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connected_components (J : Type u₁) [category.{v₁} J] : Type u₁ | quotient (zigzag.setoid J) | def | category_theory.connected_components | category_theory | src/category_theory/connected_components.lean | [
"data.list.chain",
"category_theory.is_connected",
"category_theory.sigma.basic",
"category_theory.full_subcategory"
] | [
"connected_components"
] | This type indexes the connected components of the category `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
component (j : connected_components J) : Type u₁ | full_subcategory (λ k, quotient.mk' k = j) | def | category_theory.component | category_theory | src/category_theory/connected_components.lean | [
"data.list.chain",
"category_theory.is_connected",
"category_theory.sigma.basic",
"category_theory.full_subcategory"
] | [
"connected_components",
"quotient.mk'"
] | Given an index for a connected component, produce the actual component as a full subcategory. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
component.ι (j) : component j ⥤ J | full_subcategory_inclusion _ | def | category_theory.component.ι | category_theory | src/category_theory/connected_components.lean | [
"data.list.chain",
"category_theory.is_connected",
"category_theory.sigma.basic",
"category_theory.full_subcategory"
] | [] | The inclusion functor from a connected component to the whole category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
decomposed (J : Type u₁) [category.{v₁} J] | Σ (j : connected_components J), component j | abbreviation | category_theory.decomposed | category_theory | src/category_theory/connected_components.lean | [
"data.list.chain",
"category_theory.is_connected",
"category_theory.sigma.basic",
"category_theory.full_subcategory"
] | [
"connected_components"
] | The disjoint union of `J`s connected components, written explicitly as a sigma-type with the
category structure.
This category is equivalent to `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion (j : connected_components J) : component j ⥤ decomposed J | sigma.incl _ | abbreviation | category_theory.inclusion | category_theory | src/category_theory/connected_components.lean | [
"data.list.chain",
"category_theory.is_connected",
"category_theory.sigma.basic",
"category_theory.full_subcategory"
] | [
"connected_components"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decomposed_to (J : Type u₁) [category.{v₁} J] : decomposed J ⥤ J | sigma.desc component.ι | def | category_theory.decomposed_to | category_theory | src/category_theory/connected_components.lean | [
"data.list.chain",
"category_theory.is_connected",
"category_theory.sigma.basic",
"category_theory.full_subcategory"
] | [] | The forward direction of the equivalence between the decomposed category and the original. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion_comp_decomposed_to (j : connected_components J) :
inclusion j ⋙ decomposed_to J = component.ι j | rfl | lemma | category_theory.inclusion_comp_decomposed_to | category_theory | src/category_theory/connected_components.lean | [
"data.list.chain",
"category_theory.is_connected",
"category_theory.sigma.basic",
"category_theory.full_subcategory"
] | [
"connected_components"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decomposed_equiv : decomposed J ≌ J | (decomposed_to J).as_equivalence | def | category_theory.decomposed_equiv | category_theory | src/category_theory/connected_components.lean | [
"data.list.chain",
"category_theory.is_connected",
"category_theory.sigma.basic",
"category_theory.full_subcategory"
] | [] | This gives that any category is equivalent to a disjoint union of connected categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
core (C : Type u₁) | C | def | category_theory.core | category_theory | src/category_theory/core.lean | [
"control.equiv_functor",
"category_theory.groupoid",
"category_theory.whiskering",
"category_theory.types"
] | [] | The core of a category C is the groupoid whose morphisms are all the
isomorphisms of C. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
core_category : groupoid.{v₁} (core C) | { hom := λ X Y : C, X ≅ Y,
inv := λ X Y f, iso.symm f,
id := λ X, iso.refl X,
comp := λ X Y Z f g, iso.trans f g } | instance | category_theory.core_category | category_theory | src/category_theory/core.lean | [
"control.equiv_functor",
"category_theory.groupoid",
"category_theory.whiskering",
"category_theory.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_hom (X : core C) : iso.hom (𝟙 X) = 𝟙 X | rfl | lemma | category_theory.core.id_hom | category_theory | src/category_theory/core.lean | [
"control.equiv_functor",
"category_theory.groupoid",
"category_theory.whiskering",
"category_theory.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_hom {X Y Z : core C} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g).hom = f.hom ≫ g.hom | rfl | lemma | category_theory.core.comp_hom | category_theory | src/category_theory/core.lean | [
