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lift_star {D : Type*} [category D] {Z : D} (F : C ⥤ D)
(M : Π (x : C), Z ⟶ F.obj x) (hM : ∀ (x y : C) (f : x ⟶ y), M x ≫ F.map f = M y) :
(lift F M hM).obj star ≅ Z | eq_to_iso rfl | def | category_theory.with_initial.lift_star | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"lift"
] | The isomorphism between `(lift F _ _).obj with_term.star` with `Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_star_lift_map {D : Type*} [category D] {Z : D} (F : C ⥤ D)
(M : Π (x : C), Z ⟶ F.obj x) (hM : ∀ (x y : C) (f : x ⟶ y), M x ≫ F.map f = M y) (x : C) :
(lift_star F M hM).hom ≫ (lift F M hM).map (star_initial.to (incl.obj x)) =
M x ≫ (incl_lift F M hM).hom.app x | begin
erw [category.id_comp, category.comp_id],
refl,
end | lemma | category_theory.with_initial.lift_star_lift_map | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_unique {D : Type*} [category D] {Z : D} (F : C ⥤ D)
(M : Π (x : C), Z ⟶ F.obj x) (hM : ∀ (x y : C) (f : x ⟶ y), M x ≫ F.map f = M y)
(G : with_initial C ⥤ D) (h : incl ⋙ G ≅ F) (hG : G.obj star ≅ Z)
(hh : ∀ x : C, hG.symm.hom ≫ G.map (star_initial.to (incl.obj x)) = M x ≫ h.symm.hom.app x) :
G ≅ lift F M h... | nat_iso.of_components
(λ X,
match X with
| of x := h.app x
| star := hG
end)
begin
rintro (X|X) (Y|Y) f,
{ apply h.hom.naturality },
{ cases f, },
{ cases f,
change G.map _ ≫ h.hom.app _ = hG.hom ≫ _,
symmetry,
erw [← iso.eq_inv_comp, ← category.assoc, hh],
simpa },
{ cases f,
chan... | def | category_theory.with_initial.lift_unique | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"lift",
"lift_unique"
] | The uniqueness of `lift`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_to_initial {D : Type*} [category D] {Z : D} (F : C ⥤ D) (hZ : limits.is_initial Z) :
with_initial C ⥤ D | lift F (λ x, hZ.to _) (λ x y f, hZ.hom_ext _ _) | def | category_theory.with_initial.lift_to_initial | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"lift"
] | A variant of `lift` with `Z` an initial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
incl_lift_to_initial {D : Type*} [category D] {Z : D} (F : C ⥤ D) (hZ : limits.is_initial Z) :
incl ⋙ lift_to_initial F hZ ≅ F | incl_lift _ _ _ | def | category_theory.with_initial.incl_lift_to_initial | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | A variant of `incl_lift` with `Z` an initial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_to_initial_unique {D : Type*} [category D] {Z : D} (F : C ⥤ D) (hZ : limits.is_initial Z)
(G : with_initial C ⥤ D) (h : incl ⋙ G ≅ F) (hG : G.obj star ≅ Z) :
G ≅ lift_to_initial F hZ | lift_unique F (λ z, hZ.to _) (λ x y f, hZ.hom_ext _ _) G h hG (λ x, hZ.hom_ext _ _) | def | category_theory.with_initial.lift_to_initial_unique | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [
"lift_unique"
] | A variant of `lift_unique` with `Z` an initial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_to (X : C) : star ⟶ incl.obj X | star_initial.to _ | def | category_theory.with_initial.hom_to | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | Constructs a morphism from `star` to `of X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_to_star {X : with_initial C} (f : X ⟶ star) : is_iso f | by tidy | instance | category_theory.with_initial.is_iso_of_to_star | category_theory | src/category_theory/with_terminal.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
yoneda : C ⥤ (Cᵒᵖ ⥤ Type v₁) | { obj := λ X,
{ obj := λ Y, unop Y ⟶ X,
map := λ Y Y' f g, f.unop ≫ g,
map_comp' := λ _ _ _ f g, begin ext, dsimp, erw [category.assoc] end,
map_id' := λ Y, begin ext, dsimp, erw [category.id_comp] end },
map := λ X X' f, { app := λ Y g, g ≫ f } } | def | category_theory.yoneda | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | The Yoneda embedding, as a functor from `C` into presheaves on `C`.
