statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
binary_coproduct_iso_inl_comp_hom (X Y : Type u) :
limits.coprod.inl ≫ (binary_coproduct_iso X Y).hom = sum.inl | colimit.iso_colimit_cocone_ι_hom (binary_coproduct_colimit_cocone X Y) ⟨walking_pair.left⟩ | lemma | category_theory.limits.types.binary_coproduct_iso_inl_comp_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_coproduct_iso_inr_comp_hom (X Y : Type u) :
limits.coprod.inr ≫ (binary_coproduct_iso X Y).hom = sum.inr | colimit.iso_colimit_cocone_ι_hom (binary_coproduct_colimit_cocone X Y) ⟨walking_pair.right⟩ | lemma | category_theory.limits.types.binary_coproduct_iso_inr_comp_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_coproduct_iso_inl_comp_inv (X Y : Type u) :
↾(sum.inl : X ⟶ X ⊕ Y) ≫ (binary_coproduct_iso X Y).inv = limits.coprod.inl | colimit.iso_colimit_cocone_ι_inv (binary_coproduct_colimit_cocone X Y) ⟨walking_pair.left⟩ | lemma | category_theory.limits.types.binary_coproduct_iso_inl_comp_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_coproduct_iso_inr_comp_inv (X Y : Type u) :
↾(sum.inr : Y ⟶ X ⊕ Y) ≫ (binary_coproduct_iso X Y).inv = limits.coprod.inr | colimit.iso_colimit_cocone_ι_inv (binary_coproduct_colimit_cocone X Y) ⟨walking_pair.right⟩ | lemma | category_theory.limits.types.binary_coproduct_iso_inr_comp_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan_is_colimit_iff {X Y : Type u} (c : binary_cofan X Y) :
nonempty (is_colimit c) ↔
injective c.inl ∧ injective c.inr ∧ is_compl (set.range c.inl) (set.range c.inr) | begin
classical,
split,
{ rintro ⟨h⟩,
rw [← show _ = c.inl, from h.comp_cocone_point_unique_up_to_iso_inv
(binary_coproduct_colimit X Y) ⟨walking_pair.left⟩,
← show _ = c.inr, from h.comp_cocone_point_unique_up_to_iso_inv
(binary_coproduct_colimit X Y) ⟨walking_pair.right⟩],
dsimp [binar... | lemma | category_theory.limits.types.binary_cofan_is_colimit_iff | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [
"dite_eq_right_iff",
"eq_compl_iff_is_compl",
"equiv.apply_of_injective_symm",
"equiv.of_injective",
"equiv.of_injective_symm_apply",
"forall_exists_index",
"is_compl",
"or_not",
"set.image_compl_eq",
"set.range",
"set.range_comp",
"sum.inl_injective",
"sum.inr_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprod_of_mono {X Y : Type u} (f : X ⟶ Y) [mono f] :
is_colimit (binary_cofan.mk f (subtype.val : (set.range f)ᶜ → Y)) | nonempty.some $ (binary_cofan_is_colimit_iff _).mpr
⟨(mono_iff_injective f).mp infer_instance, subtype.val_injective,
(eq_compl_iff_is_compl.mp $ subtype.range_val).symm⟩ | def | category_theory.limits.types.is_coprod_of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [
"nonempty.some",
"set.range",
"subtype.range_val",
"subtype.val_injective"
] | Any monomorphism in `Type` is an coproduct injection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
product_limit_cone {J : Type u} (F : J → Type (max u v)) :
limits.limit_cone (discrete.functor F) | { cone :=
{ X := Π j, F j,
π := { app := λ j f, f j.as }, },
is_limit :=
{ lift := λ s x j, s.π.app ⟨j⟩ x,
uniq' := λ s m w, funext $ λ x, funext $ λ j, (congr_fun (w ⟨j⟩) x : _) } } | def | category_theory.limits.types.product_limit_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [
"lift"
] | The category of types has `Π j, f j` as the product of a type family `f : J → Type`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
product_iso {J : Type u} (F : J → Type (max u v)) : ∏ F ≅ Π j, F j | limit.iso_limit_cone (product_limit_cone F) | def | category_theory.limits.types.product_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The categorical product in `Type u` is the type theoretic product `Π j, F j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
product_iso_hom_comp_eval {J : Type u} (F : J → Type (max u v)) (j : J) :
(product_iso F).hom ≫ (λ f, f j) = pi.π F j | rfl | lemma | category_theory.limits.types.product_iso_hom_comp_eval | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
product_iso_inv_comp_π {J : Type u} (F : J → Type (max u v)) (j : J) :
(product_iso F).inv ≫ pi.π F j = (λ f, f j) | limit.iso_limit_cone_inv_π (product_limit_cone F) ⟨j⟩ | lemma | category_theory.limits.types.product_iso_inv_comp_π | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coproduct_colimit_cocone {J : Type u} (F : J → Type u) :
limits.colimit_cocone (discrete.functor F) | { cocone :=
{ X := Σ j, F j,
ι :=
{ app := λ j x, ⟨j.as, x⟩ }, },
is_colimit :=
{ desc := λ s x, s.ι.app ⟨x.1⟩ x.2,
uniq' := λ s m w,
begin
ext ⟨j, x⟩,
have := congr_fun (w ⟨j⟩) x,
exact this,
