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extend (c : cone F) {X : C} (f : X ⟶ c.X) : cone F | { X := X,
π := c.extensions.app (op X) ⟨f⟩ } | def | category_theory.limits.cone.extend | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [
"extend"
] | A map to the vertex of a cone induces a cone by composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker (E : K ⥤ J) (c : cone F) : cone (E ⋙ F) | { X := c.X,
π := whisker_left E c.π } | def | category_theory.limits.cone.whisker | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Whisker a cone by precomposition of a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv (F : J ⥤ C) : cocone F ≅ Σ X, F.cocones.obj X | { hom := λ c, ⟨c.X, c.ι⟩,
inv := λ c, { X := c.1, ι := c.2 },
hom_inv_id' := by { ext1, cases x, refl },
inv_hom_id' := by { ext1, cases x, refl } } | def | category_theory.limits.cocone.equiv | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [
"equiv"
] | The isomorphism between a cocone on `F` and an element of the functor `F.cocones`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extensions (c : cocone F) : coyoneda.obj (op c.X) ⋙ ulift_functor.{u₁} ⟶ F.cocones | { app := λ X f, c.ι ≫ (const J).map f.down } | def | category_theory.limits.cocone.extensions | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | A map from the vertex of a cocone naturally induces a cocone by composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend (c : cocone F) {X : C} (f : c.X ⟶ X) : cocone F | { X := X,
ι := c.extensions.app X ⟨f⟩ } | def | category_theory.limits.cocone.extend | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [
"extend"
] | A map from the vertex of a cocone induces a cocone by composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker (E : K ⥤ J) (c : cocone F) : cocone (E ⋙ F) | { X := c.X,
ι := whisker_left E c.ι } | def | category_theory.limits.cocone.whisker | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Whisker a cocone by precomposition of a functor. See `whiskering` for a functorial
version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_morphism (A B : cone F) | (hom : A.X ⟶ B.X)
(w' : ∀ j : J, hom ≫ B.π.app j = A.π.app j . obviously) | structure | category_theory.limits.cone_morphism | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | A cone morphism between two cones for the same diagram is a morphism of the cone points which
commutes with the cone legs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited_cone_morphism (A : cone F) : inhabited (cone_morphism A A) | ⟨{ hom := 𝟙 _ }⟩ | instance | category_theory.limits.inhabited_cone_morphism | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cone.category : category (cone F) | { hom := λ A B, cone_morphism A B,
comp := λ X Y Z f g, { hom := f.hom ≫ g.hom },
id := λ B, { hom := 𝟙 B.X } } | instance | category_theory.limits.cone.category | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | The category of cones on a given diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {c c' : cone F}
(φ : c.X ≅ c'.X) (w : ∀ j, c.π.app j = φ.hom ≫ c'.π.app j) : c ≅ c' | { hom := { hom := φ.hom },
inv := { hom := φ.inv, w' := λ j, φ.inv_comp_eq.mpr (w j) } } | def | category_theory.limits.cones.ext | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | To give an isomorphism between cones, it suffices to give an
isomorphism between their vertices which commutes with the cone
maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eta (c : cone F) : c ≅ ⟨c.X, c.π⟩ | cones.ext (iso.refl _) (by tidy) | def | category_theory.limits.cones.eta | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Eta rule for cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_iso_of_hom_iso {K : J ⥤ C} {c d : cone K} (f : c ⟶ d) [i : is_iso f.hom] :
is_iso f | ⟨⟨{ hom := inv f.hom,
w' := λ j, (as_iso f.hom).inv_comp_eq.2 (f.w j).symm }, by tidy⟩⟩ | lemma | category_theory.limits.cones.cone_iso_of_hom_iso | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Given a cone morphism whose object part is an isomorphism, produce an
isomorphism of cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
postcompose {G : J ⥤ C} (α : F ⟶ G) : cone F ⥤ cone G | { obj := λ c, { X := c.X, π := c.π ≫ α },
map := λ c₁ c₂ f, { hom := f.hom } } | def | category_theory.limits.cones.postcompose | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Functorially postcompose a cone for `F` by a natural transformation `F ⟶ G` to give a cone for `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
postcompose_comp {G H : J ⥤ C} (α : F ⟶ G) (β : G ⟶ H) :
postcompose (α ≫ β) ≅ postcompose α ⋙ postcompose β | nat_iso.of_components (λ s, cones.ext (iso.refl _) (by tidy)) (by tidy) | def | category_theory.limits.cones.postcompose_comp | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Postcomposing a cone by the composite natural transformation `α ≫ β` is the same as
postcomposing by `α` and then by `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
