statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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|---|---|---|---|---|---|---|---|---|---|---|
pullback {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) [has_pullback f g] | limit (cospan f g) | abbreviation | category_theory.limits.pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `pullback f g` computes the pullback of a pair of morphisms with the same target. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) [has_pushout f g] | colimit (span f g) | abbreviation | category_theory.limits.pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `pushout f g` computes the pushout of a pair of morphisms with the same source. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback.fst {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} [has_pullback f g] :
pullback f g ⟶ X | limit.π (cospan f g) walking_cospan.left | abbreviation | category_theory.limits.pullback.fst | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The first projection of the pullback of `f` and `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback.snd {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} [has_pullback f g] :
pullback f g ⟶ Y | limit.π (cospan f g) walking_cospan.right | abbreviation | category_theory.limits.pullback.snd | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The second projection of the pullback of `f` and `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout.inl {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} [has_pushout f g] :
Y ⟶ pushout f g | colimit.ι (span f g) walking_span.left | abbreviation | category_theory.limits.pushout.inl | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The first inclusion into the pushout of `f` and `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout.inr {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} [has_pushout f g] :
Z ⟶ pushout f g | colimit.ι (span f g) walking_span.right | abbreviation | category_theory.limits.pushout.inr | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The second inclusion into the pushout of `f` and `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback.lift {W X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} [has_pullback f g]
(h : W ⟶ X) (k : W ⟶ Y) (w : h ≫ f = k ≫ g) : W ⟶ pullback f g | limit.lift _ (pullback_cone.mk h k w) | abbreviation | category_theory.limits.pullback.lift | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | A pair of morphisms `h : W ⟶ X` and `k : W ⟶ Y` satisfying `h ≫ f = k ≫ g` induces a morphism
`pullback.lift : W ⟶ pullback f g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout.desc {W X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} [has_pushout f g]
(h : Y ⟶ W) (k : Z ⟶ W) (w : f ≫ h = g ≫ k) : pushout f g ⟶ W | colimit.desc _ (pushout_cocone.mk h k w) | abbreviation | category_theory.limits.pushout.desc | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | A pair of morphisms `h : Y ⟶ W` and `k : Z ⟶ W` satisfying `f ≫ h = g ≫ k` induces a morphism
`pushout.desc : pushout f g ⟶ W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_cone.fst_colimit_cocone {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z)
[has_limit (cospan f g)] : pullback_cone.fst (limit.cone (cospan f g)) = pullback.fst | rfl | lemma | category_theory.limits.pullback_cone.fst_colimit_cocone | 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_cone.snd_colimit_cocone {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z)
[has_limit (cospan f g)] : pullback_cone.snd (limit.cone (cospan f g)) = pullback.snd | rfl | lemma | category_theory.limits.pullback_cone.snd_colimit_cocone | 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.inl_colimit_cocone {X Y Z : C} (f : Z ⟶ X) (g : Z ⟶ Y)
[has_colimit (span f g)] : pushout_cocone.inl (colimit.cocone (span f g)) = pushout.inl | rfl | lemma | category_theory.limits.pushout_cocone.inl_colimit_cocone | 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.inr_colimit_cocone {X Y Z : C} (f : Z ⟶ X) (g : Z ⟶ Y)
[has_colimit (span f g)] : pushout_cocone.inr (colimit.cocone (span f g)) = pushout.inr | rfl | lemma | category_theory.limits.pushout_cocone.inr_colimit_cocone | 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.lift_fst {W X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} [has_pullback f g]
