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coprod_comparison_inr :
coprod.inr ≫ coprod_comparison F A B = F.map coprod.inr | coprod.inr_desc _ _ | lemma | category_theory.limits.coprod_comparison_inr | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_comparison_natural (f : A ⟶ A') (g : B ⟶ B') :
coprod_comparison F A B ≫ F.map (coprod.map f g) =
coprod.map (F.map f) (F.map g) ≫ coprod_comparison F A' B' | begin
rw [coprod_comparison, coprod_comparison, coprod.map_desc, ← F.map_comp, ← F.map_comp,
coprod.desc_comp, ← F.map_comp, coprod.inl_map, ← F.map_comp, coprod.inr_map]
end | lemma | category_theory.limits.coprod_comparison_natural | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | Naturality of the coprod_comparison morphism in both arguments. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod_comparison_nat_trans [has_binary_coproducts C] [has_binary_coproducts D]
(F : C ⥤ D) (A : C) :
F ⋙ coprod.functor.obj (F.obj A) ⟶ coprod.functor.obj A ⋙ F | { app := λ B, coprod_comparison F A B,
naturality' := λ B B' f, by simp [coprod_comparison_natural] } | def | category_theory.limits.coprod_comparison_nat_trans | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | The coproduct comparison morphism from `FA ⨿ F-` to `F(A ⨿ -)`, whose components are given by
`coprod_comparison`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_inl_inv_coprod_comparison [is_iso (coprod_comparison F A B)] :
F.map coprod.inl ≫ inv (coprod_comparison F A B) = coprod.inl | by simp [is_iso.inv_comp_eq] | lemma | category_theory.limits.map_inl_inv_coprod_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inr_inv_coprod_comparison [is_iso (coprod_comparison F A B)] :
F.map coprod.inr ≫ inv (coprod_comparison F A B) = coprod.inr | by simp [is_iso.inv_comp_eq] | lemma | category_theory.limits.map_inr_inv_coprod_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_comparison_inv_natural (f : A ⟶ A') (g : B ⟶ B')
[is_iso (coprod_comparison F A B)] [is_iso (coprod_comparison F A' B')] :
inv (coprod_comparison F A B) ≫ coprod.map (F.map f) (F.map g) =
F.map (coprod.map f g) ≫ inv (coprod_comparison F A' B') | by rw [is_iso.eq_comp_inv, category.assoc, is_iso.inv_comp_eq, coprod_comparison_natural] | lemma | category_theory.limits.coprod_comparison_inv_natural | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | If the coproduct comparison morphism is an iso, its inverse is natural. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod_comparison_nat_iso [has_binary_coproducts C] [has_binary_coproducts D]
(A : C) [∀ B, is_iso (coprod_comparison F A B)] :
F ⋙ coprod.functor.obj (F.obj A) ≅ coprod.functor.obj A ⋙ F | { hom := coprod_comparison_nat_trans F A
..(@as_iso _ _ _ _ _ (nat_iso.is_iso_of_is_iso_app ⟨_, _⟩)) } | def | category_theory.limits.coprod_comparison_nat_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | The natural isomorphism `FA ⨿ F- ≅ F(A ⨿ -)`, provided each `coprod_comparison F A B` is an
isomorphism (as `B` changes). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
over.coprod_obj [has_binary_coproducts C] {A : C} : over A → over A ⥤ over A | λ f,
{ obj := λ g, over.mk (coprod.desc f.hom g.hom),
map := λ g₁ g₂ k, over.hom_mk (coprod.map (𝟙 _) k.left) } | def | category_theory.over.coprod_obj | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | Auxiliary definition for `over.coprod`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
over.coprod [has_binary_coproducts C] {A : C} : over A ⥤ over A ⥤ over A | { obj := λ f, over.coprod_obj f,
map := λ f₁ f₂ k,
{ app := λ g, over.hom_mk (coprod.map k.left (𝟙 _))
(by { dsimp, rw [coprod.map_desc, category.id_comp, over.w k] }),
naturality' := λ f g k, by ext; { dsimp, simp, }, },
map_id' := λ X, by ext; { dsimp, simp, },
map_comp' := λ X Y Z f g, by ext; { d... | def | category_theory.over.coprod | category_theory.limits.shapes | src/category_theory/limits/shapes/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.discrete_category",
"category_theory.epi_mono",
"category_theory.over"
] | [] | A category with binary coproducts has a functorial `sup` operation on over categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bicone (F : J → C) | (X : C)
(π : Π j, X ⟶ F j)
(ι : Π j, F j ⟶ X)
(ι_π : ∀ j j', ι j ≫ π j' = if h : j = j' then eq_to_hom (congr_arg F h) else 0 . obviously) | structure | category_theory.limits.bicone | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | A `c : bicone F` is:
* an object `c.X` and
* morphisms `π j : X ⟶ F j` and `ι j : F j ⟶ X` for each `j`,
* such that `ι j ≫ π j'` is the identity when `j = j'` and zero otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bicone_ι_π_self {F : J → C} (B : bicone F) (j : J) :
