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has_pushout_of_right_factors_epi (f : X ⟶ Y) : has_pushout h (h ≫ f) | by { conv { congr, rw ←category.comp_id h, }, apply_instance } | instance | category_theory.limits.has_pushout_of_right_factors_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_inr_iso_of_right_factors_epi (f : X ⟶ Y) :
is_iso (pushout.inr : _ ⟶ pushout h (h ≫ f)) | begin
convert (congr_arg is_iso (show pushout.inr ≫ _ = _,
from colimit.iso_colimit_cocone_ι_inv ⟨_, pushout_is_pushout_of_epi_comp (𝟙 _) f h⟩
walking_span.right)).mp infer_instance;
exact (category.comp_id _).symm
end | instance | category_theory.limits.pushout_inr_iso_of_right_factors_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_cocone_of_right_iso : pushout_cocone f g | pushout_cocone.mk (𝟙 _) (inv g ≫ f) $ by simp | def | category_theory.limits.pushout_cocone_of_right_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | If `f : X ⟶ Z` is iso, then `Y ⨿[X] Z ≅ Y`. This is the explicit colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout_cocone_of_right_iso_X :
(pushout_cocone_of_right_iso f g).X = Y | rfl | lemma | category_theory.limits.pushout_cocone_of_right_iso_X | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_cocone_of_right_iso_inl :
(pushout_cocone_of_right_iso f g).inl = 𝟙 _ | rfl | lemma | category_theory.limits.pushout_cocone_of_right_iso_inl | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_cocone_of_right_iso_inr :
(pushout_cocone_of_right_iso f g).inr = inv g ≫ f | rfl | lemma | category_theory.limits.pushout_cocone_of_right_iso_inr | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_cocone_of_right_iso_ι_app_none :
(pushout_cocone_of_right_iso f g).ι.app none = f | by { delta pushout_cocone_of_right_iso, simp } | lemma | category_theory.limits.pushout_cocone_of_right_iso_ι_app_none | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_cocone_of_right_iso_ι_app_left :
(pushout_cocone_of_right_iso f g).ι.app left = 𝟙 _ | rfl | lemma | category_theory.limits.pushout_cocone_of_right_iso_ι_app_left | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_cocone_of_right_iso_ι_app_right :
(pushout_cocone_of_right_iso f g).ι.app right = inv g ≫ f | rfl | lemma | category_theory.limits.pushout_cocone_of_right_iso_ι_app_right | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_cocone_of_right_iso_is_limit :
is_colimit (pushout_cocone_of_right_iso f g) | pushout_cocone.is_colimit_aux' _ (λ s, ⟨s.inl, by simp [←s.condition]⟩) | def | category_theory.limits.pushout_cocone_of_right_iso_is_limit | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Verify that the constructed cocone is indeed a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pushout_of_right_iso : has_pushout f g | ⟨⟨⟨_, pushout_cocone_of_right_iso_is_limit f g⟩⟩⟩ | lemma | category_theory.limits.has_pushout_of_right_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_inl_iso_of_right_iso : is_iso (pushout.inl : _ ⟶ pushout f g) | begin
refine ⟨⟨pushout.desc (𝟙 _) (inv g ≫ f) (by simp), (by simp), _⟩⟩,
ext,
{ simp [←pushout.condition] },
{ simp [pushout.condition] },
end | instance | category_theory.limits.pushout_inl_iso_of_right_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pushout_of_left_factors_epi (f : X ⟶ Y) : has_pushout (h ≫ f) h | by { conv { congr, skip, rw ←category.comp_id h, }, apply_instance } | instance | category_theory.limits.has_pushout_of_left_factors_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_inl_iso_of_left_factors_epi (f : X ⟶ Y) :
is_iso (pushout.inl : _ ⟶ pushout (h ≫ f) h) | begin
convert (congr_arg is_iso (show pushout.inl ≫ _ = _,
from colimit.iso_colimit_cocone_ι_inv ⟨_, pushout_is_pushout_of_epi_comp f (𝟙 _) h⟩
walking_span.left)).mp infer_instance;
exact (category.comp_id _).symm
end | instance | category_theory.limits.pushout_inl_iso_of_left_factors_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_kernel_pair_of_mono [mono f] : has_pullback f f | ⟨⟨⟨_, pullback_cone.is_limit_mk_id_id f⟩⟩⟩ | instance | category_theory.limits.has_kernel_pair_of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_eq_snd_of_mono_eq [mono f] : (pullback.fst : pullback f f ⟶ _) = pullback.snd | ((pullback_cone.is_limit_mk_id_id f).fac (get_limit_cone (cospan f f)).cone left).symm.trans
((pullback_cone.is_limit_mk_id_id f).fac (get_limit_cone (cospan f f)).cone right : _) | lemma | category_theory.limits.fst_eq_snd_of_mono_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_symmetry_hom_of_mono_eq [mono f] :
(pullback_symmetry f f).hom = 𝟙 _ | by ext; simp [fst_eq_snd_of_mono_eq] | lemma | category_theory.limits.pullback_symmetry_hom_of_mono_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_iso_of_mono_eq [mono f] : is_iso (pullback.fst : pullback f f ⟶ _) | begin
