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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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full.to_ess_image (F : C ⥤ D) [full F] : full F.to_ess_image | begin
haveI := full.of_iso F.to_ess_image_comp_essential_image_inclusion.symm,
exactI full.of_comp_faithful F.to_ess_image F.ess_image_inclusion
end | instance | category_theory.full.to_ess_image | category_theory | src/category_theory/essential_image.lean | [
"category_theory.natural_isomorphism",
"category_theory.full_subcategory"
] | [] | The induced functor of a full functor is full | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans.equifibered {F G : J ⥤ C} (α : F ⟶ G) : Prop | ∀ ⦃i j : J⦄ (f : i ⟶ j), is_pullback (F.map f) (α.app i) (α.app j) (G.map f) | def | category_theory.nat_trans.equifibered | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | A natural transformation is equifibered if every commutative square of the following form is
a pullback.
```
F(X) → F(Y)
↓ ↓
G(X) → G(Y)
``` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans.equifibered_of_is_iso {F G : J ⥤ C} (α : F ⟶ G) [is_iso α] : α.equifibered | λ _ _ f, is_pullback.of_vert_is_iso ⟨nat_trans.naturality _ f⟩ | lemma | category_theory.nat_trans.equifibered_of_is_iso | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans.equifibered.comp {F G H : J ⥤ C} {α : F ⟶ G} {β : G ⟶ H}
(hα : α.equifibered) (hβ : β.equifibered) : (α ≫ β).equifibered | λ i j f, (hα f).paste_vert (hβ f) | lemma | category_theory.nat_trans.equifibered.comp | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_universal_colimit {F : J ⥤ C} (c : cocone F) : Prop | ∀ ⦃F' : J ⥤ C⦄ (c' : cocone F') (α : F' ⟶ F) (f : c'.X ⟶ c.X)
(h : α ≫ c.ι = c'.ι ≫ (functor.const J).map f) (hα : α.equifibered),
(∀ j : J, is_pullback (c'.ι.app j) (α.app j) f (c.ι.app j)) → nonempty (is_colimit c') | def | category_theory.is_universal_colimit | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [
"functor.const"
] | A (colimit) cocone over a diagram `F : J ⥤ C` is universal if it is stable under pullbacks. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_van_kampen_colimit {F : J ⥤ C} (c : cocone F) : Prop | ∀ ⦃F' : J ⥤ C⦄ (c' : cocone F') (α : F' ⟶ F) (f : c'.X ⟶ c.X)
(h : α ≫ c.ι = c'.ι ≫ (functor.const J).map f) (hα : α.equifibered),
nonempty (is_colimit c') ↔ ∀ j : J, is_pullback (c'.ι.app j) (α.app j) f (c.ι.app j) | def | category_theory.is_van_kampen_colimit | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [
"functor.const"
] | A (colimit) cocone over a diagram `F : J ⥤ C` is van Kampen if for every cocone `c'` over the
pullback of the diagram `F' : J ⥤ C'`, `c'` is colimiting iff `c'` is the pullback of `c`.
TODO: Show that this is iff the functor `C ⥤ Catᵒᵖ` sending `x` to `C/x` preserves it.
TODO: Show that this is iff the inclusion funct... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_van_kampen_colimit.is_universal {F : J ⥤ C} {c : cocone F} (H : is_van_kampen_colimit c) :
is_universal_colimit c | λ _ c' α f h hα, (H c' α f h hα).mpr | lemma | category_theory.is_van_kampen_colimit.is_universal | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_van_kampen_colimit.is_colimit {F : J ⥤ C} {c : cocone F} (h : is_van_kampen_colimit c) :
is_colimit c | begin
refine ((h c (𝟙 F) (𝟙 c.X : _) (by rw [functor.map_id, category.comp_id, category.id_comp])
(nat_trans.equifibered_of_is_iso _)).mpr $ λ j, _).some,
haveI : is_iso (𝟙 c.X) := infer_instance,
exact is_pullback.of_vert_is_iso ⟨by erw [nat_trans.id_app, category.comp_id, category.id_comp]⟩,
end | def | category_theory.is_van_kampen_colimit.is_colimit | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [
"functor.map_id"
] | A van Kampen colimit is a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial.is_van_kampen_colimit [has_strict_initial_objects C] {X : C} (h : is_initial X) :
is_van_kampen_colimit (as_empty_cocone X) | begin
intros F' c' α f hf hα,
have : F' = functor.empty C := by apply functor.hext; rintro ⟨⟨⟩⟩,
subst this,
haveI := h.is_iso_to f,
refine ⟨by rintro _ ⟨⟨⟩⟩, λ _,
⟨is_colimit.of_iso_colimit h (cocones.ext (as_iso f).symm $ by rintro ⟨⟨⟩⟩)⟩⟩
end | lemma | category_theory.is_initial.is_van_kampen_colimit | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finitary_extensive (C : Type u) [category.{v} C] : Prop | [has_finite_coproducts : has_finite_coproducts C]
(van_kampen' : ∀ {X Y : C} (c : binary_cofan X Y), is_colimit c → is_van_kampen_colimit c) | class | category_theory.finitary_extensive | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | A category is (finitary) extensive if it has finite coproducts,
and binary coproducts are van Kampen.
