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binary_bicone.inl_cokernel_cofork_π :
(binary_bicone.inl_cokernel_cofork c).π = c.snd | rfl | lemma | category_theory.limits.binary_bicone.inl_cokernel_cofork_π | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_bicone.inr_cokernel_cofork : cokernel_cofork c.inr | cokernel_cofork.of_π c.fst c.inr_fst | def | category_theory.limits.binary_bicone.inr_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.inr`. It consists of the morphism
`binary_bicone.fst`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_bicone.inr_cokernel_cofork_π :
(binary_bicone.inr_cokernel_cofork c).π = c.fst | rfl | lemma | category_theory.limits.binary_bicone.inr_cokernel_cofork_π | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_bicone.is_limit_fst_kernel_fork (i : is_limit c.to_cone) :
is_limit c.fst_kernel_fork | fork.is_limit.mk' _ $ λ s,
⟨s.ι ≫ c.snd, by apply binary_fan.is_limit.hom_ext i; simp, λ m hm, by simp [← hm]⟩ | def | category_theory.limits.binary_bicone.is_limit_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"
] | [] | The fork defined in `binary_bicone.fst_kernel_fork` is indeed a kernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_bicone.is_limit_snd_kernel_fork (i : is_limit c.to_cone) :
is_limit c.snd_kernel_fork | fork.is_limit.mk' _ $ λ s,
⟨s.ι ≫ c.fst, by apply binary_fan.is_limit.hom_ext i; simp, λ m hm, by simp [← hm]⟩ | def | category_theory.limits.binary_bicone.is_limit_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"
] | [] | The fork defined in `binary_bicone.snd_kernel_fork` is indeed a kernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_bicone.is_colimit_inl_cokernel_cofork (i : is_colimit c.to_cocone) :
is_colimit c.inl_cokernel_cofork | cofork.is_colimit.mk' _ $ λ s, ⟨c.inr ≫ s.π, by apply binary_cofan.is_colimit.hom_ext i; simp,
λ m hm, by simp [← hm]⟩ | def | category_theory.limits.binary_bicone.is_colimit_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"
] | [] | The cofork defined in `binary_bicone.inl_cokernel_cofork` is indeed a cokernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_bicone.is_colimit_inr_cokernel_cofork (i : is_colimit c.to_cocone) :
is_colimit c.inr_cokernel_cofork | cofork.is_colimit.mk' _ $ λ s, ⟨c.inl ≫ s.π, by apply binary_cofan.is_colimit.hom_ext i; simp,
λ m hm, by simp [← hm]⟩ | def | category_theory.limits.binary_bicone.is_colimit_inr_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"
] | [] | The cofork defined in `binary_bicone.inr_cokernel_cofork` is indeed a cokernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.fst_kernel_fork : kernel_fork (biprod.fst : X ⊞ Y ⟶ X) | binary_bicone.fst_kernel_fork _ | def | category_theory.limits.biprod.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 `biprod.fst`. It consists of the
morphism `biprod.inr`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.fst_kernel_fork_ι : fork.ι (biprod.fst_kernel_fork X Y) = biprod.inr | rfl | lemma | category_theory.limits.biprod.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 | |
biprod.is_kernel_fst_kernel_fork : is_limit (biprod.fst_kernel_fork X Y) | binary_bicone.is_limit_fst_kernel_fork (binary_biproduct.is_limit _ _) | def | category_theory.limits.biprod.is_kernel_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"
] | [] | The fork `biprod.fst_kernel_fork` is indeed a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.snd_kernel_fork : kernel_fork (biprod.snd : X ⊞ Y ⟶ Y) | binary_bicone.snd_kernel_fork _ | def | category_theory.limits.biprod.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 `biprod.snd`. It consists of the
morphism `biprod.inl`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.snd_kernel_fork_ι : fork.ι (biprod.snd_kernel_fork X Y) = biprod.inl | rfl | lemma | category_theory.limits.biprod.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 | |
biprod.is_kernel_snd_kernel_fork : is_limit (biprod.snd_kernel_fork X Y) | binary_bicone.is_limit_snd_kernel_fork (binary_biproduct.is_limit _ _) | def | category_theory.limits.biprod.is_kernel_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"
] | [] | The fork `biprod.snd_kernel_fork` is indeed a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.inl_cokernel_cofork : cokernel_cofork (biprod.inl : X ⟶ X ⊞ Y) | binary_bicone.inl_cokernel_cofork _ | def | category_theory.limits.biprod.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 `biprod.inl`. It consists of the