refine ⟨⟨pullback.lift (𝟙 _) (𝟙 _) (by simp), _, by simp⟩⟩,
ext,
{ simp },
{ simp [fst_eq_snd_of_mono_eq] }
end | instance | category_theory.limits.fst_iso_of_mono_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_iso_of_mono_eq [mono f] : is_iso (pullback.snd : pullback f f ⟶ _) | by { rw ← fst_eq_snd_of_mono_eq, apply_instance } | instance | category_theory.limits.snd_iso_of_mono_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_cokernel_pair_of_epi [epi f] : has_pushout f f | ⟨⟨⟨_, pushout_cocone.is_colimit_mk_id_id f⟩⟩⟩ | instance | category_theory.limits.has_cokernel_pair_of_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_eq_inr_of_epi_eq [epi f] : (pushout.inl : _ ⟶ pushout f f) = pushout.inr | ((pushout_cocone.is_colimit_mk_id_id f).fac
(get_colimit_cocone (span f f)).cocone left).symm.trans
((pushout_cocone.is_colimit_mk_id_id f).fac
(get_colimit_cocone (span f f)).cocone right : _) | lemma | category_theory.limits.inl_eq_inr_of_epi_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_symmetry_hom_of_epi_eq [epi f] :
(pushout_symmetry f f).hom = 𝟙 _ | by ext; simp [inl_eq_inr_of_epi_eq] | lemma | category_theory.limits.pullback_symmetry_hom_of_epi_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_iso_of_epi_eq [epi f] : is_iso (pushout.inl : _ ⟶ pushout f f) | begin
refine ⟨⟨pushout.desc (𝟙 _) (𝟙 _) (by simp), by simp, _⟩⟩,
ext,
{ simp },
{ simp [inl_eq_inr_of_epi_eq] }
end | instance | category_theory.limits.inl_iso_of_epi_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_iso_of_epi_eq [epi f] : is_iso (pushout.inr : _ ⟶ pushout f f) | by { rw ← inl_eq_inr_of_epi_eq, apply_instance } | instance | category_theory.limits.inr_iso_of_epi_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
big_square_is_pullback (H : is_limit (pullback_cone.mk _ _ h₂))
(H' : is_limit (pullback_cone.mk _ _ h₁)) :
is_limit (pullback_cone.mk _ _ (show i₁ ≫ g₁ ≫ g₂ = (f₁ ≫ f₂) ≫ i₃,
by rw [← category.assoc, h₁, category.assoc, h₂, category.assoc])) | begin
fapply pullback_cone.is_limit_aux',
intro s,
have : (s.fst ≫ g₁) ≫ g₂ = s.snd ≫ i₃ := by rw [← s.condition, category.assoc],
rcases pullback_cone.is_limit.lift' H (s.fst ≫ g₁) s.snd this with ⟨l₁, hl₁, hl₁'⟩,
rcases pullback_cone.is_limit.lift' H' s.fst l₁ hl₁.symm with ⟨l₂, hl₂, hl₂'⟩,
use l₂,
use ... | def | category_theory.limits.big_square_is_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Given
X₁ - f₁ -> X₂ - f₂ -> X₃
| | |
i₁ i₂ i₃
∨ ∨ ∨
Y₁ - g₁ -> Y₂ - g₂ -> Y₃
Then the big square is a pullback if both the small squares are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
big_square_is_pushout (H : is_colimit (pushout_cocone.mk _ _ h₂))
(H' : is_colimit (pushout_cocone.mk _ _ h₁)) :
is_colimit (pushout_cocone.mk _ _ (show i₁ ≫ g₁ ≫ g₂ = (f₁ ≫ f₂) ≫ i₃,
by rw [← category.assoc, h₁, category.assoc, h₂, category.assoc])) | begin
fapply pushout_cocone.is_colimit_aux',
intro s,
have : i₁ ≫ s.inl = f₁ ≫ (f₂ ≫ s.inr) := by rw [s.condition, category.assoc],
rcases pushout_cocone.is_colimit.desc' H' s.inl (f₂ ≫ s.inr) this with ⟨l₁, hl₁, hl₁'⟩,
rcases pushout_cocone.is_colimit.desc' H l₁ s.inr hl₁' with ⟨l₂, hl₂, hl₂'⟩,
use l₂,
u... | def | category_theory.limits.big_square_is_pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Given
X₁ - f₁ -> X₂ - f₂ -> X₃
| | |
i₁ i₂ i₃
∨ ∨ ∨
Y₁ - g₁ -> Y₂ - g₂ -> Y₃
Then the big square is a pushout if both the small squares are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_square_is_pullback (H : is_limit (pullback_cone.mk _ _ h₂))
(H' : is_limit (pullback_cone.mk _ _ (show i₁ ≫ g₁ ≫ g₂ = (f₁ ≫ f₂) ≫ i₃,
by rw [← category.assoc, h₁, category.assoc, h₂, category.assoc]))) :
is_limit (pullback_cone.mk _ _ h₁) | begin
fapply pullback_cone.is_limit_aux',
intro s,
have : s.fst ≫ g₁ ≫ g₂ = (s.snd ≫ f₂) ≫ i₃ :=
by { rw [← category.assoc, s.condition, category.assoc, category.assoc, h₂] },
rcases pullback_cone.is_limit.lift' H' s.fst (s.snd ≫ f₂) this with ⟨l₁, hl₁, hl₁'⟩,
use l₁,
use hl₁,
split,
{ apply pullback_... | def | category_theory.limits.left_square_is_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Given
X₁ - f₁ -> X₂ - f₂ -> X₃
| | |
i₁ i₂ i₃
∨ ∨ ∨
Y₁ - g₁ -> Y₂ - g₂ -> Y₃
Then the left square is a pullback if the right square and the big square are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_square_is_pushout (H : is_colimit (pushout_cocone.mk _ _ h₁))
(H' : is_colimit (pushout_cocone.mk _ _ (show i₁ ≫ g₁ ≫ g₂ = (f₁ ≫ f₂) ≫ i₃,
by rw [← category.assoc, h₁, category.assoc, h₂, category.assoc]))) :
is_colimit (pushout_cocone.mk _ _ h₂) | begin
fapply pushout_cocone.is_colimit_aux',
intro s,
have : i₁ ≫ g₁ ≫ s.inl = (f₁ ≫ f₂) ≫ s.inr :=
by { rw [category.assoc, ← s.condition, ← category.assoc, ← category.assoc, h₁] },