"control.equiv_functor",
"category_theory.groupoid",
"category_theory.whiskering",
"category_theory.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion : core C ⥤ C | { obj := id,
map := λ X Y f, f.hom } | def | category_theory.core.inclusion | category_theory | src/category_theory/core.lean | [
"control.equiv_functor",
"category_theory.groupoid",
"category_theory.whiskering",
"category_theory.types"
] | [] | The core of a category is naturally included in the category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_to_core (F : G ⥤ C) : G ⥤ core C | { obj := λ X, F.obj X,
map := λ X Y f, ⟨F.map f, F.map (inv f)⟩ } | def | category_theory.core.functor_to_core | category_theory | src/category_theory/core.lean | [
"control.equiv_functor",
"category_theory.groupoid",
"category_theory.whiskering",
"category_theory.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_functor_to_core : (G ⥤ core C) ⥤ (G ⥤ C) | (whiskering_right _ _ _).obj (inclusion C) | def | category_theory.core.forget_functor_to_core | category_theory | src/category_theory/core.lean | [
"control.equiv_functor",
"category_theory.groupoid",
"category_theory.whiskering",
"category_theory.types"
] | [] | We can functorially associate to any functor from a groupoid to the core of a category `C`,
a functor from the groupoid to `C`, simply by composing with the embedding `core C ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_equiv_functor (m : Type u₁ → Type u₂) [equiv_functor m] :
core (Type u₁) ⥤ core (Type u₂) | { obj := m,
map := λ α β f, (equiv_functor.map_equiv m f.to_equiv).to_iso,
-- These are not very pretty.
map_id' := λ α, begin ext, exact (congr_fun (equiv_functor.map_refl _) x), end,
map_comp' := λ α β γ f g,
begin
ext,
simp only [equiv_functor.map_equiv_apply, equiv.to_iso_hom,
fu... | def | category_theory.of_equiv_functor | category_theory | src/category_theory/core.lean | [
"control.equiv_functor",
"category_theory.groupoid",
"category_theory.whiskering",
"category_theory.types"
] | [
"equiv.to_iso_hom",
"equiv_functor",
"equiv_functor.map_equiv",
"equiv_functor.map_equiv_apply"
] | `of_equiv_functor m` lifts a type-level `equiv_functor`
to a categorical functor `core (Type u₁) ⥤ core (Type u₂)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differential_object | (X : C)
(d : X ⟶ X⟦1⟧)
(d_squared' : d ≫ d⟦(1:ℤ)⟧' = 0 . obviously) | structure | category_theory.differential_object | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | A differential object in a category with zero morphisms and a shift is
an object `X` equipped with
a morphism `d : X ⟶ X⟦1⟧`, such that `d^2 = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom (X Y : differential_object C) | (f : X.X ⟶ Y.X)
(comm' : X.d ≫ f⟦1⟧' = f ≫ Y.d . obviously) | structure | category_theory.differential_object.hom | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | A morphism of differential objects is a morphism commuting with the differentials. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id (X : differential_object C) : hom X X | { f := 𝟙 X.X } | def | category_theory.differential_object.hom.id | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | The identity morphism of a differential object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp {X Y Z : differential_object C} (f : hom X Y) (g : hom Y Z) : hom X Z | { f := f.f ≫ g.f, } | def | category_theory.differential_object.hom.comp | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | The composition of morphisms of differential objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category_of_differential_objects : category (differential_object C) | { hom := hom,
id := hom.id,
comp := λ X Y Z f g, hom.comp f g, } | instance | category_theory.differential_object.category_of_differential_objects | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_f (X : differential_object C) : ((𝟙 X) : X ⟶ X).f = 𝟙 (X.X) | rfl | lemma | category_theory.differential_object.id_f | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_f {X Y Z : differential_object C} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).f = f.f ≫ g.f | rfl | lemma | category_theory.differential_object.comp_f | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_hom_f {X Y : differential_object C} (h : X = Y) :
hom.f (eq_to_hom h) = eq_to_hom (congr_arg _ h) | by { subst h, rw [eq_to_hom_refl, eq_to_hom_refl], refl } | lemma | category_theory.differential_object.eq_to_hom_f | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget : (differential_object C) ⥤ C | { obj := λ X, X.X,
map := λ X Y f, f.f, } | def | category_theory.differential_object.forget | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | The forgetful functor taking a differential object to its underlying object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_faithful : faithful (forget C) | { } | instance | category_theory.differential_object.forget_faithful | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zero_morphisms : has_zero_morphisms (differential_object C) | { has_zero := λ X Y,
