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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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cospan_comp_iso_app_right :
(cospan_comp_iso F f g).app walking_cospan.right = iso.refl _ | rfl | lemma | category_theory.limits.cospan_comp_iso_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 | |
cospan_comp_iso_app_one :
(cospan_comp_iso F f g).app walking_cospan.one = iso.refl _ | rfl | lemma | category_theory.limits.cospan_comp_iso_app_one | 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 | |
cospan_comp_iso_hom_app_left :
(cospan_comp_iso F f g).hom.app walking_cospan.left = 𝟙 _ | rfl | lemma | category_theory.limits.cospan_comp_iso_hom_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 | |
cospan_comp_iso_hom_app_right :
(cospan_comp_iso F f g).hom.app walking_cospan.right = 𝟙 _ | rfl | lemma | category_theory.limits.cospan_comp_iso_hom_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 | |
cospan_comp_iso_hom_app_one :
(cospan_comp_iso F f g).hom.app walking_cospan.one = 𝟙 _ | rfl | lemma | category_theory.limits.cospan_comp_iso_hom_app_one | 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 | |
cospan_comp_iso_inv_app_left :
(cospan_comp_iso F f g).inv.app walking_cospan.left = 𝟙 _ | rfl | lemma | category_theory.limits.cospan_comp_iso_inv_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 | |
cospan_comp_iso_inv_app_right :
(cospan_comp_iso F f g).inv.app walking_cospan.right = 𝟙 _ | rfl | lemma | category_theory.limits.cospan_comp_iso_inv_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 | |
cospan_comp_iso_inv_app_one :
(cospan_comp_iso F f g).inv.app walking_cospan.one = 𝟙 _ | rfl | lemma | category_theory.limits.cospan_comp_iso_inv_app_one | 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 | |
span_comp_iso (F : C ⥤ D) {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) :
span f g ⋙ F ≅ span (F.map f) (F.map g) | nat_iso.of_components (by rintros (⟨⟩|⟨⟨⟩⟩); exact iso.refl _)
(by rintros (⟨⟩|⟨⟨⟩⟩) (⟨⟩|⟨⟨⟩⟩) ⟨⟩; repeat { dsimp, simp, }) | def | category_theory.limits.span_comp_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | A functor applied to a span is a span. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_comp_iso_app_left : (span_comp_iso F f g).app walking_span.left = iso.refl _ | rfl | lemma | category_theory.limits.span_comp_iso_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 | |
span_comp_iso_app_right : (span_comp_iso F f g).app walking_span.right = iso.refl _ | rfl | lemma | category_theory.limits.span_comp_iso_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 | |
span_comp_iso_app_zero : (span_comp_iso F f g).app walking_span.zero = iso.refl _ | rfl | lemma | category_theory.limits.span_comp_iso_app_zero | 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 | |
span_comp_iso_hom_app_left : (span_comp_iso F f g).hom.app walking_span.left = 𝟙 _ | rfl | lemma | category_theory.limits.span_comp_iso_hom_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 | |
span_comp_iso_hom_app_right :
(span_comp_iso F f g).hom.app walking_span.right = 𝟙 _ | rfl | lemma | category_theory.limits.span_comp_iso_hom_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 | |
span_comp_iso_hom_app_zero : (span_comp_iso F f g).hom.app walking_span.zero = 𝟙 _ | rfl | lemma | category_theory.limits.span_comp_iso_hom_app_zero | 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 | |
span_comp_iso_inv_app_left : (span_comp_iso F f g).inv.app walking_span.left = 𝟙 _ | rfl | lemma | category_theory.limits.span_comp_iso_inv_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 | |
span_comp_iso_inv_app_right :
(span_comp_iso F f g).inv.app walking_span.right = 𝟙 _ | rfl | lemma | category_theory.limits.span_comp_iso_inv_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 | |
span_comp_iso_inv_app_zero : (span_comp_iso F f g).inv.app walking_span.zero = 𝟙 _ | rfl | lemma | category_theory.limits.span_comp_iso_inv_app_zero | 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 | |
cospan_ext (wf : iX.hom ≫ f' = f ≫ iZ.hom) (wg : iY.hom ≫ g' = g ≫ iZ.hom) :
cospan f g ≅ cospan f' g' | nat_iso.of_components (by { rintros (⟨⟩|⟨⟨⟩⟩), exacts [iZ, iX, iY], })
(by rintros (⟨⟩|⟨⟨⟩⟩) (⟨⟩|⟨⟨⟩⟩) ⟨⟩; repeat { dsimp, simp [wf, wg], }) | def | category_theory.limits.cospan_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Construct an isomorphism of cospans from components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cospan_ext_app_left : (cospan_ext iX iY iZ wf wg).app walking_cospan.left = iX | by { dsimp [cospan_ext], simp, } | lemma | category_theory.limits.cospan_ext_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 | |