See <https://stacks.math.columbia.edu/tag/001O>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coyoneda : Cᵒᵖ ⥤ (C ⥤ Type v₁) | { obj := λ X,
{ obj := λ Y, unop X ⟶ Y,
map := λ Y Y' f g, g ≫ f },
map := λ X X' f, { app := λ Y g, f.unop ≫ g } } | def | category_theory.coyoneda | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | The co-Yoneda embedding, as a functor from `Cᵒᵖ` into co-presheaves on `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj_map_id {X Y : C} (f : op X ⟶ op Y) :
(yoneda.obj X).map f (𝟙 X) = (yoneda.map f.unop).app (op Y) (𝟙 Y) | by { dsimp, simp } | lemma | category_theory.yoneda.obj_map_id | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
naturality {X Y : C} (α : yoneda.obj X ⟶ yoneda.obj Y)
{Z Z' : C} (f : Z ⟶ Z') (h : Z' ⟶ X) : f ≫ α.app (op Z') h = α.app (op Z) (f ≫ h) | (functor_to_types.naturality _ _ α f.op h).symm | lemma | category_theory.yoneda.naturality | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
yoneda_full : full (yoneda : C ⥤ Cᵒᵖ ⥤ Type v₁) | { preimage := λ X Y f, f.app (op X) (𝟙 X) } | instance | category_theory.yoneda.yoneda_full | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | The Yoneda embedding is full.
See <https://stacks.math.columbia.edu/tag/001P>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
yoneda_faithful : faithful (yoneda : C ⥤ Cᵒᵖ ⥤ Type v₁) | { map_injective' := λ X Y f g p, by convert (congr_fun (congr_app p (op X)) (𝟙 X)); dsimp; simp } | instance | category_theory.yoneda.yoneda_faithful | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | The Yoneda embedding is faithful.
See <https://stacks.math.columbia.edu/tag/001P>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext (X Y : C)
(p : Π {Z : C}, (Z ⟶ X) → (Z ⟶ Y)) (q : Π {Z : C}, (Z ⟶ Y) → (Z ⟶ X))
(h₁ : Π {Z : C} (f : Z ⟶ X), q (p f) = f) (h₂ : Π {Z : C} (f : Z ⟶ Y), p (q f) = f)
(n : Π {Z Z' : C} (f : Z' ⟶ Z) (g : Z ⟶ X), p (f ≫ g) = f ≫ p g) : X ≅ Y | yoneda.preimage_iso (nat_iso.of_components (λ Z, { hom := p, inv := q, }) (by tidy)) | def | category_theory.yoneda.ext | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | Extensionality via Yoneda. The typical usage would be
```
-- Goal is `X ≅ Y`
apply yoneda.ext,
-- Goals are now functions `(Z ⟶ X) → (Z ⟶ Y)`, `(Z ⟶ Y) → (Z ⟶ X)`, and the fact that these
functions are inverses and natural in `Z`.
``` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso {X Y : C} (f : X ⟶ Y) [is_iso (yoneda.map f)] : is_iso f | is_iso_of_fully_faithful yoneda f | lemma | category_theory.yoneda.is_iso | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | If `yoneda.map f` is an isomorphism, so was `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
naturality {X Y : Cᵒᵖ} (α : coyoneda.obj X ⟶ coyoneda.obj Y)
{Z Z' : C} (f : Z' ⟶ Z) (h : unop X ⟶ Z') : (α.app Z' h) ≫ f = α.app Z (h ≫ f) | (functor_to_types.naturality _ _ α f h).symm | lemma | category_theory.coyoneda.naturality | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coyoneda_full : full (coyoneda : Cᵒᵖ ⥤ C ⥤ Type v₁) | { preimage := λ X Y f, (f.app _ (𝟙 X.unop)).op } | instance | category_theory.coyoneda.coyoneda_full | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coyoneda_faithful : faithful (coyoneda : Cᵒᵖ ⥤ C ⥤ Type v₁) | { map_injective' := λ X Y f g p,
begin
have t := congr_fun (congr_app p X.unop) (𝟙 _),
simpa using congr_arg quiver.hom.op t,
end } | instance | category_theory.coyoneda.coyoneda_faithful | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [
"quiver.hom.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso {X Y : Cᵒᵖ} (f : X ⟶ Y) [is_iso (coyoneda.map f)] : is_iso f | is_iso_of_fully_faithful coyoneda f | lemma | category_theory.coyoneda.is_iso | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | If `coyoneda.map f` is an isomorphism, so was `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
punit_iso : coyoneda.obj (opposite.op punit) ≅ 𝟭 (Type v₁) | nat_iso.of_components
(λ X, { hom := λ f, f ⟨⟩, inv := λ x _, x })
(by tidy) | def | category_theory.coyoneda.punit_iso | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [
"opposite.op"
] | The identity functor on `Type` is isomorphic to the coyoneda functor coming from `punit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj_op_op (X : C) : coyoneda.obj (op (op X)) ≅ yoneda.obj X | nat_iso.of_components (λ Y, (op_equiv _ _).to_iso) (λ X Y f, rfl) | def | category_theory.coyoneda.obj_op_op | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | Taking the `unop` of morphisms is a natural isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
representable (F : Cᵒᵖ ⥤ Type v₁) : Prop | (has_representation : ∃ X (f : yoneda.obj X ⟶ F), is_iso f) | class | category_theory.functor.representable | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | A functor `F : Cᵒᵖ ⥤ Type v₁` is representable if there is object `X` so `F ≅ yoneda.obj X`.
