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is_unit_iff_is_iso {C : Type u} [category.{v} C] {X : C} (f : End X) :
is_unit (f : End X) ↔ is_iso f | ⟨λ h, { out := ⟨h.unit.inv, ⟨h.unit.inv_val, h.unit.val_inv⟩⟩ },
λ h, by exactI ⟨⟨f, inv f, by simp, by simp⟩, rfl⟩⟩ | lemma | category_theory.is_unit_iff_is_iso | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Aut (X : C) | X ≅ X | def | category_theory.Aut | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [] | Automorphisms of an object in a category.
The order of arguments in multiplication agrees with
`function.comp`, not with `category.comp`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited : inhabited (Aut X) | ⟨iso.refl X⟩ | instance | category_theory.Aut.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 | |
Aut_mul_def (f g : Aut X) : f * g = g.trans f | rfl | lemma | category_theory.Aut.Aut_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 | |
Aut_inv_def (f : Aut X) : f ⁻¹ = f.symm | rfl | lemma | category_theory.Aut.Aut_inv_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 | |
units_End_equiv_Aut : (End X)ˣ ≃* Aut X | { to_fun := λ f, ⟨f.1, f.2, f.4, f.3⟩,
inv_fun := λ f, ⟨f.1, f.2, f.4, f.3⟩,
left_inv := λ ⟨f₁, f₂, f₃, f₄⟩, rfl,
right_inv := λ ⟨f₁, f₂, f₃, f₄⟩, rfl,
map_mul' := λ f g, by rcases f; rcases g; refl } | def | category_theory.Aut.units_End_equiv_Aut | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [
"inv_fun"
] | Units in the monoid of endomorphisms of an object
are (multiplicatively) equivalent to automorphisms of that object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Aut_mul_equiv_of_iso {X Y : C} (h : X ≅ Y) : Aut X ≃* Aut Y | { to_fun := λ x, ⟨h.inv ≫ x.hom ≫ h.hom, h.inv ≫ x.inv ≫ h.hom⟩,
inv_fun := λ y, ⟨h.hom ≫ y.hom ≫ h.inv, h.hom ≫ y.inv ≫ h.inv⟩,
left_inv := by tidy,
right_inv := by tidy,
map_mul' := by simp [Aut_mul_def] } | def | category_theory.Aut.Aut_mul_equiv_of_iso | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [
"inv_fun"
] | Isomorphisms induce isomorphisms of the automorphism group | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_End : End X →* End (f.obj X) | { to_fun := functor.map f,
map_mul' := λ x y, f.map_comp y x,
map_one' := f.map_id X } | def | category_theory.functor.map_End | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [] | `f.map` as a monoid hom between endomorphism monoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_Aut : Aut X →* Aut (f.obj X) | { to_fun := f.map_iso,
map_mul' := λ x y, f.map_iso_trans y x,
map_one' := f.map_iso_refl X } | def | category_theory.functor.map_Aut | category_theory | src/category_theory/endomorphism.lean | [
"algebra.hom.equiv.basic",
"category_theory.groupoid",
"category_theory.opposites",
"group_theory.group_action.defs"
] | [] | `f.map_iso` as a group hom between automorphism groups. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_mono_of_epi {A B : Cᵒᵖ} (f : A ⟶ B) [epi f] : mono f.unop | ⟨λ Z g h eq, quiver.hom.op_inj ((cancel_epi f).1 (quiver.hom.unop_inj eq))⟩ | instance | category_theory.unop_mono_of_epi | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [
"quiver.hom.op_inj",
"quiver.hom.unop_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_epi_of_mono {A B : Cᵒᵖ} (f : A ⟶ B) [mono f] : epi f.unop | ⟨λ Z g h eq, quiver.hom.op_inj ((cancel_mono f).1 (quiver.hom.unop_inj eq))⟩ | instance | category_theory.unop_epi_of_mono | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [
"quiver.hom.op_inj",
"quiver.hom.unop_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_mono_of_epi {A B : C} (f : A ⟶ B) [epi f] : mono f.op | ⟨λ Z g h eq, quiver.hom.unop_inj ((cancel_epi f).1 (quiver.hom.op_inj eq))⟩ | instance | category_theory.op_mono_of_epi | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [
"quiver.hom.op_inj",
"quiver.hom.unop_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_epi_of_mono {A B : C} (f : A ⟶ B) [mono f] : epi f.op | ⟨λ Z g h eq, quiver.hom.unop_inj ((cancel_mono f).1 (quiver.hom.op_inj eq))⟩ | instance | category_theory.op_epi_of_mono | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [
"quiver.hom.op_inj",
"quiver.hom.unop_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_mono {X Y : C} (f : X ⟶ Y) | (retraction : Y ⟶ X)
(id' : f ≫ retraction = 𝟙 X . obviously) | structure | category_theory.split_mono | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | A split monomorphism is a morphism `f : X ⟶ Y` with a given retraction `retraction f : Y ⟶ X`
such that `f ≫ retraction f = 𝟙 X`.
