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map_cocone_equiv {D : Type u₄} [category.{v₄} D] {K : J ⥤ C} {F G : C ⥤ D} (h : F ≅ G)
{c : cocone K} (t : is_colimit (F.map_cocone c)) : is_colimit (G.map_cocone c) | begin
apply is_colimit.of_iso_colimit _ (precompose_whisker_left_map_cocone h c),
apply (precompose_inv_equiv (iso_whisker_left K h : _) _).symm t,
end | def | category_theory.limits.is_colimit.map_cocone_equiv | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | If `F` and `G` are naturally isomorphic, then `F.map_cone c` being a colimit implies
`G.map_cone c` is also a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_unique_cocone_morphism {t : cocone F} :
is_colimit t ≅ Π s, unique (t ⟶ s) | { hom := λ h s,
{ default := h.desc_cocone_morphism s,
uniq := λ _, h.uniq_cocone_morphism },
inv := λ h,
{ desc := λ s, (h s).default.hom,
uniq' := λ s f w, congr_arg cocone_morphism.hom ((h s).uniq ⟨f, w⟩) } } | def | category_theory.limits.is_colimit.iso_unique_cocone_morphism | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"unique"
] | A cocone is a colimit cocone exactly if
there is a unique cocone morphism from any other cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_of_hom {Y : C} (f : X ⟶ Y) : cocone F | { X := Y, ι := h.hom.app Y ⟨f⟩ } | def | category_theory.limits.is_colimit.of_nat_iso.cocone_of_hom | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | If `F.cocones` is corepresented by `X`, each morphism `f : X ⟶ Y` gives a cocone with cone
point `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_of_cocone (s : cocone F) : X ⟶ s.X | (h.inv.app s.X s.ι).down | def | category_theory.limits.is_colimit.of_nat_iso.hom_of_cocone | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | If `F.cocones` is corepresented by `X`, each cocone `s` gives a morphism `X ⟶ s.X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_of_hom_of_cocone (s : cocone F) : cocone_of_hom h (hom_of_cocone h s) = s | begin
dsimp [cocone_of_hom, hom_of_cocone], cases s, congr, dsimp,
convert congr_fun (congr_fun (congr_arg nat_trans.app h.inv_hom_id) s_X) s_ι,
exact ulift.up_down _
end | lemma | category_theory.limits.is_colimit.of_nat_iso.cocone_of_hom_of_cocone | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_of_cocone_of_hom {Y : C} (f : X ⟶ Y) : hom_of_cocone h (cocone_of_hom h f) = f | congr_arg ulift.down (congr_fun (congr_fun (congr_arg nat_trans.app h.hom_inv_id) Y) ⟨f⟩ : _) | lemma | category_theory.limits.is_colimit.of_nat_iso.hom_of_cocone_of_hom | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_cocone : cocone F | cocone_of_hom h (𝟙 X) | def | category_theory.limits.is_colimit.of_nat_iso.colimit_cocone | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | If `F.cocones` is corepresented by `X`, the cocone corresponding to the identity morphism on `X`
will be a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_of_hom_fac {Y : C} (f : X ⟶ Y) :
cocone_of_hom h f = (colimit_cocone h).extend f | begin
dsimp [cocone_of_hom, colimit_cocone, cocone.extend],
congr' with j,
have t := congr_fun (h.hom.naturality f) ⟨𝟙 X⟩,
dsimp at t,
simp only [id_comp] at t,
rw congr_fun (congr_arg nat_trans.app t) j,
refl,
end | lemma | category_theory.limits.is_colimit.of_nat_iso.cocone_of_hom_fac | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"extend"
] | If `F.cocones` is corepresented by `X`, the cocone corresponding to a morphism `f : Y ⟶ X` is
the colimit cocone extended by `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_fac (s : cocone F) : (colimit_cocone h).extend (hom_of_cocone h s) = s | begin
rw ←cocone_of_hom_of_cocone h s,
conv_lhs { simp only [hom_of_cocone_of_hom] },
apply (cocone_of_hom_fac _ _).symm,
end | lemma | category_theory.limits.is_colimit.of_nat_iso.cocone_fac | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"extend"
] | If `F.cocones` is corepresented by `X`, any cocone is the extension of the colimit cocone by the
corresponding morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_nat_iso {X : C} (h : coyoneda.obj (op X) ⋙ ulift_functor.{u₁} ≅ F.cocones) :
is_colimit (colimit_cocone h) | { desc := λ s, hom_of_cocone h s,
fac' := λ s j,
begin
have h := cocone_fac h s,
cases s,
injection h with h₁ h₂,
simp only [heq_iff_eq] at h₂,
conv_rhs { rw ← h₂ }, refl,
end,
uniq' := λ s m w,
begin
rw ←hom_of_cocone_of_hom h m,
congr,
rw cocone_of_hom_fac,
dsimp [cocone.... | def | category_theory.limits.is_colimit.of_nat_iso | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"heq_iff_eq"
] | If `F.cocones` is corepresentable, then the cocone corresponding to the identity morphism on
the representing object is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram (F : S ⥤ D) (x : L) : structured_arrow x ι ⥤ D | structured_arrow.proj x ι ⋙ F | abbreviation | category_theory.Ran.diagram | category_theory.limits | src/category_theory/limits/kan_extension.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | The diagram indexed by `Ran.index ι x` used to define `Ran`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone {F : S ⥤ D} {G : L ⥤ D} (x : L) (f : ι ⋙ G ⟶ F) :
cone (diagram ι F x) | { X := G.obj x,
π :=
{ app := λ i, G.map i.hom ≫ f.app i.right,
naturality' := begin
rintro ⟨⟨il⟩, ir, i⟩ ⟨⟨jl⟩, jr, j⟩ ⟨⟨⟨fl⟩⟩, fr, ff⟩,
dsimp at *,
simp only [category.id_comp, category.assoc] at *,
rw [ff],
have := f.naturality,
tidy,
end } } | def | category_theory.Ran.cone | category_theory.limits | src/category_theory/limits/kan_extension.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | A cone over `Ran.diagram ι F x` used to define `Ran`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
loc (F : S ⥤ D) [∀ x, has_limit (diagram ι F x)] : L ⥤ D | { obj := λ x, limit (diagram ι F x),
map := λ x y f, limit.pre (diagram _ _ _) (structured_arrow.map f : structured_arrow _ ι ⥤ _),
map_id' := begin
intro l,
ext j,
simp only [category.id_comp, limit.pre_π],
congr' 1,
simp,
end,
map_comp' := begin
intros x y z f g,
ext j,
