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app {P Q : C} (f : P ⟶ Q) (a : over P) : over Q | a.hom ≫ f | def | category_theory.abelian.app | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | This is just composition of morphisms in `C`. Another way to express this would be
`(over.map f).obj a`, but our definition has nicer definitional properties. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
app_hom {P Q : C} (f : P ⟶ Q) (a : over P) : (app f a).hom = a.hom ≫ f | rfl | lemma | category_theory.abelian.app_hom | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_equal (P : C) (f g : over P) : Prop | ∃ (R : C) (p : R ⟶ f.1) (q : R ⟶ g.1) (_ : epi p) (_ : epi q), p ≫ f.hom = q ≫ g.hom | def | category_theory.abelian.pseudo_equal | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | Two arrows `f : X ⟶ P` and `g : Y ⟶ P` are called pseudo-equal if there is some object
`R` and epimorphisms `p : R ⟶ X` and `q : R ⟶ Y` such that `p ≫ f = q ≫ g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_equal_refl {P : C} : reflexive (pseudo_equal P) | λ f, ⟨f.1, 𝟙 f.1, 𝟙 f.1, by apply_instance, by apply_instance, by simp⟩ | lemma | category_theory.abelian.pseudo_equal_refl | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_equal_symm {P : C} : symmetric (pseudo_equal P) | λ f g ⟨R, p, q, ep, eq, comm⟩, ⟨R, q, p, eq, ep, comm.symm⟩ | lemma | category_theory.abelian.pseudo_equal_symm | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_equal_trans {P : C} : transitive (pseudo_equal P) | λ f g h ⟨R, p, q, ep, eq, comm⟩ ⟨R', p', q', ep', eq', comm'⟩,
begin
refine ⟨pullback q p', pullback.fst ≫ p, pullback.snd ≫ q', _, _, _⟩,
{ resetI, exact epi_comp _ _ },
{ resetI, exact epi_comp _ _ },
{ rw [category.assoc, comm, ←category.assoc, pullback.condition,
category.assoc, comm', category.assoc]... | lemma | category_theory.abelian.pseudo_equal_trans | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [
"comm"
] | Pseudoequality is transitive: Just take the pullback. The pullback morphisms will
be epimorphisms since in an abelian category, pullbacks of epimorphisms are epimorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudoelement.setoid (P : C) : setoid (over P) | ⟨_, ⟨pseudo_equal_refl, pseudo_equal_symm, pseudo_equal_trans⟩⟩ | def | category_theory.abelian.pseudoelement.setoid | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | The arrows with codomain `P` equipped with the equivalence relation of being pseudo-equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudoelement (P : C) : Type (max u v) | quotient (pseudoelement.setoid P) | def | category_theory.abelian.pseudoelement | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | A `pseudoelement` of `P` is just an equivalence class of arrows ending in `P` by being
pseudo-equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
object_to_sort : has_coe_to_sort C (Type (max u v)) | ⟨λ P, pseudoelement P⟩ | def | category_theory.abelian.pseudoelement.object_to_sort | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | A coercion from an object of an abelian category to its pseudoelements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
over_to_sort {P : C} : has_coe (over P) (pseudoelement P) | ⟨quot.mk (pseudo_equal P)⟩ | def | category_theory.abelian.pseudoelement.over_to_sort | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | A coercion from an arrow with codomain `P` to its associated pseudoelement. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
over_coe_def {P Q : C} (a : Q ⟶ P) : (a : pseudoelement P) = ⟦a⟧ | rfl | lemma | category_theory.abelian.pseudoelement.over_coe_def | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_apply_aux {P Q : C} (f : P ⟶ Q) (a b : over P) :
a ≈ b → app f a ≈ app f b | λ ⟨R, p, q, ep, eq, comm⟩,
⟨R, p, q, ep, eq, show p ≫ a.hom ≫ f = q ≫ b.hom ≫ f, by rw reassoc_of comm⟩ | lemma | category_theory.abelian.pseudoelement.pseudo_apply_aux | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | If two elements are pseudo-equal, then their composition with a morphism is, too. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_apply {P Q : C} (f : P ⟶ Q) : P → Q | quotient.map (λ (g : over P), app f g) (pseudo_apply_aux f) | def | category_theory.abelian.pseudoelement.pseudo_apply | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [
"quotient.map"
] | A morphism `f` induces a function `pseudo_apply f` on pseudoelements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_to_fun {P Q : C} : has_coe_to_fun (P ⟶ Q) (λ _, P → Q) | ⟨pseudo_apply⟩ | def | category_theory.abelian.pseudoelement.hom_to_fun | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | A coercion from morphisms to functions on pseudoelements | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_apply_mk {P Q : C} (f : P ⟶ Q) (a : over P) : f ⟦a⟧ = ⟦a.hom ≫ f⟧ | rfl | lemma | category_theory.abelian.pseudoelement.pseudo_apply_mk | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply {P Q R : C} (f : P ⟶ Q) (g : Q ⟶ R) (a : P) : (f ≫ g) a = g (f a) | quotient.induction_on a $ λ x, quotient.sound $
by { unfold app, rw [←category.assoc, over.coe_hom] } | theorem | category_theory.abelian.pseudoelement.comp_apply | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | Applying a pseudoelement to a composition of morphisms is the same as composing
with each morphism. Sadly, this is not a definitional equality, but at least it is
true. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_comp {P Q R : C} (f : P ⟶ Q) (g : Q ⟶ R) : g ∘ f = f ≫ g | funext $ λ x, (comp_apply _ _ _).symm | theorem | category_theory.abelian.pseudoelement.comp_comp | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | Composition of functions on pseudoelements is composition of morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_zero_aux {P : C} (Q : C) (f : over P) : f ≈ (0 : Q ⟶ P) ↔ f.hom = 0 | ⟨λ ⟨R, p, q, ep, eq, comm⟩, by exactI zero_of_epi_comp p (by simp [comm]),
λ hf, ⟨biprod f.1 Q, biprod.fst, biprod.snd, by apply_instance, by apply_instance,
by rw [hf, over.coe_hom, has_zero_morphisms.comp_zero, has_zero_morphisms.comp_zero]⟩⟩ | lemma | category_theory.abelian.pseudoelement.pseudo_zero_aux | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [
"comm"
] | The arrows pseudo-equal to a zero morphism are precisely the zero morphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_eq_zero' {P Q R : C} :
⟦((0 : Q ⟶ P) : over P)⟧ = ⟦((0 : R ⟶ P) : over P)⟧ | quotient.sound $ (pseudo_zero_aux R _).2 rfl | lemma | category_theory.abelian.pseudoelement.zero_eq_zero' | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_zero {P : C} : P | ⟦(0 : P ⟶ P)⟧ | def | category_theory.abelian.pseudoelement.pseudo_zero | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | The zero pseudoelement is the class of a zero morphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_zero {P : C} : has_zero P | ⟨pseudo_zero⟩ | def | category_theory.abelian.pseudoelement.has_zero | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | We can not use `pseudo_zero` as a global `has_zero` instance,
as it would trigger on any type class search for `has_zero` applied to a `coe_sort`.
