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of {X : Type*} (to_prod : X × X) : Bipointed | ⟨X, to_prod⟩ | def | Bipointed.of | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed"
] | Turns a bipointing into a bipointed type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of {X : Type*} (to_prod : X × X) : ↥(of to_prod) = X | rfl | lemma | Bipointed.coe_of | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom (X Y : Bipointed.{u}) : Type u | (to_fun : X → Y)
(map_fst : to_fun X.to_prod.1 = Y.to_prod.1)
(map_snd : to_fun X.to_prod.2 = Y.to_prod.2) | structure | Bipointed.hom | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [] | Morphisms in `Bipointed`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id (X : Bipointed) : hom X X | ⟨id, rfl, rfl⟩ | def | Bipointed.hom.id | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed"
] | The identity morphism of `X : Bipointed`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp {X Y Z : Bipointed.{u}} (f : hom X Y) (g : hom Y Z) : hom X Z | ⟨g.to_fun ∘ f.to_fun, by rw [function.comp_apply, f.map_fst, g.map_fst],
by rw [function.comp_apply, f.map_snd, g.map_snd]⟩ | def | Bipointed.hom.comp | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"function.comp_apply"
] | Composition of morphisms of `Bipointed`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
large_category : large_category Bipointed | { hom := hom,
id := hom.id,
comp := @hom.comp,
id_comp' := λ _ _ _, hom.ext _ _ rfl,
comp_id' := λ _ _ _, hom.ext _ _ rfl,
assoc' := λ _ _ _ _ _ _ _, hom.ext _ _ rfl } | instance | Bipointed.large_category | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category : concrete_category Bipointed | { forget := { obj := Bipointed.X, map := @hom.to_fun },
forget_faithful := ⟨@hom.ext⟩ } | instance | Bipointed.concrete_category | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
swap : Bipointed ⥤ Bipointed | { obj := λ X, ⟨X, X.to_prod.swap⟩, map := λ X Y f, ⟨f.to_fun, f.map_snd, f.map_fst⟩ } | def | Bipointed.swap | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed"
] | Swaps the pointed elements of a bipointed type. `prod.swap` as a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
swap_equiv : Bipointed ≌ Bipointed | equivalence.mk swap swap
(nat_iso.of_components (λ X, { hom := ⟨id, rfl, rfl⟩, inv := ⟨id, rfl, rfl⟩ }) $ λ X Y f, rfl)
(nat_iso.of_components (λ X, { hom := ⟨id, rfl, rfl⟩, inv := ⟨id, rfl, rfl⟩ }) $ λ X Y f, rfl) | def | Bipointed.swap_equiv | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed"
] | The equivalence between `Bipointed` and itself induced by `prod.swap` both ways. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
swap_equiv_symm : swap_equiv.symm = swap_equiv | rfl | lemma | Bipointed.swap_equiv_symm | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Bipointed_to_Pointed_fst : Bipointed ⥤ Pointed | { obj := λ X, ⟨X, X.to_prod.1⟩, map := λ X Y f, ⟨f.to_fun, f.map_fst⟩ } | def | Bipointed_to_Pointed_fst | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed",
"Pointed"
] | The forgetful functor from `Bipointed` to `Pointed` which forgets about the second point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Bipointed_to_Pointed_snd : Bipointed ⥤ Pointed | { obj := λ X, ⟨X, X.to_prod.2⟩, map := λ X Y f, ⟨f.to_fun, f.map_snd⟩ } | def | Bipointed_to_Pointed_snd | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed",
"Pointed"
] | The forgetful functor from `Bipointed` to `Pointed` which forgets about the first point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Bipointed_to_Pointed_fst_comp_forget :
Bipointed_to_Pointed_fst ⋙ forget Pointed = forget Bipointed | rfl | lemma | Bipointed_to_Pointed_fst_comp_forget | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed",
"Bipointed_to_Pointed_fst",
"Pointed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Bipointed_to_Pointed_snd_comp_forget :
Bipointed_to_Pointed_snd ⋙ forget Pointed = forget Bipointed | rfl | lemma | Bipointed_to_Pointed_snd_comp_forget | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed",
"Bipointed_to_Pointed_snd",
"Pointed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
swap_comp_Bipointed_to_Pointed_fst :
Bipointed.swap ⋙ Bipointed_to_Pointed_fst = Bipointed_to_Pointed_snd | rfl | lemma | swap_comp_Bipointed_to_Pointed_fst | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed.swap",
"Bipointed_to_Pointed_fst",
"Bipointed_to_Pointed_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
swap_comp_Bipointed_to_Pointed_snd :
Bipointed.swap ⋙ Bipointed_to_Pointed_snd = Bipointed_to_Pointed_fst | rfl | lemma | swap_comp_Bipointed_to_Pointed_snd | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed.swap",
"Bipointed_to_Pointed_fst",
"Bipointed_to_Pointed_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pointed_to_Bipointed : Pointed.{u} ⥤ Bipointed | { obj := λ X, ⟨X, X.point, X.point⟩, map := λ X Y f, ⟨f.to_fun, f.map_point, f.map_point⟩ } | def | Pointed_to_Bipointed | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed"
