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biproduct.components_matrix (m : ⨁ f ⟶ ⨁ g) :
biproduct.matrix (λ j k, biproduct.components m j k) = m | by { ext, simp [biproduct.components], } | lemma | category_theory.limits.biproduct.components_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.matrix_equiv : (⨁ f ⟶ ⨁ g) ≃ (Π j k, f j ⟶ g k) | { to_fun := biproduct.components,
inv_fun := biproduct.matrix,
left_inv := biproduct.components_matrix,
right_inv := λ m, by { ext, apply biproduct.matrix_components } } | def | category_theory.limits.biproduct.matrix_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"
] | [
"inv_fun"
] | Morphisms between direct sums are matrices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.ι_mono (f : J → C) [has_biproduct f] (b : J) :
is_split_mono (biproduct.ι f b) | is_split_mono.mk'
{ retraction := biproduct.desc $ pi.single b _ } | instance | category_theory.limits.biproduct.ι_mono | 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.π_epi (f : J → C) [has_biproduct f] (b : J) :
is_split_epi (biproduct.π f b) | is_split_epi.mk'
{ section_ := biproduct.lift $ pi.single b _ } | instance | category_theory.limits.biproduct.π_epi | 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.cone_point_unique_up_to_iso_hom (f : J → C) [has_biproduct f] {b : bicone f}
(hb : b.is_bilimit) :
(hb.is_limit.cone_point_unique_up_to_iso (biproduct.is_limit _)).hom = biproduct.lift b.π | rfl | lemma | category_theory.limits.biproduct.cone_point_unique_up_to_iso_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"
] | [] | Auxiliary lemma for `biproduct.unique_up_to_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.cone_point_unique_up_to_iso_inv (f : J → C) [has_biproduct f] {b : bicone f}
(hb : b.is_bilimit) :
(hb.is_limit.cone_point_unique_up_to_iso (biproduct.is_limit _)).inv = biproduct.desc b.ι | begin
refine biproduct.hom_ext' _ _ (λ j, hb.is_limit.hom_ext (λ j', _)),
discrete_cases,
rw [category.assoc, is_limit.cone_point_unique_up_to_iso_inv_comp, bicone.to_cone_π_app,
biproduct.bicone_π, biproduct.ι_desc, biproduct.ι_π, b.to_cone_π_app, b.ι_π]
end | lemma | category_theory.limits.biproduct.cone_point_unique_up_to_iso_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"
] | [] | Auxiliary lemma for `biproduct.unique_up_to_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biproduct.unique_up_to_iso (f : J → C) [has_biproduct f] {b : bicone f} (hb : b.is_bilimit) :
b.X ≅ ⨁ f | { hom := biproduct.lift b.π,
inv := biproduct.desc b.ι,
hom_inv_id' := by rw [← biproduct.cone_point_unique_up_to_iso_hom f hb,
← biproduct.cone_point_unique_up_to_iso_inv f hb, iso.hom_inv_id],
inv_hom_id' := by rw [← biproduct.cone_point_unique_up_to_iso_hom f hb,
← biproduct.cone_point_unique_up_to_iso... | def | category_theory.limits.biproduct.unique_up_to_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"
] | [] | Biproducts are unique up to isomorphism. This already follows because bilimits are limits,
but in the case of biproducts we can give an isomorphism with particularly nice definitional
properties, namely that `biproduct.lift b.π` and `biproduct.desc b.ι` are inverses of each
other. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_zero_object_of_has_finite_biproducts [has_finite_biproducts C] : has_zero_object C | by { refine ⟨⟨biproduct empty.elim, λ X, ⟨⟨⟨0⟩, _⟩⟩, λ X, ⟨⟨⟨0⟩, _⟩⟩⟩⟩, tidy, } | instance | category_theory.limits.has_zero_object_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"
] | [
"empty.elim"
] | A category with finite biproducts has a zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_bicone_of_unique : limit_bicone f | { bicone :=
{ X := f default,
π := λ j, eq_to_hom (by congr),
ι := λ j, eq_to_hom (by congr), },
is_bilimit :=
{ is_limit := (limit_cone_of_unique f).is_limit,
is_colimit := (colimit_cocone_of_unique f).is_colimit, }, } | def | category_theory.limits.limit_bicone_of_unique | 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 limit bicone for the biproduct over an index type with exactly one term. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_biproduct_unique : has_biproduct f | has_biproduct.mk (limit_bicone_of_unique f) | instance | category_theory.limits.has_biproduct_unique | 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_unique_iso : ⨁ f ≅ f default | (biproduct.unique_up_to_iso _ (limit_bicone_of_unique f).is_bilimit).symm | def | category_theory.limits.biproduct_unique_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"