TODO: Show that this is iff all finite coproducts are van Kampen. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finitary_extensive.van_kampen [finitary_extensive C] {F : discrete walking_pair ⥤ C}
(c : cocone F) (hc : is_colimit c) : is_van_kampen_colimit c | begin
let X := F.obj ⟨walking_pair.left⟩, let Y := F.obj ⟨walking_pair.right⟩,
have : F = pair X Y,
{ apply functor.hext, { rintros ⟨⟨⟩⟩; refl }, { rintros ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩; simpa } },
clear_value X Y, subst this,
exact finitary_extensive.van_kampen' c hc
end | lemma | category_theory.finitary_extensive.van_kampen | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_pair_equifibered {F F' : discrete walking_pair ⥤ C} (α : F ⟶ F') : α.equifibered | begin
rintros ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩,
all_goals { dsimp, simp only [discrete.functor_map_id],
exact is_pullback.of_horiz_is_iso ⟨by simp only [category.comp_id, category.id_comp]⟩ }
end | lemma | category_theory.map_pair_equifibered | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan.is_van_kampen_iff (c : binary_cofan X Y) :
is_van_kampen_colimit c ↔
∀ {X' Y' : C} (c' : binary_cofan X' Y') (αX : X' ⟶ X) (αY : Y' ⟶ Y)
(f : c'.X ⟶ c.X) (hαX : αX ≫ c.inl = c'.inl ≫ f) (hαY : αY ≫ c.inr = c'.inr ≫ f),
nonempty (is_colimit c') ↔ is_pullback c'.inl αX f c.inl ∧ is_pullback c'.in... | begin
split,
{ introv H hαX hαY,
rw H c' (map_pair αX αY) f (by ext ⟨⟨⟩⟩; dsimp; assumption) (map_pair_equifibered _),
split, { intro H, exact ⟨H _, H _⟩ }, { rintros H ⟨⟨⟩⟩, exacts [H.1, H.2] } },
{ introv H F' hα h,
let X' := F'.obj ⟨walking_pair.left⟩, let Y' := F'.obj ⟨walking_pair.right⟩,
hav... | lemma | category_theory.binary_cofan.is_van_kampen_iff | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan.is_van_kampen_mk {X Y : C} (c : binary_cofan X Y)
(cofans : ∀ (X Y : C), binary_cofan X Y) (colimits : ∀ X Y, is_colimit (cofans X Y))
(cones : ∀ {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z), pullback_cone f g)
(limits : ∀ {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z), is_limit (cones f g))
(h₁ : ∀ {X' Y' : C} (αX :... | begin
rw binary_cofan.is_van_kampen_iff,
introv hX hY,
split,
{ rintros ⟨h⟩,
let e := h.cocone_point_unique_up_to_iso (colimits _ _),
obtain ⟨hl, hr⟩ := h₁ αX αY (e.inv ≫ f) (by simp [hX]) (by simp [hY]),
split,
{ rw [← category.id_comp αX, ← iso.hom_inv_id_assoc e f],
have : c'.inl ≫ e.ho... | lemma | category_theory.binary_cofan.is_van_kampen_mk | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan.mono_inr_of_is_van_kampen [has_initial C] {X Y : C} {c : binary_cofan X Y}
(h : is_van_kampen_colimit c) : mono c.inr | begin
refine pullback_cone.mono_of_is_limit_mk_id_id _ (is_pullback.is_limit _),
refine (h (binary_cofan.mk (initial.to Y) (𝟙 Y))
(map_pair (initial.to X) (𝟙 Y)) c.inr _ (map_pair_equifibered _)).mp ⟨_⟩ ⟨walking_pair.right⟩,
{ ext ⟨⟨⟩⟩; dsimp; simp },
{ exact ((binary_cofan.is_colimit_iff_is_iso_inr initi... | lemma | category_theory.binary_cofan.mono_inr_of_is_van_kampen | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finitary_extensive.mono_inr_of_is_colimit [finitary_extensive C]
{c : binary_cofan X Y} (hc : is_colimit c) : mono c.inr | binary_cofan.mono_inr_of_is_van_kampen (finitary_extensive.van_kampen c hc) | lemma | category_theory.finitary_extensive.mono_inr_of_is_colimit | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finitary_extensive.mono_inl_of_is_colimit [finitary_extensive C]
{c : binary_cofan X Y} (hc : is_colimit c) : mono c.inl | finitary_extensive.mono_inr_of_is_colimit (binary_cofan.is_colimit_flip hc) | lemma | category_theory.finitary_extensive.mono_inl_of_is_colimit | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan.is_pullback_initial_to_of_is_van_kampen [has_initial C]
{c : binary_cofan X Y}
(h : is_van_kampen_colimit c) : is_pullback (initial.to _) (initial.to _) c.inl c.inr | begin
refine ((h (binary_cofan.mk (initial.to Y) (𝟙 Y)) (map_pair (initial.to X) (𝟙 Y)) c.inr _
(map_pair_equifibered _)).mp ⟨_⟩ ⟨walking_pair.left⟩).flip,
{ ext ⟨⟨⟩⟩; dsimp; simp },
{ exact ((binary_cofan.is_colimit_iff_is_iso_inr initial_is_initial _).mpr
(by { dsimp, apply_instance })).some }
end | lemma | category_theory.binary_cofan.is_pullback_initial_to_of_is_van_kampen | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finitary_extensive.is_pullback_initial_to_binary_cofan [finitary_extensive C]
{c : binary_cofan X Y} (hc : is_colimit c) :
is_pullback (initial.to _) (initial.to _) c.inl c.inr | binary_cofan.is_pullback_initial_to_of_is_van_kampen (finitary_extensive.van_kampen c hc) | lemma | category_theory.finitary_extensive.is_pullback_initial_to_binary_cofan | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_initial_of_is_universal [has_initial C]
(H : is_universal_colimit (binary_cofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C)))) :
has_strict_initial_objects C | has_strict_initial_objects_of_initial_is_strict
begin
intros A f,
suffices : is_colimit (binary_cofan.mk (𝟙 A) (𝟙 A)),
{ obtain ⟨l, h₁, h₂⟩ := limits.binary_cofan.is_colimit.desc' this (f ≫ initial.to A) (𝟙 A),
rcases (category.id_comp _).symm.trans h₂ with rfl,
exact ⟨⟨_, ((category.id_comp _).symm.tr... | lemma | category_theory.has_strict_initial_of_is_universal | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_initial_objects_of_finitary_extensive [finitary_extensive C] :
has_strict_initial_objects C | has_strict_initial_of_is_universal (finitary_extensive.van_kampen _
((binary_cofan.is_colimit_iff_is_iso_inr initial_is_initial _).mpr
(by { dsimp, apply_instance })).some).is_universal | instance | category_theory.has_strict_initial_objects_of_finitary_extensive | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [
"is_universal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finitary_extensive_iff_of_is_terminal (C : Type u) [category.{v} C] [has_finite_coproducts C]
(T : C) (HT : is_terminal T) (c₀ : binary_cofan T T) (hc₀ : is_colimit c₀) :
finitary_extensive C ↔ is_van_kampen_colimit c₀ | begin
refine ⟨λ H, H.2 c₀ hc₀, λ H, _⟩,
constructor,
simp_rw binary_cofan.is_van_kampen_iff at H ⊢,
intros X Y c hc X' Y' c' αX αY f hX hY,
obtain ⟨d, hd, hd'⟩ := limits.binary_cofan.is_colimit.desc' hc
(HT.from _ ≫ c₀.inl) (HT.from _ ≫ c₀.inr),
rw H c' (αX ≫ HT.from _) (αY ≫ HT.from _) (f ≫ d)