This would be too expensive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_zero_def {P : C} : (0 : pseudoelement P) = ⟦(0 : P ⟶ P)⟧ | rfl | lemma | category_theory.abelian.pseudoelement.pseudo_zero_def | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_eq_zero {P Q : C} : ⟦((0 : Q ⟶ P) : over P)⟧ = (0 : pseudoelement P) | zero_eq_zero' | lemma | category_theory.abelian.pseudoelement.zero_eq_zero | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_zero_iff {P : C} (a : over P) : (a : P) = 0 ↔ a.hom = 0 | by { rw ←pseudo_zero_aux P a, exact quotient.eq } | lemma | category_theory.abelian.pseudoelement.pseudo_zero_iff | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [
"quotient.eq"
] | The pseudoelement induced by an arrow is zero precisely when that arrow is zero | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_zero {P Q : C} (f : P ⟶ Q) : f 0 = 0 | by { rw [pseudo_zero_def, pseudo_apply_mk], simp } | theorem | category_theory.abelian.pseudoelement.apply_zero | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | Morphisms map the zero pseudoelement to the zero pseudoelement | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_apply {P : C} (Q : C) (a : P) : (0 : P ⟶ Q) a = 0 | quotient.induction_on a $ λ a',
by { rw [pseudo_zero_def, pseudo_apply_mk], simp } | theorem | category_theory.abelian.pseudoelement.zero_apply | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | The zero morphism maps every pseudoelement to 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_morphism_ext {P Q : C} (f : P ⟶ Q) : (∀ a, f a = 0) → f = 0 | λ h, by { rw ←category.id_comp f, exact (pseudo_zero_iff ((𝟙 P ≫ f) : over Q)).1 (h (𝟙 P)) } | theorem | category_theory.abelian.pseudoelement.zero_morphism_ext | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | An extensionality lemma for being the zero arrow. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_morphism_ext' {P Q : C} (f : P ⟶ Q) : (∀ a, f a = 0) → 0 = f | eq.symm ∘ zero_morphism_ext f | theorem | category_theory.abelian.pseudoelement.zero_morphism_ext' | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_iff {P Q : C} (f : P ⟶ Q) : f = 0 ↔ ∀ a, f a = 0 | ⟨λ h a, by simp [h], zero_morphism_ext _⟩ | theorem | category_theory.abelian.pseudoelement.eq_zero_iff | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [
"eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_injective_of_mono {P Q : C} (f : P ⟶ Q) [mono f] : function.injective f | λ abar abar', quotient.induction_on₂ abar abar' $ λ a a' ha, quotient.sound $
have ⟦(a.hom ≫ f : over Q)⟧ = ⟦a'.hom ≫ f⟧, by convert ha,
match quotient.exact this with ⟨R, p, q, ep, eq, comm⟩ :=
⟨R, p, q, ep, eq, (cancel_mono f).1 $ by { simp only [category.assoc], exact comm }⟩
end | theorem | category_theory.abelian.pseudoelement.pseudo_injective_of_mono | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [
"comm"
] | A monomorphism is injective on pseudoelements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_of_map_zero {P Q : C} (f : P ⟶ Q) : function.injective f → ∀ a, f a = 0 → a = 0 | λ h a ha, by { rw ←apply_zero f at ha, exact h ha } | lemma | category_theory.abelian.pseudoelement.zero_of_map_zero | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | A morphism that is injective on pseudoelements only maps the zero element to zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_of_zero_of_map_zero {P Q : C} (f : P ⟶ Q) : (∀ a, f a = 0 → a = 0) → mono f | λ h, (mono_iff_cancel_zero _).2 $ λ R g hg, (pseudo_zero_iff (g : over P)).1 $ h _ $
show f g = 0, from (pseudo_zero_iff (g ≫ f : over Q)).2 hg | theorem | category_theory.abelian.pseudoelement.mono_of_zero_of_map_zero | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | A morphism that only maps the zero pseudoelement to zero is a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_surjective_of_epi {P Q : C} (f : P ⟶ Q) [epi f] : function.surjective f | λ qbar, quotient.induction_on qbar $ λ q, ⟨((pullback.fst : pullback f q.hom ⟶ P) : over P),
quotient.sound $ ⟨pullback f q.hom, 𝟙 (pullback f q.hom), pullback.snd, by apply_instance,
by apply_instance, by rw [category.id_comp, ←pullback.condition, app_hom, over.coe_hom]⟩⟩ | theorem | category_theory.abelian.pseudoelement.pseudo_surjective_of_epi | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | An epimorphism is surjective on pseudoelements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_of_pseudo_surjective {P Q : C} (f : P ⟶ Q) : function.surjective f → epi f | λ h, match h (𝟙 Q) with ⟨pbar, hpbar⟩ :=
match quotient.exists_rep pbar with ⟨p, hp⟩ :=
have ⟦(p.hom ≫ f : over Q)⟧ = ⟦𝟙 Q⟧, by { rw ←hp at hpbar, exact hpbar },
match quotient.exact this with ⟨R, x, y, ex, ey, comm⟩ :=
@epi_of_epi_fac _ _ _ _ _ (x ≫ p.hom) f y ey $
by { dsimp at comm, rw [cat... | theorem | category_theory.abelian.pseudoelement.epi_of_pseudo_surjective | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [
"comm"
] | A morphism that is surjective on pseudoelements is an epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_exact_of_exact {P Q R : C} {f : P ⟶ Q} {g : Q ⟶ R} (h : exact f g) :
(∀ a, g (f a) = 0) ∧ (∀ b, g b = 0 → ∃ a, f a = b) | ⟨λ a, by { rw [←comp_apply, h.w], exact zero_apply _ _ },
λ b', quotient.induction_on b' $ λ b hb,
have hb' : b.hom ≫ g = 0, from (pseudo_zero_iff _).1 hb,
begin
-- By exactness, b factors through im f = ker g via some c
obtain ⟨c, hc⟩ := kernel_fork.is_limit.lift' (is_limit_image f g h) _ hb',
... | theorem | category_theory.abelian.pseudoelement.pseudo_exact_of_exact | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | Two morphisms in an exact sequence are exact on pseudoelements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_eq_zero_of_comp_eq_zero {P Q R : C} (f : Q ⟶ R) (a : P ⟶ Q) : a ≫ f = 0 → f a = 0 | λ h, by simp [over_coe_def, pseudo_apply_mk, over.coe_hom, h] | lemma | category_theory.abelian.pseudoelement.apply_eq_zero_of_comp_eq_zero | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_of_pseudo_exact {P Q R : C} (f : P ⟶ Q) (g : Q ⟶ R) :
(∀ a, g (f a) = 0) ∧ (∀ b, g b = 0 → ∃ a, f a = b) → exact f g | λ ⟨h₁, h₂⟩, (abelian.exact_iff _ _).2 ⟨zero_morphism_ext _ $ λ a, by rw [comp_apply, h₁ a],
begin
-- If we apply g to the pseudoelement induced by its kernel, we get 0 (of course!).