end }, } | def | category_theory.limits.types.coproduct_colimit_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The category of types has `Σ j, f j` as the coproduct of a type family `f : J → Type`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coproduct_iso {J : Type u} (F : J → Type u) : ∐ F ≅ Σ j, F j | colimit.iso_colimit_cocone (coproduct_colimit_cocone F) | def | category_theory.limits.types.coproduct_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The categorical coproduct in `Type u` is the type theoretic coproduct `Σ j, F j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coproduct_iso_ι_comp_hom {J : Type u} (F : J → Type u) (j : J) :
sigma.ι F j ≫ (coproduct_iso F).hom = (λ x : F j, (⟨j, x⟩ : Σ j, F j)) | colimit.iso_colimit_cocone_ι_hom (coproduct_colimit_cocone F) ⟨j⟩ | lemma | category_theory.limits.types.coproduct_iso_ι_comp_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coproduct_iso_mk_comp_inv {J : Type u} (F : J → Type u) (j : J) :
↾(λ x : F j, (⟨j, x⟩ : Σ j, F j)) ≫ (coproduct_iso F).inv = sigma.ι F j | rfl | lemma | category_theory.limits.types.coproduct_iso_mk_comp_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_equalizer_of_unique (t : ∀ (y : Y), g y = h y → ∃! (x : X), f x = y) :
is_limit (fork.of_ι _ w) | fork.is_limit.mk' _ $ λ s,
begin
refine ⟨λ i, _, _, _⟩,
{ apply classical.some (t (s.ι i) _),
apply congr_fun s.condition i },
{ ext i,
apply (classical.some_spec (t (s.ι i) _)).1 },
{ intros m hm,
ext i,
apply (classical.some_spec (t (s.ι i) _)).2,
apply congr_fun hm i },
end | def | category_theory.limits.types.type_equalizer_of_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | Show the given fork in `Type u` is an equalizer given that any element in the "difference kernel"
comes from `X`.
The converse of `unique_of_type_equalizer`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_of_type_equalizer (t : is_limit (fork.of_ι _ w)) (y : Y) (hy : g y = h y) :
∃! (x : X), f x = y | begin
let y' : punit ⟶ Y := λ _, y,
have hy' : y' ≫ g = y' ≫ h := funext (λ _, hy),
refine ⟨(fork.is_limit.lift' t _ hy').1 ⟨⟩, congr_fun (fork.is_limit.lift' t y' _).2 ⟨⟩, _⟩,
intros x' hx',
suffices : (λ (_ : punit), x') = (fork.is_limit.lift' t y' hy').1,
rw ← this,
apply fork.is_limit.hom_ext t,
e... | lemma | category_theory.limits.types.unique_of_type_equalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The converse of `type_equalizer_of_unique`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
type_equalizer_iff_unique :
nonempty (is_limit (fork.of_ι _ w)) ↔ (∀ (y : Y), g y = h y → ∃! (x : X), f x = y) | ⟨λ i, unique_of_type_equalizer _ _ (classical.choice i), λ k, ⟨type_equalizer_of_unique f w k⟩⟩ | lemma | category_theory.limits.types.type_equalizer_iff_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equalizer_limit : limits.limit_cone (parallel_pair g h) | { cone := fork.of_ι (subtype.val : {x : Y // g x = h x} → Y) (funext subtype.prop),
is_limit := fork.is_limit.mk' _ $ λ s,
⟨λ i, ⟨s.ι i, by apply congr_fun s.condition i⟩,
rfl,
λ m hm, funext $ λ x, subtype.ext (congr_fun hm x)⟩ } | def | category_theory.limits.types.equalizer_limit | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [
"subtype.ext",
"subtype.prop"
] | Show that the subtype `{x : Y // g x = h x}` is an equalizer for the pair `(g,h)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer_iso : equalizer g h ≅ {x : Y // g x = h x} | limit.iso_limit_cone equalizer_limit | def | category_theory.limits.types.equalizer_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The categorical equalizer in `Type u` is `{x : Y // g x = h x}`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer_iso_hom_comp_subtype :
(equalizer_iso g h).hom ≫ subtype.val = equalizer.ι g h | rfl | lemma | category_theory.limits.types.equalizer_iso_hom_comp_subtype | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equalizer_iso_inv_comp_ι :
(equalizer_iso g h).inv ≫ equalizer.ι g h = subtype.val | limit.iso_limit_cone_inv_π equalizer_limit walking_parallel_pair.zero | lemma | category_theory.limits.types.equalizer_iso_inv_comp_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer_rel : Y → Y → Prop
| rel (x : X) : coequalizer_rel (f x) (g x) | inductive | category_theory.limits.types.coequalizer_rel | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [
"rel"
] | (Implementation) The relation to be quotiented to obtain the coequalizer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer_colimit : limits.colimit_cocone (parallel_pair f g) | { cocone := cofork.of_π (quot.mk (coequalizer_rel f g))
(funext (λ x, quot.sound (coequalizer_rel.rel x))),
is_colimit := cofork.is_colimit.mk' _ $ λ s,
⟨ quot.lift s.π (λ a b (h : coequalizer_rel f g a b),
by { cases h, exact congr_fun s.condition h_1 }),
rfl,
λ m hm, funext $ λ x, quot.i... | def | category_theory.limits.types.coequalizer_colimit | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | Show that the quotient by the relation generated by `f(x) ~ g(x)`
is a coequalizer for the pair `(f, g)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer_preimage_image_eq_of_preimage_eq (π : Y ⟶ Z)