postcompose_id : postcompose (𝟙 F) ≅ 𝟭 (cone F) | nat_iso.of_components (λ s, cones.ext (iso.refl _) (by tidy)) (by tidy) | def | category_theory.limits.cones.postcompose_id | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Postcomposing by the identity does not change the cone up to isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
postcompose_equivalence {G : J ⥤ C} (α : F ≅ G) : cone F ≌ cone G | { functor := postcompose α.hom,
inverse := postcompose α.inv,
unit_iso := nat_iso.of_components (λ s, cones.ext (iso.refl _) (by tidy)) (by tidy),
counit_iso := nat_iso.of_components (λ s, cones.ext (iso.refl _) (by tidy)) (by tidy) } | def | category_theory.limits.cones.postcompose_equivalence | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | If `F` and `G` are naturally isomorphic functors, then they have equivalent categories of
cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whiskering (E : K ⥤ J) : cone F ⥤ cone (E ⋙ F) | { obj := λ c, c.whisker E,
map := λ c c' f, { hom := f.hom } } | def | category_theory.limits.cones.whiskering | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Whiskering on the left by `E : K ⥤ J` gives a functor from `cone F` to `cone (E ⋙ F)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whiskering_equivalence (e : K ≌ J) :
cone F ≌ cone (e.functor ⋙ F) | { functor := whiskering e.functor,
inverse := whiskering e.inverse ⋙ postcompose (e.inv_fun_id_assoc F).hom,
unit_iso := nat_iso.of_components (λ s, cones.ext (iso.refl _) (by tidy)) (by tidy),
counit_iso := nat_iso.of_components (λ s, cones.ext (iso.refl _)
(begin
intro k,
dsimp, -- See library note [d... | def | category_theory.limits.cones.whiskering_equivalence | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Whiskering by an equivalence gives an equivalence between categories of cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_of_reindexing {G : K ⥤ C}
(e : K ≌ J) (α : e.functor ⋙ F ≅ G) : cone F ≌ cone G | (whiskering_equivalence e).trans (postcompose_equivalence α) | def | category_theory.limits.cones.equivalence_of_reindexing | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | The categories of cones over `F` and `G` are equivalent if `F` and `G` are naturally isomorphic
(possibly after changing the indexing category by an equivalence). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget : cone F ⥤ C | { obj := λ t, t.X, map := λ s t f, f.hom } | def | category_theory.limits.cones.forget | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Forget the cone structure and obtain just the cone point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functoriality : cone F ⥤ cone (F ⋙ G) | { obj := λ A,
{ X := G.obj A.X,
π := { app := λ j, G.map (A.π.app j), naturality' := by intros; erw ←G.map_comp; tidy } },
map := λ X Y f,
{ hom := G.map f.hom,
w' := λ j, by simp [-cone_morphism.w, ←f.w j] } } | def | category_theory.limits.cones.functoriality | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | A functor `G : C ⥤ D` sends cones over `F` to cones over `F ⋙ G` functorially. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functoriality_full [full G] [faithful G] : full (functoriality F G) | { preimage := λ X Y t,
{ hom := G.preimage t.hom,
w' := λ j, G.map_injective (by simpa using t.w j) } } | instance | category_theory.limits.cones.functoriality_full | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functoriality_faithful [faithful G] : faithful (cones.functoriality F G) | { map_injective' := λ X Y f g e, by { ext1, injection e, apply G.map_injective h_1 } } | instance | category_theory.limits.cones.functoriality_faithful | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functoriality_equivalence (e : C ≌ D) : cone F ≌ cone (F ⋙ e.functor) | let f : (F ⋙ e.functor) ⋙ e.inverse ≅ F :=
functor.associator _ _ _ ≪≫ iso_whisker_left _ (e.unit_iso).symm ≪≫ functor.right_unitor _ in
{ functor := functoriality F e.functor,
inverse := (functoriality (F ⋙ e.functor) e.inverse) ⋙
(postcompose_equivalence f).functor,
unit_iso := nat_iso.of_components (λ c, c... | def | category_theory.limits.cones.functoriality_equivalence | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | If `e : C ≌ D` is an equivalence of categories, then `functoriality F e.functor` induces an
equivalence between cones over `F` and cones over `F ⋙ e.functor`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_cone_isomorphism (F : C ⥤ D) [reflects_isomorphisms F] (K : J ⥤ C) :
reflects_isomorphisms (cones.functoriality K F) | begin
constructor,
introsI,
haveI : is_iso (F.map f.hom) :=
(cones.forget (K ⋙ F)).map_is_iso ((cones.functoriality K F).map f),
haveI := reflects_isomorphisms.reflects F f.hom,
apply cone_iso_of_hom_iso
end | instance | category_theory.limits.cones.reflects_cone_isomorphism | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | If `F` reflects isomorphisms, then `cones.functoriality F` reflects isomorphisms
as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_morphism (A B : cocone F) | (hom : A.X ⟶ B.X)
(w' : ∀ j : J, A.ι.app j ≫ hom = B.ι.app j . obviously) | structure | category_theory.limits.cocone_morphism | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | A cocone morphism between two cocones for the same diagram is a morphism of the cocone points