Every split monomorphism is a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_mono {X Y : C} (f : X ⟶ Y) : Prop | (exists_split_mono : nonempty (split_mono f)) | class | category_theory.is_split_mono | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | `is_split_mono f` is the assertion that `f` admits a retraction | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_mono.mk' {X Y : C} {f : X ⟶ Y} (sm : split_mono f) :
is_split_mono f | ⟨nonempty.intro sm⟩ | lemma | category_theory.is_split_mono.mk' | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | A constructor for `is_split_mono f` taking a `split_mono f` as an argument | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_epi {X Y : C} (f : X ⟶ Y) | (section_ : Y ⟶ X)
(id' : section_ ≫ f = 𝟙 Y . obviously) | structure | category_theory.split_epi | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | A split epimorphism is a morphism `f : X ⟶ Y` with a given section `section_ f : Y ⟶ X`
such that `section_ f ≫ f = 𝟙 Y`.
(Note that `section` is a reserved keyword, so we append an underscore.)
Every split epimorphism is an epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_epi {X Y : C} (f : X ⟶ Y) : Prop | (exists_split_epi : nonempty (split_epi f)) | class | category_theory.is_split_epi | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | `is_split_epi f` is the assertion that `f` admits a section | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_epi.mk' {X Y : C} {f : X ⟶ Y} (se : split_epi f) :
is_split_epi f | ⟨nonempty.intro se⟩ | lemma | category_theory.is_split_epi.mk' | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | A constructor for `is_split_epi f` taking a `split_epi f` as an argument | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
retraction {X Y : C} (f : X ⟶ Y) [hf : is_split_mono f] : Y ⟶ X | hf.exists_split_mono.some.retraction | def | category_theory.retraction | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | The chosen retraction of a split monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_mono.id {X Y : C} (f : X ⟶ Y) [hf : is_split_mono f] : f ≫ retraction f = 𝟙 X | hf.exists_split_mono.some.id | lemma | category_theory.is_split_mono.id | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_mono.split_epi {X Y : C} {f : X ⟶ Y} (sm : split_mono f) : split_epi (sm.retraction) | { section_ := f, } | def | category_theory.split_mono.split_epi | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | The retraction of a split monomorphism has an obvious section. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
retraction_is_split_epi {X Y : C} (f : X ⟶ Y) [hf : is_split_mono f] :
is_split_epi (retraction f) | is_split_epi.mk' (split_mono.split_epi _) | instance | category_theory.retraction_is_split_epi | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | The retraction of a split monomorphism is itself a split epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_epi_of_is_split_mono {X Y : C} (f : X ⟶ Y) [is_split_mono f] [epi f] : is_iso f | ⟨⟨retraction f, ⟨by simp, by simp [← cancel_epi f]⟩⟩⟩ | lemma | category_theory.is_iso_of_epi_of_is_split_mono | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | A split mono which is epi is an iso. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
section_ {X Y : C} (f : X ⟶ Y) [hf : is_split_epi f] : Y ⟶ X | hf.exists_split_epi.some.section_ | def | category_theory.section_ | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | The chosen section of a split epimorphism.
