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hom_of_le_le_of_hom {x y : X} (h : x ⟶ y) : h.le.hom = h | by { cases h, cases h, refl, } | lemma | category_theory.hom_of_le_le_of_hom | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_hom_of_le {x y : Xᵒᵖ} (h : unop x ≤ unop y) : y ⟶ x | h.hom.op | def | category_theory.op_hom_of_le | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [] | Construct a morphism in the opposite of a preorder category from an inequality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_of_op_hom {x y : Xᵒᵖ} (h : x ⟶ y) : unop y ≤ unop x | h.unop.le | lemma | category_theory.le_of_op_hom | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_to_top [order_top X] {x : X} : unique (x ⟶ ⊤) | by tidy | instance | category_theory.unique_to_top | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [
"order_top",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_from_bot [order_bot X] {x : X} : unique (⊥ ⟶ x) | by tidy | instance | category_theory.unique_from_bot | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [
"order_bot",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.functor {f : X → Y} (h : monotone f) : X ⥤ Y | { obj := f,
map := λ x₁ x₂ g, (h g.le).hom } | def | monotone.functor | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [
"monotone"
] | A monotone function between preorders induces a functor between the associated categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.functor_obj {f : X → Y} (h : monotone f) : h.functor.obj = f | rfl | lemma | monotone.functor_obj | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor.monotone (f : X ⥤ Y) : monotone f.obj | λ x y hxy, (f.map hxy.hom).le | lemma | category_theory.functor.monotone | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [
"monotone"
] | A functor between preorder categories is monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso.to_eq {x y : X} (f : x ≅ y) : x = y | le_antisymm f.hom.le f.inv.le | lemma | category_theory.iso.to_eq | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence.to_order_iso (e : X ≌ Y) : X ≃o Y | { to_fun := e.functor.obj,
inv_fun := e.inverse.obj,
left_inv := λ a, (e.unit_iso.app a).to_eq.symm,
right_inv := λ b, (e.counit_iso.app b).to_eq,
map_rel_iff' := λ a a',
⟨λ h, ((equivalence.unit e).app a ≫ e.inverse.map h.hom ≫ (equivalence.unit_inv e).app a').le,
λ (h : a ≤ a'), (e.functor.map h.hom)... | def | category_theory.equivalence.to_order_iso | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [
"inv_fun"
] | A categorical equivalence between partial orders is just an order isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence.to_order_iso_apply (e : X ≌ Y) (x : X) :
e.to_order_iso x = e.functor.obj x | rfl | lemma | category_theory.equivalence.to_order_iso_apply | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence.to_order_iso_symm_apply (e : X ≌ Y) (y : Y) :
e.to_order_iso.symm y = e.inverse.obj y | rfl | lemma | category_theory.equivalence.to_order_iso_symm_apply | category_theory.category | src/category_theory/category/preorder.lean | [
"category_theory.equivalence",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Quiv | bundled quiver.{(v+1) u} | def | category_theory.Quiv | category_theory.category | src/category_theory/category/Quiv.lean | [
"category_theory.adjunction.basic",
"category_theory.category.Cat",
"category_theory.path_category"
] | [] | Category of quivers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
str (C : Quiv.{v u}) : quiver.{(v+1) u} C | C.str | instance | category_theory.Quiv.str | category_theory.category | src/category_theory/category/Quiv.lean | [
"category_theory.adjunction.basic",
"category_theory.category.Cat",
"category_theory.path_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (C : Type u) [quiver.{v+1} C] : Quiv.{v u} | bundled.of C | def | category_theory.Quiv.of | category_theory.category | src/category_theory/category/Quiv.lean | [
"category_theory.adjunction.basic",
"category_theory.category.Cat",
"category_theory.path_category"
] | [] | Construct a bundled `Quiv` from the underlying type and the typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category : large_category.{max v u} Quiv.{v u} | { hom := λ C D, prefunctor C D,
id := λ C, prefunctor.id C,
comp := λ C D E F G, prefunctor.comp 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.Quiv.category | category_theory.category | src/category_theory/category/Quiv.lean | [
