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extend_cofan_is_colimit {n : ℕ} (f : fin (n+1) → C)
{c₁ : cofan (λ (i : fin n), f i.succ)} {c₂ : binary_cofan (f 0) c₁.X}
(t₁ : is_colimit c₁) (t₂ : is_colimit c₂) :
is_colimit (extend_cofan c₁ c₂) | { desc := λ s,
begin
apply (binary_cofan.is_colimit.desc' t₂ (s.ι.app ⟨0⟩) _).1,
apply t₁.desc ⟨_, discrete.nat_trans (λ i, s.ι.app ⟨i.as.succ⟩)⟩
end,
fac' := λ s,
begin
rintro ⟨j⟩,
apply fin.induction_on j,
{ apply (binary_cofan.is_colimit.desc' t₂ _ _).2.1 },
{ rintro i -,
dsimp ... | def | category_theory.extend_cofan_is_colimit | category_theory.limits.constructions | src/category_theory/limits/constructions/finite_products_of_binary_products.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.finite_products",
"logic.equiv.fin"
] | [
"fin.cases_succ",
"fin.induction_on"
] | Show that if the two given cofans in `extend_cofan` are colimits,
then the constructed cofan is also a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coproduct_fin :
Π (n : ℕ) (f : fin n → C), has_coproduct f | | 0 := λ f,
begin
letI : has_colimits_of_shape (discrete (fin 0)) C :=
has_colimits_of_shape_of_equivalence (discrete.equivalence.{0} fin_zero_equiv'.symm),
apply_instance,
end
| (n+1) := λ f,
begin
haveI := has_coproduct_fin n,
apply has_colimit.mk
⟨_, extend_cofan_is_colimit f (colim... | lemma | category_theory.has_coproduct_fin | category_theory.limits.constructions | src/category_theory/limits/constructions/finite_products_of_binary_products.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.finite_products",
"logic.equiv.fin"
] | [] | If `C` has an initial object and binary coproducts, then it has a coproduct for objects indexed by
`fin n`.
This is a helper lemma for `has_cofinite_products_of_has_binary_and_terminal`, which is more general
than this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_coproducts_of_has_binary_and_initial : has_finite_coproducts C | begin
refine ⟨λ n, ⟨λ K, _⟩⟩,
letI := has_coproduct_fin n (λ n, K.obj ⟨n⟩),
let : K ≅ discrete.functor (λ n, K.obj ⟨n⟩) := discrete.nat_iso (λ ⟨i⟩, iso.refl _),
apply has_colimit_of_iso this,
end | lemma | category_theory.has_finite_coproducts_of_has_binary_and_initial | category_theory.limits.constructions | src/category_theory/limits/constructions/finite_products_of_binary_products.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.finite_products",
"logic.equiv.fin"
] | [] | If `C` has an initial object and binary coproducts, then it has finite coproducts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_fin_of_preserves_binary_and_initial :
Π (n : ℕ) (f : fin n → C), preserves_colimit (discrete.functor f) F | | 0 := λ f,
begin
letI : preserves_colimits_of_shape (discrete (fin 0)) F :=
preserves_colimits_of_shape_of_equiv.{0 0}
(discrete.equivalence fin_zero_equiv'.symm) _,
apply_instance,
end
| (n+1) :=
begin
haveI := preserves_fin_of_preserves_binary_and_initial n,
intro f,
refine pr... | def | category_theory.preserves_fin_of_preserves_binary_and_initial | category_theory.limits.constructions | src/category_theory/limits/constructions/finite_products_of_binary_products.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.finite_products",
"logic.equiv.fin"
] | [
"fin.cases_succ",
"fin.induction_on"
] | If `F` preserves the initial object and binary coproducts, then it preserves products indexed by
`fin n` for any `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_shape_fin_of_preserves_binary_and_initial (n : ℕ) :
preserves_colimits_of_shape (discrete (fin n)) F | { preserves_colimit := λ K,
begin
let : discrete.functor (λ n, K.obj ⟨n⟩) ≅ K := discrete.nat_iso (λ ⟨i⟩, iso.refl _),
haveI := preserves_fin_of_preserves_binary_and_initial F n (λ n, K.obj ⟨n⟩),
apply preserves_colimit_of_iso_diagram F this,
end } | def | category_theory.preserves_shape_fin_of_preserves_binary_and_initial | category_theory.limits.constructions | src/category_theory/limits/constructions/finite_products_of_binary_products.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.finite_products",
"logic.equiv.fin"
] | [] | If `F` preserves the initial object and binary coproducts, then it preserves colimits of shape
`discrete (fin n)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_coproducts_of_preserves_binary_and_initial
(J : Type) [fintype J] :
preserves_colimits_of_shape (discrete J) F | begin
classical,
let e := fintype.equiv_fin J,
haveI := preserves_shape_fin_of_preserves_binary_and_initial F (fintype.card J),
apply preserves_colimits_of_shape_of_equiv.{0 0} (discrete.equivalence e).symm,
end | def | category_theory.preserves_finite_coproducts_of_preserves_binary_and_initial | category_theory.limits.constructions | src/category_theory/limits/constructions/finite_products_of_binary_products.lean | [
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.finite_products",
"logic.equiv.fin"
] | [
"fintype",
"fintype.card",
"fintype.equiv_fin"
] | If `F` preserves the initial object and binary coproducts then it preserves finite products. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
build_limit : cone F | { X := i.X,
π :=
{ app := λ j, i.ι ≫ c₁.π.app ⟨_⟩,
naturality' := λ j₁ j₂ f, begin
dsimp,
rw [category.id_comp, category.assoc, ← hs ⟨⟨_, _⟩, f⟩, i.condition_assoc, ht],
end} } | def | category_theory.limits.has_limit_of_has_products_of_has_equalizers.build_limit | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | (Implementation) Given the appropriate product and equalizer cones, build the cone for `F` which is
limiting if the given cones are also. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
build_is_limit (t₁ : is_limit c₁) (t₂ : is_limit c₂) (hi : is_limit i) :
is_limit (build_limit s t hs ht i) | { lift := λ q,
begin
refine hi.lift (fork.of_ι _ _),
{ refine t₁.lift (fan.mk _ (λ j, _)),
apply q.π.app j },
{ apply t₂.hom_ext,
intro j, discrete_cases,
simp [hs, ht] },
end,
uniq' := λ q m w, hi.hom_ext (i.equalizer_ext (t₁.hom_ext
(λ j, by { cases j, simpa using w j }))) } | def | category_theory.limits.has_limit_of_has_products_of_has_equalizers.build_is_limit | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [
"lift"
] | (Implementation) Show the cone constructed in `build_limit` is limiting, provided the cones used in
its construction are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_of_equalizer_and_product (F : J ⥤ C)
[has_limit (discrete.functor F.obj)]
[has_limit (discrete.functor (λ f : (Σ p : J × J, p.1 ⟶ p.2), F.obj f.1.2))]
[has_equalizers C] : limit_cone F | { cone := _,
is_limit :=
build_is_limit
(pi.lift (λ f, limit.π (discrete.functor F.obj) ⟨_⟩ ≫ F.map f.2))
(pi.lift (λ f, limit.π (discrete.functor F.obj) ⟨f.1.2⟩))
(by simp)
(by simp)
(limit.is_limit _)
(limit.is_limit _)
(limit.is_limit _) } | def | category_theory.limits.limit_cone_of_equalizer_and_product | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | Given the existence of the appropriate (possibly finite) products and equalizers,
we can construct a limit cone for `F`.
