statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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limit.w_apply' {F : J ⥤ Type v} {j j' : J} {x : limit F} (f : j ⟶ j') :
F.map f (limit.π F j x) = limit.π F j' x | congr_fun (limit.w F f) x | lemma | category_theory.limits.types.limit.w_apply' | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.lift_π_apply' (F : J ⥤ Type v) (s : cone F) (j : J) (x : s.X) :
limit.π F j (limit.lift F s x) = s.π.app j x | congr_fun (limit.lift_π s j) x | lemma | category_theory.limits.types.limit.lift_π_apply' | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.map_π_apply' {F G : J ⥤ Type v} (α : F ⟶ G) (j : J) (x) :
limit.π G j (lim_map α x) = α.app j (limit.π F j x) | congr_fun (lim_map_π α j) x | lemma | category_theory.limits.types.limit.map_π_apply' | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot.rel (F : J ⥤ Type (max v u)) : (Σ j, F.obj j) → (Σ j, F.obj j) → Prop | (λ p p', ∃ f : p.1 ⟶ p'.1, p'.2 = F.map f p.2) | def | category_theory.limits.types.quot.rel | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | The relation defining the quotient type which implements the colimit of a functor `F : J ⥤ Type u`.
See `category_theory.limits.types.quot`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot (F : J ⥤ Type (max v u)) : Type (max v u) | @quot (Σ j, F.obj j) (quot.rel F) | def | category_theory.limits.types.quot | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | A quotient type implementing the colimit of a functor `F : J ⥤ Type u`,
as pairs `⟨j, x⟩` where `x : F.obj j`, modulo the equivalence relation generated by
`⟨j, x⟩ ~ ⟨j', x'⟩` whenever there is a morphism `f : j ⟶ j'` so `F.map f x = x'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone (F : J ⥤ Type (max v u)) : cocone F | { X := quot F,
ι :=
{ app := λ j x, quot.mk _ ⟨j, x⟩,
naturality' := λ j j' f, funext $ λ x, eq.symm (quot.sound ⟨f, rfl⟩) } } | def | category_theory.limits.types.colimit_cocone | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | (internal implementation) the colimit cocone of a functor,
implemented as a quotient of a sigma type | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_is_colimit (F : J ⥤ Type (max v u)) : is_colimit (colimit_cocone F) | { desc := λ s, quot.lift (λ (p : Σ j, F.obj j), s.ι.app p.1 p.2)
(assume ⟨j, x⟩ ⟨j', x'⟩ ⟨f, hf⟩, by rw hf; exact (congr_fun (cocone.w s f) x).symm) } | def | category_theory.limits.types.colimit_cocone_is_colimit | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | (internal implementation) the fact that the proposed colimit cocone is the colimit | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_of_size : has_colimits_of_size.{v} (Type (max v u)) | { has_colimits_of_shape := λ J 𝒥, by exactI
{ has_colimit := λ F, has_colimit.mk
{ cocone := colimit_cocone F, is_colimit := colimit_cocone_is_colimit F } } } | instance | category_theory.limits.types.has_colimits_of_size | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | The category of types has all colimits.
See <https://stacks.math.columbia.edu/tag/002U>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_equiv_quot (F : J ⥤ Type (max v u)) : (colimit F : Type (max v u)) ≃ quot F | (is_colimit.cocone_point_unique_up_to_iso
(colimit.is_colimit F)
(colimit_cocone_is_colimit F)).to_equiv | def | category_theory.limits.types.colimit_equiv_quot | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | The equivalence between the abstract colimit of `F` in `Type u`
and the "concrete" definition as a quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_equiv_quot_symm_apply (F : J ⥤ Type (max v u)) (j : J) (x : F.obj j) :
(colimit_equiv_quot F).symm (quot.mk _ ⟨j, x⟩) = colimit.ι F j x | rfl | lemma | category_theory.limits.types.colimit_equiv_quot_symm_apply | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_equiv_quot_apply (F : J ⥤ Type (max v u)) (j : J) (x : F.obj j) :
(colimit_equiv_quot F) (colimit.ι F j x) = quot.mk _ ⟨j, x⟩ | begin
apply (colimit_equiv_quot F).symm.injective,
simp,
end | lemma | category_theory.limits.types.colimit_equiv_quot_apply | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.w_apply {F : J ⥤ Type (max v u)} {j j' : J} {x : F.obj j} (f : j ⟶ j') :
colimit.ι F j' (F.map f x) = colimit.ι F j x | congr_fun (colimit.w F f) x | lemma | category_theory.limits.types.colimit.w_apply | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.ι_desc_apply (F : J ⥤ Type (max v u)) (s : cocone F) (j : J) (x : F.obj j) :
colimit.desc F s (colimit.ι F j x) = s.ι.app j x | congr_fun (colimit.ι_desc s j) x | lemma | category_theory.limits.types.colimit.ι_desc_apply | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.ι_map_apply {F G : J ⥤ Type (max v u)} (α : F ⟶ G) (j : J) (x) :
colim.map α (colimit.ι F j x) = colimit.ι G j (α.app j x) | congr_fun (colimit.ι_map α j) x | lemma | category_theory.limits.types.colimit.ι_map_apply | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.w_apply' {F : J ⥤ Type v} {j j' : J} {x : F.obj j} (f : j ⟶ j') :
colimit.ι F j' (F.map f x) = colimit.ι F j x | congr_fun (colimit.w F f) x | lemma | category_theory.limits.types.colimit.w_apply' | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.ι_desc_apply' (F : J ⥤ Type v) (s : cocone F) (j : J) (x : F.obj j) :
colimit.desc F s (colimit.ι F j x) = s.ι.app j x | congr_fun (colimit.ι_desc s j) x | lemma | category_theory.limits.types.colimit.ι_desc_apply' | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.ι_map_apply' {F G : J ⥤ Type v} (α : F ⟶ G) (j : J) (x) :
colim.map α (colimit.ι F j x) = colimit.ι G j (α.app j x) | congr_fun (colimit.ι_map α j) x | lemma | category_theory.limits.types.colimit.ι_map_apply' | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_sound
{F : J ⥤ Type (max v u)} {j j' : J} {x : F.obj j} {x' : F.obj j'}
(f : j ⟶ j') (w : F.map f x = x') : colimit.ι F j x = colimit.ι F j' x' | begin
rw [←w],
simp,
end | lemma | category_theory.limits.types.colimit_sound | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_sound'
{F : J ⥤ Type (max v u)} {j j' : J} {x : F.obj j} {x' : F.obj j'} {j'' : J}
(f : j ⟶ j'') (f' : j' ⟶ j'') (w : F.map f x = F.map f' x') :
colimit.ι F j x = colimit.ι F j' x' | begin
rw [←colimit.w _ f, ←colimit.w _ f'],
rw [types_comp_apply, types_comp_apply, w],
end | lemma | category_theory.limits.types.colimit_sound' | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_eq {F : J ⥤ Type (max v u)} {j j' : J} {x : F.obj j} {x' : F.obj j'}
(w : colimit.ι F j x = colimit.ι F j' x') : eqv_gen (quot.rel F) ⟨j, x⟩ ⟨j', x'⟩ | begin
apply quot.eq.1,
simpa using congr_arg (colimit_equiv_quot F) w,
end | lemma | category_theory.limits.types.colimit_eq | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
jointly_surjective (F : J ⥤ Type (max v u)) {t : cocone F} (h : is_colimit t)
(x : t.X) : ∃ j y, t.ι.app j y = x | begin
suffices : (λ (x : t.X), ulift.up (∃ j y, t.ι.app j y = x)) = (λ _, ulift.up true),
{ have := congr_fun this x,
have H := congr_arg ulift.down this,
dsimp at H,
rwa eq_true_iff at H },
refine h.hom_ext _,
intro j, ext y,
erw iff_true,
exact ⟨j, y, rfl⟩
end | lemma | category_theory.limits.types.jointly_surjective | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [
"eq_true_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
jointly_surjective' {F : J ⥤ Type (max v u)}
(x : colimit F) : ∃ j y, colimit.ι F j y = x | jointly_surjective F (colimit.is_colimit _) x | lemma | category_theory.limits.types.jointly_surjective' | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | A variant of `jointly_surjective` for `x : colimit F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel (x y : Σ j, F.obj j) : Prop | ∃ k (f : x.1 ⟶ k) (g : y.1 ⟶ k), F.map f x.2 = F.map g y.2 | def | category_theory.limits.types.filtered_colimit.rel | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [
"rel"
] | An alternative relation on `Σ j, F.obj j`,
which generates the same equivalence relation as we use to define the colimit in `Type` above,
but that is more convenient when working with filtered colimits.
