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coprod.inl {X Y : C} [has_binary_coproduct X Y] : X ⟶ X ⨿ Y | colimit.ι (pair X Y) ⟨walking_pair.left⟩ | abbreviation | category_theory.limits.coprod.inl | 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 inclusion map from the first component of the coproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod.inr {X Y : C} [has_binary_coproduct X Y] : Y ⟶ X ⨿ Y | colimit.ι (pair X Y) ⟨walking_pair.right⟩ | abbreviation | category_theory.limits.coprod.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"
] | [] | The inclusion map from the second component of the coproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_is_prod (X Y : C) [has_binary_product X Y] :
is_limit (binary_fan.mk (prod.fst : X ⨯ Y ⟶ X) prod.snd) | (limit.is_limit _).of_iso_limit (cones.ext (iso.refl _) (by { rintro (_ | _), tidy })) | def | category_theory.limits.prod_is_prod | 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 binary fan constructed from the projection maps is a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod_is_coprod (X Y : C) [has_binary_coproduct X Y] :
is_colimit (binary_cofan.mk (coprod.inl : X ⟶ X ⨿ Y) coprod.inr) | (colimit.is_colimit _).of_iso_colimit (cocones.ext (iso.refl _) (by { rintro (_ | _), tidy })) | def | category_theory.limits.coprod_is_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"
] | [] | The binary cofan constructed from the coprojection maps is a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.hom_ext {W X Y : C} [has_binary_product X Y] {f g : W ⟶ X ⨯ Y}
(h₁ : f ≫ prod.fst = g ≫ prod.fst) (h₂ : f ≫ prod.snd = g ≫ prod.snd) : f = g | binary_fan.is_limit.hom_ext (limit.is_limit _) h₁ h₂ | lemma | category_theory.limits.prod.hom_ext | 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.hom_ext {W X Y : C} [has_binary_coproduct X Y] {f g : X ⨿ Y ⟶ W}
(h₁ : coprod.inl ≫ f = coprod.inl ≫ g) (h₂ : coprod.inr ≫ f = coprod.inr ≫ g) : f = g | binary_cofan.is_colimit.hom_ext (colimit.is_colimit _) h₁ h₂ | lemma | category_theory.limits.coprod.hom_ext | 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 | |
prod.lift {W X Y : C} [has_binary_product X Y] (f : W ⟶ X) (g : W ⟶ Y) : W ⟶ X ⨯ Y | limit.lift _ (binary_fan.mk f g) | abbreviation | category_theory.limits.prod.lift | 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 product of `X` and `Y` exists, then every pair of morphisms `f : W ⟶ X` and `g : W ⟶ Y`
induces a morphism `prod.lift f g : W ⟶ X ⨯ Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diag (X : C) [has_binary_product X X] : X ⟶ X ⨯ X | prod.lift (𝟙 _) (𝟙 _) | abbreviation | category_theory.limits.diag | 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"
] | [] | diagonal arrow of the binary product in the category `fam I` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod.desc {W X Y : C} [has_binary_coproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) :
X ⨿ Y ⟶ W | colimit.desc _ (binary_cofan.mk f g) | abbreviation | category_theory.limits.coprod.desc | 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 of `X` and `Y` exists, then every pair of morphisms `f : X ⟶ W` and
`g : Y ⟶ W` induces a morphism `coprod.desc f g : X ⨿ Y ⟶ W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
codiag (X : C) [has_binary_coproduct X X] : X ⨿ X ⟶ X | coprod.desc (𝟙 _) (𝟙 _) | abbreviation | category_theory.limits.codiag | 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"
] | [] | codiagonal arrow of the binary coproduct | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.lift_fst {W X Y : C} [has_binary_product X Y] (f : W ⟶ X) (g : W ⟶ Y) :
prod.lift f g ≫ prod.fst = f | limit.lift_π _ _ | lemma | category_theory.limits.prod.lift_fst | 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 | |
prod.lift_snd {W X Y : C} [has_binary_product X Y] (f : W ⟶ X) (g : W ⟶ Y) :
