory of `D'`, then we can restrict
an adjunction `L' ⊣ R'` where `L' : C' ⥤ D'` and `R' : D' ⥤ C'` to `C` and `D`.
The construction here is slightly more general, in that `C` is required only to have a full and
faithful "inclusion" functor `iC : C ⥤ C'` (and... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
counit_coequalises [∀ (X : B), regular_epi (adj₁.counit.app X)] (X : B) :
is_colimit (cofork.of_π (adj₁.counit.app X) (adj₁.counit_naturality _)) | cofork.is_colimit.mk' _ $ λ s,
begin
refine ⟨(regular_epi.desc' (adj₁.counit.app X) s.π _).1, _, _⟩,
{ rw ← cancel_epi (adj₁.counit.app (regular_epi.W (adj₁.counit.app X))),
rw ← adj₁.counit_naturality_assoc,
dsimp only [functor.comp_obj],
rw [← s.condition, ← F.map_comp_assoc, ← U.map_comp, regular_epi... | def | category_theory.lift_adjoint.counit_coequalises | category_theory.adjunction | src/category_theory/adjunction/lifting.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.reflexive",
"category_theory.monad.adjunction",
"category_theory.monad.coequalizer"
] | [] | To show that `ε_X` is a coequalizer for `(FUε_X, ε_FUX)`, it suffices to assume it's always a
coequalizer of something (i.e. a regular epi). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
other_map (X) : F'.obj (U.obj (F.obj (U.obj X))) ⟶ F'.obj (U.obj X) | F'.map (U.map (F.map (adj₂.unit.app _) ≫ adj₁.counit.app _)) ≫ adj₂.counit.app _ | def | category_theory.lift_adjoint.other_map | category_theory.adjunction | src/category_theory/adjunction/lifting.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.reflexive",
"category_theory.monad.adjunction",
"category_theory.monad.coequalizer"
] | [] | (Implementation)
To construct the left adjoint, we use the coequalizer of `F' U ε_Y` with the composite
`F' U F U X ⟶ F' U F U R F U' X ⟶ F' U R F' U X ⟶ F' U X`
where the first morphism is `F' U F ι_UX`, the second is `F' U ε_RF'UX`, and the third is `δ_F'UX`.
We will show that this coequalizer exists and that it fo... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
construct_left_adjoint_obj (Y : B) : A | coequalizer (F'.map (U.map (adj₁.counit.app Y))) (other_map _ _ adj₁ adj₂ Y) | def | category_theory.lift_adjoint.construct_left_adjoint_obj | category_theory.adjunction | src/category_theory/adjunction/lifting.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.reflexive",
"category_theory.monad.adjunction",
"category_theory.monad.coequalizer"
] | [] | Construct the object part of the desired left adjoint as the coequalizer of `F'Uε_Y` with
`other_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
construct_left_adjoint_equiv [∀ (X : B), regular_epi (adj₁.counit.app X)] (Y : A) (X : B) :
(construct_left_adjoint_obj _ _ adj₁ adj₂ X ⟶ Y) ≃ (X ⟶ R.obj Y) | calc (construct_left_adjoint_obj _ _ adj₁ adj₂ X ⟶ Y)
≃ {f : F'.obj (U.obj X) ⟶ Y //
F'.map (U.map (adj₁.counit.app X)) ≫ f = other_map _ _ adj₁ adj₂ _ ≫ f} :
cofork.is_colimit.hom_iso (colimit.is_colimit _) _
... ≃ {g : U.obj X ⟶ U.obj (R.obj Y) //
U.map (F.map g ≫ adj... | def | category_theory.lift_adjoint.construct_left_adjoint_equiv | category_theory.adjunction | src/category_theory/adjunction/lifting.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.reflexive",
"category_theory.monad.adjunction",
"category_theory.monad.coequalizer"
] | [] | The homset equivalence which helps show that `R` is a right adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
construct_left_adjoint [∀ (X : B), regular_epi (adj₁.counit.app X)] : B ⥤ A | begin
refine adjunction.left_adjoint_of_equiv (λ X Y, construct_left_adjoint_equiv R _ adj₁ adj₂ Y X) _,
intros X Y Y' g h,