have : g (kernel.ι g) = 0 := apply_eq_zero_of_comp_eq_zero _ _ (kernel.condition _),
-- By pseudo-exactness, we get a preimage.
o... | theorem | category_theory.abelian.pseudoelement.exact_of_pseudo_exact | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [
"comm"
] | If two morphisms are exact on pseudoelements, they are exact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_of_eq_image {P Q : C} (f : P ⟶ Q) (x y : P) : f x = f y →
∃ z, f z = 0 ∧ ∀ (R : C) (g : P ⟶ R), (g : P ⟶ R) y = 0 → g z = g x | quotient.induction_on₂ x y $ λ a a' h,
match quotient.exact h with ⟨R, p, q, ep, eq, comm⟩ :=
let a'' : R ⟶ P := p ≫ a.hom - q ≫ a'.hom in ⟨a'',
⟨show ⟦((p ≫ a.hom - q ≫ a'.hom) ≫ f : over Q)⟧ = ⟦(0 : Q ⟶ Q)⟧,
by { dsimp at comm, simp [sub_eq_zero.2 comm] },
λ Z g hh,
begin
obtain ⟨X, p'... | theorem | category_theory.abelian.pseudoelement.sub_of_eq_image | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [
"comm"
] | If two pseudoelements `x` and `y` have the same image under some morphism `f`, then we can form
their "difference" `z`. This pseudoelement has the properties that `f z = 0` and for all
morphisms `g`, if `g y = 0` then `g z = g x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_pullback {P Q R : C} {f : P ⟶ R} {g : Q ⟶ R} {p : P} {q : Q} : f p = g q →
∃ s, (pullback.fst : pullback f g ⟶ P) s = p ∧ (pullback.snd : pullback f g ⟶ Q) s = q | quotient.induction_on₂ p q $ λ x y h,
begin
obtain ⟨Z, a, b, ea, eb, comm⟩ := quotient.exact h,
obtain ⟨l, hl₁, hl₂⟩ := @pullback.lift' _ _ _ _ _ _ f g _ (a ≫ x.hom) (b ≫ y.hom)
(by { simp only [category.assoc], exact comm }),
exact ⟨l, ⟨quotient.sound ⟨Z, 𝟙 Z, a, by apply_instance, ea, by rwa category.id_... | theorem | category_theory.abelian.pseudoelement.pseudo_pullback | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [
"comm"
] | If `f : P ⟶ R` and `g : Q ⟶ R` are morphisms and `p : P` and `q : Q` are pseudoelements such
that `f p = g q`, then there is some `s : pullback f g` such that `fst s = p` and `snd s = q`.
Remark: Borceux claims that `s` is unique, but this is false. See
`counterexamples/pseudoelement` for details. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Module.eq_range_of_pseudoequal {R : Type*} [comm_ring R] {G : Module R} {x y : over G}
(h : pseudo_equal G x y) : x.hom.range = y.hom.range | begin
obtain ⟨P, p, q, hp, hq, H⟩ := h,
refine submodule.ext (λ a, ⟨λ ha, _, λ ha, _⟩),
{ obtain ⟨a', ha'⟩ := ha,
obtain ⟨a'', ha''⟩ := (Module.epi_iff_surjective p).1 hp a',
refine ⟨q a'', _⟩,
rw [← linear_map.comp_apply, ← Module.comp_def, ← H, Module.comp_def, linear_map.comp_apply,
ha'', ha'... | lemma | category_theory.abelian.pseudoelement.Module.eq_range_of_pseudoequal | category_theory.abelian | src/category_theory/abelian/pseudoelements.lean | [
"category_theory.abelian.exact",
"category_theory.over",
"algebra.category.Module.epi_mono"
] | [
"Module",
"Module.comp_def",
"Module.epi_iff_surjective",
"comm_ring",
"linear_map.comp_apply",
"submodule.ext"
] | In the category `Module R`, if `x` and `y` are pseudoequal, then the range of the associated
morphisms is the same. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.right_derived (F : C ⥤ D) [F.additive] (n : ℕ) : C ⥤ D | injective_resolutions C ⋙ F.map_homotopy_category _ ⋙ homotopy_category.homology_functor D _ n | def | category_theory.functor.right_derived | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [
"homotopy_category.homology_functor"
] | The right derived functors of an additive functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.right_derived_obj_iso (F : C ⥤ D) [F.additive] (n : ℕ)
{X : C} (P : InjectiveResolution X) :
(F.right_derived n).obj X ≅
(homology_functor D _ n).obj ((F.map_homological_complex _).obj P.cocomplex) | (homotopy_category.homology_functor D _ n).map_iso
(homotopy_category.iso_of_homotopy_equiv
(F.map_homotopy_equiv (InjectiveResolution.homotopy_equiv _ P)))
≪≫ (homotopy_category.homology_factors D _ n).app _ | def | category_theory.functor.right_derived_obj_iso | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [
"homology_functor",
"homotopy_category.homology_factors",
"homotopy_category.homology_functor",
"homotopy_category.iso_of_homotopy_equiv"
] | We can compute a right derived functor using a chosen injective resolution. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.right_derived_obj_injective_zero (F : C ⥤ D) [F.additive]
(X : C) [injective X] :
(F.right_derived 0).obj X ≅ F.obj X | F.right_derived_obj_iso 0 (InjectiveResolution.self X) ≪≫
(homology_functor _ _ _).map_iso ((cochain_complex.single₀_map_homological_complex F).app X) ≪≫
(cochain_complex.homology_functor_0_single₀ D).app (F.obj X) | def | category_theory.functor.right_derived_obj_injective_zero | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [
"cochain_complex.homology_functor_0_single₀",
"cochain_complex.single₀_map_homological_complex",
"homology_functor"
] | The 0-th derived functor of `F` on an injective object `X` is just `F.obj X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.right_derived_obj_injective_succ (F : C ⥤ D) [F.additive] (n : ℕ)
(X : C) [injective X] :
(F.right_derived (n+1)).obj X ≅ 0 | F.right_derived_obj_iso (n+1) (InjectiveResolution.self X) ≪≫
(homology_functor _ _ _).map_iso ((cochain_complex.single₀_map_homological_complex F).app X) ≪≫
(cochain_complex.homology_functor_succ_single₀ D n).app (F.obj X) ≪≫
(functor.zero_obj _).iso_zero | def | category_theory.functor.right_derived_obj_injective_succ | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [
"cochain_complex.homology_functor_succ_single₀",