(h : W ⟶ X) (k : W ⟶ Y) (w : h ≫ f = k ≫ g) : pullback.lift h k w ≫ pullback.fst = h | limit.lift_π _ _ | lemma | category_theory.limits.pullback.lift_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.lift_snd {W X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} [has_pullback f g]
(h : W ⟶ X) (k : W ⟶ Y) (w : h ≫ f = k ≫ g) : pullback.lift h k w ≫ pullback.snd = k | limit.lift_π _ _ | lemma | category_theory.limits.pullback.lift_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.inl_desc {W X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} [has_pushout f g]
(h : Y ⟶ W) (k : Z ⟶ W) (w : f ≫ h = g ≫ k) : pushout.inl ≫ pushout.desc h k w = h | colimit.ι_desc _ _ | lemma | category_theory.limits.pushout.inl_desc | 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_desc {W X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} [has_pushout f g]
(h : Y ⟶ W) (k : Z ⟶ W) (w : f ≫ h = g ≫ k) : pushout.inr ≫ pushout.desc h k w = k | colimit.ι_desc _ _ | lemma | category_theory.limits.pushout.inr_desc | 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.lift' {W X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} [has_pullback f g]
(h : W ⟶ X) (k : W ⟶ Y) (w : h ≫ f = k ≫ g) :
{l : W ⟶ pullback f g // l ≫ pullback.fst = h ∧ l ≫ pullback.snd = k} | ⟨pullback.lift h k w, pullback.lift_fst _ _ _, pullback.lift_snd _ _ _⟩ | def | category_theory.limits.pullback.lift' | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | A pair of morphisms `h : W ⟶ X` and `k : W ⟶ Y` satisfying `h ≫ f = k ≫ g` induces a morphism
`l : W ⟶ pullback f g` such that `l ≫ pullback.fst = h` and `l ≫ pullback.snd = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback.desc' {W X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} [has_pushout f g]
(h : Y ⟶ W) (k : Z ⟶ W) (w : f ≫ h = g ≫ k) :
{l : pushout f g ⟶ W // pushout.inl ≫ l = h ∧ pushout.inr ≫ l = k} | ⟨pushout.desc h k w, pushout.inl_desc _ _ _, pushout.inr_desc _ _ _⟩ | def | category_theory.limits.pullback.desc' | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | A pair of morphisms `h : Y ⟶ W` and `k : Z ⟶ W` satisfying `f ≫ h = g ≫ k` induces a morphism
`l : pushout f g ⟶ W` such that `pushout.inl ≫ l = h` and `pushout.inr ≫ l = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback.condition {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} [has_pullback f g] :
(pullback.fst : pullback f g ⟶ X) ≫ f = pullback.snd ≫ g | pullback_cone.condition _ | lemma | category_theory.limits.pullback.condition | 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.condition {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} [has_pushout f g] :
f ≫ (pushout.inl : Y ⟶ pushout f g) = g ≫ pushout.inr | pushout_cocone.condition _ | lemma | category_theory.limits.pushout.condition | 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.map {W X Y Z S T : C} (f₁ : W ⟶ S) (f₂ : X ⟶ S) [has_pullback f₁ f₂]
(g₁ : Y ⟶ T) (g₂ : Z ⟶ T) [has_pullback g₁ g₂] (i₁ : W ⟶ Y) (i₂ : X ⟶ Z) (i₃ : S ⟶ T)
(eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁) (eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂) : pullback f₁ f₂ ⟶ pullback g₁ g₂ | pullback.lift (pullback.fst ≫ i₁) (pullback.snd ≫ i₂)
(by simp [← eq₁, ← eq₂, pullback.condition_assoc]) | abbreviation | category_theory.limits.pullback.map | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Given such a diagram, then there is a natural morphism `W ×ₛ X ⟶ Y ×ₜ Z`.
W ⟶ Y
↘ ↘
S ⟶ T
↗ ↗
X ⟶ Z | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback.map_desc {X Y S T : C} (f : X ⟶ S) (g : Y ⟶ S) (i : S ⟶ T)
[has_pullback f g] [has_pullback (f ≫ i) (g ≫ i)] :
pullback f g ⟶ pullback (f ≫ i) (g ≫ i) | pullback.map f g (f ≫ i) (g ≫ i) (𝟙 _) (𝟙 _) i (category.id_comp _).symm (category.id_comp _).symm | abbreviation | category_theory.limits.pullback.map_desc | 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 map `X ×ₛ Y ⟶ X ×ₜ Y` given `S ⟶ T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout.map {W X Y Z S T : C} (f₁ : S ⟶ W) (f₂ : S ⟶ X) [has_pushout f₁ f₂]
(g₁ : T ⟶ Y) (g₂ : T ⟶ Z) [has_pushout g₁ g₂] (i₁ : W ⟶ Y) (i₂ : X ⟶ Z) (i₃ : S ⟶ T)
(eq₁ : f₁ ≫ i₁ = i₃ ≫ g₁) (eq₂ : f₂ ≫ i₂ = i₃ ≫ g₂) : pushout f₁ f₂ ⟶ pushout g₁ g₂ | pushout.desc (i₁ ≫ pushout.inl) (i₂ ≫ pushout.inr)
(by { simp only [← category.assoc, eq₁, eq₂], simp [pushout.condition] }) | abbreviation | category_theory.limits.pushout.map | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Given such a diagram, then there is a natural morphism `W ⨿ₛ X ⟶ Y ⨿ₜ Z`.