B.ι j ≫ B.π j = 𝟙 (F j) | by simpa using B.ι_π j j | lemma | category_theory.limits.bicone_ι_π_self | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bicone_ι_π_ne {F : J → C} (B : bicone F) {j j' : J} (h : j ≠ j') :
B.ι j ≫ B.π j' = 0 | by simpa [h] using B.ι_π j j' | lemma | category_theory.limits.bicone_ι_π_ne | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_cone (B : bicone F) : cone (discrete.functor F) | { X := B.X,
π := { app := λ j, B.π j.as }, } | def | category_theory.limits.bicone.to_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | Extract the cone from a bicone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_cone_X (B : bicone F) : B.to_cone.X = B.X | rfl | lemma | category_theory.limits.bicone.to_cone_X | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_cone_π_app (B : bicone F) (j : discrete J) : B.to_cone.π.app j = B.π j.as | rfl | lemma | category_theory.limits.bicone.to_cone_π_app | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_cone_π_app_mk (B : bicone F) (j : J) : B.to_cone.π.app ⟨j⟩ = B.π j | rfl | lemma | category_theory.limits.bicone.to_cone_π_app_mk | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_cocone (B : bicone F) : cocone (discrete.functor F) | { X := B.X,
ι := { app := λ j, B.ι j.as }, } | def | category_theory.limits.bicone.to_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | Extract the cocone from a bicone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_cocone_X (B : bicone F) : B.to_cocone.X = B.X | rfl | lemma | category_theory.limits.bicone.to_cocone_X | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_cocone_ι_app (B : bicone F) (j : discrete J) : B.to_cocone.ι.app j = B.ι j.as | rfl | lemma | category_theory.limits.bicone.to_cocone_ι_app | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_cocone_ι_app_mk (B : bicone F) (j : J) : B.to_cocone.ι.app ⟨j⟩ = B.ι j | rfl | lemma | category_theory.limits.bicone.to_cocone_ι_app_mk | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_limit_cone {f : J → C} {t : cone (discrete.functor f)} (ht : is_limit t) :
bicone f | { X := t.X,
π := λ j, t.π.app ⟨j⟩,
ι := λ j, ht.lift (fan.mk _ (λ j', if h : j = j' then eq_to_hom (congr_arg f h) else 0)),
ι_π := λ j j', by simp } | def | category_theory.limits.bicone.of_limit_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | We can turn any limit cone over a discrete collection of objects into a bicone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_of_is_limit {f : J → C} {t : bicone f} (ht : is_limit t.to_cone) (j : J) :
t.ι j = ht.lift (fan.mk _ (λ j', if h : j = j' then eq_to_hom (congr_arg f h) else 0)) | ht.hom_ext (λ j', by { rw ht.fac, discrete_cases, simp [t.ι_π] }) | lemma | category_theory.limits.bicone.ι_of_is_limit | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_colimit_cocone {f : J → C} {t : cocone (discrete.functor f)} (ht : is_colimit t) :
bicone f | { X := t.X,
π := λ j, ht.desc (cofan.mk _ (λ j', if h : j' = j then eq_to_hom (congr_arg f h) else 0)),
ι := λ j, t.ι.app ⟨j⟩,
ι_π := λ j j', by simp } | def | category_theory.limits.bicone.of_colimit_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | We can turn any colimit cocone over a discrete collection of objects into a bicone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π_of_is_colimit {f : J → C} {t : bicone f} (ht : is_colimit t.to_cocone) (j : J) :
t.π j = ht.desc (cofan.mk _ (λ j', if h : j' = j then eq_to_hom (congr_arg f h) else 0)) | ht.hom_ext (λ j', by { rw ht.fac, discrete_cases, simp [t.ι_π] }) | lemma | category_theory.limits.bicone.π_of_is_colimit | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bilimit {F : J → C} (B : bicone F) | (is_limit : is_limit B.to_cone)
(is_colimit : is_colimit B.to_cocone) | structure | category_theory.limits.bicone.is_bilimit | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | Structure witnessing that a bicone is both a limit cone and a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton_is_bilimit {f : J → C} {c : bicone f} : subsingleton c.is_bilimit | ⟨λ h h', bicone.is_bilimit.ext _ _ (subsingleton.elim _ _) (subsingleton.elim _ _)⟩ | instance | category_theory.limits.bicone.subsingleton_is_bilimit | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whisker {f : J → C} (c : bicone f) (g : K ≃ J) : bicone (f ∘ g) | { X := c.X,
π := λ k, c.π (g k),
ι := λ k, c.ι (g k),
ι_π := λ k k',
begin
simp only [c.ι_π],
split_ifs with h h' h'; simp [equiv.apply_eq_iff_eq g] at h h'; tauto
end } | def | category_theory.limits.bicone.whisker | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"equiv.apply_eq_iff_eq"