] | [] | A biproduct over a index type with exactly one term is just the object over that term. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_bicone (P Q : C) | (X : C)
(fst : X ⟶ P)
(snd : X ⟶ Q)
(inl : P ⟶ X)
(inr : Q ⟶ X)
(inl_fst' : inl ≫ fst = 𝟙 P . obviously)
(inl_snd' : inl ≫ snd = 0 . obviously)
(inr_fst' : inr ≫ fst = 0 . obviously)
(inr_snd' : inr ≫ snd = 𝟙 Q . obviously) | structure | category_theory.limits.binary_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 binary bicone for a pair of objects `P Q : C` consists of the cone point `X`,
maps from `X` to both `P` and `Q`, and maps from both `P` and `Q` to `X`,
so that `inl ≫ fst = 𝟙 P`, `inl ≫ snd = 0`, `inr ≫ fst = 0`, and `inr ≫ snd = 𝟙 Q` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_cone (c : binary_bicone P Q) : cone (pair P Q) | binary_fan.mk c.fst c.snd | def | category_theory.limits.binary_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 binary bicone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_cone_X (c : binary_bicone P Q) :
c.to_cone.X = c.X | rfl | lemma | category_theory.limits.binary_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_left (c : binary_bicone P Q) :
c.to_cone.π.app ⟨walking_pair.left⟩ = c.fst | rfl | lemma | category_theory.limits.binary_bicone.to_cone_π_app_left | 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_right (c : binary_bicone P Q) :
c.to_cone.π.app ⟨walking_pair.right⟩ = c.snd | rfl | lemma | category_theory.limits.binary_bicone.to_cone_π_app_right | 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 | |
binary_fan_fst_to_cone (c : binary_bicone P Q) : binary_fan.fst c.to_cone = c.fst | rfl | lemma | category_theory.limits.binary_bicone.binary_fan_fst_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_fan_snd_to_cone (c : binary_bicone P Q) : binary_fan.snd c.to_cone = c.snd | rfl | lemma | category_theory.limits.binary_bicone.binary_fan_snd_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_cocone (c : binary_bicone P Q) : cocone (pair P Q) | binary_cofan.mk c.inl c.inr | def | category_theory.limits.binary_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 binary bicone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_cocone_X (c : binary_bicone P Q) :
c.to_cocone.X = c.X | rfl | lemma | category_theory.limits.binary_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_left (c : binary_bicone P Q) :
c.to_cocone.ι.app ⟨walking_pair.left⟩ = c.inl | rfl | lemma | category_theory.limits.binary_bicone.to_cocone_ι_app_left | 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_right (c : binary_bicone P Q) :
c.to_cocone.ι.app ⟨walking_pair.right⟩ = c.inr | rfl | lemma | category_theory.limits.binary_bicone.to_cocone_ι_app_right | 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 | |
binary_cofan_inl_to_cocone (c : binary_bicone P Q) : binary_cofan.inl c.to_cocone = c.inl | rfl | lemma | category_theory.limits.binary_bicone.binary_cofan_inl_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan_inr_to_cocone (c : binary_bicone P Q) : binary_cofan.inr c.to_cocone = c.inr | rfl | lemma | category_theory.limits.binary_bicone.binary_cofan_inr_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_bicone {X Y : C} (b : binary_bicone X Y) : bicone (pair_function X Y) | { X := b.X,
π := λ j, walking_pair.cases_on j b.fst b.snd,
ι := λ j, walking_pair.cases_on j b.inl b.inr,
ι_π := λ j j', by { rcases j with ⟨⟩; rcases j' with ⟨⟩, tidy } } | def | category_theory.limits.binary_bicone.to_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"
] | [] | Convert a `binary_bicone` into a `bicone` over a pair. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_bicone_is_limit {X Y : C} (b : binary_bicone X Y) :
is_limit (b.to_bicone.to_cone) ≃ is_limit (b.to_cone) | is_limit.equiv_iso_limit $ cones.ext (iso.refl _) (λ j, by { cases j, tidy }) | def | category_theory.limits.binary_bicone.to_bicone_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"
] | [] | A binary bicone is a limit cone if and only if the corresponding bicone is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_bicone_is_colimit {X Y : C} (b : binary_bicone X Y) :