W ⟶ Y
↗ ↗
S ⟶ T
↘ ↘
X ⟶ Z | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout.map_lift {X Y S T : C} (f : T ⟶ X) (g : T ⟶ Y) (i : S ⟶ T)
[has_pushout f g] [has_pushout (i ≫ f) (i ≫ g)] :
pushout (i ≫ f) (i ≫ g) ⟶ pushout f g | pushout.map (i ≫ f) (i ≫ g) f g (𝟙 _) (𝟙 _) i (category.comp_id _) (category.comp_id _) | abbreviation | category_theory.limits.pushout.map_lift | 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 map `X ⨿ₛ Y ⟶ X ⨿ₜ Y` given `S ⟶ T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback.hom_ext {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} [has_pullback f g]
{W : C} {k l : W ⟶ pullback f g} (h₀ : k ≫ pullback.fst = l ≫ pullback.fst)
(h₁ : k ≫ pullback.snd = l ≫ pullback.snd) : k = l | limit.hom_ext $ pullback_cone.equalizer_ext _ h₀ h₁ | lemma | category_theory.limits.pullback.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Two morphisms into a pullback are equal if their compositions with the pullback morphisms are
equal | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_is_pullback {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) [has_pullback f g] :
is_limit (pullback_cone.mk (pullback.fst : pullback f g ⟶ _) pullback.snd pullback.condition) | pullback_cone.is_limit.mk _ (λ s, pullback.lift s.fst s.snd s.condition)
(by simp) (by simp) (by tidy) | def | category_theory.limits.pullback_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"
] | [] | The pullback cone built from the pullback projections is a pullback. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback.fst_of_mono {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} [has_pullback f g]
[mono g] : mono (pullback.fst : pullback f g ⟶ X) | pullback_cone.mono_fst_of_is_pullback_of_mono (limit.is_limit _) | instance | category_theory.limits.pullback.fst_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"
] | [] | The pullback of a monomorphism is a monomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback.snd_of_mono {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} [has_pullback f g]
[mono f] : mono (pullback.snd : pullback f g ⟶ Y) | pullback_cone.mono_snd_of_is_pullback_of_mono (limit.is_limit _) | instance | category_theory.limits.pullback.snd_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"
] | [] | The pullback of a monomorphism is a monomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_pullback_to_prod {C : Type*} [category C] {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z)
[has_pullback f g] [has_binary_product X Y] :
mono (prod.lift pullback.fst pullback.snd : pullback f g ⟶ _) | ⟨λ W i₁ i₂ h, begin
ext,
{ simpa using congr_arg (λ f, f ≫ prod.fst) h },
{ simpa using congr_arg (λ f, f ≫ prod.snd) h }
end⟩ | instance | category_theory.limits.mono_pullback_to_prod | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The map `X ×[Z] Y ⟶ X × Y` is mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout.hom_ext {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} [has_pushout f g]
{W : C} {k l : pushout f g ⟶ W} (h₀ : pushout.inl ≫ k = pushout.inl ≫ l)
(h₁ : pushout.inr ≫ k = pushout.inr ≫ l) : k = l | colimit.hom_ext $ pushout_cocone.coequalizer_ext _ h₀ h₁ | lemma | category_theory.limits.pushout.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Two morphisms out of a pushout are equal if their compositions with the pushout morphisms are
equal | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout_is_pushout {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) [has_pushout f g] :
is_colimit (pushout_cocone.mk (pushout.inl : _ ⟶ pushout f g) pushout.inr pushout.condition) | pushout_cocone.is_colimit.mk _ (λ s, pushout.desc s.inl s.inr s.condition)
(by simp) (by simp) (by tidy) | def | category_theory.limits.pushout_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"
] | [] | The pushout cocone built from the pushout coprojections is a pushout. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout.inl_of_epi {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} [has_pushout f g] [epi g] :