erw [li... | def | category_theory.Ran.loc | category_theory.limits | src/category_theory/limits/kan_extension.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | An auxiliary definition used to define `Ran`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv (F : S ⥤ D) [∀ x, has_limit (diagram ι F x)] (G : L ⥤ D) :
(G ⟶ loc ι F) ≃ (((whiskering_left _ _ _).obj ι).obj G ⟶ F) | { to_fun := λ f,
{ app := λ x, f.app _ ≫ limit.π (diagram ι F (ι.obj x)) (structured_arrow.mk (𝟙 _)),
naturality' := begin
intros x y ff,
dsimp only [whiskering_left],
simp only [functor.comp_map, nat_trans.naturality_assoc, loc_map, category.assoc],
congr' 1,
erw limit.pre_π,
change _ = _ ... | def | category_theory.Ran.equiv | category_theory.limits | src/category_theory/limits/kan_extension.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.punit",
"category_theory.structured_arrow"
] | [
"equiv",
"inv_fun"
] | An auxiliary definition used to define `Ran` and `Ran.adjunction`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Ran [∀ X, has_limits_of_shape (structured_arrow X ι) D] : (S ⥤ D) ⥤ L ⥤ D | adjunction.right_adjoint_of_equiv (λ F G, (Ran.equiv ι G F).symm) (by tidy) | def | category_theory.Ran | category_theory.limits | src/category_theory/limits/kan_extension.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | The right Kan extension of a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjunction [∀ X, has_limits_of_shape (structured_arrow X ι) D] :
(whiskering_left _ _ D).obj ι ⊣ Ran ι | adjunction.adjunction_of_equiv_right _ _ | def | category_theory.Ran.adjunction | category_theory.limits | src/category_theory/limits/kan_extension.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | The adjunction associated to `Ran`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflective [full ι] [faithful ι] [∀ X, has_limits_of_shape (structured_arrow X ι) D] :
is_iso (adjunction D ι).counit | begin
apply nat_iso.is_iso_of_is_iso_app _,
intros F,
apply nat_iso.is_iso_of_is_iso_app _,
intros X,
dsimp [adjunction],
simp only [category.id_comp],
exact is_iso.of_iso ((limit.is_limit _).cone_point_unique_up_to_iso
(limit_of_diagram_initial structured_arrow.mk_id_initial _)),
end | lemma | category_theory.Ran.reflective | category_theory.limits | src/category_theory/limits/kan_extension.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram (F : S ⥤ D) (x : L) : costructured_arrow ι x ⥤ D | costructured_arrow.proj ι x ⋙ F | abbreviation | category_theory.Lan.diagram | category_theory.limits | src/category_theory/limits/kan_extension.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | The diagram indexed by `Ran.index ι x` used to define `Ran`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone {F : S ⥤ D} {G : L ⥤ D} (x : L) (f : F ⟶ ι ⋙ G) :
cocone (diagram ι F x) | { X := G.obj x,
ι :=
{ app := λ i, f.app i.left ≫ G.map i.hom,
naturality' := begin
rintro ⟨ir, ⟨il⟩, i⟩ ⟨jl, ⟨jr⟩, j⟩ ⟨fl, ⟨⟨fl⟩⟩, ff⟩,
dsimp at *,
simp only [functor.comp_map, category.comp_id, nat_trans.naturality_assoc],
rw [← G.map_comp, ff],
tidy,
end } } | def | category_theory.Lan.cocone | category_theory.limits | src/category_theory/limits/kan_extension.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | A cocone over `Lan.diagram ι F x` used to define `Lan`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
loc (F : S ⥤ D) [I : ∀ x, has_colimit (diagram ι F x)] : L ⥤ D | { obj := λ x, colimit (diagram ι F x),
map := λ x y f,
colimit.pre (diagram _ _ _) (costructured_arrow.map f : costructured_arrow ι _ ⥤ _),
map_id' := begin
intro l,
ext j,
erw [colimit.ι_pre, category.comp_id],
congr' 1,
simp,
end,
map_comp' := begin
intros x y z f g,
ext j,
... | def | category_theory.Lan.loc | category_theory.limits | src/category_theory/limits/kan_extension.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | An auxiliary definition used to define `Lan`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv (F : S ⥤ D) [I : ∀ x, has_colimit (diagram ι F x)] (G : L ⥤ D) :
(loc ι F ⟶ G) ≃ (F ⟶ ((whiskering_left _ _ _).obj ι).obj G) | { to_fun := λ f,
{ app := λ x,
by apply colimit.ι (diagram ι F (ι.obj x)) (costructured_arrow.mk (𝟙 _)) ≫ f.app _, -- sigh
naturality' := begin
intros x y ff,
dsimp only [whiskering_left],
simp only [functor.comp_map, category.assoc],
rw [← f.naturality (ι.map ff), ← category.assoc, ← categor... | def | category_theory.Lan.equiv | category_theory.limits | src/category_theory/limits/kan_extension.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.punit",
"category_theory.structured_arrow"
] | [
"equiv",
"inv_fun"
] | An auxiliary definition used to define `Lan` and `Lan.adjunction`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Lan [∀ X, has_colimits_of_shape (costructured_arrow ι X) D] : (S ⥤ D) ⥤ L ⥤ D | adjunction.left_adjoint_of_equiv (λ F G, Lan.equiv ι F G) (by tidy) | def | category_theory.Lan | category_theory.limits | src/category_theory/limits/kan_extension.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | The left Kan extension of a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjunction [∀ X, has_colimits_of_shape (costructured_arrow ι X) D] :
Lan ι ⊣ (whiskering_left _ _ D).obj ι | adjunction.adjunction_of_equiv_left _ _ | def | category_theory.Lan.adjunction | category_theory.limits | src/category_theory/limits/kan_extension.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | The adjunction associated to `Lan`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coreflective [full ι] [faithful ι] [∀ X, has_colimits_of_shape (costructured_arrow ι X) D] :
is_iso (adjunction D ι).unit | begin
apply nat_iso.is_iso_of_is_iso_app _,
intros F,
apply nat_iso.is_iso_of_is_iso_app _,
intros X,
dsimp [adjunction],
simp only [category.comp_id],
exact is_iso.of_iso ((colimit.is_colimit _).cocone_point_unique_up_to_iso
(colimit_of_diagram_terminal costructured_arrow.mk_id_terminal _)).symm,
end | lemma | category_theory.Lan.coreflective | category_theory.limits | src/category_theory/limits/kan_extension.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.punit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_limit_cone [semilattice_inf α] [order_top α] (F : J ⥤ α) : limit_cone F | { cone :=