(e : f ≫ π = g ≫ π) (h : is_colimit (cofork.of_π π e)) (U : set Y) (H : f ⁻¹' U = g ⁻¹' U) :
π ⁻¹' (π '' U) = U | begin
have lem : ∀ x y, (coequalizer_rel f g x y) → (x ∈ U ↔ y ∈ U),
{ rintros _ _ ⟨x⟩, change x ∈ f ⁻¹' U ↔ x ∈ g ⁻¹' U, congr' 2 },
have eqv : _root_.equivalence (λ x y, x ∈ U ↔ y ∈ U) := by tidy,
ext,
split,
{ rw ← (show _ = π, from h.comp_cocone_point_unique_up_to_iso_inv
(coequalizer_colimit f g)... | lemma | category_theory.limits.types.coequalizer_preimage_image_eq_of_preimage_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [
"eqv_gen.mono"
] | If `π : Y ⟶ Z` is an equalizer for `(f, g)`, and `U ⊆ Y` such that `f ⁻¹' U = g ⁻¹' U`,
then `π ⁻¹' (π '' U) = U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer_iso : coequalizer f g ≅ _root_.quot (coequalizer_rel f g) | colimit.iso_colimit_cocone (coequalizer_colimit f g) | def | category_theory.limits.types.coequalizer_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The categorical coequalizer in `Type u` is the quotient by `f g ~ g x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer_iso_π_comp_hom :
coequalizer.π f g ≫ (coequalizer_iso f g).hom = quot.mk (coequalizer_rel f g) | colimit.iso_colimit_cocone_ι_hom (coequalizer_colimit f g) walking_parallel_pair.one | lemma | category_theory.limits.types.coequalizer_iso_π_comp_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer_iso_quot_comp_inv :
↾(quot.mk (coequalizer_rel f g)) ≫ (coequalizer_iso f g).inv = coequalizer.π f g | rfl | lemma | category_theory.limits.types.coequalizer_iso_quot_comp_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_obj : Type u | { p : X × Y // f p.1 = g p.2 } | abbreviation | category_theory.limits.types.pullback_obj | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The usual explicit pullback in the category of types, as a subtype of the product.
The full `limit_cone` data is bundled as `pullback_limit_cone f g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_cone : limits.pullback_cone f g | pullback_cone.mk (λ p : pullback_obj f g, p.1.1) (λ p, p.1.2) (funext (λ p, p.2)) | abbreviation | category_theory.limits.types.pullback_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The explicit pullback cone on `pullback_obj f g`.
This is bundled with the `is_limit` data as `pullback_limit_cone f g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_limit_cone (f : X ⟶ Z) (g : Y ⟶ Z) : limits.limit_cone (cospan f g) | { cone := pullback_cone f g,
is_limit := pullback_cone.is_limit_aux _
(λ s x, ⟨⟨s.fst x, s.snd x⟩, congr_fun s.condition x⟩)
(by tidy)
(by tidy)
(λ s m w, funext $ λ x, subtype.ext $
prod.ext (congr_fun (w walking_cospan.left) x)
(congr_fun (w walking_cospan.right) x)) } | def | category_theory.limits.types.pullback_limit_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [
"prod.ext",
"subtype.ext"
] | The explicit pullback in the category of types, bundled up as a `limit_cone`
for given `f` and `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_cone_iso_pullback : limit.cone (cospan f g) ≅ pullback_cone f g | (limit.is_limit _).unique_up_to_iso (pullback_limit_cone f g).is_limit | def | category_theory.limits.types.pullback_cone_iso_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The pullback cone given by the instance `has_pullbacks (Type u)` is isomorphic to the
explicit pullback cone given by `pullback_limit_cone`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_iso_pullback : pullback f g ≅ pullback_obj f g | (cones.forget _).map_iso $ pullback_cone_iso_pullback f g | def | category_theory.limits.types.pullback_iso_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | The pullback given by the instance `has_pullbacks (Type u)` is isomorphic to the
explicit pullback object given by `pullback_limit_obj`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_iso_pullback_hom_fst (p : pullback f g) :
((pullback_iso_pullback f g).hom p : X × Y).fst = (pullback.fst : _ ⟶ X) p | congr_fun ((pullback_cone_iso_pullback f g).hom.w left) p | lemma | category_theory.limits.types.pullback_iso_pullback_hom_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_iso_pullback_hom_snd (p : pullback f g) :
((pullback_iso_pullback f g).hom p : X × Y).snd = (pullback.snd : _ ⟶ Y) p | congr_fun ((pullback_cone_iso_pullback f g).hom.w right) p | lemma | category_theory.limits.types.pullback_iso_pullback_hom_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_iso_pullback_inv_fst :
(pullback_iso_pullback f g).inv ≫ pullback.fst = (λ p, (p : X × Y).fst) | (pullback_cone_iso_pullback f g).inv.w left | lemma | category_theory.limits.types.pullback_iso_pullback_inv_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_iso_pullback_inv_snd :
(pullback_iso_pullback f g).inv ≫ pullback.snd = (λ p, (p : X × Y).snd) | (pullback_cone_iso_pullback f g).inv.w right | lemma | category_theory.limits.types.pullback_iso_pullback_inv_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/types.lean | [