epi (pushout.inl : Y ⟶ pushout f g) | pushout_cocone.epi_inl_of_is_pushout_of_epi (colimit.is_colimit _) | instance | category_theory.limits.pushout.inl_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"
] | [] | The pushout of an epimorphism is an epimorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout.inr_of_epi {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} [has_pushout f g] [epi f] :
epi (pushout.inr : Z ⟶ pushout f g) | pushout_cocone.epi_inr_of_is_pushout_of_epi (colimit.is_colimit _) | instance | category_theory.limits.pushout.inr_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"
] | [] | The pushout of an epimorphism is an epimorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_coprod_to_pushout {C : Type*} [category C] {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z)
[has_pushout f g] [has_binary_coproduct Y Z] :
epi (coprod.desc pushout.inl pushout.inr : _ ⟶ pushout f g) | ⟨λ W i₁ i₂ h, begin
ext,
{ simpa using congr_arg (λ f, coprod.inl ≫ f) h },
{ simpa using congr_arg (λ f, coprod.inr ≫ f) h }
end⟩ | instance | category_theory.limits.epi_coprod_to_pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The map ` X ⨿ Y ⟶ X ⨿[Z] Y` is epi. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback.map_is_iso {W X Y Z S T : C} (f₁ : W ⟶ S) (f₂ : X ⟶ S) [has_pullback f₁ f₂]
(g₁ : Y ⟶ T) (g₂ : Z ⟶ T) [has_pullback g₁ g₂] (i₁ : W ⟶ Y) (i₂ : X ⟶ Z) (i₃ : S ⟶ T)
(eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁) (eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂) [is_iso i₁] [is_iso i₂] [is_iso i₃] :
is_iso (pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) | begin
refine ⟨⟨pullback.map _ _ _ _ (inv i₁) (inv i₂) (inv i₃) _ _, _, _⟩⟩,
{ rw [is_iso.comp_inv_eq, category.assoc, eq₁, is_iso.inv_hom_id_assoc] },
{ rw [is_iso.comp_inv_eq, category.assoc, eq₂, is_iso.inv_hom_id_assoc] },
tidy
end | instance | category_theory.limits.pullback.map_is_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 | |
pullback.congr_hom {X Y Z : C} {f₁ f₂ : X ⟶ Z} {g₁ g₂ : Y ⟶ Z}
(h₁ : f₁ = f₂) (h₂ : g₁ = g₂) [has_pullback f₁ g₁] [has_pullback f₂ g₂] :
pullback f₁ g₁ ≅ pullback f₂ g₂ | as_iso $ pullback.map _ _ _ _ (𝟙 _) (𝟙 _) (𝟙 _) (by simp [h₁]) (by simp [h₂]) | def | category_theory.limits.pullback.congr_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | If `f₁ = f₂` and `g₁ = g₂`, we may construct a canonical
isomorphism `pullback f₁ g₁ ≅ pullback f₂ g₂` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback.congr_hom_inv {X Y Z : C} {f₁ f₂ : X ⟶ Z} {g₁ g₂ : Y ⟶ Z}
(h₁ : f₁ = f₂) (h₂ : g₁ = g₂) [has_pullback f₁ g₁] [has_pullback f₂ g₂] :
(pullback.congr_hom h₁ h₂).inv =
pullback.map _ _ _ _ (𝟙 _) (𝟙 _) (𝟙 _) (by simp [h₁]) (by simp [h₂]) | begin
apply pullback.hom_ext,
{ erw pullback.lift_fst,
rw iso.inv_comp_eq,
erw pullback.lift_fst_assoc,
rw [category.comp_id, category.comp_id] },
{ erw pullback.lift_snd,
rw iso.inv_comp_eq,
erw pullback.lift_snd_assoc,
rw [category.comp_id, category.comp_id] },
end | lemma | category_theory.limits.pullback.congr_hom_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 | |
pushout.map_is_iso {W X Y Z S T : C} (f₁ : S ⟶ W) (f₂ : S ⟶ X) [has_pushout f₁ f₂]
(g₁ : T ⟶ Y) (g₂ : T ⟶ Z) [has_pushout g₁ g₂] (i₁ : W ⟶ Y) (i₂ : X ⟶ Z) (i₃ : S ⟶ T)
(eq₁ : f₁ ≫ i₁ = i₃ ≫ g₁) (eq₂ : f₂ ≫ i₂ = i₃ ≫ g₂) [is_iso i₁] [is_iso i₂] [is_iso i₃] :
is_iso (pushout.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) | begin
refine ⟨⟨pushout.map _ _ _ _ (inv i₁) (inv i₂) (inv i₃) _ _, _, _⟩⟩,
{ rw [is_iso.comp_inv_eq, category.assoc, eq₁, is_iso.inv_hom_id_assoc] },
{ rw [is_iso.comp_inv_eq, category.assoc, eq₂, is_iso.inv_hom_id_assoc] },
tidy
end | instance | category_theory.limits.pushout.map_is_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 | |