(Note that `section` is a reserved keyword, so we append an underscore.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_epi.id {X Y : C} (f : X ⟶ Y) [hf : is_split_epi f] : section_ f ≫ f = 𝟙 Y | hf.exists_split_epi.some.id | lemma | category_theory.is_split_epi.id | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_epi.split_mono {X Y : C} {f : X ⟶ Y} (se : split_epi f) : split_mono (se.section_) | { retraction := f, } | def | category_theory.split_epi.split_mono | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | The section of a split epimorphism has an obvious retraction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
section_is_split_mono {X Y : C} (f : X ⟶ Y) [hf : is_split_epi f] :
is_split_mono (section_ f) | is_split_mono.mk' (split_epi.split_mono _) | instance | category_theory.section_is_split_mono | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | The section of a split epimorphism is itself a split monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_mono_of_is_split_epi {X Y : C} (f : X ⟶ Y) [mono f] [is_split_epi f] : is_iso f | ⟨⟨section_ f, ⟨by simp [← cancel_mono f], by simp⟩⟩⟩ | lemma | category_theory.is_iso_of_mono_of_is_split_epi | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | A split epi which is mono is an iso. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_mono.of_iso {X Y : C} (f : X ⟶ Y) [is_iso f] : is_split_mono f | is_split_mono.mk' { retraction := inv f } | instance | category_theory.is_split_mono.of_iso | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | Every iso is a split mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_epi.of_iso {X Y : C} (f : X ⟶ Y) [is_iso f] : is_split_epi f | is_split_epi.mk' { section_ := inv f } | instance | category_theory.is_split_epi.of_iso | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | Every iso is a split epi. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_mono.mono {X Y : C} {f : X ⟶ Y} (sm : split_mono f) : mono f | { right_cancellation := λ Z g h w, begin replace w := w =≫ sm.retraction, simpa using w, end } | lemma | category_theory.split_mono.mono | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_split_mono.mono {X Y : C} (f : X ⟶ Y) [hf : is_split_mono f] : mono f | hf.exists_split_mono.some.mono | instance | category_theory.is_split_mono.mono | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | Every split mono is a mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_epi.epi {X Y : C} {f : X ⟶ Y} (se : split_epi f) : epi f | { left_cancellation := λ Z g h w, begin replace w := se.section_ ≫= w, simpa using w, end } | lemma | category_theory.split_epi.epi | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_split_epi.epi {X Y : C} (f : X ⟶ Y) [hf : is_split_epi f] : epi f | hf.exists_split_epi.some.epi | instance | category_theory.is_split_epi.epi | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | Every split epi is an epi. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso.of_mono_retraction' {X Y : C} {f : X ⟶ Y} (hf : split_mono f)
[mono $ hf.retraction] : is_iso f | ⟨⟨hf.retraction, ⟨by simp, (cancel_mono_id $ hf.retraction).mp (by simp)⟩⟩⟩ | lemma | category_theory.is_iso.of_mono_retraction' | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | Every split mono whose retraction is mono is an iso. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso.of_mono_retraction {X Y : C} (f : X ⟶ Y) [hf : is_split_mono f]
[hf' : mono $ retraction f] : is_iso f | @is_iso.of_mono_retraction' _ _ _ _ _ hf.exists_split_mono.some hf' | lemma | category_theory.is_iso.of_mono_retraction | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | Every split mono whose retraction is mono is an iso. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso.of_epi_section' {X Y : C} {f : X ⟶ Y} (hf : split_epi f)
[epi $ hf.section_] : is_iso f | ⟨⟨hf.section_, ⟨(cancel_epi_id $ hf.section_).mp (by simp), by simp⟩⟩⟩ | lemma | category_theory.is_iso.of_epi_section' | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | Every split epi whose section is epi is an iso. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso.of_epi_section {X Y : C} (f : X ⟶ Y) [hf : is_split_epi f]