"category_theory.adjunction.basic",
"category_theory.category.Cat",
"category_theory.path_category"
] | [
"prefunctor",
"prefunctor.comp",
"prefunctor.id"
] | Category structure on `Quiv` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget : Cat.{v u} ⥤ Quiv.{v u} | { obj := λ C, Quiv.of C,
map := λ C D F, F.to_prefunctor, } | def | category_theory.Quiv.forget | category_theory.category | src/category_theory/category/Quiv.lean | [
"category_theory.adjunction.basic",
"category_theory.category.Cat",
"category_theory.path_category"
] | [] | The forgetful functor from categories to quivers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free : Quiv.{v u} ⥤ Cat.{(max u v) u} | { obj := λ V, Cat.of (paths V),
map := λ V W F,
{ obj := λ X, F.obj X,
map := λ X Y f, F.map_path f,
map_comp' := λ X Y Z f g, F.map_path_comp f g, },
map_id' := λ V, by { change (show paths V ⥤ _, from _) = _, ext, apply eq_conj_eq_to_hom, refl },
map_comp' := λ U V W F G,
by { change (show paths U... | def | category_theory.Cat.free | category_theory.category | src/category_theory/category/Quiv.lean | [
"category_theory.adjunction.basic",
"category_theory.category.Cat",
"category_theory.path_category"
] | [
"free"
] | The functor sending each quiver to its path category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift {V : Type u} [quiver.{v+1} V] {C : Type*} [category C]
(F : prefunctor V C) : paths V ⥤ C | { obj := λ X, F.obj X,
map := λ X Y f, compose_path (F.map_path f), } | def | category_theory.Quiv.lift | category_theory.category | src/category_theory/category/Quiv.lean | [
"category_theory.adjunction.basic",
"category_theory.category.Cat",
"category_theory.path_category"
] | [
"lift",
"prefunctor"
] | Any prefunctor into a category lifts to a functor from the path category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adj : Cat.free ⊣ Quiv.forget | adjunction.mk_of_hom_equiv
{ hom_equiv := λ V C,
{ to_fun := λ F, paths.of.comp F.to_prefunctor,
inv_fun := λ F, lift F,
left_inv := λ F, by { ext, { erw (eq_conj_eq_to_hom _).symm, apply category.id_comp }, refl },
right_inv := begin
rintro ⟨obj,map⟩,
dsimp only [prefunctor.comp],
congr... | def | category_theory.Quiv.adj | category_theory.category | src/category_theory/category/Quiv.lean | [
"category_theory.adjunction.basic",
"category_theory.category.Cat",
"category_theory.path_category"
] | [
"adj",
"inv_fun",
"lift",
"prefunctor.comp"
] | The adjunction between forming the free category on a quiver, and forgetting a category to a quiver. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Rel | Type u | def | category_theory.Rel | category_theory.category | src/category_theory/category/Rel.lean | [
"category_theory.category.basic"
] | [] | A type synonym for `Type`, which carries the category instance for which
morphisms are binary relations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Rel.inhabited : inhabited Rel | by unfold Rel; apply_instance | instance | category_theory.Rel.inhabited | category_theory.category | src/category_theory/category/Rel.lean | [
"category_theory.category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel : large_category Rel | { hom := λ X Y, X → Y → Prop,
id := λ X, λ x y, x = y,
comp := λ X Y Z f g x z, ∃ y, f x y ∧ g y z } | instance | category_theory.rel | category_theory.category | src/category_theory/category/Rel.lean | [
"category_theory.category.basic"
] | [
"rel"
] | The category of types with binary relations as morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Twop : Type.{u+1} | (X : Type.{u})
(to_two_pointing : two_pointing X) | structure | Twop | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"two_pointing"
] | The category of two-pointed types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of {X : Type*} (to_two_pointing : two_pointing X) : Twop | ⟨X, to_two_pointing⟩ | def | Twop.of | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Twop",
"two_pointing"
] | Turns a two-pointing into a two-pointed type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of {X : Type*} (to_two_pointing : two_pointing X) :
↥(of to_two_pointing) = X | rfl | lemma | Twop.coe_of | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"two_pointing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Bipointed (X : Twop) : Bipointed | X.to_two_pointing.to_prod.Bipointed | def | Twop.to_Bipointed | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Bipointed",