pullback.map_desc_comp {X Y S T S' : C} (f : X ⟶ T) (g : Y ⟶ T) (i : T ⟶ S)
(i' : S ⟶ S') [has_pullback f g] [has_pullback (f ≫ i) (g ≫ i)]
[has_pullback (f ≫ i ≫ i') (g ≫ i ≫ i')] [has_pullback ((f ≫ i) ≫ i') ((g ≫ i) ≫ i')] :
pullback.map_desc f g (i ≫ i') = pullback.map_desc f g i ≫ pullback.map_desc _ _ i' ≫
... | by { ext; simp } | lemma | category_theory.limits.pullback.map_desc_comp | 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.congr_hom {X Y Z : C} {f₁ f₂ : X ⟶ Y} {g₁ g₂ : X ⟶ Z}
(h₁ : f₁ = f₂) (h₂ : g₁ = g₂) [has_pushout f₁ g₁] [has_pushout f₂ g₂] :
pushout f₁ g₁ ≅ pushout f₂ g₂ | as_iso $ pushout.map _ _ _ _ (𝟙 _) (𝟙 _) (𝟙 _) (by simp [h₁]) (by simp [h₂]) | def | category_theory.limits.pushout.congr_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | If `f₁ = f₂` and `g₁ = g₂`, we may construct a canonical
isomorphism `pushout f₁ g₁ ≅ pullback f₂ g₂` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout.congr_hom_inv {X Y Z : C} {f₁ f₂ : X ⟶ Y} {g₁ g₂ : X ⟶ Z}
(h₁ : f₁ = f₂) (h₂ : g₁ = g₂) [has_pushout f₁ g₁] [has_pushout f₂ g₂] :
(pushout.congr_hom h₁ h₂).inv =
pushout.map _ _ _ _ (𝟙 _) (𝟙 _) (𝟙 _) (by simp [h₁]) (by simp [h₂]) | begin
apply pushout.hom_ext,
{ erw pushout.inl_desc,
rw [iso.comp_inv_eq, category.id_comp],
erw pushout.inl_desc,
rw category.id_comp },
{ erw pushout.inr_desc,
rw [iso.comp_inv_eq, category.id_comp],
erw pushout.inr_desc,
rw category.id_comp }
end | lemma | category_theory.limits.pushout.congr_hom_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 | |
pushout.map_lift_comp {X Y S T S' : C} (f : T ⟶ X) (g : T ⟶ Y) (i : S ⟶ T)
(i' : S' ⟶ S) [has_pushout f g] [has_pushout (i ≫ f) (i ≫ g)]
[has_pushout (i' ≫ i ≫ f) (i' ≫ i ≫ g)] [has_pushout ((i' ≫ i) ≫ f) ((i' ≫ i) ≫ g)] :
pushout.map_lift f g (i' ≫ i) =
(pushout.congr_hom (category.assoc _ _ _) (category.ass... | by { ext; simp } | lemma | category_theory.limits.pushout.map_lift_comp | 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_comparison (f : X ⟶ Z) (g : Y ⟶ Z)
[has_pullback f g] [has_pullback (G.map f) (G.map g)] :
G.obj (pullback f g) ⟶ pullback (G.map f) (G.map g) | pullback.lift (G.map pullback.fst) (G.map pullback.snd)
(by simp only [←G.map_comp, pullback.condition]) | def | category_theory.limits.pullback_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The comparison morphism for the pullback of `f,g`.
This is an isomorphism iff `G` preserves the pullback of `f,g`; see
`category_theory/limits/preserves/shapes/pullbacks.lean` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_comparison_comp_fst (f : X ⟶ Z) (g : Y ⟶ Z)
[has_pullback f g] [has_pullback (G.map f) (G.map g)] :
pullback_comparison G f g ≫ pullback.fst = G.map pullback.fst | pullback.lift_fst _ _ _ | lemma | category_theory.limits.pullback_comparison_comp_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_comparison_comp_snd (f : X ⟶ Z) (g : Y ⟶ Z)
[has_pullback f g] [has_pullback (G.map f) (G.map g)] :
pullback_comparison G f g ≫ pullback.snd = G.map pullback.snd | pullback.lift_snd _ _ _ | lemma | category_theory.limits.pullback_comparison_comp_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 | |
map_lift_pullback_comparison (f : X ⟶ Z) (g : Y ⟶ Z)
[has_pullback f g] [has_pullback (G.map f) (G.map g)]
{W : C} {h : W ⟶ X} {k : W ⟶ Y} (w : h ≫ f = k ≫ g) :
G.map (pullback.lift _ _ w) ≫ pullback_comparison G f g =
pullback.lift (G.map h) (G.map k) (by simp only [←G.map_comp, w]) | by { ext; simp [← G.map_comp] } | lemma | category_theory.limits.map_lift_pullback_comparison | 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_comparison (f : X ⟶ Y) (g : X ⟶ Z)
[has_pushout f g] [has_pushout (G.map f) (G.map g)] :
pushout (G.map f) (G.map g) ⟶ G.obj (pushout f g) | pushout.desc (G.map pushout.inl) (G.map pushout.inr)
(by simp only [←G.map_comp, pushout.condition]) | def | category_theory.limits.pushout_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The comparison morphism for the pushout of `f,g`.