See <https://stacks.math.columbia.edu/tag/001Q>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
corepresentable (F : C ⥤ Type v₁) : Prop | (has_corepresentation : ∃ X (f : coyoneda.obj X ⟶ F), is_iso f) | class | category_theory.functor.corepresentable | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | A functor `F : C ⥤ Type v₁` is corepresentable if there is object `X` so `F ≅ coyoneda.obj X`.
See <https://stacks.math.columbia.edu/tag/001Q>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
repr_X : C | (representable.has_representation : ∃ X (f : _ ⟶ F), _).some | def | category_theory.functor.repr_X | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | The representing object for the representable functor `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
repr_f : yoneda.obj F.repr_X ⟶ F | representable.has_representation.some_spec.some | def | category_theory.functor.repr_f | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | The (forward direction of the) isomorphism witnessing `F` is representable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
repr_x : F.obj (op F.repr_X) | F.repr_f.app (op F.repr_X) (𝟙 F.repr_X) | def | category_theory.functor.repr_x | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | The representing element for the representable functor `F`, sometimes called the universal
element of the functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
repr_w : yoneda.obj F.repr_X ≅ F | as_iso F.repr_f | def | category_theory.functor.repr_w | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | An isomorphism between `F` and a functor of the form `C(-, F.repr_X)`. Note the components
`F.repr_w.app X` definitionally have type `(X.unop ⟶ F.repr_X) ≅ F.obj X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
repr_w_hom : F.repr_w.hom = F.repr_f | rfl | lemma | category_theory.functor.repr_w_hom | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
repr_w_app_hom (X : Cᵒᵖ) (f : unop X ⟶ F.repr_X) :
(F.repr_w.app X).hom f = F.map f.op F.repr_x | begin
change F.repr_f.app X f = (F.repr_f.app (op F.repr_X) ≫ F.map f.op) (𝟙 F.repr_X),
rw ←F.repr_f.naturality,
dsimp,
simp
end | lemma | category_theory.functor.repr_w_app_hom | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
corepr_X : C | (corepresentable.has_corepresentation : ∃ X (f : _ ⟶ F), _).some.unop | def | category_theory.functor.corepr_X | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | The representing object for the corepresentable functor `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
corepr_f : coyoneda.obj (op F.corepr_X) ⟶ F | corepresentable.has_corepresentation.some_spec.some | def | category_theory.functor.corepr_f | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | The (forward direction of the) isomorphism witnessing `F` is corepresentable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
corepr_x : F.obj F.corepr_X | F.corepr_f.app F.corepr_X (𝟙 F.corepr_X) | def | category_theory.functor.corepr_x | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | The representing element for the corepresentable functor `F`, sometimes called the universal
element of the functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
corepr_w : coyoneda.obj (op F.corepr_X) ≅ F | as_iso F.corepr_f | def | category_theory.functor.corepr_w | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | An isomorphism between `F` and a functor of the form `C(F.corepr X, -)`. Note the components
`F.corepr_w.app X` definitionally have type `F.corepr_X ⟶ X ≅ F.obj X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
corepr_w_app_hom (X : C) (f : F.corepr_X ⟶ X) :
(F.corepr_w.app X).hom f = F.map f F.corepr_x | begin
change F.corepr_f.app X f = (F.corepr_f.app F.corepr_X ≫ F.map f) (𝟙 F.corepr_X),
rw ←F.corepr_f.naturality,
dsimp,
simp
end | lemma | category_theory.functor.corepr_w_app_hom | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
representable_of_nat_iso (F : Cᵒᵖ ⥤ Type v₁) {G} (i : F ≅ G) [F.representable] :
G.representable | { has_representation := ⟨F.repr_X, F.repr_f ≫ i.hom, infer_instance⟩ } | lemma | category_theory.representable_of_nat_iso | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
corepresentable_of_nat_iso (F : C ⥤ Type v₁) {G} (i : F ≅ G) [F.corepresentable] :
G.corepresentable | { has_corepresentation := ⟨op F.corepr_X, F.corepr_f ≫ i.hom, infer_instance⟩ } | lemma | category_theory.corepresentable_of_nat_iso | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_category_instance_1 : category ((Cᵒᵖ ⥤ Type v₁) × Cᵒᵖ) | category_theory.prod.{(max u₁ v₁) v₁} (Cᵒᵖ ⥤ Type v₁) Cᵒᵖ | instance | category_theory.prod_category_instance_1 | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_category_instance_2 : category (Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁)) | category_theory.prod.{v₁ (max u₁ v₁)} Cᵒᵖ (Cᵒᵖ ⥤ Type v₁) | instance | category_theory.prod_category_instance_2 | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