] | The functor from `Pointed` to `Bipointed` which bipoints the point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pointed_to_Bipointed_fst : Pointed.{u} ⥤ Bipointed | { obj := λ X, ⟨option X, X.point, none⟩,
map := λ X Y f, ⟨option.map f.to_fun, congr_arg _ f.map_point, rfl⟩,
map_id' := λ X, Bipointed.hom.ext _ _ option.map_id,
map_comp' := λ X Y Z f g, Bipointed.hom.ext _ _ (option.map_comp_map _ _).symm } | def | Pointed_to_Bipointed_fst | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed",
"option.map_comp_map"
] | The functor from `Pointed` to `Bipointed` which adds a second point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pointed_to_Bipointed_snd : Pointed.{u} ⥤ Bipointed | { obj := λ X, ⟨option X, none, X.point⟩,
map := λ X Y f, ⟨option.map f.to_fun, rfl, congr_arg _ f.map_point⟩,
map_id' := λ X, Bipointed.hom.ext _ _ option.map_id,
map_comp' := λ X Y Z f g, Bipointed.hom.ext _ _ (option.map_comp_map _ _).symm } | def | Pointed_to_Bipointed_snd | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed",
"option.map_comp_map"
] | The functor from `Pointed` to `Bipointed` which adds a first point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pointed_to_Bipointed_fst_comp_swap :
Pointed_to_Bipointed_fst ⋙ Bipointed.swap = Pointed_to_Bipointed_snd | rfl | lemma | Pointed_to_Bipointed_fst_comp_swap | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed.swap",
"Pointed_to_Bipointed_fst",
"Pointed_to_Bipointed_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pointed_to_Bipointed_snd_comp_swap :
Pointed_to_Bipointed_snd ⋙ Bipointed.swap = Pointed_to_Bipointed_fst | rfl | lemma | Pointed_to_Bipointed_snd_comp_swap | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed.swap",
"Pointed_to_Bipointed_fst",
"Pointed_to_Bipointed_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pointed_to_Bipointed_comp_Bipointed_to_Pointed_fst :
Pointed_to_Bipointed ⋙ Bipointed_to_Pointed_fst ≅ 𝟭 _ | nat_iso.of_components (λ X, { hom := ⟨id, rfl⟩, inv := ⟨id, rfl⟩ }) $ λ X Y f, rfl | def | Pointed_to_Bipointed_comp_Bipointed_to_Pointed_fst | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed_to_Pointed_fst",
"Pointed_to_Bipointed"
] | `Bipointed_to_Pointed_fst` is inverse to `Pointed_to_Bipointed`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pointed_to_Bipointed_comp_Bipointed_to_Pointed_snd :
Pointed_to_Bipointed ⋙ Bipointed_to_Pointed_snd ≅ 𝟭 _ | nat_iso.of_components (λ X, { hom := ⟨id, rfl⟩, inv := ⟨id, rfl⟩ }) $ λ X Y f, rfl | def | Pointed_to_Bipointed_comp_Bipointed_to_Pointed_snd | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed_to_Pointed_snd",
"Pointed_to_Bipointed"
] | `Bipointed_to_Pointed_snd` is inverse to `Pointed_to_Bipointed`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pointed_to_Bipointed_fst_Bipointed_to_Pointed_fst_adjunction :
Pointed_to_Bipointed_fst ⊣ Bipointed_to_Pointed_fst | adjunction.mk_of_hom_equiv
{ hom_equiv := λ X Y, { to_fun := λ f, ⟨f.to_fun ∘ option.some, f.map_fst⟩,
inv_fun := λ f, ⟨λ o, o.elim Y.to_prod.2 f.to_fun, f.map_point, rfl⟩,
left_inv := λ f, by { ext, cases x, exact f.map_snd.symm, refl },
right_inv... | def | Pointed_to_Bipointed_fst_Bipointed_to_Pointed_fst_adjunction | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed_to_Pointed_fst",
"Pointed_to_Bipointed_fst",
"inv_fun"
] | The free/forgetful adjunction between `Pointed_to_Bipointed_fst` and `Bipointed_to_Pointed_fst`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pointed_to_Bipointed_snd_Bipointed_to_Pointed_snd_adjunction :
Pointed_to_Bipointed_snd ⊣ Bipointed_to_Pointed_snd | adjunction.mk_of_hom_equiv
{ hom_equiv := λ X Y, { to_fun := λ f, ⟨f.to_fun ∘ option.some, f.map_snd⟩,
inv_fun := λ f, ⟨λ o, o.elim Y.to_prod.1 f.to_fun, rfl, f.map_point⟩,
left_inv := λ f, by { ext, cases x, exact f.map_fst.symm, refl },
right_inv... | def | Pointed_to_Bipointed_snd_Bipointed_to_Pointed_snd_adjunction | category_theory.category | src/category_theory/category/Bipointed.lean | [
"category_theory.category.Pointed"
] | [
"Bipointed_to_Pointed_snd",
"Pointed_to_Bipointed_snd",
"inv_fun"
] | The free/forgetful adjunction between `Pointed_to_Bipointed_snd` and `Bipointed_to_Pointed_snd`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Cat | bundled category.{v u} | def | category_theory.Cat | category_theory.category | src/category_theory/category/Cat.lean | [
"category_theory.concrete_category.bundled",
"category_theory.discrete_category",
"category_theory.types",
"category_theory.bicategory.strict"
] | [] | Category of categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
str (C : Cat.{v u}) : category.{v u} C | C.str | instance | category_theory.Cat.str | category_theory.category | src/category_theory/category/Cat.lean | [
"category_theory.concrete_category.bundled",
"category_theory.discrete_category",
"category_theory.types",