is_colimit (b.to_bicone.to_cocone) ≃ is_colimit (b.to_cocone) | is_colimit.equiv_iso_colimit $ cocones.ext (iso.refl _) (λ j, by { cases j, tidy }) | def | category_theory.limits.binary_bicone.to_bicone_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"
] | [] | A binary bicone is a colimit cocone if and only if the corresponding bicone is a colimit
cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_binary_bicone {X Y : C} (b : bicone (pair_function X Y)) : binary_bicone X Y | { X := b.X,
fst := b.π walking_pair.left,
snd := b.π walking_pair.right,
inl := b.ι walking_pair.left,
inr := b.ι walking_pair.right,
inl_fst' := by { simp [bicone.ι_π], refl, },
inr_fst' := by simp [bicone.ι_π],
inl_snd' := by simp [bicone.ι_π],
inr_snd' := by { simp [bicone.ι_π], refl, }, } | def | category_theory.limits.bicone.to_binary_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"
] | [] | Convert a `bicone` over a function on `walking_pair` to a binary_bicone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_binary_bicone_is_limit {X Y : C} (b : bicone (pair_function X Y)) :
is_limit (b.to_binary_bicone.to_cone) ≃ is_limit (b.to_cone) | is_limit.equiv_iso_limit $ cones.ext (iso.refl _) (λ j, by { rcases j with ⟨⟨⟩⟩; tidy }) | def | category_theory.limits.bicone.to_binary_bicone_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"
] | [] | A bicone over a pair is a limit cone if and only if the corresponding binary bicone is a limit
cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_binary_bicone_is_colimit {X Y : C} (b : bicone (pair_function X Y)) :
is_colimit (b.to_binary_bicone.to_cocone) ≃ is_colimit (b.to_cocone) | is_colimit.equiv_iso_colimit $ cocones.ext (iso.refl _) (λ j, by { rcases j with ⟨⟨⟩⟩; tidy }) | def | category_theory.limits.bicone.to_binary_bicone_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"
] | [] | A bicone over a pair is a colimit cocone if and only if the corresponding binary bicone is a
colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_bicone.is_bilimit {P Q : C} (b : binary_bicone P Q) | (is_limit : is_limit b.to_cone)
(is_colimit : is_colimit b.to_cocone) | structure | category_theory.limits.binary_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 binary bicone is a limit cone and a limit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_bicone.to_bicone_is_bilimit {X Y : C} (b : binary_bicone X Y) :
b.to_bicone.is_bilimit ≃ b.is_bilimit | { to_fun := λ h, ⟨b.to_bicone_is_limit h.is_limit, b.to_bicone_is_colimit h.is_colimit⟩,
inv_fun := λ h, ⟨b.to_bicone_is_limit.symm h.is_limit, b.to_bicone_is_colimit.symm h.is_colimit⟩,
left_inv := λ ⟨h, h'⟩, by { dsimp only, simp },
right_inv := λ ⟨h, h'⟩, by { dsimp only, simp } } | def | category_theory.limits.binary_bicone.to_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"
] | [
"inv_fun"
] | A binary bicone is a bilimit bicone if and only if the corresponding bicone is a bilimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bicone.to_binary_bicone_is_bilimit {X Y : C} (b : bicone (pair_function X Y)) :
b.to_binary_bicone.is_bilimit ≃ b.is_bilimit | { to_fun := λ h, ⟨b.to_binary_bicone_is_limit h.is_limit,
b.to_binary_bicone_is_colimit h.is_colimit⟩,
inv_fun := λ h, ⟨b.to_binary_bicone_is_limit.symm h.is_limit,
b.to_binary_bicone_is_colimit.symm h.is_colimit⟩,
left_inv := λ ⟨h, h'⟩, by { dsimp only, simp },
right_inv := λ ⟨h, h'⟩, by { dsimp only, si... | def | category_theory.limits.bicone.to_binary_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"
] | [
"inv_fun"
] | A bicone over a pair is a bilimit bicone if and only if the corresponding binary bicone is a
bilimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_biproduct_data (P Q : C) | (bicone : binary_bicone P Q)
(is_bilimit : bicone.is_bilimit) | structure | category_theory.limits.binary_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"
] | [] | A bicone over `P Q : C`, which is both a limit cone and a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_biproduct (P Q : C) : Prop | mk' :: (exists_binary_biproduct : nonempty (binary_biproduct_data P Q)) | class | category_theory.limits.has_binary_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_binary_biproduct P Q` expresses the mere existence of a bicone which is