prod.lift f g ≫ prod.snd = g | limit.lift_π _ _ | lemma | category_theory.limits.prod.lift_snd | 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.inl_desc {W X Y : C} [has_binary_coproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) :
coprod.inl ≫ coprod.desc f g = f | colimit.ι_desc _ _ | lemma | category_theory.limits.coprod.inl_desc | 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.inr_desc {W X Y : C} [has_binary_coproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) :
coprod.inr ≫ coprod.desc f g = g | colimit.ι_desc _ _ | lemma | category_theory.limits.coprod.inr_desc | 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 | |
prod.mono_lift_of_mono_left {W X Y : C} [has_binary_product X Y] (f : W ⟶ X) (g : W ⟶ Y)
[mono f] : mono (prod.lift f g) | mono_of_mono_fac $ prod.lift_fst _ _ | instance | category_theory.limits.prod.mono_lift_of_mono_left | 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 | |
prod.mono_lift_of_mono_right {W X Y : C} [has_binary_product X Y] (f : W ⟶ X) (g : W ⟶ Y)
[mono g] : mono (prod.lift f g) | mono_of_mono_fac $ prod.lift_snd _ _ | instance | category_theory.limits.prod.mono_lift_of_mono_right | 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.epi_desc_of_epi_left {W X Y : C} [has_binary_coproduct X Y] (f : X ⟶ W) (g : Y ⟶ W)
[epi f] : epi (coprod.desc f g) | epi_of_epi_fac $ coprod.inl_desc _ _ | instance | category_theory.limits.coprod.epi_desc_of_epi_left | 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.epi_desc_of_epi_right {W X Y : C} [has_binary_coproduct X Y] (f : X ⟶ W) (g : Y ⟶ W)
[epi g] : epi (coprod.desc f g) | epi_of_epi_fac $ coprod.inr_desc _ _ | instance | category_theory.limits.coprod.epi_desc_of_epi_right | 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 | |
prod.lift' {W X Y : C} [has_binary_product X Y] (f : W ⟶ X) (g : W ⟶ Y) :
{l : W ⟶ X ⨯ Y // l ≫ prod.fst = f ∧ l ≫ prod.snd = g} | ⟨prod.lift f g, prod.lift_fst _ _, prod.lift_snd _ _⟩ | def | category_theory.limits.prod.lift' | 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 product of `X` and `Y` exists, then every pair of morphisms `f : W ⟶ X` and `g : W ⟶ Y`
induces a morphism `l : W ⟶ X ⨯ Y` satisfying `l ≫ prod.fst = f` and `l ≫ prod.snd = g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod.desc' {W X Y : C} [has_binary_coproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) :
{l : X ⨿ Y ⟶ W // coprod.inl ≫ l = f ∧ coprod.inr ≫ l = g} | ⟨coprod.desc f g, coprod.inl_desc _ _, coprod.inr_desc _ _⟩ | def | category_theory.limits.coprod.desc' | 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 of `X` and `Y` exists, then every pair of morphisms `f : X ⟶ W` and
`g : Y ⟶ W` induces a morphism `l : X ⨿ Y ⟶ W` satisfying `coprod.inl ≫ l = f` and
`coprod.inr ≫ l = g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.map {W X Y Z : C} [has_binary_product W X] [has_binary_product Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) : W ⨯ X ⟶ Y ⨯ Z | lim_map (map_pair f g) | def | category_theory.limits.prod.map | 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 products `W ⨯ X` and `Y ⨯ Z` exist, then every pair of morphisms `f : W ⟶ Y` and
`g : X ⟶ Z` induces a morphism `prod.map f g : W ⨯ X ⟶ Y ⨯ Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod.map {W X Y Z : C} [has_binary_coproduct W X] [has_binary_coproduct Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) : W ⨿ X ⟶ Y ⨿ Z | colim_map (map_pair f g) | def | category_theory.limits.coprod.map | 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 coproducts `W ⨿ X` and `Y ⨿ Z` exist, then every pair of morphisms `f : W ⟶ Y` and
`g : W ⟶ Z` induces a morphism `coprod.map f g : W ⨿ X ⟶ Y ⨿ Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.comp_lift {V W X Y : C} [has_binary_product X Y] (f : V ⟶ W) (g : W ⟶ X) (h : W ⟶ Y) :
f ≫ prod.lift g h = prod.lift (f ≫ g) (f ≫ h) | by { ext; simp } | lemma | category_theory.limits.prod.comp_lift | 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 | |