morphism `biprod.snd`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.inl_cokernel_cofork_π : cofork.π (biprod.inl_cokernel_cofork X Y) = biprod.snd | rfl | lemma | category_theory.limits.biprod.inl_cokernel_cofork_π | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biprod.is_cokernel_inl_cokernel_fork : is_colimit (biprod.inl_cokernel_cofork X Y) | binary_bicone.is_colimit_inl_cokernel_cofork (binary_biproduct.is_colimit _ _) | def | category_theory.limits.biprod.is_cokernel_inl_cokernel_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"
] | [] | The cofork `biprod.inl_cokernel_fork` is indeed a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.inr_cokernel_cofork : cokernel_cofork (biprod.inr : Y ⟶ X ⊞ Y) | binary_bicone.inr_cokernel_cofork _ | def | category_theory.limits.biprod.inr_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 `biprod.inr`. It consists of the
morphism `biprod.fst`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.inr_cokernel_cofork_π : cofork.π (biprod.inr_cokernel_cofork X Y) = biprod.fst | rfl | lemma | category_theory.limits.biprod.inr_cokernel_cofork_π | category_theory.limits.shapes | src/category_theory/limits/shapes/biproducts.lean | [
"category_theory.limits.shapes.finite_products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biprod.is_cokernel_inr_cokernel_fork : is_colimit (biprod.inr_cokernel_cofork X Y) | binary_bicone.is_colimit_inr_cokernel_cofork (binary_biproduct.is_colimit _ _) | def | category_theory.limits.biprod.is_cokernel_inr_cokernel_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"
] | [] | The cofork `biprod.inr_cokernel_fork` is indeed a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_biprod_fst_iso : kernel (biprod.fst : X ⊞ Y ⟶ X) ≅ Y | limit.iso_limit_cone ⟨_, biprod.is_kernel_fst_kernel_fork X Y⟩ | def | category_theory.limits.kernel_biprod_fst_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 kernel of `biprod.fst : X ⊞ Y ⟶ X` is `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_biprod_snd_iso : kernel (biprod.snd : X ⊞ Y ⟶ Y) ≅ X | limit.iso_limit_cone ⟨_, biprod.is_kernel_snd_kernel_fork X Y⟩ | def | category_theory.limits.kernel_biprod_snd_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 kernel of `biprod.snd : X ⊞ Y ⟶ Y` is `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_biprod_inl_iso : cokernel (biprod.inl : X ⟶ X ⊞ Y) ≅ Y | colimit.iso_colimit_cocone ⟨_, biprod.is_cokernel_inl_cokernel_fork X Y⟩ | def | category_theory.limits.cokernel_biprod_inl_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 cokernel of `biprod.inl : X ⟶ X ⊞ Y` is `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_biprod_inr_iso : cokernel (biprod.inr : Y ⟶ X ⊞ Y) ≅ X | colimit.iso_colimit_cocone ⟨_, biprod.is_cokernel_inr_cokernel_fork X Y⟩ | def | category_theory.limits.cokernel_biprod_inr_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 cokernel of `biprod.inr : Y ⟶ X ⊞ Y` is `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_biprod_zero {X Y : C} [has_binary_biproduct X Y] (hY : is_zero Y) : X ≅ X ⊞ Y | { hom := biprod.inl,
inv := biprod.fst,
inv_hom_id' := begin
apply category_theory.limits.biprod.hom_ext;
simp only [category.assoc, biprod.inl_fst, category.comp_id, category.id_comp,
biprod.inl_snd, comp_zero],
apply hY.eq_of_tgt
end } | def | category_theory.limits.iso_biprod_zero | 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"
] | [
"category_theory.limits.biprod.hom_ext"
] | If `Y` is a zero object, `X ≅ X ⊞ Y` for any `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_zero_biprod {X Y : C} [has_binary_biproduct X Y] (hY : is_zero X) : Y ≅ X ⊞ Y | { hom := biprod.inr,
inv := biprod.snd,
inv_hom_id' := begin
apply category_theory.limits.biprod.hom_ext;
simp only [category.assoc, biprod.inr_snd, category.comp_id, category.id_comp,
biprod.inr_fst, comp_zero],
apply hY.eq_of_tgt
end } | def | category_theory.limits.iso_zero_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"
] | [
"category_theory.limits.biprod.hom_ext"
] | If `X` is a zero object, `Y ≅ X ⊞ Y` for any `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.braiding (P Q : C) : P ⊞ Q ≅ Q ⊞ P | { hom := biprod.lift biprod.snd biprod.fst,
inv := biprod.lift biprod.snd biprod.fst } | def | category_theory.limits.biprod.braiding | 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 braiding isomorphism which swaps a binary biproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.braiding' (P Q : C) : P ⊞ Q ≅ Q ⊞ P | { hom := biprod.desc biprod.inr biprod.inl,