which commutes with the cocone legs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited_cocone_morphism (A : cocone F) : inhabited (cocone_morphism A A) | ⟨{ hom := 𝟙 _ }⟩ | instance | category_theory.limits.inhabited_cocone_morphism | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone.category : category (cocone F) | { hom := λ A B, cocone_morphism A B,
comp := λ _ _ _ f g,
{ hom := f.hom ≫ g.hom },
id := λ B, { hom := 𝟙 B.X } } | instance | category_theory.limits.cocone.category | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {c c' : cocone F}
(φ : c.X ≅ c'.X) (w : ∀ j, c.ι.app j ≫ φ.hom = c'.ι.app j) : c ≅ c' | { hom := { hom := φ.hom },
inv := { hom := φ.inv, w' := λ j, φ.comp_inv_eq.mpr (w j).symm } } | def | category_theory.limits.cocones.ext | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | To give an isomorphism between cocones, it suffices to give an
isomorphism between their vertices which commutes with the cocone
maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eta (c : cocone F) : c ≅ ⟨c.X, c.ι⟩ | cocones.ext (iso.refl _) (by tidy) | def | category_theory.limits.cocones.eta | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Eta rule for cocones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_iso_of_hom_iso {K : J ⥤ C} {c d : cocone K} (f : c ⟶ d) [i : is_iso f.hom] :
is_iso f | ⟨⟨{ hom := inv f.hom,
w' := λ j, (as_iso f.hom).comp_inv_eq.2 (f.w j).symm }, by tidy⟩⟩ | lemma | category_theory.limits.cocones.cocone_iso_of_hom_iso | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Given a cocone morphism whose object part is an isomorphism, produce an
isomorphism of cocones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
precompose {G : J ⥤ C} (α : G ⟶ F) : cocone F ⥤ cocone G | { obj := λ c, { X := c.X, ι := α ≫ c.ι },
map := λ c₁ c₂ f, { hom := f.hom } } | def | category_theory.limits.cocones.precompose | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Functorially precompose a cocone for `F` by a natural transformation `G ⟶ F` to give a cocone
for `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
precompose_comp {G H : J ⥤ C} (α : F ⟶ G) (β : G ⟶ H) :
precompose (α ≫ β) ≅ precompose β ⋙ precompose α | nat_iso.of_components (λ s, cocones.ext (iso.refl _) (by tidy)) (by tidy) | def | category_theory.limits.cocones.precompose_comp | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Precomposing a cocone by the composite natural transformation `α ≫ β` is the same as
precomposing by `β` and then by `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
precompose_id : precompose (𝟙 F) ≅ 𝟭 (cocone F) | nat_iso.of_components (λ s, cocones.ext (iso.refl _) (by tidy)) (by tidy) | def | category_theory.limits.cocones.precompose_id | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Precomposing by the identity does not change the cocone up to isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
precompose_equivalence {G : J ⥤ C} (α : G ≅ F) : cocone F ≌ cocone G | { functor := precompose α.hom,
inverse := precompose α.inv,
unit_iso := nat_iso.of_components (λ s, cocones.ext (iso.refl _) (by tidy)) (by tidy),
counit_iso := nat_iso.of_components (λ s, cocones.ext (iso.refl _) (by tidy)) (by tidy) } | def | category_theory.limits.cocones.precompose_equivalence | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | If `F` and `G` are naturally isomorphic functors, then they have equivalent categories of
cocones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whiskering (E : K ⥤ J) : cocone F ⥤ cocone (E ⋙ F) | { obj := λ c, c.whisker E,
map := λ c c' f, { hom := f.hom, } } | def | category_theory.limits.cocones.whiskering | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Whiskering on the left by `E : K ⥤ J` gives a functor from `cocone F` to `cocone (E ⋙ F)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whiskering_equivalence (e : K ≌ J) :
cocone F ≌ cocone (e.functor ⋙ F) | { functor := whiskering e.functor,
inverse := whiskering e.inverse ⋙
precompose ((functor.left_unitor F).inv ≫ (whisker_right (e.counit_iso).inv F) ≫
(functor.associator _ _ _).inv),
unit_iso := nat_iso.of_components (λ s, cocones.ext (iso.refl _) (by tidy)) (by tidy),
counit_iso := nat_iso.of_component... | def | category_theory.limits.cocones.whiskering_equivalence | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Whiskering by an equivalence gives an equivalence between categories of cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_of_reindexing {G : K ⥤ C}
(e : K ≌ J) (α : e.functor ⋙ F ≅ G) : cocone F ≌ cocone G | (whiskering_equivalence e).trans (precompose_equivalence α.symm) | def | category_theory.limits.cocones.equivalence_of_reindexing | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | The categories of cocones over `F` and `G` are equivalent if `F` and `G` are naturally isomorphic
(possibly after changing the indexing category by an equivalence). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget : cocone F ⥤ C | { obj := λ t, t.X, map := λ s t f, f.hom } | def | category_theory.limits.cocones.forget | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Forget the cocone structure and obtain just the cocone point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functoriality : cocone F ⥤ cocone (F ⋙ G) | { obj := λ A,