rcases pushout_cocone.is_colimit.desc' H' (g₁ ≫ s.inl) s.inr this with ⟨l₁, hl₁, hl₁'⟩,
dsimp at *,
use l₁,
refine ⟨_,_,_... | def | category_theory.limits.right_square_is_pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Given
X₁ - f₁ -> X₂ - f₂ -> X₃
| | |
i₁ i₂ i₃
∨ ∨ ∨
Y₁ - g₁ -> Y₂ - g₂ -> Y₃
Then the right square is a pushout if the left square and the big square are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_right_pullback_fst_iso :
pullback f' (pullback.fst : pullback f g ⟶ _) ≅ pullback (f' ≫ f) g | begin
let := big_square_is_pullback
(pullback.snd : pullback f' (pullback.fst : pullback f g ⟶ _) ⟶ _) pullback.snd
f' f pullback.fst pullback.fst g pullback.condition pullback.condition
(pullback_is_pullback _ _) (pullback_is_pullback _ _),
exact (this.cone_point_unique_up_to_iso (pullback_is_pullback ... | def | category_theory.limits.pullback_right_pullback_fst_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The canonical isomorphism `W ×[X] (X ×[Z] Y) ≅ W ×[Z] Y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_right_pullback_fst_iso_hom_fst :
(pullback_right_pullback_fst_iso f g f').hom ≫ pullback.fst = pullback.fst | is_limit.cone_point_unique_up_to_iso_hom_comp _ _ walking_cospan.left | lemma | category_theory.limits.pullback_right_pullback_fst_iso_hom_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_right_pullback_fst_iso_hom_snd :
(pullback_right_pullback_fst_iso f g f').hom ≫ pullback.snd = pullback.snd ≫ pullback.snd | is_limit.cone_point_unique_up_to_iso_hom_comp _ _ walking_cospan.right | lemma | category_theory.limits.pullback_right_pullback_fst_iso_hom_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_right_pullback_fst_iso_inv_fst :
(pullback_right_pullback_fst_iso f g f').inv ≫ pullback.fst = pullback.fst | is_limit.cone_point_unique_up_to_iso_inv_comp _ _ walking_cospan.left | lemma | category_theory.limits.pullback_right_pullback_fst_iso_inv_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_right_pullback_fst_iso_inv_snd_snd :
(pullback_right_pullback_fst_iso f g f').inv ≫ pullback.snd ≫ pullback.snd = pullback.snd | is_limit.cone_point_unique_up_to_iso_inv_comp _ _ walking_cospan.right | lemma | category_theory.limits.pullback_right_pullback_fst_iso_inv_snd_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_right_pullback_fst_iso_inv_snd_fst :
(pullback_right_pullback_fst_iso f g f').inv ≫ pullback.snd ≫ pullback.fst = pullback.fst ≫ f' | begin
rw ← pullback.condition,
exact pullback_right_pullback_fst_iso_inv_fst_assoc _ _ _ _
end | lemma | category_theory.limits.pullback_right_pullback_fst_iso_inv_snd_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_left_pushout_inr_iso :
pushout (pushout.inr : _ ⟶ pushout f g) g' ≅ pushout f (g ≫ g') | ((big_square_is_pushout g g' _ _ f _ _ pushout.condition pushout.condition
(pushout_is_pushout _ _) (pushout_is_pushout _ _))
.cocone_point_unique_up_to_iso (pushout_is_pushout _ _) : _) | def | category_theory.limits.pushout_left_pushout_inr_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The canonical isomorphism `(Y ⨿[X] Z) ⨿[Z] W ≅ Y ×[X] W` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl_pushout_left_pushout_inr_iso_inv :
pushout.inl ≫ (pushout_left_pushout_inr_iso f g g').inv = pushout.inl ≫ pushout.inl | ((big_square_is_pushout g g' _ _ f _ _ pushout.condition pushout.condition
(pushout_is_pushout _ _) (pushout_is_pushout _ _))
.comp_cocone_point_unique_up_to_iso_inv (pushout_is_pushout _ _) walking_span.left : _) | lemma | category_theory.limits.inl_pushout_left_pushout_inr_iso_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_pushout_left_pushout_inr_iso_hom :
pushout.inr ≫ (pushout_left_pushout_inr_iso f g g').hom = pushout.inr | ((big_square_is_pushout g g' _ _ f _ _ pushout.condition pushout.condition
(pushout_is_pushout _ _) (pushout_is_pushout _ _))
.comp_cocone_point_unique_up_to_iso_hom (pushout_is_pushout _ _) walking_span.right : _) | lemma | category_theory.limits.inr_pushout_left_pushout_inr_iso_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_pushout_left_pushout_inr_iso_inv :
pushout.inr ≫ (pushout_left_pushout_inr_iso f g g').inv = pushout.inr | by rw [iso.comp_inv_eq, inr_pushout_left_pushout_inr_iso_hom] | lemma | category_theory.limits.inr_pushout_left_pushout_inr_iso_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_inl_pushout_left_pushout_inr_iso_hom :
pushout.inl ≫ pushout.inl ≫ (pushout_left_pushout_inr_iso f g g').hom = pushout.inl | by rw [← category.assoc, ← iso.eq_comp_inv, inl_pushout_left_pushout_inr_iso_inv] | lemma | category_theory.limits.inl_inl_pushout_left_pushout_inr_iso_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_inl_pushout_left_pushout_inr_iso_hom :
pushout.inr ≫ pushout.inl ≫ (pushout_left_pushout_inr_iso f g g').hom = g' ≫ pushout.inr | by rw [← category.assoc, ← iso.eq_comp_inv, category.assoc,