"Twop"
] | Turns a two-pointed type into a bipointed type, by forgetting that the pointed elements are
distinct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_Bipointed (X : Twop) : ↥X.to_Bipointed = ↥X | rfl | lemma | Twop.coe_to_Bipointed | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Twop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
large_category : large_category Twop | induced_category.category to_Bipointed | instance | Twop.large_category | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Twop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category : concrete_category Twop | induced_category.concrete_category to_Bipointed | instance | Twop.concrete_category | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Twop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_Bipointed : has_forget₂ Twop Bipointed | induced_category.has_forget₂ to_Bipointed | instance | Twop.has_forget_to_Bipointed | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Bipointed",
"Twop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
swap : Twop ⥤ Twop | { obj := λ X, ⟨X, X.to_two_pointing.swap⟩, map := λ X Y f, ⟨f.to_fun, f.map_snd, f.map_fst⟩ } | def | Twop.swap | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Twop"
] | Swaps the pointed elements of a two-pointed type. `two_pointing.swap` as a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
swap_equiv : Twop ≌ Twop | 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 | Twop.swap_equiv | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Twop"
] | The equivalence between `Twop` and itself induced by `prod.swap` both ways. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Twop_swap_comp_forget_to_Bipointed :
Twop.swap ⋙ forget₂ Twop Bipointed = forget₂ Twop Bipointed ⋙ Bipointed.swap | rfl | lemma | Twop_swap_comp_forget_to_Bipointed | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Bipointed",
"Bipointed.swap",
"Twop",
"Twop.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pointed_to_Twop_fst : Pointed.{u} ⥤ Twop | { obj := λ X, ⟨option X, ⟨X.point, none⟩, some_ne_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_Twop_fst | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Twop",
"option.map_comp_map"
] | The functor from `Pointed` to `Twop` which adds a second point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pointed_to_Twop_snd : Pointed.{u} ⥤ Twop | { obj := λ X, ⟨option X, ⟨none, X.point⟩, (some_ne_none _).symm⟩,
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_Twop_snd | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Twop",
"option.map_comp_map"
] | The functor from `Pointed` to `Twop` which adds a first point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pointed_to_Twop_fst_comp_swap :
Pointed_to_Twop_fst ⋙ Twop.swap = Pointed_to_Twop_snd | rfl | lemma | Pointed_to_Twop_fst_comp_swap | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Pointed_to_Twop_fst",
"Pointed_to_Twop_snd",
"Twop.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pointed_to_Twop_snd_comp_swap :
Pointed_to_Twop_snd ⋙ Twop.swap = Pointed_to_Twop_fst | rfl | lemma | Pointed_to_Twop_snd_comp_swap | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Pointed_to_Twop_fst",
"Pointed_to_Twop_snd",
"Twop.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pointed_to_Twop_fst_comp_forget_to_Bipointed :
Pointed_to_Twop_fst ⋙ forget₂ Twop Bipointed = Pointed_to_Bipointed_fst | rfl | lemma | Pointed_to_Twop_fst_comp_forget_to_Bipointed | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Bipointed",
"Pointed_to_Bipointed_fst",
"Pointed_to_Twop_fst",
"Twop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pointed_to_Twop_snd_comp_forget_to_Bipointed :
Pointed_to_Twop_snd ⋙ forget₂ Twop Bipointed = Pointed_to_Bipointed_snd | rfl | lemma | Pointed_to_Twop_snd_comp_forget_to_Bipointed | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Bipointed",
"Pointed_to_Bipointed_snd",
"Pointed_to_Twop_snd",
"Twop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pointed_to_Twop_fst_forget_comp_Bipointed_to_Pointed_fst_adjunction :
Pointed_to_Twop_fst ⊣ forget₂ Twop Bipointed ⋙ 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_two_pointing.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 := λ f, Pointed.hom.ext _ _ rfl },
hom_... | def | Pointed_to_Twop_fst_forget_comp_Bipointed_to_Pointed_fst_adjunction | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Bipointed",