This is an isomorphism iff `G` preserves the pushout of `f,g`; see
`category_theory/limits/preserves/shapes/pullbacks.lean` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl_comp_pushout_comparison (f : X ⟶ Y) (g : X ⟶ Z)
[has_pushout f g] [has_pushout (G.map f) (G.map g)] :
pushout.inl ≫ pushout_comparison G f g = G.map pushout.inl | pushout.inl_desc _ _ _ | lemma | category_theory.limits.inl_comp_pushout_comparison | 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_comp_pushout_comparison (f : X ⟶ Y) (g : X ⟶ Z)
[has_pushout f g] [has_pushout (G.map f) (G.map g)] :
pushout.inr ≫ pushout_comparison G f g = G.map pushout.inr | pushout.inr_desc _ _ _ | lemma | category_theory.limits.inr_comp_pushout_comparison | 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_comparison_map_desc (f : X ⟶ Y) (g : X ⟶ Z)
[has_pushout f g] [has_pushout (G.map f) (G.map g)]
{W : C} {h : Y ⟶ W} {k : Z ⟶ W} (w : f ≫ h = g ≫ k) :
pushout_comparison G f g ≫ G.map (pushout.desc _ _ w) =
pushout.desc (G.map h) (G.map k) (by simp only [←G.map_comp, w]) | by { ext; simp [← G.map_comp] } | lemma | category_theory.limits.pushout_comparison_map_desc | 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_pullback_symmetry [has_pullback f g] : has_pullback g f | ⟨⟨⟨pullback_cone.mk _ _ pullback.condition.symm,
pullback_cone.flip_is_limit (pullback_is_pullback _ _)⟩⟩⟩ | lemma | category_theory.limits.has_pullback_symmetry | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Making this a global instance would make the typeclass seach go in an infinite loop. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_symmetry [has_pullback f g] :
pullback f g ≅ pullback g f | is_limit.cone_point_unique_up_to_iso
(pullback_cone.flip_is_limit (pullback_is_pullback f g) :
is_limit (pullback_cone.mk _ _ pullback.condition.symm))
(limit.is_limit _) | def | category_theory.limits.pullback_symmetry | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The isomorphism `X ×[Z] Y ≅ Y ×[Z] X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_symmetry_hom_comp_fst [has_pullback f g] :
(pullback_symmetry f g).hom ≫ pullback.fst = pullback.snd | by simp [pullback_symmetry] | lemma | category_theory.limits.pullback_symmetry_hom_comp_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_symmetry_hom_comp_snd [has_pullback f g] :
(pullback_symmetry f g).hom ≫ pullback.snd = pullback.fst | by simp [pullback_symmetry] | lemma | category_theory.limits.pullback_symmetry_hom_comp_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_symmetry_inv_comp_fst [has_pullback f g] :
(pullback_symmetry f g).inv ≫ pullback.fst = pullback.snd | by simp [iso.inv_comp_eq] | lemma | category_theory.limits.pullback_symmetry_inv_comp_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_symmetry_inv_comp_snd [has_pullback f g] :
(pullback_symmetry f g).inv ≫ pullback.snd = pullback.fst | by simp [iso.inv_comp_eq] | lemma | category_theory.limits.pullback_symmetry_inv_comp_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 | |
has_pushout_symmetry [has_pushout f g] : has_pushout g f | ⟨⟨⟨pushout_cocone.mk _ _ pushout.condition.symm,
pushout_cocone.flip_is_colimit (pushout_is_pushout _ _)⟩⟩⟩ | lemma | category_theory.limits.has_pushout_symmetry | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Making this a global instance would make the typeclass seach go in an infinite loop. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout_symmetry [has_pushout f g] :
pushout f g ≅ pushout g f | is_colimit.cocone_point_unique_up_to_iso
(pushout_cocone.flip_is_colimit (pushout_is_pushout f g) :
is_colimit (pushout_cocone.mk _ _ pushout.condition.symm))
(colimit.is_colimit _) | def | category_theory.limits.pushout_symmetry | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The isomorphism `Y ⨿[X] Z ≅ Z ⨿[X] Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl_comp_pushout_symmetry_hom [has_pushout f g] :
pushout.inl ≫ (pushout_symmetry f g).hom = pushout.inr | (colimit.is_colimit (span f g)).comp_cocone_point_unique_up_to_iso_hom
(pushout_cocone.flip_is_colimit (pushout_is_pushout g f)) _ | lemma | category_theory.limits.inl_comp_pushout_symmetry_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_comp_pushout_symmetry_hom [has_pushout f g] :
pushout.inr ≫ (pushout_symmetry f g).hom = pushout.inl | (colimit.is_colimit (span f g)).comp_cocone_point_unique_up_to_iso_hom
(pushout_cocone.flip_is_colimit (pushout_is_pushout g f)) _ | lemma | category_theory.limits.inr_comp_pushout_symmetry_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 | |
inl_comp_pushout_symmetry_inv [has_pushout f g] :
pushout.inl ≫ (pushout_symmetry f g).inv = pushout.inr | by simp [iso.comp_inv_eq] | lemma | category_theory.limits.inl_comp_pushout_symmetry_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_comp_pushout_symmetry_inv [has_pushout f g] :
pushout.inr ≫ (pushout_symmetry f g).inv = pushout.inl | by simp [iso.comp_inv_eq] | lemma | category_theory.limits.inr_comp_pushout_symmetry_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 | |