(This assumes the existence of all equalizers, which is technically stronger than needed.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit_of_equalizer_and_product (F : J ⥤ C)
[has_limit (discrete.functor F.obj)]
[has_limit (discrete.functor (λ f : (Σ p : J × J, p.1 ⟶ p.2), F.obj f.1.2))]
[has_equalizers C] : has_limit F | has_limit.mk (limit_cone_of_equalizer_and_product F) | lemma | category_theory.limits.has_limit_of_equalizer_and_product | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | Given the existence of the appropriate (possibly finite) products and equalizers, we know a limit of
`F` exists.
(This assumes the existence of all equalizers, which is technically stronger than needed.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_subobject_product [has_limits_of_size.{w w} C] (F : J ⥤ C) :
limit F ⟶ ∏ (λ j, F.obj j) | (limit.iso_limit_cone (limit_cone_of_equalizer_and_product F)).hom ≫ equalizer.ι _ _ | def | category_theory.limits.limit_subobject_product | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | A limit can be realised as a subobject of a product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_subobject_product_mono [has_limits_of_size.{w w} C] (F : J ⥤ C) :
mono (limit_subobject_product F) | mono_comp _ _ | instance | category_theory.limits.limit_subobject_product_mono | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits_of_has_equalizers_and_products
[has_products.{w} C] [has_equalizers C] : has_limits_of_size.{w w} C | { has_limits_of_shape := λ J 𝒥,
{ has_limit := λ F, by exactI has_limit_of_equalizer_and_product F } } | lemma | category_theory.limits.has_limits_of_has_equalizers_and_products | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | Any category with products and equalizers has all limits.
See <https://stacks.math.columbia.edu/tag/002N>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_limits_of_has_equalizers_and_finite_products
[has_finite_products C] [has_equalizers C] : has_finite_limits C | ⟨λ J _ _, { has_limit := λ F, by exactI has_limit_of_equalizer_and_product F }⟩ | lemma | category_theory.limits.has_finite_limits_of_has_equalizers_and_finite_products | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | Any category with finite products and equalizers has all finite limits.
See <https://stacks.math.columbia.edu/tag/002O>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limit_of_preserves_equalizers_and_product :
preserves_limits_of_shape J G | { preserves_limit := λ K,
begin
let P := ∏ K.obj,
let Q := ∏ (λ (f : (Σ (p : J × J), p.fst ⟶ p.snd)), K.obj f.1.2),
let s : P ⟶ Q := pi.lift (λ f, limit.π (discrete.functor K.obj) ⟨_⟩ ≫ K.map f.2),
let t : P ⟶ Q := pi.lift (λ f, limit.π (discrete.functor K.obj) ⟨f.1.2⟩),
let I := equalizer s t,
... | def | category_theory.limits.preserves_limit_of_preserves_equalizers_and_product | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | If a functor preserves equalizers and the appropriate products, it preserves limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_limits_of_preserves_equalizers_and_finite_products
[has_equalizers C] [has_finite_products C]
(G : C ⥤ D) [preserves_limits_of_shape walking_parallel_pair G]
[∀ (J : Type) [fintype J], preserves_limits_of_shape (discrete J) G] :
preserves_finite_limits G | ⟨λ _ _ _, by exactI preserves_limit_of_preserves_equalizers_and_product G⟩ | def | category_theory.limits.preserves_finite_limits_of_preserves_equalizers_and_finite_products | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [
"fintype"
] | If G preserves equalizers and finite products, it preserves finite limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_preserves_equalizers_and_products
[has_equalizers C] [has_products.{w} C]
(G : C ⥤ D) [preserves_limits_of_shape walking_parallel_pair G]
[∀ J, preserves_limits_of_shape (discrete.{w} J) G] :
preserves_limits_of_size.{w w} G | { preserves_limits_of_shape := λ J 𝒥,
by exactI preserves_limit_of_preserves_equalizers_and_product G } | def | category_theory.limits.preserves_limits_of_preserves_equalizers_and_products | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | If G preserves equalizers and products, it preserves all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_limits_of_has_terminal_and_pullbacks [has_terminal C] [has_pullbacks C] :
has_finite_limits C | @@has_finite_limits_of_has_equalizers_and_finite_products _
(@@has_finite_products_of_has_binary_and_terminal _
(has_binary_products_of_has_terminal_and_pullbacks C) infer_instance)
(@@has_equalizers_of_has_pullbacks_and_binary_products _
(has_binary_products_of_has_terminal_and_pullbacks C) infer_instance) | lemma | category_theory.limits.has_finite_limits_of_has_terminal_and_pullbacks | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [
"has_binary_products_of_has_terminal_and_pullbacks"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_finite_limits_of_preserves_terminal_and_pullbacks
[has_terminal C] [has_pullbacks C] (G : C ⥤ D)
[preserves_limits_of_shape (discrete.{0} pempty) G]
[preserves_limits_of_shape walking_cospan G] :
preserves_finite_limits G | begin