{ X := finset.univ.inf F.obj,
π := { app := λ j, hom_of_le (finset.inf_le (fintype.complete _)) } },
is_limit := { lift := λ s, hom_of_le (finset.le_inf (λ j _, (s.π.app j).down.down)) } } | def | category_theory.limits.complete_lattice.finite_limit_cone | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"finset.inf_le",
"lift",
"order_top",
"semilattice_inf"
] | The limit cone over any functor from a finite diagram into a `semilattice_inf` with `order_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_colimit_cocone [semilattice_sup α] [order_bot α] (F : J ⥤ α) : colimit_cocone F | { cocone :=
{ X := finset.univ.sup F.obj,
ι := { app := λ i, hom_of_le (finset.le_sup (fintype.complete _)) } },
is_colimit := { desc := λ s, hom_of_le (finset.sup_le (λ j _, (s.ι.app j).down.down)) } } | def | category_theory.limits.complete_lattice.finite_colimit_cocone | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"finset.le_sup",
"order_bot",
"semilattice_sup"
] | The colimit cocone over any functor from a finite diagram into a `semilattice_sup` with `order_bot`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_limits_of_semilattice_inf_order_top [semilattice_inf α] [order_top α] :
has_finite_limits α | ⟨λ J 𝒥₁ 𝒥₂, by exactI { has_limit := λ F, has_limit.mk (finite_limit_cone F) }⟩ | instance | category_theory.limits.complete_lattice.has_finite_limits_of_semilattice_inf_order_top | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"order_top",
"semilattice_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_colimits_of_semilattice_sup_order_bot [semilattice_sup α] [order_bot α] :
has_finite_colimits α | ⟨λ J 𝒥₁ 𝒥₂, by exactI { has_colimit := λ F, has_colimit.mk (finite_colimit_cocone F) }⟩ | instance | category_theory.limits.complete_lattice.has_finite_colimits_of_semilattice_sup_order_bot | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"order_bot",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_limit_eq_finset_univ_inf [semilattice_inf α] [order_top α] (F : J ⥤ α) :
limit F = finset.univ.inf F.obj | (is_limit.cone_point_unique_up_to_iso (limit.is_limit F)
(finite_limit_cone F).is_limit).to_eq | lemma | category_theory.limits.complete_lattice.finite_limit_eq_finset_univ_inf | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"order_top",
"semilattice_inf"
] | The limit of a functor from a finite diagram into a `semilattice_inf` with `order_top` is the
infimum of the objects in the image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_colimit_eq_finset_univ_sup [semilattice_sup α] [order_bot α] (F : J ⥤ α) :
colimit F = finset.univ.sup F.obj | (is_colimit.cocone_point_unique_up_to_iso (colimit.is_colimit F)
(finite_colimit_cocone F).is_colimit).to_eq | lemma | category_theory.limits.complete_lattice.finite_colimit_eq_finset_univ_sup | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"order_bot",
"semilattice_sup"
] | The colimit of a functor from a finite diagram into a `semilattice_sup` with `order_bot`
is the supremum of the objects in the image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_product_eq_finset_inf [semilattice_inf α] [order_top α] {ι : Type u}
[fintype ι] (f : ι → α) : (∏ f) = (fintype.elems ι).inf f | begin
transitivity,
exact (is_limit.cone_point_unique_up_to_iso (limit.is_limit _)
(finite_limit_cone (discrete.functor f)).is_limit).to_eq,
change finset.univ.inf (f ∘ discrete_equiv.to_embedding) = (fintype.elems ι).inf f,
simp only [←finset.inf_map, finset.univ_map_equiv_to_embedding],
refl,
end | lemma | category_theory.limits.complete_lattice.finite_product_eq_finset_inf | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"finset.univ_map_equiv_to_embedding",
"fintype",
"order_top",
"semilattice_inf"
] | A finite product in the category of a `semilattice_inf` with `order_top` is the same as the infimum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_coproduct_eq_finset_sup [semilattice_sup α] [order_bot α] {ι : Type u}
[fintype ι] (f : ι → α) : (∐ f) = (fintype.elems ι).sup f | begin
transitivity,
exact (is_colimit.cocone_point_unique_up_to_iso (colimit.is_colimit _)
(finite_colimit_cocone (discrete.functor f)).is_colimit).to_eq,
change finset.univ.sup (f ∘ discrete_equiv.to_embedding) = (fintype.elems ι).sup f,
simp only [←finset.sup_map, finset.univ_map_equiv_to_embedding],
re... | lemma | category_theory.limits.complete_lattice.finite_coproduct_eq_finset_sup | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"finset.univ_map_equiv_to_embedding",
"fintype",
"order_bot",
"semilattice_sup"
] | A finite coproduct in the category of a `semilattice_sup` with `order_bot` is the same as the
supremum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_eq_inf [semilattice_inf α] [order_top α] (x y : α) : limits.prod x y = x ⊓ y | calc limits.prod x y = limit (pair x y) : rfl
... = finset.univ.inf (pair x y).obj : by rw finite_limit_eq_finset_univ_inf (pair.{u} x y)
... = x ⊓ (y ⊓ ⊤) : rfl -- Note: finset.inf is realized as a fold, hence the definitional equality
... = x ⊓ y : by rw inf_top_eq | lemma | category_theory.limits.complete_lattice.prod_eq_inf | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"inf_top_eq",
"order_top",
"semilattice_inf"
] | The binary product in the category of a `semilattice_inf` with `order_top` is the same as the
infimum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod_eq_sup [semilattice_sup α] [order_bot α] (x y : α) : limits.coprod x y = x ⊔ y | calc limits.coprod x y = colimit (pair x y) : rfl
... = finset.univ.sup (pair x y).obj : by rw finite_colimit_eq_finset_univ_sup (pair x y)
... = x ⊔ (y ⊔ ⊥) : rfl -- Note: finset.sup is realized as a fold, hence the definitional equality
... = x ⊔ y : by rw sup_bot_eq | lemma | category_theory.limits.complete_lattice.coprod_eq_sup | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"order_bot",
"semilattice_sup",
"sup_bot_eq"
] | The binary coproduct in the category of a `semilattice_sup` with `order_bot` is the same as the