Elements in `F.obj j` and `F.obj j'` are equivalent if there is some `k : J` to the right
where their images are equ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_of_quot_rel (x y : Σ j, F.obj j) : quot.rel F x y → filtered_colimit.rel F x y | λ ⟨f, h⟩, ⟨y.1, f, 𝟙 y.1, by rw [← h, functor_to_types.map_id_apply]⟩ | lemma | category_theory.limits.types.filtered_colimit.rel_of_quot_rel | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eqv_gen_quot_rel_of_rel (x y : Σ j, F.obj j) :
filtered_colimit.rel F x y → eqv_gen (quot.rel F) x y | λ ⟨k, f, g, h⟩, eqv_gen.trans _ ⟨k, F.map f x.2⟩ _ (eqv_gen.rel _ _ ⟨f, rfl⟩)
(eqv_gen.symm _ _ (eqv_gen.rel _ _ ⟨g, h⟩)) | lemma | category_theory.limits.types.filtered_colimit.eqv_gen_quot_rel_of_rel | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit_of (t : cocone F) (hsurj : ∀ (x : t.X), ∃ i xi, x = t.ι.app i xi)
(hinj : ∀ i j xi xj, t.ι.app i xi = t.ι.app j xj →
∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f xi = F.map g xj) : is_colimit t | -- Strategy: Prove that the map from "the" colimit of F (defined above) to t.X
-- is a bijection.
begin
apply is_colimit.of_iso_colimit (colimit.is_colimit F),
refine cocones.ext (equiv.to_iso (equiv.of_bijective _ _)) _,
{ exact colimit.desc F t },
{ split,
{ show function.injective _,
intros a b h,
... | def | category_theory.limits.types.filtered_colimit.is_colimit_of | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [
"equiv.of_bijective",
"equiv.to_iso"
] | Recognizing filtered colimits of types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_equiv : equivalence (filtered_colimit.rel F) | ⟨λ x, ⟨x.1, 𝟙 x.1, 𝟙 x.1, rfl⟩,
λ x y ⟨k, f, g, h⟩, ⟨k, g, f, h.symm⟩,
λ x y z ⟨k, f, g, h⟩ ⟨k', f', g', h'⟩,
let ⟨l, fl, gl, _⟩ := is_filtered_or_empty.cocone_objs k k',
⟨m, n, hn⟩ := is_filtered_or_empty.cocone_maps (g ≫ fl) (f' ≫ gl) in
⟨m, f ≫ fl ≫ n, g' ≫ gl ≫ n, calc
F.map (f ≫ fl ≫ n) x.2
... | lemma | category_theory.limits.types.filtered_colimit.rel_equiv | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_eq_eqv_gen_quot_rel :
filtered_colimit.rel F = eqv_gen (quot.rel F) | begin
ext ⟨j, x⟩ ⟨j', y⟩,
split,
{ apply eqv_gen_quot_rel_of_rel },
{ rw ←(filtered_colimit.rel_equiv F).eqv_gen_iff,
exact eqv_gen.mono (rel_of_quot_rel F) }
end | lemma | category_theory.limits.types.filtered_colimit.rel_eq_eqv_gen_quot_rel | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [
"eqv_gen.mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_eq_iff_aux {i j : J} {xi : F.obj i} {xj : F.obj j} :
(colimit_cocone F).ι.app i xi = (colimit_cocone F).ι.app j xj ↔
filtered_colimit.rel F ⟨i, xi⟩ ⟨j, xj⟩ | begin
change quot.mk _ _ = quot.mk _ _ ↔ _,
rw [quot.eq, filtered_colimit.rel_eq_eqv_gen_quot_rel],
end | lemma | category_theory.limits.types.filtered_colimit.colimit_eq_iff_aux | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [
"quot.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit_eq_iff {t : cocone F} (ht : is_colimit t) {i j : J} {xi : F.obj i} {xj : F.obj j} :
t.ι.app i xi = t.ι.app j xj ↔ ∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f xi = F.map g xj | let t' := colimit_cocone F,
e : t' ≅ t := is_colimit.unique_up_to_iso (colimit_cocone_is_colimit F) ht,
e' : t'.X ≅ t.X := (cocones.forget _).map_iso e in
begin
refine iff.trans _ (colimit_eq_iff_aux F),
convert e'.to_equiv.apply_eq_iff_eq; rw ←e.hom.w; refl
end | lemma | category_theory.limits.types.filtered_colimit.is_colimit_eq_iff | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_eq_iff {i j : J} {xi : F.obj i} {xj : F.obj j} :
colimit.ι F i xi = colimit.ι F j xj ↔ ∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f xi = F.map g xj | is_colimit_eq_iff _ (colimit.is_colimit F) | lemma | category_theory.limits.types.filtered_colimit.colimit_eq_iff | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image : Type u | set.range f | def | category_theory.limits.types.image | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [
"set.range"
] | the image of a morphism in Type is just `set.range f` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.ι : image f ⟶ β | subtype.val | def | category_theory.limits.types.image.ι | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | the inclusion of `image f` into the target | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.lift (F' : mono_factorisation f) : image f ⟶ F'.I | (λ x, F'.e (classical.indefinite_description _ x.2).1 : image f → F'.I) | def | category_theory.limits.types.image.lift | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [] | the universal property for the image factorisation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_factorisation : mono_factorisation f | { I := image f,
m := image.ι f,
e := set.range_factorization f } | def | category_theory.limits.types.mono_factorisation | category_theory.limits | src/category_theory/limits/types.lean | [
"category_theory.limits.shapes.images",
"category_theory.filtered",
"tactic.equiv_rw"
] | [
"set.range_factorization"
] | the factorisation of any morphism in Type through a mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
punit_cone : cone F | ⟨⟨⟨⟩⟩, (functor.punit_ext _ _).hom⟩ | def | category_theory.limits.punit_cone | category_theory.limits | src/category_theory/limits/unit.lean | [
"category_theory.punit",
"category_theory.limits.has_limits"
] | [] | A trivial cone for a functor into `punit`. `punit_cone_is_limit` shows it is a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
punit_cocone : cocone F | ⟨⟨⟨⟩⟩, (functor.punit_ext _ _).hom⟩ | def | category_theory.limits.punit_cocone | category_theory.limits | src/category_theory/limits/unit.lean | [
"category_theory.punit",
"category_theory.limits.has_limits"
] | [] | A trivial cocone for a functor into `punit`. `punit_cocone_is_limit` shows it is a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
punit_cone_is_limit {c : cone F} : is_limit c | by tidy | def | category_theory.limits.punit_cone_is_limit | category_theory.limits | src/category_theory/limits/unit.lean | [
"category_theory.punit",
"category_theory.limits.has_limits"
] | [] | Any cone over a functor into `punit` is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
punit_cocone_is_colimit {c : cocone F} : is_colimit c | by tidy | def | category_theory.limits.punit_cocone_is_colimit | category_theory.limits | src/category_theory/limits/unit.lean | [