{ X := G.obj A.X,
ι := { app := λ j, G.map (A.ι.app j), naturality' := by intros; erw ←G.map_comp; tidy } },
map := λ _ _ f,
{ hom := G.map f.hom,
w' := by intros; rw [←functor.map_comp, cocone_morphism.w] } } | def | category_theory.limits.cocones.functoriality | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | A functor `G : C ⥤ D` sends cocones over `F` to cocones over `F ⋙ G` functorially. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functoriality_faithful [faithful G] : faithful (functoriality F G) | { map_injective' := λ X Y f g e, by { ext1, injection e, apply G.map_injective h_1 } } | instance | category_theory.limits.cocones.functoriality_faithful | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functoriality_equivalence (e : C ≌ D) : cocone F ≌ cocone (F ⋙ e.functor) | let f : (F ⋙ e.functor) ⋙ e.inverse ≅ F :=
functor.associator _ _ _ ≪≫ iso_whisker_left _ (e.unit_iso).symm ≪≫ functor.right_unitor _ in
{ functor := functoriality F e.functor,
inverse := (functoriality (F ⋙ e.functor) e.inverse) ⋙
(precompose_equivalence f.symm).functor,
unit_iso := nat_iso.of_components (λ ... | def | category_theory.limits.cocones.functoriality_equivalence | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [
"map_comp"
] | If `e : C ≌ D` is an equivalence of categories, then `functoriality F e.functor` induces an
equivalence between cocones over `F` and cocones over `F ⋙ e.functor`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_cocone_isomorphism (F : C ⥤ D) [reflects_isomorphisms F] (K : J ⥤ C) :
reflects_isomorphisms (cocones.functoriality K F) | begin
constructor,
introsI,
haveI : is_iso (F.map f.hom) :=
(cocones.forget (K ⋙ F)).map_is_iso ((cocones.functoriality K F).map f),
haveI := reflects_isomorphisms.reflects F f.hom,
apply cocone_iso_of_hom_iso
end | instance | category_theory.limits.cocones.reflects_cocone_isomorphism | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | If `F` reflects isomorphisms, then `cocones.functoriality F` reflects isomorphisms
as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cone (c : cone F) : cone (F ⋙ H) | (cones.functoriality F H).obj c | def | category_theory.functor.map_cone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | The image of a cone in C under a functor G : C ⥤ D is a cone in D. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cocone (c : cocone F) : cocone (F ⋙ H) | (cocones.functoriality F H).obj c | def | category_theory.functor.map_cocone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | The image of a cocone in C under a functor G : C ⥤ D is a cocone in D. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cone_morphism {c c' : cone F} (f : c ⟶ c') :
H.map_cone c ⟶ H.map_cone c' | (cones.functoriality F H).map f | def | category_theory.functor.map_cone_morphism | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Given a cone morphism `c ⟶ c'`, construct a cone morphism on the mapped cones functorially. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cocone_morphism {c c' : cocone F} (f : c ⟶ c') :
H.map_cocone c ⟶ H.map_cocone c' | (cocones.functoriality F H).map f | def | category_theory.functor.map_cocone_morphism | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Given a cocone morphism `c ⟶ c'`, construct a cocone morphism on the mapped cocones
functorially. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cone_inv [is_equivalence H]
(c : cone (F ⋙ H)) : cone F | (limits.cones.functoriality_equivalence F (as_equivalence H)).inverse.obj c | def | category_theory.functor.map_cone_inv | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | If `H` is an equivalence, we invert `H.map_cone` and get a cone for `F` from a cone
for `F ⋙ H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cone_map_cone_inv {F : J ⥤ D} (H : D ⥤ C) [is_equivalence H] (c : cone (F ⋙ H)) :
map_cone H (map_cone_inv H c) ≅ c | (limits.cones.functoriality_equivalence F (as_equivalence H)).counit_iso.app c | def | category_theory.functor.map_cone_map_cone_inv | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | `map_cone` is the left inverse to `map_cone_inv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cone_inv_map_cone {F : J ⥤ D} (H : D ⥤ C) [is_equivalence H] (c : cone F) :
map_cone_inv H (map_cone H c) ≅ c | (limits.cones.functoriality_equivalence F (as_equivalence H)).unit_iso.symm.app c | def | category_theory.functor.map_cone_inv_map_cone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | `map_cone` is the right inverse to `map_cone_inv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cocone_inv [is_equivalence H]
(c : cocone (F ⋙ H)) : cocone F | (limits.cocones.functoriality_equivalence F (as_equivalence H)).inverse.obj c | def | category_theory.functor.map_cocone_inv | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | If `H` is an equivalence, we invert `H.map_cone` and get a cone for `F` from a cone