"cochain_complex.single₀_map_homological_complex",
"homology_functor"
] | The higher derived functors vanish on injective objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor.right_derived_map_eq (F : C ⥤ D) [F.additive] (n : ℕ) {X Y : C} (f : Y ⟶ X)
{P : InjectiveResolution X} {Q : InjectiveResolution Y} (g : Q.cocomplex ⟶ P.cocomplex)
(w : Q.ι ≫ g = (cochain_complex.single₀ C).map f ≫ P.ι) :
(F.right_derived n).map f =
(F.right_derived_obj_iso n Q).hom ≫
(homology_func... | begin
dsimp only [functor.right_derived, functor.right_derived_obj_iso],
dsimp, simp only [category.comp_id, category.id_comp],
rw [←homology_functor_map, homotopy_category.homology_functor_map_factors],
simp only [←functor.map_comp],
congr' 1,
apply homotopy_category.eq_of_homotopy,
apply functor.map_hom... | lemma | category_theory.functor.right_derived_map_eq | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [
"cochain_complex.single₀",
"homology_functor",
"homotopy.trans",
"homotopy_category.eq_of_homotopy",
"homotopy_category.homology_functor_map_factors",
"homotopy_category.homotopy_out_map"
] | We can compute a right derived functor on a morphism using a descent of that morphism
to a cochain map between chosen injective resolutions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans.right_derived {F G : C ⥤ D} [F.additive] [G.additive] (α : F ⟶ G) (n : ℕ) :
F.right_derived n ⟶ G.right_derived n | whisker_left (injective_resolutions C)
(whisker_right (nat_trans.map_homotopy_category α _)
(homotopy_category.homology_functor D _ n)) | def | category_theory.nat_trans.right_derived | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [
"homotopy_category.homology_functor"
] | The natural transformation between right-derived functors induced by a natural transformation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans.right_derived_id (F : C ⥤ D) [F.additive] (n : ℕ) :
nat_trans.right_derived (𝟙 F) n = 𝟙 (F.right_derived n) | by { simp [nat_trans.right_derived], refl, } | lemma | category_theory.nat_trans.right_derived_id | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans.right_derived_comp
{F G H : C ⥤ D} [F.additive] [G.additive] [H.additive]
(α : F ⟶ G) (β : G ⟶ H) (n : ℕ) :
nat_trans.right_derived (α ≫ β) n = nat_trans.right_derived α n ≫ nat_trans.right_derived β n | by simp [nat_trans.right_derived] | lemma | category_theory.nat_trans.right_derived_comp | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans.right_derived_eq {F G : C ⥤ D} [F.additive] [G.additive] (α : F ⟶ G) (n : ℕ)
{X : C} (P : InjectiveResolution X) :
(nat_trans.right_derived α n).app X =
(F.right_derived_obj_iso n P).hom ≫
(homology_functor D _ n).map ((nat_trans.map_homological_complex α _).app P.cocomplex) ≫
(G.right_d... | begin
symmetry,
dsimp [nat_trans.right_derived, functor.right_derived_obj_iso],
simp only [category.comp_id, category.id_comp],
rw [←homology_functor_map, homotopy_category.homology_functor_map_factors],
simp only [←functor.map_comp],
congr' 1,
apply homotopy_category.eq_of_homotopy,
simp only [nat_tran... | lemma | category_theory.nat_trans.right_derived_eq | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [
"homology_functor",
"homotopy.comp_left_id",
"homotopy_category.eq_of_homotopy",
"homotopy_category.homology_functor_map_factors"
] | A component of the natural transformation between right-derived functors can be computed
using a chosen injective resolution. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_exact_of_preserves_finite_limits_of_mono [preserves_finite_limits F] [mono f]
(ex : exact f g) : exact (F.map f) (F.map g) | abelian.exact_of_is_kernel _ _ (by simp [← functor.map_comp, ex.w]) $
limits.is_limit_fork_map_of_is_limit' _ ex.w (abelian.is_limit_of_exact_of_mono _ _ ex) | lemma | category_theory.abelian.functor.preserves_exact_of_preserves_finite_limits_of_mono | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [] | If `preserves_finite_limits F` and `mono f`, then `exact (F.map f) (F.map g)` if
`exact f g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exact_of_map_injective_resolution (P: InjectiveResolution X) [preserves_finite_limits F] :
exact (F.map (P.ι.f 0))
(((F.map_homological_complex (complex_shape.up ℕ)).obj P.cocomplex).d_from 0) | preadditive.exact_of_iso_of_exact' (F.map (P.ι.f 0)) (F.map (P.cocomplex.d 0 1)) _ _
(iso.refl _) (iso.refl _)
(homological_complex.X_next_iso ((F.map_homological_complex _).obj P.cocomplex) rfl).symm
(by simp)
(by rw [iso.refl_hom, category.id_comp, iso.symm_hom, homological_complex.d_from_eq]; congr')
(pres... | lemma | category_theory.abelian.functor.exact_of_map_injective_resolution | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [
"complex_shape.up",
"homological_complex.X_next_iso",
"homological_complex.d_from_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_derived_zero_to_self_app [enough_injectives C] [preserves_finite_limits F] {X : C}
(P : InjectiveResolution X) :
(F.right_derived 0).obj X ⟶ F.obj X | (right_derived_obj_iso F 0 P).hom ≫ (homology_iso_kernel_desc _ _ _).hom ≫
kernel.map _ _ (cokernel.desc _ (𝟙 _) (by simp)) (𝟙 _) (by { ext, simp }) ≫
(as_iso (kernel.lift _ _ (exact_of_map_injective_resolution F P).w)).inv | def | category_theory.abelian.functor.right_derived_zero_to_self_app | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [
"homology_iso_kernel_desc"
] | Given `P : InjectiveResolution X`, a morphism `(F.right_derived 0).obj X ⟶ F.obj X` given
`preserves_finite_limits F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_derived_zero_to_self_app_inv [enough_injectives C] {X : C}