rw [construct_left_adjoint_equiv_apply, construct_left_adjoint_equiv_apply, function.comp_app,
function.comp_app, equiv.trans_apply, equiv.trans_apply, equiv.trans_apply, equiv.trans_a... | def | category_theory.lift_adjoint.construct_left_adjoint | category_theory.adjunction | src/category_theory/adjunction/lifting.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.reflexive",
"category_theory.monad.adjunction",
"category_theory.monad.coequalizer"
] | [
"equiv.apply_symm_apply",
"equiv.subtype_equiv_apply",
"equiv.symm_apply_eq",
"equiv.trans_apply",
"subtype.coe_mk",
"subtype.ext_iff"
] | Construct the left adjoint to `R`, with object map `construct_left_adjoint_obj`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_triangle_lift {U : B ⥤ C} {F : C ⥤ B} (R : A ⥤ B) (adj₁ : F ⊣ U)
[Π (X : B), regular_epi (adj₁.counit.app X)]
[has_reflexive_coequalizers A]
[is_right_adjoint (R ⋙ U)] : is_right_adjoint R | { left := lift_adjoint.construct_left_adjoint R _ adj₁ (adjunction.of_right_adjoint _),
adj := adjunction.adjunction_of_equiv_left _ _ } | def | category_theory.adjoint_triangle_lift | category_theory.adjunction | src/category_theory/adjunction/lifting.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.reflexive",
"category_theory.monad.adjunction",
"category_theory.monad.coequalizer"
] | [
"adj"
] | The adjoint triangle theorem: Suppose `U : B ⥤ C` has a left adjoint `F` such that each counit
`ε_X : FUX ⟶ X` is a regular epimorphism. Then if a category `A` has coequalizers of reflexive
pairs, then a functor `R : A ⥤ B` has a left adjoint if the composite `R ⋙ U` does.
Note the converse is true (with weaker assump... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monadic_adjoint_triangle_lift (U : B ⥤ C) [monadic_right_adjoint U] {R : A ⥤ B}
[has_reflexive_coequalizers A]
[is_right_adjoint (R ⋙ U)] :
is_right_adjoint R | begin
let R' : A ⥤ _ := R ⋙ monad.comparison (adjunction.of_right_adjoint U),
rsufficesI : is_right_adjoint R',
{ let : is_right_adjoint (R' ⋙ (monad.comparison (adjunction.of_right_adjoint U)).inv),
{ apply_instance },
{ let : R' ⋙ (monad.comparison (adjunction.of_right_adjoint U)).inv ≅ R :=
(is... | def | category_theory.monadic_adjoint_triangle_lift | category_theory.adjunction | src/category_theory/adjunction/lifting.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.reflexive",
"category_theory.monad.adjunction",
"category_theory.monad.coequalizer"
] | [] | If `R ⋙ U` has a left adjoint, the domain of `R` has reflexive coequalizers and `U` is a monadic
functor, then `R` has a left adjoint.
This is a special case of `adjoint_triangle_lift` which is often more useful in practice. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoint_square_lift (Q : A ⥤ B) (V : B ⥤ D) (U : A ⥤ C) (R : C ⥤ D)
(comm : U ⋙ R ≅ Q ⋙ V)
[is_right_adjoint U] [is_right_adjoint V] [is_right_adjoint R]
[∀ X, regular_epi ((adjunction.of_right_adjoint V).counit.app X)]
[has_reflexive_coequalizers A] :
is_right_adjoint Q | begin
let := adjunction.right_adjoint_of_nat_iso comm,
exactI adjoint_triangle_lift Q (adjunction.of_right_adjoint V),
end | def | category_theory.adjoint_square_lift | category_theory.adjunction | src/category_theory/adjunction/lifting.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.reflexive",
"category_theory.monad.adjunction",
"category_theory.monad.coequalizer"
] | [
"comm"
] | Suppose we have a commutative square of functors
Q
A → B
U ↓ ↓ V
C → D
R
where `U` has a left adjoint, `A` has reflexive coequalizers and `V` has a left adjoint such that
each component of the counit is a regular epi.
Then `Q` has a left adjoint if `R` has a left adjoint.