supremum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_eq_inf [semilattice_inf α] [order_top α] {x y z : α} (f : x ⟶ z) (g : y ⟶ z) :
pullback f g = x ⊓ y | calc pullback f g = limit (cospan f g) : rfl
... = finset.univ.inf (cospan f g).obj : by rw finite_limit_eq_finset_univ_inf
... = z ⊓ (x ⊓ (y ⊓ ⊤)) : rfl
... = z ⊓ (x ⊓ y) : by rw inf_top_eq
... = x ⊓ y : inf_eq_right.mpr (inf_le_of_left_le f.le) | lemma | category_theory.limits.complete_lattice.pullback_eq_inf | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"inf_le_of_left_le",
"inf_top_eq",
"order_top",
"semilattice_inf"
] | The pullback in the category of a `semilattice_inf` with `order_top` is the same as the infimum
over the objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout_eq_sup [semilattice_sup α] [order_bot α] (x y z : α) (f : z ⟶ x) (g : z ⟶ y) :
pushout f g = x ⊔ y | calc pushout f g = colimit (span f g) : rfl
... = finset.univ.sup (span f g).obj : by rw finite_colimit_eq_finset_univ_sup
... = z ⊔ (x ⊔ (y ⊔ ⊥)) : rfl
... = z ⊔ (x ⊔ y) : by rw sup_bot_eq
... = x ⊔ y : sup_eq_right.mpr (le_sup_of_le_left f.le) | lemma | category_theory.limits.complete_lattice.pushout_eq_sup | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"le_sup_of_le_left",
"order_bot",
"semilattice_sup",
"sup_bot_eq"
] | The pushout in the category of a `semilattice_sup` with `order_bot` is the same as the supremum
over the objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone (F : J ⥤ α) : limit_cone F | { cone :=
{ X := infi F.obj,
π :=
{ app := λ j, hom_of_le (complete_lattice.Inf_le _ _ (set.mem_range_self _)) } },
is_limit :=
{ lift := λ s, hom_of_le (complete_lattice.le_Inf _ _
begin rintros _ ⟨j, rfl⟩, exact (s.π.app j).le, end) } } | def | category_theory.limits.complete_lattice.limit_cone | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"infi",
"lift",
"set.mem_range_self"
] | The limit cone over any functor into a complete lattice. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone (F : J ⥤ α) : colimit_cocone F | { cocone :=
{ X := supr F.obj,
ι :=
{ app := λ j, hom_of_le (complete_lattice.le_Sup _ _ (set.mem_range_self _)) } },
is_colimit :=
{ desc := λ s, hom_of_le (complete_lattice.Sup_le _ _
begin rintros _ ⟨j, rfl⟩, exact (s.ι.app j).le, end) } } | def | category_theory.limits.complete_lattice.colimit_cocone | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"set.mem_range_self",
"supr"
] | The colimit cocone over any functor into a complete lattice. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_complete_lattice : has_limits α | { has_limits_of_shape := λ J 𝒥, by exactI
{ has_limit := λ F, has_limit.mk (limit_cone F) } } | instance | category_theory.limits.complete_lattice.has_limits_of_complete_lattice | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_complete_lattice : has_colimits α | { has_colimits_of_shape := λ J 𝒥, by exactI
{ has_colimit := λ F, has_colimit.mk (colimit_cocone F) } } | instance | category_theory.limits.complete_lattice.has_colimits_of_complete_lattice | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_eq_infi (F : J ⥤ α) : limit F = infi F.obj | (is_limit.cone_point_unique_up_to_iso (limit.is_limit F)
(limit_cone F).is_limit).to_eq | lemma | category_theory.limits.complete_lattice.limit_eq_infi | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"infi"
] | The limit of a functor into a complete lattice is the infimum of the objects in the image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_eq_supr (F : J ⥤ α) : colimit F = supr F.obj | (is_colimit.cocone_point_unique_up_to_iso (colimit.is_colimit F)
(colimit_cocone F).is_colimit).to_eq | lemma | category_theory.limits.complete_lattice.colimit_eq_supr | category_theory.limits | src/category_theory/limits/lattice.lean | [
"order.complete_lattice",
"data.fintype.lattice",
"category_theory.limits.shapes.pullbacks",
"category_theory.category.preorder",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.finite_limits"
] | [
"supr"
] | The colimit of a functor into a complete lattice is the supremum of the objects in the image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_coprod : Prop | (binary_cofan_inl : ∀ ⦃A B : C⦄ (c : binary_cofan A B) (hc : is_colimit c), mono c.inl) | class | category_theory.limits.mono_coprod | category_theory.limits | src/category_theory/limits/mono_coprod.lean | [
"category_theory.limits.shapes.regular_mono",
"category_theory.limits.shapes.zero_morphisms"
] | [] | This condition expresses that inclusion morphisms into coproducts are monomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_coprod_of_has_zero_morphisms
[has_zero_morphisms C] : mono_coprod C | ⟨λ A B c hc, begin
haveI : is_split_mono c.inl := is_split_mono.mk'
(split_mono.mk (hc.desc (binary_cofan.mk (𝟙 A) 0)) (is_colimit.fac _ _ _)),
apply_instance,
end⟩ | instance | category_theory.limits.mono_coprod_of_has_zero_morphisms | category_theory.limits | src/category_theory/limits/mono_coprod.lean | [
"category_theory.limits.shapes.regular_mono",
"category_theory.limits.shapes.zero_morphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan_inr {A B : C}[mono_coprod C] (c : binary_cofan A B) (hc : is_colimit c) :
mono c.inr | begin
have hc' : is_colimit (binary_cofan.mk c.inr c.inl) :=
binary_cofan.is_colimit_mk (λ s, hc.desc (binary_cofan.mk s.inr s.inl)) (by tidy) (by tidy)
(λ s m h₁ h₂, binary_cofan.is_colimit.hom_ext hc
(by simp only [h₂, is_colimit.fac, binary_cofan.ι_app_left, binary_cofan.mk_inl])
(by simp only ... | lemma | category_theory.limits.mono_coprod.binary_cofan_inr | category_theory.limits | src/category_theory/limits/mono_coprod.lean | [
"category_theory.limits.shapes.regular_mono",
"category_theory.limits.shapes.zero_morphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_inl_iff {A B : C} {c₁ c₂ : binary_cofan A B} (hc₁ : is_colimit c₁)
(hc₂ : is_colimit c₂) : mono c₁.inl ↔ mono c₂.inl | begin
suffices : ∀ (c₁ c₂ : binary_cofan A B) (hc₁ : is_colimit c₁) (hc₂ : is_colimit c₂)
(h : mono c₁.inl), mono c₂.inl,
{ exact ⟨λ h₁, this _ _ hc₁ hc₂ h₁, λ h₂, this _ _ hc₂ hc₁ h₂⟩, },
intros c₁ c₂ hc₁ hc₂,
introI,
