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compose_path {X : C} : Π {Y : C} (p : path X Y), X ⟶ Y | | _ path.nil := 𝟙 X
| _ (path.cons p e) := compose_path p ≫ e | def | category_theory.compose_path | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [
"path"
] | A path in a category can be composed to a single morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compose_path_to_path {X Y : C} (f : X ⟶ Y) : compose_path (f.to_path) = f | category.id_comp _ | lemma | category_theory.compose_path_to_path | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compose_path_comp {X Y Z : C} (f : path X Y) (g : path Y Z) :
compose_path (f.comp g) = compose_path f ≫ compose_path g | begin
induction g with Y' Z' g e ih,
{ simp, },
{ simp [ih], },
end | lemma | category_theory.compose_path_comp | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [
"ih",
"path"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compose_path_id {X : paths C} : compose_path (𝟙 X) = 𝟙 X | rfl | lemma | category_theory.compose_path_id | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compose_path_comp' {X Y Z : paths C} (f : X ⟶ Y) (g : Y ⟶ Z) :
compose_path (f ≫ g) = compose_path f ≫ compose_path g | compose_path_comp f g | lemma | category_theory.compose_path_comp' | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
path_composition : paths C ⥤ C | { obj := λ X, X,
map := λ X Y f, compose_path f, } | def | category_theory.path_composition | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [] | Composition of paths as functor from the path category of a category to the category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
paths_hom_rel : hom_rel (paths C) | λ X Y p q, (path_composition C).map p = (path_composition C).map q | def | category_theory.paths_hom_rel | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [
"hom_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_quotient_paths : C ⥤ quotient (paths_hom_rel C) | { obj := λ X, quotient.mk X,
map := λ X Y f, quot.mk _ f.to_path,
map_id' := λ X, quot.sound (quotient.comp_closure.of _ _ _ (by simp)),
map_comp' := λ X Y Z f g, quot.sound (quotient.comp_closure.of _ _ _ (by simp)), } | def | category_theory.to_quotient_paths | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [] | The functor from a category to the canonical quotient of its path category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_paths_to : quotient (paths_hom_rel C) ⥤ C | quotient.lift _ (path_composition C) (λ X Y p q w, w) | def | category_theory.quotient_paths_to | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [] | The functor from the canonical quotient of a path category of a category
to the original category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_paths_equiv : quotient (paths_hom_rel C) ≌ C | { functor := quotient_paths_to C,
inverse := to_quotient_paths C,
unit_iso := nat_iso.of_components (λ X, by { cases X, refl, }) begin
intros,
cases X, cases Y,
induction f,
dsimp,
simp only [category.comp_id, category.id_comp],
apply quot.sound,
apply quotient.comp_closure.of,
simp ... | def | category_theory.quotient_paths_equiv | category_theory | src/category_theory/path_category.lean | [
"category_theory.eq_to_hom",
"category_theory.quotient",
"combinatorics.quiver.path"
] | [] | The canonical quotient of the path category of a category
is equivalent to the original category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
empty_equivalence : discrete.{w} pempty ≌ discrete.{v} pempty | equivalence.mk
{ obj := pempty.elim ∘ discrete.as, map := λ x, x.as.elim }
{ obj := pempty.elim ∘ discrete.as, map := λ x, x.as.elim }
(by tidy) (by tidy) | def | category_theory.functor.empty_equivalence | category_theory | src/category_theory/pempty.lean | [
"category_theory.discrete_category"
] | [
"pempty",
"pempty.elim"
] | Equivalence between two empty categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
empty : discrete.{w} pempty ⥤ C | discrete.functor pempty.elim | def | category_theory.functor.empty | category_theory | src/category_theory/pempty.lean | [
"category_theory.discrete_category"
] | [
"pempty",
"pempty.elim"
] | The canonical functor out of the empty category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
empty_ext (F G : discrete.{w} pempty ⥤ C) : F ≅ G | discrete.nat_iso (λ x, x.as.elim) | def | category_theory.functor.empty_ext | category_theory | src/category_theory/pempty.lean | [
"category_theory.discrete_category"
] | [
"pempty"
] | Any two functors out of the empty category are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_from_empty (F : discrete.{w} pempty ⥤ C) : F ≅ empty C | empty_ext _ _ | def | category_theory.functor.unique_from_empty | category_theory | src/category_theory/pempty.lean | [
"category_theory.discrete_category"
] | [
"pempty"
] | Any functor out of the empty category is isomorphic to the canonical functor from the empty
category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
empty_ext' (F G : discrete.{w} pempty ⥤ C) : F = G | functor.ext (λ x, x.as.elim) (λ x _ _, x.as.elim) | lemma | category_theory.functor.empty_ext' | category_theory | src/category_theory/pempty.lean | [
"category_theory.discrete_category"
] | [
"functor.ext",
"pempty"
] | Any two functors out of the empty category are *equal*. You probably want to use
`empty_ext` instead of this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star : C ⥤ discrete punit | (functor.const _).obj ⟨⟨⟩⟩ | def | category_theory.functor.star | category_theory | src/category_theory/punit.lean | [
"category_theory.functor.const",
"category_theory.discrete_category"
] | [
"functor.const"
] | The constant functor sending everything to `punit.star`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
punit_ext (F G : C ⥤ discrete punit) : F ≅ G | nat_iso.of_components (λ _, eq_to_iso dec_trivial) (λ _ _ _, dec_trivial) | def | category_theory.functor.punit_ext | category_theory | src/category_theory/punit.lean | [
"category_theory.functor.const",
"category_theory.discrete_category"
] | [] | Any two functors to `discrete punit` are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
punit_ext' (F G : C ⥤ discrete punit) : F = G | functor.ext (λ _, dec_trivial) (λ _ _ _, dec_trivial) | lemma | category_theory.functor.punit_ext' | category_theory | src/category_theory/punit.lean | [
"category_theory.functor.const",
"category_theory.discrete_category"
] | [
"functor.ext"
] | Any two functors to `discrete punit` are *equal*.