"Bipointed_to_Pointed_fst",
"Pointed_to_Twop_fst",
"Twop",
"inv_fun"
] | Adding a second point is left adjoint to forgetting the second point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pointed_to_Twop_snd_forget_comp_Bipointed_to_Pointed_snd_adjunction :
Pointed_to_Twop_snd ⊣ forget₂ Twop Bipointed ⋙ 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_two_pointing.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 := λ f, Pointed.hom.ext _ _ rfl },
hom_... | def | Pointed_to_Twop_snd_forget_comp_Bipointed_to_Pointed_snd_adjunction | category_theory.category | src/category_theory/category/Twop.lean | [
"category_theory.category.Bipointed",
"data.two_pointing"
] | [
"Bipointed",
"Bipointed_to_Pointed_snd",
"Pointed_to_Twop_snd",
"Twop",
"inv_fun"
] | Adding a first point is left adjoint to forgetting the first point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift.up_functor : C ⥤ (ulift.{u₂} C) | { obj := ulift.up,
map := λ X Y f, f } | def | category_theory.ulift.up_functor | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | The functorial version of `ulift.up`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift.down_functor : (ulift.{u₂} C) ⥤ C | { obj := ulift.down,
map := λ X Y f, f } | def | category_theory.ulift.down_functor | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | The functorial version of `ulift.down`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift.equivalence : C ≌ (ulift.{u₂} C) | { functor := ulift.up_functor,
inverse := ulift.down_functor,
unit_iso :=
{ hom := 𝟙 _,
inv := 𝟙 _ },
counit_iso :=
{ hom :=
{ app := λ X, 𝟙 _,
naturality' := λ X Y f, by {change f ≫ 𝟙 _ = 𝟙 _ ≫ f, simp} },
inv :=
{ app := λ X, 𝟙 _,
naturality' := λ X Y f, by {change f ≫ 𝟙 _... | def | category_theory.ulift.equivalence | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | The categorical equivalence between `C` and `ulift C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
{w u} ulift_hom (C : Type u) | C | def | category_theory.ulift_hom | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | `ulift_hom.{w} C` is an alias for `C`, which is endowed with a category instance
whose morphisms are obtained by applying `ulift.{w}` to the morphisms from `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift_hom.obj_down {C} (A : ulift_hom C) : C | A | def | category_theory.ulift_hom.obj_down | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | The obvious function `ulift_hom C → C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift_hom.obj_up {C} (A : C) : ulift_hom C | A | def | category_theory.ulift_hom.obj_up | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | The obvious function `C → ulift_hom C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj_down_obj_up {C} (A : C) : (ulift_hom.obj_up A).obj_down = A | rfl | lemma | category_theory.obj_down_obj_up | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj_up_obj_down {C} (A : ulift_hom C) : ulift_hom.obj_up A.obj_down = A | rfl | lemma | category_theory.obj_up_obj_down | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift_hom.up : C ⥤ ulift_hom C | { obj := ulift_hom.obj_up,
map := λ X Y f, ⟨f⟩ } | def | category_theory.ulift_hom.up | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | One half of the quivalence between `C` and `ulift_hom C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift_hom.down : ulift_hom C ⥤ C | { obj := ulift_hom.obj_down,
map := λ X Y f, f.down } | def | category_theory.ulift_hom.down | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | One half of the quivalence between `C` and `ulift_hom C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift_hom.equiv : C ≌ ulift_hom C | { functor := ulift_hom.up,
inverse := ulift_hom.down,
unit_iso := nat_iso.of_components (λ A, eq_to_iso rfl) (by tidy),
counit_iso := nat_iso.of_components (λ A, eq_to_iso rfl) (by tidy) } | def | category_theory.ulift_hom.equiv | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | The equivalence between `C` and `ulift_hom C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
{w v u} as_small (C : Type u) [category.{v} C] | ulift.{max w v} C | def | category_theory.as_small | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | `as_small C` is a small category equivalent to `C`.
More specifically, if `C : Type u` is endowed with `category.{v} C`, then
`as_small.{w} C : Type (max w v u)` is endowed with an instance of a small category.