inv := biprod.desc biprod.inr biprod.inl } | def | category_theory.limits.biprod.braiding' | 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 formula for the braiding isomorphism which swaps a binary biproduct,
using the fact that the biproduct is a coproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.braiding'_eq_braiding {P Q : C} :
biprod.braiding' P Q = biprod.braiding P Q | by tidy | lemma | category_theory.limits.biprod.braiding'_eq_braiding | 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.braid_natural {W X Y Z : C} (f : X ⟶ Y) (g : Z ⟶ W) :
biprod.map f g ≫ (biprod.braiding _ _).hom = (biprod.braiding _ _).hom ≫ biprod.map g f | by tidy | lemma | category_theory.limits.biprod.braid_natural | 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 braiding isomorphism can be passed through a map by swapping the order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biprod.braiding_map_braiding {W X Y Z : C} (f : W ⟶ Y) (g : X ⟶ Z) :
(biprod.braiding X W).hom ≫ biprod.map f g ≫ (biprod.braiding Y Z).hom = biprod.map g f | by tidy | lemma | category_theory.limits.biprod.braiding_map_braiding | 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.symmetry' (P Q : C) :
biprod.lift biprod.snd biprod.fst ≫ biprod.lift biprod.snd biprod.fst = 𝟙 (P ⊞ Q) | by tidy | lemma | category_theory.limits.biprod.symmetry' | 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.symmetry (P Q : C) :
(biprod.braiding P Q).hom ≫ (biprod.braiding Q P).hom = 𝟙 _ | by simp | lemma | category_theory.limits.biprod.symmetry | 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 braiding isomorphism is symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indecomposable (X : C) : Prop | ¬ is_zero X ∧ ∀ Y Z, (X ≅ Y ⊞ Z) → is_zero Y ∨ is_zero Z | def | category_theory.indecomposable | 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 object is indecomposable if it cannot be written as the biproduct of two nonzero objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_left_of_is_iso_biprod_map
{W X Y Z : C} (f : W ⟶ Y) (g : X ⟶ Z) [is_iso (biprod.map f g)] : is_iso f | ⟨⟨biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst,
⟨begin
have t := congr_arg (λ p : W ⊞ X ⟶ W ⊞ X, biprod.inl ≫ p ≫ biprod.fst)
(is_iso.hom_inv_id (biprod.map f g)),
simp only [category.id_comp, category.assoc, biprod.inl_map_assoc] at t,
simp [t],
end,
begin
have t := congr_arg (λ p : Y ⊞ Z... | lemma | category_theory.is_iso_left_of_is_iso_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"
] | [] | If
```
(f 0)
(0 g)
```
is invertible, then `f` is invertible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_right_of_is_iso_biprod_map
{W X Y Z : C} (f : W ⟶ Y) (g : X ⟶ Z) [is_iso (biprod.map f g)] : is_iso g | begin
letI : is_iso (biprod.map g f) := by
{ rw [←biprod.braiding_map_braiding],
apply_instance, },
exact is_iso_left_of_is_iso_biprod_map g f,
end | lemma | category_theory.is_iso_right_of_is_iso_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"
] | [] | If
```
(f 0)
(0 g)
```
is invertible, then `g` is invertible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone (s : comm_sq f g h i) : pullback_cone h i | pullback_cone.mk _ _ s.w | def | category_theory.comm_sq.cone | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The (not necessarily limiting) `pullback_cone h i` implicit in the statement
that we have `comm_sq f g h i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone (s : comm_sq f g h i) : pushout_cocone f g | pushout_cocone.mk _ _ s.w | def | category_theory.comm_sq.cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The (not necessarily limiting) `pushout_cocone f g` implicit in the statement
that we have `comm_sq f g h i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_fst (s : comm_sq f g h i) : s.cone.fst = f | rfl | lemma | category_theory.comm_sq.cone_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cone_snd (s : comm_sq f g h i) : s.cone.snd = g | rfl | lemma | category_theory.comm_sq.cone_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone_inl (s : comm_sq f g h i) : s.cocone.inl = h | rfl | lemma | category_theory.comm_sq.cocone_inl | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone_inr (s : comm_sq f g h i) : s.cocone.inr = i | rfl | lemma | category_theory.comm_sq.cocone_inr | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cone_op (p : comm_sq f g h i) : p.cone.op ≅ p.flip.op.cocone | pushout_cocone.ext (iso.refl _) (by tidy) (by tidy) | def | category_theory.comm_sq.cone_op | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The pushout cocone in the opposite category associated to the cone of
