statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
inr_fst (X Y : C) [has_binary_biproduct X Y] :
is_pullback biprod.inr (0 : Y ⟶ 0) biprod.fst (0 : 0 ⟶ X) | inr_fst' (binary_biproduct.is_bilimit X Y) | 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"
] | [] | The square
```
Y --inr--> X ⊞ Y
| |
0 fst
| |
v v
0 ---0-----> X
```
is a pullback square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_bilimit' {b : binary_bicone X Y} (h : b.is_bilimit) :
is_pullback (0 : 0 ⟶ X) (0 : 0 ⟶ Y) b.inl b.inr | by { refine is_pullback.of_right _ (by simp) (is_pullback.inl_snd' h).flip, simp, } | 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_binary_biproduct (X Y : C) [has_binary_biproduct X Y] :
is_pullback (0 : 0 ⟶ X) (0 : 0 ⟶ Y) biprod.inl biprod.inr | of_is_bilimit' (binary_biproduct.is_bilimit X Y) | lemma | category_theory.is_pullback.of_has_binary_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 | |
has_pullback_biprod_fst_biprod_snd [has_binary_biproduct X Y] :
has_pullback (biprod.inl : X ⟶ _) (biprod.inr : Y ⟶ _) | has_limit.mk ⟨_, (of_has_binary_biproduct X Y).is_limit⟩ | instance | category_theory.is_pullback.has_pullback_biprod_fst_biprod_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 | |
pullback_biprod_inl_biprod_inr [has_binary_biproduct X Y] :
pullback (biprod.inl : X ⟶ _) (biprod.inr : Y ⟶ _) ≅ 0 | limit.iso_limit_cone ⟨_, (of_has_binary_biproduct X Y).is_limit⟩ | def | category_theory.is_pullback.pullback_biprod_inl_biprod_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"
] | [] | The pullback of `biprod.inl` and `biprod.inr` is the zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op (h : is_pullback fst snd f g) : is_pushout g.op f.op snd.op fst.op | is_pushout.of_is_colimit (is_colimit.of_iso_colimit
(limits.pullback_cone.is_limit_equiv_is_colimit_op h.flip.cone h.flip.is_limit)
h.to_comm_sq.flip.cone_op) | lemma | category_theory.is_pullback.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop {P X Y Z : Cᵒᵖ} {fst : P ⟶ X} {snd : P ⟶ Y} {f : X ⟶ Z} {g : Y ⟶ Z}
(h : is_pullback fst snd f g) : is_pushout g.unop f.unop snd.unop fst.unop | is_pushout.of_is_colimit (is_colimit.of_iso_colimit
(limits.pullback_cone.is_limit_equiv_is_colimit_unop h.flip.cone h.flip.is_limit)
h.to_comm_sq.flip.cone_unop) | lemma | category_theory.is_pullback.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_vert_is_iso [is_iso snd] [is_iso f] (sq : comm_sq fst snd f g) :
is_pullback fst snd f g | is_pullback.flip (of_horiz_is_iso sq.flip) | lemma | category_theory.is_pullback.of_vert_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 | |
flip (h : is_pushout f g inl inr) : is_pushout g f inr inl | of_is_colimit (@pushout_cocone.flip_is_colimit _ _ _ _ _ _ _ _ _ _ h.w.symm h.is_colimit) | lemma | category_theory.is_pushout.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_pushout f g inl inr ↔ is_pushout g f inr inl | ⟨flip, flip⟩ | lemma | category_theory.is_pushout.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_right (X : C) : is_pushout (0 : X ⟶ 0) (𝟙 X) (0 : 0 ⟶ 0) (0 : X ⟶ 0) | { w := by simp,
is_colimit' :=
⟨{ desc := λ s, 0,
fac' := λ s, begin
have c := @pushout_cocone.coequalizer_ext _ _ _ _ _ _ _ s _ 0 (𝟙 _) (by simp)
(by simpa using (pushout_cocone.condition s)),
dsimp at c,
simpa using c,
end }⟩ } | lemma | category_theory.is_pushout.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 pushout square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_bot (X : C) : is_pushout (𝟙 X) (0 : X ⟶ 0) (0 : X ⟶ 0) (0 : 0 ⟶ 0) | (zero_right X).flip | lemma | category_theory.is_pushout.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 pushout square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_left (X : C) : is_pushout (0 : 0 ⟶ X) (0 : 0 ⟶ 0) (𝟙 X) (0 : 0 ⟶ X) | of_iso_pushout (by simp) ((coprod_zero_iso X).symm ≪≫ (pushout_zero_zero_iso _ _).symm)
(by simp) (by simp) | lemma | category_theory.is_pushout.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"
] | [] | The square with `0 : 0 ⟶ 0` on the right left `𝟙 X` on the right is a pushout square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_top (X : C) : is_pushout (0 : 0 ⟶ 0) (0 : 0 ⟶ X) (0 : 0 ⟶ X) (𝟙 X) | (zero_left X).flip | lemma | category_theory.is_pushout.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 pushout 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_pushout h₁₁ v₁₁ v₁₂ h₂₁) (t : is_pushout h₂₁ v₂₁ v₂₂ h₃₁) :
is_pushout h₁₁ (v₁₁ ≫ v₂₁) (v₁₂ ≫ v₂₂) h₃₁ | (of_is_colimit
(big_square_is_pushout _ _ _ _ _ _ _ s.w t.w t.is_colimit s.is_colimit)) | lemma | category_theory.is_pushout.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_pushout h₁₁ v₁₁ v₁₂ h₂₁) (t : is_pushout h₁₂ v₁₂ v₁₃ h₂₂) :