"category_theory.bicategory.strict"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (C : Type u) [category.{v} C] : Cat.{v u} | bundled.of C | def | category_theory.Cat.of | category_theory.category | src/category_theory/category/Cat.lean | [
"category_theory.concrete_category.bundled",
"category_theory.discrete_category",
"category_theory.types",
"category_theory.bicategory.strict"
] | [] | Construct a bundled `Cat` from the underlying type and the typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bicategory : bicategory.{(max v u) (max v u)} Cat.{v u} | { hom := λ C D, C ⥤ D,
id := λ C, 𝟭 C,
comp := λ C D E F G, F ⋙ G,
hom_category := λ C D, functor.category C D,
whisker_left := λ C D E F G H η, whisker_left F η,
whisker_right := λ C D E F G η H, whisker_right η H,
associator := λ A B C D, functor.associator,
left_unitor := λ A B, functor.left_unitor,
... | instance | category_theory.Cat.bicategory | category_theory.category | src/category_theory/category/Cat.lean | [
"category_theory.concrete_category.bundled",
"category_theory.discrete_category",
"category_theory.types",
"category_theory.bicategory.strict"
] | [] | Bicategory structure on `Cat` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bicategory.strict : bicategory.strict Cat.{v u} | { id_comp' := λ C D F, by cases F; refl,
comp_id' := λ C D F, by cases F; refl,
assoc' := by intros; refl } | instance | category_theory.Cat.bicategory.strict | category_theory.category | src/category_theory/category/Cat.lean | [
"category_theory.concrete_category.bundled",
"category_theory.discrete_category",
"category_theory.types",
"category_theory.bicategory.strict"
] | [] | `Cat` is a strict bicategory. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category : large_category.{max v u} Cat.{v u} | strict_bicategory.category Cat.{v u} | instance | category_theory.Cat.category | category_theory.category | src/category_theory/category/Cat.lean | [
"category_theory.concrete_category.bundled",
"category_theory.discrete_category",
"category_theory.types",
"category_theory.bicategory.strict"
] | [] | Category structure on `Cat` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_map {C : Cat} {X Y : C} (f : X ⟶ Y) : (𝟙 C : C ⥤ C).map f = f | functor.id_map f | lemma | category_theory.Cat.id_map | category_theory.category | src/category_theory/category/Cat.lean | [
"category_theory.concrete_category.bundled",
"category_theory.discrete_category",
"category_theory.types",
"category_theory.bicategory.strict"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_obj {C D E : Cat} (F : C ⟶ D) (G : D ⟶ E) (X : C) :
(F ≫ G).obj X = G.obj (F.obj X) | functor.comp_obj F G X | lemma | category_theory.Cat.comp_obj | category_theory.category | src/category_theory/category/Cat.lean | [
"category_theory.concrete_category.bundled",
"category_theory.discrete_category",
"category_theory.types",
"category_theory.bicategory.strict"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_map {C D E : Cat} (F : C ⟶ D) (G : D ⟶ E) {X Y : C} (f : X ⟶ Y) :
(F ≫ G).map f = G.map (F.map f) | functor.comp_map F G f | lemma | category_theory.Cat.comp_map | category_theory.category | src/category_theory/category/Cat.lean | [
"category_theory.concrete_category.bundled",
"category_theory.discrete_category",
"category_theory.types",
"category_theory.bicategory.strict"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
objects : Cat.{v u} ⥤ Type u | { obj := λ C, C,
map := λ C D F, F.obj } | def | category_theory.Cat.objects | category_theory.category | src/category_theory/category/Cat.lean | [
"category_theory.concrete_category.bundled",
"category_theory.discrete_category",
"category_theory.types",
"category_theory.bicategory.strict"
] | [] | Functor that gets the set of objects of a category. It is not
called `forget`, because it is not a faithful functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_iso {C D : Cat} (γ : C ≅ D) : C ≌ D | { functor := γ.hom,
inverse := γ.inv,
unit_iso := eq_to_iso $ eq.symm γ.hom_inv_id,
counit_iso := eq_to_iso γ.inv_hom_id } | def | category_theory.Cat.equiv_of_iso | category_theory.category | src/category_theory/category/Cat.lean | [
"category_theory.concrete_category.bundled",
"category_theory.discrete_category",
"category_theory.types",
"category_theory.bicategory.strict"
] | [] | Any isomorphism in `Cat` induces an equivalence of the underlying categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Type_to_Cat : Type u ⥤ Cat | { obj := λ X, Cat.of (discrete X),
map := λ X Y f, discrete.functor (discrete.mk ∘ f),
map_id' := λ X, begin apply functor.ext, tidy, end,
map_comp' := λ X Y Z f g, begin apply functor.ext, tidy, end } | def | category_theory.Type_to_Cat | category_theory.category | src/category_theory/category/Cat.lean | [
"category_theory.concrete_category.bundled",
"category_theory.discrete_category",
"category_theory.types",
"category_theory.bicategory.strict"
] | [
"functor.ext"
] | Embedding `Type` into `Cat` as discrete categories.