yoneda_evaluation : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁) ⥤ Type (max u₁ v₁) | evaluation_uncurried Cᵒᵖ (Type v₁) ⋙ ulift_functor.{u₁} | def | category_theory.yoneda_evaluation | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | The "Yoneda evaluation" functor, which sends `X : Cᵒᵖ` and `F : Cᵒᵖ ⥤ Type`
to `F.obj X`, functorially in both `X` and `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
yoneda_evaluation_map_down
(P Q : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁)) (α : P ⟶ Q) (x : (yoneda_evaluation C).obj P) :
((yoneda_evaluation C).map α x).down = α.2.app Q.1 (P.2.map α.1 x.down) | rfl | lemma | category_theory.yoneda_evaluation_map_down | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
yoneda_pairing : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁) ⥤ Type (max u₁ v₁) | functor.prod yoneda.op (𝟭 (Cᵒᵖ ⥤ Type v₁)) ⋙ functor.hom (Cᵒᵖ ⥤ Type v₁) | def | category_theory.yoneda_pairing | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | The "Yoneda pairing" functor, which sends `X : Cᵒᵖ` and `F : Cᵒᵖ ⥤ Type`
to `yoneda.op.obj X ⟶ F`, functorially in both `X` and `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
yoneda_pairing_map
(P Q : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁)) (α : P ⟶ Q) (β : (yoneda_pairing C).obj P) :
(yoneda_pairing C).map α β = yoneda.map α.1.unop ≫ β ≫ α.2 | rfl | lemma | category_theory.yoneda_pairing_map | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
yoneda_lemma : yoneda_pairing C ≅ yoneda_evaluation C | { hom :=
{ app := λ F x, ulift.up ((x.app F.1) (𝟙 (unop F.1))),
naturality' :=
begin
intros X Y f, ext, dsimp,
erw [category.id_comp, ←functor_to_types.naturality],
simp only [category.comp_id, yoneda_obj_map],
end },
inv :=
{ app := λ F x,
{ app := λ X a, (F.2.map a.op) x.down,... | def | category_theory.yoneda_lemma | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [
"quiver.hom.unop_op"
] | The Yoneda lemma asserts that that the Yoneda pairing
`(X : Cᵒᵖ, F : Cᵒᵖ ⥤ Type) ↦ (yoneda.obj (unop X) ⟶ F)`
is naturally isomorphic to the evaluation `(X, F) ↦ F.obj X`.
See <https://stacks.math.columbia.edu/tag/001P>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
yoneda_sections (X : C) (F : Cᵒᵖ ⥤ Type v₁) :
(yoneda.obj X ⟶ F) ≅ ulift.{u₁} (F.obj (op X)) | (yoneda_lemma C).app (op X, F) | def | category_theory.yoneda_sections | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | The isomorphism between `yoneda.obj X ⟶ F` and `F.obj (op X)`
(we need to insert a `ulift` to get the universes right!)
given by the Yoneda lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
yoneda_equiv {X : C} {F : Cᵒᵖ ⥤ Type v₁} : (yoneda.obj X ⟶ F) ≃ F.obj (op X) | (yoneda_sections X F).to_equiv.trans equiv.ulift | def | category_theory.yoneda_equiv | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [
"equiv.ulift"
] | We have a type-level equivalence between natural transformations from the yoneda embedding
and elements of `F.obj X`, without any universe switching. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
yoneda_equiv_apply {X : C} {F : Cᵒᵖ ⥤ Type v₁} (f : yoneda.obj X ⟶ F) :
yoneda_equiv f = f.app (op X) (𝟙 X) | rfl | lemma | category_theory.yoneda_equiv_apply | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
yoneda_equiv_symm_app_apply {X : C} {F : Cᵒᵖ ⥤ Type v₁} (x : F.obj (op X))
(Y : Cᵒᵖ) (f : Y.unop ⟶ X) :
(yoneda_equiv.symm x).app Y f = F.map f.op x | rfl | lemma | category_theory.yoneda_equiv_symm_app_apply | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
yoneda_equiv_naturality {X Y : C} {F : Cᵒᵖ ⥤ Type v₁} (f : yoneda.obj X ⟶ F) (g : Y ⟶ X) :
F.map g.op (yoneda_equiv f) = yoneda_equiv (yoneda.map g ≫ f) | begin
change (f.app (op X) ≫ F.map g.op) (𝟙 X) = f.app (op Y) (𝟙 Y ≫ g),
rw ←f.naturality,
dsimp,
simp,
end | lemma | category_theory.yoneda_equiv_naturality | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
yoneda_sections_small {C : Type u₁} [small_category C] (X : C)
(F : Cᵒᵖ ⥤ Type u₁) :
(yoneda.obj X ⟶ F) ≅ F.obj (op X) | yoneda_sections X F ≪≫ ulift_trivial _ | def | category_theory.yoneda_sections_small | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | When `C` is a small category, we can restate the isomorphism from `yoneda_sections`
without having to change universes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
yoneda_sections_small_hom {C : Type u₁} [small_category C] (X : C)
(F : Cᵒᵖ ⥤ Type u₁) (f : yoneda.obj X ⟶ F) :
(yoneda_sections_small X F).hom f = f.app _ (𝟙 _) | rfl | lemma | category_theory.yoneda_sections_small_hom | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
yoneda_sections_small_inv_app_apply {C : Type u₁} [small_category C] (X : C)
(F : Cᵒᵖ ⥤ Type u₁) (t : F.obj (op X)) (Y : Cᵒᵖ) (f : Y.unop ⟶ X) :
((yoneda_sections_small X F).inv t).app Y f = F.map f.op t | rfl | lemma | category_theory.yoneda_sections_small_inv_app_apply | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curried_yoneda_lemma {C : Type u₁} [small_category C] :
(yoneda.op ⋙ coyoneda : Cᵒᵖ ⥤ (Cᵒᵖ ⥤ Type u₁) ⥤ Type u₁) ≅ evaluation Cᵒᵖ (Type u₁) | eq_to_iso (by tidy) ≪≫ curry.map_iso (yoneda_lemma C ≪≫
iso_whisker_left (evaluation_uncurried Cᵒᵖ (Type u₁)) ulift_functor_trivial) ≪≫