haveI : has_finite_limits C := has_finite_limits_of_has_terminal_and_pullbacks,
haveI : preserves_limits_of_shape (discrete walking_pair) G :=
preserves_binary_products_of_preserves_terminal_and_pullbacks G,
exact @@preserves_finite_limits_of_preserves_equalizers_and_finite_products _ _ _ _ G
(prese... | def | category_theory.limits.preserves_finite_limits_of_preserves_terminal_and_pullbacks | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [
"pempty",
"preserves_binary_products_of_preserves_terminal_and_pullbacks"
] | If G preserves terminal objects and pullbacks, it preserves all finite limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
build_colimit : cocone F | { X := i.X,
ι :=
{ app := λ j, c₂.ι.app ⟨_⟩ ≫ i.π,
naturality' := λ j₁ j₂ f, begin
dsimp,
rw [category.comp_id, ←reassoc_of (hs ⟨⟨_, _⟩, f⟩), i.condition, ←category.assoc, ht],
end} } | def | category_theory.limits.has_colimit_of_has_coproducts_of_has_coequalizers.build_colimit | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | (Implementation) Given the appropriate coproduct and coequalizer cocones,
build the cocone for `F` which is colimiting if the given cocones are also. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
build_is_colimit (t₁ : is_colimit c₁) (t₂ : is_colimit c₂) (hi : is_colimit i) :
is_colimit (build_colimit s t hs ht i) | { desc := λ q,
begin
refine hi.desc (cofork.of_π _ _),
{ refine t₂.desc (cofan.mk _ (λ j, _)),
apply q.ι.app j },
{ apply t₁.hom_ext,
intro j, discrete_cases,
simp [reassoc_of hs, reassoc_of ht] },
end,
uniq' := λ q m w, hi.hom_ext (i.coequalizer_ext (t₂.hom_ext
(λ j, by { cases ... | def | category_theory.limits.has_colimit_of_has_coproducts_of_has_coequalizers.build_is_colimit | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | (Implementation) Show the cocone constructed in `build_colimit` is colimiting,
provided the cocones used in its construction are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_of_coequalizer_and_coproduct (F : J ⥤ C)
[has_colimit (discrete.functor F.obj)]
[has_colimit (discrete.functor (λ f : (Σ p : J × J, p.1 ⟶ p.2), F.obj f.1.1))]
[has_coequalizers C] : colimit_cocone F | { cocone := _,
is_colimit :=
build_is_colimit
(sigma.desc (λ f, F.map f.2 ≫ colimit.ι (discrete.functor F.obj) ⟨f.1.2⟩))
(sigma.desc (λ f, colimit.ι (discrete.functor F.obj) ⟨f.1.1⟩))
(by simp)
(by simp)
(colimit.is_colimit _)
(colimit.is_colimit _)
(colimit.is_colimit _)... | def | category_theory.limits.colimit_cocone_of_coequalizer_and_coproduct | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | Given the existence of the appropriate (possibly finite) coproducts and coequalizers,
we can construct a colimit cocone for `F`.
(This assumes the existence of all coequalizers, which is technically stronger than needed.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimit_of_coequalizer_and_coproduct (F : J ⥤ C)
[has_colimit (discrete.functor F.obj)]
[has_colimit (discrete.functor (λ f : (Σ p : J × J, p.1 ⟶ p.2), F.obj f.1.1))]
[has_coequalizers C] : has_colimit F | has_colimit.mk (colimit_cocone_of_coequalizer_and_coproduct F) | lemma | category_theory.limits.has_colimit_of_coequalizer_and_coproduct | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | Given the existence of the appropriate (possibly finite) coproducts and coequalizers,
we know a colimit of `F` exists.
(This assumes the existence of all coequalizers, which is technically stronger than needed.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_quotient_coproduct [has_colimits_of_size.{w w} C] (F : J ⥤ C) :
∐ (λ j, F.obj j) ⟶ colimit F | coequalizer.π _ _ ≫ (colimit.iso_colimit_cocone (colimit_cocone_of_coequalizer_and_coproduct F)).inv | def | category_theory.limits.colimit_quotient_coproduct | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | A colimit can be realised as a quotient of a coproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_quotient_coproduct_epi [has_colimits_of_size.{w w} C] (F : J ⥤ C) :
epi (colimit_quotient_coproduct F) | epi_comp _ _ | instance | category_theory.limits.colimit_quotient_coproduct_epi | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_has_coequalizers_and_coproducts
[has_coproducts.{w} C] [has_coequalizers C] : has_colimits_of_size.{w w} C | { has_colimits_of_shape := λ J 𝒥,
{ has_colimit := λ F, by exactI has_colimit_of_coequalizer_and_coproduct F } } | lemma | category_theory.limits.has_colimits_of_has_coequalizers_and_coproducts | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | Any category with coproducts and coequalizers has all colimits.
See <https://stacks.math.columbia.edu/tag/002P>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_colimits_of_has_coequalizers_and_finite_coproducts
[has_finite_coproducts C] [has_coequalizers C] : has_finite_colimits C | ⟨λ J _ _, { has_colimit := λ F, by exactI has_colimit_of_coequalizer_and_coproduct F }⟩ | lemma | category_theory.limits.has_finite_colimits_of_has_coequalizers_and_finite_coproducts | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | Any category with finite coproducts and coequalizers has all finite colimits.