is_pushout (h₁₁ ≫ h₁₂) v₁₁ v₁₃ (h₂₁ ≫ h₂₂) | (paste_vert s.flip t.flip).flip | lemma | category_theory.is_pushout.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 pushout squares "horizontally" to obtain another pushout 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_pushout h₁₁ (v₁₁ ≫ v₂₁) (v₁₂ ≫ v₂₂) h₃₁) (p : h₂₁ ≫ v₂₂ = v₂₁ ≫ h₃₁)
(t : is_pushout h₁₁ v₁₁ v₁₂ h₂₁) :
is_pushout h₂₁ v₂₁ v₂₂ h₃₁ | of_is_colimit (right_square_is_pushout _ _ _ _ _ _ _ _ p t.is_colimit s.is_colimit) | lemma | category_theory.is_pushout.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 pushout square assembled from a pushout square on the top and
a commuting square on the bottom, the bottom square is a pushout 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_pushout (h₁₁ ≫ h₁₂) v₁₁ v₁₃ (h₂₁ ≫ h₂₂)) (p : h₁₂ ≫ v₁₃ = v₁₂ ≫ h₂₂)
(t : is_pushout h₁₁ v₁₁ v₁₂ h₂₁) :
is_pushout h₁₂ v₁₂ v₁₃ h₂₂ | (of_bot s.flip p.symm t.flip).flip | lemma | category_theory.is_pushout.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 pushout square assembled from a pushout square on the left and
a commuting square on the right, the right square is a pushout 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_pushout h₁₁ v₁₁ v₁₂ h₂₁) (e : h₂₁ ≫ v₂₂ = v₂₁ ≫ h₃₁) :
is_pushout h₁₁ (v₁₁ ≫ v₂₁) (v₁₂ ≫ v₂₂) h₃₁ ↔ is_pushout h₂₁ v₂₁ v₂₂ h₃₁ | ⟨λ h, h.of_bot e s, s.paste_vert⟩ | lemma | category_theory.is_pushout.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_pushout h₁₁ v₁₁ v₁₂ h₂₁) (e : h₁₂ ≫ v₁₃ = v₁₂ ≫ h₂₂) :
is_pushout (h₁₁ ≫ h₁₂) v₁₁ v₁₃ (h₂₁ ≫ h₂₂) ↔ is_pushout h₁₂ v₁₂ v₁₃ h₂₂ | ⟨λ h, h.of_right e s, s.paste_horiz⟩ | lemma | category_theory.is_pushout.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_pushout (0 : 0 ⟶ X) (0 : 0 ⟶ Y) b.inl b.inr | by convert is_pushout.of_is_coproduct' h.is_colimit has_zero_object.zero_is_initial | lemma | category_theory.is_pushout.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_pushout (0 : 0 ⟶ X) (0 : 0 ⟶ Y) biprod.inl biprod.inr | of_is_bilimit (binary_biproduct.is_bilimit X Y) | lemma | category_theory.is_pushout.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_pushout b.inl (0 : X ⟶ 0) b.snd (0 : 0 ⟶ Y) | by { apply flip, refine of_right _ (by simp) (of_is_bilimit h), simp, } | lemma | category_theory.is_pushout.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_pushout biprod.inl (0 : X ⟶ 0) biprod.snd (0 : 0 ⟶ Y) | inl_snd' (binary_biproduct.is_bilimit X Y) | lemma | category_theory.is_pushout.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 pushout square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inr_fst' {b : binary_bicone X Y} (h : b.is_bilimit) :
is_pushout b.inr (0 : Y ⟶ 0) b.fst (0 : 0 ⟶ X) | by { refine of_bot _ (by simp) (of_is_bilimit h), simp, } | lemma | category_theory.is_pushout.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 | |
inr_fst (X Y : C) [has_binary_biproduct X Y] :
is_pushout biprod.inr (0 : Y ⟶ 0) biprod.fst (0 : 0 ⟶ X) | inr_fst' (binary_biproduct.is_bilimit X Y) | lemma | category_theory.is_pushout.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"
] | [] | The square
```
Y --inr--> X ⊞ Y
| |
0 fst
| |
v v
0 ---0-----> X
```
is a pushout square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_bilimit' {b : binary_bicone X Y} (h : b.is_bilimit) :
is_pushout b.fst b.snd (0 : X ⟶ 0) (0 : Y ⟶ 0) | by { refine is_pushout.of_right _ (by simp) (is_pushout.inl_snd' h), simp, } | lemma | category_theory.is_pushout.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_binary_biproduct (X Y : C) [has_binary_biproduct X Y] :
is_pushout biprod.fst biprod.snd (0 : X ⟶ 0) (0 : Y ⟶ 0) | of_is_bilimit' (binary_biproduct.is_bilimit X Y) | lemma | category_theory.is_pushout.of_has_binary_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 | |
has_pushout_biprod_fst_biprod_snd [has_binary_biproduct X Y] :
has_pushout (biprod.fst : _ ⟶ X) (biprod.snd : _ ⟶ Y) | has_colimit.mk ⟨_, (of_has_binary_biproduct X Y).is_colimit⟩ | instance | category_theory.is_pushout.has_pushout_biprod_fst_biprod_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 | |
pushout_biprod_fst_biprod_snd [has_binary_biproduct X Y] :
pushout (biprod.fst : _ ⟶ X) (biprod.snd : _ ⟶ Y) ≅ 0 | colimit.iso_colimit_cocone ⟨_, (of_has_binary_biproduct X Y).is_colimit⟩ | def | category_theory.is_pushout.pushout_biprod_fst_biprod_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 pushout of `biprod.fst` and `biprod.snd` is the zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op (h : is_pushout f g inl inr) : is_pullback inr.op inl.op g.op f.op | is_pullback.of_is_limit (is_limit.of_iso_limit
(limits.pushout_cocone.is_colimit_equiv_is_limit_op h.flip.cocone h.flip.is_colimit)
h.to_comm_sq.flip.cocone_op) | lemma | category_theory.is_pushout.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop {Z X Y P : Cᵒᵖ} {f : Z ⟶ X} {g : Z ⟶ Y} {inl : X ⟶ P} {inr : Y ⟶ P}