⟨{ f := 0 }⟩} | instance | category_theory.differential_object.has_zero_morphisms | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_f (P Q : differential_object C) : (0 : P ⟶ Q).f = 0 | rfl | lemma | category_theory.differential_object.zero_f | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_app {X Y : differential_object C} (f : X ≅ Y) : X.X ≅ Y.X | ⟨f.hom.f, f.inv.f, by { dsimp, rw [← comp_f, iso.hom_inv_id, id_f] },
by { dsimp, rw [← comp_f, iso.inv_hom_id, id_f] }⟩ | def | category_theory.differential_object.iso_app | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | An isomorphism of differential objects gives an isomorphism of the underlying objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_app_refl (X : differential_object C) : iso_app (iso.refl X) = iso.refl X.X | rfl | lemma | category_theory.differential_object.iso_app_refl | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_app_symm {X Y : differential_object C} (f : X ≅ Y) :
iso_app f.symm = (iso_app f).symm | rfl | lemma | category_theory.differential_object.iso_app_symm | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_app_trans {X Y Z : differential_object C} (f : X ≅ Y) (g : Y ≅ Z) :
iso_app (f ≪≫ g) = iso_app f ≪≫ iso_app g | rfl | lemma | category_theory.differential_object.iso_app_trans | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_iso {X Y : differential_object C}
(f : X.X ≅ Y.X) (hf : X.d ≫ f.hom⟦1⟧' = f.hom ≫ Y.d) : X ≅ Y | { hom := ⟨f.hom, hf⟩,
inv := ⟨f.inv, by { dsimp, rw [← functor.map_iso_inv, iso.comp_inv_eq, category.assoc,
iso.eq_inv_comp, functor.map_iso_hom, hf] }⟩,
hom_inv_id' := by { ext1, dsimp, exact f.hom_inv_id },
inv_hom_id' := by { ext1, dsimp, exact f.inv_hom_id } } | def | category_theory.differential_object.mk_iso | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | An isomorphism of differential objects can be constructed
from an isomorphism of the underlying objects that commutes with the differentials. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_differential_object (F : C ⥤ D)
(η : (shift_functor C (1:ℤ)).comp F ⟶ F.comp (shift_functor D (1:ℤ)))
(hF : ∀ c c', F.map (0 : c ⟶ c') = 0) :
differential_object C ⥤ differential_object D | { obj := λ X, { X := F.obj X.X,
d := F.map X.d ≫ η.app X.X,
d_squared' := begin
rw [functor.map_comp, ← functor.comp_map F (shift_functor D (1:ℤ))],
slice_lhs 2 3 { rw [← η.naturality X.d] },
rw [functor.comp_map],
slice_lhs 1 2 { rw [← F.map_comp, X.d_squared, hF] },
rw [zero_comp... | def | category_theory.functor.map_differential_object | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | A functor `F : C ⥤ D` which commutes with shift functors on `C` and `D` and preserves zero morphisms
can be lifted to a functor `differential_object C ⥤ differential_object D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_zero_object : has_zero_object (differential_object C) | by { refine ⟨⟨⟨0, 0⟩, λ X, ⟨⟨⟨⟨0⟩⟩, λ f, _⟩⟩, λ X, ⟨⟨⟨⟨0⟩⟩, λ f, _⟩⟩⟩⟩; ext, } | instance | category_theory.differential_object.has_zero_object | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category_of_differential_objects :
concrete_category (differential_object C) | { forget := forget C ⋙ category_theory.forget C } | instance | category_theory.differential_object.concrete_category_of_differential_objects | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [
"category_theory.forget"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
shift_functor (n : ℤ) : differential_object C ⥤ differential_object C | { obj := λ X,
{ X := X.X⟦n⟧,
d := X.d⟦n⟧' ≫ (shift_comm _ _ _).hom,
d_squared' := by rw [functor.map_comp, category.assoc, shift_comm_hom_comp_assoc,
←functor.map_comp_assoc, X.d_squared, functor.map_zero, zero_comp] },
map := λ X Y f,
{ f := f.f⟦n⟧',
comm' := begin
dsimp,
erw [cat... | def | category_theory.differential_object.shift_functor | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [
"functor.map_id"
] | The shift functor on `differential_object C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
shift_functor_add (m n : ℤ) :
shift_functor C (m + n) ≅ shift_functor C m ⋙ shift_functor C n | begin
refine nat_iso.of_components (λ X, mk_iso (shift_add X.X _ _) _) _,
{ dsimp,
rw [← cancel_epi ((shift_functor_add C m n).inv.app X.X)],
simp only [category.assoc, iso.inv_hom_id_app_assoc],
erw [← nat_trans.naturality_assoc],
dsimp,
simp only [functor.map_comp, category.assoc,
shift_... | def | category_theory.differential_object.shift_functor_add | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | The shift functor on `differential_object C` is additive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
shift_zero : shift_functor C 0 ≅ 𝟭 (differential_object C) | begin
refine nat_iso.of_components (λ X, mk_iso ((shift_functor_zero C ℤ).app X.X) _) _,