See <https://stacks.math.columbia.edu/tag/002Q>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimit_of_preserves_coequalizers_and_coproduct :
preserves_colimits_of_shape J G | { preserves_colimit := λ K,
begin
let P := ∐ K.obj,
let Q := ∐ (λ (f : (Σ (p : J × J), p.fst ⟶ p.snd)), K.obj f.1.1),
let s : Q ⟶ P := sigma.desc (λ f, K.map f.2 ≫ colimit.ι (discrete.functor K.obj) ⟨_⟩),
let t : Q ⟶ P := sigma.desc (λ f, colimit.ι (discrete.functor K.obj) ⟨f.1.1⟩),
let I := coequ... | def | category_theory.limits.preserves_colimit_of_preserves_coequalizers_and_coproduct | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | If a functor preserves coequalizers and the appropriate coproducts, it preserves colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_colimits_of_preserves_coequalizers_and_finite_coproducts
[has_coequalizers C] [has_finite_coproducts C]
(G : C ⥤ D) [preserves_colimits_of_shape walking_parallel_pair G]
[∀ J [fintype J], preserves_colimits_of_shape (discrete.{0} J) G] :
preserves_finite_colimits G | ⟨λ _ _ _, by exactI preserves_colimit_of_preserves_coequalizers_and_coproduct G⟩ | def | category_theory.limits.preserves_finite_colimits_of_preserves_coequalizers_and_finite_coproducts | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [
"fintype"
] | If G preserves coequalizers and finite coproducts, it preserves finite colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_preserves_coequalizers_and_coproducts
[has_coequalizers C] [has_coproducts.{w} C]
(G : C ⥤ D) [preserves_colimits_of_shape walking_parallel_pair G]
[∀ J, preserves_colimits_of_shape (discrete.{w} J) G] :
preserves_colimits_of_size.{w} G | { preserves_colimits_of_shape := λ J 𝒥,
by exactI preserves_colimit_of_preserves_coequalizers_and_coproduct G } | def | category_theory.limits.preserves_colimits_of_preserves_coequalizers_and_coproducts | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [] | If G preserves coequalizers and coproducts, it preserves all colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_colimits_of_has_initial_and_pushouts [has_initial C] [has_pushouts C] :
has_finite_colimits C | @@has_finite_colimits_of_has_coequalizers_and_finite_coproducts _
(@@has_finite_coproducts_of_has_binary_and_initial _
(has_binary_coproducts_of_has_initial_and_pushouts C) infer_instance)
(@@has_coequalizers_of_has_pushouts_and_binary_coproducts _
(has_binary_coproducts_of_has_initial_and_pushouts C) infer... | lemma | category_theory.limits.has_finite_colimits_of_has_initial_and_pushouts | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [
"has_binary_coproducts_of_has_initial_and_pushouts"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_finite_colimits_of_preserves_initial_and_pushouts
[has_initial C] [has_pushouts C] (G : C ⥤ D)
[preserves_colimits_of_shape (discrete.{0} pempty) G]
[preserves_colimits_of_shape walking_span G] :
preserves_finite_colimits G | begin
haveI : has_finite_colimits C := has_finite_colimits_of_has_initial_and_pushouts,
haveI : preserves_colimits_of_shape (discrete walking_pair) G :=
preserves_binary_coproducts_of_preserves_initial_and_pushouts G,
exact @@preserves_finite_colimits_of_preserves_coequalizers_and_finite_coproducts _ _ _ _ G
... | def | category_theory.limits.preserves_finite_colimits_of_preserves_initial_and_pushouts | category_theory.limits.constructions | src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean | [
"data.fintype.prod",
"data.fintype.sigma",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.finite_products",
"category_theory.limits.preserves.shapes.products",
"category_theory.limits.preserves.shapes.equalizers",
"category_theory.limits.preserves.finite",
"category_theory.... | [
"pempty",
"preserves_binary_coproducts_of_preserves_initial_and_pushouts"
] | If G preserves initial objects and pushouts, it preserves all finite colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit_cospan_of_has_limit_pair_of_has_limit_parallel_pair
{C : Type u} [𝒞 : category.{v} C] {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) [has_limit (pair X Y)]
[has_limit (parallel_pair (prod.fst ≫ f) (prod.snd ≫ g))] : has_limit (cospan f g) | let π₁ : X ⨯ Y ⟶ X := prod.fst, π₂ : X ⨯ Y ⟶ Y := prod.snd, e := equalizer.ι (π₁ ≫ f) (π₂ ≫ g) in
has_limit.mk
{ cone := pullback_cone.mk (e ≫ π₁) (e ≫ π₂) $ by simp only [category.assoc, equalizer.condition],
is_limit := pullback_cone.is_limit.mk _
(λ s, equalizer.lift (prod.lift (s.π.app walking_cospan.left)
... | lemma | category_theory.limits.has_limit_cospan_of_has_limit_pair_of_has_limit_parallel_pair | category_theory.limits.constructions | src/category_theory/limits/constructions/pullbacks.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.pullbacks"
] | [] | If the product `X ⨯ Y` and the equalizer of `π₁ ≫ f` and `π₂ ≫ g` exist, then the
pullback of `f` and `g` exists: It is given by composing the equalizer with the projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pullbacks_of_has_binary_products_of_has_equalizers
(C : Type u) [𝒞 : category.{v} C] [has_binary_products C] [has_equalizers C] :
has_pullbacks C | { has_limit := λ F, has_limit_of_iso (diagram_iso_cospan F).symm } | lemma | category_theory.limits.has_pullbacks_of_has_binary_products_of_has_equalizers | category_theory.limits.constructions | src/category_theory/limits/constructions/pullbacks.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.pullbacks"
] | [] | If a category has all binary products and all equalizers, then it also has all pullbacks.
As usual, this is not an instance, since there may be a more direct way to construct
pullbacks. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimit_span_of_has_colimit_pair_of_has_colimit_parallel_pair
{C : Type u} [𝒞 : category.{v} C] {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) [has_colimit (pair Y Z)]
[has_colimit (parallel_pair (f ≫ coprod.inl) (g ≫ coprod.inr))] : has_colimit (span f g) | let ι₁ : Y ⟶ Y ⨿ Z := coprod.inl, ι₂ : Z ⟶ Y ⨿ Z := coprod.inr,
c := coequalizer.π (f ≫ ι₁) (g ≫ ι₂) in
has_colimit.mk
{ cocone := pushout_cocone.mk (ι₁ ≫ c) (ι₂ ≫ c) $
by rw [←category.assoc, ←category.assoc, coequalizer.condition],
is_colimit := pushout_cocone.is_colimit.mk _
(λ s, coequalizer.desc (copro... | lemma | category_theory.limits.has_colimit_span_of_has_colimit_pair_of_has_colimit_parallel_pair | category_theory.limits.constructions | src/category_theory/limits/constructions/pullbacks.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.pullbacks"
] | [] | If the coproduct `Y ⨿ Z` and the coequalizer of `f ≫ ι₁` and `g ≫ ι₂` exist, then the
pushout of `f` and `g` exists: It is given by composing the inclusions with the coequalizer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pushouts_of_has_binary_coproducts_of_has_coequalizers
(C : Type u) [𝒞 : category.{v} C] [has_binary_coproducts C] [has_coequalizers C] :
has_pushouts C | has_pushouts_of_has_colimit_span C | lemma | category_theory.limits.has_pushouts_of_has_binary_coproducts_of_has_coequalizers | category_theory.limits.constructions | src/category_theory/limits/constructions/pullbacks.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.pullbacks"
] | [] | If a category has all binary coproducts and all coequalizers, then it also has all pushouts.