a commutative square identifies to the cocone of the flipped commutative square in
the opposite category | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_op (p : comm_sq f g h i) : p.cocone.op ≅ p.flip.op.cone | pullback_cone.ext (iso.refl _) (by tidy) (by tidy) | def | category_theory.comm_sq.cocone_op | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The pullback cone in the opposite category associated to the cocone of
a commutative square identifies to the cone of the flipped commutative square in
the opposite category | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_unop {W X Y Z : Cᵒᵖ} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z}
(p : comm_sq f g h i) : p.cone.unop ≅ p.flip.unop.cocone | pushout_cocone.ext (iso.refl _) (by tidy) (by tidy) | def | category_theory.comm_sq.cone_unop | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The pushout cocone obtained from the pullback cone associated to a
commutative square in the opposite category identifies to the cocone associated
to the flipped square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_unop {W X Y Z : Cᵒᵖ} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z}
(p : comm_sq f g h i) : p.cocone.unop ≅ p.flip.unop.cone | pullback_cone.ext (iso.refl _) (by tidy) (by tidy) | def | category_theory.comm_sq.cocone_unop | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The pullback cone obtained from the pushout cone associated to a
commutative square in the opposite category identifies to the cone associated
to the flipped square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_pullback {P X Y Z : C} (fst : P ⟶ X) (snd : P ⟶ Y) (f : X ⟶ Z) (g : Y ⟶ Z)
extends comm_sq fst snd f g : Prop | (is_limit' : nonempty (is_limit (pullback_cone.mk _ _ w))) | structure | category_theory.is_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The proposition that a square
```
P --fst--> X
| |
snd f
| |
v v
Y ---g---> Z
```
is a pullback square. (Also known as a fibered product or cartesian square.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_pushout {Z X Y P : C} (f : Z ⟶ X) (g : Z ⟶ Y) (inl : X ⟶ P) (inr : Y ⟶ P)
extends comm_sq f g inl inr : Prop | (is_colimit' : nonempty (is_colimit (pushout_cocone.mk _ _ w))) | structure | category_theory.is_pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The proposition that a square
```
Z ---f---> X
| |
g inl
| |
v v
Y --inr--> P
```
is a pushout square. (Also known as a fiber coproduct or cocartesian square.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bicartesian_sq {W X Y Z : C} (f : W ⟶ X) (g : W ⟶ Y) (h : X ⟶ Z) (i : Y ⟶ Z)
extends is_pullback f g h i, is_pushout f g h i : Prop | structure | category_theory.bicartesian_sq | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | A *bicartesian* square is a commutative square
```
W ---f---> X
| |
g h
| |
v v
Y ---i---> Z
```
that is both a pullback square and a pushout square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cone (h : is_pullback fst snd f g) : pullback_cone f g | h.to_comm_sq.cone | def | category_theory.is_pullback.cone | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The (limiting) `pullback_cone f g` implicit in the statement
that we have a `is_pullback fst snd f g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_fst (h : is_pullback fst snd f g) : h.cone.fst = fst | rfl | lemma | category_theory.is_pullback.cone_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cone_snd (h : is_pullback fst snd f g) : h.cone.snd = snd | rfl | lemma | category_theory.is_pullback.cone_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit (h : is_pullback fst snd f g) : is_limit h.cone | h.is_limit'.some | def | category_theory.is_pullback.is_limit | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The cone obtained from `is_pullback fst snd f g` is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_limit {c : pullback_cone f g} (h : limits.is_limit c) :
is_pullback c.fst c.snd f g | { w := c.condition,
is_limit' := ⟨is_limit.of_iso_limit h