{ erw [← nat_trans.naturality],
dsimp,
simp only [shift_functor_zero_hom_app_shift, category.assoc], },
{ tidy, },
end | def | category_theory.differential_object.shift_zero | category_theory | src/category_theory/differential_object.lean | [
"data.int.basic",
"category_theory.shift.basic",
"category_theory.concrete_category.basic"
] | [] | The shift by zero is naturally isomorphic to the identity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discrete (α : Type u₁) | (as : α) | structure | category_theory.discrete | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete.mk_as {α : Type u₁} (X : discrete α) : discrete.mk X.as = X | by { ext, refl, } | lemma | category_theory.discrete.mk_as | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_equiv {α : Type u₁} : discrete α ≃ α | { to_fun := discrete.as,
inv_fun := discrete.mk,
left_inv := by tidy,
right_inv := by tidy, } | def | category_theory.discrete_equiv | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [
"inv_fun"
] | `discrete α` is equivalent to the original type `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discrete_category (α : Type u₁) : small_category (discrete α) | { hom := λ X Y, ulift (plift (X.as = Y.as)),
id := λ X, ulift.up (plift.up rfl),
comp := λ X Y Z g f, by { cases X, cases Y, cases Z, rcases f with ⟨⟨⟨⟩⟩⟩, exact g } } | instance | category_theory.discrete_category | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | The "discrete" category on a type, whose morphisms are equalities.
Because we do not allow morphisms in `Prop` (only in `Type`),
somewhat annoyingly we have to define `X ⟶ Y` as `ulift (plift (X = Y))`.
See <https://stacks.math.columbia.edu/tag/001A> | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.tactic.discrete_cases : tactic unit | `[cases_matching* [discrete _, (_ : discrete _) ⟶ (_ : discrete _), plift _]] | def | tactic.discrete_cases | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | A simple tactic to run `cases` on any `discrete α` hypotheses. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_hom {X Y : discrete α} (i : X ⟶ Y) : X.as = Y.as | i.down.down | lemma | category_theory.discrete.eq_of_hom | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | Extract the equation from a morphism in a discrete category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_to_hom {X Y : discrete α} (h : X.as = Y.as) : X ⟶ Y | eq_to_hom (by { ext, exact h, }) | abbreviation | category_theory.discrete.eq_to_hom | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | Promote an equation between the wrapped terms in `X Y : discrete α` to a morphism `X ⟶ Y`
in the discrete category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_to_iso {X Y : discrete α} (h : X.as = Y.as) : X ≅ Y | eq_to_iso (by { ext, exact h, }) | abbreviation | category_theory.discrete.eq_to_iso | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | Promote an equation between the wrapped terms in `X Y : discrete α` to an isomorphism `X ≅ Y`
in the discrete category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_to_hom' {a b : α} (h : a = b) : discrete.mk a ⟶ discrete.mk b | eq_to_hom h | abbreviation | category_theory.discrete.eq_to_hom' | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | A variant of `eq_to_hom` that lifts terms to the discrete category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_to_iso' {a b : α} (h : a = b) : discrete.mk a ≅ discrete.mk b | eq_to_iso h | abbreviation | category_theory.discrete.eq_to_iso' | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | A variant of `eq_to_iso` that lifts terms to the discrete category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_def (X : discrete α) : ulift.up (plift.up (eq.refl X.as)) = 𝟙 X | rfl | lemma | category_theory.discrete.id_def | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor {I : Type u₁} (F : I → C) : discrete I ⥤ C | { obj := F ∘ discrete.as,
map := λ X Y f, by { discrete_cases, cases f, exact 𝟙 (F X), } } | def | category_theory.discrete.functor | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | Any function `I → C` gives a functor `discrete I ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_obj {I : Type u₁} (F : I → C) (i : I) :
(discrete.functor F).obj (discrete.mk i) = F i | rfl | lemma | category_theory.discrete.functor_obj | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_map {I : Type u₁} (F : I → C) {i : discrete I} (f : i ⟶ i) :
(discrete.functor F).map f = 𝟙 (F i.as) | by tidy | lemma | category_theory.discrete.functor_map | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_comp {I : Type u₁} {J : Type u₁'} (f : J → C) (g : I → J) :
discrete.functor (f ∘ g) ≅ discrete.functor (discrete.mk ∘ g) ⋙ discrete.functor f | nat_iso.of_components (λ X, iso.refl _) (by tidy) | def | category_theory.discrete.functor_comp | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | The discrete functor induced by a composition of maps can be written as a