inr_pushout_left_pushout_inr_iso_inv, pushout.condition] | lemma | category_theory.limits.inr_inl_pushout_left_pushout_inr_iso_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_pullback_left_is_pullback [has_pullback (g₂ ≫ f₃) f₄] :
is_limit (pullback_cone.mk l₁ l₂ (show l₁ ≫ g₂ = l₂ ≫ g₃, from (pullback.lift_fst _ _ _).symm)) | begin
apply left_square_is_pullback,
exact pullback_is_pullback f₃ f₄,
convert pullback_is_pullback (g₂ ≫ f₃) f₄,
rw pullback.lift_snd
end | def | category_theory.limits.pullback_pullback_left_is_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `(X₁ ×[Y₁] X₂) ×[Y₂] X₃` is the pullback `(X₁ ×[Y₁] X₂) ×[X₂] (X₂ ×[Y₂] X₃)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_assoc_is_pullback [has_pullback (g₂ ≫ f₃) f₄] :
is_limit (pullback_cone.mk (l₁ ≫ g₁) l₂ (show (l₁ ≫ g₁) ≫ f₁ = l₂ ≫ (g₃ ≫ f₂),
by rw [pullback.lift_fst_assoc, category.assoc, category.assoc, pullback.condition])) | begin
apply pullback_cone.flip_is_limit,
apply big_square_is_pullback,
{ apply pullback_cone.flip_is_limit,
exact pullback_is_pullback f₁ f₂ },
{ apply pullback_cone.flip_is_limit,
apply pullback_pullback_left_is_pullback },
{ exact pullback.lift_fst _ _ _ },
{ exact pullback.condition.symm }
end | def | category_theory.limits.pullback_assoc_is_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `(X₁ ×[Y₁] X₂) ×[Y₂] X₃` is the pullback `X₁ ×[Y₁] (X₂ ×[Y₂] X₃)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pullback_assoc [has_pullback (g₂ ≫ f₃) f₄] :
has_pullback f₁ (g₃ ≫ f₂) | ⟨⟨⟨_, pullback_assoc_is_pullback f₁ f₂ f₃ f₄⟩⟩⟩ | lemma | category_theory.limits.has_pullback_assoc | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_pullback_right_is_pullback [has_pullback f₁ (g₃ ≫ f₂)] :
is_limit (pullback_cone.mk l₁' l₂' (show l₁' ≫ g₂ = l₂' ≫ g₃, from pullback.lift_snd _ _ _)) | begin
apply pullback_cone.flip_is_limit,
apply left_square_is_pullback,
{ apply pullback_cone.flip_is_limit,
exact pullback_is_pullback f₁ f₂ },
{ apply pullback_cone.flip_is_limit,
convert pullback_is_pullback f₁ (g₃ ≫ f₂),
rw pullback.lift_fst },
{ exact pullback.condition.symm }
end | def | category_theory.limits.pullback_pullback_right_is_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `X₁ ×[Y₁] (X₂ ×[Y₂] X₃)` is the pullback `(X₁ ×[Y₁] X₂) ×[X₂] (X₂ ×[Y₂] X₃)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_assoc_symm_is_pullback [has_pullback f₁ (g₃ ≫ f₂)] :
is_limit (pullback_cone.mk l₁' (l₂' ≫ g₄) (show l₁' ≫ (g₂ ≫ f₃) = (l₂' ≫ g₄) ≫ f₄,
by rw [pullback.lift_snd_assoc, category.assoc, category.assoc, pullback.condition])) | begin
apply big_square_is_pullback,
exact pullback_is_pullback f₃ f₄,
apply pullback_pullback_right_is_pullback
end | def | category_theory.limits.pullback_assoc_symm_is_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `X₁ ×[Y₁] (X₂ ×[Y₂] X₃)` is the pullback `(X₁ ×[Y₁] X₂) ×[Y₂] X₃`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pullback_assoc_symm [has_pullback f₁ (g₃ ≫ f₂)] :
has_pullback (g₂ ≫ f₃) f₄ | ⟨⟨⟨_, pullback_assoc_symm_is_pullback f₁ f₂ f₃ f₄⟩⟩⟩ | lemma | category_theory.limits.has_pullback_assoc_symm | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_assoc :
pullback (pullback.snd ≫ f₃ : pullback f₁ f₂ ⟶ _) f₄ ≅
pullback f₁ (pullback.fst ≫ f₂ : pullback f₃ f₄ ⟶ _) | (pullback_pullback_left_is_pullback f₁ f₂ f₃ f₄).cone_point_unique_up_to_iso
(pullback_pullback_right_is_pullback f₁ f₂ f₃ f₄) | def | category_theory.limits.pullback_assoc | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The canonical isomorphism `(X₁ ×[Y₁] X₂) ×[Y₂] X₃ ≅ X₁ ×[Y₁] (X₂ ×[Y₂] X₃)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_assoc_inv_fst_fst :
(pullback_assoc f₁ f₂ f₃ f₄).inv ≫ pullback.fst ≫ pullback.fst = pullback.fst | begin
transitivity l₁' ≫ pullback.fst,
rw ← category.assoc,
congr' 1,
exact is_limit.cone_point_unique_up_to_iso_inv_comp _ _ walking_cospan.left,
exact pullback.lift_fst _ _ _,
end | lemma | category_theory.limits.pullback_assoc_inv_fst_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_assoc_hom_fst :
(pullback_assoc f₁ f₂ f₃ f₄).hom ≫ pullback.fst = pullback.fst ≫ pullback.fst | by rw [← iso.eq_inv_comp, pullback_assoc_inv_fst_fst] | lemma | category_theory.limits.pullback_assoc_hom_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_assoc_hom_snd_fst :
(pullback_assoc f₁ f₂ f₃ f₄).hom ≫ pullback.snd ≫ pullback.fst = pullback.fst ≫ pullback.snd | begin
transitivity l₂ ≫ pullback.fst,
rw ← category.assoc,
congr' 1,
exact is_limit.cone_point_unique_up_to_iso_hom_comp _ _ walking_cospan.right,
exact pullback.lift_fst _ _ _,
end | lemma | category_theory.limits.pullback_assoc_hom_snd_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_assoc_hom_snd_snd :