You probably want to use `punit_ext` instead of this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_punit (X : C) : discrete punit.{v+1} ⥤ C | (functor.const _).obj X | abbreviation | category_theory.functor.from_punit | category_theory | src/category_theory/punit.lean | [
"category_theory.functor.const",
"category_theory.discrete_category"
] | [
"functor.const"
] | The functor from `discrete punit` sending everything to the given object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv : (discrete punit ⥤ C) ≌ C | { functor :=
{ obj := λ F, F.obj ⟨⟨⟩⟩,
map := λ F G θ, θ.app ⟨⟨⟩⟩ },
inverse := functor.const _,
unit_iso :=
begin
apply nat_iso.of_components _ _,
intro X,
apply discrete.nat_iso,
rintro ⟨⟨⟩⟩,
apply iso.refl _,
intros,
ext ⟨⟨⟩⟩,
simp,
end,
counit_iso :=
begin
refin... | def | category_theory.functor.equiv | category_theory | src/category_theory/punit.lean | [
"category_theory.functor.const",
"category_theory.discrete_category"
] | [
"equiv",
"functor.const"
] | Functors from `discrete punit` are equivalent to the category itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_punit_iff_unique :
nonempty (C ≌ discrete punit) ↔ (nonempty C) ∧ (∀ x y : C, nonempty $ unique (x ⟶ y)) | begin
split,
{ rintro ⟨h⟩,
refine ⟨⟨h.inverse.obj ⟨⟨⟩⟩⟩, λ x y, nonempty.intro _⟩,
apply (unique_of_subsingleton _), swap,
{ have hx : x ⟶ h.inverse.obj ⟨⟨⟩⟩ := by convert h.unit.app x,
have hy : h.inverse.obj ⟨⟨⟩⟩ ⟶ y := by convert h.unit_inv.app y,
exact hx ≫ hy, },
have : ∀ z, z = h.u... | theorem | category_theory.equiv_punit_iff_unique | category_theory | src/category_theory/punit.lean | [
"category_theory.functor.const",
"category_theory.discrete_category"
] | [
"category_theory.equivalence.mk",
"functor.const",
"unique",
"unique_of_subsingleton"
] | A category being equivalent to `punit` is equivalent to it having a unique morphism between
any two objects. (In fact, such a category is also a groupoid; see `groupoid.of_hom_unique`) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_rel (C) [quiver C] | Π ⦃X Y : C⦄, (X ⟶ Y) → (X ⟶ Y) → Prop | def | hom_rel | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [
"quiver"
] | A `hom_rel` on `C` consists of a relation on every hom-set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congruence : Prop | (is_equiv : ∀ {X Y}, is_equiv _ (@r X Y))
(comp_left : ∀ {X Y Z} (f : X ⟶ Y) {g g' : Y ⟶ Z}, r g g' → r (f ≫ g) (f ≫ g'))
(comp_right : ∀ {X Y Z} {f f' : X ⟶ Y} (g : Y ⟶ Z), r f f' → r (f ≫ g) (f' ≫ g)) | class | category_theory.congruence | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [] | A `hom_rel` is a congruence when it's an equivalence on every hom-set, and it can be composed
from left and right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient | (as : C) | structure | category_theory.quotient | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [] | A type synonym for `C`, thought of as the objects of the quotient category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_closure ⦃s t : C⦄ : (s ⟶ t) → (s ⟶ t) → Prop
| intro {a b} (f : s ⟶ a) (m₁ m₂ : a ⟶ b) (g : b ⟶ t) (h : r m₁ m₂) :
comp_closure (f ≫ m₁ ≫ g) (f ≫ m₂ ≫ g) | inductive | category_theory.quotient.comp_closure | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [] | Generates the closure of a family of relations w.r.t. composition from left and right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_closure.of {a b} (m₁ m₂ : a ⟶ b) (h : r m₁ m₂) : comp_closure r m₁ m₂ | by simpa using comp_closure.intro (𝟙 _) m₁ m₂ (𝟙 _) h | lemma | category_theory.quotient.comp_closure.of | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_left {a b c : C} (f : a ⟶ b) : Π (g₁ g₂ : b ⟶ c) (h : comp_closure r g₁ g₂),
comp_closure r (f ≫ g₁) (f ≫ g₂) | | _ _ ⟨x, m₁, m₂, y, h⟩ := by simpa using comp_closure.intro (f ≫ x) m₁ m₂ y h | lemma | category_theory.quotient.comp_left | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_right {a b c : C} (g : b ⟶ c) : Π (f₁ f₂ : a ⟶ b) (h : comp_closure r f₁ f₂),