simultaneously a limit and a colimit of the diagram `pair P Q`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_biproduct.mk {P Q : C} (d : binary_biproduct_data P Q) :
has_binary_biproduct P Q | ⟨nonempty.intro d⟩ | lemma | category_theory.limits.has_binary_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_binary_biproduct_data (P Q : C) [has_binary_biproduct P Q] : binary_biproduct_data P Q | classical.choice has_binary_biproduct.exists_binary_biproduct | def | category_theory.limits.get_binary_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 `binary_biproduct_data F` from `has_binary_biproduct F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_biproduct.bicone (P Q : C) [has_binary_biproduct P Q] : binary_bicone P Q | (get_binary_biproduct_data P Q).bicone | def | category_theory.limits.binary_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 `P Q ` which is both a limit cone and a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_biproduct.is_bilimit (P Q : C) [has_binary_biproduct P Q] :
(binary_biproduct.bicone P Q).is_bilimit | (get_binary_biproduct_data P Q).is_bilimit | def | category_theory.limits.binary_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"
] | [] | `binary_biproduct.bicone P Q` is a limit bicone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_biproduct.is_limit (P Q : C) [has_binary_biproduct P Q] :
is_limit (binary_biproduct.bicone P Q).to_cone | (get_binary_biproduct_data P Q).is_bilimit.is_limit | def | category_theory.limits.binary_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"
] | [] | `binary_biproduct.bicone P Q` is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_biproduct.is_colimit (P Q : C) [has_binary_biproduct P Q] :
is_colimit (binary_biproduct.bicone P Q).to_cocone | (get_binary_biproduct_data P Q).is_bilimit.is_colimit | def | category_theory.limits.binary_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"
] | [] | `binary_biproduct.bicone P Q` is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_biproducts : Prop | (has_binary_biproduct : Π (P Q : C), has_binary_biproduct P Q) | class | category_theory.limits.has_binary_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_binary_biproducts C` represents the existence of a bicone which is
simultaneously a limit and a colimit of the diagram `pair P Q`, for every `P Q : C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_biproducts_of_finite_biproducts [has_finite_biproducts C] :
has_binary_biproducts C | { has_binary_biproduct := λ P Q, has_binary_biproduct.mk
{ bicone := (biproduct.bicone (pair_function P Q)).to_binary_bicone,
is_bilimit := (bicone.to_binary_bicone_is_bilimit _).symm (biproduct.is_bilimit _) } } | lemma | category_theory.limits.has_binary_biproducts_of_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"
] | [] | A category with finite biproducts has binary biproducts.
This is not an instance as typically in concrete categories there will be
an alternative construction with nicer definitional properties. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_biproduct.has_limit_pair [has_binary_biproduct P Q] :
has_limit (pair P Q) | has_limit.mk ⟨_, binary_biproduct.is_limit P Q⟩ | instance | category_theory.limits.has_binary_biproduct.has_limit_pair | 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_binary_biproduct.has_colimit_pair [has_binary_biproduct P Q] :
has_colimit (pair P Q) | has_colimit.mk ⟨_, binary_biproduct.is_colimit P Q⟩ | instance | category_theory.limits.has_binary_biproduct.has_colimit_pair | 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_binary_products_of_has_binary_biproducts [has_binary_biproducts C] :
has_binary_products C | { has_limit := λ F, has_limit_of_iso (diagram_iso_pair F).symm } | instance | category_theory.limits.has_binary_products_of_has_binary_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_binary_coproducts_of_has_binary_biproducts [has_binary_biproducts C] :
has_binary_coproducts C | { has_colimit := λ F, has_colimit_of_iso (diagram_iso_pair F) } | instance | category_theory.limits.has_binary_coproducts_of_has_binary_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 | |
biprod_iso (X Y : C) [has_binary_biproduct X Y] :
limits.prod X Y ≅ limits.coprod X Y | (is_limit.cone_point_unique_up_to_iso (limit.is_limit _) (binary_biproduct.is_limit X Y)).trans $