[hf' : epi $ section_ f] : is_iso f | @is_iso.of_epi_section' _ _ _ _ _ hf.exists_split_epi.some hf' | lemma | category_theory.is_iso.of_epi_section | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | Every split epi whose section is epi is an iso. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
groupoid.of_trunc_split_mono
(all_split_mono : ∀ {X Y : C} (f : X ⟶ Y), trunc (is_split_mono f)) :
groupoid.{v₁} C | begin
apply groupoid.of_is_iso,
intros X Y f,
trunc_cases all_split_mono f,
trunc_cases all_split_mono (retraction f),
apply is_iso.of_mono_retraction,
end | def | category_theory.groupoid.of_trunc_split_mono | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [
"trunc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_mono_category | (is_split_mono_of_mono : ∀ {X Y : C} (f : X ⟶ Y) [mono f], is_split_mono f) | class | category_theory.split_mono_category | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | A split mono category is a category in which every monomorphism is split. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_epi_category | (is_split_epi_of_epi : ∀ {X Y : C} (f : X ⟶ Y) [epi f], is_split_epi f) | class | category_theory.split_epi_category | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | A split epi category is a category in which every epimorphism is split. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_mono_of_mono [split_mono_category C] {X Y : C} (f : X ⟶ Y) [mono f] :
is_split_mono f | split_mono_category.is_split_mono_of_mono _ | lemma | category_theory.is_split_mono_of_mono | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | In a category in which every monomorphism is split, every monomorphism splits. This is not an
instance because it would create an instance loop. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_epi_of_epi [split_epi_category C] {X Y : C} (f : X ⟶ Y) [epi f] :
is_split_epi f | split_epi_category.is_split_epi_of_epi _ | lemma | category_theory.is_split_epi_of_epi | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [] | In a category in which every epimorphism is split, every epimorphism splits. This is not an
instance because it would create an instance loop. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_mono.map {X Y : C} {f : X ⟶ Y} (sm : split_mono f) (F : C ⥤ D ) :
split_mono (F.map f) | { retraction := F.map (sm.retraction),
id' := by { rw [←functor.map_comp, split_mono.id, functor.map_id], } } | def | category_theory.split_mono.map | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [
"functor.map_id"
] | Split monomorphisms are also absolute monomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_epi.map {X Y : C} {f : X ⟶ Y} (se : split_epi f) (F : C ⥤ D ) :
split_epi (F.map f) | { section_ := F.map (se.section_),
id' := by { rw [←functor.map_comp, split_epi.id, functor.map_id], } } | def | category_theory.split_epi.map | category_theory | src/category_theory/epi_mono.lean | [
"category_theory.opposites",
"category_theory.groupoid"
] | [
"functor.map_id"
] | Split epimorphisms are also absolute epimorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence (C : Type u₁) [category.{v₁} C] (D : Type u₂) [category.{v₂} D] | mk' ::
(functor : C ⥤ D)
(inverse : D ⥤ C)
(unit_iso : 𝟭 C ≅ functor ⋙ inverse)
(counit_iso : inverse ⋙ functor ≅ 𝟭 D)
(functor_unit_iso_comp' : ∀(X : C), functor.map ((unit_iso.hom : 𝟭 C ⟶ functor ⋙ inverse).app X) ≫
counit_iso.hom.app (functor.obj X) = 𝟙 (functor.obj X) . obviously) | structure | category_theory.equivalence | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [
"mk'"
] | We define an equivalence as a (half)-adjoint equivalence, a pair of functors with
a unit and counit which are natural isomorphisms and the triangle law `Fη ≫ εF = 1`, or in other
words the composite `F ⟶ FGF ⟶ F` is the identity.
In `unit_inverse_comp`, we show that this is actually an adjoint equivalence, i.e.,... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit (e : C ≌ D) : 𝟭 C ⟶ e.functor ⋙ e.inverse | e.unit_iso.hom | abbreviation | category_theory.equivalence.unit | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | The unit of an equivalence of categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