"category_theory.punit",
"category_theory.limits.has_limits"
] | [] | Any cocone over a functor into `punit` is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone (X : Cᵒᵖ) : cocone (coyoneda.obj X) | { X := punit,
ι := { app := by tidy, } } | def | category_theory.coyoneda.colimit_cocone | category_theory.limits | src/category_theory/limits/yoneda.lean | [
"category_theory.limits.functor_category",
"tactic.assert_exists"
] | [] | The colimit cocone over `coyoneda.obj X`, with cocone point `punit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_is_colimit (X : Cᵒᵖ) : is_colimit (colimit_cocone X) | { desc := λ s x, s.ι.app (unop X) (𝟙 _),
fac' := λ s Y, by { ext f, convert congr_fun (s.w f).symm (𝟙 (unop X)), simp, },
uniq' := λ s m w, by { ext ⟨⟩, rw ← w, simp, } } | def | category_theory.coyoneda.colimit_cocone_is_colimit | category_theory.limits | src/category_theory/limits/yoneda.lean | [
"category_theory.limits.functor_category",
"tactic.assert_exists"
] | [] | The proposed colimit cocone over `coyoneda.obj X` is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_coyoneda_iso (X : Cᵒᵖ) : colimit (coyoneda.obj X) ≅ punit | colimit.iso_colimit_cocone { cocone := _, is_colimit := colimit_cocone_is_colimit X } | def | category_theory.coyoneda.colimit_coyoneda_iso | category_theory.limits | src/category_theory/limits/yoneda.lean | [
"category_theory.limits.functor_category",
"tactic.assert_exists"
] | [] | The colimit of `coyoneda.obj X` is isomorphic to `punit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
yoneda_preserves_limits (X : C) : preserves_limits (yoneda.obj X) | { preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ K,
{ preserves := λ c t,
{ lift := λ s x, quiver.hom.unop (t.lift ⟨op X, λ j, (s.π.app j x).op, λ j₁ j₂ α, _⟩),
fac' := λ s j, funext $ λ x, quiver.hom.op_inj (t.fac _ _),
uniq' := λ s m w, funext $ λ x,
begin
... | instance | category_theory.yoneda_preserves_limits | category_theory.limits | src/category_theory/limits/yoneda.lean | [
"category_theory.limits.functor_category",
"tactic.assert_exists"
] | [
"lift",
"quiver.hom.op_inj",
"quiver.hom.unop",
"quiver.hom.unop_inj"
] | The yoneda embedding `yoneda.obj X : Cᵒᵖ ⥤ Type v` for `X : C` preserves limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coyoneda_preserves_limits (X : Cᵒᵖ) : preserves_limits (coyoneda.obj X) | { preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ K,
{ preserves := λ c t,
{ lift := λ s x, t.lift ⟨unop X, λ j, s.π.app j x, λ j₁ j₂ α, by { dsimp, simp [← s.w α]}⟩,
-- See library note [dsimp, simp]
fac' := λ s j, funext $ λ x, t.fac _ _,
uniq' := λ s m w, f... | instance | category_theory.coyoneda_preserves_limits | category_theory.limits | src/category_theory/limits/yoneda.lean | [
"category_theory.limits.functor_category",
"tactic.assert_exists"
] | [
"lift"
] | The coyoneda embedding `coyoneda.obj X : C ⥤ Type v` for `X : Cᵒᵖ` preserves limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
yoneda_jointly_reflects_limits (J : Type w) [small_category J] (K : J ⥤ Cᵒᵖ) (c : cone K)
(t : Π (X : C), is_limit ((yoneda.obj X).map_cone c)) : is_limit c | let s' : Π (s : cone K), cone (K ⋙ yoneda.obj s.X.unop) :=
λ s, ⟨punit, λ j _, (s.π.app j).unop, λ j₁ j₂ α, funext $ λ _, quiver.hom.op_inj (s.w α).symm⟩
in
{ lift := λ s, ((t s.X.unop).lift (s' s) punit.star).op,
fac' := λ s j, quiver.hom.unop_inj (congr_fun ((t s.X.unop).fac (s' s) j) punit.star),
uniq' := λ s ... | def | category_theory.yoneda_jointly_reflects_limits | category_theory.limits | src/category_theory/limits/yoneda.lean | [
"category_theory.limits.functor_category",
"tactic.assert_exists"
] | [
"lift",
"quiver.hom.op_inj",
"quiver.hom.unop_inj"
] | The yoneda embeddings jointly reflect limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coyoneda_jointly_reflects_limits (J : Type w) [small_category J] (K : J ⥤ C) (c : cone K)
(t : Π (X : Cᵒᵖ), is_limit ((coyoneda.obj X).map_cone c)) : is_limit c | let s' : Π (s : cone K), cone (K ⋙ coyoneda.obj (op s.X)) :=
λ s, ⟨punit, λ j _, s.π.app j, λ j₁ j₂ α, funext $ λ _, (s.w α).symm⟩
in
{ lift := λ s, (t (op s.X)).lift (s' s) punit.star,
fac' := λ s j, congr_fun ((t _).fac (s' s) j) punit.star,
uniq' := λ s m w,
begin
suffices : (λ (x : punit), m) = (t _).li... | def | category_theory.coyoneda_jointly_reflects_limits | category_theory.limits | src/category_theory/limits/yoneda.lean | [
"category_theory.limits.functor_category",
"tactic.assert_exists"
] | [
"lift"
] | The coyoneda embeddings jointly reflect limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
yoneda_functor_preserves_limits : preserves_limits (@yoneda D _) | begin
apply preserves_limits_of_evaluation,
intro K,
change preserves_limits (coyoneda.obj K),
apply_instance
end | instance | category_theory.yoneda_functor_preserves_limits | category_theory.limits | src/category_theory/limits/yoneda.lean | [
"category_theory.limits.functor_category",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coyoneda_functor_preserves_limits : preserves_limits (@coyoneda D _) | begin
apply preserves_limits_of_evaluation,
intro K,
change preserves_limits (yoneda.obj K),
apply_instance
end | instance | category_theory.coyoneda_functor_preserves_limits | category_theory.limits | src/category_theory/limits/yoneda.lean | [
"category_theory.limits.functor_category",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
yoneda_functor_reflects_limits : reflects_limits (@yoneda D _) | limits.fully_faithful_reflects_limits _ | instance | category_theory.yoneda_functor_reflects_limits | category_theory.limits | src/category_theory/limits/yoneda.lean | [
"category_theory.limits.functor_category",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coyoneda_functor_reflects_limits : reflects_limits (@coyoneda D _) | limits.fully_faithful_reflects_limits _ | instance | category_theory.coyoneda_functor_reflects_limits | category_theory.limits | src/category_theory/limits/yoneda.lean | [
"category_theory.limits.functor_category",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_binary_product_of_is_terminal_is_pullback (F : discrete walking_pair ⥤ C) (c : cone F)
{X : C} (hX : is_terminal X)
(f : F.obj ⟨walking_pair.left⟩ ⟶ X) (g : F.obj ⟨walking_pair.right⟩ ⟶ X)