(P : InjectiveResolution X) :
F.obj X ⟶ (F.right_derived 0).obj X | homology.lift _ _ _ (F.map (P.ι.f 0) ≫ cokernel.π _) begin
have : (complex_shape.up ℕ).rel 0 1 := rfl,
rw [category.assoc, cokernel.π_desc, homological_complex.d_from_eq _ this,
map_homological_complex_obj_d, ← category.assoc, ← functor.map_comp],
simp only [InjectiveResolution.ι_f_zero_comp_complex_d, functo... | def | category_theory.abelian.functor.right_derived_zero_to_self_app_inv | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [
"complex_shape.up",
"homological_complex.d_from_eq",
"homology.lift",
"rel"
] | Given `P : InjectiveResolution X`, a morphism `F.obj X ⟶ (F.right_derived 0).obj X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_derived_zero_to_self_app_comp_inv [enough_injectives C] [preserves_finite_limits F]
{X : C} (P : InjectiveResolution X) : right_derived_zero_to_self_app F P ≫
right_derived_zero_to_self_app_inv F P = 𝟙 _ | begin
dsimp [right_derived_zero_to_self_app, right_derived_zero_to_self_app_inv],
rw [← category.assoc, iso.comp_inv_eq, category.id_comp, category.assoc, category.assoc,
← iso.eq_inv_comp, iso.inv_hom_id],
ext,
rw [category.assoc, category.assoc, homology.lift_ι, category.id_comp,
homology.π'_ι, catego... | lemma | category_theory.abelian.functor.right_derived_zero_to_self_app_comp_inv | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [
"homology.lift_ι",
"homology.ι",
"homology.π'_ι",
"homology_iso_kernel_desc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_derived_zero_to_self_app_inv_comp [enough_injectives C] [preserves_finite_limits F]
{X : C} (P : InjectiveResolution X) : right_derived_zero_to_self_app_inv F P ≫
right_derived_zero_to_self_app F P = 𝟙 _ | begin
dsimp [right_derived_zero_to_self_app, right_derived_zero_to_self_app_inv],
rw [← category.assoc _ (F.right_derived_obj_iso 0 P).hom,
category.assoc _ _ (F.right_derived_obj_iso 0 P).hom, iso.inv_hom_id, category.comp_id,
← category.assoc, ← category.assoc, is_iso.comp_inv_eq, category.id_comp],
ext... | lemma | category_theory.abelian.functor.right_derived_zero_to_self_app_inv_comp | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [
"homology.lift",
"homology_iso_kernel_desc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_derived_zero_to_self_app_iso [enough_injectives C] [preserves_finite_limits F]
{X : C} (P : InjectiveResolution X) : (F.right_derived 0).obj X ≅ F.obj X | { hom := right_derived_zero_to_self_app _ P,
inv := right_derived_zero_to_self_app_inv _ P,
hom_inv_id' := right_derived_zero_to_self_app_comp_inv _ P,
inv_hom_id' := right_derived_zero_to_self_app_inv_comp _ P } | def | category_theory.abelian.functor.right_derived_zero_to_self_app_iso | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [] | Given `P : InjectiveResolution X`, the isomorphism `(F.right_derived 0).obj X ≅ F.obj X` if
`preserves_finite_limits F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_derived_zero_to_self_natural [enough_injectives C]
{X : C} {Y : C} (f : X ⟶ Y)
(P : InjectiveResolution X) (Q : InjectiveResolution Y) :
F.map f ≫ right_derived_zero_to_self_app_inv F Q =
right_derived_zero_to_self_app_inv F P ≫ (F.right_derived 0).map f | begin
dsimp [right_derived_zero_to_self_app_inv],
simp only [category_theory.functor.map_id, category.id_comp, ← category.assoc],
rw [iso.comp_inv_eq, right_derived_map_eq F 0 f (InjectiveResolution.desc f Q P) (by simp),
category.assoc, category.assoc, category.assoc, category.assoc, iso.inv_hom_id,
cate... | lemma | category_theory.abelian.functor.right_derived_zero_to_self_natural | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [
"homological_complex.congr_hom",
"homological_complex.hom.sq_from_left",
"homology.lift",
"homology.lift_ι",
"homology.map_ι"
] | Given `P : InjectiveResolution X` and `Q : InjectiveResolution Y` and a morphism `f : X ⟶ Y`,
naturality of the square given by `right_derived_zero_to_self_natural`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_derived_zero_iso_self [enough_injectives C] [preserves_finite_limits F] :
(F.right_derived 0) ≅ F | iso.symm $
nat_iso.of_components (λ X, (right_derived_zero_to_self_app_iso _ (InjectiveResolution.of X)).symm)
(λ X Y f, right_derived_zero_to_self_natural _ _ _ _) | def | category_theory.abelian.functor.right_derived_zero_iso_self | category_theory.abelian | src/category_theory/abelian/right_derived.lean | [
"category_theory.abelian.injective_resolution",
"algebra.homology.additive",
"category_theory.limits.constructions.epi_mono",
"category_theory.abelian.homology",
"category_theory.abelian.exact"
] | [] | Given `preserves_finite_limits F`, the natural isomorphism `(F.right_derived 0) ≅ F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subobject_iso_subobject_op [abelian C] (X : C) : subobject X ≃o (subobject (op X))ᵒᵈ | begin
refine order_iso.of_hom_inv (cokernel_order_hom X) (kernel_order_hom X) _ _,
{ change (cokernel_order_hom X).comp (kernel_order_hom X) = _,
refine order_hom.ext _ _ (funext (subobject.ind _ _)),
introsI A f hf,
dsimp only [order_hom.comp_coe, function.comp_app, kernel_order_hom_coe, subobject.lift... | def | category_theory.abelian.subobject_iso_subobject_op | category_theory.abelian | src/category_theory/abelian/subobject.lean | [
"category_theory.subobject.limits",
"category_theory.abelian.basic"
] | [
"order_hom.ext",
"order_iso.of_hom_inv",
"quiver.hom.unop_inj",
"quiver.hom.unop_op"
] | In an abelian category, the subobjects and quotient objects of an object `X` are
order-isomorphic via taking kernels and cokernels.