for `F ⋙ H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cocone_map_cocone_inv {F : J ⥤ D} (H : D ⥤ C) [is_equivalence H] (c : cocone (F ⋙ H)) :
map_cocone H (map_cocone_inv H c) ≅ c | (limits.cocones.functoriality_equivalence F (as_equivalence H)).counit_iso.app c | def | category_theory.functor.map_cocone_map_cocone_inv | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | `map_cocone` is the left inverse to `map_cocone_inv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cocone_inv_map_cocone {F : J ⥤ D} (H : D ⥤ C) [is_equivalence H] (c : cocone F) :
map_cocone_inv H (map_cocone H c) ≅ c | (limits.cocones.functoriality_equivalence F (as_equivalence H)).unit_iso.symm.app c | def | category_theory.functor.map_cocone_inv_map_cocone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | `map_cocone` is the right inverse to `map_cocone_inv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functoriality_comp_postcompose {H H' : C ⥤ D} (α : H ≅ H') :
cones.functoriality F H ⋙ cones.postcompose (whisker_left F α.hom) ≅ cones.functoriality F H' | nat_iso.of_components (λ c, cones.ext (α.app _) (by tidy)) (by tidy) | def | category_theory.functor.functoriality_comp_postcompose | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | `functoriality F _ ⋙ postcompose (whisker_left F _)` simplifies to `functoriality F _`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
postcompose_whisker_left_map_cone {H H' : C ⥤ D} (α : H ≅ H') (c : cone F) :
(cones.postcompose (whisker_left F α.hom : _)).obj (H.map_cone c) ≅ H'.map_cone c | (functoriality_comp_postcompose α).app c | def | category_theory.functor.postcompose_whisker_left_map_cone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | For `F : J ⥤ C`, given a cone `c : cone F`, and a natural isomorphism `α : H ≅ H'` for functors
`H H' : C ⥤ D`, the postcomposition of the cone `H.map_cone` using the isomorphism `α` is
isomorphic to the cone `H'.map_cone`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cone_postcompose {α : F ⟶ G} {c} :
H.map_cone ((cones.postcompose α).obj c) ≅
(cones.postcompose (whisker_right α H : _)).obj (H.map_cone c) | cones.ext (iso.refl _) (by tidy) | def | category_theory.functor.map_cone_postcompose | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | `map_cone` commutes with `postcompose`. In particular, for `F : J ⥤ C`, given a cone `c : cone F`, a
natural transformation `α : F ⟶ G` and a functor `H : C ⥤ D`, we have two obvious ways of producing
a cone over `G ⋙ H`, and they are both isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cone_postcompose_equivalence_functor {α : F ≅ G} {c} :
H.map_cone ((cones.postcompose_equivalence α).functor.obj c) ≅
(cones.postcompose_equivalence (iso_whisker_right α H : _)).functor.obj (H.map_cone c) | cones.ext (iso.refl _) (by tidy) | def | category_theory.functor.map_cone_postcompose_equivalence_functor | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | `map_cone` commutes with `postcompose_equivalence` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functoriality_comp_precompose {H H' : C ⥤ D} (α : H ≅ H') :
cocones.functoriality F H ⋙ cocones.precompose (whisker_left F α.inv)
≅ cocones.functoriality F H' | nat_iso.of_components (λ c, cocones.ext (α.app _) (by tidy)) (by tidy) | def | category_theory.functor.functoriality_comp_precompose | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | `functoriality F _ ⋙ precompose (whisker_left F _)` simplifies to `functoriality F _`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
precompose_whisker_left_map_cocone {H H' : C ⥤ D} (α : H ≅ H') (c : cocone F) :
(cocones.precompose (whisker_left F α.inv : _)).obj (H.map_cocone c) ≅ H'.map_cocone c | (functoriality_comp_precompose α).app c | def | category_theory.functor.precompose_whisker_left_map_cocone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | For `F : J ⥤ C`, given a cocone `c : cocone F`, and a natural isomorphism `α : H ≅ H'` for functors
`H H' : C ⥤ D`, the precomposition of the cocone `H.map_cocone` using the isomorphism `α` is
isomorphic to the cocone `H'.map_cocone`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cocone_precompose {α : F ⟶ G} {c} :
H.map_cocone ((cocones.precompose α).obj c) ≅
(cocones.precompose (whisker_right α H : _)).obj (H.map_cocone c) | cocones.ext (iso.refl _) (by tidy) | def | category_theory.functor.map_cocone_precompose | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | `map_cocone` commutes with `precompose`. In particular, for `F : J ⥤ C`, given a cocone
`c : cocone F`, a natural transformation `α : F ⟶ G` and a functor `H : C ⥤ D`, we have two obvious
ways of producing a cocone over `G ⋙ H`, and they are both isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cocone_precompose_equivalence_functor {α : F ≅ G} {c} :
H.map_cocone ((cocones.precompose_equivalence α).functor.obj c) ≅
(cocones.precompose_equivalence (iso_whisker_right α H : _)).functor.obj (H.map_cocone c) | cocones.ext (iso.refl _) (by tidy) | def | category_theory.functor.map_cocone_precompose_equivalence_functor | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | `map_cocone` commutes with `precompose_equivalence` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cone_whisker {E : K ⥤ J} {c : cone F} :