counit (e : C ≌ D) : e.inverse ⋙ e.functor ⟶ 𝟭 D | e.counit_iso.hom | abbreviation | category_theory.equivalence.counit | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | The counit of an equivalence of categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_inv (e : C ≌ D) : e.functor ⋙ e.inverse ⟶ 𝟭 C | e.unit_iso.inv | abbreviation | category_theory.equivalence.unit_inv | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | The inverse of the unit of an equivalence of categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
counit_inv (e : C ≌ D) : 𝟭 D ⟶ e.inverse ⋙ e.functor | e.counit_iso.inv | abbreviation | category_theory.equivalence.counit_inv | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | The inverse of the counit of an equivalence of categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_mk'_unit (functor inverse unit_iso counit_iso f) :
(⟨functor, inverse, unit_iso, counit_iso, f⟩ : C ≌ D).unit = unit_iso.hom | rfl | lemma | category_theory.equivalence.equivalence_mk'_unit | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence_mk'_counit (functor inverse unit_iso counit_iso f) :
(⟨functor, inverse, unit_iso, counit_iso, f⟩ : C ≌ D).counit = counit_iso.hom | rfl | lemma | category_theory.equivalence.equivalence_mk'_counit | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence_mk'_unit_inv (functor inverse unit_iso counit_iso f) :
(⟨functor, inverse, unit_iso, counit_iso, f⟩ : C ≌ D).unit_inv = unit_iso.inv | rfl | lemma | category_theory.equivalence.equivalence_mk'_unit_inv | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence_mk'_counit_inv (functor inverse unit_iso counit_iso f) :
(⟨functor, inverse, unit_iso, counit_iso, f⟩ : C ≌ D).counit_inv = counit_iso.inv | rfl | lemma | category_theory.equivalence.equivalence_mk'_counit_inv | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_unit_comp (e : C ≌ D) (X : C) :
e.functor.map (e.unit.app X) ≫ e.counit.app (e.functor.obj X) = 𝟙 (e.functor.obj X) | e.functor_unit_iso_comp X | lemma | category_theory.equivalence.functor_unit_comp | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
counit_inv_functor_comp (e : C ≌ D) (X : C) :
e.counit_inv.app (e.functor.obj X) ≫ e.functor.map (e.unit_inv.app X) = 𝟙 (e.functor.obj X) | begin
erw [iso.inv_eq_inv
(e.functor.map_iso (e.unit_iso.app X) ≪≫ e.counit_iso.app (e.functor.obj X)) (iso.refl _)],
exact e.functor_unit_comp X
end | lemma | category_theory.equivalence.counit_inv_functor_comp | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
counit_inv_app_functor (e : C ≌ D) (X : C) :
e.counit_inv.app (e.functor.obj X) = e.functor.map (e.unit.app X) | by { symmetry, erw [←iso.comp_hom_eq_id (e.counit_iso.app _), functor_unit_comp], refl } | lemma | category_theory.equivalence.counit_inv_app_functor | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
counit_app_functor (e : C ≌ D) (X : C) :
e.counit.app (e.functor.obj X) = e.functor.map (e.unit_inv.app X) | by { erw [←iso.hom_comp_eq_id (e.functor.map_iso (e.unit_iso.app X)), functor_unit_comp], refl } | lemma | category_theory.equivalence.counit_app_functor | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_inverse_comp (e : C ≌ D) (Y : D) :
e.unit.app (e.inverse.obj Y) ≫ e.inverse.map (e.counit.app Y) = 𝟙 (e.inverse.obj Y) | begin
rw [←id_comp (e.inverse.map _), ←map_id e.inverse, ←counit_inv_functor_comp, map_comp],
dsimp,
rw [←iso.hom_inv_id_assoc (e.unit_iso.app _) (e.inverse.map (e.functor.map _)),
app_hom, app_inv],
slice_lhs 2 3 { erw [e.unit.naturality] },
slice_lhs 1 2 { erw [e.unit.naturality] },
slice_lhs 4 4
... | lemma | category_theory.equivalence.unit_inverse_comp | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [
"map_comp",
"map_id"
] | The other triangle equality. The proof follows the following proof in Globular:
http://globular.science/1905.001 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse_counit_inv_comp (e : C ≌ D) (Y : D) :
e.inverse.map (e.counit_inv.app Y) ≫ e.unit_inv.app (e.inverse.obj Y) = 𝟙 (e.inverse.obj Y) | begin
erw [iso.inv_eq_inv
(e.unit_iso.app (e.inverse.obj Y) ≪≫ e.inverse.map_iso (e.counit_iso.app Y)) (iso.refl _)],
exact e.unit_inverse_comp Y
end | lemma | category_theory.equivalence.inverse_counit_inv_comp | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_app_inverse (e : C ≌ D) (Y : D) :
e.unit.app (e.inverse.obj Y) = e.inverse.map (e.counit_inv.app Y) | by { erw [←iso.comp_hom_eq_id (e.inverse.map_iso (e.counit_iso.app Y)), unit_inverse_comp], refl } | lemma | category_theory.equivalence.unit_app_inverse | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_inv_app_inverse (e : C ≌ D) (Y : D) :