cospan_ext_app_right : (cospan_ext iX iY iZ wf wg).app walking_cospan.right = iY | by { dsimp [cospan_ext], simp, } | lemma | category_theory.limits.cospan_ext_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 | |
cospan_ext_app_one : (cospan_ext iX iY iZ wf wg).app walking_cospan.one = iZ | by { dsimp [cospan_ext], simp, } | lemma | category_theory.limits.cospan_ext_app_one | 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 | |
cospan_ext_hom_app_left :
(cospan_ext iX iY iZ wf wg).hom.app walking_cospan.left = iX.hom | by { dsimp [cospan_ext], simp, } | lemma | category_theory.limits.cospan_ext_hom_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 | |
cospan_ext_hom_app_right :
(cospan_ext iX iY iZ wf wg).hom.app walking_cospan.right = iY.hom | by { dsimp [cospan_ext], simp, } | lemma | category_theory.limits.cospan_ext_hom_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 | |
cospan_ext_hom_app_one :
(cospan_ext iX iY iZ wf wg).hom.app walking_cospan.one = iZ.hom | by { dsimp [cospan_ext], simp, } | lemma | category_theory.limits.cospan_ext_hom_app_one | 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 | |
cospan_ext_inv_app_left :
(cospan_ext iX iY iZ wf wg).inv.app walking_cospan.left = iX.inv | by { dsimp [cospan_ext], simp, } | lemma | category_theory.limits.cospan_ext_inv_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 | |
cospan_ext_inv_app_right :
(cospan_ext iX iY iZ wf wg).inv.app walking_cospan.right = iY.inv | by { dsimp [cospan_ext], simp, } | lemma | category_theory.limits.cospan_ext_inv_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 | |
cospan_ext_inv_app_one :
(cospan_ext iX iY iZ wf wg).inv.app walking_cospan.one = iZ.inv | by { dsimp [cospan_ext], simp, } | lemma | category_theory.limits.cospan_ext_inv_app_one | 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 | |
span_ext (wf : iX.hom ≫ f' = f ≫ iY.hom) (wg : iX.hom ≫ g' = g ≫ iZ.hom) :
span f g ≅ span f' g' | nat_iso.of_components (by { rintros (⟨⟩|⟨⟨⟩⟩), exacts [iX, iY, iZ], })
(by rintros (⟨⟩|⟨⟨⟩⟩) (⟨⟩|⟨⟨⟩⟩) ⟨⟩; repeat { dsimp, simp [wf, wg], }) | def | category_theory.limits.span_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Construct an isomorphism of spans from components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_ext_app_left : (span_ext iX iY iZ wf wg).app walking_span.left = iY | by { dsimp [span_ext], simp, } | lemma | category_theory.limits.span_ext_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 | |
span_ext_app_right : (span_ext iX iY iZ wf wg).app walking_span.right = iZ | by { dsimp [span_ext], simp, } | lemma | category_theory.limits.span_ext_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 | |
span_ext_app_one : (span_ext iX iY iZ wf wg).app walking_span.zero = iX | by { dsimp [span_ext], simp, } | lemma | category_theory.limits.span_ext_app_one | 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 | |
span_ext_hom_app_left :
(span_ext iX iY iZ wf wg).hom.app walking_span.left = iY.hom | by { dsimp [span_ext], simp, } | lemma | category_theory.limits.span_ext_hom_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 | |
span_ext_hom_app_right :
(span_ext iX iY iZ wf wg).hom.app walking_span.right = iZ.hom | by { dsimp [span_ext], simp, } | lemma | category_theory.limits.span_ext_hom_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 | |
span_ext_hom_app_zero :
(span_ext iX iY iZ wf wg).hom.app walking_span.zero = iX.hom | by { dsimp [span_ext], simp, } | lemma | category_theory.limits.span_ext_hom_app_zero | 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 | |
span_ext_inv_app_left :
(span_ext iX iY iZ wf wg).inv.app walking_span.left = iY.inv | by { dsimp [span_ext], simp, } | lemma | category_theory.limits.span_ext_inv_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 | |
span_ext_inv_app_right :
(span_ext iX iY iZ wf wg).inv.app walking_span.right = iZ.inv | by { dsimp [span_ext], simp, } | lemma | category_theory.limits.span_ext_inv_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 | |
span_ext_inv_app_zero :
(span_ext iX iY iZ wf wg).inv.app walking_span.zero = iX.inv | by { dsimp [span_ext], simp, } | lemma | category_theory.limits.span_ext_inv_app_zero | 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 (f : X ⟶ Z) (g : Y ⟶ Z) | cone (cospan f g) | abbreviation | category_theory.limits.pullback_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | A pullback cone is just a cone on the cospan formed by two morphisms `f : X ⟶ Z` and