] | Whisker a bicone with an equivalence between the indexing types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_to_cone {f : J → C} (c : bicone f) (g : K ≃ J) :
(c.whisker g).to_cone ≅ (cones.postcompose (discrete.functor_comp f g).inv).obj
(c.to_cone.whisker (discrete.functor (discrete.mk ∘ g))) | cones.ext (iso.refl _) (by tidy) | def | category_theory.limits.bicone.whisker_to_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | Taking the cone of a whiskered bicone results in a cone isomorphic to one gained
by whiskering the cone and postcomposing with a suitable isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_to_cocone {f : J → C} (c : bicone f) (g : K ≃ J) :
(c.whisker g).to_cocone ≅ (cocones.precompose (discrete.functor_comp f g).hom).obj
(c.to_cocone.whisker (discrete.functor (discrete.mk ∘ g))) | cocones.ext (iso.refl _) (by tidy) | def | category_theory.limits.bicone.whisker_to_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | Taking the cocone of a whiskered bicone results in a cone isomorphic to one gained
by whiskering the cocone and precomposing with a suitable isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whisker_is_bilimit_iff {f : J → C} (c : bicone f) (g : K ≃ J) :
(c.whisker g).is_bilimit ≃ c.is_bilimit | begin
refine equiv_of_subsingleton_of_subsingleton (λ hc, ⟨_, _⟩) (λ hc, ⟨_, _⟩),
{ let := is_limit.of_iso_limit hc.is_limit (bicone.whisker_to_cone c g),
let := (is_limit.postcompose_hom_equiv (discrete.functor_comp f g).symm _) this,
exact is_limit.of_whisker_equivalence (discrete.equivalence g) this },
... | def | category_theory.limits.bicone.whisker_is_bilimit_iff | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"equiv_of_subsingleton_of_subsingleton"
] | Whiskering a bicone with an equivalence between types preserves being a bilimit bicone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_bicone (F : J → C) | (bicone : bicone F)
(is_bilimit : bicone.is_bilimit) | structure | category_theory.limits.limit_bicone | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | A bicone over `F : J → C`, which is both a limit cone and a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_biproduct (F : J → C) : Prop | mk' :: (exists_biproduct : nonempty (limit_bicone F)) | class | category_theory.limits.has_biproduct | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"mk'"
] | `has_biproduct F` expresses the mere existence of a bicone which is
simultaneously a limit and a colimit of the diagram `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_biproduct.mk {F : J → C} (d : limit_bicone F) : has_biproduct F | ⟨nonempty.intro d⟩ | lemma | category_theory.limits.has_biproduct.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
get_biproduct_data (F : J → C) [has_biproduct F] : limit_bicone F | classical.choice has_biproduct.exists_biproduct | def | category_theory.limits.get_biproduct_data | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | Use the axiom of choice to extract explicit `biproduct_data F` from `has_biproduct F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.bicone (F : J → C) [has_biproduct F] : bicone F | (get_biproduct_data F).bicone | def | category_theory.limits.biproduct.bicone | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | A bicone for `F` which is both a limit cone and a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.is_bilimit (F : J → C) [has_biproduct F] : (biproduct.bicone F).is_bilimit | (get_biproduct_data F).is_bilimit | def | category_theory.limits.biproduct.is_bilimit | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | `biproduct.bicone F` is a bilimit bicone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.is_limit (F : J → C) [has_biproduct F] : is_limit (biproduct.bicone F).to_cone | (get_biproduct_data F).is_bilimit.is_limit | def | category_theory.limits.biproduct.is_limit | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | `biproduct.bicone F` is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.is_colimit (F : J → C) [has_biproduct F] :
is_colimit (biproduct.bicone F).to_cocone | (get_biproduct_data F).is_bilimit.is_colimit | def | category_theory.limits.biproduct.is_colimit | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | `biproduct.bicone F` is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_product_of_has_biproduct [has_biproduct F] : has_product F | has_limit.mk { cone := (biproduct.bicone F).to_cone,
is_limit := biproduct.is_limit F, } | instance | category_theory.limits.has_product_of_has_biproduct | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coproduct_of_has_biproduct [has_biproduct F] : has_coproduct F | has_colimit.mk { cocone := (biproduct.bicone F).to_cocone,