The objects and morphisms of `as_small C` are defined by applying `ulift` to the
objects and morphis... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_small.up : C ⥤ as_small C | { obj := λ X, ⟨X⟩,
map := λ X Y f, ⟨f⟩ } | def | category_theory.as_small.up | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | One half of the equivalence between `C` and `as_small C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_small.down : as_small C ⥤ C | { obj := λ X, X.down,
map := λ X Y f, f.down } | def | category_theory.as_small.down | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | One half of the equivalence between `C` and `as_small C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_small.equiv : C ≌ as_small C | { functor := as_small.up,
inverse := as_small.down,
unit_iso := nat_iso.of_components (λ X, eq_to_iso rfl) (by tidy),
counit_iso := nat_iso.of_components (λ X, eq_to_iso $ by { ext, refl }) (by tidy) } | def | category_theory.as_small.equiv | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | The equivalence between `C` and `as_small C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
{v' u' v u} ulift_hom_ulift_category.equiv (C : Type u) [category.{v} C] :
C ≌ ulift_hom.{v'} (ulift.{u'} C) | ulift.equivalence.trans ulift_hom.equiv | def | category_theory.ulift_hom_ulift_category.equiv | category_theory.category | src/category_theory/category/ulift.lean | [
"category_theory.category.basic",
"category_theory.equivalence",
"category_theory.eq_to_hom"
] | [] | The equivalence between `C` and `ulift_hom (ulift C)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category_objects {F : J ⥤ Cat.{u u}} {j} :
small_category ((F ⋙ Cat.objects.{u u}).obj j) | (F.obj j).str | instance | category_theory.Cat.has_limits.category_objects | category_theory.category.Cat | src/category_theory/category/Cat/limit.lean | [
"category_theory.category.Cat",
"category_theory.limits.types",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_diagram {F : J ⥤ Cat.{v v}} (X Y : limit (F ⋙ Cat.objects.{v v})) : J ⥤ Type v | { obj := λ j, limit.π (F ⋙ Cat.objects) j X ⟶ limit.π (F ⋙ Cat.objects) j Y,
map := λ j j' f g,
begin
refine eq_to_hom _ ≫ (F.map f).map g ≫ eq_to_hom _,
exact (congr_fun (limit.w (F ⋙ Cat.objects) f) X).symm,
exact (congr_fun (limit.w (F ⋙ Cat.objects) f) Y),
end,
map_id' := λ X, begin
ext f, d... | def | category_theory.Cat.has_limits.hom_diagram | category_theory.category.Cat | src/category_theory/category/Cat/limit.lean | [
"category_theory.category.Cat",
"category_theory.limits.types",
"category_theory.limits.preserves.basic"
] | [] | Auxiliary definition:
the diagram whose limit gives the morphism space between two objects of the limit category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_X (F : J ⥤ Cat.{v v}) : Cat.{v v} | { α := limit (F ⋙ Cat.objects), }. | def | category_theory.Cat.has_limits.limit_cone_X | category_theory.category.Cat | src/category_theory/category/Cat/limit.lean | [
"category_theory.category.Cat",
"category_theory.limits.types",
"category_theory.limits.preserves.basic"
] | [] | Auxiliary definition: the limit category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone (F : J ⥤ Cat.{v v}) : cone F | { X := limit_cone_X F,
π :=
{ app := λ j,
{ obj := limit.π (F ⋙ Cat.objects) j,
map := λ X Y, limit.π (hom_diagram X Y) j, },
naturality' := λ j j' f, category_theory.functor.ext
(λ X, (congr_fun (limit.w (F ⋙ Cat.objects) f) X).symm)
(λ X Y h, (congr_fun (limit.w (hom_diagram X Y) f) h).s... | def | category_theory.Cat.has_limits.limit_cone | category_theory.category.Cat | src/category_theory/category/Cat/limit.lean | [
"category_theory.category.Cat",
"category_theory.limits.types",
"category_theory.limits.preserves.basic"