`g : Y ⟶ Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst (t : pullback_cone f g) : t.X ⟶ X | t.π.app walking_cospan.left | abbreviation | category_theory.limits.pullback_cone.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 a pullback cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd (t : pullback_cone f g) : t.X ⟶ Y | t.π.app walking_cospan.right | abbreviation | category_theory.limits.pullback_cone.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 a pullback cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π_app_left (c : pullback_cone f g) : c.π.app walking_cospan.left = c.fst | rfl | lemma | category_theory.limits.pullback_cone.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 | |
π_app_right (c : pullback_cone f g) : c.π.app walking_cospan.right = c.snd | rfl | lemma | category_theory.limits.pullback_cone.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 | |
condition_one (t : pullback_cone f g) : t.π.app walking_cospan.one = t.fst ≫ f | begin
have w := t.π.naturality walking_cospan.hom.inl,
dsimp at w, simpa using w,
end | lemma | category_theory.limits.pullback_cone.condition_one | 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 | |
is_limit_aux (t : pullback_cone f g) (lift : Π (s : pullback_cone f g), s.X ⟶ t.X)
(fac_left : ∀ (s : pullback_cone f g), lift s ≫ t.fst = s.fst)
(fac_right : ∀ (s : pullback_cone f g), lift s ≫ t.snd = s.snd)
(uniq : ∀ (s : pullback_cone f g) (m : s.X ⟶ t.X)
(w : ∀ j : walking_cospan, m ≫ t.π.app j = s.π.app... | { lift := lift,
fac' := λ s j, option.cases_on j
(by { rw [← s.w inl, ← t.w inl, ←category.assoc], congr, exact fac_left s, } )
(λ j', walking_pair.cases_on j' (fac_left s) (fac_right s)),
uniq' := uniq } | def | category_theory.limits.pullback_cone.is_limit_aux | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [
"lift"
] | This is a slightly more convenient method to verify that a pullback cone is a limit cone. It
only asks for a proof of facts that carry any mathematical content | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_aux' (t : pullback_cone f g)
(create : Π (s : pullback_cone f g),
{l // l ≫ t.fst = s.fst ∧ l ≫ t.snd = s.snd ∧
∀ {m}, m ≫ t.fst = s.fst → m ≫ t.snd = s.snd → m = l}) :
limits.is_limit t | pullback_cone.is_limit_aux t
(λ s, (create s).1)
(λ s, (create s).2.1)
(λ s, (create s).2.2.1)
(λ s m w, (create s).2.2.2 (w walking_cospan.left) (w walking_cospan.right)) | def | category_theory.limits.pullback_cone.is_limit_aux' | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | This is another convenient method to verify that a pullback cone is a limit cone. It
only asks for a proof of facts that carry any mathematical content, and allows access to the
same `s` for all parts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk {W : C} (fst : W ⟶ X) (snd : W ⟶ Y) (eq : fst ≫ f = snd ≫ g) : pullback_cone f g | { X := W,
π := { app := λ j, option.cases_on j (fst ≫ f) (λ j', walking_pair.cases_on j' fst snd) } } | def | category_theory.limits.pullback_cone.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | A pullback cone on `f` and `g` is determined by morphisms `fst : W ⟶ X` and `snd : W ⟶ Y`
such that `fst ≫ f = snd ≫ g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_π_app_left {W : C} (fst : W ⟶ X) (snd : W ⟶ Y) (eq : fst ≫ f = snd ≫ g) :
(mk fst snd eq).π.app walking_cospan.left = fst | rfl | lemma | category_theory.limits.pullback_cone.mk_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 | |
mk_π_app_right {W : C} (fst : W ⟶ X) (snd : W ⟶ Y) (eq : fst ≫ f = snd ≫ g) :
(mk fst snd eq).π.app walking_cospan.right = snd | rfl | lemma | category_theory.limits.pullback_cone.mk_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 | |
mk_π_app_one {W : C} (fst : W ⟶ X) (snd : W ⟶ Y) (eq : fst ≫ f = snd ≫ g) :
(mk fst snd eq).π.app walking_cospan.one = fst ≫ f | rfl | lemma | category_theory.limits.pullback_cone.mk_π_app_one | 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 | |
mk_fst {W : C} (fst : W ⟶ X) (snd : W ⟶ Y) (eq : fst ≫ f = snd ≫ g) :
(mk fst snd eq).fst = fst | rfl | lemma | category_theory.limits.pullback_cone.mk_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 | |
mk_snd {W : C} (fst : W ⟶ X) (snd : W ⟶ Y) (eq : fst ≫ f = snd ≫ g) :
(mk fst snd eq).snd = snd | rfl | lemma | category_theory.limits.pullback_cone.mk_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 | |