(pullback_assoc f₁ f₂ f₃ f₄).hom ≫ pullback.snd ≫ pullback.snd = pullback.snd | begin
transitivity l₂ ≫ pullback.snd,
rw ← category.assoc,
congr' 1,
exact is_limit.cone_point_unique_up_to_iso_hom_comp _ _ walking_cospan.right,
exact pullback.lift_snd _ _ _,
end | lemma | category_theory.limits.pullback_assoc_hom_snd_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_assoc_inv_fst_snd :
(pullback_assoc f₁ f₂ f₃ f₄).inv ≫ pullback.fst ≫ pullback.snd = pullback.snd ≫ pullback.fst | by rw [iso.inv_comp_eq, pullback_assoc_hom_snd_fst] | lemma | category_theory.limits.pullback_assoc_inv_fst_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_assoc_inv_snd :
(pullback_assoc f₁ f₂ f₃ f₄).inv ≫ pullback.snd = pullback.snd ≫ pullback.snd | by rw [iso.inv_comp_eq, pullback_assoc_hom_snd_snd] | lemma | category_theory.limits.pullback_assoc_inv_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_pushout_left_is_pushout [has_pushout (g₃ ≫ f₂) g₄] :
is_colimit (pushout_cocone.mk l₁' l₂'
(show f₂ ≫ l₁' = f₃ ≫ l₂', from (pushout.inl_desc _ _ _).symm)) | begin
apply pushout_cocone.flip_is_colimit,
apply right_square_is_pushout,
{ apply pushout_cocone.flip_is_colimit,
exact pushout_is_pushout _ _ },
{ apply pushout_cocone.flip_is_colimit,
convert pushout_is_pushout (g₃ ≫ f₂) g₄,
exact pushout.inr_desc _ _ _ },
{ exact pushout.condition.symm }
end | def | category_theory.limits.pushout_pushout_left_is_pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `(X₁ ⨿[Z₁] X₂) ⨿[Z₂] X₃` is the pushout `(X₁ ⨿[Z₁] X₂) ×[X₂] (X₂ ⨿[Z₂] X₃)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout_assoc_is_pushout [has_pushout (g₃ ≫ f₂) g₄] :
is_colimit (pushout_cocone.mk (f₁ ≫ l₁') l₂' (show g₁ ≫ (f₁ ≫ l₁') = (g₂ ≫ f₃) ≫ l₂',
by rw [category.assoc, pushout.inl_desc, pushout.condition_assoc])) | begin
apply big_square_is_pushout,
{ apply pushout_pushout_left_is_pushout },
{ exact pushout_is_pushout _ _ }
end | def | category_theory.limits.pushout_assoc_is_pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `(X₁ ⨿[Z₁] X₂) ⨿[Z₂] X₃` is the pushout `X₁ ⨿[Z₁] (X₂ ⨿[Z₂] X₃)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pushout_assoc [has_pushout (g₃ ≫ f₂) g₄] :
has_pushout g₁ (g₂ ≫ f₃) | ⟨⟨⟨_, pushout_assoc_is_pushout g₁ g₂ g₃ g₄⟩⟩⟩ | lemma | category_theory.limits.has_pushout_assoc | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_pushout_right_is_pushout [has_pushout g₁ (g₂ ≫ f₃)] :
is_colimit (pushout_cocone.mk l₁ l₂ (show f₂ ≫ l₁ = f₃ ≫ l₂, from pushout.inr_desc _ _ _)) | begin
apply right_square_is_pushout,
{ exact pushout_is_pushout _ _ },
{ convert pushout_is_pushout g₁ (g₂ ≫ f₃),
rw pushout.inl_desc }
end | def | category_theory.limits.pushout_pushout_right_is_pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `X₁ ⨿[Z₁] (X₂ ⨿[Z₂] X₃)` is the pushout `(X₁ ⨿[Z₁] X₂) ×[X₂] (X₂ ⨿[Z₂] X₃)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout_assoc_symm_is_pushout [has_pushout g₁ (g₂ ≫ f₃)] :
is_colimit (pushout_cocone.mk l₁ (f₄ ≫ l₂) ((show (g₃ ≫ f₂) ≫ l₁ = g₄ ≫ (f₄ ≫ l₂),
by rw [category.assoc, pushout.inr_desc, pushout.condition_assoc]))) | begin
apply pushout_cocone.flip_is_colimit,
apply big_square_is_pushout,
{ apply pushout_cocone.flip_is_colimit,
apply pushout_pushout_right_is_pushout },
{ apply pushout_cocone.flip_is_colimit,
exact pushout_is_pushout _ _ },
{ exact pushout.condition.symm },
{ exact (pushout.inr_desc _ _ _).symm }... | def | category_theory.limits.pushout_assoc_symm_is_pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `X₁ ⨿[Z₁] (X₂ ⨿[Z₂] X₃)` is the pushout `(X₁ ⨿[Z₁] X₂) ⨿[Z₂] X₃`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pushout_assoc_symm [has_pushout g₁ (g₂ ≫ f₃)] :
has_pushout (g₃ ≫ f₂) g₄ | ⟨⟨⟨_, pushout_assoc_symm_is_pushout g₁ g₂ g₃ g₄⟩⟩⟩ | lemma | category_theory.limits.has_pushout_assoc_symm | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_assoc :
pushout (g₃ ≫ pushout.inr : _ ⟶ pushout g₁ g₂) g₄ ≅
pushout g₁ (g₂ ≫ pushout.inl : _ ⟶ pushout g₃ g₄) | (pushout_pushout_left_is_pushout g₁ g₂ g₃ g₄).cocone_point_unique_up_to_iso
(pushout_pushout_right_is_pushout g₁ g₂ g₃ g₄) | def | category_theory.limits.pushout_assoc | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The canonical isomorphism `(X₁ ⨿[Z₁] X₂) ⨿[Z₂] X₃ ≅ X₁ ⨿[Z₁] (X₂ ⨿[Z₂] X₃)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl_inl_pushout_assoc_hom :
pushout.inl ≫ pushout.inl ≫ (pushout_assoc g₁ g₂ g₃ g₄).hom = pushout.inl | begin
transitivity f₁ ≫ l₁,
{ congr' 1,
exact (pushout_pushout_left_is_pushout g₁ g₂ g₃ g₄)
.comp_cocone_point_unique_up_to_iso_hom _ walking_cospan.left },
{ exact pushout.inl_desc _ _ _ }
end | lemma | category_theory.limits.inl_inl_pushout_assoc_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_inl_pushout_assoc_hom :