Implemented here using subobjects in the opposite category,
since mathlib does not have a notion of quotient objects at the time of writing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
well_powered_opposite [abelian C] [well_powered C] : well_powered Cᵒᵖ | { subobject_small := λ X,
(small_congr (subobject_iso_subobject_op (unop X)).to_equiv).1 infer_instance } | instance | category_theory.abelian.well_powered_opposite | category_theory.abelian | src/category_theory/abelian/subobject.lean | [
"category_theory.subobject.limits",
"category_theory.abelian.basic"
] | [
"small_congr"
] | A well-powered abelian category is also well-copowered. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_kernels [preserves_finite_limits G] : has_kernels C | { has_limit := λ X Y f, begin
have := nat_iso.naturality_1 i f,
simp at this,
rw ←this,
haveI : has_kernel (G.map (F.map f) ≫ i.hom.app _) := limits.has_kernel_comp_mono _ _,
apply limits.has_kernel_iso_comp,
end } | lemma | category_theory.abelian_of_adjunction.has_kernels | category_theory.abelian | src/category_theory/abelian/transfer.lean | [
"category_theory.abelian.basic",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.adjunction.limits"
] | [] | No point making this an instance, as it requires `i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_cokernels : has_cokernels C | { has_colimit := λ X Y f, begin
haveI : preserves_colimits G := adj.left_adjoint_preserves_colimits,
have := nat_iso.naturality_1 i f,
simp at this,
rw ←this,
haveI : has_cokernel (G.map (F.map f) ≫ i.hom.app _) := limits.has_cokernel_comp_iso _ _,
apply limits.has_cokernel_epi_comp,
end } | lemma | category_theory.abelian_of_adjunction.has_cokernels | category_theory.abelian | src/category_theory/abelian/transfer.lean | [
"category_theory.abelian.basic",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.adjunction.limits"
] | [] | No point making this an instance, as it requires `i` and `adj`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_iso {X Y : C} (f : X ⟶ Y) : G.obj (cokernel (F.map f)) ≅ cokernel f | begin
-- We have to write an explicit `preserves_colimits` type here,
-- as `left_adjoint_preserves_colimits` has universe variables.
haveI : preserves_colimits G := adj.left_adjoint_preserves_colimits,
calc G.obj (cokernel (F.map f))
≅ cokernel (G.map (F.map f)) : (as_iso (cokernel_comparison _ G)).symm
... | def | category_theory.abelian_of_adjunction.cokernel_iso | category_theory.abelian | src/category_theory/abelian/transfer.lean | [
"category_theory.abelian.basic",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.adjunction.limits"
] | [] | Auxiliary construction for `coimage_iso_image` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coimage_iso_image_aux {X Y : C} (f : X ⟶ Y) :
kernel (G.map (cokernel.π (F.map f))) ≅ kernel (cokernel.π f) | begin
haveI : preserves_colimits G := adj.left_adjoint_preserves_colimits,
calc kernel (G.map (cokernel.π (F.map f)))
≅ kernel (cokernel.π (G.map (F.map f)) ≫ cokernel_comparison (F.map f) G)
: kernel_iso_of_eq (π_comp_cokernel_comparison _ _).symm
... ≅ kernel (cokernel.π (G.map (F.map f))) : ker... | def | category_theory.abelian_of_adjunction.coimage_iso_image_aux | category_theory.abelian | src/category_theory/abelian/transfer.lean | [
"category_theory.abelian.basic",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.adjunction.limits"
] | [] | Auxiliary construction for `coimage_iso_image` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coimage_iso_image {X Y : C} (f : X ⟶ Y) : abelian.coimage f ≅ abelian.image f | begin
haveI : preserves_limits F := adj.right_adjoint_preserves_limits,
haveI : preserves_colimits G := adj.left_adjoint_preserves_colimits,
calc abelian.coimage f
≅ cokernel (kernel.ι f) : iso.refl _
... ≅ G.obj (cokernel (F.map (kernel.ι f))) : (cokernel_iso _ _ i adj _).symm
... ≅ G.o... | def | category_theory.abelian_of_adjunction.coimage_iso_image | category_theory.abelian | src/category_theory/abelian/transfer.lean | [
"category_theory.abelian.basic",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.adjunction.limits"
] | [
"adj"
] | Auxiliary definition: the abelian coimage and abelian image agree.
We still need to check that this agrees with the canonical morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coimage_iso_image_hom {X Y : C} (f : X ⟶ Y) :
(coimage_iso_image F G i adj f).hom = abelian.coimage_image_comparison f | begin
ext,
simpa only [←G.map_comp_assoc, coimage_iso_image, nat_iso.inv_inv_app, cokernel_iso,
coimage_iso_image_aux, iso.trans_symm, iso.symm_symm_eq, iso.refl_trans, iso.trans_refl,
iso.trans_hom, iso.symm_hom, cokernel_comp_is_iso_inv, cokernel_epi_comp_inv, as_iso_hom,
functor.map_iso_hom, cokerne... | lemma | category_theory.abelian_of_adjunction.coimage_iso_image_hom | category_theory.abelian | src/category_theory/abelian/transfer.lean | [
"category_theory.abelian.basic",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.adjunction.limits"
] | [
"adj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abelian_of_adjunction
{C : Type u₁} [category.{v} C] [preadditive C] [has_finite_products C]
{D : Type u₂} [category.{v} D] [abelian D]
(F : C ⥤ D) [functor.preserves_zero_morphisms F]
(G : D ⥤ C) [functor.preserves_zero_morphisms G] [preserves_finite_limits G]
(i : F ⋙ G ≅ 𝟭 C) (adj : G ⊣ F) : abelian C | begin
haveI := has_kernels F G i, haveI := has_cokernels F G i adj,
haveI : ∀ {X Y : C} (f : X ⟶ Y), is_iso (abelian.coimage_image_comparison f),
{ intros X Y f, rw ←coimage_iso_image_hom F G i adj f, apply_instance, },
apply abelian.of_coimage_image_comparison_is_iso,
end | def | category_theory.abelian_of_adjunction | category_theory.abelian | src/category_theory/abelian/transfer.lean | [
"category_theory.abelian.basic",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.adjunction.limits"
] | [
"adj"
] | If `C` is an additive category, `D` is an abelian category,
we have `F : C ⥤ D` `G : D ⥤ C` (both preserving zero morphisms),
`G` is left exact (that is, preserves finite limits),
and further we have `adj : G ⊣ F` and `i : F ⋙ G ≅ 𝟭 C`,
then `C` is also abelian.