simpa only [is_colimit.comp_cocone_point_unique_up_to_iso_hom]
using mono_comp c₁.i... | lemma | category_theory.limits.mono_coprod.mono_inl_iff | category_theory.limits | src/category_theory/limits/mono_coprod.lean | [
"category_theory.limits.shapes.regular_mono",
"category_theory.limits.shapes.zero_morphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' (h : ∀ (A B : C), ∃ (c : binary_cofan A B) (hc : is_colimit c), mono c.inl) :
mono_coprod C | ⟨λ A B c' hc', begin
obtain ⟨c, hc₁, hc₂⟩ := h A B,
simpa only [mono_inl_iff hc' hc₁] using hc₂,
end⟩ | lemma | category_theory.limits.mono_coprod.mk' | category_theory.limits | src/category_theory/limits/mono_coprod.lean | [
"category_theory.limits.shapes.regular_mono",
"category_theory.limits.shapes.zero_morphisms"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_coprod_type : mono_coprod (Type u) | mono_coprod.mk' (λ A B, begin
refine ⟨binary_cofan.mk (sum.inl : A ⟶ A ⊕ B) sum.inr, _, _⟩,
{ refine binary_cofan.is_colimit.mk _ (λ Y f₁ f₂ x, by { cases x, exacts [f₁ x, f₂ x], })
(λ Y f₁ f₂, rfl) (λ Y f₁ f₂, rfl) _,
intros Y f₁ f₂ m h₁ h₂,
ext x,
cases x,
{ dsimp, exact congr_fun h₁ x, },
... | instance | category_theory.limits.mono_coprod.mono_coprod_type | category_theory.limits | src/category_theory/limits/mono_coprod.lean | [
"category_theory.limits.shapes.regular_mono",
"category_theory.limits.shapes.zero_morphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit_cocone_op (F : J ⥤ C) {c : cocone F} (hc : is_colimit c) :
is_limit c.op | { lift := λ s, (hc.desc s.unop).op,
fac' := λ s j, quiver.hom.unop_inj (by simpa),
uniq' := λ s m w,
begin
refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),
simpa only [quiver.hom.unop_op, is_colimit.fac] using w (op j)
end } | def | category_theory.limits.is_limit_cocone_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"lift",
"quiver.hom.op_inj",
"quiver.hom.unop_inj",
"quiver.hom.unop_op"
] | Turn a colimit for `F : J ⥤ C` into a limit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_cone_op (F : J ⥤ C) {c : cone F} (hc : is_limit c) :
is_colimit c.op | { desc := λ s, (hc.lift s.unop).op,
fac' := λ s j, quiver.hom.unop_inj (by simpa),
uniq' := λ s m w,
begin
refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),
simpa only [quiver.hom.unop_op, is_limit.fac] using w (op j)
end } | def | category_theory.limits.is_colimit_cone_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"quiver.hom.op_inj",
"quiver.hom.unop_inj",
"quiver.hom.unop_op"
] | Turn a limit for `F : J ⥤ C` into a colimit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_cone_left_op_of_cocone (F : J ⥤ Cᵒᵖ) {c : cocone F} (hc : is_colimit c) :
is_limit (cone_left_op_of_cocone c) | { lift := λ s, (hc.desc (cocone_of_cone_left_op s)).unop,
fac' := λ s j, quiver.hom.op_inj $ by simpa only [cone_left_op_of_cocone_π_app, op_comp,
quiver.hom.op_unop, is_colimit.fac, cocone_of_cone_left_op_ι_app],
uniq' := λ s m w,
begin
refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),... | def | category_theory.limits.is_limit_cone_left_op_of_cocone | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"lift",
"quiver.hom.op_inj",
"quiver.hom.op_unop",
"quiver.hom.unop_inj"
] | Turn a colimit for `F : J ⥤ Cᵒᵖ` into a limit for `F.left_op : Jᵒᵖ ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_cocone_left_op_of_cone (F : J ⥤ Cᵒᵖ) {c : cone F} (hc : is_limit c) :
is_colimit (cocone_left_op_of_cone c) | { desc := λ s, (hc.lift (cone_of_cocone_left_op s)).unop,
fac' := λ s j, quiver.hom.op_inj $ by simpa only [cocone_left_op_of_cone_ι_app, op_comp,
quiver.hom.op_unop, is_limit.fac, cone_of_cocone_left_op_π_app],
uniq' := λ s m w,
begin
refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),
... | def | category_theory.limits.is_colimit_cocone_left_op_of_cone | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"quiver.hom.op_inj",
"quiver.hom.op_unop",
"quiver.hom.unop_inj"
] | Turn a limit of `F : J ⥤ Cᵒᵖ` into a colimit of `F.left_op : Jᵒᵖ ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_cone_right_op_of_cocone (F : Jᵒᵖ ⥤ C) {c : cocone F} (hc : is_colimit c) :
is_limit (cone_right_op_of_cocone c) | { lift := λ s, (hc.desc (cocone_of_cone_right_op s)).op,
fac' := λ s j, quiver.hom.unop_inj (by simpa),
uniq' := λ s m w,
begin
refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),
simpa only [quiver.hom.unop_op, is_colimit.fac] using w (unop j)
end } | def | category_theory.limits.is_limit_cone_right_op_of_cocone | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"lift",
"quiver.hom.op_inj",
"quiver.hom.unop_inj",
"quiver.hom.unop_op"
] | Turn a colimit for `F : Jᵒᵖ ⥤ C` into a limit for `F.right_op : J ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_cocone_right_op_of_cone (F : Jᵒᵖ ⥤ C) {c : cone F} (hc : is_limit c) :
is_colimit (cocone_right_op_of_cone c) | { desc := λ s, (hc.lift (cone_of_cocone_right_op s)).op,
fac' := λ s j, quiver.hom.unop_inj (by simpa),
uniq' := λ s m w,
begin
refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),
simpa only [quiver.hom.unop_op, is_limit.fac] using w (unop j)
end } | def | category_theory.limits.is_colimit_cocone_right_op_of_cone | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"quiver.hom.op_inj",
"quiver.hom.unop_inj",
"quiver.hom.unop_op"
] | Turn a limit for `F : Jᵒᵖ ⥤ C` into a colimit for `F.right_op : J ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_cone_unop_of_cocone (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cocone F} (hc : is_colimit c) :
is_limit (cone_unop_of_cocone c) | { lift := λ s, (hc.desc (cocone_of_cone_unop s)).unop,
fac' := λ s j, quiver.hom.op_inj (by simpa),
uniq' := λ s m w,
begin
refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),
simpa only [quiver.hom.op_unop, is_colimit.fac] using w (unop j)
end } | def | category_theory.limits.is_limit_cone_unop_of_cocone | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"lift",
"quiver.hom.op_inj",
"quiver.hom.op_unop",
"quiver.hom.unop_inj"
] | Turn a colimit for `F : Jᵒᵖ ⥤ Cᵒᵖ` into a limit for `F.unop : J ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_cocone_unop_of_cone (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cone F} (hc : is_limit c) :