prod.comp_diag {X Y : C} [has_binary_product Y Y] (f : X ⟶ Y) :
f ≫ diag Y = prod.lift f f | by simp | lemma | category_theory.limits.prod.comp_diag | 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 | |
prod.map_fst {W X Y Z : C} [has_binary_product W X] [has_binary_product Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) : prod.map f g ≫ prod.fst = prod.fst ≫ f | lim_map_π _ _ | lemma | category_theory.limits.prod.map_fst | 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"
] | [
"prod.map_fst"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.map_snd {W X Y Z : C} [has_binary_product W X] [has_binary_product Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) : prod.map f g ≫ prod.snd = prod.snd ≫ g | lim_map_π _ _ | lemma | category_theory.limits.prod.map_snd | 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"
] | [
"prod.map_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.map_id_id {X Y : C} [has_binary_product X Y] :
prod.map (𝟙 X) (𝟙 Y) = 𝟙 _ | by { ext; simp } | lemma | category_theory.limits.prod.map_id_id | 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 | |
prod.lift_fst_snd {X Y : C} [has_binary_product X Y] :
prod.lift prod.fst prod.snd = 𝟙 (X ⨯ Y) | by { ext; simp } | lemma | category_theory.limits.prod.lift_fst_snd | 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 | |
prod.lift_map {V W X Y Z : C} [has_binary_product W X]
[has_binary_product Y Z] (f : V ⟶ W) (g : V ⟶ X) (h : W ⟶ Y) (k : X ⟶ Z) :
prod.lift f g ≫ prod.map h k = prod.lift (f ≫ h) (g ≫ k) | by { ext; simp } | lemma | category_theory.limits.prod.lift_map | 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 | |
prod.lift_fst_comp_snd_comp {W X Y Z : C} [has_binary_product W Y]
[has_binary_product X Z] (g : W ⟶ X) (g' : Y ⟶ Z) :
prod.lift (prod.fst ≫ g) (prod.snd ≫ g') = prod.map g g' | by { rw ← prod.lift_map, simp } | lemma | category_theory.limits.prod.lift_fst_comp_snd_comp | 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 | |
prod.map_map {A₁ A₂ A₃ B₁ B₂ B₃ : C}
[has_binary_product A₁ B₁] [has_binary_product A₂ B₂] [has_binary_product A₃ B₃]
(f : A₁ ⟶ A₂) (g : B₁ ⟶ B₂) (h : A₂ ⟶ A₃) (k : B₂ ⟶ B₃) :
prod.map f g ≫ prod.map h k = prod.map (f ≫ h) (g ≫ k) | by { ext; simp } | lemma | category_theory.limits.prod.map_map | 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"
] | [
"prod.map_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.map_swap {A B X Y : C} (f : A ⟶ B) (g : X ⟶ Y)
[has_limits_of_shape (discrete walking_pair) C] :
prod.map (𝟙 X) f ≫ prod.map g (𝟙 B) = prod.map g (𝟙 A) ≫ prod.map (𝟙 Y) f | by simp | lemma | category_theory.limits.prod.map_swap | 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 | |
prod.map_comp_id {X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z)
[has_binary_product X W] [has_binary_product Z W] [has_binary_product Y W] :
prod.map (f ≫ g) (𝟙 W) = prod.map f (𝟙 W) ≫ prod.map g (𝟙 W) | by simp | lemma | category_theory.limits.prod.map_comp_id | 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 | |
prod.map_id_comp {X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z)
[has_binary_product W X] [has_binary_product W Y] [has_binary_product W Z] :
prod.map (𝟙 W) (f ≫ g) = prod.map (𝟙 W) f ≫ prod.map (𝟙 W) g | by simp | lemma | category_theory.limits.prod.map_id_comp | 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 | |
prod.map_iso {W X Y Z : C} [has_binary_product W X] [has_binary_product Y Z]
(f : W ≅ Y) (g : X ≅ Z) : W ⨯ X ≅ Y ⨯ Z | { hom := prod.map f.hom g.hom,
inv := prod.map f.inv g.inv } | def | category_theory.limits.prod.map_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"
] | [] | If the products `W ⨯ X` and `Y ⨯ Z` exist, then every pair of isomorphisms `f : W ≅ Y` and