"category_theory.limits.types",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.terminal",
"category_theory.concrete_category.basic",
"tactic.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_parallel_family (J : Type w) : Type w
| zero : walking_parallel_family
| one : walking_parallel_family | inductive | category_theory.limits.walking_parallel_family | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | The type of objects for the diagram indexing a wide (co)equalizer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_parallel_family.hom (J : Type w) :
walking_parallel_family J → walking_parallel_family J → Type w
| id : Π X : walking_parallel_family.{w} J, walking_parallel_family.hom X X
| line : Π (j : J), walking_parallel_family.hom zero one | inductive | category_theory.limits.walking_parallel_family.hom | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | The type family of morphisms for the diagram indexing a wide (co)equalizer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_parallel_family.hom.comp :
Π (X Y Z : walking_parallel_family J)
(f : walking_parallel_family.hom J X Y) (g : walking_parallel_family.hom J Y Z),
walking_parallel_family.hom J X Z | | _ _ _ (id _) h := h
| _ _ _ (line j) (id one) := line j. | def | category_theory.limits.walking_parallel_family.hom.comp | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | Composition of morphisms in the indexing diagram for wide (co)equalizers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_parallel_family.category : small_category (walking_parallel_family J) | { hom := walking_parallel_family.hom J,
id := walking_parallel_family.hom.id,
comp := walking_parallel_family.hom.comp } | instance | category_theory.limits.walking_parallel_family.category | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_parallel_family.hom_id (X : walking_parallel_family J) :
walking_parallel_family.hom.id X = 𝟙 X | rfl | lemma | category_theory.limits.walking_parallel_family.hom_id | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallel_family : walking_parallel_family J ⥤ C | { obj := λ x, walking_parallel_family.cases_on x X Y,
map := λ x y h, match x, y, h with
| _, _, (id _) := 𝟙 _
| _, _, (line j) := f j
end,
map_comp' :=
begin
rintro _ _ _ ⟨⟩ ⟨⟩;
{ unfold_aux, simp; refl },
end } | def | category_theory.limits.parallel_family | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | `parallel_family f` is the diagram in `C` consisting of the given family of morphisms, each with
common domain and codomain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
parallel_family_obj_zero : (parallel_family f).obj zero = X | rfl | lemma | category_theory.limits.parallel_family_obj_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallel_family_obj_one : (parallel_family f).obj one = Y | rfl | lemma | category_theory.limits.parallel_family_obj_one | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallel_family_map_left {j : J} : (parallel_family f).map (line j) = f j | rfl | lemma | category_theory.limits.parallel_family_map_left | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_iso_parallel_family (F : walking_parallel_family J ⥤ C) :
F ≅ parallel_family (λ j, F.map (line j)) | nat_iso.of_components (λ j, eq_to_iso $ by cases j; tidy) $ by tidy | def | category_theory.limits.diagram_iso_parallel_family | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | Every functor indexing a wide (co)equalizer is naturally isomorphic (actually, equal) to a
`parallel_family` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_parallel_family_equiv_walking_parallel_pair :
walking_parallel_family.{w} (ulift bool) ≌ walking_parallel_pair | { functor := parallel_family
(λ p, cond p.down walking_parallel_pair_hom.left walking_parallel_pair_hom.right),
inverse := parallel_pair (line (ulift.up tt)) (line (ulift.up ff)),
unit_iso := nat_iso.of_components (λ X, eq_to_iso (by cases X; refl)) (by tidy),
counit_iso := nat_iso.of_components (λ X, eq_to... | def | category_theory.limits.walking_parallel_family_equiv_walking_parallel_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | `walking_parallel_pair` as a category is equivalent to a special case of
`walking_parallel_family`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trident | cone (parallel_family f) | abbreviation | category_theory.limits.trident | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | A trident on `f` is just a `cone (parallel_family f)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cotrident | cocone (parallel_family f) | abbreviation | category_theory.limits.cotrident | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | A cotrident on `f` and `g` is just a `cocone (parallel_family f)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trident.ι (t : trident f) | t.π.app zero | abbreviation | category_theory.limits.trident.ι | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | A trident `t` on the parallel family `f : J → (X ⟶ Y)` consists of two morphisms