This ought to be modelled as a 2-functor! | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
galois_connection.adjunction {l : X → Y} {u : Y → X} (gc : galois_connection l u) :
gc.monotone_l.functor ⊣ gc.monotone_u.functor | category_theory.adjunction.mk_of_hom_equiv
{ hom_equiv := λ X Y, ⟨λ f, (gc.le_u f.le).hom, λ f, (gc.l_le f.le).hom, by tidy, by tidy⟩ } | def | galois_connection.adjunction | category_theory.category | src/category_theory/category/galois_connection.lean | [
"category_theory.category.preorder",
"category_theory.adjunction.basic",
"order.galois_connection"
] | [
"category_theory.adjunction.mk_of_hom_equiv",
"galois_connection"
] | A galois connection between preorders induces an adjunction between the associated categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjunction.gc {L : X ⥤ Y} {R : Y ⥤ X} (adj : L ⊣ R) :
galois_connection L.obj R.obj | λ x y, ⟨λ h, ((adj.hom_equiv x y).to_fun h.hom).le, λ h, ((adj.hom_equiv x y).inv_fun h.hom).le⟩ | lemma | category_theory.adjunction.gc | category_theory.category | src/category_theory/category/galois_connection.lean | [
"category_theory.category.preorder",
"category_theory.adjunction.basic",
"order.galois_connection"
] | [
"adj",
"galois_connection",
"inv_fun"
] | An adjunction between preorder categories induces a galois connection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Groupoid | bundled groupoid.{v u} | def | category_theory.Groupoid | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [] | Category of groupoids | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
str (C : Groupoid.{v u}) : groupoid.{v u} C.α | C.str | instance | category_theory.Groupoid.str | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (C : Type u) [groupoid.{v} C] : Groupoid.{v u} | bundled.of C | def | category_theory.Groupoid.of | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [] | Construct a bundled `Groupoid` from the underlying type and the typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of (C : Type u) [groupoid C] : (of C : Type u) = C | rfl | lemma | category_theory.Groupoid.coe_of | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
category : large_category.{max v u} Groupoid.{v u} | { hom := λ C D, C ⥤ D,
id := λ C, 𝟭 C,
comp := λ C D E F G, F ⋙ G,
id_comp' := λ C D F, by cases F; refl,
comp_id' := λ C D F, by cases F; refl,
assoc' := by intros; refl } | instance | category_theory.Groupoid.category | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [] | Category structure on `Groupoid` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
objects : Groupoid.{v u} ⥤ Type u | { obj := bundled.α,
map := λ C D F, F.obj } | def | category_theory.Groupoid.objects | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [] | Functor that gets the set of objects of a groupoid. It is not
called `forget`, because it is not a faithful functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_to_Cat : Groupoid.{v u} ⥤ Cat.{v u} | { obj := λ C, Cat.of C,
map := λ C D, id } | def | category_theory.Groupoid.forget_to_Cat | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [] | Forgetting functor to `Cat` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_to_Cat_full : full forget_to_Cat | { preimage := λ C D, id } | instance | category_theory.Groupoid.forget_to_Cat_full | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_to_Cat_faithful : faithful forget_to_Cat | { } | instance | category_theory.Groupoid.forget_to_Cat_faithful | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_to_functor {C D E : Groupoid.{v u}} (f : C ⟶ D) (g : D ⟶ E) : f ≫ g = f ⋙ g | rfl | lemma | category_theory.Groupoid.hom_to_functor | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [] | Convert arrows in the category of groupoids to functors,
which sometimes helps in applying simp lemmas | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_to_functor {C : Groupoid.{v u}} : 𝟭 C = 𝟙 C | rfl | lemma | category_theory.Groupoid.id_to_functor | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [] | Converts identity in the category of groupoids to the functor identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_limit_fan ⦃J : Type u⦄ (F : J → Groupoid.{u u}) : limits.fan F | limits.fan.mk (@of (Π j : J, F j) _) (λ j, category_theory.pi.eval _ j) | def | category_theory.Groupoid.pi_limit_fan | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [
"category_theory.pi.eval"
] | Construct the product over an indexed family of groupoids, as a fan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_limit_fan_is_limit ⦃J : Type u⦄ (F : J → Groupoid.{u u}) :
limits.is_limit (pi_limit_fan F) | limits.mk_fan_limit (pi_limit_fan F)
(λ s, functor.pi' (λ j, s.proj j))
(by { intros, dunfold pi_limit_fan, simp [hom_to_functor], })
begin
intros s m w,
apply functor.pi_ext,
intro j, specialize w j,
simpa,
end | def | category_theory.Groupoid.pi_limit_fan_is_limit | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [] | The product fan over an indexed family of groupoids, is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pi : limits.has_products Groupoid.{u u} | limits.has_products_of_limit_fans pi_limit_fan pi_limit_fan_is_limit | instance | category_theory.Groupoid.has_pi | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_iso_pi (J : Type u) (f : J → Groupoid.{u u}) : @of (Π j, f j) _ ≅ ∏ f | limits.is_limit.cone_point_unique_up_to_iso
(pi_limit_fan_is_limit f)
(limits.limit.is_limit (discrete.functor f)) | def | category_theory.Groupoid.pi_iso_pi | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [] | The product of a family of groupoids is isomorphic
to the product object in the category of Groupoids | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_iso_pi_hom_π (J : Type u) (f : J → Groupoid.{u u}) (j : J) :