H.map_cone (c.whisker E) ≅ (H.map_cone c).whisker E | cones.ext (iso.refl _) (by tidy) | def | category_theory.functor.map_cone_whisker | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | `map_cone` commutes with `whisker` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cocone_whisker {E : K ⥤ J} {c : cocone F} :
H.map_cocone (c.whisker E) ≅ (H.map_cocone c).whisker E | cocones.ext (iso.refl _) (by tidy) | def | category_theory.functor.map_cocone_whisker | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | `map_cocone` commutes with `whisker` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone.op (c : cocone F) : cone F.op | { X := op c.X,
π := nat_trans.op c.ι } | def | category_theory.limits.cocone.op | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Change a `cocone F` into a `cone F.op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone.op (c : cone F) : cocone F.op | { X := op c.X,
ι := nat_trans.op c.π } | def | category_theory.limits.cone.op | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Change a `cone F` into a `cocone F.op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone.unop (c : cocone F.op) : cone F | { X := unop c.X,
π := nat_trans.remove_op c.ι } | def | category_theory.limits.cocone.unop | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Change a `cocone F.op` into a `cone F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone.unop (c : cone F.op) : cocone F | { X := unop c.X,
ι := nat_trans.remove_op c.π } | def | category_theory.limits.cone.unop | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Change a `cone F.op` into a `cocone F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_equivalence_op_cone_op : cocone F ≌ (cone F.op)ᵒᵖ | { functor :=
{ obj := λ c, op (cocone.op c),
map := λ X Y f, quiver.hom.op
{ hom := f.hom.op,
w' := λ j, by { apply quiver.hom.unop_inj, dsimp, apply cocone_morphism.w }, } },
inverse :=
{ obj := λ c, cone.unop (unop c),
map := λ X Y f,
{ hom := f.unop.hom.unop,
w' := λ j, by { apply q... | def | category_theory.limits.cocone_equivalence_op_cone_op | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [
"opposite.rec",
"quiver.hom.op",
"quiver.hom.op_inj",
"quiver.hom.unop_inj"
] | The category of cocones on `F`
is equivalent to the opposite category of
the category of cones on the opposite of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_of_cocone_left_op (c : cocone F.left_op) : cone F | { X := op c.X,
π := nat_trans.remove_left_op c.ι } | def | category_theory.limits.cone_of_cocone_left_op | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone_left_op_of_cone (c : cone F) : cocone (F.left_op) | { X := unop c.X,
ι := nat_trans.left_op c.π } | def | category_theory.limits.cocone_left_op_of_cone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Change a cone on `F : J ⥤ Cᵒᵖ` to a cocone on `F.left_op : Jᵒᵖ ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_of_cone_left_op (c : cone F.left_op) : cocone F | { X := op c.X,
ι := nat_trans.remove_left_op c.π } | def | category_theory.limits.cocone_of_cone_left_op | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone_of_cone_left_op_ι_app (c : cone F.left_op) (j) :
(cocone_of_cone_left_op c).ι.app j = (c.π.app (op j)).op | by { dsimp only [cocone_of_cone_left_op], simp } | lemma | category_theory.limits.cocone_of_cone_left_op_ι_app | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cone_left_op_of_cocone (c : cocone F) : cone (F.left_op) | { X := unop c.X,
π := nat_trans.left_op c.ι } | def | category_theory.limits.cone_left_op_of_cocone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Change a cocone on `F : J ⥤ Cᵒᵖ` to a cone on `F.left_op : Jᵒᵖ ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_of_cocone_right_op (c : cocone F.right_op) : cone F | { X := unop c.X,
π := nat_trans.remove_right_op c.ι } | def | category_theory.limits.cone_of_cocone_right_op | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Change a cocone on `F.right_op : J ⥤ Cᵒᵖ` to a cone on `F : Jᵒᵖ ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_right_op_of_cone (c : cone F) : cocone (F.right_op) | { X := op c.X,
ι := nat_trans.right_op c.π } | def | category_theory.limits.cocone_right_op_of_cone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Change a cone on `F : Jᵒᵖ ⥤ C` to a cocone on `F.right_op : Jᵒᵖ ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_of_cone_right_op (c : cone F.right_op) : cocone F | { X := unop c.X,
ι := nat_trans.remove_right_op c.π } | def | category_theory.limits.cocone_of_cone_right_op | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Change a cone on `F.right_op : J ⥤ Cᵒᵖ` to a cocone on `F : Jᵒᵖ ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_right_op_of_cocone (c : cocone F) : cone (F.right_op) | { X := op c.X,
π := nat_trans.right_op c.ι } | def | category_theory.limits.cone_right_op_of_cocone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Change a cocone on `F : Jᵒᵖ ⥤ C` to a cone on `F.right_op : J ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_of_cocone_unop (c : cocone F.unop) : cone F | { X := op c.X,