is_colimit := biproduct.is_colimit F, } | instance | category_theory.limits.has_coproduct_of_has_biproduct | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_biproducts_of_shape : Prop | (has_biproduct : ∀ F : J → C, has_biproduct F) | class | category_theory.limits.has_biproducts_of_shape | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | `C` has biproducts of shape `J` if we have
a limit and a colimit, with the same cone points,
of every function `F : J → C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_biproducts : Prop | (out [] : ∀ n, has_biproducts_of_shape (fin n) C) | class | category_theory.limits.has_finite_biproducts | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | `has_finite_biproducts C` represents a choice of biproduct for every family of objects in `C`
indexed by a finite type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_biproducts_of_shape_of_equiv {K : Type w'} [has_biproducts_of_shape K C] (e : J ≃ K) :
has_biproducts_of_shape J C | ⟨λ F, let ⟨⟨h⟩⟩ := has_biproducts_of_shape.has_biproduct (F ∘ e.symm), ⟨c, hc⟩ := h
in has_biproduct.mk $ by simpa only [(∘), e.symm_apply_apply]
using limit_bicone.mk (c.whisker e) ((c.whisker_is_bilimit_iff _).2 hc)⟩ | lemma | category_theory.limits.has_biproducts_of_shape_of_equiv | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_biproducts_of_shape_finite [has_finite_biproducts C] [finite J] :
has_biproducts_of_shape J C | begin
rcases finite.exists_equiv_fin J with ⟨n, ⟨e⟩⟩,
haveI := has_finite_biproducts.out C n,
exact has_biproducts_of_shape_of_equiv C e
end | instance | category_theory.limits.has_biproducts_of_shape_finite | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"finite",
"finite.exists_equiv_fin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_products_of_has_finite_biproducts [has_finite_biproducts C] :
has_finite_products C | { out := λ n, ⟨λ F, has_limit_of_iso discrete.nat_iso_functor.symm⟩ } | instance | category_theory.limits.has_finite_products_of_has_finite_biproducts | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_coproducts_of_has_finite_biproducts [has_finite_biproducts C] :
has_finite_coproducts C | { out := λ n, ⟨λ F, has_colimit_of_iso discrete.nat_iso_functor⟩ } | instance | category_theory.limits.has_finite_coproducts_of_has_finite_biproducts | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct_iso (F : J → C) [has_biproduct F] :
limits.pi_obj F ≅ limits.sigma_obj F | (is_limit.cone_point_unique_up_to_iso (limit.is_limit _) (biproduct.is_limit F)).trans $
is_colimit.cocone_point_unique_up_to_iso (biproduct.is_colimit F) (colimit.is_colimit _) | def | category_theory.limits.biproduct_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | The isomorphism between the specified limit and the specified colimit for
a functor with a bilimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct (f : J → C) [has_biproduct f] : C | (biproduct.bicone f).X | abbreviation | category_theory.limits.biproduct | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | `biproduct f` computes the biproduct of a family of elements `f`. (It is defined as an
abbreviation for `limit (discrete.functor f)`, so for most facts about `biproduct f`, you will
just use general facts about limits and colimits.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.π (f : J → C) [has_biproduct f] (b : J) : ⨁ f ⟶ f b | (biproduct.bicone f).π b | abbreviation | category_theory.limits.biproduct.π | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | The projection onto a summand of a biproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.bicone_π (f : J → C) [has_biproduct f] (b : J) :
(biproduct.bicone f).π b = biproduct.π f b | rfl | lemma | category_theory.limits.biproduct.bicone_π | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.ι (f : J → C) [has_biproduct f] (b : J) : f b ⟶ ⨁ f | (biproduct.bicone f).ι b | abbreviation | category_theory.limits.biproduct.ι | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | The inclusion into a summand of a biproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.bicone_ι (f : J → C) [has_biproduct f] (b : J) :
(biproduct.bicone f).ι b = biproduct.ι f b | rfl | lemma | category_theory.limits.biproduct.bicone_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.ι_π [decidable_eq J] (f : J → C) [has_biproduct f] (j j' : J) :
biproduct.ι f j ≫ biproduct.π f j' = if h : j = j' then eq_to_hom (congr_arg f h) else 0 | by convert (biproduct.bicone f).ι_π j j' | lemma | category_theory.limits.biproduct.ι_π | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | Note that as this lemma has a `if` in the statement, we include a `decidable_eq` argument.