(pi_iso_pi J f).hom ≫ (limits.pi.π f j) = category_theory.pi.eval _ j | by { simp [pi_iso_pi], refl, } | lemma | category_theory.Groupoid.pi_iso_pi_hom_π | category_theory.category | src/category_theory/category/Groupoid.lean | [
"category_theory.single_obj",
"category_theory.limits.shapes.products",
"category_theory.pi.basic",
"category_theory.limits.is_limit"
] | [
"category_theory.pi.eval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Kleisli (m : Type u → Type v) | Type u | def | category_theory.Kleisli | category_theory.category | src/category_theory/category/Kleisli.lean | [
"category_theory.category.basic"
] | [] | The Kleisli category on the (type-)monad `m`. Note that the monad is not assumed to be lawful
yet. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Kleisli.mk (m) (α : Type u) : Kleisli m | α | def | category_theory.Kleisli.mk | category_theory.category | src/category_theory/category/Kleisli.lean | [
"category_theory.category.basic"
] | [] | Construct an object of the Kleisli category from a type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Kleisli.category_struct {m} [monad.{u v} m] : category_struct (Kleisli m) | { hom := λ α β, α → m β,
id := λ α x, pure x,
comp := λ X Y Z f g, f >=> g } | instance | category_theory.Kleisli.category_struct | category_theory.category | src/category_theory/category/Kleisli.lean | [
"category_theory.category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Kleisli.category {m} [monad.{u v} m] [is_lawful_monad m] : category (Kleisli m) | by refine { id_comp' := _, comp_id' := _, assoc' := _ };
intros; ext; unfold_projs; simp only [(>=>)] with functor_norm | instance | category_theory.Kleisli.category | category_theory.category | src/category_theory/category/Kleisli.lean | [
"category_theory.category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Kleisli.id_def {m} [monad m] (α : Kleisli m) :
𝟙 α = @pure m _ α | rfl | lemma | category_theory.Kleisli.id_def | category_theory.category | src/category_theory/category/Kleisli.lean | [
"category_theory.category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Kleisli.comp_def {m} [monad m] (α β γ : Kleisli m)
(xs : α ⟶ β) (ys : β ⟶ γ) (a : α) :
(xs ≫ ys) a = xs a >>= ys | rfl | lemma | category_theory.Kleisli.comp_def | category_theory.category | src/category_theory/category/Kleisli.lean | [
"category_theory.category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pairwise (ι : Type v)
| single : ι → pairwise
| pair : ι → ι → pairwise | inductive | category_theory.pairwise | category_theory.category | src/category_theory/category/pairwise.lean | [
"order.complete_lattice",
"category_theory.category.preorder",
"category_theory.limits.is_limit"
] | [
"pairwise"
] | An inductive type representing either a single term of a type `ι`, or a pair of terms.
We use this as the objects of a category to describe the sheaf condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pairwise_inhabited [inhabited ι] : inhabited (pairwise ι) | ⟨single default⟩ | instance | category_theory.pairwise.pairwise_inhabited | category_theory.category | src/category_theory/category/pairwise.lean | [
"order.complete_lattice",
"category_theory.category.preorder",
"category_theory.limits.is_limit"
] | [
"pairwise"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom : pairwise ι → pairwise ι → Type v
| id_single : Π i, hom (single i) (single i)
| id_pair : Π i j, hom (pair i j) (pair i j)
| left : Π i j, hom (pair i j) (single i)
| right : Π i j, hom (pair i j) (single j) | inductive | category_theory.pairwise.hom | category_theory.category | src/category_theory/category/pairwise.lean | [
"order.complete_lattice",
"category_theory.category.preorder",
"category_theory.limits.is_limit"
] | [
"pairwise"
] | Morphisms in the category `pairwise ι`. The only non-identity morphisms are
`left i j : single i ⟶ pair i j` and `right i j : single j ⟶ pair i j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_inhabited [inhabited ι] : inhabited (hom (single (default : ι)) (single default)) | ⟨id_single default⟩ | instance | category_theory.pairwise.hom_inhabited | category_theory.category | src/category_theory/category/pairwise.lean | [
"order.complete_lattice",
"category_theory.category.preorder",
"category_theory.limits.is_limit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : Π (o : pairwise ι), hom o o | | (single i) := id_single i
| (pair i j) := id_pair i j | def | category_theory.pairwise.id | category_theory.category | src/category_theory/category/pairwise.lean | [
"order.complete_lattice",
"category_theory.category.preorder",
"category_theory.limits.is_limit"
] | [
"pairwise"
] | The identity morphism in `pairwise ι`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp : Π {o₁ o₂ o₃ : pairwise ι} (f : hom o₁ o₂) (g : hom o₂ o₃), hom o₁ o₃ | | _ _ _ (id_single i) g := g
| _ _ _ (id_pair i j) g := g
| _ _ _ (left i j) (id_single _) := left i j
| _ _ _ (right i j) (id_single _) := right i j | def | category_theory.pairwise.comp | category_theory.category | src/category_theory/category/pairwise.lean | [
"order.complete_lattice",
"category_theory.category.preorder",
"category_theory.limits.is_limit"
] | [
"pairwise"
] | Composition of morphisms in `pairwise ι`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram_obj : pairwise ι → α | | (single i) := U i
| (pair i j) := U i ⊓ U j | def | category_theory.pairwise.diagram_obj | category_theory.category | src/category_theory/category/pairwise.lean | [
"order.complete_lattice",
"category_theory.category.preorder",
"category_theory.limits.is_limit"
] | [
"pairwise"