condition (t : pullback_cone f g) : fst t ≫ f = snd t ≫ g | (t.w inl).trans (t.w inr).symm | lemma | category_theory.limits.pullback_cone.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 | |
equalizer_ext (t : pullback_cone f g) {W : C} {k l : W ⟶ t.X}
(h₀ : k ≫ fst t = l ≫ fst t) (h₁ : k ≫ snd t = l ≫ snd t) :
∀ (j : walking_cospan), k ≫ t.π.app j = l ≫ t.π.app j | | (some walking_pair.left) := h₀
| (some walking_pair.right) := h₁
| none := by rw [← t.w inl, reassoc_of h₀] | lemma | category_theory.limits.pullback_cone.equalizer_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | To check whether a morphism is equalized by the maps of a pullback cone, it suffices to check
it for `fst t` and `snd t` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit.hom_ext {t : pullback_cone f g} (ht : is_limit t) {W : C} {k l : W ⟶ t.X}
(h₀ : k ≫ fst t = l ≫ fst t) (h₁ : k ≫ snd t = l ≫ snd t) : k = l | ht.hom_ext $ equalizer_ext _ h₀ h₁ | lemma | category_theory.limits.pullback_cone.is_limit.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_snd_of_is_pullback_of_mono {t : pullback_cone f g} (ht : is_limit t) [mono f] :
mono t.snd | ⟨λ W h k i, is_limit.hom_ext ht (by simp [←cancel_mono f, t.condition, reassoc_of i]) i⟩ | lemma | category_theory.limits.pullback_cone.mono_snd_of_is_pullback_of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_fst_of_is_pullback_of_mono {t : pullback_cone f g} (ht : is_limit t) [mono g] :
mono t.fst | ⟨λ W h k i, is_limit.hom_ext ht i (by simp [←cancel_mono g, ←t.condition, reassoc_of i])⟩ | lemma | category_theory.limits.pullback_cone.mono_fst_of_is_pullback_of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {s t : pullback_cone f g} (i : s.X ≅ t.X)
(w₁ : s.fst = i.hom ≫ t.fst) (w₂ : s.snd = i.hom ≫ t.snd) :
s ≅ t | walking_cospan.ext i w₁ w₂ | def | category_theory.limits.pullback_cone.ext | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | To construct an isomorphism of pullback cones, it suffices to construct an isomorphism
of the cone points and check it commutes with `fst` and `snd`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit.lift' {t : pullback_cone f g} (ht : is_limit t) {W : C} (h : W ⟶ X) (k : W ⟶ Y)
(w : h ≫ f = k ≫ g) : {l : W ⟶ t.X // l ≫ fst t = h ∧ l ≫ snd t = k} | ⟨ht.lift $ pullback_cone.mk _ _ w, ht.fac _ _, ht.fac _ _⟩ | def | category_theory.limits.pullback_cone.is_limit.lift' | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | If `t` is a limit pullback cone over `f` and `g` and `h : W ⟶ X` and `k : W ⟶ Y` are such that
`h ≫ f = k ≫ g`, then we have `l : W ⟶ t.X` satisfying `l ≫ fst t = h` and `l ≫ snd t = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit.mk {W : C} {fst : W ⟶ X} {snd : W ⟶ Y} (eq : fst ≫ f = snd ≫ g)
(lift : Π (s : pullback_cone f g), s.X ⟶ W)
(fac_left : ∀ (s : pullback_cone f g), lift s ≫ fst = s.fst)
(fac_right : ∀ (s : pullback_cone f g), lift s ≫ snd = s.snd)
(uniq : ∀ (s : pullback_cone f g) (m : s.X ⟶ W)
(w_fst : m ≫ fst = s... | is_limit_aux _ lift fac_left fac_right
(λ s m w, uniq s m (w walking_cospan.left) (w walking_cospan.right)) | def | category_theory.limits.pullback_cone.is_limit.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [
"lift"
] | This is a more convenient formulation to show that a `pullback_cone` constructed using
`pullback_cone.mk` is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
flip_is_limit {W : C} {h : W ⟶ X} {k : W ⟶ Y}
{comm : h ≫ f = k ≫ g} (t : is_limit (mk _ _ comm.symm)) :
is_limit (mk _ _ comm) | is_limit_aux' _ $ λ s,
begin
refine ⟨(is_limit.lift' t _ _ s.condition.symm).1,
(is_limit.lift' t _ _ _).2.2,
(is_limit.lift' t _ _ _).2.1, λ m m₁ m₂, t.hom_ext _⟩,
apply (mk k h _).equalizer_ext,
{ rwa (is_limit.lift' t _ _ _).2.1 },
{ rwa (is_limit.lift' t _ _ _).2.2 },
end | def | category_theory.limits.pullback_cone.flip_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"
] | [
"comm"
] | The flip of a pullback square is a pullback square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_mk_id_id (f : X ⟶ Y) [mono f] :
is_limit (mk (𝟙 X) (𝟙 X) rfl : pullback_cone f f) | is_limit.mk _
(λ s, s.fst)
(λ s, category.comp_id _)
(λ s, by rw [←cancel_mono f, category.comp_id, s.condition])
(λ s m m₁ m₂, by simpa using m₁) | def | category_theory.limits.pullback_cone.is_limit_mk_id_id | 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 `(𝟙 X, 𝟙 X)` for the pair `(f, f)` is a limit if `f` is a mono. The converse is
shown in `mono_of_pullback_is_id`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_of_is_limit_mk_id_id (f : X ⟶ Y)