π := nat_trans.remove_unop c.ι } | def | category_theory.limits.cone_of_cocone_unop | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Change a cocone on `F.unop : J ⥤ C` into a cone on `F : Jᵒᵖ ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_unop_of_cone (c : cone F) : cocone F.unop | { X := unop c.X,
ι := nat_trans.unop c.π } | def | category_theory.limits.cocone_unop_of_cone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Change a cone on `F : Jᵒᵖ ⥤ Cᵒᵖ` into a cocone on `F.unop : J ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_of_cone_unop (c : cone F.unop) : cocone F | { X := op c.X,
ι := nat_trans.remove_unop c.π } | def | category_theory.limits.cocone_of_cone_unop | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Change a cone on `F.unop : J ⥤ C` into a cocone on `F : Jᵒᵖ ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_unop_of_cocone (c : cocone F) : cone F.unop | { X := unop c.X,
π := nat_trans.unop c.ι } | def | category_theory.limits.cone_unop_of_cocone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Change a cocone on `F : Jᵒᵖ ⥤ Cᵒᵖ` into a cone on `F.unop : J ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cone_op (t : cone F) : (G.map_cone t).op ≅ (G.op.map_cocone t.op) | cocones.ext (iso.refl _) (by tidy) | def | category_theory.functor.map_cone_op | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | The opposite cocone of the image of a cone is the image of the opposite cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_cocone_op {t : cocone F} : (G.map_cocone t).op ≅ (G.op.map_cone t.op) | cones.ext (iso.refl _) (by tidy) | def | category_theory.functor.map_cocone_op | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | The opposite cone of the image of a cocone is the image of the opposite cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone.to_costructured_arrow (F : J ⥤ C) : cone F ⥤ costructured_arrow (const J) F | { obj := λ c, costructured_arrow.mk c.π,
map := λ c d f, costructured_arrow.hom_mk f.hom $ by { ext, simp } } | def | category_theory.limits.cone.to_costructured_arrow | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | Construct an object of the category `(Δ ↓ F)` from a cone on `F`. This is part of an
equivalence, see `cone.equiv_costructured_arrow`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone.from_costructured_arrow (F : J ⥤ C) : costructured_arrow (const J) F ⥤ cone F | { obj := λ c, ⟨c.left, c.hom⟩,
map := λ c d f,
{ hom := f.left,
w' := λ j, by { convert (congr_fun (congr_arg nat_trans.app f.w) j), dsimp, simp } } } | def | category_theory.limits.cone.from_costructured_arrow | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | Construct a cone on `F` from an object of the category `(Δ ↓ F)`. This is part of an
equivalence, see `cone.equiv_costructured_arrow`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone.equiv_costructured_arrow (F : J ⥤ C) : cone F ≌ costructured_arrow (const J) F | equivalence.mk (cone.to_costructured_arrow F) (cone.from_costructured_arrow F)
(nat_iso.of_components cones.eta (by tidy))
(nat_iso.of_components (λ c, (costructured_arrow.eta _).symm) (by tidy)) | def | category_theory.limits.cone.equiv_costructured_arrow | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | The category of cones on `F` is just the comma category `(Δ ↓ F)`, where `Δ` is the constant
functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone.is_limit_equiv_is_terminal {F : J ⥤ C} (c : cone F) : is_limit c ≃ is_terminal c | is_limit.iso_unique_cone_morphism.to_equiv.trans
{ to_fun := λ h, by exactI is_terminal.of_unique _,
inv_fun := λ h s, ⟨⟨is_terminal.from h s⟩, λ a, is_terminal.hom_ext h a _⟩,
left_inv := by tidy,
right_inv := by tidy } | def | category_theory.limits.cone.is_limit_equiv_is_terminal | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [
"inv_fun"
] | A cone is a limit cone iff it is terminal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit_iff_has_terminal_cone (F : J ⥤ C) : has_limit F ↔ has_terminal (cone F) | ⟨λ h, by exactI (cone.is_limit_equiv_is_terminal _ (limit.is_limit F)).has_terminal,
λ h, ⟨⟨by exactI ⟨⊤_ _, (cone.is_limit_equiv_is_terminal _).symm terminal_is_terminal⟩⟩⟩⟩ | lemma | category_theory.limits.has_limit_iff_has_terminal_cone | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits_of_shape_iff_is_left_adjoint_const :
has_limits_of_shape J C ↔ nonempty (is_left_adjoint (const J : C ⥤ _)) | calc has_limits_of_shape J C
↔ ∀ F : J ⥤ C, has_limit F : ⟨λ h, h.has_limit, λ h, by exactI has_limits_of_shape.mk⟩
... ↔ ∀ F : J ⥤ C, has_terminal (cone F) : forall_congr has_limit_iff_has_terminal_cone
... ↔ ∀ F : J ⥤ C, has_terminal (costructured_arrow (const J) F) :
forall_congr $ λ F, (cone.equiv_cos... | lemma | category_theory.limits.has_limits_of_shape_iff_is_left_adjoint_const | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit.lift_cone_morphism_eq_is_terminal_from {F : J ⥤ C} {c : cone F} (hc : is_limit c)