is_colimit (cocone_unop_of_cone c) | { desc := λ s, (hc.lift (cone_of_cocone_unop s)).unop,
fac' := λ s j, quiver.hom.op_inj (by simpa),
uniq' := λ s m w,
begin
refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),
simpa only [quiver.hom.op_unop, is_limit.fac] using w (unop j)
end } | def | category_theory.limits.is_colimit_cocone_unop_of_cone | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"quiver.hom.op_inj",
"quiver.hom.op_unop",
"quiver.hom.unop_inj"
] | Turn a limit of `F : Jᵒᵖ ⥤ Cᵒᵖ` into a colimit of `F.unop : J ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_cocone_unop (F : J ⥤ C) {c : cocone F.op} (hc : is_colimit c) :
is_limit c.unop | { lift := λ s, (hc.desc s.op).unop,
fac' := λ s j, quiver.hom.op_inj (by simpa),
uniq' := λ s m w,
begin
refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),
simpa only [quiver.hom.op_unop, is_colimit.fac] using w (unop j)
end } | def | category_theory.limits.is_limit_cocone_unop | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"lift",
"quiver.hom.op_inj",
"quiver.hom.op_unop",
"quiver.hom.unop_inj"
] | Turn a colimit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ` into a limit for `F : J ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_cone_unop (F : J ⥤ C) {c : cone F.op} (hc : is_limit c) :
is_colimit c.unop | { desc := λ s, (hc.lift s.op).unop,
fac' := λ s j, quiver.hom.op_inj (by simpa),
uniq' := λ s m w,
begin
refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),
simpa only [quiver.hom.op_unop, is_limit.fac] using w (unop j)
end } | def | category_theory.limits.is_colimit_cone_unop | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"quiver.hom.op_inj",
"quiver.hom.op_unop",
"quiver.hom.unop_inj"
] | Turn a limit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ` into a colimit for `F : J ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_cone_of_cocone_left_op (F : J ⥤ Cᵒᵖ) {c : cocone F.left_op}
(hc : is_colimit c) : is_limit (cone_of_cocone_left_op c) | { lift := λ s, (hc.desc (cocone_left_op_of_cone s)).op,
fac' := λ s j, quiver.hom.unop_inj $ by simpa only [cone_of_cocone_left_op_π_app, unop_comp,
quiver.hom.unop_op, is_colimit.fac, cocone_left_op_of_cone_ι_app],
uniq' := λ s m w,
begin
refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _))... | def | category_theory.limits.is_limit_cone_of_cocone_left_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"lift",
"quiver.hom.op_inj",
"quiver.hom.unop_inj",
"quiver.hom.unop_op"
] | Turn a colimit for `F.left_op : Jᵒᵖ ⥤ C` into a limit for `F : J ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_cocone_of_cone_left_op (F : J ⥤ Cᵒᵖ) {c : cone (F.left_op)}
(hc : is_limit c) : is_colimit (cocone_of_cone_left_op c) | { desc := λ s, (hc.lift (cone_left_op_of_cocone s)).op,
fac' := λ s j, quiver.hom.unop_inj $ by simpa only [cocone_of_cone_left_op_ι_app, unop_comp,
quiver.hom.unop_op, is_limit.fac, cone_left_op_of_cocone_π_app],
uniq' := λ s m w,
begin
refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),
... | def | category_theory.limits.is_colimit_cocone_of_cone_left_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"quiver.hom.op_inj",
"quiver.hom.unop_inj",
"quiver.hom.unop_op"
] | Turn a limit of `F.left_op : Jᵒᵖ ⥤ C` into a colimit of `F : J ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_cone_of_cocone_right_op (F : Jᵒᵖ ⥤ C) {c : cocone F.right_op}
(hc : is_colimit c) : is_limit (cone_of_cocone_right_op c) | { lift := λ s, (hc.desc (cocone_right_op_of_cone s)).unop,
fac' := λ s j, quiver.hom.op_inj (by simpa),
uniq' := λ s m w,
begin
refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),
simpa only [quiver.hom.op_unop, is_colimit.fac] using w (op j)
end } | def | category_theory.limits.is_limit_cone_of_cocone_right_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"lift",
"quiver.hom.op_inj",
"quiver.hom.op_unop",
"quiver.hom.unop_inj"
] | Turn a colimit for `F.right_op : J ⥤ Cᵒᵖ` into a limit for `F : Jᵒᵖ ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_cocone_of_cone_right_op (F : Jᵒᵖ ⥤ C) {c : cone F.right_op}
(hc : is_limit c) : is_colimit (cocone_of_cone_right_op c) | { desc := λ s, (hc.lift (cone_right_op_of_cocone s)).unop,
fac' := λ s j, quiver.hom.op_inj (by simpa),
uniq' := λ s m w,
begin
refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),
simpa only [quiver.hom.op_unop, is_limit.fac] using w (op j)
end } | def | category_theory.limits.is_colimit_cocone_of_cone_right_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"quiver.hom.op_inj",
"quiver.hom.op_unop",
"quiver.hom.unop_inj"
] | Turn a limit for `F.right_op : J ⥤ Cᵒᵖ` into a limit for `F : Jᵒᵖ ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_cone_of_cocone_unop (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cocone F.unop} (hc : is_colimit c) :
is_limit (cone_of_cocone_unop c) | { lift := λ s, (hc.desc (cocone_unop_of_cone s)).op,
fac' := λ s j, quiver.hom.unop_inj (by simpa),
uniq' := λ s m w,
begin
refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),
simpa only [quiver.hom.unop_op, is_colimit.fac] using w (op j)
end } | def | category_theory.limits.is_limit_cone_of_cocone_unop | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"lift",
"quiver.hom.op_inj",
"quiver.hom.unop_inj",
"quiver.hom.unop_op"
] | Turn a colimit for `F.unop : J ⥤ C` into a limit for `F : Jᵒᵖ ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_cone_of_cocone_unop (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cone F.unop} (hc : is_limit c) :
is_colimit (cocone_of_cone_unop c) | { desc := λ s, (hc.lift (cone_unop_of_cocone s)).op,
fac' := λ s j, quiver.hom.unop_inj (by simpa),
uniq' := λ s m w,
begin
refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),
simpa only [quiver.hom.unop_op, is_limit.fac] using w (op j)
end } | def | category_theory.limits.is_colimit_cone_of_cocone_unop | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [
"quiver.hom.op_inj",
"quiver.hom.unop_inj",
"quiver.hom.unop_op"