See https://ncatlab.... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monadic_adjoint_square_lift (Q : A ⥤ B) (V : B ⥤ D) (U : A ⥤ C) (R : C ⥤ D)
(comm : U ⋙ R ≅ Q ⋙ V)
[is_right_adjoint U] [monadic_right_adjoint V] [is_right_adjoint R]
[has_reflexive_coequalizers A] :
is_right_adjoint Q | begin
let := adjunction.right_adjoint_of_nat_iso comm,
exactI monadic_adjoint_triangle_lift V,
end | def | category_theory.monadic_adjoint_square_lift | category_theory.adjunction | src/category_theory/adjunction/lifting.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.reflexive",
"category_theory.monad.adjunction",
"category_theory.monad.coequalizer"
] | [
"comm"
] | Suppose we have a commutative square of functors
Q
A → B
U ↓ ↓ V
C → D
R
where `U` has a left adjoint, `A` has reflexive coequalizers and `V` is monadic.
Then `Q` has a left adjoint if `R` has a left adjoint.
See https://ncatlab.org/nlab/show/adjoint+lifting+theorem | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functoriality_right_adjoint : cocone (K ⋙ F) ⥤ cocone K | (cocones.functoriality _ G) ⋙
(cocones.precompose (K.right_unitor.inv ≫ (whisker_left K adj.unit) ≫ (associator _ _ _).inv)) | def | category_theory.adjunction.functoriality_right_adjoint | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | The right adjoint of `cocones.functoriality K F : cocone K ⥤ cocone (K ⋙ F)`.
Auxiliary definition for `functoriality_is_left_adjoint`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functoriality_unit :
𝟭 (cocone K) ⟶ cocones.functoriality _ F ⋙ functoriality_right_adjoint adj K | { app := λ c, { hom := adj.unit.app c.X } } | def | category_theory.adjunction.functoriality_unit | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [
"adj"
] | The unit for the adjunction for `cocones.functoriality K F : cocone K ⥤ cocone (K ⋙ F)`.
Auxiliary definition for `functoriality_is_left_adjoint`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functoriality_counit :
functoriality_right_adjoint adj K ⋙ cocones.functoriality _ F ⟶ 𝟭 (cocone (K ⋙ F)) | { app := λ c, { hom := adj.counit.app c.X } } | def | category_theory.adjunction.functoriality_counit | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [
"adj"
] | The counit for the adjunction for `cocones.functoriality K F : cocone K ⥤ cocone (K ⋙ F)`.
Auxiliary definition for `functoriality_is_left_adjoint`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functoriality_is_left_adjoint :
is_left_adjoint (cocones.functoriality K F) | { right := functoriality_right_adjoint adj K,
adj := mk_of_unit_counit
{ unit := functoriality_unit adj K,
counit := functoriality_counit adj K } } | def | category_theory.adjunction.functoriality_is_left_adjoint | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [
"adj"
] | The functor `cocones.functoriality K F : cocone K ⥤ cocone (K ⋙ F)` is a left adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_adjoint_preserves_colimits : preserves_colimits_of_size.{v u} F | { preserves_colimits_of_shape := λ J 𝒥,
{ preserves_colimit := λ F,
by exactI
{ preserves := λ c hc, is_colimit.iso_unique_cocone_morphism.inv
(λ s, @equiv.unique _ _ (is_colimit.iso_unique_cocone_morphism.hom hc _)
(((adj.functoriality_is_left_adjoint _).adj).hom_equiv _ _)) } } }. | def | category_theory.adjunction.left_adjoint_preserves_colimits | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [
"adj",
"equiv.unique"
] | A left adjoint preserves colimits.