`g : X ≅ Z` induces an isomorphism `prod.map_iso f g : W ⨯ X ≅ Y ⨯ Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_prod {W X Y Z : C} [has_binary_product W X] [has_binary_product Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) [is_iso f] [is_iso g] : is_iso (prod.map f g) | is_iso.of_iso (prod.map_iso (as_iso f) (as_iso g)) | instance | category_theory.limits.is_iso_prod | 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 | |
prod.map_mono {C : Type*} [category C] {W X Y Z : C} (f : W ⟶ Y) (g : X ⟶ Z) [mono f]
[mono g] [has_binary_product W X] [has_binary_product Y Z] : mono (prod.map f g) | ⟨λ A i₁ i₂ h, begin
ext,
{ rw ← cancel_mono f, simpa using congr_arg (λ f, f ≫ prod.fst) h },
{ rw ← cancel_mono g, simpa using congr_arg (λ f, f ≫ prod.snd) h }
end⟩ | instance | category_theory.limits.prod.map_mono | 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 | |
prod.diag_map {X Y : C} (f : X ⟶ Y) [has_binary_product X X] [has_binary_product Y Y] :
diag X ≫ prod.map f f = f ≫ diag Y | by simp | lemma | category_theory.limits.prod.diag_map | 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 | |
prod.diag_map_fst_snd {X Y : C} [has_binary_product X Y]
[has_binary_product (X ⨯ Y) (X ⨯ Y)] :
diag (X ⨯ Y) ≫ prod.map prod.fst prod.snd = 𝟙 (X ⨯ Y) | by simp | lemma | category_theory.limits.prod.diag_map_fst_snd | 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 | |
prod.diag_map_fst_snd_comp [has_limits_of_shape (discrete walking_pair) C]
{X X' Y Y' : C} (g : X ⟶ Y) (g' : X' ⟶ Y') :
diag (X ⨯ X') ≫ prod.map (prod.fst ≫ g) (prod.snd ≫ g') = prod.map g g' | by simp | lemma | category_theory.limits.prod.diag_map_fst_snd_comp | 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.desc_comp {V W X Y : C} [has_binary_coproduct X Y] (f : V ⟶ W) (g : X ⟶ V)
(h : Y ⟶ V) :
coprod.desc g h ≫ f = coprod.desc (g ≫ f) (h ≫ f) | by { ext; simp } | lemma | category_theory.limits.coprod.desc_comp | 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.diag_comp {X Y : C} [has_binary_coproduct X X] (f : X ⟶ Y) :
codiag X ≫ f = coprod.desc f f | by simp | lemma | category_theory.limits.coprod.diag_comp | 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.inl_map {W X Y Z : C} [has_binary_coproduct W X] [has_binary_coproduct Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) : coprod.inl ≫ coprod.map f g = f ≫ coprod.inl | ι_colim_map _ _ | lemma | category_theory.limits.coprod.inl_map | 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.inr_map {W X Y Z : C} [has_binary_coproduct W X] [has_binary_coproduct Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) : coprod.inr ≫ coprod.map f g = g ≫ coprod.inr | ι_colim_map _ _ | lemma | category_theory.limits.coprod.inr_map | 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.map_id_id {X Y : C} [has_binary_coproduct X Y] :
coprod.map (𝟙 X) (𝟙 Y) = 𝟙 _ | by { ext; simp } | lemma | category_theory.limits.coprod.map_id_id | 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.desc_inl_inr {X Y : C} [has_binary_coproduct X Y] :
coprod.desc coprod.inl coprod.inr = 𝟙 (X ⨿ Y) | by { ext; simp } | lemma | category_theory.limits.coprod.desc_inl_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.map_desc {S T U V W : C} [has_binary_coproduct U W] [has_binary_coproduct T V]
(f : U ⟶ S) (g : W ⟶ S) (h : T ⟶ U) (k : V ⟶ W) :
coprod.map h k ≫ coprod.desc f g = coprod.desc (h ≫ f) (k ≫ g) | by { ext; simp } | lemma | category_theory.limits.coprod.map_desc | 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.desc_comp_inl_comp_inr {W X Y Z : C}
[has_binary_coproduct W Y] [has_binary_coproduct X Z]
(g : W ⟶ X) (g' : Y ⟶ Z) :
coprod.desc (g ≫ coprod.inl) (g' ≫ coprod.inr) = coprod.map g g' | by { rw ← coprod.map_desc, simp } | lemma | category_theory.limits.coprod.desc_comp_inl_comp_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.map_map {A₁ A₂ A₃ B₁ B₂ B₃ : C}
[has_binary_coproduct A₁ B₁] [has_binary_coproduct A₂ B₂] [has_binary_coproduct A₃ B₃]
(f : A₁ ⟶ A₂) (g : B₁ ⟶ B₂) (h : A₂ ⟶ A₃) (k : B₂ ⟶ B₃) :