composition of two discrete functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans {I : Type u₁} {F G : discrete I ⥤ C}
(f : Π i : discrete I, F.obj i ⟶ G.obj i) : F ⟶ G | { app := f,
naturality' := λ X Y g, by { discrete_cases, cases g, simp, } } | def | category_theory.discrete.nat_trans | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | For functors out of a discrete category,
a natural transformation is just a collection of maps,
as the naturality squares are trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_iso {I : Type u₁} {F G : discrete I ⥤ C}
(f : Π i : discrete I, F.obj i ≅ G.obj i) : F ≅ G | nat_iso.of_components f (λ X Y g, by { discrete_cases, cases g, simp, }) | def | category_theory.discrete.nat_iso | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | For functors out of a discrete category,
a natural isomorphism is just a collection of isomorphisms,
as the naturality squares are trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_iso_app {I : Type u₁} {F G : discrete I ⥤ C}
(f : Π i : discrete I, F.obj i ≅ G.obj i) (i : discrete I) :
(discrete.nat_iso f).app i = f i | by tidy | lemma | category_theory.discrete.nat_iso_app | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_iso_functor {I : Type u₁} {F : discrete I ⥤ C} :
F ≅ discrete.functor (F.obj ∘ discrete.mk) | nat_iso $ λ i, by { discrete_cases, refl, } | def | category_theory.discrete.nat_iso_functor | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | Every functor `F` from a discrete category is naturally isomorphic (actually, equal) to
`discrete.functor (F.obj)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_nat_iso_discrete {I : Type u₁} {D : Type u₃} [category.{v₃} D]
(F : I → C) (G : C ⥤ D) : discrete.functor F ⋙ G ≅ discrete.functor (G.obj ∘ F) | nat_iso $ λ i, iso.refl _ | def | category_theory.discrete.comp_nat_iso_discrete | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | Composing `discrete.functor F` with another functor `G` amounts to composing `F` with `G.obj` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence {I : Type u₁} {J : Type u₂} (e : I ≃ J) : discrete I ≌ discrete J | { functor := discrete.functor (discrete.mk ∘ (e : I → J)),
inverse := discrete.functor (discrete.mk ∘ (e.symm : J → I)),
unit_iso := discrete.nat_iso (λ i, eq_to_iso (by { discrete_cases, simp })),
counit_iso := discrete.nat_iso (λ j, eq_to_iso (by { discrete_cases, simp })), } | def | category_theory.discrete.equivalence | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | We can promote a type-level `equiv` to
an equivalence between the corresponding `discrete` categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_equivalence {α : Type u₁} {β : Type u₂} (h : discrete α ≌ discrete β) : α ≃ β | { to_fun := discrete.as ∘ h.functor.obj ∘ discrete.mk,
inv_fun := discrete.as ∘ h.inverse.obj ∘ discrete.mk,
left_inv := λ a, by simpa using eq_of_hom (h.unit_iso.app (discrete.mk a)).2,
right_inv := λ a, by simpa using eq_of_hom (h.counit_iso.app (discrete.mk a)).1, } | def | category_theory.discrete.equiv_of_equivalence | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [
"inv_fun"
] | We can convert an equivalence of `discrete` categories to a type-level `equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
opposite (α : Type u₁) : (discrete α)ᵒᵖ ≌ discrete α | let F : discrete α ⥤ (discrete α)ᵒᵖ := discrete.functor (λ x, op (discrete.mk x)) in
begin
refine equivalence.mk (functor.left_op F) F _
(discrete.nat_iso $ λ X, by { discrete_cases, simp [F] }),
refine nat_iso.of_components (λ X, by { tactic.op_induction', discrete_cases, simp [F], }) _,
tidy
end | def | category_theory.discrete.opposite | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [
"opposite",
"tactic.op_induction'"
] | A discrete category is equivalent to its opposite category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_map_id
(F : discrete J ⥤ C) {j : discrete J} (f : j ⟶ j) : F.map f = 𝟙 (F.obj j) | begin
have h : f = 𝟙 j, { cases f, cases f, ext, },
rw h,
simp,
end | lemma | category_theory.discrete.functor_map_id | category_theory | src/category_theory/discrete_category.lean | [
"category_theory.eq_to_hom",
"data.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor.elements (F : C ⥤ Type w) | (Σ c : C, F.obj c) | def | category_theory.functor.elements | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | The type of objects for the category of elements of a functor `F : C ⥤ Type`
is a pair `(X : C, x : F.obj X)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category_of_elements (F : C ⥤ Type w) : category.{v} F.elements | { hom := λ p q, { f : p.1 ⟶ q.1 // (F.map f) p.2 = q.2 },
id := λ p, ⟨𝟙 p.1, by obviously⟩,
comp := λ p q r f g, ⟨f.val ≫ g.val, by obviously⟩ } | instance | category_theory.category_of_elements | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | The category structure on `F.elements`, for `F : C ⥤ Type`.