(limits.pullback_cone.ext (iso.refl _) (by tidy) (by tidy))⟩, } | lemma | category_theory.is_pullback.of_is_limit | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | If `c` is a limiting pullback cone, then we have a `is_pullback c.fst c.snd f g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_limit' (w : comm_sq fst snd f g) (h : limits.is_limit w.cone) :
is_pullback fst snd f g | of_is_limit h | lemma | category_theory.is_pullback.of_is_limit' | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | A variant of `of_is_limit` that is more useful with `apply`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_has_pullback (f : X ⟶ Z) (g : Y ⟶ Z) [has_pullback f g] :
is_pullback (pullback.fst : pullback f g ⟶ X) (pullback.snd : pullback f g ⟶ Y) f g | of_is_limit (limit.is_limit (cospan f g)) | lemma | category_theory.is_pullback.of_has_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The pullback provided by `has_pullback f g` fits into a `is_pullback`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_product {c : binary_fan X Y} (h : limits.is_limit c) (t : is_terminal Z) :
is_pullback c.fst c.snd (t.from _) (t.from _) | of_is_limit (is_pullback_of_is_terminal_is_product _ _ _ _ t
(is_limit.of_iso_limit h (limits.cones.ext (iso.refl c.X) (by rintro ⟨⟨⟩⟩; { dsimp, simp, })))) | lemma | category_theory.is_pullback.of_is_product | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [
"is_pullback_of_is_terminal_is_product"
] | If `c` is a limiting binary product cone, and we have a terminal object,
then we have `is_pullback c.fst c.snd 0 0`
(where each `0` is the unique morphism to the terminal object). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_product' (h : limits.is_limit (binary_fan.mk fst snd)) (t : is_terminal Z) :
is_pullback fst snd (t.from _) (t.from _) | of_is_product h t | lemma | category_theory.is_pullback.of_is_product' | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | A variant of `of_is_product` that is more useful with `apply`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_has_binary_product' [has_binary_product X Y] [has_terminal C] :
is_pullback limits.prod.fst limits.prod.snd (terminal.from X) (terminal.from Y) | of_is_product (limit.is_limit _) terminal_is_terminal | lemma | category_theory.is_pullback.of_has_binary_product' | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_has_binary_product [has_binary_product X Y] [has_zero_object C] [has_zero_morphisms C] :
is_pullback limits.prod.fst limits.prod.snd (0 : X ⟶ 0) (0 : Y ⟶ 0) | by convert of_is_product (limit.is_limit _) has_zero_object.zero_is_terminal | lemma | category_theory.is_pullback.of_has_binary_product | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_pullback (h : is_pullback fst snd f g) [has_pullback f g] : P ≅ pullback f g | (limit.iso_limit_cone ⟨_, h.is_limit⟩).symm | def | category_theory.is_pullback.iso_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | Any object at the top left of a pullback square is
isomorphic to the pullback provided by the `has_limit` API. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_pullback_hom_fst (h : is_pullback fst snd f g) [has_pullback f g] :
h.iso_pullback.hom ≫ pullback.fst = fst | by { dsimp [iso_pullback, cone, comm_sq.cone], simp, } | lemma | category_theory.is_pullback.iso_pullback_hom_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_pullback_hom_snd (h : is_pullback fst snd f g) [has_pullback f g] :
h.iso_pullback.hom ≫ pullback.snd = snd | by { dsimp [iso_pullback, cone, comm_sq.cone], simp, } | lemma | category_theory.is_pullback.iso_pullback_hom_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_pullback_inv_fst (h : is_pullback fst snd f g) [has_pullback f g] :
h.iso_pullback.inv ≫ fst = pullback.fst | by simp [iso.inv_comp_eq] | lemma | category_theory.is_pullback.iso_pullback_inv_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_pullback_inv_snd (h : is_pullback fst snd f g) [has_pullback f g] :
h.iso_pullback.inv ≫ snd = pullback.snd | by simp [iso.inv_comp_eq] | lemma | category_theory.is_pullback.iso_pullback_inv_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_iso_pullback (h : comm_sq fst snd f g) [has_pullback f g] (i : P ≅ pullback f g)
(w₁ : i.hom ≫ pullback.fst = fst) (w₂ : i.hom ≫ pullback.snd = snd) : is_pullback fst snd f g | of_is_limit' h (limits.is_limit.of_iso_limit (limit.is_limit _)
(@pullback_cone.ext _ _ _ _ _ _ _ (pullback_cone.mk _ _ _) _ i w₁.symm w₂.symm).symm) | lemma | category_theory.is_pullback.of_iso_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_horiz_is_iso [is_iso fst] [is_iso g] (sq : comm_sq fst snd f g) :