(by ... | lemma | category_theory.finitary_extensive_iff_of_is_terminal | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
types.finitary_extensive : finitary_extensive (Type u) | begin
rw [finitary_extensive_iff_of_is_terminal (Type u) punit types.is_terminal_punit _
(types.binary_coproduct_colimit _ _)],
apply binary_cofan.is_van_kampen_mk _ _ (λ X Y, types.binary_coproduct_colimit X Y) _
(λ X Y Z f g, (limits.types.pullback_limit_cone f g).2),
{ intros,
split,
{ refine ⟨... | instance | category_theory.types.finitary_extensive | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [
"exists_unique_eq'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finitary_extensive_Top_aux (Z : Top.{u}) (f : Z ⟶ Top.of (punit.{u+1} ⊕ punit.{u+1})) :
is_colimit (binary_cofan.mk
(Top.pullback_fst f (Top.binary_cofan (Top.of punit) (Top.of punit)).inl)
(Top.pullback_fst f (Top.binary_cofan (Top.of punit) (Top.of punit)).inr)) | begin
have : ∀ x, f x = sum.inl punit.star ∨ f x = sum.inr punit.star,
{ intro x, rcases f x with (⟨⟨⟩⟩|⟨⟨⟩⟩), exacts [or.inl rfl, or.inr rfl] },
let eX : {p : Z × punit // f p.fst = sum.inl p.snd} ≃ { x : Z // f x = sum.inl punit.star } :=
⟨λ p, ⟨p.1.1, p.2.trans (congr_arg sum.inl $ subsingleton.elim _ _)⟩,... | def | category_theory.finitary_extensive_Top_aux | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [
"Top.binary_cofan",
"Top.of",
"Top.pullback_fst",
"continuity",
"continuous",
"continuous_iff_continuous_at",
"continuous_on_iff_continuous_restrict",
"is_open.continuous_on_iff",
"open_embedding_inl",
"open_embedding_inr",
"set.range"
] | (Implementation) An auxiliary lemma for the proof that `Top` is finitary extensive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans.equifibered.whisker_right {F G : J ⥤ C} {α : F ⟶ G} (hα : α.equifibered)
(H : C ⥤ D) [preserves_limits_of_shape walking_cospan H] : (whisker_right α H).equifibered | λ i j f, (hα f).map H | lemma | category_theory.nat_trans.equifibered.whisker_right | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_van_kampen_colimit.of_iso {F : J ⥤ C} {c c' : cocone F} (H : is_van_kampen_colimit c)
(e : c ≅ c') : is_van_kampen_colimit c' | begin
intros F' c'' α f h hα,
have : c'.ι ≫ (functor.const J).map e.inv.hom = c.ι,
{ ext j, exact e.inv.2 j },
rw H c'' α (f ≫ e.inv.1) (by rw [functor.map_comp, ← reassoc_of h, this]) hα,
apply forall_congr,
intro j,
conv_lhs { rw [← category.comp_id (α.app j)] },
haveI : is_iso e.inv.hom := functor.ma... | lemma | category_theory.is_van_kampen_colimit.of_iso | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [
"functor.const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_van_kampen_colimit.of_map {D : Type*} [category D] (G : C ⥤ D) {F : J ⥤ C} {c : cocone F}
[preserves_limits_of_shape walking_cospan G] [reflects_limits_of_shape walking_cospan G]
[preserves_colimits_of_shape J G] [reflects_colimits_of_shape J G]
(H : is_van_kampen_colimit (G.map_cocone c)) : is_van_kampen_c... | begin
intros F' c' α f h hα,
refine (iff.trans _ (H (G.map_cocone c') (whisker_right α G) (G.map f)
(by { ext j, simpa using G.congr_map (nat_trans.congr_app h j) })
(hα.whisker_right G))).trans (forall_congr $ λ j, _),
{ exact ⟨λ h, ⟨is_colimit_of_preserves G h.some⟩, λ h, ⟨is_colimit_of_reflects G h.som... | lemma | category_theory.is_van_kampen_colimit.of_map | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_van_kampen_colimit_of_evaluation [has_pullbacks D] [has_colimits_of_shape J D]
(F : J ⥤ C ⥤ D) (c : cocone F)
(hc : ∀ x : C, is_van_kampen_colimit (((evaluation C D).obj x).map_cocone c)) :
is_van_kampen_colimit c | begin
intros F' c' α f e hα,
have := λ x, hc x (((evaluation C D).obj x).map_cocone c') (whisker_right α _)
(((evaluation C D).obj x).map f)
(by { ext y, dsimp, exact nat_trans.congr_app (nat_trans.congr_app e y) x })
(hα.whisker_right _),
split,
{ rintros ⟨hc'⟩ j,
refine ⟨⟨(nat_trans.congr_app ... | lemma | category_theory.is_van_kampen_colimit_of_evaluation | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finitary_extensive_of_preserves_and_reflects (F : C ⥤ D)
[finitary_extensive D] [has_finite_coproducts C]
[preserves_limits_of_shape walking_cospan F]
[reflects_limits_of_shape walking_cospan F]
[preserves_colimits_of_shape (discrete walking_pair) F]
[reflects_colimits_of_shape (discrete walking_pair)... | ⟨λ X Y c hc, (finitary_extensive.van_kampen _ (is_colimit_of_preserves F hc)).of_map F⟩ | lemma | category_theory.finitary_extensive_of_preserves_and_reflects | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finitary_extensive_of_preserves_and_reflects_isomorphism (F : C ⥤ D)
[finitary_extensive D] [has_finite_coproducts C] [has_pullbacks C]
[preserves_limits_of_shape walking_cospan F]
[preserves_colimits_of_shape (discrete walking_pair) F]
[reflects_isomorphisms F] :
finitary_extensive C | begin
haveI : reflects_limits_of_shape walking_cospan F :=
reflects_limits_of_shape_of_reflects_isomorphisms,
haveI : reflects_colimits_of_shape (discrete walking_pair) F :=
reflects_colimits_of_shape_of_reflects_isomorphisms,
exact finitary_extensive_of_preserves_and_reflects F,
end | lemma | category_theory.finitary_extensive_of_preserves_and_reflects_isomorphism | category_theory | src/category_theory/extensive.lean | [
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.strict_initial",
"category_theory.limits.shapes.types",
"topology.category.Top.limits.pullbacks",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_filtered_or_empty : Prop | (cocone_objs : ∀ (X Y : C), ∃ Z (f : X ⟶ Z) (g : Y ⟶ Z), true)
(cocone_maps : ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), ∃ Z (h : Y ⟶ Z), f ≫ h = g ≫ h) | class | category_theory.is_filtered_or_empty | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | A category `is_filtered_or_empty` if
1. for every pair of objects there exists another object "to the right", and
2. for every pair of parallel morphisms there exists a morphism to the right so the compositions
are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_filtered extends is_filtered_or_empty C : Prop | [nonempty : nonempty C] | class | category_theory.is_filtered | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | A category `is_filtered` if
1. for every pair of objects there exists another object "to the right",
2. for every pair of parallel morphisms there exists a morphism to the right so the compositions
are equal, and
3. there exists some object.