] | Auxiliary definition for `diagram`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram_map : Π {o₁ o₂ : pairwise ι} (f : o₁ ⟶ o₂), diagram_obj U o₁ ⟶ diagram_obj U o₂ | | _ _ (id_single i) := 𝟙 _
| _ _ (id_pair i j) := 𝟙 _
| _ _ (left i j) := hom_of_le inf_le_left
| _ _ (right i j) := hom_of_le inf_le_right | def | category_theory.pairwise.diagram_map | category_theory.category | src/category_theory/category/pairwise.lean | [
"order.complete_lattice",
"category_theory.category.preorder",
"category_theory.limits.is_limit"
] | [
"inf_le_left",
"inf_le_right",
"pairwise"
] | Auxiliary definition for `diagram`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram : pairwise ι ⥤ α | { obj := diagram_obj U,
map := λ X Y f, diagram_map U f, } | def | category_theory.pairwise.diagram | category_theory.category | src/category_theory/category/pairwise.lean | [
"order.complete_lattice",
"category_theory.category.preorder",
"category_theory.limits.is_limit"
] | [
"pairwise"
] | Given a function `U : ι → α` for `[semilattice_inf α]`, we obtain a functor `pairwise ι ⥤ α`,
sending `single i` to `U i` and `pair i j` to `U i ⊓ U j`,
and the morphisms to the obvious inequalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_ι_app : Π (o : pairwise ι), diagram_obj U o ⟶ supr U | | (single i) := hom_of_le (le_supr U i)
| (pair i j) := hom_of_le inf_le_left ≫ hom_of_le (le_supr U i) | def | category_theory.pairwise.cocone_ι_app | category_theory.category | src/category_theory/category/pairwise.lean | [
"order.complete_lattice",
"category_theory.category.preorder",
"category_theory.limits.is_limit"
] | [
"inf_le_left",
"le_supr",
"pairwise",
"supr"
] | Auxiliary definition for `cocone`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone : cocone (diagram U) | { X := supr U,
ι := { app := cocone_ι_app U, } } | def | category_theory.pairwise.cocone | category_theory.category | src/category_theory/category/pairwise.lean | [
"order.complete_lattice",
"category_theory.category.preorder",
"category_theory.limits.is_limit"
] | [
"supr"
] | Given a function `U : ι → α` for `[complete_lattice α]`,
`supr U` provides a cocone over `diagram U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_is_colimit : is_colimit (cocone U) | { desc := λ s, hom_of_le
begin
apply complete_lattice.Sup_le,
rintros _ ⟨j, rfl⟩,
exact (s.ι.app (single j)).le
end } | def | category_theory.pairwise.cocone_is_colimit | category_theory.category | src/category_theory/category/pairwise.lean | [
"order.complete_lattice",
"category_theory.category.preorder",
"category_theory.limits.is_limit"
] | [] | Given a function `U : ι → α` for `[complete_lattice α]`,
`infi U` provides a limit cone over `diagram U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
PartialFun : Type* | Type* | def | PartialFun | category_theory.category | src/category_theory/category/PartialFun.lean | [
"category_theory.category.Pointed",
"data.pfun"
] | [] | The category of types equipped with partial functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of : Type* → PartialFun | id | def | PartialFun.of | category_theory.category | src/category_theory/category/PartialFun.lean | [
"category_theory.category.Pointed",
"data.pfun"
] | [
"PartialFun"
] | Turns a type into a `PartialFun`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of (X : Type*) : ↥(of X) = X | rfl | lemma | PartialFun.coe_of | category_theory.category | src/category_theory/category/PartialFun.lean | [
"category_theory.category.Pointed",
"data.pfun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
large_category : large_category.{u} PartialFun | { hom := pfun,
id := pfun.id,
comp := λ X Y Z f g, g.comp f,
id_comp' := @pfun.comp_id,
comp_id' := @pfun.id_comp,
assoc' := λ W X Y Z _ _ _, (pfun.comp_assoc _ _ _).symm } | instance | PartialFun.large_category | category_theory.category | src/category_theory/category/PartialFun.lean | [
"category_theory.category.Pointed",
"data.pfun"
] | [
"PartialFun",
"pfun",
"pfun.comp_assoc",
"pfun.comp_id",
"pfun.id",
"pfun.id_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso.mk {α β : PartialFun.{u}} (e : α ≃ β) : α ≅ β | { hom := e,
inv := e.symm,
hom_inv_id' := (pfun.coe_comp _ _).symm.trans $ congr_arg coe e.symm_comp_self,
inv_hom_id' := (pfun.coe_comp _ _).symm.trans $ congr_arg coe e.self_comp_symm } | def | PartialFun.iso.mk | category_theory.category | src/category_theory/category/PartialFun.lean | [
"category_theory.category.Pointed",
"data.pfun"
] | [
"pfun.coe_comp"
] | Constructs a partial function isomorphism between types from an equivalence between them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Type_to_PartialFun : Type.{u} ⥤ PartialFun | { obj := id,
map := @pfun.lift,
map_comp' := λ _ _ _ _ _, pfun.coe_comp _ _ } | def | Type_to_PartialFun | category_theory.category | src/category_theory/category/PartialFun.lean | [
"category_theory.category.Pointed",
"data.pfun"
] | [
"PartialFun",
"pfun.coe_comp",
"pfun.lift"
] | The forgetful functor from `Type` to `PartialFun` which forgets that the maps are total. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pointed_to_PartialFun : Pointed.{u} ⥤ PartialFun | { obj := λ X, {x : X // x ≠ X.point},
map := λ X Y f, pfun.to_subtype _ f.to_fun ∘ subtype.val,
map_id' := λ X, pfun.ext $ λ a b,
pfun.mem_to_subtype_iff.trans (subtype.coe_inj.trans part.mem_some_iff.symm),
map_comp' := λ X Y Z f g, pfun.ext $ λ a c, begin
refine (pfun.mem_to_subtype_iff.trans _).trans p... | def | Pointed_to_PartialFun | category_theory.category | src/category_theory/category/PartialFun.lean | [
"category_theory.category.Pointed",
"data.pfun"
] | [
"PartialFun",
"pfun.ext",
"pfun.mem_to_subtype_iff",
"pfun.to_subtype"
] | The functor which deletes the point of a pointed type. In return, this makes the maps partial.