This means you may not be able to `simp` using this lemma unless you `open_locale classical`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.ι_π_self (f : J → C) [has_biproduct f] (j : J) :
biproduct.ι f j ≫ biproduct.π f j = 𝟙 _ | by simp [biproduct.ι_π] | lemma | category_theory.limits.biproduct.ι_π_self | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.ι_π_ne (f : J → C) [has_biproduct f] {j j' : J} (h : j ≠ j') :
biproduct.ι f j ≫ biproduct.π f j' = 0 | by simp [biproduct.ι_π, h] | lemma | category_theory.limits.biproduct.ι_π_ne | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.lift
{f : J → C} [has_biproduct f] {P : C} (p : Π b, P ⟶ f b) : P ⟶ ⨁ f | (biproduct.is_limit f).lift (fan.mk P p) | abbreviation | category_theory.limits.biproduct.lift | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"lift"
] | Given a collection of maps into the summands, we obtain a map into the biproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.desc
{f : J → C} [has_biproduct f] {P : C} (p : Π b, f b ⟶ P) : ⨁ f ⟶ P | (biproduct.is_colimit f).desc (cofan.mk P p) | abbreviation | category_theory.limits.biproduct.desc | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | Given a collection of maps out of the summands, we obtain a map out of the biproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.lift_π {f : J → C} [has_biproduct f] {P : C} (p : Π b, P ⟶ f b) (j : J) :
biproduct.lift p ≫ biproduct.π f j = p j | (biproduct.is_limit f).fac _ ⟨j⟩ | lemma | category_theory.limits.biproduct.lift_π | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.ι_desc {f : J → C} [has_biproduct f] {P : C} (p : Π b, f b ⟶ P) (j : J) :
biproduct.ι f j ≫ biproduct.desc p = p j | (biproduct.is_colimit f).fac _ ⟨j⟩ | lemma | category_theory.limits.biproduct.ι_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.map {f g : J → C} [has_biproduct f] [has_biproduct g]
(p : Π b, f b ⟶ g b) : ⨁ f ⟶ ⨁ g | is_limit.map (biproduct.bicone f).to_cone (biproduct.is_limit g)
(discrete.nat_trans (λ j, p j.as)) | abbreviation | category_theory.limits.biproduct.map | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | Given a collection of maps between corresponding summands of a pair of biproducts
indexed by the same type, we obtain a map between the biproducts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.map' {f g : J → C} [has_biproduct f] [has_biproduct g]
(p : Π b, f b ⟶ g b) : ⨁ f ⟶ ⨁ g | is_colimit.map (biproduct.is_colimit f) (biproduct.bicone g).to_cocone
(discrete.nat_trans (λ j, p j.as)) | abbreviation | category_theory.limits.biproduct.map' | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | An alternative to `biproduct.map` constructed via colimits.
This construction only exists in order to show it is equal to `biproduct.map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.hom_ext {f : J → C} [has_biproduct f]
{Z : C} (g h : Z ⟶ ⨁ f)
(w : ∀ j, g ≫ biproduct.π f j = h ≫ biproduct.π f j) : g = h | (biproduct.is_limit f).hom_ext (λ j, w j.as) | lemma | category_theory.limits.biproduct.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.hom_ext' {f : J → C} [has_biproduct f]
{Z : C} (g h : ⨁ f ⟶ Z)
(w : ∀ j, biproduct.ι f j ≫ g = biproduct.ι f j ≫ h) : g = h | (biproduct.is_colimit f).hom_ext (λ j, w j.as) | lemma | category_theory.limits.biproduct.hom_ext' | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.iso_product (f : J → C) [has_biproduct f] : ⨁ f ≅ ∏ f | is_limit.cone_point_unique_up_to_iso (biproduct.is_limit f) (limit.is_limit _) | def | category_theory.limits.biproduct.iso_product | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | The canonical isomorphism between the chosen biproduct and the chosen product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.iso_product_hom {f : J → C} [has_biproduct f] :
(biproduct.iso_product f).hom = pi.lift (biproduct.π f) | limit.hom_ext $ λ j, by simp [biproduct.iso_product] | lemma | category_theory.limits.biproduct.iso_product_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.iso_product_inv {f : J → C} [has_biproduct f] :
(biproduct.iso_product f).inv = biproduct.lift (pi.π f) | biproduct.hom_ext _ _ $ λ j, by simp [iso.inv_comp_eq] | lemma | category_theory.limits.biproduct.iso_product_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.iso_coproduct (f : J → C) [has_biproduct f] : ⨁ f ≅ ∐ f | is_colimit.cocone_point_unique_up_to_iso (biproduct.is_colimit f) (colimit.is_colimit _) | def | category_theory.limits.biproduct.iso_coproduct | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | The canonical isomorphism between the chosen biproduct and the chosen coproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.iso_coproduct_inv {f : J → C} [has_biproduct f] :