See <https://stacks.math.columbia.edu/tag/03A3> | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abelian_of_equivalence
{C : Type u₁} [category.{v} C] [preadditive C] [has_finite_products C]
{D : Type u₂} [category.{v} D] [abelian D]
(F : C ⥤ D) [functor.preserves_zero_morphisms F] [is_equivalence F] : abelian C | abelian_of_adjunction F F.inv F.as_equivalence.unit_iso.symm F.as_equivalence.symm.to_adjunction | def | category_theory.abelian_of_equivalence | category_theory.abelian | src/category_theory/abelian/transfer.lean | [
"category_theory.abelian.basic",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.adjunction.limits"
] | [] | If `C` is an additive category equivalent to an abelian category `D`
via a functor that preserves zero morphisms,
then `C` is also abelian. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_of_epi_of_mono_of_mono (hα : epi α) (hβ : mono β) (hδ : mono δ) : mono γ | mono_of_zero_of_map_zero _ $ λ c hc,
have h c = 0, from
suffices δ (h c) = 0, from zero_of_map_zero _ (pseudo_injective_of_mono _) _ this,
calc δ (h c) = h' (γ c) : by rw [←comp_apply, ←comm₃, comp_apply]
... = h' 0 : by rw hc
... = 0 : apply_zero _,
exists.elim ((pseudo... | lemma | category_theory.abelian.mono_of_epi_of_mono_of_mono | category_theory.abelian.diagram_lemmas | src/category_theory/abelian/diagram_lemmas/four.lean | [
"category_theory.abelian.pseudoelements"
] | [] | The four lemma, mono version. For names of objects and morphisms, refer to the following
diagram:
```
A ---f--> B ---g--> C ---h--> D
| | | |
α β γ δ
| | | |
v v v v
A' --f'-> B' --g'-> C' --h'-> D'
``` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_of_epi_of_epi_of_mono (hα : epi α) (hγ : epi γ) (hδ : mono δ) : epi β | preadditive.epi_of_cancel_zero _ $ λ R r hβr,
have hf'r : f' ≫ r = 0, from limits.zero_of_epi_comp α $
calc α ≫ f' ≫ r = f ≫ β ≫ r : by rw reassoc_of comm₁
... = f ≫ 0 : by rw hβr
... = 0 : has_zero_morphisms.comp_zero _ _,
let y : R ⟶ pushout r g' := pushout.inl... | lemma | category_theory.abelian.epi_of_epi_of_epi_of_mono | category_theory.abelian.diagram_lemmas | src/category_theory/abelian/diagram_lemmas/four.lean | [
"category_theory.abelian.pseudoelements"
] | [] | The four lemma, epi version. For names of objects and morphisms, refer to the following
diagram:
```
A ---f--> B ---g--> C ---h--> D
| | | |
α β γ δ
| | | |
v v v v
A' --f'-> B' --g'-> C' --h'-> D'
``` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso : is_iso γ | have mono γ, by apply mono_of_epi_of_mono_of_mono comm₁ comm₂ comm₃ hfg hgh hf'g'; apply_instance,
have epi γ, by apply epi_of_epi_of_epi_of_mono comm₂ comm₃ comm₄ hhi hg'h' hh'i'; apply_instance,
by exactI is_iso_of_mono_of_epi _ | lemma | category_theory.abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso | category_theory.abelian.diagram_lemmas | src/category_theory/abelian/diagram_lemmas/four.lean | [
"category_theory.abelian.pseudoelements"
] | [] | The five lemma. For names of objects and morphisms, refer to the following diagram:
```
A ---f--> B ---g--> C ---h--> D ---i--> E
| | | | |
α β γ δ ε
| | | | |
v v v v v
A' --f'-> B' --g'-> C... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
solution_set_condition {D : Type u} [category.{v} D] (G : D ⥤ C) : Prop | ∀ (A : C), ∃ (ι : Type v) (B : ι → D) (f : Π (i : ι), A ⟶ G.obj (B i)),
∀ X (h : A ⟶ G.obj X), ∃ (i : ι) (g : B i ⟶ X), f i ≫ G.map g = h | def | category_theory.solution_set_condition | category_theory.adjunction | src/category_theory/adjunction/adjoint_functor_theorems.lean | [
"category_theory.generator",
"category_theory.limits.cone_category",
"category_theory.limits.constructions.weakly_initial",
"category_theory.limits.functor_category",
"category_theory.subobject.comma"
] | [] | The functor `G : D ⥤ C` satisfies the *solution set condition* if for every `A : C`, there is a
family of morphisms `{f_i : A ⟶ G (B_i) // i ∈ ι}` such that given any morphism `h : A ⟶ G X`,
there is some `i ∈ ι` such that `h` factors through `f_i`.
The key part of this definition is that the indexing set `ι` lives in... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
solution_set_condition_of_is_right_adjoint [is_right_adjoint G] :
solution_set_condition G | begin
intros A,
refine ⟨punit, λ _, (left_adjoint G).obj A, λ _, (adjunction.of_right_adjoint G).unit.app A, _⟩,
intros B h,
refine ⟨punit.star, ((adjunction.of_right_adjoint G).hom_equiv _ _).symm h, _⟩,
rw [←adjunction.hom_equiv_unit, equiv.apply_symm_apply],
end | lemma | category_theory.solution_set_condition_of_is_right_adjoint | category_theory.adjunction | src/category_theory/adjunction/adjoint_functor_theorems.lean | [
"category_theory.generator",
"category_theory.limits.cone_category",
"category_theory.limits.constructions.weakly_initial",
"category_theory.limits.functor_category",
"category_theory.subobject.comma"
] | [
"equiv.apply_symm_apply"
] | If `G : D ⥤ C` is a right adjoint it satisfies the solution set condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_adjoint_of_preserves_limits_of_solution_set_condition
[has_limits D] [preserves_limits G] (hG : solution_set_condition G) :
is_right_adjoint G | begin
apply is_right_adjoint_of_structured_arrow_initials _,
intro A,
specialize hG A,
choose ι B f g using hG,
let B' : ι → structured_arrow A G := λ i, structured_arrow.mk (f i),
have hB' : ∀ (A' : structured_arrow A G), ∃ i, nonempty (B' i ⟶ A'),
{ intros A',
obtain ⟨i, _, t⟩ := g _ A'.hom,
exa... | def | category_theory.is_right_adjoint_of_preserves_limits_of_solution_set_condition | category_theory.adjunction | src/category_theory/adjunction/adjoint_functor_theorems.lean | [
"category_theory.generator",
"category_theory.limits.cone_category",
"category_theory.limits.constructions.weakly_initial",
"category_theory.limits.functor_category",
"category_theory.subobject.comma"
] | [] | The general adjoint functor theorem says that if `G : D ⥤ C` preserves limits and `D` has them,
if `G` satisfies the solution set condition then `G` is a right adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_adjoint_of_preserves_limits_of_is_coseparating [has_limits D]
[well_powered D] {𝒢 : set D} [small.{v} 𝒢] (h𝒢 : is_coseparating 𝒢) (G : D ⥤ C)
[preserves_limits G] : is_right_adjoint G | have ∀ A, has_initial (structured_arrow A G),