is_colimit.cocone_point_unique_up_to_iso (binary_biproduct.is_colimit X Y) (colimit.is_colimit _) | def | category_theory.limits.biprod_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 binary product and the specified binary coproduct for
a pair for a binary biproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod (X Y : C) [has_binary_biproduct X Y] | (binary_biproduct.bicone X Y).X | abbreviation | category_theory.limits.biprod | 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 arbitrary choice of biproduct of a pair of objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.fst {X Y : C} [has_binary_biproduct X Y] : X ⊞ Y ⟶ X | (binary_biproduct.bicone X Y).fst | abbreviation | category_theory.limits.biprod.fst | 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 the first summand of a binary biproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.snd {X Y : C} [has_binary_biproduct X Y] : X ⊞ Y ⟶ Y | (binary_biproduct.bicone X Y).snd | abbreviation | category_theory.limits.biprod.snd | 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 the second summand of a binary biproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.inl {X Y : C} [has_binary_biproduct X Y] : X ⟶ X ⊞ Y | (binary_biproduct.bicone X Y).inl | abbreviation | category_theory.limits.biprod.inl | 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 the first summand of a binary biproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.inr {X Y : C} [has_binary_biproduct X Y] : Y ⟶ X ⊞ Y | (binary_biproduct.bicone X Y).inr | abbreviation | category_theory.limits.biprod.inr | 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 the second summand of a binary biproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_biproduct.bicone_fst : (binary_biproduct.bicone X Y).fst = biprod.fst | rfl | lemma | category_theory.limits.binary_biproduct.bicone_fst | 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 | |
binary_biproduct.bicone_snd : (binary_biproduct.bicone X Y).snd = biprod.snd | rfl | lemma | category_theory.limits.binary_biproduct.bicone_snd | 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 | |
binary_biproduct.bicone_inl : (binary_biproduct.bicone X Y).inl = biprod.inl | rfl | lemma | category_theory.limits.binary_biproduct.bicone_inl | 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 | |
binary_biproduct.bicone_inr : (binary_biproduct.bicone X Y).inr = biprod.inr | rfl | lemma | category_theory.limits.binary_biproduct.bicone_inr | 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 | |
biprod.inl_fst {X Y : C} [has_binary_biproduct X Y] :
(biprod.inl : X ⟶ X ⊞ Y) ≫ (biprod.fst : X ⊞ Y ⟶ X) = 𝟙 X | (binary_biproduct.bicone X Y).inl_fst | lemma | category_theory.limits.biprod.inl_fst | 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 | |
biprod.inl_snd {X Y : C} [has_binary_biproduct X Y] :
(biprod.inl : X ⟶ X ⊞ Y) ≫ (biprod.snd : X ⊞ Y ⟶ Y) = 0 | (binary_biproduct.bicone X Y).inl_snd | lemma | category_theory.limits.biprod.inl_snd | 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 | |
biprod.inr_fst {X Y : C} [has_binary_biproduct X Y] :
(biprod.inr : Y ⟶ X ⊞ Y) ≫ (biprod.fst : X ⊞ Y ⟶ X) = 0 | (binary_biproduct.bicone X Y).inr_fst | lemma | category_theory.limits.biprod.inr_fst | 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 | |
biprod.inr_snd {X Y : C} [has_binary_biproduct X Y] :
(biprod.inr : Y ⟶ X ⊞ Y) ≫ (biprod.snd : X ⊞ Y ⟶ Y) = 𝟙 Y | (binary_biproduct.bicone X Y).inr_snd | lemma | category_theory.limits.biprod.inr_snd | 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 | |
biprod.lift {W X Y : C} [has_binary_biproduct X Y] (f : W ⟶ X) (g : W ⟶ Y) :
W ⟶ X ⊞ Y | (binary_biproduct.is_limit X Y).lift (binary_fan.mk f g) | abbreviation | category_theory.limits.biprod.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 pair of maps into the summands of a binary biproduct,
we obtain a map into the binary biproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.desc {W X Y : C} [has_binary_biproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) :
X ⊞ Y ⟶ W | (binary_biproduct.is_colimit X Y).desc (binary_cofan.mk f g) | abbreviation | category_theory.limits.biprod.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 pair of maps out of the summands of a binary biproduct,