(biproduct.iso_coproduct f).inv = sigma.desc (biproduct.ι f) | colimit.hom_ext $ λ j, by simp [biproduct.iso_coproduct] | lemma | category_theory.limits.biproduct.iso_coproduct_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.iso_coproduct_hom {f : J → C} [has_biproduct f] :
(biproduct.iso_coproduct f).hom = biproduct.desc (sigma.ι f) | biproduct.hom_ext' _ _ $ λ j, by simp [← iso.eq_comp_inv] | lemma | category_theory.limits.biproduct.iso_coproduct_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.map_eq_map' {f g : J → C} [has_biproduct f] [has_biproduct g]
(p : Π b, f b ⟶ g b) : biproduct.map p = biproduct.map' p | begin
ext j j',
simp only [discrete.nat_trans_app, limits.is_colimit.ι_map, limits.is_limit.map_π, category.assoc,
←bicone.to_cone_π_app_mk, ←biproduct.bicone_π, ←bicone.to_cocone_ι_app_mk, ←biproduct.bicone_ι],
simp only [biproduct.bicone_ι, biproduct.bicone_π, bicone.to_cocone_ι_app, bicone.to_cone_π_app],
... | lemma | category_theory.limits.biproduct.map_eq_map' | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.map_π {f g : J → C} [has_biproduct f] [has_biproduct g]
(p : Π j, f j ⟶ g j) (j : J) :
biproduct.map p ≫ biproduct.π g j = biproduct.π f j ≫ p j | limits.is_limit.map_π _ _ _ (discrete.mk j) | lemma | category_theory.limits.biproduct.map_π | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.ι_map {f g : J → C} [has_biproduct f] [has_biproduct g]
(p : Π j, f j ⟶ g j) (j : J) :
biproduct.ι f j ≫ biproduct.map p = p j ≫ biproduct.ι g j | begin
rw biproduct.map_eq_map',
convert limits.is_colimit.ι_map _ _ _ (discrete.mk j); refl
end | lemma | category_theory.limits.biproduct.ι_map | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.map_desc {f g : J → C} [has_biproduct f] [has_biproduct g]
(p : Π j, f j ⟶ g j) {P : C} (k : Π j, g j ⟶ P) :
biproduct.map p ≫ biproduct.desc k = biproduct.desc (λ j, p j ≫ k j) | by { ext, simp, } | lemma | category_theory.limits.biproduct.map_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.lift_map {f g : J → C} [has_biproduct f] [has_biproduct g]
{P : C} (k : Π j, P ⟶ f j) (p : Π j, f j ⟶ g j) :
biproduct.lift k ≫ biproduct.map p = biproduct.lift (λ j, k j ≫ p j) | by { ext, simp, } | lemma | category_theory.limits.biproduct.lift_map | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.map_iso {f g : J → C} [has_biproduct f] [has_biproduct g]
(p : Π b, f b ≅ g b) : ⨁ f ≅ ⨁ g | { hom := biproduct.map (λ b, (p b).hom),
inv := biproduct.map (λ b, (p b).inv), } | def | category_theory.limits.biproduct.map_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | Given a collection of isomorphisms between corresponding summands of a pair of biproducts
indexed by the same type, we obtain an isomorphism between the biproducts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.from_subtype : ⨁ subtype.restrict p f ⟶ ⨁ f | biproduct.desc $ λ j, biproduct.ι _ _ | def | category_theory.limits.biproduct.from_subtype | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"subtype.restrict"
] | The canonical morphism from the biproduct over a restricted index type to the biproduct of
the full index type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.to_subtype : ⨁ f ⟶ ⨁ subtype.restrict p f | biproduct.lift $ λ j, biproduct.π _ _ | def | category_theory.limits.biproduct.to_subtype | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"subtype.restrict"
] | The canonical morphism from a biproduct to the biproduct over a restriction of its index
type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.from_subtype_π [decidable_pred p] (j : J) :
biproduct.from_subtype f p ≫ biproduct.π f j =
if h : p j then biproduct.π (subtype.restrict p f) ⟨j, h⟩ else 0 | begin
ext i,
rw [biproduct.from_subtype, biproduct.ι_desc_assoc, biproduct.ι_π],
by_cases h : p j,
{ rw [dif_pos h, biproduct.ι_π],
split_ifs with h₁ h₂ h₂,
exacts [rfl, false.elim (h₂ (subtype.ext h₁)),
false.elim (h₁ (congr_arg subtype.val h₂)), rfl] },
{ rw [dif_neg h, dif_neg (show (i : J) ≠... | lemma | category_theory.limits.biproduct.from_subtype_π | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"subtype.ext",
"subtype.restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.from_subtype_eq_lift [decidable_pred p] : biproduct.from_subtype f p =
biproduct.lift (λ j, if h : p j then biproduct.π (subtype.restrict p f) ⟨j, h⟩ else 0) | biproduct.hom_ext _ _ (by simp) | lemma | category_theory.limits.biproduct.from_subtype_eq_lift | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"subtype.restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.from_subtype_π_subtype (j : subtype p) :
biproduct.from_subtype f p ≫ biproduct.π f j = biproduct.π (subtype.restrict p f) j | begin