See <https://stacks.math.columbia.edu/tag/002V>. (They also define a diagr... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_filtered_or_empty_of_semilattice_sup
(α : Type u) [semilattice_sup α] : is_filtered_or_empty α | { cocone_objs := λ X Y, ⟨X ⊔ Y, hom_of_le le_sup_left, hom_of_le le_sup_right, trivial⟩,
cocone_maps := λ X Y f g, ⟨Y, 𝟙 _, (by ext)⟩, } | instance | category_theory.is_filtered_or_empty_of_semilattice_sup | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [
"le_sup_left",
"le_sup_right",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_filtered_of_semilattice_sup_nonempty
(α : Type u) [semilattice_sup α] [nonempty α] : is_filtered α | {} | instance | category_theory.is_filtered_of_semilattice_sup_nonempty | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_filtered_or_empty_of_directed_le (α : Type u) [preorder α] [is_directed α (≤)] :
is_filtered_or_empty α | { cocone_objs := λ X Y, let ⟨Z, h1, h2⟩ := exists_ge_ge X Y in
⟨Z, hom_of_le h1, hom_of_le h2, trivial⟩,
cocone_maps := λ X Y f g, ⟨Y, 𝟙 _, by simp⟩ } | instance | category_theory.is_filtered_or_empty_of_directed_le | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [
"exists_ge_ge",
"is_directed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_filtered_of_directed_le_nonempty (α : Type u) [preorder α] [is_directed α (≤)]
[nonempty α] :
is_filtered α | {} | instance | category_theory.is_filtered_of_directed_le_nonempty | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [
"is_directed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone_objs : ∀ (X Y : C), ∃ Z (f : X ⟶ Z) (g : Y ⟶ Z), true | is_filtered_or_empty.cocone_objs | lemma | category_theory.is_filtered.cocone_objs | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone_maps : ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), ∃ Z (h : Y ⟶ Z), f ≫ h = g ≫ h | is_filtered_or_empty.cocone_maps | lemma | category_theory.is_filtered.cocone_maps | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max (j j' : C) : C | (cocone_objs j j').some | def | category_theory.is_filtered.max | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `max j j'` is an arbitrary choice of object to the right of both `j` and `j'`,
whose existence is ensured by `is_filtered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_to_max (j j' : C) : j ⟶ max j j' | (cocone_objs j j').some_spec.some | def | category_theory.is_filtered.left_to_max | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `left_to_max j j'` is an arbitrary choice of morphism from `j` to `max j j'`,
whose existence is ensured by `is_filtered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_to_max (j j' : C) : j' ⟶ max j j' | (cocone_objs j j').some_spec.some_spec.some | def | category_theory.is_filtered.right_to_max | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `right_to_max j j'` is an arbitrary choice of morphism from `j'` to `max j j'`,
whose existence is ensured by `is_filtered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeq {j j' : C} (f f' : j ⟶ j') : C | (cocone_maps f f').some | def | category_theory.is_filtered.coeq | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `coeq f f'`, for morphisms `f f' : j ⟶ j'`, is an arbitrary choice of object
which admits a morphism `coeq_hom f f' : j' ⟶ coeq f f'` such that
`coeq_condition : f ≫ coeq_hom f f' = f' ≫ coeq_hom f f'`.
Its existence is ensured by `is_filtered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeq_hom {j j' : C} (f f' : j ⟶ j') : j' ⟶ coeq f f' | (cocone_maps f f').some_spec.some | def | category_theory.is_filtered.coeq_hom | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `coeq_hom f f'`, for morphisms `f f' : j ⟶ j'`, is an arbitrary choice of morphism
`coeq_hom f f' : j' ⟶ coeq f f'` such that
`coeq_condition : f ≫ coeq_hom f f' = f' ≫ coeq_hom f f'`.
Its existence is ensured by `is_filtered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeq_condition {j j' : C} (f f' : j ⟶ j') : f ≫ coeq_hom f f' = f' ≫ coeq_hom f f' | (cocone_maps f f').some_spec.some_spec | lemma | category_theory.is_filtered.coeq_condition | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `coeq_condition f f'`, for morphisms `f f' : j ⟶ j'`, is the proof that
`f ≫ coeq_hom f f' = f' ≫ coeq_hom f f'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sup_objs_exists (O : finset C) : ∃ (S : C), ∀ {X}, X ∈ O → _root_.nonempty (X ⟶ S) | begin
classical,
apply finset.induction_on O,
{ exact ⟨is_filtered.nonempty.some, (by rintros - ⟨⟩)⟩, },
{ rintros X O' nm ⟨S', w'⟩,
use max X S',
rintros Y mY,
obtain rfl|h := eq_or_ne Y X,
{ exact ⟨left_to_max _ _⟩, },
{ exact ⟨(w' (finset.mem_of_mem_insert_of_ne mY h)).some ≫ right_to_max... | lemma | category_theory.is_filtered.sup_objs_exists | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [
"eq_or_ne",
"finset",
"finset.induction_on",
"finset.mem_of_mem_insert_of_ne"
] | Any finite collection of objects in a filtered category has an object "to the right". | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sup_exists :
∃ (S : C) (T : Π {X : C}, X ∈ O → (X ⟶ S)), ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},
(⟨X, Y, mX, mY, f⟩ : (Σ' (X Y : C) (mX : X ∈ O) (mY : Y ∈ O), X ⟶ Y)) ∈ H → f ≫ T mY = T mX | begin
classical,
apply finset.induction_on H,
{ obtain ⟨S, f⟩ := sup_objs_exists O,
refine ⟨S, λ X mX, (f mX).some, _⟩,
rintros - - - - - ⟨⟩, },
{ rintros ⟨X, Y, mX, mY, f⟩ H' nmf ⟨S', T', w'⟩,
refine ⟨coeq (f ≫ T' mY) (T' mX), λ Z mZ, T' mZ ≫ coeq_hom (f ≫ T' mY) (T' mX), _⟩,
intros X' Y' mX' m... | lemma | category_theory.is_filtered.sup_exists | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [
"finset.induction_on",