(t : is_limit (mk (𝟙 X) (𝟙 X) rfl : pullback_cone f f)) :
mono f | ⟨λ Z g h eq, by { rcases pullback_cone.is_limit.lift' t _ _ eq with ⟨_, rfl, rfl⟩, refl } ⟩ | lemma | category_theory.limits.pullback_cone.mono_of_is_limit_mk_id_id | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `f` is a mono if the pullback cone `(𝟙 X, 𝟙 X)` is a limit for the pair `(f, f)`. The converse is
given in `pullback_cone.is_id_of_mono`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_of_factors (f : X ⟶ Z) (g : Y ⟶ Z) (h : W ⟶ Z) [mono h]
(x : X ⟶ W) (y : Y ⟶ W) (hxh : x ≫ h = f) (hyh : y ≫ h = g) (s : pullback_cone f g)
(hs : is_limit s) : is_limit (pullback_cone.mk _ _ (show s.fst ≫ x = s.snd ≫ y,
from (cancel_mono h).1 $ by simp only [category.assoc, hxh, hyh, s.condition])) | pullback_cone.is_limit_aux' _ $ λ t,
⟨hs.lift (pullback_cone.mk t.fst t.snd $ by rw [←hxh, ←hyh, reassoc_of t.condition]),
⟨hs.fac _ walking_cospan.left, hs.fac _ walking_cospan.right, λ r hr hr',
begin
apply pullback_cone.is_limit.hom_ext hs;
simp only [pullback_cone.mk_fst, pullback_cone.mk_snd] at ⊢ hr... | def | category_theory.limits.pullback_cone.is_limit_of_factors | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Suppose `f` and `g` are two morphisms with a common codomain and `s` is a limit cone over the
diagram formed by `f` and `g`. Suppose `f` and `g` both factor through a monomorphism `h` via
`x` and `y`, respectively. Then `s` is also a limit cone over the diagram formed by `x` and
`y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_of_comp_mono (f : X ⟶ W) (g : Y ⟶ W) (i : W ⟶ Z) [mono i]
(s : pullback_cone f g) (H : is_limit s) :
is_limit (pullback_cone.mk _ _ (show s.fst ≫ f ≫ i = s.snd ≫ g ≫ i,
by rw [← category.assoc, ← category.assoc, s.condition])) | begin
apply pullback_cone.is_limit_aux',
intro s,
rcases pullback_cone.is_limit.lift' H s.fst s.snd
((cancel_mono i).mp (by simpa using s.condition)) with ⟨l, h₁, h₂⟩,
refine ⟨l,h₁,h₂,_⟩,
intros m hm₁ hm₂,
exact (pullback_cone.is_limit.hom_ext H (hm₁.trans h₁.symm) (hm₂.trans h₂.symm) : _)
end | def | category_theory.limits.pullback_cone.is_limit_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"
] | [] | If `W` is the pullback of `f, g`,
it is also the pullback of `f ≫ i, g ≫ i` for any mono `i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout_cocone (f : X ⟶ Y) (g : X ⟶ Z) | cocone (span f g) | abbreviation | category_theory.limits.pushout_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | A pushout cocone is just a cocone on the span formed by two morphisms `f : X ⟶ Y` and
`g : X ⟶ Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl (t : pushout_cocone f g) : Y ⟶ t.X | t.ι.app walking_span.left | abbreviation | category_theory.limits.pushout_cocone.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 of a pushout cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inr (t : pushout_cocone f g) : Z ⟶ t.X | t.ι.app walking_span.right | abbreviation | category_theory.limits.pushout_cocone.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 of a pushout cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_app_left (c : pushout_cocone f g) : c.ι.app walking_span.left = c.inl | rfl | lemma | category_theory.limits.pushout_cocone.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 | |
ι_app_right (c : pushout_cocone f g) : c.ι.app walking_span.right = c.inr | rfl | lemma | category_theory.limits.pushout_cocone.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 | |
condition_zero (t : pushout_cocone f g) : t.ι.app walking_span.zero = f ≫ t.inl | begin
have w := t.ι.naturality walking_span.hom.fst,
dsimp at w, simpa using w.symm,
end | lemma | category_theory.limits.pushout_cocone.condition_zero | 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 | |
is_colimit_aux (t : pushout_cocone f g) (desc : Π (s : pushout_cocone f g), t.X ⟶ s.X)
(fac_left : ∀ (s : pushout_cocone f g), t.inl ≫ desc s = s.inl)
(fac_right : ∀ (s : pushout_cocone f g), t.inr ≫ desc s = s.inr)
(uniq : ∀ (s : pushout_cocone f g) (m : t.X ⟶ s.X)
(w : ∀ j : walking_span, t.ι.app j ≫ m = s.... | { desc := desc,
fac' := λ s j, option.cases_on j (by { simp [← s.w fst, ← t.w fst, fac_left s] } )
(λ j', walking_pair.cases_on j' (fac_left s) (fac_right s)),
uniq' := uniq } | def | category_theory.limits.pushout_cocone.is_colimit_aux | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | This is a slightly more convenient method to verify that a pushout cocone is a colimit cocone.