ext i,
rw [biproduct.from_subtype, biproduct.ι_desc_assoc, biproduct.ι_π, biproduct.ι_π],
split_ifs with h₁ h₂ h₂,
exacts [rfl, false.elim (h₂ (subtype.ext h₁)), false.elim (h₁ (congr_arg subtype.val h₂)), rfl]
end | lemma | category_theory.limits.biproduct.from_subtype_π_subtype | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"subtype.ext",
"subtype.restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.to_subtype_π (j : subtype p) :
biproduct.to_subtype f p ≫ biproduct.π (subtype.restrict p f) j = biproduct.π f j | biproduct.lift_π _ _ | lemma | category_theory.limits.biproduct.to_subtype_π | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"subtype.restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.ι_to_subtype [decidable_pred p] (j : J) :
biproduct.ι f j ≫ biproduct.to_subtype f p =
if h : p j then biproduct.ι (subtype.restrict p f) ⟨j, h⟩ else 0 | begin
ext i,
rw [biproduct.to_subtype, category.assoc, biproduct.lift_π, biproduct.ι_π],
by_cases h : p j,
{ rw [dif_pos h, biproduct.ι_π],
split_ifs with h₁ h₂ h₂,
exacts [rfl, false.elim (h₂ (subtype.ext h₁)),
false.elim (h₁ (congr_arg subtype.val h₂)), rfl] },
{ rw [dif_neg h, dif_neg (show j... | lemma | category_theory.limits.biproduct.ι_to_subtype | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"subtype.ext",
"subtype.restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.to_subtype_eq_desc [decidable_pred p] : biproduct.to_subtype f p =
biproduct.desc (λ j, if h : p j then biproduct.ι (subtype.restrict p f) ⟨j, h⟩ else 0) | biproduct.hom_ext' _ _ (by simp) | lemma | category_theory.limits.biproduct.to_subtype_eq_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"subtype.restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.ι_to_subtype_subtype (j : subtype p) :
biproduct.ι f j ≫ biproduct.to_subtype f p = biproduct.ι (subtype.restrict p f) j | begin
ext i,
rw [biproduct.to_subtype, category.assoc, biproduct.lift_π, biproduct.ι_π, biproduct.ι_π],
split_ifs with h₁ h₂ h₂,
exacts [rfl, false.elim (h₂ (subtype.ext h₁)), false.elim (h₁ (congr_arg subtype.val h₂)), rfl]
end | lemma | category_theory.limits.biproduct.ι_to_subtype_subtype | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"subtype.ext",
"subtype.restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.ι_from_subtype (j : subtype p) :
biproduct.ι (subtype.restrict p f) j ≫ biproduct.from_subtype f p = biproduct.ι f j | biproduct.ι_desc _ _ | lemma | category_theory.limits.biproduct.ι_from_subtype | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"subtype.restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.from_subtype_to_subtype :
biproduct.from_subtype f p ≫ biproduct.to_subtype f p = 𝟙 (⨁ subtype.restrict p f) | begin
refine biproduct.hom_ext _ _ (λ j, _),
rw [category.assoc, biproduct.to_subtype_π, biproduct.from_subtype_π_subtype, category.id_comp]
end | lemma | category_theory.limits.biproduct.from_subtype_to_subtype | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"subtype.restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.to_subtype_from_subtype [decidable_pred p] :
biproduct.to_subtype f p ≫ biproduct.from_subtype f p =
biproduct.map (λ j, if p j then 𝟙 (f j) else 0) | begin
ext1 i,
by_cases h : p i,
{ simp [h], congr },
{ simp [h] }
end | lemma | category_theory.limits.biproduct.to_subtype_from_subtype | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.is_limit_from_subtype : is_limit
(kernel_fork.of_ι (biproduct.from_subtype f (λ j, j ≠ i))
(by simp) : kernel_fork (biproduct.π f i)) | fork.is_limit.mk' _ $ λ s,
⟨s.ι ≫ biproduct.to_subtype _ _,
begin
ext j,
rw [kernel_fork.ι_of_ι, category.assoc, category.assoc,
biproduct.to_subtype_from_subtype_assoc, biproduct.map_π],
rcases em (i = j) with (rfl|h),
{ rw [if_neg (not_not.2 rfl), comp_zero, comp_zero, kernel_fork.condition] },
{... | def | category_theory.limits.biproduct.is_limit_from_subtype | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"em"
] | The kernel of `biproduct.π f i` is the inclusion from the biproduct which omits `i`
from the index set `J` into the biproduct over `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_biproduct_π_iso : kernel (biproduct.π f i) ≅ ⨁ subtype.restrict (λ j, j ≠ i) f | limit.iso_limit_cone ⟨_, biproduct.is_limit_from_subtype f i⟩ | def | category_theory.limits.kernel_biproduct_π_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"subtype.restrict"
] | The kernel of `biproduct.π f i` is `⨁ subtype.restrict {i}ᶜ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.is_colimit_to_subtype : is_colimit
(cokernel_cofork.of_π (biproduct.to_subtype f (λ j, j ≠ i))
(by simp) : cokernel_cofork (biproduct.ι f i)) | cofork.is_colimit.mk' _ $ λ s,