comp_closure r (f₁ ≫ g) (f₂ ≫ g) | | _ _ ⟨x, m₁, m₂, y, h⟩ := by simpa using comp_closure.intro x m₁ m₂ (y ≫ g) h | lemma | category_theory.quotient.comp_right | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom (s t : quotient r) | quot $ @comp_closure C _ r s.as t.as | def | category_theory.quotient.hom | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [] | Hom-sets of the quotient category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp ⦃a b c : quotient r⦄ : hom r a b → hom r b c → hom r a c | λ hf hg, quot.lift_on hf ( λ f, quot.lift_on hg (λ g, quot.mk _ (f ≫ g))
(λ g₁ g₂ h, quot.sound $ comp_left r f g₁ g₂ h) )
(λ f₁ f₂ h, quot.induction_on hg $ λ g, quot.sound $ comp_right r g f₁ f₂ h) | def | category_theory.quotient.comp | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [] | Composition in the quotient category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_mk {a b c : quotient r} (f : a.as ⟶ b.as) (g : b.as ⟶ c.as) :
comp r (quot.mk _ f) (quot.mk _ g) = quot.mk _ (f ≫ g) | rfl | lemma | category_theory.quotient.comp_mk | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
category : category (quotient r) | { hom := hom r,
id := λ a, quot.mk _ (𝟙 a.as),
comp := comp r } | instance | category_theory.quotient.category | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor : C ⥤ quotient r | { obj := λ a, { as := a },
map := λ _ _ f, quot.mk _ f } | def | category_theory.quotient.functor | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [] | The functor from a category to its quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induction {P : Π {a b : quotient r}, (a ⟶ b) → Prop}
(h : ∀ {x y : C} (f : x ⟶ y), P ((functor r).map f)) :
∀ {a b : quotient r} (f : a ⟶ b), P f | by { rintros ⟨x⟩ ⟨y⟩ ⟨f⟩, exact h f, } | lemma | category_theory.quotient.induction | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sound {a b : C} {f₁ f₂ : a ⟶ b} (h : r f₁ f₂) :
(functor r).map f₁ = (functor r).map f₂ | by simpa using quot.sound (comp_closure.intro (𝟙 a) f₁ f₂ (𝟙 b) h) | lemma | category_theory.quotient.sound | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_map_eq_iff [congruence r] {X Y : C} (f f' : X ⟶ Y) :
(functor r).map f = (functor r).map f' ↔ r f f' | begin
split,
{ erw quot.eq,
intro h,
induction h with m m' hm,
{ cases hm, apply congruence.comp_left, apply congruence.comp_right, assumption, },
{ apply refl },
{ apply symm, assumption },
{ apply trans; assumption }, },
{ apply quotient.sound },
end | lemma | category_theory.quotient.functor_map_eq_iff | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [
"quot.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift : quotient r ⥤ D | { obj := λ a, F.obj a.as,
map := λ a b hf, quot.lift_on hf (λ f, F.map f)
(by { rintro _ _ ⟨_, _, _, _, h⟩, simp [H _ _ _ _ h], }),
map_id' := λ a, F.map_id a.as,
map_comp' := by { rintros a b c ⟨f⟩ ⟨g⟩, exact F.map_comp f g, } } | def | category_theory.quotient.lift | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [
"lift"
] | The induced functor on the quotient category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_spec : (functor r) ⋙ lift r F H = F | begin
apply functor.ext, rotate,
{ rintro X, refl, },
{ rintro X Y f, simp, },
end | lemma | category_theory.quotient.lift_spec | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [
"functor.ext",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_unique (Φ : quotient r ⥤ D) (hΦ : (functor r) ⋙ Φ = F) : Φ = lift r F H | begin
subst_vars,
apply functor.hext,
{ rintro X, dsimp [lift, functor], congr, ext, refl, },
{ rintro X Y f,
dsimp [lift, functor],
apply quot.induction_on f,
rintro ff,
simp only [quot.lift_on_mk, functor.comp_map],
congr; ext; refl, },
end | lemma | category_theory.quotient.lift_unique | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [
"lift",
"lift_unique",
"quot.lift_on_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift.is_lift : (functor r) ⋙ lift r F H ≅ F | nat_iso.of_components (λ X, iso.refl _) (by tidy) | def | category_theory.quotient.lift.is_lift | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [
"lift"