`t.π.app zero : t.X ⟶ X` and `t.π.app one : t.X ⟶ Y`. Of these, only the first one is
interesting, and we give it the shorter name `trident.ι t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cotrident.π (t : cotrident f) | t.ι.app one | abbreviation | category_theory.limits.cotrident.π | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | A cotrident `t` on the parallel family `f : J → (X ⟶ Y)` consists of two morphisms
`t.ι.app zero : X ⟶ t.X` and `t.ι.app one : Y ⟶ t.X`. Of these, only the second one is
interesting, and we give it the shorter name `cotrident.π t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trident.ι_eq_app_zero (t : trident f) : t.ι = t.π.app zero | rfl | lemma | category_theory.limits.trident.ι_eq_app_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cotrident.π_eq_app_one (t : cotrident f) : t.π = t.ι.app one | rfl | lemma | category_theory.limits.cotrident.π_eq_app_one | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trident.app_zero (s : trident f) (j : J) :
s.π.app zero ≫ f j = s.π.app one | by rw [←s.w (line j), parallel_family_map_left] | lemma | category_theory.limits.trident.app_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cotrident.app_one (s : cotrident f) (j : J) :
f j ≫ s.ι.app one = s.ι.app zero | by rw [←s.w (line j), parallel_family_map_left] | lemma | category_theory.limits.cotrident.app_one | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trident.of_ι [nonempty J] {P : C} (ι : P ⟶ X) (w : ∀ j₁ j₂, ι ≫ f j₁ = ι ≫ f j₂) :
trident f | { X := P,
π :=
{ app := λ X, walking_parallel_family.cases_on X ι (ι ≫ f (classical.arbitrary J)),
naturality' := λ i j f,
begin
dsimp,
cases f with _ k,
{ simp },
{ simp [w (classical.arbitrary J) k] },
end } } | def | category_theory.limits.trident.of_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [
"classical.arbitrary"
] | A trident on `f : J → (X ⟶ Y)` is determined by the morphism `ι : P ⟶ X` satisfying
`∀ j₁ j₂, ι ≫ f j₁ = ι ≫ f j₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cotrident.of_π [nonempty J] {P : C} (π : Y ⟶ P) (w : ∀ j₁ j₂, f j₁ ≫ π = f j₂ ≫ π) :
cotrident f | { X := P,
ι :=
{ app := λ X, walking_parallel_family.cases_on X (f (classical.arbitrary J) ≫ π) π,
naturality' := λ i j f,
begin
dsimp,
cases f with _ k,
{ simp },
{ simp [w (classical.arbitrary J) k] }
end } } | def | category_theory.limits.cotrident.of_π | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [
"classical.arbitrary"
] | A cotrident on `f : J → (X ⟶ Y)` is determined by the morphism `π : Y ⟶ P` satisfying
`∀ j₁ j₂, f j₁ ≫ π = f j₂ ≫ π`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trident.ι_of_ι [nonempty J] {P : C} (ι : P ⟶ X) (w : ∀ j₁ j₂, ι ≫ f j₁ = ι ≫ f j₂) :
(trident.of_ι ι w).ι = ι | rfl | lemma | category_theory.limits.trident.ι_of_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cotrident.π_of_π [nonempty J] {P : C} (π : Y ⟶ P) (w : ∀ j₁ j₂, f j₁ ≫ π = f j₂ ≫ π) :
(cotrident.of_π π w).π = π | rfl | lemma | category_theory.limits.cotrident.π_of_π | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trident.condition (j₁ j₂ : J) (t : trident f) : t.ι ≫ f j₁ = t.ι ≫ f j₂ | by rw [t.app_zero, t.app_zero] | lemma | category_theory.limits.trident.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cotrident.condition (j₁ j₂ : J) (t : cotrident f) : f j₁ ≫ t.π = f j₂ ≫ t.π | by rw [t.app_one, t.app_one] | lemma | category_theory.limits.cotrident.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trident.equalizer_ext [nonempty J] (s : trident f) {W : C} {k l : W ⟶ s.X}
(h : k ≫ s.ι = l ≫ s.ι) : ∀ (j : walking_parallel_family J),
k ≫ s.π.app j = l ≫ s.π.app j | | zero := h
| one := by rw [←s.app_zero (classical.arbitrary J), reassoc_of h] | lemma | category_theory.limits.trident.equalizer_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [
"classical.arbitrary"
] | To check whether two maps are equalized by both maps of a trident, it suffices to check it for
the first map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cotrident.coequalizer_ext [nonempty J] (s : cotrident f) {W : C} {k l : s.X ⟶ W}
(h : s.π ≫ k = s.π ≫ l) : ∀ (j : walking_parallel_family J),
s.ι.app j ≫ k = s.ι.app j ≫ l | | zero := by rw [←s.app_one (classical.arbitrary J), category.assoc, category.assoc, h]
| one := h | lemma | category_theory.limits.cotrident.coequalizer_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [
"classical.arbitrary"
] | To check whether two maps are coequalized by both maps of a cotrident, it suffices to check it
for the second map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trident.is_limit.hom_ext [nonempty J] {s : trident f} (hs : is_limit s)