(hc : is_limit (pullback_cone.mk (c.π.app ⟨walking_pair.left⟩) (c.π.app ⟨walking_pair.right⟩ : _)
$ hX.hom_ext (_ ≫ f)... | { lift := λ s, hc.lift
(pullback_cone.mk (s.π.app ⟨walking_pair.left⟩) (s.π.app ⟨walking_pair.right⟩)
(hX.hom_ext _ _)),
fac' := λ s j, discrete.cases_on j
(λ j, walking_pair.cases_on j (hc.fac _ walking_cospan.left) (hc.fac _ walking_cospan.right)),
uniq' := λ s m J,
begin
let c' := pullbac... | def | is_binary_product_of_is_terminal_is_pullback | category_theory.limits.constructions | src/category_theory/limits/constructions/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.terminal"
] | [
"lift"
] | If a span is the pullback span over the terminal object, then it is a binary product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_product_of_is_terminal_is_pullback {W X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) (h : W ⟶ X)
(k : W ⟶ Y) (H₁ : is_terminal Z)
(H₂ : is_limit (pullback_cone.mk _ _ (show h ≫ f = k ≫ g, from H₁.hom_ext _ _))) :
is_limit (binary_fan.mk h k) | begin
apply is_binary_product_of_is_terminal_is_pullback _ _ H₁,
exact H₂
end | def | is_product_of_is_terminal_is_pullback | category_theory.limits.constructions | src/category_theory/limits/constructions/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.terminal"
] | [
"is_binary_product_of_is_terminal_is_pullback"
] | The pullback over the terminal object is the product | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_pullback_of_is_terminal_is_product {W X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) (h : W ⟶ X)
(k : W ⟶ Y) (H₁ : is_terminal Z)
(H₂ : is_limit (binary_fan.mk h k)) :
is_limit (pullback_cone.mk _ _ (show h ≫ f = k ≫ g, from H₁.hom_ext _ _)) | begin
apply pullback_cone.is_limit_aux',
intro s,
use H₂.lift (binary_fan.mk s.fst s.snd),
use H₂.fac (binary_fan.mk s.fst s.snd) ⟨walking_pair.left⟩,
use H₂.fac (binary_fan.mk s.fst s.snd) ⟨walking_pair.right⟩,
intros m h₁ h₂,
apply H₂.hom_ext,
rintro ⟨⟨⟩⟩,
{ exact h₁.trans (H₂.fac (binary_fan.mk s.f... | def | is_pullback_of_is_terminal_is_product | category_theory.limits.constructions | src/category_theory/limits/constructions/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.terminal"
] | [] | The product is the pullback over the terminal object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_of_terminal_and_pullbacks [has_terminal C] [has_pullbacks C]
(F : discrete walking_pair ⥤ C) : limit_cone F | { cone :=
{ X := pullback (terminal.from (F.obj ⟨walking_pair.left⟩))
(terminal.from (F.obj ⟨walking_pair.right⟩)),
π := discrete.nat_trans (λ x, discrete.cases_on x
(λ x, walking_pair.cases_on x pullback.fst pullback.snd)) },
is_limit := is_binary_product_of_is_terminal_is_pullback
... | def | limit_cone_of_terminal_and_pullbacks | category_theory.limits.constructions | src/category_theory/limits/constructions/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.terminal"
] | [
"is_binary_product_of_is_terminal_is_pullback"
] | Any category with pullbacks and a terminal object has a limit cone for each walking pair. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_products_of_has_terminal_and_pullbacks
[has_terminal C] [has_pullbacks C] :
has_binary_products C | { has_limit := λ F, has_limit.mk (limit_cone_of_terminal_and_pullbacks F) } | lemma | has_binary_products_of_has_terminal_and_pullbacks | category_theory.limits.constructions | src/category_theory/limits/constructions/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.terminal"
] | [
"limit_cone_of_terminal_and_pullbacks"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_binary_products_of_preserves_terminal_and_pullbacks
[has_terminal C] [has_pullbacks C]
[preserves_limits_of_shape (discrete.{0} pempty) F]
[preserves_limits_of_shape walking_cospan F] :
preserves_limits_of_shape (discrete walking_pair) F | ⟨λ K, preserves_limit_of_preserves_limit_cone (limit_cone_of_terminal_and_pullbacks K).2
begin
apply is_binary_product_of_is_terminal_is_pullback _ _
(is_limit_of_has_terminal_of_preserves_limit F),
apply is_limit_of_has_pullback_of_preserves_limit,
end⟩ | def | preserves_binary_products_of_preserves_terminal_and_pullbacks | category_theory.limits.constructions | src/category_theory/limits/constructions/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.terminal"
] | [
"is_binary_product_of_is_terminal_is_pullback",
"limit_cone_of_terminal_and_pullbacks",
"pempty"
] | A functor that preserves terminal objects and pullbacks preserves binary products. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_iso_pullback [has_terminal C] [has_pullbacks C] (X Y : C) [has_binary_product X Y] :
X ⨯ Y ≅ pullback (terminal.from X) (terminal.from Y) | limit.iso_limit_cone (limit_cone_of_terminal_and_pullbacks _) | def | prod_iso_pullback | category_theory.limits.constructions | src/category_theory/limits/constructions/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.terminal"
] | [
"limit_cone_of_terminal_and_pullbacks"
] | In a category with a terminal object and pullbacks,
a product of objects `X` and `Y` is isomorphic to a pullback. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_binary_coproduct_of_is_initial_is_pushout (F : discrete walking_pair ⥤ C) (c : cocone F)
{X : C} (hX : is_initial X)
(f : X ⟶ F.obj ⟨walking_pair.left⟩) (g : X ⟶ F.obj ⟨walking_pair.right⟩)
(hc : is_colimit (pushout_cocone.mk (c.ι.app ⟨walking_pair.left⟩)
(c.ι.app ⟨walking_pair.right⟩ : _) $ hX.hom_ext (f ... | { desc := λ s, hc.desc
(pushout_cocone.mk (s.ι.app ⟨walking_pair.left⟩) (s.ι.app ⟨walking_pair.right⟩)
(hX.hom_ext _ _)),
fac' := λ s j, discrete.cases_on j
(λ j, walking_pair.cases_on j (hc.fac _ walking_span.left) (hc.fac _ walking_span.right)),
uniq' := λ s m J,
begin
let c' := pushout_co... | def | is_binary_coproduct_of_is_initial_is_pushout | category_theory.limits.constructions | src/category_theory/limits/constructions/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.terminal"
] | [] | If a cospan is the pushout cospan under the initial object, then it is a binary coproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_coproduct_of_is_initial_is_pushout {W X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) (h : W ⟶ X)
(k : W ⟶ Y) (H₁ : is_initial W)
(H₂ : is_colimit (pushout_cocone.mk _ _ (show h ≫ f = k ≫ g, from H₁.hom_ext _ _))) :
is_colimit (binary_cofan.mk f g) | begin