A morphism `(X, x) ⟶ (Y, y)` is a morphism `f : X ⟶ Y` in `C`, so `F.map f` takes `x` to `y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext (F : C ⥤ Type w) {x y : F.elements} (f g : x ⟶ y) (w : f.val = g.val) : f = g | subtype.ext_val w | lemma | category_theory.category_of_elements.ext | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [
"subtype.ext_val"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_val {F : C ⥤ Type w} {p q r : F.elements} {f : p ⟶ q} {g : q ⟶ r} :
(f ≫ g).val = f.val ≫ g.val | rfl | lemma | category_theory.category_of_elements.comp_val | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_val {F : C ⥤ Type w} {p : F.elements} : (𝟙 p : p ⟶ p).val = 𝟙 p.1 | rfl | lemma | category_theory.category_of_elements.id_val | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
groupoid_of_elements {G : Type u} [groupoid.{v} G] (F : G ⥤ Type w) :
groupoid F.elements | { inv := λ p q f, ⟨inv f.val,
calc F.map (inv f.val) q.2 = F.map (inv f.val) (F.map f.val p.2) : by rw f.2
... = (F.map f.val ≫ F.map (inv f.val)) p.2 : rfl
... = p.2 : by {rw ← F.map_comp, simp} ⟩,
inv_comp' := λ _ _ _, by { ext, simp },
comp_inv' := λ _ _ ... | instance | category_theory.groupoid_of_elements | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
π : F.elements ⥤ C | { obj := λ X, X.1,
map := λ X Y f, f.val } | def | category_theory.category_of_elements.π | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | The functor out of the category of elements which forgets the element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map {F₁ F₂ : C ⥤ Type w} (α : F₁ ⟶ F₂) : F₁.elements ⥤ F₂.elements | { obj := λ t, ⟨t.1, α.app t.1 t.2⟩,
map := λ t₁ t₂ k, ⟨k.1, by simpa [←k.2] using (functor_to_types.naturality _ _ α k.1 t₁.2).symm⟩ } | def | category_theory.category_of_elements.map | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | A natural transformation between functors induces a functor between the categories of elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_π {F₁ F₂ : C ⥤ Type w} (α : F₁ ⟶ F₂) : map α ⋙ π F₂ = π F₁ | rfl | lemma | category_theory.category_of_elements.map_π | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_structured_arrow : F.elements ⥤ structured_arrow punit F | { obj := λ X, structured_arrow.mk (λ _, X.2),
map := λ X Y f, structured_arrow.hom_mk f.val (by tidy) } | def | category_theory.category_of_elements.to_structured_arrow | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | The forward direction of the equivalence `F.elements ≅ (*, F)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_structured_arrow_obj (X) :
(to_structured_arrow F).obj X = { left := ⟨⟨⟩⟩, right := X.1, hom := λ _, X.2 } | rfl | lemma | category_theory.category_of_elements.to_structured_arrow_obj | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_comma_map_right {X Y} (f : X ⟶ Y) :
((to_structured_arrow F).map f).right = f.val | rfl | lemma | category_theory.category_of_elements.to_comma_map_right | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_structured_arrow : structured_arrow punit F ⥤ F.elements | { obj := λ X, ⟨X.right, X.hom (punit.star)⟩,
map := λ X Y f, ⟨f.right, congr_fun f.w'.symm punit.star⟩ } | def | category_theory.category_of_elements.from_structured_arrow | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | The reverse direction of the equivalence `F.elements ≅ (*, F)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_structured_arrow_obj (X) :
(from_structured_arrow F).obj X = ⟨X.right, X.hom (punit.star)⟩ | rfl | lemma | category_theory.category_of_elements.from_structured_arrow_obj | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_structured_arrow_map {X Y} (f : X ⟶ Y) :
(from_structured_arrow F).map f = ⟨f.right, congr_fun f.w'.symm punit.star⟩ | rfl | lemma | category_theory.category_of_elements.from_structured_arrow_map | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
structured_arrow_equivalence : F.elements ≌ structured_arrow punit F | equivalence.mk (to_structured_arrow F) (from_structured_arrow F)
(nat_iso.of_components (λ X, eq_to_iso (by tidy)) (by tidy))
(nat_iso.of_components