coprod.map f g ≫ coprod.map h k = coprod.map (f ≫ h) (g ≫ k) | by { ext; simp } | lemma | category_theory.limits.coprod.map_map | 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.map_swap {A B X Y : C} (f : A ⟶ B) (g : X ⟶ Y)
[has_colimits_of_shape (discrete walking_pair) C] :
coprod.map (𝟙 X) f ≫ coprod.map g (𝟙 B) = coprod.map g (𝟙 A) ≫ coprod.map (𝟙 Y) f | by simp | lemma | category_theory.limits.coprod.map_swap | 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.map_comp_id {X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z)
[has_binary_coproduct Z W] [has_binary_coproduct Y W] [has_binary_coproduct X W] :
coprod.map (f ≫ g) (𝟙 W) = coprod.map f (𝟙 W) ≫ coprod.map g (𝟙 W) | by simp | lemma | category_theory.limits.coprod.map_comp_id | 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.map_id_comp {X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z)
[has_binary_coproduct W X] [has_binary_coproduct W Y] [has_binary_coproduct W Z] :
coprod.map (𝟙 W) (f ≫ g) = coprod.map (𝟙 W) f ≫ coprod.map (𝟙 W) g | by simp | lemma | category_theory.limits.coprod.map_id_comp | 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.map_iso {W X Y Z : C} [has_binary_coproduct W X] [has_binary_coproduct Y Z]
(f : W ≅ Y) (g : X ≅ Z) : W ⨿ X ≅ Y ⨿ Z | { hom := coprod.map f.hom g.hom,
inv := coprod.map f.inv g.inv } | def | category_theory.limits.coprod.map_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"
] | [] | If the coproducts `W ⨿ X` and `Y ⨿ Z` exist, then every pair of isomorphisms `f : W ≅ Y` and
`g : W ≅ Z` induces a isomorphism `coprod.map_iso f g : W ⨿ X ≅ Y ⨿ Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_coprod {W X Y Z : C} [has_binary_coproduct W X] [has_binary_coproduct Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) [is_iso f] [is_iso g] : is_iso (coprod.map f g) | is_iso.of_iso (coprod.map_iso (as_iso f) (as_iso g)) | instance | category_theory.limits.is_iso_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod.map_epi {C : Type*} [category C] {W X Y Z : C} (f : W ⟶ Y) (g : X ⟶ Z) [epi f]
[epi g] [has_binary_coproduct W X] [has_binary_coproduct Y Z] : epi (coprod.map f g) | ⟨λ A i₁ i₂ h, begin
ext,
{ rw ← cancel_epi f, simpa using congr_arg (λ f, coprod.inl ≫ f) h },
{ rw ← cancel_epi g, simpa using congr_arg (λ f, coprod.inr ≫ f) h }
end⟩ | instance | category_theory.limits.coprod.map_epi | 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.map_codiag {X Y : C} (f : X ⟶ Y) [has_binary_coproduct X X]
[has_binary_coproduct Y Y] :
coprod.map f f ≫ codiag Y = codiag X ≫ f | by simp | lemma | category_theory.limits.coprod.map_codiag | 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.map_inl_inr_codiag {X Y : C} [has_binary_coproduct X Y]
[has_binary_coproduct (X ⨿ Y) (X ⨿ Y)] :
coprod.map coprod.inl coprod.inr ≫ codiag (X ⨿ Y) = 𝟙 (X ⨿ Y) | by simp | lemma | category_theory.limits.coprod.map_inl_inr_codiag | 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.map_comp_inl_inr_codiag [has_colimits_of_shape (discrete walking_pair) C]
{X X' Y Y' : C} (g : X ⟶ Y) (g' : X' ⟶ Y') :
coprod.map (g ≫ coprod.inl) (g' ≫ coprod.inr) ≫ codiag (Y ⨿ Y') = coprod.map g g' | by simp | lemma | category_theory.limits.coprod.map_comp_inl_inr_codiag | 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 | |
has_binary_products | has_limits_of_shape (discrete walking_pair) C | abbreviation | category_theory.limits.has_binary_products | 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"
] | [] | `has_binary_products` represents a choice of product for every pair of objects.
See <https://stacks.math.columbia.edu/tag/001T>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_coproducts | has_colimits_of_shape (discrete walking_pair) C | abbreviation | category_theory.limits.has_binary_coproducts | 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"
] | [] | `has_binary_coproducts` represents a choice of coproduct for every pair of objects.