] | The original functor factors through the induced functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift.is_lift_hom (X : C) : (lift.is_lift r F H).hom.app X = 𝟙 (F.obj X) | rfl | lemma | category_theory.quotient.lift.is_lift_hom | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift.is_lift_inv (X : C) : (lift.is_lift r F H).inv.app X = 𝟙 (F.obj X) | rfl | lemma | category_theory.quotient.lift.is_lift_inv | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_map_functor_map {X Y : C} (f : X ⟶ Y) :
(lift r F H).map ((functor r).map f) = F.map f | by { rw ←(nat_iso.naturality_1 (lift.is_lift r F H)), dsimp, simp, } | lemma | category_theory.quotient.lift_map_functor_map | category_theory | src/category_theory/quotient.lean | [
"category_theory.natural_isomorphism",
"category_theory.eq_to_hom"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simple (X : C) : Prop | (mono_is_iso_iff_nonzero : ∀ {Y : C} (f : Y ⟶ X) [mono f], is_iso f ↔ (f ≠ 0)) | class | category_theory.simple | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [] | An object is simple if monomorphisms into it are (exclusively) either isomorphisms or zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_mono_of_nonzero {X Y : C} [simple Y] {f : X ⟶ Y} [mono f] (w : f ≠ 0) :
is_iso f | (simple.mono_is_iso_iff_nonzero f).mpr w | lemma | category_theory.is_iso_of_mono_of_nonzero | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [] | A nonzero monomorphism to a simple object is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simple.of_iso {X Y : C} [simple Y] (i : X ≅ Y) : simple X | { mono_is_iso_iff_nonzero := λ Z f m, begin
resetI,
haveI : mono (f ≫ i.hom) := mono_comp _ _,
split,
{ introsI h w,
haveI j : is_iso (f ≫ i.hom), apply_instance,
rw simple.mono_is_iso_iff_nonzero at j,
unfreezingI { subst w, },
simpa using j, },
{ intro h,
haveI j : is... | lemma | category_theory.simple.of_iso | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simple.iff_of_iso {X Y : C} (i : X ≅ Y) : simple X ↔ simple Y | ⟨λ h, by exactI simple.of_iso i.symm, λ h, by exactI simple.of_iso i⟩ | lemma | category_theory.simple.iff_of_iso | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_zero_of_nonzero_from_simple
{X Y : C} [simple X] {f : X ⟶ Y} [has_kernel f] (w : f ≠ 0) :
kernel.ι f = 0 | begin
classical,
by_contra,
haveI := is_iso_of_mono_of_nonzero h,
exact w (eq_zero_of_epi_kernel f),
end | lemma | category_theory.kernel_zero_of_nonzero_from_simple | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [
"by_contra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_of_nonzero_to_simple [has_equalizers C] {X Y : C} [simple Y]
{f : X ⟶ Y} [has_image f] (w : f ≠ 0) : epi f | begin
rw ←image.fac f,
haveI : is_iso (image.ι f) := is_iso_of_mono_of_nonzero (λ h, w (eq_zero_of_image_eq_zero h)),
apply epi_comp,
end | lemma | category_theory.epi_of_nonzero_to_simple | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_to_simple_zero_of_not_iso
{X Y : C} [simple Y] {f : X ⟶ Y} [mono f] (w : is_iso f → false) : f = 0 | begin
classical,
by_contra,
exact w (is_iso_of_mono_of_nonzero h)
end | lemma | category_theory.mono_to_simple_zero_of_not_iso | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [
"by_contra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_nonzero (X : C) [simple.{v} X] : 𝟙 X ≠ 0 | (simple.mono_is_iso_iff_nonzero (𝟙 X)).mp (by apply_instance) | lemma | category_theory.id_nonzero | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simple.not_is_zero (X : C) [simple X] : ¬ is_zero X | by simpa [limits.is_zero.iff_id_eq_zero] using id_nonzero X | lemma | category_theory.simple.not_is_zero | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_not_simple [simple (0 : C)] : false | (simple.mono_is_iso_iff_nonzero (0 : (0 : C) ⟶ (0 : C))).mp ⟨⟨0, by tidy⟩⟩ rfl | lemma | category_theory.zero_not_simple | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [] | We don't want the definition of 'simple' to include the zero object, so we check that here. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simple_of_cosimple (X : C) (h : ∀ {Z : C} (f : X ⟶ Z) [epi f], is_iso f ↔ (f ≠ 0)) :
simple X | ⟨λ Y f I,
begin
classical,
fsplit,
{ introsI,
have hx := cokernel.π_of_epi f,
by_contra,
substI h,
exact (h _).mp (cokernel.π_of_zero _ _) hx },
{ intro hf,
suffices : epi f,