See <https://stacks.math.columbia.edu/tag/0038>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_equivalence_preserves_colimits (E : C ⥤ D) [is_equivalence E] :
preserves_colimits_of_size.{v u} E | left_adjoint_preserves_colimits E.adjunction | instance | category_theory.adjunction.is_equivalence_preserves_colimits | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalence_reflects_colimits (E : D ⥤ C) [is_equivalence E] :
reflects_colimits_of_size.{v u} E | { reflects_colimits_of_shape := λ J 𝒥, by exactI
{ reflects_colimit := λ K,
{ reflects := λ c t,
begin
have l := (is_colimit_of_preserves E.inv t).map_cocone_equiv E.as_equivalence.unit_iso.symm,
refine (((is_colimit.precompose_inv_equiv K.right_unitor _).symm) l).of_iso_colimit _,
... | instance | category_theory.adjunction.is_equivalence_reflects_colimits | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalence_creates_colimits (H : D ⥤ C) [is_equivalence H] :
creates_colimits_of_size.{v u} H | { creates_colimits_of_shape := λ J 𝒥, by exactI
{ creates_colimit := λ F,
{ lifts := λ c t,
{ lifted_cocone := H.map_cocone_inv c,
valid_lift := H.map_cocone_map_cocone_inv c } } } } | instance | category_theory.adjunction.is_equivalence_creates_colimits | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit_comp_equivalence (E : C ⥤ D) [is_equivalence E] [has_colimit K] :
has_colimit (K ⋙ E) | has_colimit.mk
{ cocone := E.map_cocone (colimit.cocone K),
is_colimit := preserves_colimit.preserves (colimit.is_colimit K) } | lemma | category_theory.adjunction.has_colimit_comp_equivalence | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit_of_comp_equivalence (E : C ⥤ D) [is_equivalence E] [has_colimit (K ⋙ E)] :
has_colimit K | @has_colimit_of_iso _ _ _ _ (K ⋙ E ⋙ inv E) K
(@has_colimit_comp_equivalence _ _ _ _ _ _ (K ⋙ E) (inv E) _ _)
((functor.right_unitor _).symm ≪≫ iso_whisker_left K (E.as_equivalence.unit_iso)) | lemma | category_theory.adjunction.has_colimit_of_comp_equivalence | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_shape_of_equivalence (E : C ⥤ D) [is_equivalence E]
[has_colimits_of_shape J D] : has_colimits_of_shape J C | ⟨λ F, by exactI has_colimit_of_comp_equivalence F E⟩ | lemma | category_theory.adjunction.has_colimits_of_shape_of_equivalence | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | Transport a `has_colimits_of_shape` instance across an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_of_equivalence (E : C ⥤ D) [is_equivalence E] [has_colimits_of_size.{v u} D] :
has_colimits_of_size.{v u} C | ⟨λ J hJ, by { exactI has_colimits_of_shape_of_equivalence E }⟩ | lemma | category_theory.adjunction.has_colimits_of_equivalence | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | Transport a `has_colimits` instance across an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functoriality_left_adjoint : cone (K ⋙ G) ⥤ cone K | (cones.functoriality _ F) ⋙ (cones.postcompose
((associator _ _ _).hom ≫ (whisker_left K adj.counit) ≫ K.right_unitor.hom)) | def | category_theory.adjunction.functoriality_left_adjoint | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | The left adjoint of `cones.functoriality K G : cone K ⥤ cone (K ⋙ G)`.
Auxiliary definition for `functoriality_is_right_adjoint`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functoriality_unit' :
𝟭 (cone (K ⋙ G)) ⟶ functoriality_left_adjoint adj K ⋙ cones.functoriality _ G | { app := λ c, { hom := adj.unit.app c.X, } } | def | category_theory.adjunction.functoriality_unit' | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [
"adj"
] | The unit for the adjunction for`cones.functoriality K G : cone K ⥤ cone (K ⋙ G)`.
Auxiliary definition for `functoriality_is_right_adjoint`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functoriality_counit' :
cones.functoriality _ G ⋙ functoriality_left_adjoint adj K ⟶ 𝟭 (cone K) | { app := λ c, { hom := adj.counit.app c.X, } } | def | category_theory.adjunction.functoriality_counit' | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [
"adj"
] | The counit for the adjunction for`cones.functoriality K G : cone K ⥤ cone (K ⋙ G)`.