(h : is_pushout f g inl inr) : is_pullback inr.unop inl.unop g.unop f.unop | is_pullback.of_is_limit (is_limit.of_iso_limit
(limits.pushout_cocone.is_colimit_equiv_is_limit_unop h.flip.cocone h.flip.is_colimit)
h.to_comm_sq.flip.cocone_unop) | lemma | category_theory.is_pushout.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_horiz_is_iso [is_iso f] [is_iso inr] (sq : comm_sq f g inl inr) :
is_pushout f g inl inr | of_is_colimit' sq
begin
refine pushout_cocone.is_colimit.mk _ (λ s, inv inr ≫ s.inr) (λ s, _) (by tidy) (by tidy),
simp only [← cancel_epi f, s.condition, sq.w_assoc, is_iso.hom_inv_id_assoc],
end | lemma | category_theory.is_pushout.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 | |
of_vert_is_iso [is_iso g] [is_iso inl] (sq : comm_sq f g inl inr) :
is_pushout f g inl inr | (of_horiz_is_iso sq.flip).flip | lemma | category_theory.is_pushout.of_vert_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 | |
is_pullback.is_limit_fork (H : is_pullback f f g g') :
is_limit (fork.of_ι f H.w) | begin
fapply fork.is_limit.mk,
{ exact λ s, H.is_limit.lift (pullback_cone.mk s.ι s.ι s.condition) },
{ exact λ s, H.is_limit.fac _ walking_cospan.left },
{ intros s m e, apply pullback_cone.is_limit.hom_ext H.is_limit; refine e.trans _;
symmetry; exact H.is_limit.fac _ _ }
end | def | category_theory.is_pullback.is_limit_fork | 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 `f : X ⟶ Y`, `g g' : Y ⟶ Z` forms a pullback square, then `f` is the equalizer of
`g` and `g'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_pushout.is_limit_fork (H : is_pushout f f' g g) :
is_colimit (cofork.of_π g H.w) | begin
fapply cofork.is_colimit.mk,
{ exact λ s, H.is_colimit.desc (pushout_cocone.mk s.π s.π s.condition) },
{ exact λ s, H.is_colimit.fac _ walking_span.left },
{ intros s m e, apply pushout_cocone.is_colimit.hom_ext H.is_colimit; refine e.trans _;
symmetry; exact H.is_colimit.fac _ _ }
end | def | category_theory.is_pushout.is_limit_fork | 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 `f f' : X ⟶ Y`, `g : Y ⟶ Z` forms a pushout square, then `g` is the coequalizer of
`f` and `f'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_pullback_is_pushout (p₁ : is_pullback f g h i) (p₂ : is_pushout f g h i) :
bicartesian_sq f g h i | bicartesian_sq.mk p₁.to_comm_sq ⟨p₁.is_limit⟩ ⟨p₂.is_colimit⟩ | lemma | category_theory.bicartesian_sq.of_is_pullback_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flip (p : bicartesian_sq f g h i) : bicartesian_sq g f i h | of_is_pullback_is_pushout p.to_is_pullback.flip p.to_is_pushout.flip | lemma | category_theory.bicartesian_sq.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 | |
of_is_biproduct₁ {b : binary_bicone X Y} (h : b.is_bilimit) :
bicartesian_sq b.fst b.snd (0 : X ⟶ 0) (0 : Y ⟶ 0) | of_is_pullback_is_pushout (is_pullback.of_is_bilimit h) (is_pushout.of_is_bilimit' h) | lemma | category_theory.bicartesian_sq.of_is_biproduct₁ | 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"
] | [] | ```
X ⊞ Y --fst--> X
| |
snd 0
| |
v v
Y -----0---> 0
```
is a bicartesian square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_biproduct₂ {b : binary_bicone X Y} (h : b.is_bilimit) :
bicartesian_sq (0 : 0 ⟶ X) (0 : 0 ⟶ Y) b.inl b.inr | of_is_pullback_is_pushout (is_pullback.of_is_bilimit' h) (is_pushout.of_is_bilimit h) | lemma | category_theory.bicartesian_sq.of_is_biproduct₂ | 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"
] | [] | ```
0 -----0---> X
| |
0 inl
| |
v v
Y --inr--> X ⊞ Y
```
is a bicartesian square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_has_biproduct₁ [has_binary_biproduct X Y] :
bicartesian_sq biprod.fst biprod.snd (0 : X ⟶ 0) (0 : Y ⟶ 0) | by convert of_is_biproduct₁ (binary_biproduct.is_bilimit X Y) | lemma | category_theory.bicartesian_sq.of_has_biproduct₁ | 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"
] | [] | ```
X ⊞ Y --fst--> X
| |
snd 0
| |
v v
Y -----0---> 0
```
is a bicartesian square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_has_biproduct₂ [has_binary_biproduct X Y] :
bicartesian_sq (0 : 0 ⟶ X) (0 : 0 ⟶ Y) biprod.inl biprod.inr | by convert of_is_biproduct₂ (binary_biproduct.is_bilimit X Y) | lemma | category_theory.bicartesian_sq.of_has_biproduct₂ | 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"
] | [] | ```
0 -----0---> X
| |
0 inl
| |
v v
Y --inr--> X ⊞ Y
```
is a bicartesian square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.map_is_pullback [preserves_limit (cospan h i) F] (s : is_pullback f g h i) :
is_pullback (F.map f) (F.map g) (F.map h) (F.map i) | -- This is made slightly awkward because `C` and `D` have different universes,
-- and so the relevant `walking_cospan` diagrams live in different universes too!