⟨biproduct.from_subtype _ _ ≫ s.π,
begin
ext j,
rw [cokernel_cofork.π_of_π, biproduct.to_subtype_from_subtype_assoc,
biproduct.ι_map_assoc],
rcases em (i = j) with (rfl|h),
{ rw [if_neg (not_not.2 rfl), zero_comp, cokernel_cofork.condition] },
{ rw [if_pos (ne.symm h)... | def | category_theory.limits.biproduct.is_colimit_to_subtype | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"em"
] | The cokernel of `biproduct.ι f i` is the projection from the biproduct over the index set `J`
onto the biproduct omitting `i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_biproduct_ι_iso : cokernel (biproduct.ι f i) ≅ ⨁ subtype.restrict (λ j, j ≠ i) f | colimit.iso_colimit_cocone ⟨_, biproduct.is_colimit_to_subtype f i⟩ | def | category_theory.limits.cokernel_biproduct_ι_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"subtype.restrict"
] | The cokernel of `biproduct.ι f i` is `⨁ subtype.restrict {i}ᶜ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_fork_biproduct_to_subtype (p : set K) :
limit_cone (parallel_pair (biproduct.to_subtype f p) 0) | { cone := kernel_fork.of_ι (biproduct.from_subtype f pᶜ) begin
ext j k,
simp only [biproduct.ι_from_subtype_assoc, biproduct.ι_to_subtype, comp_zero, zero_comp],
erw [dif_neg j.2],
simp only [zero_comp],
end,
is_limit := kernel_fork.is_limit.of_ι _ _ (λ W g h, g ≫ biproduct.to_subtype f pᶜ)
begi... | def | category_theory.limits.kernel_fork_biproduct_to_subtype | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"pi.compl_apply"
] | The limit cone exhibiting `⨁ subtype.restrict pᶜ f` as the kernel of
`biproduct.to_subtype f p` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_biproduct_to_subtype_iso (p : set K) :
kernel (biproduct.to_subtype f p) ≅ ⨁ subtype.restrict pᶜ f | limit.iso_limit_cone (kernel_fork_biproduct_to_subtype f p) | def | category_theory.limits.kernel_biproduct_to_subtype_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"subtype.restrict"
] | The kernel of `biproduct.to_subtype f p` is `⨁ subtype.restrict pᶜ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_cofork_biproduct_from_subtype (p : set K) :
colimit_cocone (parallel_pair (biproduct.from_subtype f p) 0) | { cocone := cokernel_cofork.of_π (biproduct.to_subtype f pᶜ) begin
ext j k,
simp only [pi.compl_apply, biproduct.ι_from_subtype_assoc, biproduct.ι_to_subtype,
comp_zero, zero_comp],
rw [dif_neg],
simp only [zero_comp],
exact not_not.mpr j.2,
end,
is_colimit := cokernel_cofork.is_colimit.of... | def | category_theory.limits.cokernel_cofork_biproduct_from_subtype | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"pi.compl_apply"
] | The colimit cocone exhibiting `⨁ subtype.restrict pᶜ f` as the cokernel of
`biproduct.from_subtype f p` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_biproduct_from_subtype_iso (p : set K) :
cokernel (biproduct.from_subtype f p) ≅ ⨁ subtype.restrict pᶜ f | colimit.iso_colimit_cocone (cokernel_cofork_biproduct_from_subtype f p) | def | category_theory.limits.cokernel_biproduct_from_subtype_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [
"subtype.restrict"
] | The cokernel of `biproduct.from_subtype f p` is `⨁ subtype.restrict pᶜ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.matrix (m : Π j k, f j ⟶ g k) : ⨁ f ⟶ ⨁ g | biproduct.desc (λ j, biproduct.lift (λ k, m j k)) | def | category_theory.limits.biproduct.matrix | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | Convert a (dependently typed) matrix to a morphism of biproducts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.matrix_π (m : Π j k, f j ⟶ g k) (k : K) :
biproduct.matrix m ≫ biproduct.π g k = biproduct.desc (λ j, m j k) | by { ext, simp [biproduct.matrix], } | lemma | category_theory.limits.biproduct.matrix_π | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.ι_matrix (m : Π j k, f j ⟶ g k) (j : J) :
biproduct.ι f j ≫ biproduct.matrix m = biproduct.lift (λ k, m j k) | by { ext, simp [biproduct.matrix], } | lemma | category_theory.limits.biproduct.ι_matrix | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biproduct.components (m : ⨁ f ⟶ ⨁ g) (j : J) (k : K) : f j ⟶ g k | biproduct.ι f j ≫ m ≫ biproduct.π g k | def | category_theory.limits.biproduct.components | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | Extract the matrix components from a morphism of biproducts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.matrix_components (m : Π j k, f j ⟶ g k) (j : J) (k : K) :
biproduct.components (biproduct.matrix m) j k = m j k | by simp [biproduct.components] | lemma | category_theory.limits.biproduct.matrix_components | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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