As usual, this is not an instance, since there may be a more direct way to construct
pushouts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_weakly_initial_of_weakly_initial_set_and_has_products [has_products.{v} C]
{ι : Type v} {B : ι → C} (hB : ∀ (A : C), ∃ i, nonempty (B i ⟶ A)) :
∃ (T : C), ∀ X, nonempty (T ⟶ X) | ⟨∏ B, λ X, ⟨pi.π _ _ ≫ (hB X).some_spec.some⟩⟩ | lemma | category_theory.has_weakly_initial_of_weakly_initial_set_and_has_products | category_theory.limits.constructions | src/category_theory/limits/constructions/weakly_initial.lean | [
"category_theory.limits.shapes.wide_equalizers",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.terminal"
] | [] | If `C` has (small) products and a small weakly initial set of objects, then it has a weakly initial
object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_initial_of_weakly_initial_and_has_wide_equalizers [has_wide_equalizers.{v} C]
{T : C} (hT : ∀ X, nonempty (T ⟶ X)) :
has_initial C | begin
let endos := T ⟶ T,
let i := wide_equalizer.ι (id : endos → endos),
haveI : nonempty endos := ⟨𝟙 _⟩,
have : ∀ (X : C), unique (wide_equalizer (id : endos → endos) ⟶ X),
{ intro X,
refine ⟨⟨i ≫ classical.choice (hT X)⟩, λ a, _⟩,
let E := equalizer a (i ≫ classical.choice (hT _)),
let e : E ⟶... | lemma | category_theory.has_initial_of_weakly_initial_and_has_wide_equalizers | category_theory.limits.constructions | src/category_theory/limits/constructions/weakly_initial.lean | [
"category_theory.limits.shapes.wide_equalizers",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.terminal"
] | [
"unique"
] | If `C` has (small) wide equalizers and a weakly initial object, then it has an initial object.
The initial object is constructed as the wide equalizer of all endomorphisms on the given weakly
initial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_fan_zero_left (X : C) : binary_fan (0 : C) X | binary_fan.mk 0 (𝟙 X) | def | category_theory.limits.binary_fan_zero_left | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | The limit cone for the product with a zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_fan_zero_left_is_limit (X : C) : is_limit (binary_fan_zero_left X) | binary_fan.is_limit_mk (λ s, binary_fan.snd s) (by tidy) (by tidy) (by tidy) | def | category_theory.limits.binary_fan_zero_left_is_limit | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | The limit cone for the product with a zero object is limiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_product_zero_left (X : C) : has_binary_product (0 : C) X | has_limit.mk ⟨_, binary_fan_zero_left_is_limit X⟩ | instance | category_theory.limits.has_binary_product_zero_left | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_prod_iso (X : C) : (0 : C) ⨯ X ≅ X | limit.iso_limit_cone ⟨_, binary_fan_zero_left_is_limit X⟩ | def | category_theory.limits.zero_prod_iso | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | A zero object is a left unit for categorical product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_prod_iso_hom (X : C) : (zero_prod_iso X).hom = prod.snd | rfl | lemma | category_theory.limits.zero_prod_iso_hom | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_prod_iso_inv_snd (X : C) : (zero_prod_iso X).inv ≫ prod.snd = 𝟙 X | by { dsimp [zero_prod_iso, binary_fan_zero_left], simp, } | lemma | category_theory.limits.zero_prod_iso_inv_snd | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_fan_zero_right (X : C) : binary_fan X (0 : C) | binary_fan.mk (𝟙 X) 0 | def | category_theory.limits.binary_fan_zero_right | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | The limit cone for the product with a zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_fan_zero_right_is_limit (X : C) : is_limit (binary_fan_zero_right X) | binary_fan.is_limit_mk (λ s, binary_fan.fst s) (by tidy) (by tidy) (by tidy) | def | category_theory.limits.binary_fan_zero_right_is_limit | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | The limit cone for the product with a zero object is limiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_product_zero_right (X : C) : has_binary_product X (0 : C) | has_limit.mk ⟨_, binary_fan_zero_right_is_limit X⟩ | instance | category_theory.limits.has_binary_product_zero_right | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_zero_iso (X : C) : X ⨯ (0 : C) ≅ X | limit.iso_limit_cone ⟨_, binary_fan_zero_right_is_limit X⟩ | def | category_theory.limits.prod_zero_iso | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | A zero object is a right unit for categorical product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_zero_iso_hom (X : C) : (prod_zero_iso X).hom = prod.fst | rfl | lemma | category_theory.limits.prod_zero_iso_hom | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_zero_iso_iso_inv_snd (X : C) : (prod_zero_iso X).inv ≫ prod.fst = 𝟙 X | by { dsimp [prod_zero_iso, binary_fan_zero_right], simp, } | lemma | category_theory.limits.prod_zero_iso_iso_inv_snd | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan_zero_left (X : C) : binary_cofan (0 : C) X | binary_cofan.mk 0 (𝟙 X) | def | category_theory.limits.binary_cofan_zero_left | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | The colimit cocone for the coproduct with a zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_cofan_zero_left_is_colimit (X : C) : is_colimit (binary_cofan_zero_left X) | binary_cofan.is_colimit_mk (λ s, binary_cofan.inr s) (by tidy) (by tidy) (by tidy) | def | category_theory.limits.binary_cofan_zero_left_is_colimit | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | The colimit cocone for the coproduct with a zero object is colimiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_coproduct_zero_left (X : C) : has_binary_coproduct (0 : C) X | has_colimit.mk ⟨_, binary_cofan_zero_left_is_colimit X⟩ | instance | category_theory.limits.has_binary_coproduct_zero_left | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_coprod_iso (X : C) : (0 : C) ⨿ X ≅ X | colimit.iso_colimit_cocone ⟨_, binary_cofan_zero_left_is_colimit X⟩ | def | category_theory.limits.zero_coprod_iso | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | A zero object is a left unit for categorical coproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inr_zero_coprod_iso_hom (X : C) : coprod.inr ≫ (zero_coprod_iso X).hom = 𝟙 X | by { dsimp [zero_coprod_iso, binary_cofan_zero_left], simp, } | lemma | category_theory.limits.inr_zero_coprod_iso_hom | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_coprod_iso_inv (X : C) : (zero_coprod_iso X).inv = coprod.inr | rfl | lemma | category_theory.limits.zero_coprod_iso_inv | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan_zero_right (X : C) : binary_cofan X (0 : C) | binary_cofan.mk (𝟙 X) 0 | def | category_theory.limits.binary_cofan_zero_right | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | The colimit cocone for the coproduct with a zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_cofan_zero_right_is_colimit (X : C) : is_colimit (binary_cofan_zero_right X) | binary_cofan.is_colimit_mk (λ s, binary_cofan.inl s) (by tidy) (by tidy) (by tidy) | def | category_theory.limits.binary_cofan_zero_right_is_colimit | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | The colimit cocone for the coproduct with a zero object is colimiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_coproduct_zero_right (X : C) : has_binary_coproduct X (0 : C) | has_colimit.mk ⟨_, binary_cofan_zero_right_is_colimit X⟩ | instance | category_theory.limits.has_binary_coproduct_zero_right | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_zero_iso (X : C) : X ⨿ (0 : C) ≅ X | colimit.iso_colimit_cocone ⟨_, binary_cofan_zero_right_is_colimit X⟩ | def | category_theory.limits.coprod_zero_iso | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | A zero object is a right unit for categorical coproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inr_coprod_zeroiso_hom (X : C) : coprod.inl ≫ (coprod_zero_iso X).hom = 𝟙 X | by { dsimp [coprod_zero_iso, binary_cofan_zero_right], simp, } | lemma | category_theory.limits.inr_coprod_zeroiso_hom | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_zero_iso_inv (X : C) : (coprod_zero_iso X).inv = coprod.inl | rfl | lemma | category_theory.limits.coprod_zero_iso_inv | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pullback_over_zero
(X Y : C) [has_binary_product X Y] : has_pullback (0 : X ⟶ 0) (0 : Y ⟶ 0) | has_limit.mk ⟨_, is_pullback_of_is_terminal_is_product _ _ _ _
has_zero_object.zero_is_terminal (prod_is_prod X Y)⟩ | instance | category_theory.limits.has_pullback_over_zero | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [
"is_pullback_of_is_terminal_is_product"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_zero_zero_iso (X Y : C) [has_binary_product X Y] :
pullback (0 : X ⟶ 0) (0 : Y ⟶ 0) ≅ X ⨯ Y | limit.iso_limit_cone ⟨_, is_pullback_of_is_terminal_is_product _ _ _ _
has_zero_object.zero_is_terminal (prod_is_prod X Y)⟩ | def | category_theory.limits.pullback_zero_zero_iso | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [
"is_pullback_of_is_terminal_is_product"
] | The pullback over the zeron object is the product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_zero_zero_iso_inv_fst (X Y : C) [has_binary_product X Y] :
(pullback_zero_zero_iso X Y).inv ≫ pullback.fst = prod.fst | by { dsimp [pullback_zero_zero_iso], simp, } | lemma | category_theory.limits.pullback_zero_zero_iso_inv_fst | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_zero_zero_iso_inv_snd (X Y : C) [has_binary_product X Y] :
(pullback_zero_zero_iso X Y).inv ≫ pullback.snd = prod.snd | by { dsimp [pullback_zero_zero_iso], simp, } | lemma | category_theory.limits.pullback_zero_zero_iso_inv_snd | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_zero_zero_iso_hom_fst (X Y : C) [has_binary_product X Y] :
(pullback_zero_zero_iso X Y).hom ≫ prod.fst = pullback.fst | by { simp [←iso.eq_inv_comp], } | lemma | category_theory.limits.pullback_zero_zero_iso_hom_fst | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_zero_zero_iso_hom_snd (X Y : C) [has_binary_product X Y] :
(pullback_zero_zero_iso X Y).hom ≫ prod.snd = pullback.snd | by { simp [←iso.eq_inv_comp], } | lemma | category_theory.limits.pullback_zero_zero_iso_hom_snd | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pushout_over_zero
(X Y : C) [has_binary_coproduct X Y] : has_pushout (0 : 0 ⟶ X) (0 : 0 ⟶ Y) | has_colimit.mk ⟨_, is_pushout_of_is_initial_is_coproduct _ _ _ _
has_zero_object.zero_is_initial (coprod_is_coprod X Y)⟩ | instance | category_theory.limits.has_pushout_over_zero | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [
"is_pushout_of_is_initial_is_coproduct"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pushout_zero_zero_iso
(X Y : C) [has_binary_coproduct X Y] : pushout (0 : 0 ⟶ X) (0 : 0 ⟶ Y) ≅ X ⨿ Y | colimit.iso_colimit_cocone ⟨_, is_pushout_of_is_initial_is_coproduct _ _ _ _
has_zero_object.zero_is_initial (coprod_is_coprod X Y)⟩ | def | category_theory.limits.pushout_zero_zero_iso | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [
"is_pushout_of_is_initial_is_coproduct"
] | The pushout over the zero object is the coproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl_pushout_zero_zero_iso_hom (X Y : C) [has_binary_coproduct X Y] :
pushout.inl ≫ (pushout_zero_zero_iso X Y).hom = coprod.inl | by { dsimp [pushout_zero_zero_iso], simp, } | lemma | category_theory.limits.inl_pushout_zero_zero_iso_hom | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_pushout_zero_zero_iso_hom (X Y : C) [has_binary_coproduct X Y] :
pushout.inr ≫ (pushout_zero_zero_iso X Y).hom = coprod.inr | by { dsimp [pushout_zero_zero_iso], simp, } | lemma | category_theory.limits.inr_pushout_zero_zero_iso_hom | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_pushout_zero_zero_iso_inv (X Y : C) [has_binary_coproduct X Y] :
coprod.inl ≫ (pushout_zero_zero_iso X Y).inv = pushout.inl | by { simp [iso.comp_inv_eq], } | lemma | category_theory.limits.inl_pushout_zero_zero_iso_inv | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_pushout_zero_zero_iso_inv (X Y : C) [has_binary_coproduct X Y] :
coprod.inr ≫ (pushout_zero_zero_iso X Y).inv = pushout.inr | by { simp [iso.comp_inv_eq], } | lemma | category_theory.limits.inr_pushout_zero_zero_iso_inv | category_theory.limits.constructions | src/category_theory/limits/constructions/zero_objects.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.constructions.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_finite_limits {B : C} [has_finite_wide_pullbacks C] : has_finite_limits (over B) | begin
apply @has_finite_limits_of_has_equalizers_and_finite_products _ _ _ _,
{ exact construct_products.over_finite_products_of_finite_wide_pullbacks, },
{ apply @has_equalizers_of_has_pullbacks_and_binary_products _ _ _ _,
{ haveI : has_pullbacks C := ⟨by apply_instance⟩,
exact construct_products.over... | instance | category_theory.over.has_finite_limits | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/basic.lean | [
"category_theory.limits.connected",
"category_theory.limits.constructions.over.products",
"category_theory.limits.constructions.over.connected",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.limits.constructions.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits {B : C} [has_wide_pullbacks.{w} C] : has_limits_of_size.{w} (over B) | begin
apply @has_limits_of_has_equalizers_and_products _ _ _ _,
{ exact construct_products.over_products_of_wide_pullbacks },
{ apply @has_equalizers_of_has_pullbacks_and_binary_products _ _ _ _,
{ haveI : has_pullbacks C := ⟨infer_instance⟩,
exact construct_products.over_binary_product_of_pullback },
... | instance | category_theory.over.has_limits | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/basic.lean | [
"category_theory.limits.connected",
"category_theory.limits.constructions.over.products",