begin
refine is_pullback.of_is_limit' (F.map_comm_sq s.to_comm_sq)
(is_limit.equiv_of_nat_iso_of_iso (cospan_comp_iso F h i) _ _ (walking_cospan.ext _ _ ... | lemma | category_theory.functor.map_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor.map_is_pushout [preserves_colimit (span f g) F] (s : is_pushout f g h i) :
is_pushout (F.map f) (F.map g) (F.map h) (F.map i) | begin
refine is_pushout.of_is_colimit' (F.map_comm_sq s.to_comm_sq)
(is_colimit.equiv_of_nat_iso_of_iso (span_comp_iso F f g) _ _ (walking_span.ext _ _ _)
(is_colimit_of_preserves F s.is_colimit)),
{ refl, },
{ dsimp, simp, },
{ dsimp, simp, },
end | lemma | category_theory.functor.map_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_pullback.of_map [reflects_limit (cospan h i) F] (e : f ≫ h = g ≫ i)
(H : is_pullback (F.map f) (F.map g) (F.map h) (F.map i)) : is_pullback f g h i | begin
refine ⟨⟨e⟩, ⟨is_limit_of_reflects F $ _⟩⟩,
refine (is_limit.equiv_of_nat_iso_of_iso (cospan_comp_iso F h i) _ _
(walking_cospan.ext _ _ _)).symm H.is_limit,
exacts [iso.refl _, (category.comp_id _).trans (category.id_comp _).symm,
(category.comp_id _).trans (category.id_comp _).symm]
end | lemma | category_theory.is_pullback.of_map | 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_pullback.of_map_of_faithful [reflects_limit (cospan h i) F] [faithful F]
(H : is_pullback (F.map f) (F.map g) (F.map h) (F.map i)) : is_pullback f g h i | H.of_map F (F.map_injective $ by simpa only [F.map_comp] using H.w) | lemma | category_theory.is_pullback.of_map_of_faithful | 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_pullback.map_iff {D : Type*} [category D] (F : C ⥤ D)
[preserves_limit (cospan h i) F] [reflects_limit (cospan h i) F] (e : f ≫ h = g ≫ i) :
is_pullback (F.map f) (F.map g) (F.map h) (F.map i) ↔ is_pullback f g h i | ⟨λ h, h.of_map F e, λ h, h.map F⟩ | lemma | category_theory.is_pullback.map_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 | |
is_pushout.of_map [reflects_colimit (span f g) F] (e : f ≫ h = g ≫ i)
(H : is_pushout (F.map f) (F.map g) (F.map h) (F.map i)) : is_pushout f g h i | begin
refine ⟨⟨e⟩, ⟨is_colimit_of_reflects F $ _⟩⟩,
refine (is_colimit.equiv_of_nat_iso_of_iso (span_comp_iso F f g) _ _
(walking_span.ext _ _ _)).symm H.is_colimit,
exacts [iso.refl _, (category.comp_id _).trans (category.id_comp _),
(category.comp_id _).trans (category.id_comp _)]
end | lemma | category_theory.is_pushout.of_map | 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_pushout.of_map_of_faithful [reflects_colimit (span f g) F] [faithful F]
(H : is_pushout (F.map f) (F.map g) (F.map h) (F.map i)) : is_pushout f g h i | H.of_map F (F.map_injective $ by simpa only [F.map_comp] using H.w) | lemma | category_theory.is_pushout.of_map_of_faithful | 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_pushout.map_iff {D : Type*} [category D] (F : C ⥤ D)
[preserves_colimit (span f g) F] [reflects_colimit (span f g) F] (e : f ≫ h = g ≫ i) :
is_pushout (F.map f) (F.map g) (F.map h) (F.map i) ↔ is_pushout f g h i | ⟨λ h, h.of_map F e, λ h, h.map F⟩ | lemma | category_theory.is_pushout.map_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 | |
diagonal_obj : C | pullback f f | abbreviation | category_theory.limits.pullback.diagonal_obj | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | The diagonal object of a morphism `f : X ⟶ Y` is `Δ_{X/Y} := pullback f f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonal : X ⟶ diagonal_obj f | pullback.lift (𝟙 _) (𝟙 _) rfl | def | category_theory.limits.pullback.diagonal | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | The diagonal morphism `X ⟶ Δ_{X/Y}` for a morphism `f : X ⟶ Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonal_fst : diagonal f ≫ pullback.fst = 𝟙 _ | pullback.lift_fst _ _ _ | lemma | category_theory.limits.pullback.diagonal_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_snd : diagonal f ≫ pullback.snd = 𝟙 _ | pullback.lift_snd _ _ _ | lemma | category_theory.limits.pullback.diagonal_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_is_kernel_pair :
is_kernel_pair f (pullback.fst : diagonal_obj f ⟶ _) pullback.snd | is_pullback.of_has_pullback f f | lemma | category_theory.limits.pullback.diagonal_is_kernel_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | The two projections `Δ_{X/Y} ⟶ X` form a kernel pair for `f : X ⟶ Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_diagonal_map_snd_fst_fst :
(pullback.snd : pullback (diagonal f) (map (i₁ ≫ snd) (i₂ ≫ snd) f f (i₁ ≫ fst) (i₂ ≫ fst) i
(by simp [condition]) (by simp [condition])) ⟶ _) ≫ fst ≫ i₁ ≫ fst = pullback.fst | begin
conv_rhs { rw ← category.comp_id pullback.fst },
rw [← diagonal_fst f, pullback.condition_assoc, pullback.lift_fst]
end | lemma | category_theory.limits.pullback_diagonal_map_snd_fst_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_diagonal_map_snd_snd_fst :
(pullback.snd : pullback (diagonal f) (map (i₁ ≫ snd) (i₂ ≫ snd) f f (i₁ ≫ fst) (i₂ ≫ fst) i
(by simp [condition]) (by simp [condition])) ⟶ _) ≫ snd ≫ i₂ ≫ fst = pullback.fst | begin
conv_rhs { rw ← category.comp_id pullback.fst },
rw [← diagonal_snd f, pullback.condition_assoc, pullback.lift_snd]
end | lemma | category_theory.limits.pullback_diagonal_map_snd_snd_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_diagonal_map_iso :
pullback (diagonal f) (map (i₁ ≫ snd) (i₂ ≫ snd) f f (i₁ ≫ fst) (i₂ ≫ fst) i
(by simp [condition]) (by simp [condition])) ≅ pullback i₁ i₂ | { hom := pullback.lift (pullback.snd ≫ pullback.fst) (pullback.snd ≫ pullback.snd)
begin
ext; simp only [category.assoc, pullback.condition, pullback_diagonal_map_snd_fst_fst,
pullback_diagonal_map_snd_snd_fst],
end,
inv := pullback.lift (pullback.fst ≫ i₁ ≫ pullback.fst) (pullback.map _ _ _ _ (... | def | category_theory.limits.pullback_diagonal_map_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | This iso witnesses the fact that
given `f : X ⟶ Y`, `i : U ⟶ Y`, and `i₁ : V₁ ⟶ X ×[Y] U`, `i₂ : V₂ ⟶ X ×[Y] U`, the diagram
V₁ ×[X ×[Y] U] V₂ ⟶ V₁ ×[U] V₂
| |
| |
↓ ↓
X ⟶ X ×[Y] X
is a pullback square.