pushout.inr ≫ pushout.inl ≫ (pushout_assoc g₁ g₂ g₃ g₄).hom = pushout.inl ≫ pushout.inr | begin
transitivity f₂ ≫ l₁,
{ congr' 1,
exact (pushout_pushout_left_is_pushout g₁ g₂ g₃ g₄)
.comp_cocone_point_unique_up_to_iso_hom _ walking_cospan.left },
{ exact pushout.inr_desc _ _ _ }
end | lemma | category_theory.limits.inr_inl_pushout_assoc_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_inr_pushout_assoc_inv :
pushout.inr ≫ pushout.inr ≫ (pushout_assoc g₁ g₂ g₃ g₄).inv = pushout.inr | begin
transitivity f₄ ≫ l₂',
{ congr' 1,
exact (pushout_pushout_left_is_pushout g₁ g₂ g₃ g₄).comp_cocone_point_unique_up_to_iso_inv
(pushout_pushout_right_is_pushout g₁ g₂ g₃ g₄) walking_cospan.right },
{ exact pushout.inr_desc _ _ _ }
end | lemma | category_theory.limits.inr_inr_pushout_assoc_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_pushout_assoc_inv :
pushout.inl ≫ (pushout_assoc g₁ g₂ g₃ g₄).inv = pushout.inl ≫ pushout.inl | by rw [iso.comp_inv_eq, category.assoc, inl_inl_pushout_assoc_hom] | lemma | category_theory.limits.inl_pushout_assoc_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_inr_pushout_assoc_inv :
pushout.inl ≫ pushout.inr ≫ (pushout_assoc g₁ g₂ g₃ g₄).inv = pushout.inr ≫ pushout.inl | by rw [← category.assoc, iso.comp_inv_eq, category.assoc, inr_inl_pushout_assoc_hom] | lemma | category_theory.limits.inl_inr_pushout_assoc_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_pushout_assoc_hom :
pushout.inr ≫ (pushout_assoc g₁ g₂ g₃ g₄).hom = pushout.inr ≫ pushout.inr | by rw [← iso.eq_comp_inv, category.assoc, inr_inr_pushout_assoc_inv] | lemma | category_theory.limits.inr_pushout_assoc_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pullbacks | has_limits_of_shape walking_cospan C | abbreviation | category_theory.limits.has_pullbacks | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `has_pullbacks` represents a choice of pullback for every pair of morphisms
See <https://stacks.math.columbia.edu/tag/001W> | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pushouts | has_colimits_of_shape walking_span C | abbreviation | category_theory.limits.has_pushouts | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `has_pushouts` represents a choice of pushout for every pair of morphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pullbacks_of_has_limit_cospan
[Π {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z}, has_limit (cospan f g)] :
has_pullbacks C | { has_limit := λ F, has_limit_of_iso (diagram_iso_cospan F).symm } | lemma | category_theory.limits.has_pullbacks_of_has_limit_cospan | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | If `C` has all limits of diagrams `cospan f g`, then it has all pullbacks | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pushouts_of_has_colimit_span
[Π {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z}, has_colimit (span f g)] :
has_pushouts C | { has_colimit := λ F, has_colimit_of_iso (diagram_iso_span F) } | lemma | category_theory.limits.has_pushouts_of_has_colimit_span | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | If `C` has all colimits of diagrams `span f g`, then it has all pushouts | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_span_op_equiv : walking_spanᵒᵖ ≌ walking_cospan | wide_pushout_shape_op_equiv _ | def | category_theory.limits.walking_span_op_equiv | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The duality equivalence `walking_spanᵒᵖ ≌ walking_cospan` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_cospan_op_equiv : walking_cospanᵒᵖ ≌ walking_span | wide_pullback_shape_op_equiv _ | def | category_theory.limits.walking_cospan_op_equiv | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The duality equivalence `walking_cospanᵒᵖ ≌ walking_span` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pullbacks_of_has_wide_pullbacks [has_wide_pullbacks.{w} C] : has_pullbacks C | begin
haveI := has_wide_pullbacks_shrink.{0 w} C,
apply_instance
end | instance | category_theory.limits.has_pullbacks_of_has_wide_pullbacks | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Having wide pullback at any universe level implies having binary pullbacks. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
base_change [has_pullbacks C] {X Y : C} (f : X ⟶ Y) : over Y ⥤ over X | { obj := λ g, over.mk (pullback.snd : pullback g.hom f ⟶ _),
map := λ g₁ g₂ i, over.hom_mk (pullback.map _ _ _ _ i.left (𝟙 _) (𝟙 _) (by simp) (by simp))
(by simp) } | def | category_theory.limits.base_change | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Given a morphism `f : X ⟶ Y`, we can take morphisms over `Y` to morphisms over `X` via