we obtain a map out of the binary biproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.lift_fst {W X Y : C} [has_binary_biproduct X Y] (f : W ⟶ X) (g : W ⟶ Y) :
biprod.lift f g ≫ biprod.fst = f | (binary_biproduct.is_limit X Y).fac _ ⟨walking_pair.left⟩ | lemma | category_theory.limits.biprod.lift_fst | 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 | |
biprod.lift_snd {W X Y : C} [has_binary_biproduct X Y] (f : W ⟶ X) (g : W ⟶ Y) :
biprod.lift f g ≫ biprod.snd = g | (binary_biproduct.is_limit X Y).fac _ ⟨walking_pair.right⟩ | lemma | category_theory.limits.biprod.lift_snd | 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 | |
biprod.inl_desc {W X Y : C} [has_binary_biproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) :
biprod.inl ≫ biprod.desc f g = f | (binary_biproduct.is_colimit X Y).fac _ ⟨walking_pair.left⟩ | lemma | category_theory.limits.biprod.inl_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 | |
biprod.inr_desc {W X Y : C} [has_binary_biproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) :
biprod.inr ≫ biprod.desc f g = g | (binary_biproduct.is_colimit X Y).fac _ ⟨walking_pair.right⟩ | lemma | category_theory.limits.biprod.inr_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 | |
biprod.mono_lift_of_mono_left {W X Y : C} [has_binary_biproduct X Y] (f : W ⟶ X)
(g : W ⟶ Y) [mono f] : mono (biprod.lift f g) | mono_of_mono_fac $ biprod.lift_fst _ _ | instance | category_theory.limits.biprod.mono_lift_of_mono_left | 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 | |
biprod.mono_lift_of_mono_right {W X Y : C} [has_binary_biproduct X Y] (f : W ⟶ X)
(g : W ⟶ Y) [mono g] : mono (biprod.lift f g) | mono_of_mono_fac $ biprod.lift_snd _ _ | instance | category_theory.limits.biprod.mono_lift_of_mono_right | 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 | |
biprod.epi_desc_of_epi_left {W X Y : C} [has_binary_biproduct X Y] (f : X ⟶ W) (g : Y ⟶ W)
[epi f] : epi (biprod.desc f g) | epi_of_epi_fac $ biprod.inl_desc _ _ | instance | category_theory.limits.biprod.epi_desc_of_epi_left | 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 | |
biprod.epi_desc_of_epi_right {W X Y : C} [has_binary_biproduct X Y] (f : X ⟶ W) (g : Y ⟶ W)
[epi g] : epi (biprod.desc f g) | epi_of_epi_fac $ biprod.inr_desc _ _ | instance | category_theory.limits.biprod.epi_desc_of_epi_right | 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 | |
biprod.map {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) : W ⊞ X ⟶ Y ⊞ Z | is_limit.map (binary_biproduct.bicone W X).to_cone (binary_biproduct.is_limit Y Z)
(@map_pair _ _ (pair W X) (pair Y Z) f g) | abbreviation | category_theory.limits.biprod.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 pair of maps between the summands of a pair of binary biproducts,
we obtain a map between the binary biproducts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.map' {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) : W ⊞ X ⟶ Y ⊞ Z | is_colimit.map (binary_biproduct.is_colimit W X) (binary_biproduct.bicone Y Z).to_cocone
(@map_pair _ _ (pair W X) (pair Y Z) f g) | abbreviation | category_theory.limits.biprod.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 `biprod.map` constructed via colimits.
This construction only exists in order to show it is equal to `biprod.map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.hom_ext {X Y Z : C} [has_binary_biproduct X Y] (f g : Z ⟶ X ⊞ Y)
(h₀ : f ≫ biprod.fst = g ≫ biprod.fst) (h₁ : f ≫ biprod.snd = g ≫ biprod.snd) : f = g | binary_fan.is_limit.hom_ext (binary_biproduct.is_limit X Y) h₀ h₁ | lemma | category_theory.limits.biprod.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biprod.hom_ext' {X Y Z : C} [has_binary_biproduct X Y] (f g : X ⊞ Y ⟶ Z)
(h₀ : biprod.inl ≫ f = biprod.inl ≫ g) (h₁ : biprod.inr ≫ f = biprod.inr ≫ g) : f = g | binary_cofan.is_colimit.hom_ext (binary_biproduct.is_colimit X Y) h₀ h₁ | lemma | category_theory.limits.biprod.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biprod.iso_prod (X Y : C) [has_binary_biproduct X Y] : X ⊞ Y ≅ X ⨯ Y | is_limit.cone_point_unique_up_to_iso (binary_biproduct.is_limit X Y) (limit.is_limit _) | def | category_theory.limits.biprod.iso_prod | 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 |
biprod.iso_prod_hom {X Y : C} [has_binary_biproduct X Y] :