See <https://stacks.math.columbia.edu/tag/04AP>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_products_of_has_limit_pair [Π {X Y : C}, has_limit (pair X Y)] :
has_binary_products C | { has_limit := λ F, has_limit_of_iso (diagram_iso_pair F).symm } | lemma | category_theory.limits.has_binary_products_of_has_limit_pair | 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 `C` has all limits of diagrams `pair X Y`, then it has all binary products | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_coproducts_of_has_colimit_pair [Π {X Y : C}, has_colimit (pair X Y)] :
has_binary_coproducts C | { has_colimit := λ F, has_colimit_of_iso (diagram_iso_pair F) } | lemma | category_theory.limits.has_binary_coproducts_of_has_colimit_pair | 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 `C` has all colimits of diagrams `pair X Y`, then it has all binary coproducts | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.braiding (P Q : C) [has_binary_product P Q] [has_binary_product Q P] :
P ⨯ Q ≅ Q ⨯ P | { hom := prod.lift prod.snd prod.fst,
inv := prod.lift prod.snd prod.fst } | def | category_theory.limits.prod.braiding | 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 braiding isomorphism which swaps a binary product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
braid_natural [has_binary_products C] {W X Y Z : C} (f : X ⟶ Y) (g : Z ⟶ W) :
prod.map f g ≫ (prod.braiding _ _).hom = (prod.braiding _ _).hom ≫ prod.map g f | by simp | lemma | category_theory.limits.braid_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"
] | [] | The braiding isomorphism can be passed through a map by swapping the order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.symmetry' (P Q : C) [has_binary_product P Q] [has_binary_product Q P] :
prod.lift prod.snd prod.fst ≫ prod.lift prod.snd prod.fst = 𝟙 (P ⨯ Q) | (prod.braiding _ _).hom_inv_id | lemma | category_theory.limits.prod.symmetry' | 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 | |
prod.symmetry (P Q : C) [has_binary_product P Q] [has_binary_product Q P] :
(prod.braiding P Q).hom ≫ (prod.braiding Q P).hom = 𝟙 _ | (prod.braiding _ _).hom_inv_id | lemma | category_theory.limits.prod.symmetry | 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 braiding isomorphism is symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.associator [has_binary_products C] (P Q R : C) :
(P ⨯ Q) ⨯ R ≅ P ⨯ (Q ⨯ R) | { hom :=
prod.lift
(prod.fst ≫ prod.fst)
(prod.lift (prod.fst ≫ prod.snd) prod.snd),
inv :=
prod.lift
(prod.lift prod.fst (prod.snd ≫ prod.fst))
(prod.snd ≫ prod.snd) } | def | category_theory.limits.prod.associator | 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 associator isomorphism for binary products. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.pentagon [has_binary_products C] (W X Y Z : C) :
prod.map ((prod.associator W X Y).hom) (𝟙 Z) ≫
(prod.associator W (X ⨯ Y) Z).hom ≫ prod.map (𝟙 W) ((prod.associator X Y Z).hom) =
(prod.associator (W ⨯ X) Y Z).hom ≫ (prod.associator W X (Y ⨯ Z)).hom | by simp | lemma | category_theory.limits.prod.pentagon | 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 | |
prod.associator_naturality [has_binary_products C] {X₁ X₂ X₃ Y₁ Y₂ Y₃ : C}
(f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (f₃ : X₃ ⟶ Y₃) :
prod.map (prod.map f₁ f₂) f₃ ≫ (prod.associator Y₁ Y₂ Y₃).hom =
(prod.associator X₁ X₂ X₃).hom ≫ prod.map f₁ (prod.map f₂ f₃) | by simp | lemma | category_theory.limits.prod.associator_naturality | 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 | |
prod.left_unitor (P : C) [has_binary_product (⊤_ C) P] :
⊤_ C ⨯ P ≅ P | { hom := prod.snd,
inv := prod.lift (terminal.from P) (𝟙 _) } | def | category_theory.limits.prod.left_unitor | 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 left unitor isomorphism for binary products with the terminal object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.right_unitor (P : C) [has_binary_product P (⊤_ C)] :
P ⨯ ⊤_ C ≅ P | { hom := prod.fst,
inv := prod.lift (𝟙 _) (terminal.from P) } | def | category_theory.limits.prod.right_unitor | 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 right unitor isomorphism for binary products with the terminal object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.left_unitor_hom_naturality [has_binary_products C] (f : X ⟶ Y) :
prod.map (𝟙 _) f ≫ (prod.left_unitor Y).hom = (prod.left_unitor X).hom ≫ f | prod.map_snd _ _ | lemma | category_theory.limits.prod.left_unitor_hom_naturality | 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"
] | [
"prod.map_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.left_unitor_inv_naturality [has_binary_products C] (f : X ⟶ Y) :
(prod.left_unitor X).inv ≫ prod.map (𝟙 _) f = f ≫ (prod.left_unitor Y).inv | by rw [iso.inv_comp_eq, ← category.assoc, iso.eq_comp_inv, prod.left_unitor_hom_naturality] | lemma | category_theory.limits.prod.left_unitor_inv_naturality | 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 | |
prod.right_unitor_hom_naturality [has_binary_products C] (f : X ⟶ Y) :
prod.map f (𝟙 _) ≫ (prod.right_unitor Y).hom = (prod.right_unitor X).hom ≫ f | prod.map_fst _ _ | lemma | category_theory.limits.prod.right_unitor_hom_naturality | 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"
] | [