apply is_binary_coproduct_of_is_initial_is_pushout _ _ H₁,
exact H₂
end | def | is_coproduct_of_is_initial_is_pushout | category_theory.limits.constructions | src/category_theory/limits/constructions/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.terminal"
] | [
"is_binary_coproduct_of_is_initial_is_pushout"
] | The pushout under the initial object is the coproduct | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_pushout_of_is_initial_is_coproduct {W X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) (h : W ⟶ X)
(k : W ⟶ Y) (H₁ : is_initial W)
(H₂ : is_colimit (binary_cofan.mk f g)) :
is_colimit (pushout_cocone.mk _ _ (show h ≫ f = k ≫ g, from H₁.hom_ext _ _)) | begin
apply pushout_cocone.is_colimit_aux',
intro s,
use H₂.desc (binary_cofan.mk s.inl s.inr),
use H₂.fac (binary_cofan.mk s.inl s.inr) ⟨walking_pair.left⟩,
use H₂.fac (binary_cofan.mk s.inl s.inr) ⟨walking_pair.right⟩,
intros m h₁ h₂,
apply H₂.hom_ext,
rintro ⟨⟨⟩⟩,
{ exact h₁.trans (H₂.fac (binary_c... | def | is_pushout_of_is_initial_is_coproduct | category_theory.limits.constructions | src/category_theory/limits/constructions/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.terminal"
] | [] | The coproduct is the pushout under the initial object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_of_initial_and_pushouts [has_initial C] [has_pushouts C]
(F : discrete walking_pair ⥤ C) : colimit_cocone F | { cocone :=
{ X := pushout (initial.to (F.obj ⟨walking_pair.left⟩))
(initial.to (F.obj ⟨walking_pair.right⟩)),
ι := discrete.nat_trans (λ x, discrete.cases_on x
(λ x, walking_pair.cases_on x pushout.inl pushout.inr)) },
is_colimit := is_binary_coproduct_of_is_initial_is_pushout
F _ i... | def | colimit_cocone_of_initial_and_pushouts | category_theory.limits.constructions | src/category_theory/limits/constructions/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.terminal"
] | [
"is_binary_coproduct_of_is_initial_is_pushout"
] | Any category with pushouts and an initial object has a colimit cocone for each walking pair. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_binary_coproducts_of_has_initial_and_pushouts
[has_initial C] [has_pushouts C] :
has_binary_coproducts C | { has_colimit := λ F, has_colimit.mk (colimit_cocone_of_initial_and_pushouts F) } | lemma | has_binary_coproducts_of_has_initial_and_pushouts | category_theory.limits.constructions | src/category_theory/limits/constructions/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.terminal"
] | [
"colimit_cocone_of_initial_and_pushouts"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_binary_coproducts_of_preserves_initial_and_pushouts
[has_initial C] [has_pushouts C]
[preserves_colimits_of_shape (discrete.{0} pempty) F]
[preserves_colimits_of_shape walking_span F] :
preserves_colimits_of_shape (discrete walking_pair) F | ⟨λ K, preserves_colimit_of_preserves_colimit_cocone (colimit_cocone_of_initial_and_pushouts K).2
begin
apply is_binary_coproduct_of_is_initial_is_pushout _ _
(is_colimit_of_has_initial_of_preserves_colimit F),
apply is_colimit_of_has_pushout_of_preserves_colimit,
end⟩ | def | preserves_binary_coproducts_of_preserves_initial_and_pushouts | category_theory.limits.constructions | src/category_theory/limits/constructions/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.terminal"
] | [
"colimit_cocone_of_initial_and_pushouts",
"is_binary_coproduct_of_is_initial_is_pushout",
"pempty"
] | A functor that preserves initial objects and pushouts preserves binary coproducts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod_iso_pushout [has_initial C] [has_pushouts C] (X Y : C) [has_binary_coproduct X Y] :
X ⨿ Y ≅ pushout (initial.to X) (initial.to Y) | colimit.iso_colimit_cocone (colimit_cocone_of_initial_and_pushouts _) | def | coprod_iso_pushout | category_theory.limits.constructions | src/category_theory/limits/constructions/binary_products.lean | [
"category_theory.limits.shapes.terminal",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.terminal"
] | [
"colimit_cocone_of_initial_and_pushouts"
] | In a category with an initial object and pushouts,
a coproduct of objects `X` and `Y` is isomorphic to a pushout. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_mono_of_preserves_limit {X Y : C} (f : X ⟶ Y) [preserves_limit (cospan f f) F]
[mono f] : mono (F.map f) | begin
have := is_limit_pullback_cone_map_of_is_limit F _ (pullback_cone.is_limit_mk_id_id f),
simp_rw [F.map_id] at this,
apply pullback_cone.mono_of_is_limit_mk_id_id _ this,
end | lemma | category_theory.preserves_mono_of_preserves_limit | category_theory.limits.constructions | src/category_theory/limits/constructions/epi_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks"
] | [] | If `F` preserves pullbacks, then it preserves monomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_monomorphisms_of_preserves_limits_of_shape
[preserves_limits_of_shape walking_cospan F] : F.preserves_monomorphisms | { preserves := λ X Y f hf, by exactI preserves_mono_of_preserves_limit F f } | instance | category_theory.preserves_monomorphisms_of_preserves_limits_of_shape | category_theory.limits.constructions | src/category_theory/limits/constructions/epi_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_mono_of_reflects_limit {X Y : C} (f : X ⟶ Y) [reflects_limit (cospan f f) F]
[mono (F.map f)] : mono f | begin
have := pullback_cone.is_limit_mk_id_id (F.map f),
simp_rw [←F.map_id] at this,
apply pullback_cone.mono_of_is_limit_mk_id_id _ (is_limit_of_is_limit_pullback_cone_map F _ this),
end | lemma | category_theory.reflects_mono_of_reflects_limit | category_theory.limits.constructions | src/category_theory/limits/constructions/epi_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks"
] | [] | If `F` reflects pullbacks, then it reflects monomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_monomorphisms_of_reflects_limits_of_shape
[reflects_limits_of_shape walking_cospan F] : F.reflects_monomorphisms | { reflects := λ X Y f hf, by exactI reflects_mono_of_reflects_limit F f } | instance | category_theory.reflects_monomorphisms_of_reflects_limits_of_shape | category_theory.limits.constructions | src/category_theory/limits/constructions/epi_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_epi_of_preserves_colimit {X Y : C} (f : X ⟶ Y) [preserves_colimit (span f f) F]
[epi f] : epi (F.map f) | begin