It only asks for a proof of facts that carry any mathematical content | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_aux' (t : pushout_cocone f g)
(create : Π (s : pushout_cocone f g),
{l // t.inl ≫ l = s.inl ∧ t.inr ≫ l = s.inr ∧
∀ {m}, t.inl ≫ m = s.inl → t.inr ≫ m = s.inr → m = l}) :
is_colimit t | is_colimit_aux t
(λ s, (create s).1)
(λ s, (create s).2.1)
(λ s, (create s).2.2.1)
(λ s m w, (create s).2.2.2 (w walking_cospan.left) (w walking_cospan.right)) | def | category_theory.limits.pushout_cocone.is_colimit_aux' | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | This is another convenient method to verify that a pushout cocone is a colimit cocone. It
only asks for a proof of facts that carry any mathematical content, and allows access to the
same `s` for all parts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk {W : C} (inl : Y ⟶ W) (inr : Z ⟶ W) (eq : f ≫ inl = g ≫ inr) : pushout_cocone f g | { X := W,
ι := { app := λ j, option.cases_on j (f ≫ inl) (λ j', walking_pair.cases_on j' inl inr) } } | def | category_theory.limits.pushout_cocone.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | A pushout cocone on `f` and `g` is determined by morphisms `inl : Y ⟶ W` and `inr : Z ⟶ W` such
that `f ≫ inl = g ↠ inr`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_ι_app_left {W : C} (inl : Y ⟶ W) (inr : Z ⟶ W) (eq : f ≫ inl = g ≫ inr) :
(mk inl inr eq).ι.app walking_span.left = inl | rfl | lemma | category_theory.limits.pushout_cocone.mk_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 | |
mk_ι_app_right {W : C} (inl : Y ⟶ W) (inr : Z ⟶ W) (eq : f ≫ inl = g ≫ inr) :
(mk inl inr eq).ι.app walking_span.right = inr | rfl | lemma | category_theory.limits.pushout_cocone.mk_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 | |
mk_ι_app_zero {W : C} (inl : Y ⟶ W) (inr : Z ⟶ W) (eq : f ≫ inl = g ≫ inr) :
(mk inl inr eq).ι.app walking_span.zero = f ≫ inl | rfl | lemma | category_theory.limits.pushout_cocone.mk_ι_app_zero | 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 | |
mk_inl {W : C} (inl : Y ⟶ W) (inr : Z ⟶ W) (eq : f ≫ inl = g ≫ inr) :
(mk inl inr eq).inl = inl | rfl | lemma | category_theory.limits.pushout_cocone.mk_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 | |
mk_inr {W : C} (inl : Y ⟶ W) (inr : Z ⟶ W) (eq : f ≫ inl = g ≫ inr) :
(mk inl inr eq).inr = inr | rfl | lemma | category_theory.limits.pushout_cocone.mk_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 | |
condition (t : pushout_cocone f g) : f ≫ (inl t) = g ≫ (inr t) | (t.w fst).trans (t.w snd).symm | lemma | category_theory.limits.pushout_cocone.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 | |
coequalizer_ext (t : pushout_cocone f g) {W : C} {k l : t.X ⟶ W}
(h₀ : inl t ≫ k = inl t ≫ l) (h₁ : inr t ≫ k = inr t ≫ l) :
∀ (j : walking_span), t.ι.app j ≫ k = t.ι.app j ≫ l | | (some walking_pair.left) := h₀
| (some walking_pair.right) := h₁
| none := by rw [← t.w fst, category.assoc, category.assoc, h₀] | lemma | category_theory.limits.pushout_cocone.coequalizer_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | To check whether a morphism is coequalized by the maps of a pushout cocone, it suffices to check
it for `inl t` and `inr t` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit.hom_ext {t : pushout_cocone f g} (ht : is_colimit t) {W : C} {k l : t.X ⟶ W}
(h₀ : inl t ≫ k = inl t ≫ l) (h₁ : inr t ≫ k = inr t ≫ l) : k = l | ht.hom_ext $ coequalizer_ext _ h₀ h₁ | lemma | category_theory.limits.pushout_cocone.is_colimit.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit.desc' {t : pushout_cocone f g} (ht : is_colimit t) {W : C} (h : Y ⟶ W) (k : Z ⟶ W)