eq_to_iso (by tidy) | def | category_theory.curried_yoneda_lemma | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [] | The curried version of yoneda lemma when `C` is small. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
curried_yoneda_lemma' {C : Type u₁} [small_category C] :
yoneda ⋙ (whiskering_left Cᵒᵖ (Cᵒᵖ ⥤ Type u₁)ᵒᵖ (Type u₁)).obj yoneda.op ≅ 𝟭 (Cᵒᵖ ⥤ Type u₁) | eq_to_iso (by tidy) ≪≫ curry.map_iso (iso_whisker_left (prod.swap _ _)
(yoneda_lemma C ≪≫ iso_whisker_left
(evaluation_uncurried Cᵒᵖ (Type u₁)) ulift_functor_trivial : _)) ≪≫ eq_to_iso (by tidy) | def | category_theory.curried_yoneda_lemma' | category_theory | src/category_theory/yoneda.lean | [
"category_theory.functor.hom",
"category_theory.functor.currying",
"category_theory.products.basic"
] | [
"prod.swap"
] | The curried version of yoneda lemma when `C` is small. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abelian extends preadditive C, normal_mono_category C, normal_epi_category C | [has_finite_products : has_finite_products C]
[has_kernels : has_kernels C]
[has_cokernels : has_cokernels C] | class | category_theory.abelian | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | A (preadditive) category `C` is called abelian if it has all finite products,
all kernels and cokernels, and if every monomorphism is the kernel of some morphism
and every epimorphism is the cokernel of some morphism.
(This definition implies the existence of zero objects:
finite products give a terminal object, and i... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_mono_factorisation {X Y : C} (f : X ⟶ Y) : mono_factorisation f | { I := abelian.image f,
m := kernel.ι _,
m_mono := infer_instance,
e := kernel.lift _ f (cokernel.condition _),
fac' := kernel.lift_ι _ _ _ } | def | category_theory.abelian.of_coimage_image_comparison_is_iso.image_mono_factorisation | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | The factorisation of a morphism through its abelian image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_mono_factorisation_e' {X Y : C} (f : X ⟶ Y) :
(image_mono_factorisation f).e = cokernel.π _ ≫ abelian.coimage_image_comparison f | begin
ext,
simp only [abelian.coimage_image_comparison, image_mono_factorisation_e,
category.assoc, cokernel.π_desc_assoc],
end | lemma | category_theory.abelian.of_coimage_image_comparison_is_iso.image_mono_factorisation_e' | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_factorisation {X Y : C} (f : X ⟶ Y) [is_iso (abelian.coimage_image_comparison f)] :
image_factorisation f | { F := image_mono_factorisation f,
is_image :=
{ lift := λ F, inv (abelian.coimage_image_comparison f) ≫ cokernel.desc _ F.e F.kernel_ι_comp,
lift_fac' := λ F, begin
simp only [image_mono_factorisation_m, is_iso.inv_comp_eq, category.assoc,
abelian.coimage_image_comparison],
ext,
simp ... | def | category_theory.abelian.of_coimage_image_comparison_is_iso.image_factorisation | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [
"lift"
] | If the coimage-image comparison morphism for a morphism `f` is an isomorphism,
we obtain an image factorisation of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_images : has_images C | { has_image := λ X Y f,
{ exists_image := ⟨image_factorisation f⟩ } } | lemma | category_theory.abelian.of_coimage_image_comparison_is_iso.has_images | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | A category in which coimage-image comparisons are all isomorphisms has images. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_mono_category : normal_mono_category C | { normal_mono_of_mono := λ X Y f m,
{ Z := _,
g := cokernel.π f,
w := by simp,
is_limit := begin
haveI : limits.has_images C := has_images,
haveI : has_equalizers C := preadditive.has_equalizers_of_has_kernels,
haveI : has_zero_object C := limits.has_zero_object_of_has_finite_biproducts ... | def | category_theory.abelian.of_coimage_image_comparison_is_iso.normal_mono_category | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [
"aux"
] | A category with finite products in which coimage-image comparisons are all isomorphisms
is a normal mono category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_epi_category : normal_epi_category C | { normal_epi_of_epi := λ X Y f m,
{ W := kernel f,
g := kernel.ι _,
w := kernel.condition _,
is_colimit := begin
haveI : limits.has_images C := has_images,
haveI : has_equalizers C := preadditive.has_equalizers_of_has_kernels,
haveI : has_zero_object C := limits.has_zero_object_of_has_fi... | def | category_theory.abelian.of_coimage_image_comparison_is_iso.normal_epi_category | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [
"aux"
] | A category with finite products in which coimage-image comparisons are all isomorphisms
is a normal epi category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_coimage_image_comparison_is_iso : abelian C | {} | def | category_theory.abelian.of_coimage_image_comparison_is_iso | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | A preadditive category with kernels, cokernels, and finite products,
in which the coimage-image comparison morphism is always an isomorphism,
is an abelian category.