Auxiliary definition for `functoriality_is_right_adjoint`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functoriality_is_right_adjoint :
is_right_adjoint (cones.functoriality K G) | { left := functoriality_left_adjoint adj K,
adj := mk_of_unit_counit
{ unit := functoriality_unit' adj K,
counit := functoriality_counit' adj K } } | def | category_theory.adjunction.functoriality_is_right_adjoint | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [
"adj"
] | The functor `cones.functoriality K G : cone K ⥤ cone (K ⋙ G)` is a right adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_adjoint_preserves_limits : preserves_limits_of_size.{v u} G | { preserves_limits_of_shape := λ J 𝒥,
{ preserves_limit := λ K,
by exactI
{ preserves := λ c hc, is_limit.iso_unique_cone_morphism.inv
(λ s, @equiv.unique _ _ (is_limit.iso_unique_cone_morphism.hom hc _)
(((adj.functoriality_is_right_adjoint _).adj).hom_equiv _ _).symm) } } }. | def | category_theory.adjunction.right_adjoint_preserves_limits | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [
"adj",
"equiv.unique"
] | A right adjoint preserves limits.
See <https://stacks.math.columbia.edu/tag/0038>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_equivalence_preserves_limits (E : D ⥤ C) [is_equivalence E] :
preserves_limits_of_size.{v u} E | right_adjoint_preserves_limits E.inv.adjunction | instance | category_theory.adjunction.is_equivalence_preserves_limits | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalence_reflects_limits (E : D ⥤ C) [is_equivalence E] :
reflects_limits_of_size.{v u} E | { reflects_limits_of_shape := λ J 𝒥, by exactI
{ reflects_limit := λ K,
{ reflects := λ c t,
begin
have := (is_limit_of_preserves E.inv t).map_cone_equiv E.as_equivalence.unit_iso.symm,
refine (((is_limit.postcompose_hom_equiv K.left_unitor _).symm) this).of_iso_limit _,
tidy,
... | instance | category_theory.adjunction.is_equivalence_reflects_limits | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalence_creates_limits (H : D ⥤ C) [is_equivalence H] :
creates_limits_of_size.{v u} H | { creates_limits_of_shape := λ J 𝒥, by exactI
{ creates_limit := λ F,
{ lifts := λ c t,
{ lifted_cone := H.map_cone_inv c,
valid_lift := H.map_cone_map_cone_inv c } } } } | instance | category_theory.adjunction.is_equivalence_creates_limits | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limit_comp_equivalence (E : D ⥤ C) [is_equivalence E] [has_limit K] :
has_limit (K ⋙ E) | has_limit.mk
{ cone := E.map_cone (limit.cone K),
is_limit := preserves_limit.preserves (limit.is_limit K) } | lemma | category_theory.adjunction.has_limit_comp_equivalence | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limit_of_comp_equivalence (E : D ⥤ C) [is_equivalence E] [has_limit (K ⋙ E)] :
has_limit K | @has_limit_of_iso _ _ _ _ (K ⋙ E ⋙ inv E) K
(@has_limit_comp_equivalence _ _ _ _ _ _ (K ⋙ E) (inv E) _ _)
((iso_whisker_left K E.as_equivalence.unit_iso.symm) ≪≫ (functor.right_unitor _)) | lemma | category_theory.adjunction.has_limit_of_comp_equivalence | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits_of_shape_of_equivalence (E : D ⥤ C) [is_equivalence E] [has_limits_of_shape J C] :
has_limits_of_shape J D | ⟨λ F, by exactI has_limit_of_comp_equivalence F E⟩ | lemma | category_theory.adjunction.has_limits_of_shape_of_equivalence | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | Transport a `has_limits_of_shape` instance across an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_equivalence (E : D ⥤ C) [is_equivalence E] [has_limits_of_size.{v u} C] :
has_limits_of_size.{v u} D | ⟨λ J hJ, by exactI has_limits_of_shape_of_equivalence E⟩ | lemma | category_theory.adjunction.has_limits_of_equivalence | category_theory.adjunction | src/category_theory/adjunction/limits.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.creates"
] | [] | Transport a `has_limits` instance across an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocones_iso_component_hom {J : Type u} [category.{v} J] {K : J ⥤ C}
(Y : D) (t : ((cocones J D).obj (op (K ⋙ F))).obj Y) :
(G ⋙ (cocones J C).obj (op K)).obj Y | { app := λ j, (adj.hom_equiv (K.obj j) Y) (t.app j),
naturality' := λ j j' f, by { erw [← adj.hom_equiv_naturality_left, t.naturality], dsimp, simp } } | def | category_theory.adjunction.cocones_iso_component_hom | 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 |
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