(biprod.iso_prod X Y).hom = prod.lift biprod.fst biprod.snd | by ext; simp [biprod.iso_prod] | lemma | category_theory.limits.biprod.iso_prod_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 | |
biprod.iso_prod_inv {X Y : C} [has_binary_biproduct X Y] :
(biprod.iso_prod X Y).inv = biprod.lift prod.fst prod.snd | by apply biprod.hom_ext; simp [iso.inv_comp_eq] | lemma | category_theory.limits.biprod.iso_prod_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 | |
biprod.iso_coprod (X Y : C) [has_binary_biproduct X Y] : X ⊞ Y ≅ X ⨿ Y | is_colimit.cocone_point_unique_up_to_iso (binary_biproduct.is_colimit X Y) (colimit.is_colimit _) | def | category_theory.limits.biprod.iso_coprod | 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 |
biprod.iso_coprod_inv {X Y : C} [has_binary_biproduct X Y] :
(biprod.iso_coprod X Y).inv = coprod.desc biprod.inl biprod.inr | by ext; simp [biprod.iso_coprod]; refl | lemma | category_theory.limits.biprod.iso_coprod_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 | |
biprod_iso_coprod_hom {X Y : C} [has_binary_biproduct X Y] :
(biprod.iso_coprod X Y).hom = biprod.desc coprod.inl coprod.inr | by apply biprod.hom_ext'; simp [← iso.eq_comp_inv] | lemma | category_theory.limits.biprod_iso_coprod_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 | |
biprod.map_eq_map' {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) : biprod.map f g = biprod.map' f g | begin
ext,
{ simp only [map_pair_left, is_colimit.ι_map, is_limit.map_π, biprod.inl_fst_assoc,
category.assoc, ←binary_bicone.to_cone_π_app_left, ←binary_biproduct.bicone_fst,
←binary_bicone.to_cocone_ι_app_left, ←binary_biproduct.bicone_inl],
simp },
{ simp only [map_pair_left, is_colimit.ι_map, is_l... | lemma | category_theory.limits.biprod.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 | |
biprod.inl_mono {X Y : C} [has_binary_biproduct X Y] :
is_split_mono (biprod.inl : X ⟶ X ⊞ Y) | is_split_mono.mk' { retraction := biprod.fst } | instance | category_theory.limits.biprod.inl_mono | 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 | |
biprod.inr_mono {X Y : C} [has_binary_biproduct X Y] :
is_split_mono (biprod.inr : Y ⟶ X ⊞ Y) | is_split_mono.mk' { retraction := biprod.snd } | instance | category_theory.limits.biprod.inr_mono | 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 | |
biprod.fst_epi {X Y : C} [has_binary_biproduct X Y] :
is_split_epi (biprod.fst : X ⊞ Y ⟶ X) | is_split_epi.mk' { section_ := biprod.inl } | instance | category_theory.limits.biprod.fst_epi | 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 | |
biprod.snd_epi {X Y : C} [has_binary_biproduct X Y] :
is_split_epi (biprod.snd : X ⊞ Y ⟶ Y) | is_split_epi.mk' { section_ := biprod.inr } | instance | category_theory.limits.biprod.snd_epi | 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 | |
biprod.map_fst {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) :
biprod.map f g ≫ biprod.fst = biprod.fst ≫ f | is_limit.map_π _ _ _ (⟨walking_pair.left⟩ : discrete walking_pair) | lemma | category_theory.limits.biprod.map_fst | 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 | |
biprod.map_snd {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) :
biprod.map f g ≫ biprod.snd = biprod.snd ≫ g | is_limit.map_π _ _ _ (⟨walking_pair.right⟩ : discrete walking_pair) | lemma | category_theory.limits.biprod.map_snd | 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 | |
biprod.inl_map {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) :
biprod.inl ≫ biprod.map f g = f ≫ biprod.inl | begin
rw biprod.map_eq_map',
exact is_colimit.ι_map (binary_biproduct.is_colimit W X) _ _ ⟨walking_pair.left⟩
end | lemma | category_theory.limits.biprod.inl_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 | |
biprod.inr_map {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) :
biprod.inr ≫ biprod.map f g = g ≫ biprod.inr | begin
rw biprod.map_eq_map',
exact is_colimit.ι_map (binary_biproduct.is_colimit W X) _ _ ⟨walking_pair.right⟩
end | lemma | category_theory.limits.biprod.inr_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 | |
biprod.map_iso {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]
(f : W ≅ Y) (g : X ≅ Z) : W ⊞ X ≅ Y ⊞ Z | { hom := biprod.map f.hom g.hom,