"prod.map_fst"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_right_unitor_inv_naturality [has_binary_products C] (f : X ⟶ Y) :
(prod.right_unitor X).inv ≫ prod.map f (𝟙 _) = f ≫ (prod.right_unitor Y).inv | by rw [iso.inv_comp_eq, ← category.assoc, iso.eq_comp_inv, prod.right_unitor_hom_naturality] | lemma | category_theory.limits.prod_right_unitor_inv_naturality | 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 | |
prod.triangle [has_binary_products C] (X Y : C) :
(prod.associator X (⊤_ C) Y).hom ≫ prod.map (𝟙 X) ((prod.left_unitor Y).hom) =
prod.map ((prod.right_unitor X).hom) (𝟙 Y) | by tidy | lemma | category_theory.limits.prod.triangle | 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.braiding (P Q : C) : P ⨿ Q ≅ Q ⨿ P | { hom := coprod.desc coprod.inr coprod.inl,
inv := coprod.desc coprod.inr coprod.inl } | def | category_theory.limits.coprod.braiding | 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 braiding isomorphism which swaps a binary coproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod.symmetry' (P Q : C) :
coprod.desc coprod.inr coprod.inl ≫ coprod.desc coprod.inr coprod.inl = 𝟙 (P ⨿ Q) | (coprod.braiding _ _).hom_inv_id | lemma | category_theory.limits.coprod.symmetry' | 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.symmetry (P Q : C) :
(coprod.braiding P Q).hom ≫ (coprod.braiding Q P).hom = 𝟙 _ | coprod.symmetry' _ _ | lemma | category_theory.limits.coprod.symmetry | 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 braiding isomorphism is symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod.associator
(P Q R : C) : (P ⨿ Q) ⨿ R ≅ P ⨿ (Q ⨿ R) | { hom :=
coprod.desc
(coprod.desc coprod.inl (coprod.inl ≫ coprod.inr))
(coprod.inr ≫ coprod.inr),
inv :=
coprod.desc
(coprod.inl ≫ coprod.inl)
(coprod.desc (coprod.inr ≫ coprod.inl) coprod.inr) } | def | category_theory.limits.coprod.associator | 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 associator isomorphism for binary coproducts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod.pentagon (W X Y Z : C) :
coprod.map ((coprod.associator W X Y).hom) (𝟙 Z) ≫
(coprod.associator W (X ⨿ Y) Z).hom ≫ coprod.map (𝟙 W) ((coprod.associator X Y Z).hom) =
(coprod.associator (W ⨿ X) Y Z).hom ≫ (coprod.associator W X (Y ⨿ Z)).hom | by simp | lemma | category_theory.limits.coprod.pentagon | 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.associator_naturality {X₁ X₂ X₃ Y₁ Y₂ Y₃ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂)
(f₃ : X₃ ⟶ Y₃) :
coprod.map (coprod.map f₁ f₂) f₃ ≫ (coprod.associator Y₁ Y₂ Y₃).hom =
(coprod.associator X₁ X₂ X₃).hom ≫ coprod.map f₁ (coprod.map f₂ f₃) | by simp | lemma | category_theory.limits.coprod.associator_naturality | 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.left_unitor
(P : C) : ⊥_ C ⨿ P ≅ P | { hom := coprod.desc (initial.to P) (𝟙 _),
inv := coprod.inr } | def | category_theory.limits.coprod.left_unitor | 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 left unitor isomorphism for binary coproducts with the initial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod.right_unitor
(P : C) : P ⨿ ⊥_ C ≅ P | { hom := coprod.desc (𝟙 _) (initial.to P),
inv := coprod.inl } | def | category_theory.limits.coprod.right_unitor | 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 right unitor isomorphism for binary coproducts with the initial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod.triangle (X Y : C) :
(coprod.associator X (⊥_ C) Y).hom ≫ coprod.map (𝟙 X) ((coprod.left_unitor Y).hom) =
coprod.map ((coprod.right_unitor X).hom) (𝟙 Y) | by tidy | lemma | category_theory.limits.coprod.triangle | 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 | |
prod.functor : C ⥤ C ⥤ C | { obj := λ X, { obj := λ Y, X ⨯ Y, map := λ Y Z, prod.map (𝟙 X) },
map := λ Y Z f, { app := λ T, prod.map f (𝟙 T) }} | def | category_theory.limits.prod.functor | 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 binary product functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.functor_left_comp (X Y : C) :
prod.functor.obj (X ⨯ Y) ≅ prod.functor.obj Y ⋙ prod.functor.obj X | nat_iso.of_components (prod.associator _ _) (by tidy) | def | category_theory.limits.prod.functor_left_comp | 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 product functor can be decomposed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod.functor : C ⥤ C ⥤ C | { obj := λ X, { obj := λ Y, X ⨿ Y, map := λ Y Z, coprod.map (𝟙 X) },
map := λ Y Z f, { app := λ T, coprod.map f (𝟙 T) }} | def | category_theory.limits.coprod.functor | 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 binary coproduct functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod.functor_left_comp (X Y : C) :
coprod.functor.obj (X ⨿ Y) ≅ coprod.functor.obj Y ⋙ coprod.functor.obj X | nat_iso.of_components (coprod.associator _ _) (by tidy) | def | category_theory.limits.coprod.functor_left_comp | 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 functor can be decomposed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_comparison (F : C ⥤ D) (A B : C)
[has_binary_product A B] [has_binary_product (F.obj A) (F.obj B)] :
F.obj (A ⨯ B) ⟶ F.obj A ⨯ F.obj B | prod.lift (F.map prod.fst) (F.map prod.snd) | def | category_theory.limits.prod_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"
] | [] | The product comparison morphism.