e.unit_inv.app (e.inverse.obj Y) = e.inverse.map (e.counit.app Y) | by { symmetry, erw [←iso.hom_comp_eq_id (e.unit_iso.app _), unit_inverse_comp], refl } | lemma | category_theory.equivalence.unit_inv_app_inverse | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fun_inv_map (e : C ≌ D) (X Y : D) (f : X ⟶ Y) :
e.functor.map (e.inverse.map f) = e.counit.app X ≫ f ≫ e.counit_inv.app Y | (nat_iso.naturality_2 (e.counit_iso) f).symm | lemma | category_theory.equivalence.fun_inv_map | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_fun_map (e : C ≌ D) (X Y : C) (f : X ⟶ Y) :
e.inverse.map (e.functor.map f) = e.unit_inv.app X ≫ f ≫ e.unit.app Y | (nat_iso.naturality_1 (e.unit_iso) f).symm | lemma | category_theory.equivalence.inv_fun_map | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjointify_η : 𝟭 C ≅ F ⋙ G | calc
𝟭 C ≅ F ⋙ G : η
... ≅ F ⋙ (𝟭 D ⋙ G) : iso_whisker_left F (left_unitor G).symm
... ≅ F ⋙ ((G ⋙ F) ⋙ G) : iso_whisker_left F (iso_whisker_right ε.symm G)
... ≅ F ⋙ (G ⋙ (F ⋙ G)) : iso_whisker_left F (associator G F G)
... ≅ (F ⋙ G) ⋙ (F ⋙ G) : (associator F G (F ⋙ G)).symm
... ≅ 𝟭 C... | def | category_theory.equivalence.adjointify_η | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | If `η : 𝟭 C ≅ F ⋙ G` is part of a (not necessarily half-adjoint) equivalence, we can upgrade it
to a refined natural isomorphism `adjointify_η η : 𝟭 C ≅ F ⋙ G` which exhibits the properties
required for a half-adjoint equivalence. See `equivalence.mk`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjointify_η_ε (X : C) :
F.map ((adjointify_η η ε).hom.app X) ≫ ε.hom.app (F.obj X) = 𝟙 (F.obj X) | begin
dsimp [adjointify_η], simp,
have := ε.hom.naturality (F.map (η.inv.app X)), dsimp at this, rw [this], clear this,
rw [←assoc _ _ (F.map _)],
have := ε.hom.naturality (ε.inv.app $ F.obj X), dsimp at this, rw [this], clear this,
have := (ε.app $ F.obj X).hom_inv_id, dsimp at this, rw [this], clear this,
... | lemma | category_theory.equivalence.adjointify_η_ε | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk (F : C ⥤ D) (G : D ⥤ C)
(η : 𝟭 C ≅ F ⋙ G) (ε : G ⋙ F ≅ 𝟭 D) : C ≌ D | ⟨F, G, adjointify_η η ε, ε, adjointify_η_ε η ε⟩ | definition | category_theory.equivalence.mk | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | Every equivalence of categories consisting of functors `F` and `G` such that `F ⋙ G` and
`G ⋙ F` are naturally isomorphic to identity functors can be transformed into a half-adjoint
equivalence without changing `F` or `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl : C ≌ C | ⟨𝟭 C, 𝟭 C, iso.refl _, iso.refl _, λ X, category.id_comp _⟩ | def | category_theory.equivalence.refl | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | Equivalence of categories is reflexive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm (e : C ≌ D) : D ≌ C | ⟨e.inverse, e.functor, e.counit_iso.symm, e.unit_iso.symm, e.inverse_counit_inv_comp⟩ | def | category_theory.equivalence.symm | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | Equivalence of categories is symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans (e : C ≌ D) (f : D ≌ E) : C ≌ E | { functor := e.functor ⋙ f.functor,
inverse := f.inverse ⋙ e.inverse,
unit_iso :=
begin
refine iso.trans e.unit_iso _,
exact iso_whisker_left e.functor (iso_whisker_right f.unit_iso e.inverse) ,
end,
counit_iso :=
begin
refine iso.trans _ f.counit_iso,
exact iso_whisker_left f.inverse (iso_w... | def | category_theory.equivalence.trans | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [
"functor.map_id"
] | Equivalence of categories is transitive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fun_inv_id_assoc (e : C ≌ D) (F : C ⥤ E) : e.functor ⋙ e.inverse ⋙ F ≅ F | (functor.associator _ _ _).symm ≪≫ iso_whisker_right e.unit_iso.symm F ≪≫ F.left_unitor | def | category_theory.equivalence.fun_inv_id_assoc | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | Composing a functor with both functors of an equivalence yields a naturally isomorphic
functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fun_inv_id_assoc_hom_app (e : C ≌ D) (F : C ⥤ E) (X : C) :