have := is_colimit_pushout_cocone_map_of_is_colimit F _ (pushout_cocone.is_colimit_mk_id_id f),
simp_rw [F.map_id] at this,
apply pushout_cocone.epi_of_is_colimit_mk_id_id _ this,
end | lemma | category_theory.preserves_epi_of_preserves_colimit | category_theory.limits.constructions | src/category_theory/limits/constructions/epi_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks"
] | [] | If `F` preserves pushouts, then it preserves epimorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_epimorphisms_of_preserves_colimits_of_shape
[preserves_colimits_of_shape walking_span F] : F.preserves_epimorphisms | { preserves := λ X Y f hf, by exactI preserves_epi_of_preserves_colimit F f } | instance | category_theory.preserves_epimorphisms_of_preserves_colimits_of_shape | category_theory.limits.constructions | src/category_theory/limits/constructions/epi_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_epi_of_reflects_colimit {X Y : C} (f : X ⟶ Y) [reflects_colimit (span f f) F]
[epi (F.map f)] : epi f | begin
have := pushout_cocone.is_colimit_mk_id_id (F.map f),
simp_rw [← F.map_id] at this,
apply pushout_cocone.epi_of_is_colimit_mk_id_id _
(is_colimit_of_is_colimit_pushout_cocone_map F _ this)
end | lemma | category_theory.reflects_epi_of_reflects_colimit | category_theory.limits.constructions | src/category_theory/limits/constructions/epi_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks"
] | [] | If `F` reflects pushouts, then it reflects epimorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_epimorphisms_of_reflects_colimits_of_shape
[reflects_colimits_of_shape walking_span F] : F.reflects_epimorphisms | { reflects := λ X Y f hf, by exactI reflects_epi_of_reflects_colimit F f } | instance | category_theory.reflects_epimorphisms_of_reflects_colimits_of_shape | category_theory.limits.constructions | src/category_theory/limits/constructions/epi_mono.lean | [
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.preserves.shapes.pullbacks"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
construct_equalizer (F : walking_parallel_pair ⥤ C) : C | pullback (prod.lift (𝟙 _) (F.map walking_parallel_pair_hom.left))
(prod.lift (𝟙 _) (F.map walking_parallel_pair_hom.right)) | def | category_theory.limits.has_equalizers_of_has_pullbacks_and_binary_products.construct_equalizer | category_theory.limits.constructions | src/category_theory/limits/constructions/equalizers.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.binary_products"
] | [] | Define the equalizing object | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_fst (F : walking_parallel_pair ⥤ C) :
construct_equalizer F ⟶ F.obj walking_parallel_pair.zero | pullback.fst | abbreviation | category_theory.limits.has_equalizers_of_has_pullbacks_and_binary_products.pullback_fst | category_theory.limits.constructions | src/category_theory/limits/constructions/equalizers.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.binary_products"
] | [] | Define the equalizing morphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_fst_eq_pullback_snd (F : walking_parallel_pair ⥤ C) :
pullback_fst F = pullback.snd | by convert pullback.condition =≫ limits.prod.fst; simp | lemma | category_theory.limits.has_equalizers_of_has_pullbacks_and_binary_products.pullback_fst_eq_pullback_snd | category_theory.limits.constructions | src/category_theory/limits/constructions/equalizers.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equalizer_cone (F : walking_parallel_pair ⥤ C) : cone F | cone.of_fork
(fork.of_ι (pullback_fst F)
(begin
conv_rhs { rw pullback_fst_eq_pullback_snd, },
convert pullback.condition =≫ limits.prod.snd using 1; simp
end)) | def | category_theory.limits.has_equalizers_of_has_pullbacks_and_binary_products.equalizer_cone | category_theory.limits.constructions | src/category_theory/limits/constructions/equalizers.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.binary_products"
] | [] | Define the equalizing cone | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer_cone_is_limit (F : walking_parallel_pair ⥤ C) : is_limit (equalizer_cone F) | { lift :=
begin
intro c, apply pullback.lift (c.π.app _) (c.π.app _),
apply limit.hom_ext,
rintro (_ | _); simp
end,
fac' := by rintros c (_ | _); simp,
uniq' :=
begin
intros c _ J,
have J0 := J walking_parallel_pair.zero, simp at J0,
apply pullback.hom_ext,
{ rwa limit.lift_π },
... | def | category_theory.limits.has_equalizers_of_has_pullbacks_and_binary_products.equalizer_cone_is_limit | category_theory.limits.constructions | src/category_theory/limits/constructions/equalizers.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.binary_products"
] | [
"lift"
] | Show the equalizing cone is a limit | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_equalizers_of_has_pullbacks_and_binary_products
[has_binary_products C] [has_pullbacks C] :
has_equalizers C | { has_limit := λ F, has_limit.mk
{ cone := equalizer_cone F,
is_limit := equalizer_cone_is_limit F } } | lemma | category_theory.limits.has_equalizers_of_has_pullbacks_and_binary_products | category_theory.limits.constructions | src/category_theory/limits/constructions/equalizers.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_equalizers_of_preserves_pullbacks_and_binary_products
[has_binary_products C] [has_pullbacks C]
[preserves_limits_of_shape (discrete walking_pair) G]
[preserves_limits_of_shape walking_cospan G] :
preserves_limits_of_shape walking_parallel_pair G | ⟨λ K, preserves_limit_of_preserves_limit_cone (equalizer_cone_is_limit K) $
{ lift := λ c, begin
refine pullback.lift _ _ _ ≫ (@@preserves_pullback.iso _ _ _ _ _ _ _ _).inv,
{ exact c.π.app walking_parallel_pair.zero },
{ exact c.π.app walking_parallel_pair.zero },
apply (map_is_limit_of_preserves_of_is... | def | category_theory.limits.preserves_equalizers_of_preserves_pullbacks_and_binary_products | category_theory.limits.constructions | src/category_theory/limits/constructions/equalizers.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.binary_products"
] | [
"hom_ext",
"lift"
] | A functor that preserves pullbacks and binary products also presrves equalizers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
construct_coequalizer (F : walking_parallel_pair ⥤ C) : C | pushout (coprod.desc (𝟙 _) (F.map walking_parallel_pair_hom.left))