(w : f ≫ h = g ≫ k) : {l : t.X ⟶ W // inl t ≫ l = h ∧ inr t ≫ l = k } | ⟨ht.desc $ pushout_cocone.mk _ _ w, ht.fac _ _, ht.fac _ _⟩ | def | category_theory.limits.pushout_cocone.is_colimit.desc' | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | If `t` is a colimit pushout cocone over `f` and `g` and `h : Y ⟶ W` and `k : Z ⟶ W` are
morphisms satisfying `f ≫ h = g ≫ k`, then we have a factorization `l : t.X ⟶ W` such that
`inl t ≫ l = h` and `inr t ≫ l = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_inr_of_is_pushout_of_epi {t : pushout_cocone f g} (ht : is_colimit t) [epi f] :
epi t.inr | ⟨λ W h k i, is_colimit.hom_ext ht (by simp [←cancel_epi f, t.condition_assoc, i]) i⟩ | lemma | category_theory.limits.pushout_cocone.epi_inr_of_is_pushout_of_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_inl_of_is_pushout_of_epi {t : pushout_cocone f g} (ht : is_colimit t) [epi g] :
epi t.inl | ⟨λ W h k i, is_colimit.hom_ext ht i (by simp [←cancel_epi g, ←t.condition_assoc, i])⟩ | lemma | category_theory.limits.pushout_cocone.epi_inl_of_is_pushout_of_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {s t : pushout_cocone f g} (i : s.X ≅ t.X)
(w₁ : s.inl ≫ i.hom = t.inl) (w₂ : s.inr ≫ i.hom = t.inr) :
s ≅ t | walking_span.ext i w₁ w₂ | def | category_theory.limits.pushout_cocone.ext | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | To construct an isomorphism of pushout cocones, it suffices to construct an isomorphism
of the cocone points and check it commutes with `inl` and `inr`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit.mk {W : C} {inl : Y ⟶ W} {inr : Z ⟶ W} (eq : f ≫ inl = g ≫ inr)
(desc : Π (s : pushout_cocone f g), W ⟶ s.X)
(fac_left : ∀ (s : pushout_cocone f g), inl ≫ desc s = s.inl)
(fac_right : ∀ (s : pushout_cocone f g), inr ≫ desc s = s.inr)
(uniq : ∀ (s : pushout_cocone f g) (m : W ⟶ s.X)
(w_inl : inl ≫... | is_colimit_aux _ desc fac_left fac_right
(λ s m w, uniq s m (w walking_cospan.left) (w walking_cospan.right)) | def | category_theory.limits.pushout_cocone.is_colimit.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | This is a more convenient formulation to show that a `pushout_cocone` constructed using
`pushout_cocone.mk` is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
flip_is_colimit {W : C} {h : Y ⟶ W} {k : Z ⟶ W}
{comm : f ≫ h = g ≫ k} (t : is_colimit (mk _ _ comm.symm)) :
is_colimit (mk _ _ comm) | is_colimit_aux' _ $ λ s,
begin
refine ⟨(is_colimit.desc' t _ _ s.condition.symm).1,
(is_colimit.desc' t _ _ _).2.2,
(is_colimit.desc' t _ _ _).2.1, λ m m₁ m₂, t.hom_ext _⟩,
apply (mk k h _).coequalizer_ext,
{ rwa (is_colimit.desc' t _ _ _).2.1 },
{ rwa (is_colimit.desc' t _ _ _).2.2 },
end | def | category_theory.limits.pushout_cocone.flip_is_colimit | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [
"comm"
] | The flip of a pushout square is a pushout square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_mk_id_id (f : X ⟶ Y) [epi f] :
is_colimit (mk (𝟙 Y) (𝟙 Y) rfl : pushout_cocone f f) | is_colimit.mk _
(λ s, s.inl)
(λ s, category.id_comp _)
(λ s, by rw [←cancel_epi f, category.id_comp, s.condition])
(λ s m m₁ m₂, by simpa using m₁) | def | category_theory.limits.pushout_cocone.is_colimit_mk_id_id | 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 `(𝟙 X, 𝟙 X)` for the pair `(f, f)` is a colimit if `f` is an epi. The converse is
shown in `epi_of_is_colimit_mk_id_id`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_of_is_colimit_mk_id_id (f : X ⟶ Y)
(t : is_colimit (mk (𝟙 Y) (𝟙 Y) rfl : pushout_cocone f f)) :
epi f | ⟨λ Z g h eq, by { rcases pushout_cocone.is_colimit.desc' t _ _ eq with ⟨_, rfl, rfl⟩, refl }⟩ | lemma | category_theory.limits.pushout_cocone.epi_of_is_colimit_mk_id_id | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `f` is an epi if the pushout cocone `(𝟙 X, 𝟙 X)` is a colimit for the pair `(f, f)`.