"finset.mem_of_mem_insert_of_ne",
"heq_iff_eq",
"psigma.mk.inj_iff"
] | Given any `finset` of objects `{X, ...}` and
indexed collection of `finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : X ⟶ S` from each `X`,
such that the triangles commute: `f ≫ T Y = T X`, for `f : X ⟶ Y` in the `finset`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sup : C | (sup_exists O H).some | def | category_theory.is_filtered.sup | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | An arbitrary choice of object "to the right"
of a finite collection of objects `O` and morphisms `H`,
making all the triangles commute. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_sup {X : C} (m : X ∈ O) :
X ⟶ sup O H | (sup_exists O H).some_spec.some m | def | category_theory.is_filtered.to_sup | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | The morphisms to `sup O H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_sup_commutes
{X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y}
(mf : (⟨X, Y, mX, mY, f⟩ : Σ' (X Y : C) (mX : X ∈ O) (mY : Y ∈ O), X ⟶ Y) ∈ H) :
f ≫ to_sup O H mY = to_sup O H mX | (sup_exists O H).some_spec.some_spec mX mY mf | lemma | category_theory.is_filtered.to_sup_commutes | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | The triangles of consisting of a morphism in `H` and the maps to `sup O H` commute. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_nonempty (F : J ⥤ C) : _root_.nonempty (cocone F) | begin
classical,
let O := (finset.univ.image F.obj),
let H : finset (Σ' (X Y : C) (mX : X ∈ O) (mY : Y ∈ O), X ⟶ Y) :=
finset.univ.bUnion (λ X : J, finset.univ.bUnion (λ Y : J, finset.univ.image (λ f : X ⟶ Y,
⟨F.obj X, F.obj Y, by simp, by simp, F.map f⟩))),
obtain ⟨Z, f, w⟩ := sup_exists O H,
refin... | lemma | category_theory.is_filtered.cocone_nonempty | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [
"exists_and_distrib_left",
"exists_prop_of_true",
"finset",
"finset.mem_bUnion",
"finset.mem_image",
"finset.mem_univ"
] | If we have `is_filtered C`, then for any functor `F : J ⥤ C` with `fin_category J`,
there exists a cocone over `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone (F : J ⥤ C) : cocone F | (cocone_nonempty F).some | def | category_theory.is_filtered.cocone | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | An arbitrary choice of cocone over `F : J ⥤ C`, for `fin_category J` and `is_filtered C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_right_adjoint {L : D ⥤ C} {R : C ⥤ D} (h : L ⊣ R) : is_filtered D | { cocone_objs := λ X Y,
⟨_, h.hom_equiv _ _ (left_to_max _ _), h.hom_equiv _ _ (right_to_max _ _), ⟨⟩⟩,
cocone_maps := λ X Y f g,
⟨_, h.hom_equiv _ _ (coeq_hom _ _),
by rw [← h.hom_equiv_naturality_left, ← h.hom_equiv_naturality_left, coeq_condition]⟩,
nonempty := is_filtered.nonempty.map R.obj } | lemma | category_theory.is_filtered.of_right_adjoint | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | If `C` is filtered, and we have a functor `R : C ⥤ D` with a left adjoint, then `D` is filtered. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_right_adjoint (R : C ⥤ D) [is_right_adjoint R] : is_filtered D | of_right_adjoint (adjunction.of_right_adjoint R) | lemma | category_theory.is_filtered.of_is_right_adjoint | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | If `C` is filtered, and we have a right adjoint functor `R : C ⥤ D`, then `D` is filtered. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_equivalence (h : C ≌ D) : is_filtered D | of_right_adjoint h.symm.to_adjunction | lemma | category_theory.is_filtered.of_equivalence | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | Being filtered is preserved by equivalence of categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
max₃ (j₁ j₂ j₃ : C) : C | max (max j₁ j₂) j₃ | def | category_theory.is_filtered.max₃ | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `max₃ j₁ j₂ j₃` is an arbitrary choice of object to the right of `j₁`, `j₂` and `j₃`,
whose existence is ensured by `is_filtered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
first_to_max₃ (j₁ j₂ j₃ : C) : j₁ ⟶ max₃ j₁ j₂ j₃ | left_to_max j₁ j₂ ≫ left_to_max (max j₁ j₂) j₃ | def | category_theory.is_filtered.first_to_max₃ | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `first_to_max₃ j₁ j₂ j₃` is an arbitrary choice of morphism from `j₁` to `max₃ j₁ j₂ j₃`,
whose existence is ensured by `is_filtered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
second_to_max₃ (j₁ j₂ j₃ : C) : j₂ ⟶ max₃ j₁ j₂ j₃ | right_to_max j₁ j₂ ≫ left_to_max (max j₁ j₂) j₃ | def | category_theory.is_filtered.second_to_max₃ | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `second_to_max₃ j₁ j₂ j₃` is an arbitrary choice of morphism from `j₂` to `max₃ j₁ j₂ j₃`,
whose existence is ensured by `is_filtered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
third_to_max₃ (j₁ j₂ j₃ : C) : j₃ ⟶ max₃ j₁ j₂ j₃ | right_to_max (max j₁ j₂) j₃ | def | category_theory.is_filtered.third_to_max₃ | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `third_to_max₃ j₁ j₂ j₃` is an arbitrary choice of morphism from `j₃` to `max₃ j₁ j₂ j₃`,
whose existence is ensured by `is_filtered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeq₃ {j₁ j₂ : C} (f g h : j₁ ⟶ j₂) : C | coeq (coeq_hom f g ≫ left_to_max (coeq f g) (coeq g h))
(coeq_hom g h ≫ right_to_max (coeq f g) (coeq g h)) | def | category_theory.is_filtered.coeq₃ | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `coeq₃ f g h`, for morphisms `f g h : j₁ ⟶ j₂`, is an arbitrary choice of object
which admits a morphism `coeq₃_hom f g h : j₂ ⟶ coeq₃ f g h` such that
`coeq₃_condition₁`, `coeq₃_condition₂` and `coeq₃_condition₃` are satisfied.