(λ X, structured_arrow.iso_mk (iso.refl _) (by tidy))
(by tidy)) | def | category_theory.category_of_elements.structured_arrow_equivalence | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | The equivalence between the category of elements `F.elements`
and the comma category `(*, F)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_costructured_arrow (F : Cᵒᵖ ⥤ Type v) : (F.elements)ᵒᵖ ⥤ costructured_arrow yoneda F | { obj := λ X, costructured_arrow.mk
((yoneda_sections (unop (unop X).fst) F).inv (ulift.up (unop X).2)),
map := λ X Y f,
begin
fapply costructured_arrow.hom_mk,
exact f.unop.val.unop,
ext y,
simp only [costructured_arrow.mk_hom_eq_self, yoneda_map_app, functor_to_types.comp, op_comp,
yoned... | def | category_theory.category_of_elements.to_costructured_arrow | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [
"quiver.hom.op_unop",
"subtype.val_eq_coe"
] | The forward direction of the equivalence `F.elementsᵒᵖ ≅ (yoneda, F)`,
given by `category_theory.yoneda_sections`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_costructured_arrow (F : Cᵒᵖ ⥤ Type v) : (costructured_arrow yoneda F)ᵒᵖ ⥤ F.elements | { obj := λ X, ⟨op (unop X).1, yoneda_equiv.1 (unop X).3⟩,
map := λ X Y f, ⟨f.unop.1.op,
begin
convert (congr_fun ((unop X).hom.naturality f.unop.left.op) (𝟙 _)).symm,
simp only [equiv.to_fun_as_coe, quiver.hom.unop_op, yoneda_equiv_apply,
types_comp_apply, category.comp_id, yoneda_obj_map],
have ... | def | category_theory.category_of_elements.from_costructured_arrow | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [
"equiv.to_fun_as_coe",
"quiver.hom.unop_op"
] | The reverse direction of the equivalence `F.elementsᵒᵖ ≅ (yoneda, F)`,
given by `category_theory.yoneda_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_costructured_arrow_obj_mk (F : Cᵒᵖ ⥤ Type v) {X : C} (f : yoneda.obj X ⟶ F) :
(from_costructured_arrow F).obj (op (costructured_arrow.mk f)) = ⟨op X, yoneda_equiv.1 f⟩ | rfl | lemma | category_theory.category_of_elements.from_costructured_arrow_obj_mk | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_to_costructured_arrow_eq (F : Cᵒᵖ ⥤ Type v) :
(to_costructured_arrow F).right_op ⋙ from_costructured_arrow F = 𝟭 _ | begin
apply functor.ext,
intros X Y f,
have : ∀ {a b : F.elements} (H : a = b),
↑(eq_to_hom H) = eq_to_hom (show a.fst = b.fst, by { cases H, refl }) :=
λ _ _ H, by { cases H, refl },
ext, simp[this],
tidy
end | lemma | category_theory.category_of_elements.from_to_costructured_arrow_eq | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [
"functor.ext"
] | The unit of the equivalence `F.elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_from_costructured_arrow_eq (F : Cᵒᵖ ⥤ Type v) :
(from_costructured_arrow F).right_op ⋙ to_costructured_arrow F = 𝟭 _ | begin
apply functor.hext,
{ intro X, cases X, cases X_right,
simp only [functor.id_obj, functor.right_op_obj,
to_costructured_arrow_obj, functor.comp_obj, costructured_arrow.mk],
congr,
ext x f,
convert congr_fun (X_hom.naturality f.op).symm (𝟙 X_left),
simp only [quiver.hom.unop_op, yone... | lemma | category_theory.category_of_elements.to_from_costructured_arrow_eq | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [
"proof_irrel_heq",
"quiver.hom.unop_op"
] | The counit of the equivalence `F.elementsᵒᵖ ≅ (yoneda, F)` is indeed iso. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
costructured_arrow_yoneda_equivalence (F : Cᵒᵖ ⥤ Type v) :
(F.elements)ᵒᵖ ≌ costructured_arrow yoneda F | equivalence.mk (to_costructured_arrow F) (from_costructured_arrow F).right_op
(nat_iso.op (eq_to_iso (from_to_costructured_arrow_eq F)))
(eq_to_iso $ to_from_costructured_arrow_eq F) | def | category_theory.category_of_elements.costructured_arrow_yoneda_equivalence | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [] | The equivalence `F.elementsᵒᵖ ≅ (yoneda, F)` given by yoneda lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
costructured_arrow_yoneda_equivalence_naturality {F₁ F₂ : Cᵒᵖ ⥤ Type v}
(α : F₁ ⟶ F₂) : (map α).op ⋙ to_costructured_arrow F₂ =
to_costructured_arrow F₁ ⋙ costructured_arrow.map α | begin
fapply functor.ext,
{ intro X,