(fun_inv_id_assoc e F).hom.app X = F.map (e.unit_inv.app X) | by { dsimp [fun_inv_id_assoc], tidy } | lemma | category_theory.equivalence.fun_inv_id_assoc_hom_app | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fun_inv_id_assoc_inv_app (e : C ≌ D) (F : C ⥤ E) (X : C) :
(fun_inv_id_assoc e F).inv.app X = F.map (e.unit.app X) | by { dsimp [fun_inv_id_assoc], tidy } | lemma | category_theory.equivalence.fun_inv_id_assoc_inv_app | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_fun_id_assoc (e : C ≌ D) (F : D ⥤ E) : e.inverse ⋙ e.functor ⋙ F ≅ F | (functor.associator _ _ _).symm ≪≫ iso_whisker_right e.counit_iso F ≪≫ F.left_unitor | def | category_theory.equivalence.inv_fun_id_assoc | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | Composing a functor with both functors of an equivalence yields a naturally isomorphic
functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_fun_id_assoc_hom_app (e : C ≌ D) (F : D ⥤ E) (X : D) :
(inv_fun_id_assoc e F).hom.app X = F.map (e.counit.app X) | by { dsimp [inv_fun_id_assoc], tidy } | lemma | category_theory.equivalence.inv_fun_id_assoc_hom_app | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_fun_id_assoc_inv_app (e : C ≌ D) (F : D ⥤ E) (X : D) :
(inv_fun_id_assoc e F).inv.app X = F.map (e.counit_inv.app X) | by { dsimp [inv_fun_id_assoc], tidy } | lemma | category_theory.equivalence.inv_fun_id_assoc_inv_app | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_left (e : C ≌ D) : (C ⥤ E) ≌ (D ⥤ E) | equivalence.mk
((whiskering_left _ _ _).obj e.inverse)
((whiskering_left _ _ _).obj e.functor)
(nat_iso.of_components (λ F, (e.fun_inv_id_assoc F).symm) (by tidy))
(nat_iso.of_components (λ F, e.inv_fun_id_assoc F) (by tidy)) | def | category_theory.equivalence.congr_left | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | If `C` is equivalent to `D`, then `C ⥤ E` is equivalent to `D ⥤ E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_right (e : C ≌ D) : (E ⥤ C) ≌ (E ⥤ D) | equivalence.mk
((whiskering_right _ _ _).obj e.functor)
((whiskering_right _ _ _).obj e.inverse)
(nat_iso.of_components
(λ F, F.right_unitor.symm ≪≫ iso_whisker_left F e.unit_iso ≪≫ functor.associator _ _ _)
(by tidy))
(nat_iso.of_components
(λ F, functor.associator _ _ _ ≪≫ iso_whisker_left F e.cou... | def | category_theory.equivalence.congr_right | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | If `C` is equivalent to `D`, then `E ⥤ C` is equivalent to `E ⥤ D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cancel_unit_right {X Y : C}
(f f' : X ⟶ Y) :
f ≫ e.unit.app Y = f' ≫ e.unit.app Y ↔ f = f' | by simp only [cancel_mono] | lemma | category_theory.equivalence.cancel_unit_right | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_unit_inv_right {X Y : C}
(f f' : X ⟶ e.inverse.obj (e.functor.obj Y)) :
f ≫ e.unit_inv.app Y = f' ≫ e.unit_inv.app Y ↔ f = f' | by simp only [cancel_mono] | lemma | category_theory.equivalence.cancel_unit_inv_right | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_counit_right {X Y : D}
(f f' : X ⟶ e.functor.obj (e.inverse.obj Y)) :
f ≫ e.counit.app Y = f' ≫ e.counit.app Y ↔ f = f' | by simp only [cancel_mono] | lemma | category_theory.equivalence.cancel_counit_right | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_counit_inv_right {X Y : D}
(f f' : X ⟶ Y) :
f ≫ e.counit_inv.app Y = f' ≫ e.counit_inv.app Y ↔ f = f' | by simp only [cancel_mono] | lemma | category_theory.equivalence.cancel_counit_inv_right | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_unit_right_assoc {W X X' Y : C}
(f : W ⟶ X) (g : X ⟶ Y) (f' : W ⟶ X') (g' : X' ⟶ Y) :
f ≫ g ≫ e.unit.app Y = f' ≫ g' ≫ e.unit.app Y ↔ f ≫ g = f' ≫ g' | by simp only [←category.assoc, cancel_mono] | lemma | category_theory.equivalence.cancel_unit_right_assoc | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_counit_inv_right_assoc {W X X' Y : D}
(f : W ⟶ X) (g : X ⟶ Y) (f' : W ⟶ X') (g' : X' ⟶ Y) :
f ≫ g ≫ e.counit_inv.app Y = f' ≫ g' ≫ e.counit_inv.app Y ↔ f ≫ g = f' ≫ g' | by simp only [←category.assoc, cancel_mono] | lemma | category_theory.equivalence.cancel_counit_inv_right_assoc | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_unit_right_assoc' {W X X' Y Y' Z : C}
(f : W ⟶ X) (g : X ⟶ Y) (h : Y ⟶ Z) (f' : W ⟶ X') (g' : X' ⟶ Y') (h' : Y' ⟶ Z) :