Its existence is ensured by `is_filtered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeq₃_hom {j₁ j₂ : C} (f g h : j₁ ⟶ j₂) : j₂ ⟶ coeq₃ f g h | coeq_hom f g ≫ left_to_max (coeq f g) (coeq g h) ≫
coeq_hom (coeq_hom f g ≫ left_to_max (coeq f g) (coeq g h))
(coeq_hom g h ≫ right_to_max (coeq f g) (coeq g h)) | def | category_theory.is_filtered.coeq₃_hom | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `coeq₃_hom f g h`, for morphisms `f g h : j₁ ⟶ j₂`, is an arbitrary choice of morphism
`j₂ ⟶ coeq₃ f g h` such that `coeq₃_condition₁`, `coeq₃_condition₂` and `coeq₃_condition₃`
are satisfied. Its existence is ensured by `is_filtered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeq₃_condition₁ {j₁ j₂ : C} (f g h : j₁ ⟶ j₂) :
f ≫ coeq₃_hom f g h = g ≫ coeq₃_hom f g h | by rw [coeq₃_hom, reassoc_of (coeq_condition f g)] | lemma | category_theory.is_filtered.coeq₃_condition₁ | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeq₃_condition₂ {j₁ j₂ : C} (f g h : j₁ ⟶ j₂) :
g ≫ coeq₃_hom f g h = h ≫ coeq₃_hom f g h | begin
dsimp [coeq₃_hom],
slice_lhs 2 4 { rw [← category.assoc, coeq_condition _ _] },
slice_rhs 2 4 { rw [← category.assoc, coeq_condition _ _] },
slice_lhs 1 3 { rw [← category.assoc, coeq_condition _ _] },
simp only [category.assoc],
end | lemma | category_theory.is_filtered.coeq₃_condition₂ | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeq₃_condition₃ {j₁ j₂ : C} (f g h : j₁ ⟶ j₂) :
f ≫ coeq₃_hom f g h = h ≫ coeq₃_hom f g h | eq.trans (coeq₃_condition₁ f g h) (coeq₃_condition₂ f g h) | lemma | category_theory.is_filtered.coeq₃_condition₃ | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span {i j j' : C} (f : i ⟶ j) (f' : i ⟶ j') :
∃ (k : C) (g : j ⟶ k) (g' : j' ⟶ k), f ≫ g = f' ≫ g' | let ⟨K, G, G', _⟩ := cocone_objs j j', ⟨k, e, he⟩ := cocone_maps (f ≫ G) (f' ≫ G') in
⟨k, G ≫ e, G' ≫ e, by simpa only [← category.assoc]⟩ | lemma | category_theory.is_filtered.span | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | For every span `j ⟵ i ⟶ j'`, there
exists a cocone `j ⟶ k ⟵ j'` such that the square commutes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bowtie {j₁ j₂ k₁ k₂ : C}
(f₁ : j₁ ⟶ k₁) (g₁ : j₁ ⟶ k₂) (f₂ : j₂ ⟶ k₁) (g₂ : j₂ ⟶ k₂) :
∃ (s : C) (α : k₁ ⟶ s) (β : k₂ ⟶ s), f₁ ≫ α = g₁ ≫ β ∧ f₂ ≫ α = g₂ ≫ β | begin
obtain ⟨t, k₁t, k₂t, ht⟩ := span f₁ g₁,
obtain ⟨s, ts, hs⟩ := cocone_maps (f₂ ≫ k₁t) (g₂ ≫ k₂t),
simp_rw category.assoc at hs,
exact ⟨s, k₁t ≫ ts, k₂t ≫ ts, by rw reassoc_of ht, hs⟩,
end | lemma | category_theory.is_filtered.bowtie | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | Given a "bowtie" of morphisms
```
j₁ j₂
|\ /|
| \/ |
| /\ |
|/ \∣
vv vv
k₁ k₂
```
in a filtered category, we can construct an object `s` and two morphisms from `k₁` and `k₂` to `s`,
making the resulting squares commute. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tulip {j₁ j₂ j₃ k₁ k₂ l : C} (f₁ : j₁ ⟶ k₁) (f₂ : j₂ ⟶ k₁) (f₃ : j₂ ⟶ k₂) (f₄ : j₃ ⟶ k₂)
(g₁ : j₁ ⟶ l) (g₂ : j₃ ⟶ l) :
∃ (s : C) (α : k₁ ⟶ s) (β : l ⟶ s) (γ : k₂ ⟶ s),
f₁ ≫ α = g₁ ≫ β ∧ f₂ ≫ α = f₃ ≫ γ ∧ f₄ ≫ γ = g₂ ≫ β | begin
obtain ⟨l', k₁l, k₂l, hl⟩ := span f₂ f₃,
obtain ⟨s, ls, l's, hs₁, hs₂⟩ := bowtie g₁ (f₁ ≫ k₁l) g₂ (f₄ ≫ k₂l),
refine ⟨s, k₁l ≫ l's, ls, k₂l ≫ l's, _, by rw reassoc_of hl, _⟩;
simp only [hs₁, hs₂, category.assoc],
end | lemma | category_theory.is_filtered.tulip | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | Given a "tulip" of morphisms
```
j₁ j₂ j₃
|\ / \ / |
| \ / \ / |
| vv vv |
\ k₁ k₂ /
\ /
\ /
\ /
\ /
v v
l
```
in a filtered category, we can construct an object `s` and three morphisms from `k₁`, `k₂` and `l`
to `s`, making the resulting squares... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_cofiltered_or_empty : Prop | (cone_objs : ∀ (X Y : C), ∃ W (f : W ⟶ X) (g : W ⟶ Y), true)
(cone_maps : ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), ∃ W (h : W ⟶ X), h ≫ f = h ≫ g) | class | category_theory.is_cofiltered_or_empty | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | A category `is_cofiltered_or_empty` if
1. for every pair of objects there exists another object "to the left", and
2. for every pair of parallel morphisms there exists a morphism to the left so the compositions
are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_cofiltered extends is_cofiltered_or_empty C : Prop | [nonempty : nonempty C] | class | category_theory.is_cofiltered | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | A category `is_cofiltered` if
1. for every pair of objects there exists another object "to the left",
2. for every pair of parallel morphisms there exists a morphism to the left so the compositions
are equal, and
3. there exists some object.