{ exactI is_iso_of_mono_of_epi _ },
apply preadditive.epi_of_cokernel_zero,
by_contra h',
exact coke... | lemma | category_theory.simple_of_cosimple | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [
"by_contra"
] | In an abelian category, an object satisfying the dual of the definition of a simple object is
simple. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_epi_of_nonzero {X Y : C} [simple X] {f : X ⟶ Y} [epi f] (w : f ≠ 0) :
is_iso f | begin
-- `f ≠ 0` means that `kernel.ι f` is not an iso, and hence zero, and hence `f` is a mono.
haveI : mono f :=
preadditive.mono_of_kernel_zero (mono_to_simple_zero_of_not_iso (kernel_not_iso_of_nonzero w)),
exact is_iso_of_mono_of_epi f,
end | lemma | category_theory.is_iso_of_epi_of_nonzero | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [] | A nonzero epimorphism from a simple object is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_zero_of_nonzero_to_simple
{X Y : C} [simple Y] {f : X ⟶ Y} (w : f ≠ 0) :
cokernel.π f = 0 | begin
classical,
by_contradiction h,
haveI := is_iso_of_epi_of_nonzero h,
exact w (eq_zero_of_mono_cokernel f),
end | lemma | category_theory.cokernel_zero_of_nonzero_to_simple | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [
"by_contradiction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_from_simple_zero_of_not_iso
{X Y : C} [simple X] {f : X ⟶ Y} [epi f] (w : is_iso f → false) : f = 0 | begin
classical,
by_contra,
exact w (is_iso_of_epi_of_nonzero h),
end | lemma | category_theory.epi_from_simple_zero_of_not_iso | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [
"by_contra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biprod.is_iso_inl_iff_is_zero (X Y : C) : is_iso (biprod.inl : X ⟶ X ⊞ Y) ↔ is_zero Y | begin
rw [biprod.is_iso_inl_iff_id_eq_fst_comp_inl, ←biprod.total, add_right_eq_self],
split,
{ intro h, replace h := h =≫ biprod.snd,
simpa [←is_zero.iff_is_split_epi_eq_zero (biprod.snd : X ⊞ Y ⟶ Y)] using h, },
{ intro h, rw is_zero.iff_is_split_epi_eq_zero (biprod.snd : X ⊞ Y ⟶ Y) at h,
rw [h, zero_... | lemma | category_theory.biprod.is_iso_inl_iff_is_zero | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indecomposable_of_simple (X : C) [simple X] : indecomposable X | ⟨simple.not_is_zero X,
λ Y Z i, begin
refine or_iff_not_imp_left.mpr (λ h, _),
rw is_zero.iff_is_split_mono_eq_zero (biprod.inl : Y ⟶ Y ⊞ Z) at h,
change biprod.inl ≠ 0 at h,
rw ←(simple.mono_is_iso_iff_nonzero biprod.inl) at h,
{ rwa biprod.is_iso_inl_iff_is_zero at h, },
{ exact simple.of_iso i.symm, },
... | lemma | category_theory.indecomposable_of_simple | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [] | Any simple object in a preadditive category is indecomposable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simple_of_is_simple_order_subobject (X : C) [is_simple_order (subobject X)] : simple X | begin
split, introsI, split,
{ introI i,
rw subobject.is_iso_iff_mk_eq_top at i,
intro w,
rw ←subobject.mk_eq_bot_iff_zero at w,
exact is_simple_order.bot_ne_top (w.symm.trans i), },
{ intro i,
rcases is_simple_order.eq_bot_or_eq_top (subobject.mk f) with h|h,
{ rw subobject.mk_eq_bot_iff_... | lemma | category_theory.simple_of_is_simple_order_subobject | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [
"is_simple_order",
"is_simple_order.bot_ne_top"
] | If `X` has subobject lattice `{⊥, ⊤}`, then `X` is simple. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simple_iff_subobject_is_simple_order (X : C) : simple X ↔ is_simple_order (subobject X) | ⟨by { introI h, apply_instance, },
by { introI h, exact simple_of_is_simple_order_subobject X, }⟩ | lemma | category_theory.simple_iff_subobject_is_simple_order | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [
"is_simple_order"
] | `X` is simple iff it has subobject lattice `{⊥, ⊤}`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subobject_simple_iff_is_atom {X : C} (Y : subobject X) : simple (Y : C) ↔ is_atom Y | (simple_iff_subobject_is_simple_order _).trans
((order_iso.is_simple_order_iff (subobject_order_iso Y)).trans
set.is_simple_order_Iic_iff_is_atom) | lemma | category_theory.subobject_simple_iff_is_atom | category_theory | src/category_theory/simple.lean | [