This the computable part of the equivalence `PartialFun_equiv_Pointed`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
PartialFun_to_Pointed : PartialFun ⥤ Pointed | by classical; exact
{ obj := λ X, ⟨option X, none⟩,
map := λ X Y f, ⟨option.elim none (λ a, (f a).to_option), rfl⟩,
map_id' := λ X, Pointed.hom.ext _ _ $ funext $ λ o,
option.rec_on o rfl $ λ a, part.some_to_option _,
map_comp' := λ X Y Z f g, Pointed.hom.ext _ _ $ funext $ λ o, option.rec_on o rfl $ λ a,
... | def | PartialFun_to_Pointed | category_theory.category | src/category_theory/category/PartialFun.lean | [
"category_theory.category.Pointed",
"data.pfun"
] | [
"PartialFun",
"Pointed",
"part.bind_to_option",
"part.some_to_option"
] | The functor which maps undefined values to a new point. This makes the maps total and creates
pointed types. This the noncomputable part of the equivalence `PartialFun_equiv_Pointed`. It can't
be computable because `= option.none` is decidable while the domain of a general `part` isn't. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
PartialFun_equiv_Pointed : PartialFun.{u} ≌ Pointed | by classical; exact
equivalence.mk PartialFun_to_Pointed Pointed_to_PartialFun
(nat_iso.of_components (λ X, PartialFun.iso.mk
{ to_fun := λ a, ⟨some a, some_ne_none a⟩,
inv_fun := λ a, get $ ne_none_iff_is_some.1 a.2,
left_inv := λ a, get_some _ _,
right_inv := λ a, by simp only [subtype.val_eq_... | def | PartialFun_equiv_Pointed | category_theory.category | src/category_theory/category/PartialFun.lean | [
"category_theory.category.Pointed",
"data.pfun"
] | [
"PartialFun.iso.mk",
"PartialFun_to_Pointed",
"Pointed",
"Pointed.iso.mk",
"Pointed_to_PartialFun",
"exists_eq_right'",
"exists_prop",
"inv_fun",
"of_not_not",
"option.elim",
"part.bind_some",
"part.elim_to_option",
"part.mem_some_iff",
"pfun.ext",
"subtype.coe_eta",
"subtype.mk_eq_mk"... | The equivalence induced by `PartialFun_to_Pointed` and `Pointed_to_PartialFun`.
`part.equiv_option` made functorial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Type_to_PartialFun_iso_PartialFun_to_Pointed :
Type_to_PartialFun ⋙ PartialFun_to_Pointed ≅ Type_to_Pointed | nat_iso.of_components (λ X, { hom := ⟨id, rfl⟩,
inv := ⟨id, rfl⟩,
hom_inv_id' := rfl,
inv_hom_id' := rfl }) $ λ X Y f,
Pointed.hom.ext _ _ $ funext $ λ a, option.rec_on a rfl $ λ a, by convert part.some_to_option _ | def | Type_to_PartialFun_iso_PartialFun_to_Pointed | category_theory.category | src/category_theory/category/PartialFun.lean | [
"category_theory.category.Pointed",
"data.pfun"
] | [
"PartialFun_to_Pointed",
"Type_to_PartialFun",
"Type_to_Pointed",
"part.some_to_option"
] | Forgetting that maps are total and making them total again by adding a point is the same as just
adding a point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pointed : Type.{u + 1} | (X : Type.{u})
(point : X) | structure | Pointed | category_theory.category | src/category_theory/category/Pointed.lean | [
"category_theory.concrete_category.basic"
] | [] | The category of pointed types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of {X : Type*} (point : X) : Pointed | ⟨X, point⟩ | def | Pointed.of | category_theory.category | src/category_theory/category/Pointed.lean | [
"category_theory.concrete_category.basic"
] | [
"Pointed"
] | Turns a point into a pointed type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of {X : Type*} (point : X) : ↥(of point) = X | rfl | lemma | Pointed.coe_of | category_theory.category | src/category_theory/category/Pointed.lean | [
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom (X Y : Pointed.{u}) : Type u | (to_fun : X → Y)
(map_point : to_fun X.point = Y.point) | structure | Pointed.hom | category_theory.category | src/category_theory/category/Pointed.lean | [
"category_theory.concrete_category.basic"
] | [] | Morphisms in `Pointed`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id (X : Pointed) : hom X X | ⟨id, rfl⟩ | def | Pointed.hom.id | category_theory.category | src/category_theory/category/Pointed.lean | [
"category_theory.concrete_category.basic"
] | [
"Pointed"
] | The identity morphism of `X : Pointed`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp {X Y Z : Pointed.{u}} (f : hom X Y) (g : hom Y Z) : hom X Z | ⟨g.to_fun ∘ f.to_fun, by rw [function.comp_apply, f.map_point, g.map_point]⟩ | def | Pointed.hom.comp | category_theory.category | src/category_theory/category/Pointed.lean | [