Also see `pullback_fst_map... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_diagonal_map_iso_hom_fst :
(pullback_diagonal_map_iso f i i₁ i₂).hom ≫ pullback.fst = pullback.snd ≫ pullback.fst | by { delta pullback_diagonal_map_iso, simp } | lemma | category_theory.limits.pullback_diagonal_map_iso_hom_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_diagonal_map_iso_hom_snd :
(pullback_diagonal_map_iso f i i₁ i₂).hom ≫ pullback.snd = pullback.snd ≫ pullback.snd | by { delta pullback_diagonal_map_iso, simp } | lemma | category_theory.limits.pullback_diagonal_map_iso_hom_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_diagonal_map_iso_inv_fst :
(pullback_diagonal_map_iso f i i₁ i₂).inv ≫ pullback.fst = pullback.fst ≫ i₁ ≫ pullback.fst | by { delta pullback_diagonal_map_iso, simp } | lemma | category_theory.limits.pullback_diagonal_map_iso_inv_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_diagonal_map_iso_inv_snd_fst :
(pullback_diagonal_map_iso f i i₁ i₂).inv ≫ pullback.snd ≫ pullback.fst = pullback.fst | by { delta pullback_diagonal_map_iso, simp } | lemma | category_theory.limits.pullback_diagonal_map_iso_inv_snd_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_diagonal_map_iso_inv_snd_snd :
(pullback_diagonal_map_iso f i i₁ i₂).inv ≫ pullback.snd ≫ pullback.snd = pullback.snd | by { delta pullback_diagonal_map_iso, simp } | lemma | category_theory.limits.pullback_diagonal_map_iso_inv_snd_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_fst_map_snd_is_pullback :
is_pullback
(fst ≫ i₁ ≫ fst)
(map i₁ i₂ (i₁ ≫ snd) (i₂ ≫ snd) _ _ _ (category.id_comp _).symm (category.id_comp _).symm)
(diagonal f)
(map (i₁ ≫ snd) (i₂ ≫ snd) f f (i₁ ≫ fst) (i₂ ≫ fst) i
(by simp [condition]) (by simp [condition])) | is_pullback.of_iso_pullback ⟨by ext; simp [condition_assoc]⟩
(pullback_diagonal_map_iso f i i₁ i₂).symm (pullback_diagonal_map_iso_inv_fst f i i₁ i₂)
(by ext1; simp) | lemma | category_theory.limits.pullback_fst_map_snd_is_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_diagonal_map_id_iso :
pullback (diagonal i) (pullback.map (f ≫ i) (g ≫ i) i i f g (𝟙 _)
(category.comp_id _) (category.comp_id _)) ≅ pullback f g | begin
refine (as_iso $ pullback.map _ _ _ _ (𝟙 _) (pullback.congr_hom _ _).hom (𝟙 _) _ _) ≪≫
pullback_diagonal_map_iso i (𝟙 _) (f ≫ inv pullback.fst) (g ≫ inv pullback.fst) ≪≫
(as_iso $ pullback.map _ _ _ _ (𝟙 _) (𝟙 _) pullback.fst _ _),
{ rw [← category.comp_id pullback.snd, ← condition, category.as... | def | category_theory.limits.pullback_diagonal_map_id_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | This iso witnesses the fact that
given `f : X ⟶ T`, `g : Y ⟶ T`, and `i : T ⟶ S`, the diagram
X ×ₜ Y ⟶ X ×ₛ Y
| |
| |
↓ ↓
T ⟶ T ×ₛ T
is a pullback square.