simp only [costructured_arrow.map_mk, to_costructured_arrow_obj,
functor.op_obj, functor.comp_obj],
congr,
ext x f,
simpa using congr_fun (α.naturality f.op).symm (unop X).snd },
{ intros X Y f, ext,
have : ∀ {F : Cᵒᵖ ⥤ Type v} {a b : costructured_arr... | lemma | category_theory.category_of_elements.costructured_arrow_yoneda_equivalence_naturality | category_theory | src/category_theory/elements.lean | [
"category_theory.structured_arrow",
"category_theory.groupoid",
"category_theory.punit"
] | [
"functor.ext"
] | The equivalence `(-.elements)ᵒᵖ ≅ (yoneda, -)` of is actually a natural isomorphism of functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
End {C : Type u} [category_struct.{v} C] (X : C) | X ⟶ X | def | category_theory.End | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [] | Endomorphisms of an object in a category. Arguments order in multiplication agrees with
`function.comp`, not with `category.comp`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_one : has_one (End X) | ⟨𝟙 X⟩ | instance | category_theory.End.has_one | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inhabited : inhabited (End X) | ⟨𝟙 X⟩ | instance | category_theory.End.inhabited | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_mul : has_mul (End X) | ⟨λ x y, y ≫ x⟩ | instance | category_theory.End.has_mul | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [] | Multiplication of endomorphisms agrees with `function.comp`, not `category_struct.comp`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (f : X ⟶ X) : End X | f | def | category_theory.End.of | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [] | Assist the typechecker by expressing a morphism `X ⟶ X` as a term of `End X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_hom (f : End X) : X ⟶ X | f | def | category_theory.End.as_hom | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [] | Assist the typechecker by expressing an endomorphism `f : End X` as a term of `X ⟶ X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_def : (1 : End X) = 𝟙 X | rfl | lemma | category_theory.End.one_def | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_def (xs ys : End X) : xs * ys = ys ≫ xs | rfl | lemma | category_theory.End.mul_def | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid {C : Type u} [category.{v} C] {X : C} : monoid (End X) | { mul_one := category.id_comp,
one_mul := category.comp_id,
mul_assoc := λ x y z, (category.assoc z y x).symm,
..End.has_mul X, ..End.has_one X } | instance | category_theory.End.monoid | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [
"monoid",
"mul_assoc",
"mul_one",
"one_mul"
] | Endomorphisms of an object form a monoid | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_action_right {X Y : C} : mul_action (End Y) (X ⟶ Y) | { smul := λ r f, f ≫ r,
one_smul := category.comp_id,
mul_smul := λ r s f, eq.symm $ category.assoc _ _ _ } | instance | category_theory.End.mul_action_right | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [
"mul_action",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action_left {X : Cᵒᵖ} {Y : C} : mul_action (End X) (unop X ⟶ Y) | { smul := λ r f, r.unop ≫ f,
one_smul := category.id_comp,
mul_smul := λ r s f, category.assoc _ _ _ } | instance | category_theory.End.mul_action_left | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [
"mul_action",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_right {X Y : C} {r : End Y} {f : X ⟶ Y} : r • f = f ≫ r | rfl | lemma | category_theory.End.smul_right | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_left {X : Cᵒᵖ} {Y : C} {r : (End X)} {f : unop X ⟶ Y} : r • f = r.unop ≫ f | rfl | lemma | category_theory.End.smul_left | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group {C : Type u} [groupoid.{v} C] (X : C) : group (End X) | { mul_left_inv := groupoid.comp_inv, inv := groupoid.inv, ..End.monoid } | instance | category_theory.End.group | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [
"group",
"mul_left_inv"
] | In a groupoid, endomorphisms form a group | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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