"category_theory.limits.shapes.zero_morphisms",
"category_theory.limits.shapes.kernels",
"category_theory.abelian.basic",
"category_theory.subobject.lattice",
"order.atoms"
] | [
"is_atom",
"order_iso.is_simple_order_iff",
"set.is_simple_order_Iic_iff_is_atom"
] | A subobject is simple iff it is an atom in the subobject lattice. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
single_obj | quiver.single_obj | abbreviation | category_theory.single_obj | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"quiver.single_obj"
] | Abbreviation that allows writing `category_theory.single_obj` rather than `quiver.single_obj`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category_struct [has_one α] [has_mul α] : category_struct (single_obj α) | { hom := λ _ _, α,
comp := λ _ _ _ x y, y * x,
id := λ _, 1 } | instance | category_theory.single_obj.category_struct | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [] | One and `flip (*)` become `id` and `comp` for morphisms of the single object category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category [monoid α] : category (single_obj α) | { comp_id' := λ _ _, one_mul,
id_comp' := λ _ _, mul_one,
assoc' := λ _ _ _ _ x y z, (mul_assoc z y x).symm } | instance | category_theory.single_obj.category | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"monoid",
"mul_assoc",
"mul_one",
"one_mul"
] | Monoid laws become category laws for the single object category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_as_one [monoid α] (x : single_obj α) : 𝟙 x = 1 | rfl | lemma | category_theory.single_obj.id_as_one | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_as_mul [monoid α] {x y z : single_obj α} (f : x ⟶ y) (g : y ⟶ z) :
f ≫ g = g * f | rfl | lemma | category_theory.single_obj.comp_as_mul | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
groupoid [group α] : groupoid (single_obj α) | { inv := λ _ _ x, x⁻¹,
inv_comp' := λ _ _, mul_right_inv,
comp_inv' := λ _ _, mul_left_inv } | instance | category_theory.single_obj.groupoid | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"group",
"mul_left_inv",
"mul_right_inv"
] | Groupoid structure on `single_obj α`.
See <https://stacks.math.columbia.edu/tag/0019>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_as_inv [group α] {x y : single_obj α} (f : x ⟶ y) : inv f = f⁻¹ | by { ext, rw [comp_as_mul, inv_mul_self, id_as_one] } | lemma | category_theory.single_obj.inv_as_inv | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"group",
"inv_mul_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star : single_obj α | quiver.single_obj.star α | abbreviation | category_theory.single_obj.star | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"quiver.single_obj.star"
] | Abbreviation that allows writing `category_theory.single_obj.star` rather than
`quiver.single_obj.star`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_End [monoid α] : α ≃* End (single_obj.star α) | { map_mul' := λ x y, rfl,
.. equiv.refl α } | def | category_theory.single_obj.to_End | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"equiv.refl",
"monoid"
] | The endomorphisms monoid of the only object in `single_obj α` is equivalent to the original
monoid α. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_End_def [monoid α] (x : α) : to_End α x = x | rfl | lemma | category_theory.single_obj.to_End_def | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_hom (α : Type u) (β : Type v) [monoid α] [monoid β] :
(α →* β) ≃ (single_obj α) ⥤ (single_obj β) | { to_fun := λ f,
{ obj := id,
map := λ _ _, ⇑f,
map_id' := λ _, f.map_one,
map_comp' := λ _ _ _ x y, f.map_mul y x },
inv_fun := λ f,
{ to_fun := @functor.map _ _ _ _ f (single_obj.star α) (single_obj.star α),
map_one' := f.map_id _,
map_mul' := λ x y, f.map_comp y x },
left_inv := λ ⟨... | def | category_theory.single_obj.map_hom | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"inv_fun",
"monoid"
] | There is a 1-1 correspondence between monoid homomorphisms `α → β` and functors between the
corresponding single-object categories. It means that `single_obj` is a fully faithful
functor.