Also see `pullback_map_diagonal_is_pullback`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_diagonal_map_id_iso_hom_fst :
(pullback_diagonal_map_id_iso f g i).hom ≫ pullback.fst = pullback.snd ≫ pullback.fst | by { delta pullback_diagonal_map_id_iso, simp } | lemma | category_theory.limits.pullback_diagonal_map_id_iso_hom_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_diagonal_map_id_iso_hom_snd :
(pullback_diagonal_map_id_iso f g i).hom ≫ pullback.snd = pullback.snd ≫ pullback.snd | by { delta pullback_diagonal_map_id_iso, simp } | lemma | category_theory.limits.pullback_diagonal_map_id_iso_hom_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_diagonal_map_id_iso_inv_fst :
(pullback_diagonal_map_id_iso f g i).inv ≫ pullback.fst = pullback.fst ≫ f | begin
rw [iso.inv_comp_eq, ← category.comp_id pullback.fst, ← diagonal_fst i, pullback.condition_assoc],
simp,
end | lemma | category_theory.limits.pullback_diagonal_map_id_iso_inv_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_diagonal_map_id_iso_inv_snd_fst :
(pullback_diagonal_map_id_iso f g i).inv ≫ pullback.snd ≫ pullback.fst = pullback.fst | by { rw iso.inv_comp_eq, simp } | lemma | category_theory.limits.pullback_diagonal_map_id_iso_inv_snd_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_diagonal_map_id_iso_inv_snd_snd :
(pullback_diagonal_map_id_iso f g i).inv ≫ pullback.snd ≫ pullback.snd = pullback.snd | by { rw iso.inv_comp_eq, simp } | lemma | category_theory.limits.pullback_diagonal_map_id_iso_inv_snd_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback.diagonal_comp (f : X ⟶ Y) (g : Y ⟶ Z) [has_pullback f f] [has_pullback g g]
[has_pullback (f ≫ g) (f ≫ g)] :
diagonal (f ≫ g) = diagonal f ≫ (pullback_diagonal_map_id_iso f f g).inv ≫ pullback.snd | by ext; simp | lemma | category_theory.limits.pullback.diagonal_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_map_diagonal_is_pullback : is_pullback (pullback.fst ≫ f)
(pullback.map f g (f ≫ i) (g ≫ i) _ _ i (category.id_comp _).symm (category.id_comp _).symm)
(diagonal i)
(pullback.map (f ≫ i) (g ≫ i) i i f g (𝟙 _) (category.comp_id _) (category.comp_id _)) | begin
apply is_pullback.of_iso_pullback _ (pullback_diagonal_map_id_iso f g i).symm,
{ simp },
{ ext; simp },
{ constructor, ext; simp [condition] },
end | lemma | category_theory.limits.pullback_map_diagonal_is_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_obj_pullback_fst_iso {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :
diagonal_obj (pullback.fst : pullback f g ⟶ X) ≅
pullback (pullback.snd ≫ g : diagonal_obj g ⟶ Z) f | pullback_right_pullback_fst_iso _ _ _ ≪≫ pullback.congr_hom pullback.condition rfl ≪≫
pullback_assoc _ _ _ _ ≪≫ pullback_symmetry _ _ ≪≫ pullback.congr_hom pullback.condition rfl | def | category_theory.limits.diagonal_obj_pullback_fst_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | The diagonal object of `X ×[Z] Y ⟶ X` is isomorphic to `Δ_{Y/Z} ×[Z] X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonal_obj_pullback_fst_iso_hom_fst_fst {X Y Z : C} (f : X ⟶ Z)
(g : Y ⟶ Z) :
(diagonal_obj_pullback_fst_iso f g).hom ≫ pullback.fst ≫ pullback.fst =
pullback.fst ≫ pullback.snd | by { delta diagonal_obj_pullback_fst_iso, simp } | lemma | category_theory.limits.diagonal_obj_pullback_fst_iso_hom_fst_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_obj_pullback_fst_iso_hom_fst_snd {X Y Z : C} (f : X ⟶ Z)
(g : Y ⟶ Z) :
(diagonal_obj_pullback_fst_iso f g).hom ≫ pullback.fst ≫ pullback.snd =
pullback.snd ≫ pullback.snd | by { delta diagonal_obj_pullback_fst_iso, simp } | lemma | category_theory.limits.diagonal_obj_pullback_fst_iso_hom_fst_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_obj_pullback_fst_iso_hom_snd {X Y Z : C} (f : X ⟶ Z)
(g : Y ⟶ Z) :
(diagonal_obj_pullback_fst_iso f g).hom ≫ pullback.snd = pullback.fst ≫ pullback.fst | by { delta diagonal_obj_pullback_fst_iso, simp } | lemma | category_theory.limits.diagonal_obj_pullback_fst_iso_hom_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_obj_pullback_fst_iso_inv_fst_fst {X Y Z : C} (f : X ⟶ Z)
(g : Y ⟶ Z) :
(diagonal_obj_pullback_fst_iso f g).inv ≫ pullback.fst ≫ pullback.fst =
pullback.snd | by { delta diagonal_obj_pullback_fst_iso, simp } | lemma | category_theory.limits.diagonal_obj_pullback_fst_iso_inv_fst_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_obj_pullback_fst_iso_inv_fst_snd {X Y Z : C} (f : X ⟶ Z)
(g : Y ⟶ Z) :
(diagonal_obj_pullback_fst_iso f g).inv ≫ pullback.fst ≫ pullback.snd =
pullback.fst ≫ pullback.fst | by { delta diagonal_obj_pullback_fst_iso, simp } | lemma | category_theory.limits.diagonal_obj_pullback_fst_iso_inv_fst_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_obj_pullback_fst_iso_inv_snd_fst {X Y Z : C} (f : X ⟶ Z)
(g : Y ⟶ Z) :
(diagonal_obj_pullback_fst_iso f g).inv ≫ pullback.snd ≫ pullback.fst = pullback.snd | by { delta diagonal_obj_pullback_fst_iso, simp } | lemma | category_theory.limits.diagonal_obj_pullback_fst_iso_inv_snd_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_obj_pullback_fst_iso_inv_snd_snd {X Y Z : C} (f : X ⟶ Z)
(g : Y ⟶ Z) :
(diagonal_obj_pullback_fst_iso f g).inv ≫ pullback.snd ≫ pullback.snd =
pullback.fst ≫ pullback.snd | by { delta diagonal_obj_pullback_fst_iso, simp } | lemma | category_theory.limits.diagonal_obj_pullback_fst_iso_inv_snd_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal_pullback_fst {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :
diagonal (pullback.fst : pullback f g ⟶ _) =
(pullback_symmetry _ _).hom ≫ ((base_change f).map
(over.hom_mk (diagonal g) (by simp) : over.mk g ⟶ over.mk (pullback.snd ≫ g))).left ≫
(diagonal_obj_pullback_fst_iso f g).inv | by ext; simp | lemma | category_theory.limits.diagonal_pullback_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_fst_fst_iso {X Y S X' Y' S' : C} (f : X ⟶ S) (g : Y ⟶ S) (f' : X' ⟶ S')
(g' : Y' ⟶ S') (i₁ : X ⟶ X') (i₂ : Y ⟶ Y') (i₃ : S ⟶ S') (e₁ : f ≫ i₃ = i₁ ≫ f')
(e₂ : g ≫ i₃ = i₂ ≫ g') [mono i₃] :
pullback (pullback.fst : pullback (pullback.fst : pullback f' g' ⟶ _) i₁ ⟶ _)
(pullback.fst : pullback (pull... | { hom := pullback.lift (pullback.fst ≫ pullback.snd) (pullback.snd ≫ pullback.snd)
begin
rw [← cancel_mono i₃, category.assoc, category.assoc, category.assoc, category.assoc, e₁, e₂,
← pullback.condition_assoc, pullback.condition_assoc, pullback.condition,
pullback.condition_assoc]
end,
... | def | category_theory.limits.pullback_fst_fst_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | Given the following diagram with `S ⟶ S'` a monomorphism,
X ⟶ X'
↘ ↘
S ⟶ S'
↗ ↗
Y ⟶ Y'
This iso witnesses the fact that
X ×[S] Y ⟶ (X' ×[S'] Y') ×[Y'] Y
| |
| |
↓ ↓
(X' ×[S'] Y') ×[X'] X ⟶ ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_map_eq_pullback_fst_fst_iso_inv {X Y S X' Y' S' : C} (f : X ⟶ S) (g : Y ⟶ S)
(f' : X' ⟶ S')
(g' : Y' ⟶ S') (i₁ : X ⟶ X') (i₂ : Y ⟶ Y') (i₃ : S ⟶ S') (e₁ : f ≫ i₃ = i₁ ≫ f')
(e₂ : g ≫ i₃ = i₂ ≫ g') [mono i₃] :
pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂ =
(pullback_fst_fst_iso f g f' g' i₁ i₂ i₃ e₁ e₂).in... | begin
ext; simp only [category.assoc, category.id_comp, lift_fst, lift_snd, lift_fst_assoc,
lift_snd_assoc, pullback_fst_fst_iso_inv, ← pullback.condition, ← pullback.condition_assoc],
end | lemma | category_theory.limits.pullback_map_eq_pullback_fst_fst_iso_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_lift_map_is_pullback {X Y S X' Y' S' : C} (f : X ⟶ S) (g : Y ⟶ S) (f' : X' ⟶ S')
(g' : Y' ⟶ S') (i₁ : X ⟶ X') (i₂ : Y ⟶ Y') (i₃ : S ⟶ S') (e₁ : f ≫ i₃ = i₁ ≫ f')
(e₂ : g ≫ i₃ = i₂ ≫ g') [mono i₃] :
is_pullback
(pullback.lift (pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂) fst (lift_fst _ _ _))
(pullback.... | is_pullback.of_iso_pullback ⟨by rw [lift_fst, lift_fst]⟩
(pullback_fst_fst_iso f g f' g' i₁ i₂ i₃ e₁ e₂).symm (by simp) (by simp) | lemma | category_theory.limits.pullback_lift_map_is_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/diagonal.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.kernel_pair",
"category_theory.limits.shapes.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coproduct_disjoint (X₁ X₂ : C) | (is_initial_of_is_pullback_of_is_coproduct :
∀ {X Z} {pX₁ : X₁ ⟶ X} {pX₂ : X₂ ⟶ X} {f : Z ⟶ X₁} {g : Z ⟶ X₂}
(cX : is_colimit (binary_cofan.mk pX₁ pX₂)) {comm : f ≫ pX₁ = g ≫ pX₂},
is_limit (pullback_cone.mk _ _ comm) → is_initial Z)
(mono_inl : ∀ X (X₁ : X₁ ⟶ X) (X₂ : X₂ ⟶ X) (cX : is_colimit (binary_cofan.m... | class | category_theory.limits.coproduct_disjoint | category_theory.limits.shapes | src/category_theory/limits/shapes/disjoint_coproduct.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks"
] | [
"comm"
] | Given any pullback diagram of the form
Z ⟶ X₁
↓ ↓
X₂ ⟶ X
where `X₁ ⟶ X ← X₂` is a coproduct diagram, then `Z` is initial, and both `X₁ ⟶ X` and `X₂ ⟶ X`
are mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial_of_is_pullback_of_is_coproduct {Z X₁ X₂ X : C} [coproduct_disjoint X₁ X₂]
{pX₁ : X₁ ⟶ X} {pX₂ : X₂ ⟶ X} (cX : is_colimit (binary_cofan.mk pX₁ pX₂))
{f : Z ⟶ X₁} {g : Z ⟶ X₂} {comm : f ≫ pX₁ = g ≫ pX₂}
(cZ : is_limit (pullback_cone.mk _ _ comm)) :
is_initial Z | coproduct_disjoint.is_initial_of_is_pullback_of_is_coproduct cX cZ | def | category_theory.limits.is_initial_of_is_pullback_of_is_coproduct | category_theory.limits.shapes | src/category_theory/limits/shapes/disjoint_coproduct.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks"
] | [
"comm"
] | If the coproduct of `X₁` and `X₂` is disjoint, then given any pullback square
Z ⟶ X₁
↓ ↓
X₂ ⟶ X
where `X₁ ⟶ X ← X₂` is a coproduct, then `Z` is initial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial_of_is_pullback_of_coproduct {Z X₁ X₂ : C}
[has_binary_coproduct X₁ X₂] [coproduct_disjoint X₁ X₂]
{f : Z ⟶ X₁} {g : Z ⟶ X₂} {comm : f ≫ (coprod.inl : X₁ ⟶ _ ⨿ X₂) = g ≫ coprod.inr}
(cZ : is_limit (pullback_cone.mk _ _ comm)) :
is_initial Z | coproduct_disjoint.is_initial_of_is_pullback_of_is_coproduct (coprod_is_coprod _ _) cZ | def | category_theory.limits.is_initial_of_is_pullback_of_coproduct | category_theory.limits.shapes | src/category_theory/limits/shapes/disjoint_coproduct.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks"
] | [
"comm"
] | If the coproduct of `X₁` and `X₂` is disjoint, then given any pullback square
Z ⟶ X₁
↓ ↓
X₂ ⟶ X₁ ⨿ X₂
`Z` is initial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial_of_pullback_of_is_coproduct {X X₁ X₂ : C} [coproduct_disjoint X₁ X₂]
{pX₁ : X₁ ⟶ X} {pX₂ : X₂ ⟶ X} [has_pullback pX₁ pX₂]
(cX : is_colimit (binary_cofan.mk pX₁ pX₂)) :
is_initial (pullback pX₁ pX₂) | coproduct_disjoint.is_initial_of_is_pullback_of_is_coproduct cX (pullback_is_pullback _ _) | def | category_theory.limits.is_initial_of_pullback_of_is_coproduct | category_theory.limits.shapes | src/category_theory/limits/shapes/disjoint_coproduct.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks"