(coprod.desc (𝟙 _) (F.map walking_parallel_pair_hom.right)) | def | category_theory.limits.has_coequalizers_of_has_pushouts_and_binary_coproducts.construct_coequalizer | category_theory.limits.constructions | src/category_theory/limits/constructions/equalizers.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.binary_products"
] | [] | Define the equalizing object | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout_inl (F : walking_parallel_pair ⥤ C) :
F.obj walking_parallel_pair.one ⟶ construct_coequalizer F | pushout.inl | abbreviation | category_theory.limits.has_coequalizers_of_has_pushouts_and_binary_coproducts.pushout_inl | category_theory.limits.constructions | src/category_theory/limits/constructions/equalizers.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.binary_products"
] | [] | Define the equalizing morphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout_inl_eq_pushout_inr (F : walking_parallel_pair ⥤ C) :
pushout_inl F = pushout.inr | by convert limits.coprod.inl ≫= pushout.condition; simp | lemma | category_theory.limits.has_coequalizers_of_has_pushouts_and_binary_coproducts.pushout_inl_eq_pushout_inr | category_theory.limits.constructions | src/category_theory/limits/constructions/equalizers.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer_cocone (F : walking_parallel_pair ⥤ C) : cocone F | cocone.of_cofork
(cofork.of_π (pushout_inl F)
(begin
conv_rhs { rw pushout_inl_eq_pushout_inr, },
convert limits.coprod.inr ≫= pushout.condition using 1; simp
end)) | def | category_theory.limits.has_coequalizers_of_has_pushouts_and_binary_coproducts.coequalizer_cocone | category_theory.limits.constructions | src/category_theory/limits/constructions/equalizers.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.binary_products"
] | [] | Define the equalizing cocone | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer_cocone_is_colimit (F : walking_parallel_pair ⥤ C) :
is_colimit (coequalizer_cocone F) | { desc :=
begin
intro c, apply pushout.desc (c.ι.app _) (c.ι.app _),
apply colimit.hom_ext,
rintro (_ | _); simp
end,
fac' := by rintros c (_ | _); simp,
uniq' :=
begin
intros c _ J,
have J1 : pushout_inl F ≫ m = c.ι.app walking_parallel_pair.one :=
by simpa using J walking_parallel_... | def | category_theory.limits.has_coequalizers_of_has_pushouts_and_binary_coproducts.coequalizer_cocone_is_colimit | category_theory.limits.constructions | src/category_theory/limits/constructions/equalizers.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.binary_products"
] | [] | Show the equalizing cocone is a colimit | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coequalizers_of_has_pushouts_and_binary_coproducts
[has_binary_coproducts C] [has_pushouts C] : has_coequalizers C | { has_colimit := λ F, has_colimit.mk
{ cocone := coequalizer_cocone F,
is_colimit := coequalizer_cocone_is_colimit F } } | lemma | category_theory.limits.has_coequalizers_of_has_pushouts_and_binary_coproducts | category_theory.limits.constructions | src/category_theory/limits/constructions/equalizers.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_coequalizers_of_preserves_pushouts_and_binary_coproducts
[has_binary_coproducts C] [has_pushouts C]
[preserves_colimits_of_shape (discrete walking_pair) G]
[preserves_colimits_of_shape walking_span G] :
preserves_colimits_of_shape walking_parallel_pair G | ⟨λ K, preserves_colimit_of_preserves_colimit_cocone (coequalizer_cocone_is_colimit K) $
{ desc := λ c, begin
refine (@@preserves_pushout.iso _ _ _ _ _ _ _ _).inv ≫ pushout.desc _ _ _,
{ exact c.ι.app walking_parallel_pair.one },
{ exact c.ι.app walking_parallel_pair.one },
apply (map_is_colimit_of_prese... | def | category_theory.limits.preserves_coequalizers_of_preserves_pushouts_and_binary_coproducts | category_theory.limits.constructions | src/category_theory/limits/constructions/equalizers.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.pullbacks",
"category_theory.limits.preserves.shapes.pullbacks",
"category_theory.limits.preserves.shapes.binary_products"
] | [
"hom_ext"
] | A functor that preserves pushouts and binary coproducts also presrves coequalizers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_to_finset [has_finite_coproducts C] (F : discrete α ⥤ C) : finset (discrete α) ⥤ C | { obj := λ s, ∐ λ x : s, F.obj x,
map := λ s t h, sigma.desc (λ y, sigma.ι (λ x : t, F.obj x) ⟨y, h.down.down y.2⟩) } | def | category_theory.limits.coproducts_from_finite_filtered.lift_to_finset | category_theory.limits.constructions | src/category_theory/limits/constructions/filtered.lean | [
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.limits.opposites"
] | [
"finset"
] | If `C` has finite coproducts, a functor `discrete α ⥤ C` lifts to a functor
`finset (discrete α) ⥤ C` by taking coproducts. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_to_finset_colimit_cocone [has_finite_coproducts C] [has_filtered_colimits_of_size.{w w} C]
[decidable_eq α] (F : discrete α ⥤ C) : colimit_cocone F | { cocone :=
{ X := colimit (lift_to_finset F),
ι := discrete.nat_trans $ λ j,
@sigma.ι _ _ _ (λ x : ({j} : finset (discrete α)), F.obj x) _ ⟨j, by simp⟩ ≫
colimit.ι (lift_to_finset F) {j} },
is_colimit :=
{ desc := λ s, colimit.desc (lift_to_finset F)
{ X := s.X,
ι := { app := λ t, sigma... | def | category_theory.limits.coproducts_from_finite_filtered.lift_to_finset_colimit_cocone | category_theory.limits.constructions | src/category_theory/limits/constructions/filtered.lean | [
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.limits.opposites"
] | [
"finset"
] | If `C` has finite coproducts and filtered colimits, we can construct arbitrary coproducts by
taking the colimit of the diagram formed by the coproducts of finite sets over the indexing
type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coproducts_of_finite_and_filtered [has_finite_coproducts C]
[has_filtered_colimits_of_size.{w w} C] : has_coproducts.{w} C | λ α, by { classical, exactI ⟨λ F, has_colimit.mk (lift_to_finset_colimit_cocone F)⟩ } | lemma | category_theory.limits.has_coproducts_of_finite_and_filtered | category_theory.limits.constructions | src/category_theory/limits/constructions/filtered.lean | [
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.limits.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_finite_and_filtered [has_finite_colimits C]