See <https://stacks.math.columbia.edu/tag/04AZ>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_cofiltered_or_empty_of_semilattice_inf
(α : Type u) [semilattice_inf α] : is_cofiltered_or_empty α | { cone_objs := λ X Y, ⟨X ⊓ Y, hom_of_le inf_le_left, hom_of_le inf_le_right, trivial⟩,
cone_maps := λ X Y f g, ⟨X, 𝟙 _, (by ext)⟩, } | instance | category_theory.is_cofiltered_or_empty_of_semilattice_inf | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [
"inf_le_left",
"inf_le_right",
"semilattice_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cofiltered_of_semilattice_inf_nonempty
(α : Type u) [semilattice_inf α] [nonempty α] : is_cofiltered α | {} | instance | category_theory.is_cofiltered_of_semilattice_inf_nonempty | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [
"semilattice_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cofiltered_or_empty_of_directed_ge (α : Type u) [preorder α]
[is_directed α (≥)] :
is_cofiltered_or_empty α | { cone_objs := λ X Y, let ⟨Z, hX, hY⟩ := exists_le_le X Y in
⟨Z, hom_of_le hX, hom_of_le hY, trivial⟩,
cone_maps := λ X Y f g, ⟨X, 𝟙 _, by simp⟩ } | instance | category_theory.is_cofiltered_or_empty_of_directed_ge | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [
"exists_le_le",
"is_directed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cofiltered_of_directed_ge_nonempty (α : Type u) [preorder α] [is_directed α (≥)]
[nonempty α] :
is_cofiltered α | {} | instance | category_theory.is_cofiltered_of_directed_ge_nonempty | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [
"is_directed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cone_objs : ∀ (X Y : C), ∃ W (f : W ⟶ X) (g : W ⟶ Y), true | is_cofiltered_or_empty.cone_objs | lemma | category_theory.is_cofiltered.cone_objs | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cone_maps : ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), ∃ W (h : W ⟶ X), h ≫ f = h ≫ g | is_cofiltered_or_empty.cone_maps | lemma | category_theory.is_cofiltered.cone_maps | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min (j j' : C) : C | (cone_objs j j').some | def | category_theory.is_cofiltered.min | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `min j j'` is an arbitrary choice of object to the left of both `j` and `j'`,
whose existence is ensured by `is_cofiltered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
min_to_left (j j' : C) : min j j' ⟶ j | (cone_objs j j').some_spec.some | def | category_theory.is_cofiltered.min_to_left | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `min_to_left j j'` is an arbitrary choice of morphism from `min j j'` to `j`,
whose existence is ensured by `is_cofiltered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
min_to_right (j j' : C) : min j j' ⟶ j' | (cone_objs j j').some_spec.some_spec.some | def | category_theory.is_cofiltered.min_to_right | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `min_to_right j j'` is an arbitrary choice of morphism from `min j j'` to `j'`,
whose existence is ensured by `is_cofiltered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq {j j' : C} (f f' : j ⟶ j') : C | (cone_maps f f').some | def | category_theory.is_cofiltered.eq | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `eq f f'`, for morphisms `f f' : j ⟶ j'`, is an arbitrary choice of object
which admits a morphism `eq_hom f f' : eq f f' ⟶ j` such that
`eq_condition : eq_hom f f' ≫ f = eq_hom f f' ≫ f'`.
Its existence is ensured by `is_cofiltered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_hom {j j' : C} (f f' : j ⟶ j') : eq f f' ⟶ j | (cone_maps f f').some_spec.some | def | category_theory.is_cofiltered.eq_hom | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `eq_hom f f'`, for morphisms `f f' : j ⟶ j'`, is an arbitrary choice of morphism
`eq_hom f f' : eq f f' ⟶ j` such that
`eq_condition : eq_hom f f' ≫ f = eq_hom f f' ≫ f'`.
Its existence is ensured by `is_cofiltered`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_condition {j j' : C} (f f' : j ⟶ j') : eq_hom f f' ≫ f = eq_hom f f' ≫ f' | (cone_maps f f').some_spec.some_spec | lemma | category_theory.is_cofiltered.eq_condition | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | `eq_condition f f'`, for morphisms `f f' : j ⟶ j'`, is the proof that
`eq_hom f f' ≫ f = eq_hom f f' ≫ f'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cospan {i j j' : C} (f : j ⟶ i) (f' : j' ⟶ i) :
∃ (k : C) (g : k ⟶ j) (g' : k ⟶ j'), g ≫ f = g' ≫ f' | let ⟨K, G, G', _⟩ := cone_objs j j', ⟨k, e, he⟩ := cone_maps (G ≫ f) (G' ≫ f') in
⟨k, e ≫ G, e ≫ G', by simpa only [category.assoc] using he⟩ | lemma | category_theory.is_cofiltered.cospan | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | For every cospan `j ⟶ i ⟵ j'`,
there exists a cone `j ⟵ k ⟶ j'` such that the square commutes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.category_theory.functor.ranges_directed (F : C ⥤ Type*) (j : C) :
directed (⊇) (λ (f : Σ' i, i ⟶ j), set.range (F.map f.2)) | λ ⟨i, ij⟩ ⟨k, kj⟩, let ⟨l, li, lk, e⟩ := cospan ij kj in
by refine ⟨⟨l, lk ≫ kj⟩, e ▸ _, _⟩; simp_rw F.map_comp; apply set.range_comp_subset_range | lemma | category_theory.functor.ranges_directed | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [
"directed",
"set.range",
"set.range_comp_subset_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_objs_exists (O : finset C) : ∃ (S : C), ∀ {X}, X ∈ O → _root_.nonempty (S ⟶ X) | begin
classical,
apply finset.induction_on O,
{ exact ⟨is_cofiltered.nonempty.some, (by rintros - ⟨⟩)⟩, },
{ rintros X O' nm ⟨S', w'⟩,
use min X S',
rintros Y mY,
obtain rfl|h := eq_or_ne Y X,
{ exact ⟨min_to_left _ _⟩, },
{ exact ⟨min_to_right _ _ ≫ (w' (finset.mem_of_mem_insert_of_ne mY h)... | lemma | category_theory.is_cofiltered.inf_objs_exists | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [
"eq_or_ne",
"finset",
"finset.induction_on",
"finset.mem_of_mem_insert_of_ne"
] | Any finite collection of objects in a cofiltered category has an object "to the left". | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_exists :
∃ (S : C) (T : Π {X : C}, X ∈ O → (S ⟶ X)), ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y},
(⟨X, Y, mX, mY, f⟩ : (Σ' (X Y : C) (mX : X ∈ O) (mY : Y ∈ O), X ⟶ Y)) ∈ H → T mX ≫ f = T mY | begin
classical,
apply finset.induction_on H,
{ obtain ⟨S, f⟩ := inf_objs_exists O,
refine ⟨S, λ X mX, (f mX).some, _⟩,
rintros - - - - - ⟨⟩, },
{ rintros ⟨X, Y, mX, mY, f⟩ H' nmf ⟨S', T', w'⟩,
refine ⟨eq (T' mX ≫ f) (T' mY), λ Z mZ, eq_hom (T' mX ≫ f) (T' mY) ≫ T' mZ, _⟩,