The Stacks project uses this characterisation at the definition of an abelian category.
See <https://stacks.math.columbia.edu/tag/0109>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_biproducts : has_finite_biproducts C | limits.has_finite_biproducts.of_has_finite_products | theorem | category_theory.abelian.has_finite_biproducts | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_binary_biproducts : has_binary_biproducts C | limits.has_binary_biproducts_of_finite_biproducts _ | instance | category_theory.abelian.has_binary_biproducts | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zero_object : has_zero_object C | has_zero_object_of_has_initial_object | instance | category_theory.abelian.has_zero_object | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_preadditive_abelian : non_preadditive_abelian C | { ..‹abelian C› } | def | category_theory.abelian.non_preadditive_abelian | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | Every abelian category is, in particular, `non_preadditive_abelian`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_factor_thru_image [mono f] : is_iso (abelian.factor_thru_image f) | by apply_instance | instance | category_theory.abelian.is_iso_factor_thru_image | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_factor_thru_coimage [epi f] : is_iso (abelian.factor_thru_coimage f) | by apply_instance | instance | category_theory.abelian.is_iso_factor_thru_coimage | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_of_kernel_ι_eq_zero (h : kernel.ι f = 0) : mono f | mono_of_kernel_zero h | lemma | category_theory.abelian.mono_of_kernel_ι_eq_zero | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_of_cokernel_π_eq_zero (h : cokernel.π f = 0) : epi f | begin
apply normal_mono_category.epi_of_zero_cokernel _ (cokernel f),
simp_rw ←h,
exact is_colimit.of_iso_colimit (colimit.is_colimit (parallel_pair f 0)) (iso_of_π _)
end | lemma | category_theory.abelian.epi_of_cokernel_π_eq_zero | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_ι_comp_eq_zero {R : C} {g : Q ⟶ R} (h : f ≫ g = 0) : abelian.image.ι f ≫ g = 0 | zero_of_epi_comp (abelian.factor_thru_image f) $ by simp [h] | lemma | category_theory.abelian.image_ι_comp_eq_zero | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_coimage_π_eq_zero {R : C} {g : Q ⟶ R} (h : f ≫ g = 0) : f ≫ abelian.coimage.π g = 0 | zero_of_comp_mono (abelian.factor_thru_coimage g) $ by simp [h] | lemma | category_theory.abelian.comp_coimage_π_eq_zero | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_strong_epi_mono_factorisation : strong_epi_mono_factorisation f | { I := abelian.image f,
m := image.ι f,
m_mono := by apply_instance,
e := abelian.factor_thru_image f,
e_strong_epi := strong_epi_of_epi _ } | def | category_theory.abelian.image_strong_epi_mono_factorisation | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | Factoring through the image is a strong epi-mono factorisation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coimage_strong_epi_mono_factorisation : strong_epi_mono_factorisation f | { I := abelian.coimage f,
m := abelian.factor_thru_coimage f,
m_mono := by apply_instance,
e := coimage.π f,
e_strong_epi := strong_epi_of_epi _ } | def | category_theory.abelian.coimage_strong_epi_mono_factorisation | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | Factoring through the coimage is a strong epi-mono factorisation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coimage_iso_image : abelian.coimage f ≅ abelian.image f | as_iso (coimage_image_comparison f) | abbreviation | category_theory.abelian.coimage_iso_image | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | There is a canonical isomorphism between the abelian coimage and the abelian image of a
morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coimage_iso_image' : abelian.coimage f ≅ image f | is_image.iso_ext (coimage_strong_epi_mono_factorisation f).to_mono_is_image
(image.is_image f) | abbreviation | category_theory.abelian.coimage_iso_image' | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | There is a canonical isomorphism between the abelian coimage and the categorical image of a
morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coimage_iso_image'_hom :
(coimage_iso_image' f).hom = cokernel.desc _ (factor_thru_image f)
(by simp [←cancel_mono (limits.image.ι f)]) | begin
ext,
simp only [←cancel_mono (limits.image.ι f), is_image.iso_ext_hom, cokernel.π_desc, category.assoc,
is_image.lift_ι, coimage_strong_epi_mono_factorisation_to_mono_factorisation_m,
limits.image.fac],
end | lemma | category_theory.abelian.coimage_iso_image'_hom | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