{W : C} {k l : W ⟶ s.X} (h : k ≫ s.ι = l ≫ s.ι) :
k = l | hs.hom_ext $ trident.equalizer_ext _ h | lemma | category_theory.limits.trident.is_limit.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cotrident.is_colimit.hom_ext [nonempty J] {s : cotrident f} (hs : is_colimit s)
{W : C} {k l : s.X ⟶ W} (h : s.π ≫ k = s.π ≫ l) :
k = l | hs.hom_ext $ cotrident.coequalizer_ext _ h | lemma | category_theory.limits.cotrident.is_colimit.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trident.is_limit.lift' [nonempty J] {s : trident f} (hs : is_limit s) {W : C} (k : W ⟶ X)
(h : ∀ j₁ j₂, k ≫ f j₁ = k ≫ f j₂) :
{l : W ⟶ s.X // l ≫ trident.ι s = k} | ⟨hs.lift $ trident.of_ι _ h, hs.fac _ _⟩ | def | category_theory.limits.trident.is_limit.lift' | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | If `s` is a limit trident over `f`, then a morphism `k : W ⟶ X` satisfying
`∀ j₁ j₂, k ≫ f j₁ = k ≫ f j₂` induces a morphism `l : W ⟶ s.X` such that
`l ≫ trident.ι s = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cotrident.is_colimit.desc' [nonempty J] {s : cotrident f} (hs : is_colimit s) {W : C}
(k : Y ⟶ W) (h : ∀ j₁ j₂, f j₁ ≫ k = f j₂ ≫ k) :
{l : s.X ⟶ W // cotrident.π s ≫ l = k} | ⟨hs.desc $ cotrident.of_π _ h, hs.fac _ _⟩ | def | category_theory.limits.cotrident.is_colimit.desc' | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | If `s` is a colimit cotrident over `f`, then a morphism `k : Y ⟶ W` satisfying
`∀ j₁ j₂, f j₁ ≫ k = f j₂ ≫ k` induces a morphism `l : s.X ⟶ W` such that
`cotrident.π s ≫ l = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trident.is_limit.mk [nonempty J] (t : trident f)
(lift : Π (s : trident f), s.X ⟶ t.X)
(fac : ∀ (s : trident f), lift s ≫ t.ι = s.ι)
(uniq : ∀ (s : trident f) (m : s.X ⟶ t.X)
(w : ∀ j : walking_parallel_family J, m ≫ t.π.app j = s.π.app j), m = lift s) :
is_limit t | { lift := lift,
fac' := λ s j, walking_parallel_family.cases_on j (fac s)
(by rw [←t.w (line (classical.arbitrary J)), reassoc_of fac, s.w]),
uniq' := uniq } | def | category_theory.limits.trident.is_limit.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [
"classical.arbitrary",
"lift"
] | This is a slightly more convenient method to verify that a trident is a limit cone. It
only asks for a proof of facts that carry any mathematical content | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trident.is_limit.mk' [nonempty J] (t : trident f)
(create : Π (s : trident f), {l // l ≫ t.ι = s.ι ∧ ∀ {m}, m ≫ t.ι = s.ι → m = l}) :
is_limit t | trident.is_limit.mk t
(λ s, (create s).1)
(λ s, (create s).2.1)
(λ s m w, (create s).2.2 (w zero)) | def | category_theory.limits.trident.is_limit.mk' | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | This is another convenient method to verify that a trident 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 |
cotrident.is_colimit.mk [nonempty J] (t : cotrident f)
(desc : Π (s : cotrident f), t.X ⟶ s.X)
(fac : ∀ (s : cotrident f), t.π ≫ desc s = s.π)
(uniq : ∀ (s : cotrident f) (m : t.X ⟶ s.X)
(w : ∀ j : walking_parallel_family J, t.ι.app j ≫ m = s.ι.app j), m = desc s) :
is_colimit t | { desc := desc,
fac' := λ s j, walking_parallel_family.cases_on j
(by rw [←t.w_assoc (line (classical.arbitrary J)), fac, s.w]) (fac s),
uniq' := uniq } | def | category_theory.limits.cotrident.is_colimit.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [
"classical.arbitrary"
] | This is a slightly more convenient method to verify that a cotrident is a colimit cocone. It
only asks for a proof of facts that carry any mathematical content | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cotrident.is_colimit.mk' [nonempty J] (t : cotrident f)
(create : Π (s : cotrident f), {l : t.X ⟶ s.X // t.π ≫ l = s.π ∧ ∀ {m}, t.π ≫ m = s.π → m = l}) :
is_colimit t | cotrident.is_colimit.mk t
(λ s, (create s).1)
(λ s, (create s).2.1)
(λ s m w, (create s).2.2 (w one)) | def | category_theory.limits.cotrident.is_colimit.mk' | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | This is another convenient method to verify that a cotrident 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 |
trident.is_limit.hom_iso [nonempty J] {t : trident f} (ht : is_limit t) (Z : C) :
(Z ⟶ t.X) ≃ {h : Z ⟶ X // ∀ j₁ j₂, h ≫ f j₁ = h ≫ f j₂} | { to_fun := λ k, ⟨k ≫ t.ι, by simp⟩,
inv_fun := λ h, (trident.is_limit.lift' ht _ h.prop).1,
left_inv := λ k, trident.is_limit.hom_ext ht (trident.is_limit.lift' _ _ _).prop,
right_inv := λ h, subtype.ext (trident.is_limit.lift' ht _ _).prop } | def | category_theory.limits.trident.is_limit.hom_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [
"inv_fun",
"subtype.ext"
] | Given a limit cone for the family `f : J → (X ⟶ Y)`, for any `Z`, morphisms from `Z` to its point
are in bijection with morphisms `h : Z ⟶ X` such that `∀ j₁ j₂, h ≫ f j₁ = h ≫ f j₂`.