pullbacks. This is right adjoint to `over.map` (TODO) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_reflexive_pair (f g : A ⟶ B) : Prop | (common_section [] : ∃ (s : B ⟶ A), s ≫ f = 𝟙 B ∧ s ≫ g = 𝟙 B) | class | category_theory.is_reflexive_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | The pair `f g : A ⟶ B` is reflexive if there is a morphism `B ⟶ A` which is a section for both. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_coreflexive_pair (f g : A ⟶ B) : Prop | (common_retraction [] : ∃ (s : B ⟶ A), f ≫ s = 𝟙 A ∧ g ≫ s = 𝟙 A) | class | category_theory.is_coreflexive_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | The pair `f g : A ⟶ B` is coreflexive if there is a morphism `B ⟶ A` which is a retraction for both. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_reflexive_pair.mk' (s : B ⟶ A) (sf : s ≫ f = 𝟙 B) (sg : s ≫ g = 𝟙 B) :
is_reflexive_pair f g | ⟨⟨s, sf, sg⟩⟩ | lemma | category_theory.is_reflexive_pair.mk' | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coreflexive_pair.mk' (s : B ⟶ A) (fs : f ≫ s = 𝟙 A) (gs : g ≫ s = 𝟙 A) :
is_coreflexive_pair f g | ⟨⟨s, fs, gs⟩⟩ | lemma | category_theory.is_coreflexive_pair.mk' | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
common_section (f g : A ⟶ B) [is_reflexive_pair f g] : B ⟶ A | (is_reflexive_pair.common_section f g).some | def | category_theory.common_section | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | Get the common section for a reflexive pair. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
section_comp_left (f g : A ⟶ B) [is_reflexive_pair f g] :
common_section f g ≫ f = 𝟙 B | (is_reflexive_pair.common_section f g).some_spec.1 | lemma | category_theory.section_comp_left | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
section_comp_right (f g : A ⟶ B) [is_reflexive_pair f g] :
common_section f g ≫ g = 𝟙 B | (is_reflexive_pair.common_section f g).some_spec.2 | lemma | category_theory.section_comp_right | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
common_retraction (f g : A ⟶ B) [is_coreflexive_pair f g] :
B ⟶ A | (is_coreflexive_pair.common_retraction f g).some | def | category_theory.common_retraction | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | Get the common retraction for a coreflexive pair. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_comp_retraction (f g : A ⟶ B) [is_coreflexive_pair f g] :
f ≫ common_retraction f g = 𝟙 A | (is_coreflexive_pair.common_retraction f g).some_spec.1 | lemma | category_theory.left_comp_retraction | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_comp_retraction (f g : A ⟶ B) [is_coreflexive_pair f g] :
g ≫ common_retraction f g = 𝟙 A | (is_coreflexive_pair.common_retraction f g).some_spec.2 | lemma | category_theory.right_comp_retraction | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_kernel_pair.is_reflexive_pair {R : C} {f g : R ⟶ A} {q : A ⟶ B}
(h : is_kernel_pair q f g) :
is_reflexive_pair f g | is_reflexive_pair.mk' _ (h.lift' _ _ rfl).2.1 (h.lift' _ _ _).2.2 | lemma | category_theory.is_kernel_pair.is_reflexive_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | If `f,g` is a kernel pair for some morphism `q`, then it is reflexive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_reflexive_pair.swap [is_reflexive_pair f g] :
is_reflexive_pair g f | is_reflexive_pair.mk' _ (section_comp_right f g) (section_comp_left f g) | lemma | category_theory.is_reflexive_pair.swap | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coreflexive_pair.swap [is_coreflexive_pair f g] :
is_coreflexive_pair g f | is_coreflexive_pair.mk' _ (right_comp_retraction f g) (left_comp_retraction f g) | lemma | category_theory.is_coreflexive_pair.swap | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_reflexive_coequalizers : Prop | (has_coeq : ∀ ⦃A B : C⦄ (f g : A ⟶ B) [is_reflexive_pair f g], has_coequalizer f g) | class | category_theory.limits.has_reflexive_coequalizers | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | `C` has reflexive coequalizers if it has coequalizers for every reflexive pair. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coreflexive_equalizers : Prop | (has_eq : ∀ ⦃A B : C⦄ (f g : A ⟶ B) [is_coreflexive_pair f g], has_equalizer f g) | class | category_theory.limits.has_coreflexive_equalizers | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | `C` has coreflexive equalizers if it has equalizers for every coreflexive pair. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coequalizer_of_common_section [has_reflexive_coequalizers C]
{A B : C} {f g : A ⟶ B} (r : B ⟶ A)
(rf : r ≫ f = 𝟙 _) (rg : r ≫ g = 𝟙 _) :
has_coequalizer f g | begin
letI := is_reflexive_pair.mk' r rf rg,
apply_instance
end | lemma | category_theory.limits.has_coequalizer_of_common_section | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_equalizer_of_common_retraction [has_coreflexive_equalizers C]
{A B : C} {f g : A ⟶ B} (r : B ⟶ A)
(fr : f ≫ r = 𝟙 _) (gr : g ≫ r = 𝟙 _) :