] | [
"category_theory.functor.ext"
] | Auxiliary definition: the cone over the limit category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_lift (F : J ⥤ Cat.{v v}) (s : cone F) : s.X ⟶ limit_cone_X F | { obj := limit.lift (F ⋙ Cat.objects)
{ X := s.X,
π :=
{ app := λ j, (s.π.app j).obj,
naturality' := λ j j' f, (congr_arg functor.obj (s.π.naturality f) : _), } },
map := λ X Y f,
begin
fapply types.limit.mk.{v v},
{ intro j,
refine eq_to_hom _ ≫ (s.π.app j).map f ≫ eq_to_hom _;
... | def | category_theory.Cat.has_limits.limit_cone_lift | category_theory.category.Cat | src/category_theory/category/Cat/limit.lean | [
"category_theory.category.Cat",
"category_theory.limits.types",
"category_theory.limits.preserves.basic"
] | [] | Auxiliary definition: the universal morphism to the proposed limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_π_hom_diagram_eq_to_hom {F : J ⥤ Cat.{v v}}
(X Y : limit (F ⋙ Cat.objects.{v v})) (j : J) (h : X = Y) :
limit.π (hom_diagram X Y) j (eq_to_hom h) =
eq_to_hom (congr_arg (limit.π (F ⋙ Cat.objects.{v v}) j) h) | by { subst h, simp, } | lemma | category_theory.Cat.has_limits.limit_π_hom_diagram_eq_to_hom | category_theory.category.Cat | src/category_theory/category/Cat/limit.lean | [
"category_theory.category.Cat",
"category_theory.limits.types",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_cone_is_limit (F : J ⥤ Cat.{v v}) : is_limit (limit_cone F) | { lift := limit_cone_lift F,
fac' := λ s j, category_theory.functor.ext (by tidy) (λ X Y f, types.limit.π_mk _ _ _ _),
uniq' := λ s m w,
begin
symmetry,
fapply category_theory.functor.ext,
{ intro X,
ext,
dsimp, simp only [types.limit.lift_π_apply', ←w j],
refl, },
{ intros X Y f... | def | category_theory.Cat.has_limits.limit_cone_is_limit | category_theory.category.Cat | src/category_theory/category/Cat/limit.lean | [
"category_theory.category.Cat",
"category_theory.limits.types",
"category_theory.limits.preserves.basic"
] | [
"category_theory.functor.ext",
"lift"
] | Auxiliary definition: the proposed cone is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exponentiable {C : Type u} [category.{v} C] [has_finite_products C] (X : C) | closed X | abbreviation | category_theory.exponentiable | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | An object `X` is *exponentiable* if `(X × -)` is a left adjoint.
We define this as being `closed` in the cartesian monoidal structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_product_exponentiable {C : Type u} [category.{v} C] [has_finite_products C] {X Y : C}
(hX : exponentiable X) (hY : exponentiable Y) : exponentiable (X ⨯ Y) | tensor_closed hX hY | def | category_theory.binary_product_exponentiable | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | If `X` and `Y` are exponentiable then `X ⨯ Y` is.
This isn't an instance because it's not usually how we want to construct exponentials, we'll usually
prove all objects are exponential uniformly. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminal_exponentiable {C : Type u} [category.{v} C] [has_finite_products C] :
exponentiable ⊤_ C | unit_closed | def | category_theory.terminal_exponentiable | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | The terminal object is always exponentiable.
This isn't an instance because most of the time we'll prove cartesian closed for all objects
at once, rather than just for this one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cartesian_closed (C : Type u) [category.{v} C] [has_finite_products C] | monoidal_closed C | abbreviation | category_theory.cartesian_closed | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | A category `C` is cartesian closed if it has finite products and every object is exponentiable.