"category_theory.limits.constructions.over.connected",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.limits.constructions.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans_in_over {B : C} (F : J ⥤ over B) :
F ⋙ forget B ⟶ (category_theory.functor.const J).obj B | { app := λ j, (F.obj j).hom } | def | category_theory.over.creates_connected.nat_trans_in_over | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/connected.lean | [
"category_theory.limits.creates",
"category_theory.over",
"category_theory.is_connected"
] | [
"category_theory.functor.const"
] | (Impl) Given a diagram in the over category, produce a natural transformation from the
diagram legs to the specific object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
raise_cone [is_connected J] {B : C} {F : J ⥤ over B} (c : cone (F ⋙ forget B)) :
cone F | { X := over.mk (c.π.app (classical.arbitrary J) ≫ (F.obj (classical.arbitrary J)).hom),
π :=
{ app := λ j,
over.hom_mk (c.π.app j) (nat_trans_from_is_connected (c.π ≫ nat_trans_in_over F) j _) } } | def | category_theory.over.creates_connected.raise_cone | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/connected.lean | [
"category_theory.limits.creates",
"category_theory.over",
"category_theory.is_connected"
] | [
"classical.arbitrary",
"is_connected"
] | (Impl) Given a cone in the base category, raise it to a cone in the over category. Note this is
where the connected assumption is used. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
raised_cone_lowers_to_original [is_connected J] {B : C} {F : J ⥤ over B}
(c : cone (F ⋙ forget B)) (t : is_limit c) :
(forget B).map_cone (raise_cone c) = c | by tidy | lemma | category_theory.over.creates_connected.raised_cone_lowers_to_original | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/connected.lean | [
"category_theory.limits.creates",
"category_theory.over",
"category_theory.is_connected"
] | [
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
raised_cone_is_limit [is_connected J] {B : C} {F : J ⥤ over B}
{c : cone (F ⋙ forget B)} (t : is_limit c) :
is_limit (raise_cone c) | { lift := λ s, over.hom_mk (t.lift ((forget B).map_cone s)) (by { dsimp, simp }),
uniq' := λ s m K, by { ext1, apply t.hom_ext, intro j, simp [← K j] } } | def | category_theory.over.creates_connected.raised_cone_is_limit | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/connected.lean | [
"category_theory.limits.creates",
"category_theory.over",
"category_theory.is_connected"
] | [
"is_connected",
"lift"
] | (Impl) Show that the raised cone is a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_creates_connected_limits
[is_connected J] {B : C} : creates_limits_of_shape J (forget B) | { creates_limit := λ K,
creates_limit_of_reflects_iso (λ c t,
{ lifted_cone := creates_connected.raise_cone c,
valid_lift := eq_to_iso (creates_connected.raised_cone_lowers_to_original c t),
makes_limit := creates_connected.raised_cone_is_limit t } ) } | instance | category_theory.over.forget_creates_connected_limits | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/connected.lean | [
"category_theory.limits.creates",
"category_theory.over",
"category_theory.is_connected"
] | [
"is_connected"
] | The forgetful functor from the over category creates any connected limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_connected_limits
{B : C} [is_connected J] [has_limits_of_shape J C] : has_limits_of_shape J (over B) | { has_limit := λ F, has_limit_of_created F (forget B) } | instance | category_theory.over.has_connected_limits | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/connected.lean | [
"category_theory.limits.creates",
"category_theory.over",
"category_theory.is_connected"
] | [
"is_connected"
] | The over category has any connected limit which the original category has. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pullback_diagram_of_diagram_over (B : C) {J : Type w} (F : discrete J ⥤ over B) :
wide_pullback_shape J ⥤ C | wide_pullback_shape.wide_cospan B (λ j, (F.obj ⟨j⟩).left) (λ j, (F.obj ⟨j⟩).hom) | def | category_theory.over.construct_products.wide_pullback_diagram_of_diagram_over | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/products.lean | [
"category_theory.over",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products"
] | [] | (Implementation)
Given a product diagram in `C/B`, construct the corresponding wide pullback diagram
in `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cones_equiv_inverse_obj (B : C) {J : Type w} (F : discrete J ⥤ over B) (c : cone F) :
cone (wide_pullback_diagram_of_diagram_over B F) | { X := c.X.left,
π :=
{ app := λ X, option.cases_on X c.X.hom (λ (j : J), (c.π.app ⟨j⟩).left),
-- `tidy` can do this using `case_bash`, but let's try to be a good `-T50000` citizen:
naturality' := λ X Y f,
begin
dsimp, cases X; cases Y; cases f,
{ rw [category.id_comp, category.comp_id], },
... | def | category_theory.over.construct_products.cones_equiv_inverse_obj | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/products.lean | [
"category_theory.over",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products"
] | [] | (Impl) A preliminary definition to avoid timeouts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cones_equiv_inverse (B : C) {J : Type w} (F : discrete J ⥤ over B) :
cone F ⥤ cone (wide_pullback_diagram_of_diagram_over B F) | { obj := cones_equiv_inverse_obj B F,
map := λ c₁ c₂ f,
{ hom := f.hom.left,
w' := λ j,
begin
cases j,
{ simp },
{ dsimp,
rw ← f.w ⟨j⟩,
refl }
end } } | def | category_theory.over.construct_products.cones_equiv_inverse | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/products.lean | [
"category_theory.over",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products"
] | [] | (Impl) A preliminary definition to avoid timeouts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cones_equiv_functor (B : C) {J : Type w} (F : discrete J ⥤ over B) :
cone (wide_pullback_diagram_of_diagram_over B F) ⥤ cone F | { obj := λ c,
{ X := over.mk (c.π.app none),
π :=
{ app := λ ⟨j⟩, over.hom_mk (c.π.app (some j))
(by apply c.w (wide_pullback_shape.hom.term j)) } },
map := λ c₁ c₂ f,
{ hom := over.hom_mk f.hom } } | def | category_theory.over.construct_products.cones_equiv_functor | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/products.lean | [
"category_theory.over",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products"
] | [] | (Impl) A preliminary definition to avoid timeouts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cones_equiv_unit_iso (B : C) (F : discrete J ⥤ over B) :
𝟭 (cone (wide_pullback_diagram_of_diagram_over B F)) ≅
cones_equiv_functor B F ⋙ cones_equiv_inverse B F | nat_iso.of_components (λ _, cones.ext {hom := 𝟙 _, inv := 𝟙 _} (by tidy)) (by tidy) | def | category_theory.over.construct_products.cones_equiv_unit_iso | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/products.lean | [
"category_theory.over",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products"
] | [] | (Impl) A preliminary definition to avoid timeouts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cones_equiv_counit_iso (B : C) (F : discrete J ⥤ over B) :