has_equalizer f g | begin
letI := is_coreflexive_pair.mk' r fr gr,
apply_instance
end | lemma | category_theory.limits.has_equalizer_of_common_retraction | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_reflexive_coequalizers_of_has_coequalizers [has_coequalizers C] :
has_reflexive_coequalizers C | { has_coeq := λ A B f g i, by apply_instance } | instance | category_theory.limits.has_reflexive_coequalizers_of_has_coequalizers | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | If `C` has coequalizers, then it has reflexive coequalizers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coreflexive_equalizers_of_has_equalizers [has_equalizers C] :
has_coreflexive_equalizers C | { has_eq := λ A B f g i, by apply_instance } | instance | category_theory.limits.has_coreflexive_equalizers_of_has_equalizers | category_theory.limits.shapes | src/category_theory/limits/shapes/reflexive.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.kernel_pair"
] | [] | If `C` has equalizers, then it has coreflexive equalizers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_mono (f : X ⟶ Y) | (Z : C)
(left right : Y ⟶ Z)
(w : f ≫ left = f ≫ right)
(is_limit : is_limit (fork.of_ι f w)) | class | category_theory.regular_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | A regular monomorphism is a morphism which is the equalizer of some parallel pair. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_mono.mono (f : X ⟶ Y) [regular_mono f] : mono f | mono_of_is_limit_fork regular_mono.is_limit | instance | category_theory.regular_mono.mono | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | Every regular monomorphism is a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer_regular (g h : X ⟶ Y) [has_limit (parallel_pair g h)] :
regular_mono (equalizer.ι g h) | { Z := Y,
left := g,
right := h,
w := equalizer.condition g h,
is_limit := fork.is_limit.mk _ (λ s, limit.lift _ s) (by simp) (λ s m w, by { ext1, simp [←w] }) } | instance | category_theory.equalizer_regular | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
regular_mono.of_is_split_mono (f : X ⟶ Y) [is_split_mono f] : regular_mono f | { Z := Y,
left := 𝟙 Y,
right := retraction f ≫ f,
w := by tidy,
is_limit := is_split_mono_equalizes f } | instance | category_theory.regular_mono.of_is_split_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | Every split monomorphism is a regular monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_mono.lift' {W : C} (f : X ⟶ Y) [regular_mono f] (k : W ⟶ Y)
(h : k ≫ (regular_mono.left : Y ⟶ @regular_mono.Z _ _ _ _ f _) = k ≫ regular_mono.right) :
{l : W ⟶ X // l ≫ f = k} | fork.is_limit.lift' regular_mono.is_limit _ h | def | category_theory.regular_mono.lift' | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [] | If `f` is a regular mono, then any map `k : W ⟶ Y` equalizing `regular_mono.left` and
`regular_mono.right` induces a morphism `l : W ⟶ X` such that `l ≫ f = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_of_is_pullback_snd_of_regular {P Q R S : C} {f : P ⟶ Q} {g : P ⟶ R} {h : Q ⟶ S}
{k : R ⟶ S} [hr : regular_mono h] (comm : f ≫ h = g ≫ k)
(t : is_limit (pullback_cone.mk _ _ comm)) :
regular_mono g | { Z := hr.Z,
left := k ≫ hr.left,
right := k ≫ hr.right,
w := by rw [← reassoc_of comm, ← reassoc_of comm, hr.w],
is_limit :=
begin
apply fork.is_limit.mk' _ _,
intro s,
have l₁ : (fork.ι s ≫ k) ≫ regular_mono.left = (fork.ι s ≫ k) ≫ regular_mono.right,
rw [category.assoc, s.condition, categ... | def | category_theory.regular_of_is_pullback_snd_of_regular | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [
"comm"
] | The second leg of a pullback cone is a regular monomorphism if the right component is too.
See also `pullback.snd_of_mono` for the basic monomorphism version, and
`regular_of_is_pullback_fst_of_regular` for the flipped version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_of_is_pullback_fst_of_regular {P Q R S : C} {f : P ⟶ Q} {g : P ⟶ R} {h : Q ⟶ S}
{k : R ⟶ S} [hr : regular_mono k] (comm : f ≫ h = g ≫ k)
(t : is_limit (pullback_cone.mk _ _ comm)) :
regular_mono f | regular_of_is_pullback_snd_of_regular comm.symm (pullback_cone.flip_is_limit t) | def | category_theory.regular_of_is_pullback_fst_of_regular | category_theory.limits.shapes | src/category_theory/limits/shapes/regular_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.strong_epi",
"category_theory.limits.shapes.equalizers"
] | [
"comm"
] | The first leg of a pullback cone is a regular monomorphism if the left component is too.
See also `pullback.fst_of_mono` for the basic monomorphism version, and
`regular_of_is_pullback_snd_of_regular` for the flipped version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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