See <https://stacks.math.columbia.edu/tag/001F> --
although we do not characterize when the functor is full or faithful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_hom_id (α : Type u) [monoid α] : map_hom α α (monoid_hom.id α) = 𝟭 _ | rfl | lemma | category_theory.single_obj.map_hom_id | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"monoid",
"monoid_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_hom_comp {α : Type u} {β : Type v} [monoid α] [monoid β] (f : α →* β)
{γ : Type w} [monoid γ] (g : β →* γ) :
map_hom α γ (g.comp f) = map_hom α β f ⋙ map_hom β γ g | rfl | lemma | category_theory.single_obj.map_hom_comp | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
difference_functor {C G} [category C] [group G] (f : C → G) : C ⥤ single_obj G | { obj := λ _, (),
map := λ x y _, f y * (f x)⁻¹,
map_id' := by { intro, rw [single_obj.id_as_one, mul_right_inv] },
map_comp' := by { intros, rw [single_obj.comp_as_mul, ←mul_assoc,
mul_left_inj, mul_assoc, inv_mul_self, mul_one] } } | def | category_theory.single_obj.difference_functor | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"group",
"inv_mul_self",
"mul_assoc",
"mul_left_inj",
"mul_one",
"mul_right_inv"
] | Given a function `f : C → G` from a category to a group, we get a functor
`C ⥤ G` sending any morphism `x ⟶ y` to `f y * (f x)⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_functor {α : Type u} {β : Type v} [monoid α] [monoid β] (f : α →* β) :
(single_obj α) ⥤ (single_obj β) | single_obj.map_hom α β f | def | monoid_hom.to_functor | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"monoid"
] | Reinterpret a monoid homomorphism `f : α → β` as a functor `(single_obj α) ⥤ (single_obj β)`.
See also `category_theory.single_obj.map_hom` for an equivalence between these types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_to_functor (α : Type u) [monoid α] : (id α).to_functor = 𝟭 _ | rfl | lemma | monoid_hom.id_to_functor | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_to_functor {α : Type u} {β : Type v} [monoid α] [monoid β] (f : α →* β)
{γ : Type w} [monoid γ] (g : β →* γ) :
(g.comp f).to_functor = f.to_functor ⋙ g.to_functor | rfl | lemma | monoid_hom.comp_to_functor | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Aut : αˣ ≃* Aut (single_obj.star α) | (units.map_equiv (single_obj.to_End α)).trans $
Aut.units_End_equiv_Aut _ | def | units.to_Aut | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"units.map_equiv"
] | The units in a monoid are (multiplicatively) equivalent to
the automorphisms of `star` when we think of the monoid as a single-object category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Aut_hom (x : αˣ) : (to_Aut α x).hom = single_obj.to_End α x | rfl | lemma | units.to_Aut_hom | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Aut_inv (x : αˣ) :
(to_Aut α x).inv = single_obj.to_End α (x⁻¹ : αˣ) | rfl | lemma | units.to_Aut_inv | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Cat : Mon ⥤ Cat | { obj := λ x, Cat.of (single_obj x),
map := λ x y f, single_obj.map_hom x y f } | def | Mon.to_Cat | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"Mon"
] | The fully faithful functor from `Mon` to `Cat`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Cat_full : full to_Cat | { preimage := λ x y, (single_obj.map_hom x y).inv_fun,
witness' := λ x y, by apply equiv.right_inv } | instance | Mon.to_Cat_full | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Cat_faithful : faithful to_Cat | { map_injective' := λ x y, by apply equiv.injective } | instance | Mon.to_Cat_faithful | category_theory | src/category_theory/single_obj.lean | [
"category_theory.endomorphism",
"category_theory.category.Cat",
"algebra.category.Mon.basic",
"combinatorics.quiver.single_obj"
] | [
"equiv.injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skeletal : Prop | ∀ ⦃X Y : C⦄, is_isomorphic X Y → X = Y | def | category_theory.skeletal | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | A category is skeletal if isomorphic objects are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_skeleton_of (F : D ⥤ C) | (skel : skeletal D)
(eqv : is_equivalence F) | structure | category_theory.is_skeleton_of | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | `is_skeleton_of C D F` says that `F : D ⥤ C` exhibits `D` as a skeletal full subcategory of `C`,
in particular `F` is a (strong) equivalence and `D` is skeletal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.eq_of_iso {F₁ F₂ : D ⥤ C} [quiver.is_thin C] (hC : skeletal C)
(hF : F₁ ≅ F₂) : F₁ = F₂ | functor.ext (λ X, hC ⟨hF.app X⟩) (λ _ _ _, subsingleton.elim _ _) | lemma | category_theory.functor.eq_of_iso | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [
"functor.ext",
"quiver.is_thin"
] | If `C` is thin and skeletal, then any naturally isomorphic functors to `C` are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_skeletal [quiver.is_thin C] (hC : skeletal C) : skeletal (D ⥤ C) | λ F₁ F₂ h, h.elim (functor.eq_of_iso hC) | lemma | category_theory.functor_skeletal | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [
"quiver.is_thin"
] | If `C` is thin and skeletal, `D ⥤ C` is skeletal.