] | [] | If the coproduct of `X₁` and `X₂` is disjoint, then provided `X₁ ⟶ X ← X₂` is a coproduct the
pullback is an initial object:
X₁
↓
X₂ ⟶ X | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial_of_pullback_of_coproduct {X₁ X₂ : C}
[has_binary_coproduct X₁ X₂] [coproduct_disjoint X₁ X₂]
[has_pullback (coprod.inl : X₁ ⟶ _ ⨿ X₂) coprod.inr] :
is_initial (pullback (coprod.inl : X₁ ⟶ _ ⨿ X₂) coprod.inr) | is_initial_of_is_pullback_of_coproduct (pullback_is_pullback _ _) | def | category_theory.limits.is_initial_of_pullback_of_coproduct | category_theory.limits.shapes | src/category_theory/limits/shapes/disjoint_coproduct.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks"
] | [] | If the coproduct of `X₁` and `X₂` is disjoint, the pullback of `X₁ ⟶ X₁ ⨿ X₂` and `X₂ ⟶ X₁ ⨿ X₂`
is initial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coproducts_disjoint (C : Type u) [category.{v} C] | (coproduct_disjoint : ∀ (X Y : C), coproduct_disjoint X Y) | class | category_theory.limits.coproducts_disjoint | category_theory.limits.shapes | src/category_theory/limits/shapes/disjoint_coproduct.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks"
] | [] | `C` has disjoint coproducts if every coproduct is disjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial_mono_class_of_disjoint_coproducts [coproducts_disjoint C] : initial_mono_class C | { is_initial_mono_from := λ I X hI,
coproduct_disjoint.mono_inl _ _ (𝟙 X)
{ desc := λ (s : binary_cofan _ _), s.inr,
fac' := λ s j, discrete.cases_on j
(λ j, walking_pair.cases_on j (hI.hom_ext _ _) (id_comp _)),
uniq' := λ (s : binary_cofan _ _) m w, (id_comp _).symm.trans (w ⟨walk... | lemma | category_theory.limits.initial_mono_class_of_disjoint_coproducts | category_theory.limits.shapes | src/category_theory/limits/shapes/disjoint_coproduct.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks"
] | [] | If `C` has disjoint coproducts, any morphism out of initial is mono. Note it isn't true in
general that `C` has strict initial objects, for instance consider the category of types and
partial functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_parallel_pair : Type
| zero | one | inductive | category_theory.limits.walking_parallel_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The type of objects for the diagram indexing a (co)equalizer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_parallel_pair_hom :
walking_parallel_pair → walking_parallel_pair → Type
| left : walking_parallel_pair_hom zero one
| right : walking_parallel_pair_hom zero one
| id : Π X : walking_parallel_pair, walking_parallel_pair_hom X X | inductive | category_theory.limits.walking_parallel_pair_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | The type family of morphisms for the diagram indexing a (co)equalizer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_parallel_pair_hom.comp :
Π (X Y Z : walking_parallel_pair)
(f : walking_parallel_pair_hom X Y) (g : walking_parallel_pair_hom Y Z),
walking_parallel_pair_hom X Z | | _ _ _ (id _) h := h
| _ _ _ left (id one) := left
| _ _ _ right (id one) := right | def | category_theory.limits.walking_parallel_pair_hom.comp | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | Composition of morphisms in the indexing diagram for (co)equalizers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_parallel_pair_hom_category : small_category walking_parallel_pair | { hom := walking_parallel_pair_hom,
id := walking_parallel_pair_hom.id,
comp := walking_parallel_pair_hom.comp } | instance | category_theory.limits.walking_parallel_pair_hom_category | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_parallel_pair_hom_id (X : walking_parallel_pair) :
walking_parallel_pair_hom.id X = 𝟙 X | rfl | lemma | category_theory.limits.walking_parallel_pair_hom_id | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_parallel_pair_op : walking_parallel_pair ⥤ walking_parallel_pairᵒᵖ | { obj := (λ x, op $ by { cases x, exacts [one, zero] }),
map := λ i j f, by { cases f; apply quiver.hom.op, exacts [left, right,
walking_parallel_pair_hom.id _] },
map_comp' := by { rintros (_|_) (_|_) (_|_) (_|_|_) (_|_|_); refl } } | def | category_theory.limits.walking_parallel_pair_op | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [
"quiver.hom.op"
] | The functor `walking_parallel_pair ⥤ walking_parallel_pairᵒᵖ` sending left to left and right to
right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_parallel_pair_op_zero :
walking_parallel_pair_op.obj zero = op one | rfl | lemma | category_theory.limits.walking_parallel_pair_op_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_parallel_pair_op_one :
walking_parallel_pair_op.obj one = op zero | rfl | lemma | category_theory.limits.walking_parallel_pair_op_one | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_parallel_pair_op_left :
walking_parallel_pair_op.map left = @quiver.hom.op _ _ zero one left | rfl | lemma | category_theory.limits.walking_parallel_pair_op_left | category_theory.limits.shapes | src/category_theory/limits/shapes/equalizers.lean | [
"category_theory.epi_mono",
"category_theory.limits.has_limits"
] | [
"quiver.hom.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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