factor_thru_image_comp_coimage_iso_image'_inv :
factor_thru_image f ≫ (coimage_iso_image' f).inv = cokernel.π _ | by simp only [is_image.iso_ext_inv, image.is_image_lift, image.fac_lift,
coimage_strong_epi_mono_factorisation_to_mono_factorisation_e] | lemma | category_theory.abelian.factor_thru_image_comp_coimage_iso_image'_inv | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_iso_image : abelian.image f ≅ image f | is_image.iso_ext (image_strong_epi_mono_factorisation f).to_mono_is_image (image.is_image f) | abbreviation | category_theory.abelian.image_iso_image | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | There is a canonical isomorphism between the abelian image and the categorical image of a
morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_iso_image_hom_comp_image_ι :
(image_iso_image f).hom ≫ limits.image.ι _ = kernel.ι _ | by simp only [is_image.iso_ext_hom, is_image.lift_ι,
image_strong_epi_mono_factorisation_to_mono_factorisation_m] | lemma | category_theory.abelian.image_iso_image_hom_comp_image_ι | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_iso_image_inv :
(image_iso_image f).inv = kernel.lift _ (limits.image.ι f)
(by simp [←cancel_epi (factor_thru_image f)]) | begin
ext,
simp only [is_image.iso_ext_inv, image.is_image_lift, limits.image.fac_lift,
image_strong_epi_mono_factorisation_to_mono_factorisation_e, category.assoc,
kernel.lift_ι, limits.image.fac],
end | lemma | category_theory.abelian.image_iso_image_inv | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_is_cokernel_of_kernel [epi f] (s : fork f 0) (h : is_limit s) :
is_colimit (cokernel_cofork.of_π f (kernel_fork.condition s)) | non_preadditive_abelian.epi_is_cokernel_of_kernel s h | def | category_theory.abelian.epi_is_cokernel_of_kernel | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | In an abelian category, an epi is the cokernel of its kernel. More precisely:
If `f` is an epimorphism and `s` is some limit kernel cone on `f`, then `f` is a cokernel
of `fork.ι s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_is_kernel_of_cokernel [mono f] (s : cofork f 0) (h : is_colimit s) :
is_limit (kernel_fork.of_ι f (cokernel_cofork.condition s)) | non_preadditive_abelian.mono_is_kernel_of_cokernel s h | def | category_theory.abelian.mono_is_kernel_of_cokernel | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | In an abelian category, a mono is the kernel of its cokernel. More precisely:
If `f` is a monomorphism and `s` is some colimit cokernel cocone on `f`, then `f` is a kernel
of `cofork.π s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_desc [epi f] {T : C} (g : X ⟶ T) (hg : kernel.ι f ≫ g = 0) : Y ⟶ T | (epi_is_cokernel_of_kernel _ (limit.is_limit _)).desc (cokernel_cofork.of_π _ hg) | def | category_theory.abelian.epi_desc | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | In an abelian category, any morphism that turns to zero when precomposed with the kernel of an
epimorphism factors through that epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_epi_desc [epi f] {T : C} (g : X ⟶ T) (hg : kernel.ι f ≫ g = 0) :
f ≫ epi_desc f g hg = g | (epi_is_cokernel_of_kernel _ (limit.is_limit _)).fac (cokernel_cofork.of_π _ hg)
walking_parallel_pair.one | lemma | category_theory.abelian.comp_epi_desc | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_lift [mono f] {T : C} (g : T ⟶ Y) (hg : g ≫ cokernel.π f = 0) : T ⟶ X | (mono_is_kernel_of_cokernel _ (colimit.is_colimit _)).lift (kernel_fork.of_ι _ hg) | def | category_theory.abelian.mono_lift | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [
"lift"
] | In an abelian category, any morphism that turns to zero when postcomposed with the cokernel of a
monomorphism factors through that monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_lift_comp [mono f] {T : C} (g : T ⟶ Y) (hg : g ≫ cokernel.π f = 0) :
mono_lift f g hg ≫ f = g | (mono_is_kernel_of_cokernel _ (colimit.is_colimit _)).fac (kernel_fork.of_ι _ hg)