In `category_theory/limits/preserves` we show this is always an iso iff F preserves binary products. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_comparison_fst :
prod_comparison F A B ≫ prod.fst = F.map prod.fst | prod.lift_fst _ _ | lemma | category_theory.limits.prod_comparison_fst | 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 | |
prod_comparison_snd :
prod_comparison F A B ≫ prod.snd = F.map prod.snd | prod.lift_snd _ _ | lemma | category_theory.limits.prod_comparison_snd | 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 | |
prod_comparison_natural (f : A ⟶ A') (g : B ⟶ B') :
F.map (prod.map f g) ≫ prod_comparison F A' B' =
prod_comparison F A B ≫ prod.map (F.map f) (F.map g) | begin
rw [prod_comparison, prod_comparison, prod.lift_map, ← F.map_comp, ← F.map_comp,
prod.comp_lift, ← F.map_comp, prod.map_fst, ← F.map_comp, prod.map_snd]
end | lemma | category_theory.limits.prod_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"
] | [
"prod.map_fst",
"prod.map_snd"
] | Naturality of the prod_comparison morphism in both arguments. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_comparison_nat_trans [has_binary_products C] [has_binary_products D]
(F : C ⥤ D) (A : C) :
prod.functor.obj A ⋙ F ⟶ F ⋙ prod.functor.obj (F.obj A) | { app := λ B, prod_comparison F A B,
naturality' := λ B B' f, by simp [prod_comparison_natural] } | def | category_theory.limits.prod_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 product comparison morphism from `F(A ⨯ -)` to `FA ⨯ F-`, whose components are given by
`prod_comparison`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_prod_comparison_map_fst [is_iso (prod_comparison F A B)] :
inv (prod_comparison F A B) ≫ F.map prod.fst = prod.fst | by simp [is_iso.inv_comp_eq] | lemma | category_theory.limits.inv_prod_comparison_map_fst | 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 | |
inv_prod_comparison_map_snd [is_iso (prod_comparison F A B)] :
inv (prod_comparison F A B) ≫ F.map prod.snd = prod.snd | by simp [is_iso.inv_comp_eq] | lemma | category_theory.limits.inv_prod_comparison_map_snd | 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 | |
prod_comparison_inv_natural (f : A ⟶ A') (g : B ⟶ B')
[is_iso (prod_comparison F A B)] [is_iso (prod_comparison F A' B')] :
inv (prod_comparison F A B) ≫ F.map (prod.map f g) =
prod.map (F.map f) (F.map g) ≫ inv (prod_comparison F A' B') | by rw [is_iso.eq_comp_inv, category.assoc, is_iso.inv_comp_eq, prod_comparison_natural] | lemma | category_theory.limits.prod_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 product comparison morphism is an iso, its inverse is natural. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_comparison_nat_iso [has_binary_products C] [has_binary_products D]
(A : C) [∀ B, is_iso (prod_comparison F A B)] :
prod.functor.obj A ⋙ F ≅ F ⋙ prod.functor.obj (F.obj A) | { hom := prod_comparison_nat_trans F A
..(@as_iso _ _ _ _ _ (nat_iso.is_iso_of_is_iso_app ⟨_, _⟩)) } | def | category_theory.limits.prod_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 `F(A ⨯ -) ≅ FA ⨯ F-`, provided each `prod_comparison F A B` is an
isomorphism (as `B` changes). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod_comparison (F : C ⥤ D) (A B : C)
[has_binary_coproduct A B] [has_binary_coproduct (F.obj A) (F.obj B)] :
F.obj A ⨿ F.obj B ⟶ F.obj (A ⨿ B) | coprod.desc (F.map coprod.inl) (F.map coprod.inr) | def | category_theory.limits.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"
] | [] | The coproduct comparison morphism.
In `category_theory/limits/preserves` we show
this is always an iso iff F preserves binary coproducts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod_comparison_inl :
coprod.inl ≫ coprod_comparison F A B = F.map coprod.inl | coprod.inl_desc _ _ | lemma | category_theory.limits.coprod_comparison_inl | 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 |
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