The converse is given in `pushout_cocone.is_colimit_mk_id_id`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_of_factors (f : X ⟶ Y) (g : X ⟶ Z) (h : X ⟶ W) [epi h]
(x : W ⟶ Y) (y : W ⟶ Z) (hhx : h ≫ x = f) (hhy : h ≫ y = g) (s : pushout_cocone f g)
(hs : is_colimit s) : is_colimit (pushout_cocone.mk _ _ (show x ≫ s.inl = y ≫ s.inr,
from (cancel_epi h).1 $ by rw [reassoc_of hhx, reassoc_of hhy, s.condition])... | pushout_cocone.is_colimit_aux' _ $ λ t,
⟨hs.desc (pushout_cocone.mk t.inl t.inr $
by rw [←hhx, ←hhy, category.assoc, category.assoc, t.condition]),
⟨hs.fac _ walking_span.left, hs.fac _ walking_span.right, λ r hr hr',
begin
apply pushout_cocone.is_colimit.hom_ext hs;
simp only [pushout_cocone.mk_inl, ... | def | category_theory.limits.pushout_cocone.is_colimit_of_factors | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Suppose `f` and `g` are two morphisms with a common domain and `s` is a colimit cocone over the
diagram formed by `f` and `g`. Suppose `f` and `g` both factor through an epimorphism `h` via
`x` and `y`, respectively. Then `s` is also a colimit cocone over the diagram formed by `x` and
`y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_of_epi_comp (f : X ⟶ Y) (g : X ⟶ Z) (h : W ⟶ X) [epi h]
(s : pushout_cocone f g) (H : is_colimit s) :
is_colimit (pushout_cocone.mk _ _ (show (h ≫ f) ≫ s.inl = (h ≫ g) ≫ s.inr,
by rw [category.assoc, category.assoc, s.condition])) | begin
apply pushout_cocone.is_colimit_aux',
intro s,
rcases pushout_cocone.is_colimit.desc' H s.inl s.inr
((cancel_epi h).mp (by simpa using s.condition)) with ⟨l, h₁, h₂⟩,
refine ⟨l,h₁,h₂,_⟩,
intros m hm₁ hm₂,
exact (pushout_cocone.is_colimit.hom_ext H (hm₁.trans h₁.symm) (hm₂.trans h₂.symm) : _)
end | def | category_theory.limits.pushout_cocone.is_colimit_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"
] | [] | If `W` is the pushout of `f, g`,
it is also the pushout of `h ≫ f, h ≫ g` for any epi `h`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone.of_pullback_cone
{F : walking_cospan ⥤ C} (t : pullback_cone (F.map inl) (F.map inr)) : cone F | { X := t.X,
π := t.π ≫ (diagram_iso_cospan F).inv } | def | category_theory.limits.cone.of_pullback_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | This is a helper construction that can be useful when verifying that a category has all
pullbacks. Given `F : walking_cospan ⥤ C`, which is really the same as
`cospan (F.map inl) (F.map inr)`, and a pullback cone on `F.map inl` and `F.map inr`, we
get a cone on `F`.
If you're thinking about using this,... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone.of_pushout_cocone
{F : walking_span ⥤ C} (t : pushout_cocone (F.map fst) (F.map snd)) : cocone F | { X := t.X,
ι := (diagram_iso_span F).hom ≫ t.ι } | def | category_theory.limits.cocone.of_pushout_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | This is a helper construction that can be useful when verifying that a category has all
pushout. Given `F : walking_span ⥤ C`, which is really the same as
`span (F.map fst) (F.mal snd)`, and a pushout cocone on `F.map fst` and `F.map snd`,
we get a cocone on `F`.
If you're thinking about using this, ha... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_cone.of_cone
{F : walking_cospan ⥤ C} (t : cone F) : pullback_cone (F.map inl) (F.map inr) | { X := t.X,
π := t.π ≫ (diagram_iso_cospan F).hom } | def | category_theory.limits.pullback_cone.of_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Given `F : walking_cospan ⥤ C`, which is really the same as `cospan (F.map inl) (F.map inr)`,
and a cone on `F`, we get a pullback cone on `F.map inl` and `F.map inr`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_cone.iso_mk {F : walking_cospan ⥤ C} (t : cone F) :
(cones.postcompose (diagram_iso_cospan.{v} _).hom).obj t ≅
pullback_cone.mk (t.π.app walking_cospan.left) (t.π.app walking_cospan.right)
((t.π.naturality inl).symm.trans (t.π.naturality inr : _)) | cones.ext (iso.refl _) $ by rintro (_|(_|_)); { dsimp, simp } | def | category_theory.limits.pullback_cone.iso_mk | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | A diagram `walking_cospan ⥤ C` is isomorphic to some `pullback_cone.mk` after
composing with `diagram_iso_cospan`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout_cocone.of_cocone
{F : walking_span ⥤ C} (t : cocone F) : pushout_cocone (F.map fst) (F.map snd) | { X := t.X,
ι := (diagram_iso_span F).inv ≫ t.ι } | def | category_theory.limits.pushout_cocone.of_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Given `F : walking_span ⥤ C`, which is really the same as `span (F.map fst) (F.map snd)`,
and a cocone on `F`, we get a pushout cocone on `F.map fst` and `F.map snd`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout_cocone.iso_mk {F : walking_span ⥤ C} (t : cocone F) :
(cocones.precompose (diagram_iso_span.{v} _).inv).obj t ≅
pushout_cocone.mk (t.ι.app walking_span.left) (t.ι.app walking_span.right)
((t.ι.naturality fst).trans (t.ι.naturality snd).symm) | cocones.ext (iso.refl _) $ by rintro (_|(_|_)); { dsimp, simp } | def | category_theory.limits.pushout_cocone.iso_mk | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | A diagram `walking_span ⥤ C` is isomorphic to some `pushout_cocone.mk` after composing with
`diagram_iso_span`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pullback {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) | has_limit (cospan f g) | abbreviation | category_theory.limits.has_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `has_pullback f g` represents a particular choice of limiting cone
for the pair of morphisms `f : X ⟶ Z` and `g : Y ⟶ Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pushout {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) | has_colimit (span f g) | abbreviation | category_theory.limits.has_pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `has_pushout f g` represents a particular choice of colimiting cocone
for the pair of morphisms `f : X ⟶ Y` and `g : X ⟶ Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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