"category_theory.concrete_category.basic"
] | [
"function.comp_apply"
] | Composition of morphisms of `Pointed`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
large_category : large_category Pointed | { hom := hom,
id := hom.id,
comp := @hom.comp,
id_comp' := λ _ _ _, hom.ext _ _ rfl,
comp_id' := λ _ _ _, hom.ext _ _ rfl,
assoc' := λ _ _ _ _ _ _ _, hom.ext _ _ rfl } | instance | Pointed.large_category | category_theory.category | src/category_theory/category/Pointed.lean | [
"category_theory.concrete_category.basic"
] | [
"Pointed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category : concrete_category Pointed | { forget := { obj := Pointed.X, map := @hom.to_fun },
forget_faithful := ⟨@hom.ext⟩ } | instance | Pointed.concrete_category | category_theory.category | src/category_theory/category/Pointed.lean | [
"category_theory.concrete_category.basic"
] | [
"Pointed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso.mk {α β : Pointed} (e : α ≃ β) (he : e α.point = β.point) : α ≅ β | { hom := ⟨e, he⟩,
inv := ⟨e.symm, e.symm_apply_eq.2 he.symm⟩,
hom_inv_id' := Pointed.hom.ext _ _ e.symm_comp_self,
inv_hom_id' := Pointed.hom.ext _ _ e.self_comp_symm } | def | Pointed.iso.mk | category_theory.category | src/category_theory/category/Pointed.lean | [
"category_theory.concrete_category.basic"
] | [
"Pointed"
] | Constructs a isomorphism between pointed types from an equivalence that preserves the point
between them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Type_to_Pointed : Type.{u} ⥤ Pointed.{u} | { obj := λ X, ⟨option X, none⟩,
map := λ X Y f, ⟨option.map f, rfl⟩,
map_id' := λ X, Pointed.hom.ext _ _ option.map_id,
map_comp' := λ X Y Z f g, Pointed.hom.ext _ _ (option.map_comp_map _ _).symm } | def | Type_to_Pointed | category_theory.category | src/category_theory/category/Pointed.lean | [
"category_theory.concrete_category.basic"
] | [
"option.map_comp_map"
] | `option` as a functor from types to pointed types. This is the free functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Type_to_Pointed_forget_adjunction : Type_to_Pointed ⊣ forget Pointed | adjunction.mk_of_hom_equiv
{ hom_equiv := λ X Y, { to_fun := λ f, f.to_fun ∘ option.some,
inv_fun := λ f, ⟨λ o, o.elim Y.point f, rfl⟩,
left_inv := λ f, by { ext, cases x, exact f.map_point.symm, refl },
right_inv := λ f, funext $ λ _, rfl },
hom... | def | Type_to_Pointed_forget_adjunction | category_theory.category | src/category_theory/category/Pointed.lean | [
"category_theory.concrete_category.basic"
] | [
"Pointed",
"Type_to_Pointed",
"inv_fun"
] | `Type_to_Pointed` is the free functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
small_category (α : Type u) [preorder α] : small_category α | { hom := λ U V, ulift (plift (U ≤ V)),
id := λ X, ⟨ ⟨ le_refl X ⟩ ⟩,
comp := λ X Y Z f g, ⟨ ⟨ le_trans _ _ _ f.down.down g.down.down ⟩ ⟩ } | instance | preorder.small_category | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [] | The category structure coming from a preorder. There is a morphism `X ⟶ Y` if and only if `X ≤ Y`.
Because we don't allow morphisms to live in `Prop`,
we have to define `X ⟶ Y` as `ulift (plift (X ≤ Y))`.
See `category_theory.hom_of_le` and `category_theory.le_of_hom`.
See <https://stacks.math.columbia.edu/tag/00D3>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_of_le {x y : X} (h : x ≤ y) : x ⟶ y | ulift.up (plift.up h) | def | category_theory.hom_of_le | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [] | Express an inequality as a morphism in the corresponding preorder category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_of_le_refl {x : X} : (le_refl x).hom = 𝟙 x | rfl | lemma | category_theory.hom_of_le_refl | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_of_le_comp {x y z : X} (h : x ≤ y) (k : y ≤ z) :
h.hom ≫ k.hom = (h.trans k).hom | rfl | lemma | category_theory.hom_of_le_comp | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_hom {x y : X} (h : x ⟶ y) : x ≤ y | h.down.down | lemma | category_theory.le_of_hom | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [] | Extract the underlying inequality from a morphism in a preorder category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_of_hom_hom_of_le {x y : X} (h : x ≤ y) : h.hom.le = h | rfl | lemma | category_theory.le_of_hom_hom_of_le | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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