(s : cone F) : hc.lift_cone_morphism s =
is_terminal.from (cone.is_limit_equiv_is_terminal _ hc) _ | rfl | lemma | category_theory.limits.is_limit.lift_cone_morphism_eq_is_terminal_from | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_terminal.from_eq_lift_cone_morphism {F : J ⥤ C} {c : cone F} (hc : is_terminal c)
(s : cone F) : is_terminal.from hc s =
((cone.is_limit_equiv_is_terminal _).symm hc).lift_cone_morphism s | by convert (is_limit.lift_cone_morphism_eq_is_terminal_from _ s).symm | lemma | category_theory.limits.is_terminal.from_eq_lift_cone_morphism | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit.of_preserves_cone_terminal {F : J ⥤ C} {F' : K ⥤ D} (G : cone F ⥤ cone F')
[preserves_limit (functor.empty.{0} _) G] {c : cone F} (hc : is_limit c) :
is_limit (G.obj c) | (cone.is_limit_equiv_is_terminal _).symm $
(cone.is_limit_equiv_is_terminal _ hc).is_terminal_obj _ _ | def | category_theory.limits.is_limit.of_preserves_cone_terminal | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | If `G : cone F ⥤ cone F'` preserves terminal objects, it preserves limit cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit.of_reflects_cone_terminal {F : J ⥤ C} {F' : K ⥤ D} (G : cone F ⥤ cone F')
[reflects_limit (functor.empty.{0} _) G] {c : cone F} (hc : is_limit (G.obj c)) :
is_limit c | (cone.is_limit_equiv_is_terminal _).symm $
(cone.is_limit_equiv_is_terminal _ hc).is_terminal_of_obj _ _ | def | category_theory.limits.is_limit.of_reflects_cone_terminal | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | If `G : cone F ⥤ cone F'` reflects terminal objects, it reflects limit cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone.to_structured_arrow (F : J ⥤ C) : cocone F ⥤ structured_arrow F (const J) | { obj := λ c, structured_arrow.mk c.ι,
map := λ c d f, structured_arrow.hom_mk f.hom $ by { ext, simp } } | def | category_theory.limits.cocone.to_structured_arrow | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | Construct an object of the category `(F ↓ Δ)` from a cocone on `F`. This is part of an
equivalence, see `cocone.equiv_structured_arrow`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone.from_structured_arrow (F : J ⥤ C) : structured_arrow F (const J) ⥤ cocone F | { obj := λ c, ⟨c.right, c.hom⟩,
map := λ c d f,
{ hom := f.right,
w' := λ j, by { convert (congr_fun (congr_arg nat_trans.app f.w) j).symm, dsimp, simp } } } | def | category_theory.limits.cocone.from_structured_arrow | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | Construct a cocone on `F` from an object of the category `(F ↓ Δ)`. This is part of an
equivalence, see `cocone.equiv_structured_arrow`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone.equiv_structured_arrow (F : J ⥤ C) : cocone F ≌ structured_arrow F (const J) | equivalence.mk (cocone.to_structured_arrow F) (cocone.from_structured_arrow F)
(nat_iso.of_components cocones.eta (by tidy))
(nat_iso.of_components (λ c, (structured_arrow.eta _).symm) (by tidy)) | def | category_theory.limits.cocone.equiv_structured_arrow | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | The category of cocones on `F` is just the comma category `(F ↓ Δ)`, where `Δ` is the constant
functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone.is_colimit_equiv_is_initial {F : J ⥤ C} (c : cocone F) : is_colimit c ≃ is_initial c | is_colimit.iso_unique_cocone_morphism.to_equiv.trans
{ to_fun := λ h, by exactI is_initial.of_unique _,
inv_fun := λ h s, ⟨⟨is_initial.to h s⟩, λ a, is_initial.hom_ext h a _⟩,
left_inv := by tidy,
right_inv := by tidy } | def | category_theory.limits.cocone.is_colimit_equiv_is_initial | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [
"inv_fun"
] | A cocone is a colimit cocone iff it is initial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimit_iff_has_initial_cocone (F : J ⥤ C) : has_colimit F ↔ has_initial (cocone F) | ⟨λ h, by exactI (cocone.is_colimit_equiv_is_initial _ (colimit.is_colimit F)).has_initial,
λ h, ⟨⟨by exactI ⟨⊥_ _, (cocone.is_colimit_equiv_is_initial _).symm initial_is_initial⟩⟩⟩⟩ | lemma | category_theory.limits.has_colimit_iff_has_initial_cocone | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_shape_iff_is_right_adjoint_const :
has_colimits_of_shape J C ↔ nonempty (is_right_adjoint (const J : C ⥤ _)) | calc has_colimits_of_shape J C
↔ ∀ F : J ⥤ C, has_colimit F : ⟨λ h, h.has_colimit, λ h, by exactI has_colimits_of_shape.mk⟩
... ↔ ∀ F : J ⥤ C, has_initial (cocone F) : forall_congr has_colimit_iff_has_initial_cocone
... ↔ ∀ F : J ⥤ C, has_initial (structured_arrow F (const J)) :
forall_congr $ λ F, (cocon... | lemma | category_theory.limits.has_colimits_of_shape_iff_is_right_adjoint_const | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit.desc_cocone_morphism_eq_is_initial_to {F : J ⥤ C} {c : cocone F}
(hc : is_colimit c) (s : cocone F) :
hc.desc_cocone_morphism s =
is_initial.to (cocone.is_colimit_equiv_is_initial _ hc) _ | rfl | lemma | category_theory.limits.is_colimit.desc_cocone_morphism_eq_is_initial_to | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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