We define this as `monoidal_closed` with respect to the cartesian monoidal structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp : C ⥤ C | ihom A | abbreviation | category_theory.exp | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"exp"
] | This is (-)^A. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjunction : prod.functor.obj A ⊣ exp A | ihom.adjunction A | abbreviation | category_theory.exp.adjunction | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"exp"
] | The adjunction between A ⨯ - and (-)^A. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ev : exp A ⋙ prod.functor.obj A ⟶ 𝟭 C | ihom.ev A | abbreviation | category_theory.exp.ev | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"exp"
] | The evaluation natural transformation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coev : 𝟭 C ⟶ prod.functor.obj A ⋙ exp A | ihom.coev A | abbreviation | category_theory.exp.coev | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"exp"
] | The coevaluation natural transformation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ev_coev :
limits.prod.map (𝟙 A) ((coev A).app B) ≫ (ev A).app (A ⨯ B) = 𝟙 (A ⨯ B) | ihom.ev_coev A B | lemma | category_theory.exp.ev_coev | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coev_ev : (coev A).app (A ⟹ B) ≫ (exp A).map ((ev A).app B) = 𝟙 (A ⟹ B) | ihom.coev_ev A B | lemma | category_theory.exp.coev_ev | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry : (A ⨯ Y ⟶ X) → (Y ⟶ A ⟹ X) | (exp.adjunction A).hom_equiv _ _ | def | category_theory.cartesian_closed.curry | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | Currying in a cartesian closed category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uncurry : (Y ⟶ A ⟹ X) → (A ⨯ Y ⟶ X) | ((exp.adjunction A).hom_equiv _ _).symm | def | category_theory.cartesian_closed.uncurry | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | Uncurrying in a cartesian closed category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_equiv_apply_eq (f : A ⨯ Y ⟶ X) :
(exp.adjunction A).hom_equiv _ _ f = curry f | rfl | lemma | category_theory.cartesian_closed.hom_equiv_apply_eq | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_equiv_symm_apply_eq (f : Y ⟶ A ⟹ X) :
((exp.adjunction A).hom_equiv _ _).symm f = uncurry f | rfl | lemma | category_theory.cartesian_closed.hom_equiv_symm_apply_eq | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_natural_left (f : X ⟶ X') (g : A ⨯ X' ⟶ Y) :
curry (limits.prod.map (𝟙 _) f ≫ g) = f ≫ curry g | adjunction.hom_equiv_naturality_left _ _ _ | lemma | category_theory.cartesian_closed.curry_natural_left | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_natural_right (f : A ⨯ X ⟶ Y) (g : Y ⟶ Y') :
curry (f ≫ g) = curry f ≫ (exp _).map g | adjunction.hom_equiv_naturality_right _ _ _ | lemma | category_theory.cartesian_closed.curry_natural_right | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_natural_right (f : X ⟶ A⟹Y) (g : Y ⟶ Y') :
uncurry (f ≫ (exp _).map g) = uncurry f ≫ g | adjunction.hom_equiv_naturality_right_symm _ _ _ | lemma | category_theory.cartesian_closed.uncurry_natural_right | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_natural_left (f : X ⟶ X') (g : X' ⟶ A⟹Y) :
uncurry (f ≫ g) = limits.prod.map (𝟙 _) f ≫ uncurry g | adjunction.hom_equiv_naturality_left_symm _ _ _ | lemma | category_theory.cartesian_closed.uncurry_natural_left | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_curry (f : A ⨯ X ⟶ Y) : uncurry (curry f) = f | (closed.is_adj.adj.hom_equiv _ _).left_inv f | lemma | category_theory.cartesian_closed.uncurry_curry | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_uncurry (f : X ⟶ A⟹Y) : curry (uncurry f) = f | (closed.is_adj.adj.hom_equiv _ _).right_inv f | lemma | category_theory.cartesian_closed.curry_uncurry | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_eq_iff (f : A ⨯ Y ⟶ X) (g : Y ⟶ A ⟹ X) :
curry f = g ↔ f = uncurry g | adjunction.hom_equiv_apply_eq _ f g | lemma | category_theory.cartesian_closed.curry_eq_iff | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_curry_iff (f : A ⨯ Y ⟶ X) (g : Y ⟶ A ⟹ X) :