pullback_is_pullback_of_comp_mono (f : X ⟶ W) (g : Y ⟶ W) (i : W ⟶ Z)
[mono i] [has_pullback f g] :
is_limit (pullback_cone.mk pullback.fst pullback.snd _) | pullback_cone.is_limit_of_comp_mono f g i _ (limit.is_limit (cospan f g)) | def | category_theory.limits.pullback_is_pullback_of_comp_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The pullback of `f, g` is also the pullback of `f ≫ i, g ≫ i` for any mono `i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pullback_of_comp_mono (f : X ⟶ W) (g : Y ⟶ W) (i : W ⟶ Z)
[mono i] [has_pullback f g] : has_pullback (f ≫ i) (g ≫ i) | ⟨⟨⟨_,pullback_is_pullback_of_comp_mono f g i⟩⟩⟩ | instance | category_theory.limits.has_pullback_of_comp_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 | |
pullback_cone_of_left_iso : pullback_cone f g | pullback_cone.mk (g ≫ inv f) (𝟙 _) $ by simp | def | category_theory.limits.pullback_cone_of_left_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 `X ×[Z] Y ≅ Y`. This is the explicit limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_cone_of_left_iso_X :
(pullback_cone_of_left_iso f g).X = Y | rfl | lemma | category_theory.limits.pullback_cone_of_left_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 | |
pullback_cone_of_left_iso_fst :
(pullback_cone_of_left_iso f g).fst = g ≫ inv f | rfl | lemma | category_theory.limits.pullback_cone_of_left_iso_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_cone_of_left_iso_snd :
(pullback_cone_of_left_iso f g).snd = 𝟙 _ | rfl | lemma | category_theory.limits.pullback_cone_of_left_iso_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_cone_of_left_iso_π_app_none :
(pullback_cone_of_left_iso f g).π.app none = g | by { delta pullback_cone_of_left_iso, simp } | lemma | category_theory.limits.pullback_cone_of_left_iso_π_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 | |
pullback_cone_of_left_iso_π_app_left :
(pullback_cone_of_left_iso f g).π.app left = g ≫ inv f | rfl | lemma | category_theory.limits.pullback_cone_of_left_iso_π_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 | |
pullback_cone_of_left_iso_π_app_right :
(pullback_cone_of_left_iso f g).π.app right = 𝟙 _ | rfl | lemma | category_theory.limits.pullback_cone_of_left_iso_π_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 | |
pullback_cone_of_left_iso_is_limit :
is_limit (pullback_cone_of_left_iso f g) | pullback_cone.is_limit_aux' _ (λ s, ⟨s.snd, by simp [← s.condition_assoc]⟩) | def | category_theory.limits.pullback_cone_of_left_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 limit cone is indeed a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pullback_of_left_iso : has_pullback f g | ⟨⟨⟨_, pullback_cone_of_left_iso_is_limit f g⟩⟩⟩ | lemma | category_theory.limits.has_pullback_of_left_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 | |
pullback_snd_iso_of_left_iso : is_iso (pullback.snd : pullback f g ⟶ _) | begin
refine ⟨⟨pullback.lift (g ≫ inv f) (𝟙 _) (by simp), _, by simp⟩⟩,
ext,
{ simp [← pullback.condition_assoc] },
{ simp [pullback.condition_assoc] },
end | instance | category_theory.limits.pullback_snd_iso_of_left_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_pullback_of_right_factors_mono (f : X ⟶ Z) : has_pullback i (f ≫ i) | by { conv { congr, rw ←category.id_comp i, }, apply_instance } | instance | category_theory.limits.has_pullback_of_right_factors_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 | |
pullback_snd_iso_of_right_factors_mono (f : X ⟶ Z) :
is_iso (pullback.snd : pullback i (f ≫ i) ⟶ _) | begin
convert (congr_arg is_iso (show _ ≫ pullback.snd = _,
from limit.iso_limit_cone_hom_π ⟨_,pullback_is_pullback_of_comp_mono (𝟙 _) f i⟩
walking_cospan.right)).mp infer_instance;
exact (category.id_comp _).symm
end | instance | category_theory.limits.pullback_snd_iso_of_right_factors_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 | |
pullback_cone_of_right_iso : pullback_cone f g | pullback_cone.mk (𝟙 _) (f ≫ inv g) $ by simp | def | category_theory.limits.pullback_cone_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 `g : Y ⟶ Z` is iso, then `X ×[Z] Y ≅ X`. This is the explicit limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_cone_of_right_iso_X :
(pullback_cone_of_right_iso f g).X = X | rfl | lemma | category_theory.limits.pullback_cone_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 | |
pullback_cone_of_right_iso_fst :
(pullback_cone_of_right_iso f g).fst = 𝟙 _ | rfl | lemma | category_theory.limits.pullback_cone_of_right_iso_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_cone_of_right_iso_snd :
(pullback_cone_of_right_iso f g).snd = f ≫ inv g | rfl | lemma | category_theory.limits.pullback_cone_of_right_iso_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_cone_of_right_iso_π_app_none :
(pullback_cone_of_right_iso f g).π.app none = f | category.id_comp _ | lemma | category_theory.limits.pullback_cone_of_right_iso_π_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 | |