] | Turn a limit for `F.unop : J ⥤ C` into a colimit for `F : Jᵒᵖ ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit_of_has_colimit_left_op (F : J ⥤ Cᵒᵖ) [has_colimit F.left_op] : has_limit F | has_limit.mk
{ cone := cone_of_cocone_left_op (colimit.cocone F.left_op),
is_limit := is_limit_cone_of_cocone_left_op _ (colimit.is_colimit _) } | lemma | category_theory.limits.has_limit_of_has_colimit_left_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | If `F.left_op : Jᵒᵖ ⥤ C` has a colimit, we can construct a limit for `F : J ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit_of_has_colimit_op (F : J ⥤ C) [has_colimit F.op] : has_limit F | has_limit.mk
{ cone := (colimit.cocone F.op).unop,
is_limit := is_limit_cocone_unop _ (colimit.is_colimit _) } | lemma | category_theory.limits.has_limit_of_has_colimit_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits_of_shape_op_of_has_colimits_of_shape [has_colimits_of_shape Jᵒᵖ C] :
has_limits_of_shape J Cᵒᵖ | { has_limit := λ F, has_limit_of_has_colimit_left_op F } | lemma | category_theory.limits.has_limits_of_shape_op_of_has_colimits_of_shape | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | If `C` has colimits of shape `Jᵒᵖ`, we can construct limits in `Cᵒᵖ` of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_shape_of_has_colimits_of_shape_op [has_colimits_of_shape Jᵒᵖ Cᵒᵖ] :
has_limits_of_shape J C | { has_limit := λ F, has_limit_of_has_colimit_op F } | lemma | category_theory.limits.has_limits_of_shape_of_has_colimits_of_shape_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits_op_of_has_colimits [has_colimits C] : has_limits Cᵒᵖ | ⟨infer_instance⟩ | instance | category_theory.limits.has_limits_op_of_has_colimits | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | If `C` has colimits, we can construct limits for `Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_has_colimits_op [has_colimits Cᵒᵖ] : has_limits C | { has_limits_of_shape := λ J hJ, by exactI has_limits_of_shape_of_has_colimits_of_shape_op } | lemma | category_theory.limits.has_limits_of_has_colimits_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_cofiltered_limits_op_of_has_filtered_colimits
[has_filtered_colimits_of_size.{v₂ u₂} C] : has_cofiltered_limits_of_size.{v₂ u₂} Cᵒᵖ | { has_limits_of_shape := λ I hI₁ hI₂, by exactI has_limits_of_shape_op_of_has_colimits_of_shape } | instance | category_theory.limits.has_cofiltered_limits_op_of_has_filtered_colimits | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_cofiltered_limits_of_has_filtered_colimits_op
[has_filtered_colimits_of_size.{v₂ u₂} Cᵒᵖ] : has_cofiltered_limits_of_size.{v₂ u₂} C | { has_limits_of_shape := λ I hI₂ hI₂, by exactI has_limits_of_shape_of_has_colimits_of_shape_op } | lemma | category_theory.limits.has_cofiltered_limits_of_has_filtered_colimits_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit_of_has_limit_left_op (F : J ⥤ Cᵒᵖ) [has_limit F.left_op] : has_colimit F | has_colimit.mk
{ cocone := cocone_of_cone_left_op (limit.cone F.left_op),
is_colimit := is_colimit_cocone_of_cone_left_op _ (limit.is_limit _) } | lemma | category_theory.limits.has_colimit_of_has_limit_left_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | If `F.left_op : Jᵒᵖ ⥤ C` has a limit, we can construct a colimit for `F : J ⥤ Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimit_of_has_limit_op (F : J ⥤ C) [has_limit F.op] : has_colimit F | has_colimit.mk
{ cocone := (limit.cone F.op).unop,
is_colimit := is_colimit_cone_unop _ (limit.is_limit _) } | lemma | category_theory.limits.has_colimit_of_has_limit_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_shape_op_of_has_limits_of_shape [has_limits_of_shape Jᵒᵖ C] :
has_colimits_of_shape J Cᵒᵖ | { has_colimit := λ F, has_colimit_of_has_limit_left_op F } | instance | category_theory.limits.has_colimits_of_shape_op_of_has_limits_of_shape | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | If `C` has colimits of shape `Jᵒᵖ`, we can construct limits in `Cᵒᵖ` of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_of_shape_of_has_limits_of_shape_op [has_limits_of_shape Jᵒᵖ Cᵒᵖ] :
has_colimits_of_shape J C | { has_colimit := λ F, has_colimit_of_has_limit_op F } | lemma | category_theory.limits.has_colimits_of_shape_of_has_limits_of_shape_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_op_of_has_limits [has_limits C] : has_colimits Cᵒᵖ | ⟨infer_instance⟩ | instance | category_theory.limits.has_colimits_op_of_has_limits | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | If `C` has limits, we can construct colimits for `Cᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_of_has_limits_op [has_limits Cᵒᵖ] : has_colimits C | { has_colimits_of_shape := λ J hJ, by exactI has_colimits_of_shape_of_has_limits_of_shape_op } | lemma | category_theory.limits.has_colimits_of_has_limits_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_filtered_colimits_op_of_has_cofiltered_limits
[has_cofiltered_limits_of_size.{v₂ u₂} C] : has_filtered_colimits_of_size.{v₂ u₂} Cᵒᵖ | { has_colimits_of_shape := λ I hI₁ hI₂, by exactI infer_instance } | instance | category_theory.limits.has_filtered_colimits_op_of_has_cofiltered_limits | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_filtered_colimits_of_has_cofiltered_limits_op
[has_cofiltered_limits_of_size.{v₂ u₂} Cᵒᵖ] : has_filtered_colimits_of_size.{v₂ u₂} C | { has_colimits_of_shape := λ I hI₁ hI₂, by exactI has_colimits_of_shape_of_has_limits_of_shape_op } | lemma | category_theory.limits.has_filtered_colimits_of_has_cofiltered_limits_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coproducts_of_shape_opposite [has_products_of_shape X C] :
has_coproducts_of_shape X Cᵒᵖ | begin
haveI : has_limits_of_shape (discrete X)ᵒᵖ C :=
has_limits_of_shape_of_equivalence (discrete.opposite X).symm,
apply_instance
end | instance | category_theory.limits.has_coproducts_of_shape_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | If `C` has products indexed by `X`, then `Cᵒᵖ` has coproducts indexed by `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coproducts_of_shape_of_opposite [has_products_of_shape X Cᵒᵖ] :