Further, this bijection is natural in `Z`: see `trident.is_limit.hom_iso_natural`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trident.is_limit.hom_iso_natural [nonempty J] {t : trident f} (ht : is_limit t)
{Z Z' : C} (q : Z' ⟶ Z) (k : Z ⟶ t.X) :
(trident.is_limit.hom_iso ht _ (q ≫ k) : Z' ⟶ X) =
q ≫ (trident.is_limit.hom_iso ht _ k : Z ⟶ X) | category.assoc _ _ _ | lemma | category_theory.limits.trident.is_limit.hom_iso_natural | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | The bijection of `trident.is_limit.hom_iso` is natural in `Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cotrident.is_colimit.hom_iso [nonempty J] {t : cotrident f} (ht : is_colimit t) (Z : C) :
(t.X ⟶ Z) ≃ {h : Y ⟶ Z // ∀ j₁ j₂, f j₁ ≫ h = f j₂ ≫ h} | { to_fun := λ k, ⟨t.π ≫ k, by simp⟩,
inv_fun := λ h, (cotrident.is_colimit.desc' ht _ h.prop).1,
left_inv := λ k, cotrident.is_colimit.hom_ext ht (cotrident.is_colimit.desc' _ _ _).prop,
right_inv := λ h, subtype.ext (cotrident.is_colimit.desc' ht _ _).prop } | def | category_theory.limits.cotrident.is_colimit.hom_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [
"inv_fun",
"subtype.ext"
] | Given a colimit cocone for the family `f : J → (X ⟶ Y)`, for any `Z`, morphisms from the cocone
point to `Z` are in bijection with morphisms `h : Z ⟶ X` such that
`∀ j₁ j₂, f j₁ ≫ h = f j₂ ≫ h`. Further, this bijection is natural in `Z`: see
`cotrident.is_colimit.hom_iso_natural`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cotrident.is_colimit.hom_iso_natural [nonempty J] {t : cotrident f} {Z Z' : C}
(q : Z ⟶ Z') (ht : is_colimit t) (k : t.X ⟶ Z) :
(cotrident.is_colimit.hom_iso ht _ (k ≫ q) : Y ⟶ Z') =
(cotrident.is_colimit.hom_iso ht _ k : Y ⟶ Z) ≫ q | (category.assoc _ _ _).symm | lemma | category_theory.limits.cotrident.is_colimit.hom_iso_natural | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | The bijection of `cotrident.is_colimit.hom_iso` is natural in `Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone.of_trident
{F : walking_parallel_family J ⥤ C} (t : trident (λ j, F.map (line j))) : cone F | { X := t.X,
π :=
{ app := λ X, t.π.app X ≫ eq_to_hom (by tidy),
naturality' := λ j j' g, by { cases g; { dsimp, simp } } } } | def | category_theory.limits.cone.of_trident | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | This is a helper construction that can be useful when verifying that a category has certain wide
equalizers. Given `F : walking_parallel_family ⥤ C`, which is really the same as
`parallel_family (λ j, F.map (line j))`, and a trident on `λ j, F.map (line j)`, we get a cone
on `F`.
If you're thinking abo... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone.of_cotrident
{F : walking_parallel_family J ⥤ C} (t : cotrident (λ j, F.map (line j))) : cocone F | { X := t.X,
ι :=
{ app := λ X, eq_to_hom (by tidy) ≫ t.ι.app X,
naturality' := λ j j' g, by { cases g; dsimp; simp [cotrident.app_one t] } } } | def | category_theory.limits.cocone.of_cotrident | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | This is a helper construction that can be useful when verifying that a category has all
coequalizers. Given `F : walking_parallel_family ⥤ C`, which is really the same as
`parallel_family (λ j, F.map (line j))`, and a cotrident on `λ j, F.map (line j)` we get a
cocone on `F`.
If you're thinking about u... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone.of_trident_π
{F : walking_parallel_family J ⥤ C} (t : trident (λ j, F.map (line j))) (j) :
(cone.of_trident t).π.app j = t.π.app j ≫ eq_to_hom (by tidy) | rfl | lemma | category_theory.limits.cone.of_trident_π | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone.of_cotrident_ι
{F : walking_parallel_family J ⥤ C} (t : cotrident (λ j, F.map (line j))) (j) :
(cocone.of_cotrident t).ι.app j = eq_to_hom (by tidy) ≫ t.ι.app j | rfl | lemma | category_theory.limits.cocone.of_cotrident_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trident.of_cone
{F : walking_parallel_family J ⥤ C} (t : cone F) : trident (λ j, F.map (line j)) | { X := t.X,
π := { app := λ X, t.π.app X ≫ eq_to_hom (by tidy) } } | def | category_theory.limits.trident.of_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | Given `F : walking_parallel_family ⥤ C`, which is really the same as
`parallel_family (λ j, F.map (line j))` and a cone on `F`, we get a trident on
`λ j, F.map (line j)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cotrident.of_cocone
{F : walking_parallel_family J ⥤ C} (t : cocone F) : cotrident (λ j, F.map (line j)) | { X := t.X,
ι := { app := λ X, eq_to_hom (by tidy) ≫ t.ι.app X } } | def | category_theory.limits.cotrident.of_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | Given `F : walking_parallel_family ⥤ C`, which is really the same as
`parallel_family (F.map left) (F.map right)` and a cocone on `F`, we get a cotrident on
`λ j, F.map (line j)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trident.of_cone_π {F : walking_parallel_family J ⥤ C} (t : cone F) (j) :
(trident.of_cone t).π.app j = t.π.app j ≫ eq_to_hom (by tidy) | rfl | lemma | category_theory.limits.trident.of_cone_π | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cotrident.of_cocone_ι {F : walking_parallel_family J ⥤ C} (t : cocone F) (j) :
(cotrident.of_cocone t).ι.app j = eq_to_hom (by tidy) ≫ t.ι.app j | rfl | lemma | category_theory.limits.cotrident.of_cocone_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trident.mk_hom [nonempty J] {s t : trident f} (k : s.X ⟶ t.X) (w : k ≫ t.ι = s.ι) : s ⟶ t | { hom := k,
w' :=
begin
rintro ⟨_|_⟩,
{ exact w },
{ simpa using w =≫ f (classical.arbitrary J) },
end } | def | category_theory.limits.trident.mk_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [
"classical.arbitrary"