cones_equiv_inverse B F ⋙ cones_equiv_functor B F ≅ 𝟭 (cone F) | nat_iso.of_components
(λ _, cones.ext {hom := over.hom_mk (𝟙 _), inv := over.hom_mk (𝟙 _)} (by tidy)) (by tidy) | def | category_theory.over.construct_products.cones_equiv_counit_iso | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/products.lean | [
"category_theory.over",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products"
] | [] | (Impl) A preliminary definition to avoid timeouts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cones_equiv (B : C) (F : discrete J ⥤ over B) :
cone (wide_pullback_diagram_of_diagram_over B F) ≌ cone F | { functor := cones_equiv_functor B F,
inverse := cones_equiv_inverse B F,
unit_iso := cones_equiv_unit_iso B F,
counit_iso := cones_equiv_counit_iso B F, } | def | category_theory.over.construct_products.cones_equiv | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/products.lean | [
"category_theory.over",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products"
] | [] | (Impl) Establish an equivalence between the category of cones for `F` and for the "grown" `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_over_limit_discrete_of_wide_pullback_limit {B : C} (F : discrete J ⥤ over B)
[has_limit (wide_pullback_diagram_of_diagram_over B F)] :
has_limit F | has_limit.mk
{ cone := _,
is_limit := is_limit.of_right_adjoint
(cones_equiv B F).functor (limit.is_limit (wide_pullback_diagram_of_diagram_over B F)) } | lemma | category_theory.over.construct_products.has_over_limit_discrete_of_wide_pullback_limit | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/products.lean | [
"category_theory.over",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products"
] | [] | Use the above equivalence to prove we have a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
over_product_of_wide_pullback [has_limits_of_shape (wide_pullback_shape J) C] {B : C} :
has_limits_of_shape (discrete J) (over B) | { has_limit := λ F, has_over_limit_discrete_of_wide_pullback_limit F } | lemma | category_theory.over.construct_products.over_product_of_wide_pullback | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/products.lean | [
"category_theory.over",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products"
] | [] | Given a wide pullback in `C`, construct a product in `C/B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
over_binary_product_of_pullback [has_pullbacks C] {B : C} :
has_binary_products (over B) | over_product_of_wide_pullback | lemma | category_theory.over.construct_products.over_binary_product_of_pullback | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/products.lean | [
"category_theory.over",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products"
] | [] | Given a pullback in `C`, construct a binary product in `C/B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
over_products_of_wide_pullbacks [has_wide_pullbacks.{w} C] {B : C} :
has_products.{w} (over B) | λ J, over_product_of_wide_pullback | lemma | category_theory.over.construct_products.over_products_of_wide_pullbacks | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/products.lean | [
"category_theory.over",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products"
] | [] | Given all wide pullbacks in `C`, construct products in `C/B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
over_finite_products_of_finite_wide_pullbacks [has_finite_wide_pullbacks C] {B : C} :
has_finite_products (over B) | ⟨λ n, over_product_of_wide_pullback⟩ | lemma | category_theory.over.construct_products.over_finite_products_of_finite_wide_pullbacks | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/products.lean | [
"category_theory.over",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products"
] | [] | Given all finite wide pullbacks in `C`, construct finite products in `C/B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
over_has_terminal (B : C) : has_terminal (over B) | { has_limit := λ F, has_limit.mk
{ cone :=
{ X := over.mk (𝟙 _),
π := { app := λ p, p.as.elim } },
is_limit :=
{ lift := λ s, over.hom_mk _,
fac' := λ _ j, j.as.elim,
uniq' := λ s m _,
begin
ext,
rw over.hom_mk_left,
have := m.w,
... | lemma | category_theory.over.over_has_terminal | category_theory.limits.constructions.over | src/category_theory/limits/constructions/over/products.lean | [
"category_theory.over",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.finite_products"
] | [
"lift"
] | Construct terminal object in the over category. This isn't an instance as it's not typically the
way we want to define terminal objects.
(For instance, this gives a terminal object which is different from the generic one given by
`over_product_of_wide_pullback` above.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limit (K : J ⥤ C) (F : C ⥤ D) | (preserves : Π {c : cone K}, is_limit c → is_limit (F.map_cone c)) | class | category_theory.limits.preserves_limit | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A functor `F` preserves limits of `K` (written as `preserves_limit K F`)
if `F` maps any limit cone over `K` to a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimit (K : J ⥤ C) (F : C ⥤ D) | (preserves : Π {c : cocone K}, is_colimit c → is_colimit (F.map_cocone c)) | class | category_theory.limits.preserves_colimit | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A functor `F` preserves colimits of `K` (written as `preserves_colimit K F`)
if `F` maps any colimit cocone over `K` to a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_shape (J : Type w) [category.{w'} J] (F : C ⥤ D) | (preserves_limit : Π {K : J ⥤ C}, preserves_limit K F . tactic.apply_instance) | class | category_theory.limits.preserves_limits_of_shape | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | We say that `F` preserves limits of shape `J` if `F` preserves limits for every diagram
`K : J ⥤ C`, i.e., `F` maps limit cones over `K` to limit cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_shape (J : Type w) [category.{w'} J] (F : C ⥤ D) | (preserves_colimit : Π {K : J ⥤ C}, preserves_colimit K F . tactic.apply_instance) | class | category_theory.limits.preserves_colimits_of_shape | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | We say that `F` preserves colimits of shape `J` if `F` preserves colimits for every diagram
`K : J ⥤ C`, i.e., `F` maps colimit cocones over `K` to colimit cocones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_size (F : C ⥤ D) | (preserves_limits_of_shape : Π {J : Type w} [category.{w'} J],
preserves_limits_of_shape J F . tactic.apply_instance) | class | category_theory.limits.preserves_limits_of_size | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | `preserves_limits_of_size.{v u} F` means that `F` sends all limit cones over any
diagram `J ⥤ C` to limit cones, where `J : Type u` with `[category.{v} J]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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