f ≫ g ≫ h ≫ e.unit.app Z = f' ≫ g' ≫ h' ≫ e.unit.app Z ↔ f ≫ g ≫ h = f' ≫ g' ≫ h' | by simp only [←category.assoc, cancel_mono] | lemma | category_theory.equivalence.cancel_unit_right_assoc' | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_counit_inv_right_assoc' {W X X' Y Y' Z : D}
(f : W ⟶ X) (g : X ⟶ Y) (h : Y ⟶ Z) (f' : W ⟶ X') (g' : X' ⟶ Y') (h' : Y' ⟶ Z) :
f ≫ g ≫ h ≫ e.counit_inv.app Z = f' ≫ g' ≫ h' ≫ e.counit_inv.app Z ↔ f ≫ g ≫ h = f' ≫ g' ≫ h' | by simp only [←category.assoc, cancel_mono] | lemma | category_theory.equivalence.cancel_counit_inv_right_assoc' | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_nat (e : C ≌ C) : ℕ → (C ≌ C) | | 0 := equivalence.refl
| 1 := e
| (n+2) := e.trans (pow_nat (n+1)) | def | category_theory.equivalence.pow_nat | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | Natural number powers of an auto-equivalence. Use `(^)` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow (e : C ≌ C) : ℤ → (C ≌ C) | | (int.of_nat n) := e.pow_nat n
| (int.neg_succ_of_nat n) := e.symm.pow_nat (n+1) | def | category_theory.equivalence.pow | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | Powers of an auto-equivalence. Use `(^)` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_zero (e : C ≌ C) : e^(0 : ℤ) = equivalence.refl | rfl | lemma | category_theory.equivalence.pow_zero | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_one (e : C ≌ C) : e^(1 : ℤ) = e | rfl | lemma | category_theory.equivalence.pow_one | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_neg_one (e : C ≌ C) : e^(-1 : ℤ) = e.symm | rfl | lemma | category_theory.equivalence.pow_neg_one | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalence (F : C ⥤ D) | mk' ::
(inverse : D ⥤ C)
(unit_iso : 𝟭 C ≅ F ⋙ inverse)
(counit_iso : inverse ⋙ F ≅ 𝟭 D)
(functor_unit_iso_comp' : ∀ (X : C), F.map ((unit_iso.hom : 𝟭 C ⟶ F ⋙ inverse).app X) ≫
counit_iso.hom.app (F.obj X) = 𝟙 (F.obj X) . obviously) | class | category_theory.is_equivalence | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [
"mk'"
] | A functor that is part of a (half) adjoint equivalence | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_equivalence (F : C ≌ D) : is_equivalence F.functor | { ..F } | instance | category_theory.is_equivalence.of_equivalence | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_equivalence_inverse (F : C ≌ D) : is_equivalence F.inverse | is_equivalence.of_equivalence F.symm | instance | category_theory.is_equivalence.of_equivalence_inverse | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk {F : C ⥤ D} (G : D ⥤ C)
(η : 𝟭 C ≅ F ⋙ G) (ε : G ⋙ F ≅ 𝟭 D) : is_equivalence F | ⟨G, adjointify_η η ε, ε, adjointify_η_ε η ε⟩ | definition | category_theory.is_equivalence.mk | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | To see that a functor is an equivalence, it suffices to provide an inverse functor `G` such that
`F ⋙ G` and `G ⋙ F` are naturally isomorphic to identity functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_equivalence (F : C ⥤ D) [is_equivalence F] : C ≌ D | ⟨F, is_equivalence.inverse F, is_equivalence.unit_iso, is_equivalence.counit_iso,
is_equivalence.functor_unit_iso_comp⟩ | def | category_theory.functor.as_equivalence | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | Interpret a functor that is an equivalence as an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_equivalence_refl : is_equivalence (𝟭 C) | is_equivalence.of_equivalence equivalence.refl | instance | category_theory.functor.is_equivalence_refl | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv (F : C ⥤ D) [is_equivalence F] : D ⥤ C | is_equivalence.inverse F | def | category_theory.functor.inv | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | The inverse functor of a functor that is an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_equivalence_inv (F : C ⥤ D) [is_equivalence F] : is_equivalence F.inv | is_equivalence.of_equivalence F.as_equivalence.symm | instance | category_theory.functor.is_equivalence_inv | category_theory | src/category_theory/equivalence.lean | [
"category_theory.functor.fully_faithful",
"category_theory.full_subcategory",
"category_theory.whiskering",
"category_theory.essential_image",
"tactic.slice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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