] | Helper function for constructing morphisms between wide equalizer tridents. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trident.ext [nonempty J] {s t : trident f} (i : s.X ≅ t.X) (w : i.hom ≫ t.ι = s.ι) : s ≅ t | { hom := trident.mk_hom i.hom w,
inv := trident.mk_hom i.inv (by rw [← w, iso.inv_hom_id_assoc]) } | def | category_theory.limits.trident.ext | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | To construct an isomorphism between tridents,
it suffices to give an isomorphism between the cone points
and check that it commutes with the `ι` morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cotrident.mk_hom [nonempty J] {s t : cotrident f} (k : s.X ⟶ t.X) (w : s.π ≫ k = t.π) :
s ⟶ t | { hom := k,
w' :=
begin
rintro ⟨_|_⟩,
{ simpa using f (classical.arbitrary J) ≫= w },
{ exact w },
end } | def | category_theory.limits.cotrident.mk_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [
"classical.arbitrary"
] | Helper function for constructing morphisms between coequalizer cotridents. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cotrident.ext [nonempty J] {s t : cotrident f} (i : s.X ≅ t.X) (w : s.π ≫ i.hom = t.π) :
s ≅ t | { hom := cotrident.mk_hom i.hom w,
inv := cotrident.mk_hom i.inv (by rw [iso.comp_inv_eq, w]) } | def | category_theory.limits.cotrident.ext | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | To construct an isomorphism between cotridents,
it suffices to give an isomorphism between the cocone points
and check that it commutes with the `π` morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_wide_equalizer | has_limit (parallel_family f) | abbreviation | category_theory.limits.has_wide_equalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | `has_wide_equalizer f` represents a particular choice of limiting cone for the parallel family of
morphisms `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_equalizer : C | limit (parallel_family f) | abbreviation | category_theory.limits.wide_equalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | If a wide equalizer of `f` exists, we can access an arbitrary choice of such by
saying `wide_equalizer f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_equalizer.ι : wide_equalizer f ⟶ X | limit.π (parallel_family f) zero | abbreviation | category_theory.limits.wide_equalizer.ι | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | If a wide equalizer of `f` exists, we can access the inclusion `wide_equalizer f ⟶ X` by
saying `wide_equalizer.ι f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_equalizer.trident : trident f | limit.cone (parallel_family f) | abbreviation | category_theory.limits.wide_equalizer.trident | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | A wide equalizer cone for a parallel family `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_equalizer.trident_ι :
(wide_equalizer.trident f).ι = wide_equalizer.ι f | rfl | lemma | category_theory.limits.wide_equalizer.trident_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wide_equalizer.trident_π_app_zero :
(wide_equalizer.trident f).π.app zero = wide_equalizer.ι f | rfl | lemma | category_theory.limits.wide_equalizer.trident_π_app_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wide_equalizer.condition (j₁ j₂ : J) :
wide_equalizer.ι f ≫ f j₁ = wide_equalizer.ι f ≫ f j₂ | trident.condition j₁ j₂ $ limit.cone $ parallel_family f | lemma | category_theory.limits.wide_equalizer.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wide_equalizer_is_wide_equalizer [nonempty J] :
is_limit (trident.of_ι (wide_equalizer.ι f) (wide_equalizer.condition f)) | is_limit.of_iso_limit (limit.is_limit _) (trident.ext (iso.refl _) (by tidy)) | def | category_theory.limits.wide_equalizer_is_wide_equalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | The wide_equalizer built from `wide_equalizer.ι f` is limiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_equalizer.lift [nonempty J] {W : C} (k : W ⟶ X)
(h : ∀ j₁ j₂, k ≫ f j₁ = k ≫ f j₂) :
W ⟶ wide_equalizer f | limit.lift (parallel_family f) (trident.of_ι k h) | abbreviation | category_theory.limits.wide_equalizer.lift | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | A morphism `k : W ⟶ X` satisfying `∀ j₁ j₂, k ≫ f j₁ = k ≫ f j₂` factors through the
wide equalizer of `f` via `wide_equalizer.lift : W ⟶ wide_equalizer f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_equalizer.lift_ι [nonempty J] {W : C} (k : W ⟶ X) (h : ∀ j₁ j₂, k ≫ f j₁ = k ≫ f j₂) :
wide_equalizer.lift k h ≫ wide_equalizer.ι f = k | limit.lift_π _ _ | lemma | category_theory.limits.wide_equalizer.lift_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wide_equalizer.lift' [nonempty J] {W : C} (k : W ⟶ X) (h : ∀ j₁ j₂, k ≫ f j₁ = k ≫ f j₂) :
{l : W ⟶ wide_equalizer f // l ≫ wide_equalizer.ι f = k} | ⟨wide_equalizer.lift k h, wide_equalizer.lift_ι _ _⟩ | def | category_theory.limits.wide_equalizer.lift' | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | A morphism `k : W ⟶ X` satisfying `∀ j₁ j₂, k ≫ f j₁ = k ≫ f j₂` induces a morphism
`l : W ⟶ wide_equalizer f` satisfying `l ≫ wide_equalizer.ι f = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_equalizer.hom_ext [nonempty J] {W : C} {k l : W ⟶ wide_equalizer f}
(h : k ≫ wide_equalizer.ι f = l ≫ wide_equalizer.ι f) : k = l | trident.is_limit.hom_ext (limit.is_limit _) h | lemma | category_theory.limits.wide_equalizer.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | Two maps into a wide equalizer are equal if they are are equal when composed with the wide
equalizer map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_equalizer.ι_mono [nonempty J] : mono (wide_equalizer.ι f) | { right_cancellation := λ Z h k w, wide_equalizer.hom_ext w } | instance | category_theory.limits.wide_equalizer.ι_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | A wide equalizer morphism is a monomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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