has_coproducts_of_shape X C | begin
haveI : has_limits_of_shape (discrete X)ᵒᵖ Cᵒᵖ :=
has_limits_of_shape_of_equivalence (discrete.opposite X).symm,
exact has_colimits_of_shape_of_has_limits_of_shape_op
end | lemma | category_theory.limits.has_coproducts_of_shape_of_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_products_of_shape_opposite [has_coproducts_of_shape X C] :
has_products_of_shape X Cᵒᵖ | begin
haveI : has_colimits_of_shape (discrete X)ᵒᵖ C :=
has_colimits_of_shape_of_equivalence (discrete.opposite X).symm,
apply_instance
end | instance | category_theory.limits.has_products_of_shape_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | If `C` has coproducts indexed by `X`, then `Cᵒᵖ` has products indexed by `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_products_of_shape_of_opposite [has_coproducts_of_shape X Cᵒᵖ] :
has_products_of_shape X C | begin
haveI : has_colimits_of_shape (discrete X)ᵒᵖ Cᵒᵖ :=
has_colimits_of_shape_of_equivalence (discrete.opposite X).symm,
exact has_limits_of_shape_of_has_colimits_of_shape_op
end | lemma | category_theory.limits.has_products_of_shape_of_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_products_opposite [has_coproducts.{v₂} C] : has_products.{v₂} Cᵒᵖ | λ X, infer_instance | instance | category_theory.limits.has_products_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_products_of_opposite [has_coproducts.{v₂} Cᵒᵖ] : has_products.{v₂} C | λ X, has_products_of_shape_of_opposite X | lemma | category_theory.limits.has_products_of_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coproducts_opposite [has_products.{v₂} C] : has_coproducts.{v₂} Cᵒᵖ | λ X, infer_instance | instance | category_theory.limits.has_coproducts_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coproducts_of_opposite [has_products.{v₂} Cᵒᵖ] : has_coproducts.{v₂} C | λ X, has_coproducts_of_shape_of_opposite X | lemma | category_theory.limits.has_coproducts_of_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_coproducts_opposite [has_finite_products C] : has_finite_coproducts Cᵒᵖ | { out := λ n, limits.has_coproducts_of_shape_opposite _ } | instance | category_theory.limits.has_finite_coproducts_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_coproducts_of_opposite [has_finite_products Cᵒᵖ] : has_finite_coproducts C | { out := λ n, has_coproducts_of_shape_of_opposite _ } | lemma | category_theory.limits.has_finite_coproducts_of_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_products_opposite [has_finite_coproducts C] : has_finite_products Cᵒᵖ | { out := λ n, infer_instance } | instance | category_theory.limits.has_finite_products_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_products_of_opposite [has_finite_coproducts Cᵒᵖ] : has_finite_products C | { out := λ n, has_products_of_shape_of_opposite _ } | lemma | category_theory.limits.has_finite_products_of_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_equalizers_opposite [has_coequalizers C] : has_equalizers Cᵒᵖ | begin
haveI : has_colimits_of_shape walking_parallel_pairᵒᵖ C :=
has_colimits_of_shape_of_equivalence walking_parallel_pair_op_equiv,
apply_instance
end | instance | category_theory.limits.has_equalizers_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coequalizers_opposite [has_equalizers C] : has_coequalizers Cᵒᵖ | begin
haveI : has_limits_of_shape walking_parallel_pairᵒᵖ C :=
has_limits_of_shape_of_equivalence walking_parallel_pair_op_equiv,
apply_instance
end | instance | category_theory.limits.has_coequalizers_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_colimits_opposite [has_finite_limits C] :
has_finite_colimits Cᵒᵖ | { out := λ J 𝒟 𝒥, by { resetI, apply_instance, }, } | instance | category_theory.limits.has_finite_colimits_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_limits_opposite [has_finite_colimits C] :
has_finite_limits Cᵒᵖ | { out := λ J 𝒟 𝒥, by { resetI, apply_instance, }, } | instance | category_theory.limits.has_finite_limits_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pullbacks_opposite [has_pushouts C] : has_pullbacks Cᵒᵖ | begin
haveI : has_colimits_of_shape walking_cospanᵒᵖ C :=
has_colimits_of_shape_of_equivalence walking_cospan_op_equiv.symm,
apply has_limits_of_shape_op_of_has_colimits_of_shape,
end | instance | category_theory.limits.has_pullbacks_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pushouts_opposite [has_pullbacks C] : has_pushouts Cᵒᵖ | begin
haveI : has_limits_of_shape walking_spanᵒᵖ C :=
has_limits_of_shape_of_equivalence walking_span_op_equiv.symm,
apply_instance
end | instance | category_theory.limits.has_pushouts_opposite | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_op {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :
span f.op g.op ≅ walking_cospan_op_equiv.inverse ⋙ (cospan f g).op | nat_iso.of_components (by { rintro (_|_|_); refl, })
(by { rintros (_|_|_) (_|_|_) f; cases f; tidy, }) | def | category_theory.limits.span_op | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | The canonical isomorphism relating `span f.op g.op` and `(cospan f g).op` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_cospan {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :
(cospan f g).op ≅ walking_cospan_op_equiv.functor ⋙ span f.op g.op | calc (cospan f g).op ≅ 𝟭 _ ⋙ (cospan f g).op : by refl
... ≅ (walking_cospan_op_equiv.functor ⋙ walking_cospan_op_equiv.inverse) ⋙ (cospan f g).op :
iso_whisker_right walking_cospan_op_equiv.unit_iso _
... ≅ walking_cospan_op_equiv.functor ⋙ (walking_cospan_op_equiv.inverse ⋙ (cospan f g).op) :
functor.associator ... | def | category_theory.limits.op_cospan | category_theory.limits | src/category_theory/limits/opposites.lean | [
"category_theory.limits.filtered",
"category_theory.limits.shapes.finite_products",
"category_theory.discrete_category",
"tactic.equiv_rw"
] | [] | The canonical isomorphism relating `(cospan f g).op` and `span f.op g.op` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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