walking_parallel_pair.zero | lemma | category_theory.abelian.mono_lift_comp | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_equalizers : has_equalizers C | preadditive.has_equalizers_of_has_kernels | instance | category_theory.abelian.has_equalizers | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pullbacks : has_pullbacks C | has_pullbacks_of_has_binary_products_of_has_equalizers C | instance | category_theory.abelian.has_pullbacks | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | Any abelian category has pullbacks | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coequalizers : has_coequalizers C | preadditive.has_coequalizers_of_has_cokernels | instance | category_theory.abelian.has_coequalizers | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pushouts : has_pushouts C | has_pushouts_of_has_binary_coproducts_of_has_coequalizers C | instance | category_theory.abelian.has_pushouts | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | Any abelian category has pushouts | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_limits : has_finite_limits C | limits.has_finite_limits_of_has_equalizers_and_finite_products | instance | category_theory.abelian.has_finite_limits | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_colimits : has_finite_colimits C | limits.has_finite_colimits_of_has_coequalizers_and_finite_coproducts | instance | category_theory.abelian.has_finite_colimits | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_to_biproduct : pullback f g ⟶ X ⊞ Y | biprod.lift pullback.fst pullback.snd | abbreviation | category_theory.abelian.pullback_to_biproduct_is_kernel.pullback_to_biproduct | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | The canonical map `pullback f g ⟶ X ⊞ Y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_to_biproduct_fork : kernel_fork (biprod.desc f (-g)) | kernel_fork.of_ι (pullback_to_biproduct f g) $
by rw [biprod.lift_desc, comp_neg, pullback.condition, add_right_neg] | abbreviation | category_theory.abelian.pullback_to_biproduct_is_kernel.pullback_to_biproduct_fork | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | The canonical map `pullback f g ⟶ X ⊞ Y` induces a kernel cone on the map
`biproduct X Y ⟶ Z` induced by `f` and `g`. A slightly more intuitive way to think of
this may be that it induces an equalizer fork on the maps induced by `(f, 0)` and
`(0, g)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_pullback_to_biproduct : is_limit (pullback_to_biproduct_fork f g) | fork.is_limit.mk _
(λ s, pullback.lift (fork.ι s ≫ biprod.fst) (fork.ι s ≫ biprod.snd) $
sub_eq_zero.1 $ by rw [category.assoc, category.assoc, ←comp_sub, sub_eq_add_neg, ←comp_neg,
←biprod.desc_eq, kernel_fork.condition s])
(λ s,
begin
ext; rw [fork.ι_of_ι, category.assoc],
{ rw [biprod.lift_fs... | def | category_theory.abelian.pullback_to_biproduct_is_kernel.is_limit_pullback_to_biproduct | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | The canonical map `pullback f g ⟶ X ⊞ Y` is a kernel of the map induced by
`(f, -g)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct_to_pushout : Y ⊞ Z ⟶ pushout f g | biprod.desc pushout.inl pushout.inr | abbreviation | category_theory.abelian.biproduct_to_pushout_is_cokernel.biproduct_to_pushout | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | The canonical map `Y ⊞ Z ⟶ pushout f g` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct_to_pushout_cofork : cokernel_cofork (biprod.lift f (-g)) | cokernel_cofork.of_π (biproduct_to_pushout f g) $
by rw [biprod.lift_desc, neg_comp, pushout.condition, add_right_neg] | abbreviation | category_theory.abelian.biproduct_to_pushout_is_cokernel.biproduct_to_pushout_cofork | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | The canonical map `Y ⊞ Z ⟶ pushout f g` induces a cokernel cofork on the map
`X ⟶ Y ⊞ Z` induced by `f` and `-g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_biproduct_to_pushout : is_colimit (biproduct_to_pushout_cofork f g) | cofork.is_colimit.mk _
(λ s, pushout.desc (biprod.inl ≫ cofork.π s) (biprod.inr ≫ cofork.π s) $
sub_eq_zero.1 $ by rw [←category.assoc, ←category.assoc, ←sub_comp, sub_eq_add_neg, ←neg_comp,
←biprod.lift_eq, cofork.condition s, zero_comp])
(λ s, by ext; simp)
(λ s m h, by ext; simp [←h] ) | def | category_theory.abelian.biproduct_to_pushout_is_cokernel.is_colimit_biproduct_to_pushout | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | The cofork induced by the canonical map `Y ⊞ Z ⟶ pushout f g` is in fact a colimit cokernel
cofork. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_pullback_of_epi_f [epi f] : epi (pullback.snd : pullback f g ⟶ Y) | -- It will suffice to consider some morphism e : Y ⟶ R such that
-- pullback.snd ≫ e = 0 and show that e = 0.
epi_of_cancel_zero _ $ λ R e h,
begin
-- Consider the morphism u := (0, e) : X ⊞ Y⟶ R.
let u := biprod.desc (0 : X ⟶ R) e,
-- The composite pullback f g ⟶ X ⊞ Y ⟶ R is zero by assumption.
have hu : pull... | instance | category_theory.abelian.epi_pullback_of_epi_f | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | In an abelian category, the pullback of an epimorphism is an epimorphism.
Proof from [aluffi2016, IX.2.3], cf. [borceux-vol2, 1.7.6] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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