pullback_cone_of_right_iso_π_app_left :
(pullback_cone_of_right_iso f g).π.app left = 𝟙 _ | rfl | lemma | category_theory.limits.pullback_cone_of_right_iso_π_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 | |
pullback_cone_of_right_iso_π_app_right :
(pullback_cone_of_right_iso f g).π.app right = f ≫ inv g | rfl | lemma | category_theory.limits.pullback_cone_of_right_iso_π_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 | |
pullback_cone_of_right_iso_is_limit :
is_limit (pullback_cone_of_right_iso f g) | pullback_cone.is_limit_aux' _ (λ s, ⟨s.fst, by simp [s.condition_assoc]⟩) | def | category_theory.limits.pullback_cone_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 limit cone is indeed a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pullback_of_right_iso : has_pullback f g | ⟨⟨⟨_, pullback_cone_of_right_iso_is_limit f g⟩⟩⟩ | lemma | category_theory.limits.has_pullback_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 | |
pullback_snd_iso_of_right_iso : is_iso (pullback.fst : pullback f g ⟶ _) | begin
refine ⟨⟨pullback.lift (𝟙 _) (f ≫ inv g) (by simp), _, by simp⟩⟩,
ext,
{ simp },
{ simp [pullback.condition_assoc] },
end | instance | category_theory.limits.pullback_snd_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_pullback_of_left_factors_mono (f : X ⟶ Z) : has_pullback (f ≫ i) i | by { conv { congr, skip, rw ←category.id_comp i, }, apply_instance } | instance | category_theory.limits.has_pullback_of_left_factors_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 | |
pullback_snd_iso_of_left_factors_mono (f : X ⟶ Z) :
is_iso (pullback.fst : pullback (f ≫ i) i ⟶ _) | begin
convert (congr_arg is_iso (show _ ≫ pullback.fst = _,
from limit.iso_limit_cone_hom_π ⟨_,pullback_is_pullback_of_comp_mono f (𝟙 _) i⟩
walking_cospan.left)).mp infer_instance;
exact (category.id_comp _).symm
end | instance | category_theory.limits.pullback_snd_iso_of_left_factors_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 | |
pushout_is_pushout_of_epi_comp (f : X ⟶ Y) (g : X ⟶ Z) (h : W ⟶ X)
[epi h] [has_pushout f g] :
is_colimit (pushout_cocone.mk pushout.inl pushout.inr _) | pushout_cocone.is_colimit_of_epi_comp f g h _ (colimit.is_colimit (span f g)) | def | category_theory.limits.pushout_is_pushout_of_epi_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The pushout of `f, g` is also the pullback of `h ≫ f, h ≫ g` for any epi `h`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pushout_of_epi_comp (f : X ⟶ Y) (g : X ⟶ Z) (h : W ⟶ X)
[epi h] [has_pushout f g] : has_pushout (h ≫ f) (h ≫ g) | ⟨⟨⟨_,pushout_is_pushout_of_epi_comp f g h⟩⟩⟩ | instance | category_theory.limits.has_pushout_of_epi_comp | 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_left_iso : pushout_cocone f g | pushout_cocone.mk (inv f ≫ g) (𝟙 _) $ by simp | def | category_theory.limits.pushout_cocone_of_left_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 ⟶ Y` is iso, then `Y ⨿[X] Z ≅ Z`. This is the explicit colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout_cocone_of_left_iso_X :
(pushout_cocone_of_left_iso f g).X = Z | rfl | lemma | category_theory.limits.pushout_cocone_of_left_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_left_iso_inl :
(pushout_cocone_of_left_iso f g).inl = inv f ≫ g | rfl | lemma | category_theory.limits.pushout_cocone_of_left_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_left_iso_inr :
(pushout_cocone_of_left_iso f g).inr = 𝟙 _ | rfl | lemma | category_theory.limits.pushout_cocone_of_left_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_left_iso_ι_app_none :
(pushout_cocone_of_left_iso f g).ι.app none = g | by { delta pushout_cocone_of_left_iso, simp } | lemma | category_theory.limits.pushout_cocone_of_left_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_left_iso_ι_app_left :
(pushout_cocone_of_left_iso f g).ι.app left = inv f ≫ g | rfl | lemma | category_theory.limits.pushout_cocone_of_left_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_left_iso_ι_app_right :
(pushout_cocone_of_left_iso f g).ι.app right = 𝟙 _ | rfl | lemma | category_theory.limits.pushout_cocone_of_left_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_left_iso_is_limit :
is_colimit (pushout_cocone_of_left_iso f g) | pushout_cocone.is_colimit_aux' _ (λ s, ⟨s.inr, by simp [← s.condition]⟩) | def | category_theory.limits.pushout_cocone_of_left_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_left_iso : has_pushout f g | ⟨⟨⟨_, pushout_cocone_of_left_iso_is_limit f g⟩⟩⟩ | lemma | category_theory.limits.has_pushout_of_left_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_inr_iso_of_left_iso : is_iso (pushout.inr : _ ⟶ pushout f g) | begin
refine ⟨⟨pushout.desc (inv f ≫ g) (𝟙 _) (by simp), (by simp), _⟩⟩,
ext,
{ simp [← pushout.condition] },
{ simp [pushout.condition_assoc] },
end | instance | category_theory.limits.pushout_inr_iso_of_left_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 |
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