[has_filtered_colimits_of_size.{w w} C] : has_colimits_of_size.{w w} C | have has_coproducts.{w} C, from has_coproducts_of_finite_and_filtered,
by exactI has_colimits_of_has_coequalizers_and_coproducts | lemma | category_theory.limits.has_colimits_of_finite_and_filtered | category_theory.limits.constructions | src/category_theory/limits/constructions/filtered.lean | [
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.limits.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_products_of_finite_and_cofiltered [has_finite_products C]
[has_cofiltered_limits_of_size.{w w} C] : has_products.{w} C | have has_coproducts.{w} Cᵒᵖ, from has_coproducts_of_finite_and_filtered,
by exactI has_products_of_opposite | lemma | category_theory.limits.has_products_of_finite_and_cofiltered | category_theory.limits.constructions | src/category_theory/limits/constructions/filtered.lean | [
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.limits.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits_of_finite_and_cofiltered [has_finite_limits C]
[has_cofiltered_limits_of_size.{w w} C] : has_limits_of_size.{w w} C | have has_products.{w} C, from has_products_of_finite_and_cofiltered,
by exactI has_limits_of_has_equalizers_and_products | lemma | category_theory.limits.has_limits_of_finite_and_cofiltered | category_theory.limits.constructions | src/category_theory/limits/constructions/filtered.lean | [
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.limits.opposites"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_fan {n : ℕ} {f : fin (n+1) → C}
(c₁ : fan (λ (i : fin n), f i.succ))
(c₂ : binary_fan (f 0) c₁.X) :
fan f | fan.mk c₂.X
begin
refine fin.cases _ _,
{ apply c₂.fst },
{ intro i, apply c₂.snd ≫ c₁.π.app ⟨i⟩ },
end | def | category_theory.extend_fan | 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"
] | Given `n+1` objects of `C`, a fan for the last `n` with point `c₁.X` and a binary fan on `c₁.X` and
`f 0`, we can build a fan for all `n+1`.
In `extend_fan_is_limit` we show that if the two given fans are limits, then this fan is also a
limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_fan_is_limit {n : ℕ} (f : fin (n+1) → C)
{c₁ : fan (λ (i : fin n), f i.succ)} {c₂ : binary_fan (f 0) c₁.X}
(t₁ : is_limit c₁) (t₂ : is_limit c₂) :
is_limit (extend_fan c₁ c₂) | { lift := λ s,
begin
apply (binary_fan.is_limit.lift' t₂ (s.π.app ⟨0⟩) _).1,
apply t₁.lift ⟨_, discrete.nat_trans (λ ⟨i⟩, s.π.app ⟨i.succ⟩)⟩
end,
fac' := λ s ⟨j⟩,
begin
apply fin.induction_on j,
{ apply (binary_fan.is_limit.lift' t₂ _ _).2.1 },
{ rintro i -,
dsimp only [extend_fan_π_ap... | def | category_theory.extend_fan_is_limit | 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",
"lift"
] | Show that if the two given fans in `extend_fan` are limits, then the constructed fan is also a
limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_product_fin :
Π (n : ℕ) (f : fin n → C), has_product f | | 0 := λ f,
begin
letI : has_limits_of_shape (discrete (fin 0)) C :=
has_limits_of_shape_of_equivalence (discrete.equivalence.{0} fin_zero_equiv'.symm),
apply_instance,
end
| (n+1) := λ f,
begin
haveI := has_product_fin n,
apply has_limit.mk ⟨_, extend_fan_is_limit f (limit.is_limit _) (limi... | lemma | category_theory.has_product_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 a terminal object and binary products, then it has a product for objects indexed by
`fin n`.
This is a helper lemma for `has_finite_products_of_has_binary_and_terminal`, which is more general
than this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_products_of_has_binary_and_terminal : has_finite_products C | begin
refine ⟨λ n, ⟨λ K, _⟩⟩,
letI := has_product_fin n (λ n, K.obj ⟨n⟩),
let : discrete.functor (λ n, K.obj ⟨n⟩) ≅ K := discrete.nat_iso (λ ⟨i⟩, iso.refl _),
apply has_limit_of_iso this,
end | lemma | category_theory.has_finite_products_of_has_binary_and_terminal | 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 a terminal object and binary products, then it has finite products. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_fin_of_preserves_binary_and_terminal :
Π (n : ℕ) (f : fin n → C), preserves_limit (discrete.functor f) F | | 0 := λ f,
begin
letI : preserves_limits_of_shape (discrete (fin 0)) F :=
preserves_limits_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_terminal n,
intro f,
refine prese... | def | category_theory.preserves_fin_of_preserves_binary_and_terminal | 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 terminal object and binary products, 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_terminal (n : ℕ) :
preserves_limits_of_shape (discrete (fin n)) F | { preserves_limit := 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_terminal F n (λ n, K.obj ⟨n⟩),
apply preserves_limit_of_iso_diagram F this,
end } | def | category_theory.preserves_shape_fin_of_preserves_binary_and_terminal | 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 terminal object and binary products, then it preserves limits of shape
`discrete (fin n)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_products_of_preserves_binary_and_terminal
(J : Type) [fintype J] :
preserves_limits_of_shape (discrete J) F | begin
classical,
let e := fintype.equiv_fin J,
haveI := preserves_shape_fin_of_preserves_binary_and_terminal F (fintype.card J),
apply preserves_limits_of_shape_of_equiv.{0 0}
(discrete.equivalence e).symm,
end | def | category_theory.preserves_finite_products_of_preserves_binary_and_terminal | 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 terminal object and binary products then it preserves finite products. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_cofan {n : ℕ} {f : fin (n+1) → C}
(c₁ : cofan (λ (i : fin n), f i.succ))
(c₂ : binary_cofan (f 0) c₁.X) :
cofan f | cofan.mk c₂.X
begin
refine fin.cases _ _,
{ apply c₂.inl },
{ intro i,
apply c₁.ι.app ⟨i⟩ ≫ c₂.inr },
end | def | category_theory.extend_cofan | 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"
] | Given `n+1` objects of `C`, a cofan for the last `n` with point `c₁.X`
and a binary cofan on `c₁.X` and `f 0`, we can build a cofan for all `n+1`.
In `extend_cofan_is_colimit` we show that if the two given cofans are colimits,
then this cofan is also a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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