is_pullback fst snd f g | of_is_limit' sq
begin
refine pullback_cone.is_limit.mk _ (λ s, s.fst ≫ inv fst) (by tidy) (λ s, _) (by tidy),
simp only [← cancel_mono g, category.assoc, ← sq.w, is_iso.inv_hom_id_assoc, s.condition],
end | lemma | category_theory.is_pullback.of_horiz_is_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone (h : is_pushout f g inl inr) : pushout_cocone f g | h.to_comm_sq.cocone | def | category_theory.is_pushout.cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The (colimiting) `pushout_cocone f g` implicit in the statement
that we have a `is_pushout f g inl inr`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_inl (h : is_pushout f g inl inr) : h.cocone.inl = inl | rfl | lemma | category_theory.is_pushout.cocone_inl | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone_inr (h : is_pushout f g inl inr) : h.cocone.inr = inr | rfl | lemma | category_theory.is_pushout.cocone_inr | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit (h : is_pushout f g inl inr) : is_colimit h.cocone | h.is_colimit'.some | def | category_theory.is_pushout.is_colimit | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The cocone obtained from `is_pushout f g inl inr` is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_colimit {c : pushout_cocone f g} (h : limits.is_colimit c) :
is_pushout f g c.inl c.inr | { w := c.condition,
is_colimit' := ⟨is_colimit.of_iso_colimit h
(limits.pushout_cocone.ext (iso.refl _) (by tidy) (by tidy))⟩, } | lemma | category_theory.is_pushout.of_is_colimit | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | If `c` is a colimiting pushout cocone, then we have a `is_pushout f g c.inl c.inr`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_colimit' (w : comm_sq f g inl inr) (h : limits.is_colimit w.cocone) :
is_pushout f g inl inr | of_is_colimit h | lemma | category_theory.is_pushout.of_is_colimit' | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | A variant of `of_is_colimit` that is more useful with `apply`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_has_pushout (f : Z ⟶ X) (g : Z ⟶ Y) [has_pushout f g] :
is_pushout f g (pushout.inl : X ⟶ pushout f g) (pushout.inr : Y ⟶ pushout f g) | of_is_colimit (colimit.is_colimit (span f g)) | lemma | category_theory.is_pushout.of_has_pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The pushout provided by `has_pushout f g` fits into a `is_pushout`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_coproduct {c : binary_cofan X Y} (h : limits.is_colimit c) (t : is_initial Z) :
is_pushout (t.to _) (t.to _) c.inl c.inr | of_is_colimit (is_pushout_of_is_initial_is_coproduct _ _ _ _ t
(is_colimit.of_iso_colimit h
(limits.cocones.ext (iso.refl c.X) (by rintro ⟨⟨⟩⟩; { dsimp, simp, })))) | lemma | category_theory.is_pushout.of_is_coproduct | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [
"is_pushout_of_is_initial_is_coproduct"
] | If `c` is a colimiting binary coproduct cocone, and we have an initial object,
then we have `is_pushout 0 0 c.inl c.inr`
(where each `0` is the unique morphism from the initial object). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_coproduct' (h : limits.is_colimit (binary_cofan.mk inl inr)) (t : is_initial Z) :
is_pushout (t.to _) (t.to _) inl inr | of_is_coproduct h t | lemma | category_theory.is_pushout.of_is_coproduct' | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | A variant of `of_is_coproduct` that is more useful with `apply`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_has_binary_coproduct' [has_binary_coproduct X Y] [has_initial C] :
is_pushout (initial.to _) (initial.to _) (coprod.inl : X ⟶ _) (coprod.inr : Y ⟶ _) | of_is_coproduct (colimit.is_colimit _) initial_is_initial | lemma | category_theory.is_pushout.of_has_binary_coproduct' | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_has_binary_coproduct
[has_binary_coproduct X Y] [has_zero_object C] [has_zero_morphisms C] :
is_pushout (0 : 0 ⟶ X) (0 : 0 ⟶ Y) coprod.inl coprod.inr | by convert of_is_coproduct (colimit.is_colimit _) has_zero_object.zero_is_initial | lemma | category_theory.is_pushout.of_has_binary_coproduct | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_pushout (h : is_pushout f g inl inr) [has_pushout f g] : P ≅ pushout f g | (colimit.iso_colimit_cocone ⟨_, h.is_colimit⟩).symm | def | category_theory.is_pushout.iso_pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | Any object at the top left of a pullback square is