inv := biprod.map f.inv g.inv } | def | category_theory.limits.biprod.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 pair of isomorphisms between the summands of a pair of binary biproducts,
we obtain an isomorphism between the binary biproducts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.cone_point_unique_up_to_iso_hom (X Y : C) [has_binary_biproduct X Y]
{b : binary_bicone X Y} (hb : b.is_bilimit) :
(hb.is_limit.cone_point_unique_up_to_iso (binary_biproduct.is_limit _ _)).hom
= biprod.lift b.fst b.snd | rfl | lemma | category_theory.limits.biprod.cone_point_unique_up_to_iso_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"
] | [] | Auxiliary lemma for `biprod.unique_up_to_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.cone_point_unique_up_to_iso_inv (X Y : C) [has_binary_biproduct X Y]
{b : binary_bicone X Y} (hb : b.is_bilimit) :
(hb.is_limit.cone_point_unique_up_to_iso (binary_biproduct.is_limit _ _)).inv
= biprod.desc b.inl b.inr | begin
refine biprod.hom_ext' _ _ (hb.is_limit.hom_ext (λ j, _)) (hb.is_limit.hom_ext (λ j, _)),
all_goals { simp only [category.assoc, is_limit.cone_point_unique_up_to_iso_inv_comp],
rcases j with ⟨⟨⟩⟩ },
all_goals { simp }
end | lemma | category_theory.limits.biprod.cone_point_unique_up_to_iso_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"
] | [] | Auxiliary lemma for `biprod.unique_up_to_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.unique_up_to_iso (X Y : C) [has_binary_biproduct X Y] {b : binary_bicone X Y}
(hb : b.is_bilimit) : b.X ≅ X ⊞ Y | { hom := biprod.lift b.fst b.snd,
inv := biprod.desc b.inl b.inr,
hom_inv_id' := by rw [← biprod.cone_point_unique_up_to_iso_hom X Y hb,
← biprod.cone_point_unique_up_to_iso_inv X Y hb, iso.hom_inv_id],
inv_hom_id' := by rw [← biprod.cone_point_unique_up_to_iso_hom X Y hb,
← biprod.cone_point_unique_up_to... | def | category_theory.limits.biprod.unique_up_to_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"
] | [] | Binary biproducts are unique up to isomorphism. This already follows because bilimits are
limits, but in the case of biproducts we can give an isomorphism with particularly nice
definitional properties, namely that `biprod.lift b.fst b.snd` and `biprod.desc b.inl b.inr`
are inverses of each other. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.is_iso_inl_iff_id_eq_fst_comp_inl (X Y : C) [has_binary_biproduct X Y] :
is_iso (biprod.inl : X ⟶ X ⊞ Y) ↔ 𝟙 (X ⊞ Y) = biprod.fst ≫ biprod.inl | begin
split,
{ introI h,
have := (cancel_epi (inv biprod.inl : X ⊞ Y ⟶ X)).2 biprod.inl_fst,
rw [is_iso.inv_hom_id_assoc, category.comp_id] at this,
rw [this, is_iso.inv_hom_id], },
{ intro h, exact ⟨⟨biprod.fst, biprod.inl_fst, h.symm⟩⟩, },
end | lemma | category_theory.limits.biprod.is_iso_inl_iff_id_eq_fst_comp_inl | 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 | |
binary_bicone.fst_kernel_fork : kernel_fork c.fst | kernel_fork.of_ι c.inr c.inr_fst | def | category_theory.limits.binary_bicone.fst_kernel_fork | 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 kernel fork for the kernel of `binary_bicone.fst`. It consists of the morphism
`binary_bicone.inr`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_bicone.fst_kernel_fork_ι : (binary_bicone.fst_kernel_fork c).ι = c.inr | rfl | lemma | category_theory.limits.binary_bicone.fst_kernel_fork_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 | |
binary_bicone.snd_kernel_fork : kernel_fork c.snd | kernel_fork.of_ι c.inl c.inl_snd | def | category_theory.limits.binary_bicone.snd_kernel_fork | 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 kernel fork for the kernel of `binary_bicone.snd`. It consists of the morphism
`binary_bicone.inl`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_bicone.snd_kernel_fork_ι : (binary_bicone.snd_kernel_fork c).ι = c.inl | rfl | lemma | category_theory.limits.binary_bicone.snd_kernel_fork_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 | |
binary_bicone.inl_cokernel_cofork : cokernel_cofork c.inl | cokernel_cofork.of_π c.snd c.inl_snd | def | category_theory.limits.binary_bicone.inl_cokernel_cofork | 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 cokernel cofork for the cokernel of `binary_bicone.inl`. It consists of the morphism
`binary_bicone.snd`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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