intros X' Y' mX' mY' f... | lemma | category_theory.is_cofiltered.inf_exists | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [
"finset.induction_on",
"finset.mem_of_mem_insert_of_ne",
"heq_iff_eq",
"psigma.mk.inj_iff"
] | Given any `finset` of objects `{X, ...}` and
indexed collection of `finset`s of morphisms `{f, ...}` in `C`,
there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `finset`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf : C | (inf_exists O H).some | def | category_theory.is_cofiltered.inf | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | An arbitrary choice of object "to the left"
of a finite collection of objects `O` and morphisms `H`,
making all the triangles commute. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_to {X : C} (m : X ∈ O) :
inf O H ⟶ X | (inf_exists O H).some_spec.some m | def | category_theory.is_cofiltered.inf_to | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | The morphisms from `inf O H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_to_commutes
{X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y}
(mf : (⟨X, Y, mX, mY, f⟩ : Σ' (X Y : C) (mX : X ∈ O) (mY : Y ∈ O), X ⟶ Y) ∈ H) :
inf_to O H mX ≫ f = inf_to O H mY | (inf_exists O H).some_spec.some_spec mX mY mf | lemma | category_theory.is_cofiltered.inf_to_commutes | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | The triangles consisting of a morphism in `H` and the maps from `inf O H` commute. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_nonempty (F : J ⥤ C) : _root_.nonempty (cone F) | begin
classical,
let O := (finset.univ.image F.obj),
let H : finset (Σ' (X Y : C) (mX : X ∈ O) (mY : Y ∈ O), X ⟶ Y) :=
finset.univ.bUnion (λ X : J, finset.univ.bUnion (λ Y : J, finset.univ.image (λ f : X ⟶ Y,
⟨F.obj X, F.obj Y, by simp, by simp, F.map f⟩))),
obtain ⟨Z, f, w⟩ := inf_exists O H,
refin... | lemma | category_theory.is_cofiltered.cone_nonempty | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [
"exists_and_distrib_left",
"exists_prop_of_true",
"finset",
"finset.mem_bUnion",
"finset.mem_image",
"finset.mem_univ"
] | If we have `is_cofiltered C`, then for any functor `F : J ⥤ C` with `fin_category J`,
there exists a cone over `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone (F : J ⥤ C) : cone F | (cone_nonempty F).some | def | category_theory.is_cofiltered.cone | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | An arbitrary choice of cone over `F : J ⥤ C`, for `fin_category J` and `is_cofiltered C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_left_adjoint {L : C ⥤ D} {R : D ⥤ C} (h : L ⊣ R) : is_cofiltered D | { cone_objs := λ X Y,
⟨L.obj (min (R.obj X) (R.obj Y)),
(h.hom_equiv _ X).symm (min_to_left _ _), (h.hom_equiv _ Y).symm (min_to_right _ _), ⟨⟩⟩,
cone_maps := λ X Y f g,
⟨L.obj (eq (R.map f) (R.map g)), (h.hom_equiv _ _).symm (eq_hom _ _),
by rw [← h.hom_equiv_naturality_right_symm, ← h.hom_equiv_n... | lemma | category_theory.is_cofiltered.of_left_adjoint | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | If `C` is cofiltered, and we have a functor `L : C ⥤ D` with a right adjoint,
then `D` is cofiltered. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_left_adjoint (L : C ⥤ D) [is_left_adjoint L] : is_cofiltered D | of_left_adjoint (adjunction.of_left_adjoint L) | lemma | category_theory.is_cofiltered.of_is_left_adjoint | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | If `C` is cofiltered, and we have a left adjoint functor `L : C ⥤ D`, then `D` is cofiltered. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_equivalence (h : C ≌ D) : is_cofiltered D | of_left_adjoint h.to_adjunction | lemma | category_theory.is_cofiltered.of_equivalence | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | Being cofiltered is preserved by equivalence of categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_cofiltered_op_of_is_filtered [is_filtered C] : is_cofiltered Cᵒᵖ | { cone_objs := λ X Y, ⟨op (is_filtered.max X.unop Y.unop),
(is_filtered.left_to_max _ _).op, (is_filtered.right_to_max _ _).op, trivial⟩,
cone_maps := λ X Y f g, ⟨op (is_filtered.coeq f.unop g.unop),
(is_filtered.coeq_hom _ _).op, begin
rw [(show f = f.unop.op, by simp), (show g = g.unop.op, by simp),
... | instance | category_theory.is_cofiltered_op_of_is_filtered | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_filtered_op_of_is_cofiltered [is_cofiltered C] : is_filtered Cᵒᵖ | { cocone_objs := λ X Y, ⟨op (is_cofiltered.min X.unop Y.unop),
(is_cofiltered.min_to_left X.unop Y.unop).op,
(is_cofiltered.min_to_right X.unop Y.unop).op, trivial⟩,
cocone_maps := λ X Y f g, ⟨op (is_cofiltered.eq f.unop g.unop),
(is_cofiltered.eq_hom f.unop g.unop).op, begin
rw [(show f = f.unop.op... | instance | category_theory.is_filtered_op_of_is_cofiltered | category_theory | src/category_theory/filtered.lean | [
"category_theory.fin_category",
"category_theory.limits.cones",
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Fintype | bundled fintype | def | Fintype | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [
"fintype"
] | The category of finite types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (X : Type*) [fintype X] : Fintype | bundled.of X | def | Fintype.of | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [
"Fintype",
"fintype"
] | Construct a bundled `Fintype` from the underlying type and typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
incl : Fintype ⥤ Type* | induced_functor _ | def | Fintype.incl | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [
"Fintype"
] | The fully faithful embedding of `Fintype` into the category of types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concrete_category_Fintype : concrete_category Fintype | ⟨incl⟩ | instance | Fintype.concrete_category_Fintype | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [
"Fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (X : Fintype) (x : X) : (𝟙 X : X → X) x = x | rfl | lemma | Fintype.id_apply | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [
"Fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply {X Y Z : Fintype} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
(f ≫ g) x = g (f x) | rfl | lemma | Fintype.comp_apply | category_theory | src/category_theory/Fintype.lean | [
"category_theory.concrete_category.basic",
"category_theory.full_subcategory",
"category_theory.skeletal",
"category_theory.elementwise",
"data.fintype.card"
] | [
"Fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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