isomorphic to the pullback provided by the `has_limit` API. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl_iso_pushout_inv (h : is_pushout f g inl inr) [has_pushout f g] :
pushout.inl ≫ h.iso_pushout.inv = inl | by { dsimp [iso_pushout, cocone, comm_sq.cocone], simp, } | lemma | category_theory.is_pushout.inl_iso_pushout_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_iso_pushout_inv (h : is_pushout f g inl inr) [has_pushout f g] :
pushout.inr ≫ h.iso_pushout.inv = inr | by { dsimp [iso_pushout, cocone, comm_sq.cocone], simp, } | lemma | category_theory.is_pushout.inr_iso_pushout_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_iso_pushout_hom (h : is_pushout f g inl inr) [has_pushout f g] :
inl ≫ h.iso_pushout.hom = pushout.inl | by simp [←iso.eq_comp_inv] | lemma | category_theory.is_pushout.inl_iso_pushout_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_iso_pushout_hom (h : is_pushout f g inl inr) [has_pushout f g] :
inr ≫ h.iso_pushout.hom = pushout.inr | by simp [←iso.eq_comp_inv] | lemma | category_theory.is_pushout.inr_iso_pushout_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_iso_pushout (h : comm_sq f g inl inr) [has_pushout f g] (i : P ≅ pushout f g)
(w₁ : inl ≫ i.hom = pushout.inl) (w₂ : inr ≫ i.hom = pushout.inr) : is_pushout f g inl inr | of_is_colimit' h (limits.is_colimit.of_iso_colimit (colimit.is_colimit _)
(@pushout_cocone.ext _ _ _ _ _ _ _ (pushout_cocone.mk _ _ _) _ i w₁ w₂).symm) | lemma | category_theory.is_pushout.of_iso_pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flip (h : is_pullback fst snd f g) : is_pullback snd fst g f | of_is_limit (@pullback_cone.flip_is_limit _ _ _ _ _ _ _ _ _ _ h.w.symm h.is_limit) | lemma | category_theory.is_pullback.flip | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flip_iff : is_pullback fst snd f g ↔ is_pullback snd fst g f | ⟨flip, flip⟩ | lemma | category_theory.is_pullback.flip_iff | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_left (X : C) : is_pullback (0 : 0 ⟶ X) (0 : 0 ⟶ 0) (𝟙 X) (0 : 0 ⟶ X) | { w := by simp,
is_limit' :=
⟨{ lift := λ s, 0,
fac' := λ s, by simpa using @pullback_cone.equalizer_ext _ _ _ _ _ _ _ s _ 0 (𝟙 _)
(by simpa using (pullback_cone.condition s).symm), }⟩ } | lemma | category_theory.is_pullback.zero_left | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [
"lift"
] | The square with `0 : 0 ⟶ 0` on the left and `𝟙 X` on the right is a pullback square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_top (X : C) : is_pullback (0 : 0 ⟶ 0) (0 : 0 ⟶ X) (0 : 0 ⟶ X) (𝟙 X) | (zero_left X).flip | lemma | category_theory.is_pullback.zero_top | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The square with `0 : 0 ⟶ 0` on the top and `𝟙 X` on the bottom is a pullback square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_right (X : C) : is_pullback (0 : X ⟶ 0) (𝟙 X) (0 : 0 ⟶ 0) (0 : X ⟶ 0) | of_iso_pullback (by simp) ((zero_prod_iso X).symm ≪≫ (pullback_zero_zero_iso _ _).symm)
(by simp) (by simp) | lemma | category_theory.is_pullback.zero_right | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The square with `0 : 0 ⟶ 0` on the right and `𝟙 X` on the left is a pullback square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_bot (X : C) : is_pullback (𝟙 X) (0 : X ⟶ 0) (0 : X ⟶ 0) (0 : 0 ⟶ 0) | (zero_right X).flip | lemma | category_theory.is_pullback.zero_bot | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The square with `0 : 0 ⟶ 0` on the bottom and `𝟙 X` on the top is a pullback square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
paste_vert {X₁₁ X₁₂ X₂₁ X₂₂ X₃₁ X₃₂ : C}
{h₁₁ : X₁₁ ⟶ X₁₂} {h₂₁ : X₂₁ ⟶ X₂₂} {h₃₁ : X₃₁ ⟶ X₃₂}
{v₁₁ : X₁₁ ⟶ X₂₁} {v₁₂ : X₁₂ ⟶ X₂₂} {v₂₁ : X₂₁ ⟶ X₃₁} {v₂₂ : X₂₂ ⟶ X₃₂}
(s : is_pullback h₁₁ v₁₁ v₁₂ h₂₁) (t : is_pullback h₂₁ v₂₁ v₂₂ h₃₁) :
is_pullback h₁₁ (v₁₁ ≫ v₂₁) (v₁₂ ≫ v₂₂) h₃₁ | (of_is_limit
(big_square_is_pullback _ _ _ _ _ _ _ s.w t.w t.is_limit s.is_limit)) | lemma | category_theory.is_pullback.paste_vert | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
paste_horiz {X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C}
{h₁₁ : X₁₁ ⟶ X₁₂} {h₁₂ : X₁₂ ⟶ X₁₃} {h₂₁ : X₂₁ ⟶ X₂₂} {h₂₂ : X₂₂ ⟶ X₂₃}
{v₁₁ : X₁₁ ⟶ X₂₁} {v₁₂ : X₁₂ ⟶ X₂₂} {v₁₃ : X₁₃ ⟶ X₂₃}
(s : is_pullback h₁₁ v₁₁ v₁₂ h₂₁) (t : is_pullback h₁₂ v₁₂ v₁₃ h₂₂) :