g = curry f ↔ uncurry g = f | adjunction.eq_hom_equiv_apply _ f g | lemma | category_theory.cartesian_closed.eq_curry_iff | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_eq (g : Y ⟶ A ⟹ X) : uncurry g = limits.prod.map (𝟙 A) g ≫ (exp.ev A).app X | adjunction.hom_equiv_counit _ | lemma | category_theory.cartesian_closed.uncurry_eq | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_eq (g : A ⨯ Y ⟶ X) : curry g = (exp.coev A).app Y ≫ (exp A).map g | adjunction.hom_equiv_unit _ | lemma | category_theory.cartesian_closed.curry_eq | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_id_eq_ev (A X : C) [exponentiable A] : uncurry (𝟙 (A ⟹ X)) = (exp.ev A).app X | by rw [uncurry_eq, prod.map_id_id, id_comp] | lemma | category_theory.cartesian_closed.uncurry_id_eq_ev | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_id_eq_coev (A X : C) [exponentiable A] : curry (𝟙 _) = (exp.coev A).app X | by { rw [curry_eq, (exp A).map_id (A ⨯ _)], apply comp_id } | lemma | category_theory.cartesian_closed.curry_id_eq_coev | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"exp",
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry_injective : function.injective (curry : (A ⨯ Y ⟶ X) → (Y ⟶ A ⟹ X)) | (closed.is_adj.adj.hom_equiv _ _).injective | lemma | category_theory.cartesian_closed.curry_injective | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_injective : function.injective (uncurry : (Y ⟶ A ⟹ X) → (A ⨯ Y ⟶ X)) | (closed.is_adj.adj.hom_equiv _ _).symm.injective | lemma | category_theory.cartesian_closed.uncurry_injective | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_terminal_iso_self [exponentiable ⊤_ C] : (⊤_ C ⟹ X) ≅ X | yoneda.ext (⊤_ C ⟹ X) X
(λ Y f, (prod.left_unitor Y).inv ≫ cartesian_closed.uncurry f)
(λ Y f, cartesian_closed.curry ((prod.left_unitor Y).hom ≫ f))
(λ Z g, by rw [curry_eq_iff, iso.hom_inv_id_assoc] )
(λ Z g, by simp)
(λ Z W f g, by rw [uncurry_natural_left, prod.left_unitor_inv_naturality_assoc f] ) | def | category_theory.exp_terminal_iso_self | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | Show that the exponential of the terminal object is isomorphic to itself, i.e. `X^1 ≅ X`.
The typeclass argument is explicit: any instance can be used. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
internalize_hom (f : A ⟶ Y) : ⊤_ C ⟶ (A ⟹ Y) | cartesian_closed.curry (limits.prod.fst ≫ f) | def | category_theory.internalize_hom | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | The internal element which points at the given morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pre (f : B ⟶ A) [exponentiable B] : exp A ⟶ exp B | transfer_nat_trans_self (exp.adjunction _) (exp.adjunction _) (prod.functor.map f) | def | category_theory.pre | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"exp"
] | Pre-compose an internal hom with an external hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_map_pre_app_comp_ev (f : B ⟶ A) [exponentiable B] (X : C) :
limits.prod.map (𝟙 B) ((pre f).app X) ≫ (exp.ev B).app X =
limits.prod.map f (𝟙 (A ⟹ X)) ≫ (exp.ev A).app X | transfer_nat_trans_self_counit _ _ (prod.functor.map f) X | lemma | category_theory.prod_map_pre_app_comp_ev | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uncurry_pre (f : B ⟶ A) [exponentiable B] (X : C) :
cartesian_closed.uncurry ((pre f).app X) = limits.prod.map f (𝟙 _) ≫ (exp.ev A).app X | begin
rw [uncurry_eq, prod_map_pre_app_comp_ev]
end | lemma | category_theory.uncurry_pre | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coev_app_comp_pre_app (f : B ⟶ A) [exponentiable B] :
(exp.coev A).app X ≫ (pre f).app (A ⨯ X) =
(exp.coev B).app X ≫ (exp B).map (limits.prod.map f (𝟙 _)) | unit_transfer_nat_trans_self _ _ (prod.functor.map f) X | lemma | category_theory.coev_app_comp_pre_app | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pre_id (A : C) [exponentiable A] : pre (𝟙 A) = 𝟙 _ | by simp [pre] | lemma | category_theory.pre_id | category_theory.closed | src/category_theory/closed/cartesian.lean | [
"category_theory.epi_mono",
"category_theory.limits.shapes.finite_products",
"category_theory.monoidal.of_has_finite_products",
"category_theory.limits.preserves.shapes.binary_products",
"category_theory.adjunction.limits",
"category_theory.adjunction.mates",
"category_theory.closed.monoidal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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