`category_theory.functor_thin` shows it is thin also. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skeleton : Type u₁ | induced_category C quotient.out | def | category_theory.skeleton | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [
"quotient.out"
] | Construct the skeleton category as the induced category on the isomorphism classes, and derive
its category structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_skeleton : skeleton C ⥤ C | induced_functor _ | def | category_theory.from_skeleton | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | The functor from the skeleton of `C` to `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skeleton_equivalence : skeleton C ≌ C | (from_skeleton C).as_equivalence | def | category_theory.skeleton_equivalence | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | The equivalence between the skeleton and the category itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skeleton_skeletal : skeletal (skeleton C) | begin
rintro X Y ⟨h⟩,
have : X.out ≈ Y.out := ⟨(from_skeleton C).map_iso h⟩,
simpa using quotient.sound this,
end | lemma | category_theory.skeleton_skeletal | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skeleton_is_skeleton : is_skeleton_of C (skeleton C) (from_skeleton C) | { skel := skeleton_skeletal C,
eqv := from_skeleton.is_equivalence C } | def | category_theory.skeleton_is_skeleton | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | The `skeleton` of `C` given by choice is a skeleton of `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence.skeleton_equiv (e : C ≌ D) : skeleton C ≃ skeleton D | let f := ((skeleton_equivalence C).trans e).trans (skeleton_equivalence D).symm in
{ to_fun := f.functor.obj,
inv_fun := f.inverse.obj,
left_inv := λ X, skeleton_skeletal C ⟨(f.unit_iso.app X).symm⟩,
right_inv := λ Y, skeleton_skeletal D ⟨(f.counit_iso.app Y)⟩, } | def | category_theory.equivalence.skeleton_equiv | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [
"inv_fun"
] | Two categories which are categorically equivalent have skeletons with equivalent objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thin_skeleton : Type u₁ | quotient (is_isomorphic_setoid C) | def | category_theory.thin_skeleton | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | Construct the skeleton category by taking the quotient of objects. This construction gives a
preorder with nice definitional properties, but is only really appropriate for thin categories.
If your original category is not thin, you probably want to be using `skeleton` instead of this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited_thin_skeleton [inhabited C] : inhabited (thin_skeleton C) | ⟨quotient.mk default⟩ | instance | category_theory.inhabited_thin_skeleton | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
thin_skeleton.preorder : preorder (thin_skeleton C) | { le := quotient.lift₂ (λ X Y, nonempty (X ⟶ Y))
begin
rintros _ _ _ _ ⟨i₁⟩ ⟨i₂⟩,
exact propext ⟨nonempty.map (λ f, i₁.inv ≫ f ≫ i₂.hom),
nonempty.map (λ f, i₁.hom ≫ f ≫ i₂.inv)⟩,
end,
le_refl :=
begin
refine quotient.ind (λ a, _),
exact ⟨𝟙 _⟩,
end,
le_trans := λ a b c, quotient.induc... | instance | category_theory.thin_skeleton.preorder | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [
"nonempty.map",
"nonempty.map2"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_thin_skeleton : C ⥤ thin_skeleton C | { obj := quotient.mk,
map := λ X Y f, hom_of_le (nonempty.intro f) } | def | category_theory.to_thin_skeleton | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [] | The functor from a category to its thin skeleton. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
thin : quiver.is_thin (thin_skeleton C) | λ _ _, ⟨by { rintros ⟨⟨f₁⟩⟩ ⟨⟨f₂⟩⟩, refl }⟩ | instance | category_theory.thin_skeleton.thin | category_theory | src/category_theory/skeletal.lean | [
"category_theory.adjunction.basic",
"category_theory.category.preorder",
"category_theory.isomorphism_classes",
"category_theory.thin"
] | [
"quiver.is_thin"
] | The thin skeleton is thin. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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