is_pullback (h₁₁ ≫ h₁₂) v₁₁ v₁₃ (h₂₁ ≫ h₂₂) | (paste_vert s.flip t.flip).flip | lemma | category_theory.is_pullback.paste_horiz | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | Paste two pullback squares "horizontally" to obtain another pullback square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_bot {X₁₁ X₁₂ X₂₁ X₂₂ X₃₁ X₃₂ : C}
{h₁₁ : X₁₁ ⟶ X₁₂} {h₂₁ : X₂₁ ⟶ X₂₂} {h₃₁ : X₃₁ ⟶ X₃₂}
{v₁₁ : X₁₁ ⟶ X₂₁} {v₁₂ : X₁₂ ⟶ X₂₂} {v₂₁ : X₂₁ ⟶ X₃₁} {v₂₂ : X₂₂ ⟶ X₃₂}
(s : is_pullback h₁₁ (v₁₁ ≫ v₂₁) (v₁₂ ≫ v₂₂) h₃₁) (p : h₁₁ ≫ v₁₂ = v₁₁ ≫ h₂₁)
(t : is_pullback h₂₁ v₂₁ v₂₂ h₃₁) :
is_pullback h₁₁ v₁₁ v₁₂ h₂₁ | of_is_limit (left_square_is_pullback _ _ _ _ _ _ _ p _ t.is_limit s.is_limit) | lemma | category_theory.is_pullback.of_bot | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | Given a pullback square assembled from a commuting square on the top and
a pullback square on the bottom, the top square is a pullback square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_right {X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C}
{h₁₁ : X₁₁ ⟶ X₁₂} {h₁₂ : X₁₂ ⟶ X₁₃} {h₂₁ : X₂₁ ⟶ X₂₂} {h₂₂ : X₂₂ ⟶ X₂₃}
{v₁₁ : X₁₁ ⟶ X₂₁} {v₁₂ : X₁₂ ⟶ X₂₂} {v₁₃ : X₁₃ ⟶ X₂₃}
(s : is_pullback (h₁₁ ≫ h₁₂) v₁₁ v₁₃ (h₂₁ ≫ h₂₂)) (p : h₁₁ ≫ v₁₂ = v₁₁ ≫ h₂₁)
(t : is_pullback h₁₂ v₁₂ v₁₃ h₂₂) :
is_pullback h₁₁ v₁₁ v₁₂ h₂₁ | (of_bot s.flip p.symm t.flip).flip | lemma | category_theory.is_pullback.of_right | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | Given a pullback square assembled from a commuting square on the left and
a pullback square on the right, the left square is a pullback square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
paste_vert_iff {X₁₁ X₁₂ X₂₁ X₂₂ X₃₁ X₃₂ : C}
{h₁₁ : X₁₁ ⟶ X₁₂} {h₂₁ : X₂₁ ⟶ X₂₂} {h₃₁ : X₃₁ ⟶ X₃₂}
{v₁₁ : X₁₁ ⟶ X₂₁} {v₁₂ : X₁₂ ⟶ X₂₂} {v₂₁ : X₂₁ ⟶ X₃₁} {v₂₂ : X₂₂ ⟶ X₃₂}
(s : is_pullback h₂₁ v₂₁ v₂₂ h₃₁) (e : h₁₁ ≫ v₁₂ = v₁₁ ≫ h₂₁) :
is_pullback h₁₁ (v₁₁ ≫ v₂₁) (v₁₂ ≫ v₂₂) h₃₁ ↔ is_pullback h₁₁ v₁₁ v₁₂ h₂₁ | ⟨λ h, h.of_bot e s, λ h, h.paste_vert s⟩ | lemma | category_theory.is_pullback.paste_vert_iff | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
paste_horiz_iff {X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C}
{h₁₁ : X₁₁ ⟶ X₁₂} {h₁₂ : X₁₂ ⟶ X₁₃} {h₂₁ : X₂₁ ⟶ X₂₂} {h₂₂ : X₂₂ ⟶ X₂₃}
{v₁₁ : X₁₁ ⟶ X₂₁} {v₁₂ : X₁₂ ⟶ X₂₂} {v₁₃ : X₁₃ ⟶ X₂₃}
(s : is_pullback h₁₂ v₁₂ v₁₃ h₂₂) (e : h₁₁ ≫ v₁₂ = v₁₁ ≫ h₂₁) :
is_pullback (h₁₁ ≫ h₁₂) v₁₁ v₁₃ (h₂₁ ≫ h₂₂) ↔ is_pullback h₁₁ v₁₁ v₁₂ h₂₁ | ⟨λ h, h.of_right e s, λ h, h.paste_horiz s⟩ | lemma | category_theory.is_pullback.paste_horiz_iff | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_is_bilimit {b : binary_bicone X Y} (h : b.is_bilimit) :
is_pullback b.fst b.snd (0 : X ⟶ 0) (0 : Y ⟶ 0) | by convert is_pullback.of_is_product' h.is_limit has_zero_object.zero_is_terminal | lemma | category_theory.is_pullback.of_is_bilimit | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_has_biproduct (X Y : C) [has_binary_biproduct X Y] :
is_pullback biprod.fst biprod.snd (0 : X ⟶ 0) (0 : Y ⟶ 0) | of_is_bilimit (binary_biproduct.is_bilimit X Y) | lemma | category_theory.is_pullback.of_has_biproduct | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_snd' {b : binary_bicone X Y} (h : b.is_bilimit) :
is_pullback b.inl (0 : X ⟶ 0) b.snd (0 : 0 ⟶ Y) | by { refine of_right _ (by simp) (of_is_bilimit h), simp, } | lemma | category_theory.is_pullback.inl_snd' | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_snd (X Y : C) [has_binary_biproduct X Y] :
is_pullback biprod.inl (0 : X ⟶ 0) biprod.snd (0 : 0 ⟶ Y) | inl_snd' (binary_biproduct.is_bilimit X Y) | lemma | category_theory.is_pullback.inl_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | The square
```
X --inl--> X ⊞ Y
| |
0 snd
| |
v v
0 ---0-----> Y
```
is a pullback square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inr_fst' {b : binary_bicone X Y} (h : b.is_bilimit) :
is_pullback b.inr (0 : Y ⟶ 0) b.fst (0 : 0 ⟶ X) | by { apply flip, refine of_bot _ (by simp) (of_is_bilimit h), simp, } | lemma | category_theory.is_pullback.inr_fst' | category_theory.limits.shapes | src/category_theory/limits/shapes/comm_sq.lean | [
"category_theory.comm_sq",
"category_theory.limits.opposites",
"category_theory.limits.shapes.biproducts",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products",
"category_theory.limits.constructions.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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