from λ A, has_initial_of_is_coseparating (structured_arrow.is_coseparating_proj_preimage A G h𝒢),
by exactI is_right_adjoint_of_structured_arrow_initials _ | def | category_theory.is_right_adjoint_of_preserves_limits_of_is_coseparating | category_theory.adjunction | src/category_theory/adjunction/adjoint_functor_theorems.lean | [
"category_theory.generator",
"category_theory.limits.cone_category",
"category_theory.limits.constructions.weakly_initial",
"category_theory.limits.functor_category",
"category_theory.subobject.comma"
] | [] | The special adjoint functor theorem: if `G : D ⥤ C` preserves limits and `D` is complete,
well-powered and has a small coseparating set, then `G` has a left adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_adjoint_of_preserves_colimits_of_is_separatig [has_colimits C]
[well_powered Cᵒᵖ] {𝒢 : set C} [small.{v} 𝒢] (h𝒢 : is_separating 𝒢) (F : C ⥤ D)
[preserves_colimits F] : is_left_adjoint F | have ∀ A, has_terminal (costructured_arrow F A),
from λ A, has_terminal_of_is_separating (costructured_arrow.is_separating_proj_preimage F A h𝒢),
by exactI is_left_adjoint_of_costructured_arrow_terminals _ | def | category_theory.is_left_adjoint_of_preserves_colimits_of_is_separatig | category_theory.adjunction | src/category_theory/adjunction/adjoint_functor_theorems.lean | [
"category_theory.generator",
"category_theory.limits.cone_category",
"category_theory.limits.constructions.weakly_initial",
"category_theory.limits.functor_category",
"category_theory.subobject.comma"
] | [] | The special adjoint functor theorem: if `F : C ⥤ D` preserves colimits and `C` is cocomplete,
well-copowered and has a small separating set, then `F` has a right adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_of_has_limits_of_is_coseparating [has_limits C] [well_powered C]
{𝒢 : set C} [small.{v} 𝒢] (h𝒢 : is_coseparating 𝒢) : has_colimits C | { has_colimits_of_shape := λ J hJ, by exactI has_colimits_of_shape_iff_is_right_adjoint_const.2
⟨is_right_adjoint_of_preserves_limits_of_is_coseparating h𝒢 _⟩ } | lemma | category_theory.limits.has_colimits_of_has_limits_of_is_coseparating | category_theory.adjunction | src/category_theory/adjunction/adjoint_functor_theorems.lean | [
"category_theory.generator",
"category_theory.limits.cone_category",
"category_theory.limits.constructions.weakly_initial",
"category_theory.limits.functor_category",
"category_theory.subobject.comma"
] | [] | A consequence of the special adjoint functor theorem: if `C` is complete, well-powered and
has a small coseparating set, then it is cocomplete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_has_colimits_of_is_separating [has_colimits C] [well_powered Cᵒᵖ]
{𝒢 : set C} [small.{v} 𝒢] (h𝒢 : is_separating 𝒢) : has_limits C | { has_limits_of_shape := λ J hJ, by exactI has_limits_of_shape_iff_is_left_adjoint_const.2
⟨is_left_adjoint_of_preserves_colimits_of_is_separatig h𝒢 _⟩ } | lemma | category_theory.limits.has_limits_of_has_colimits_of_is_separating | category_theory.adjunction | src/category_theory/adjunction/adjoint_functor_theorems.lean | [
"category_theory.generator",
"category_theory.limits.cone_category",
"category_theory.limits.constructions.weakly_initial",
"category_theory.limits.functor_category",
"category_theory.subobject.comma"
] | [] | A consequence of the special adjoint functor theorem: if `C` is cocomplete, well-copowered and
has a small separating set, then it is complete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjunction (F : C ⥤ D) (G : D ⥤ C) | (hom_equiv : Π (X Y), (F.obj X ⟶ Y) ≃ (X ⟶ G.obj Y))
(unit : 𝟭 C ⟶ F.comp G)
(counit : G.comp F ⟶ 𝟭 D)
(hom_equiv_unit' : Π {X Y f}, (hom_equiv X Y) f = (unit : _ ⟶ _).app X ≫ G.map f . obviously)
(hom_equiv_counit' : Π {X Y g}, (hom_equiv X Y).symm g = F.map g ≫ counit.app Y . obviously) | structure | category_theory.adjunction | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | `F ⊣ G` represents the data of an adjunction between two functors
`F : C ⥤ D` and `G : D ⥤ C`. `F` is the left adjoint and `G` is the right adjoint.
To construct an `adjunction` between two functors, it's often easier to instead use the
constructors `mk_of_hom_equiv` or `mk_of_unit_counit`. To construct a left adjoint... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_adjoint (left : C ⥤ D) | (right : D ⥤ C)
(adj : left ⊣ right) | class | category_theory.is_left_adjoint | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"adj"
] | A class giving a chosen right adjoint to the functor `left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_adjoint (right : D ⥤ C) | (left : C ⥤ D)
(adj : left ⊣ right) | class | category_theory.is_right_adjoint | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"adj"
] | A class giving a chosen left adjoint to the functor `right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_adjoint (R : D ⥤ C) [is_right_adjoint R] : C ⥤ D | is_right_adjoint.left R | def | category_theory.left_adjoint | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | Extract the left adjoint from the instance giving the chosen adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_adjoint (L : C ⥤ D) [is_left_adjoint L] : D ⥤ C | is_left_adjoint.right L | def | category_theory.right_adjoint | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | Extract the right adjoint from the instance giving the chosen adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjunction.of_left_adjoint (left : C ⥤ D) [is_left_adjoint left] :
adjunction left (right_adjoint left) | is_left_adjoint.adj | def | category_theory.adjunction.of_left_adjoint | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | The adjunction associated to a functor known to be a left adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjunction.of_right_adjoint (right : C ⥤ D) [is_right_adjoint right] :
adjunction (left_adjoint right) right | is_right_adjoint.adj | def | category_theory.adjunction.of_right_adjoint | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | The adjunction associated to a functor known to be a right adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_equiv_id (X : C) : adj.hom_equiv X _ (𝟙 _) = adj.unit.app X | by simp | lemma | category_theory.adjunction.hom_equiv_id | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_equiv_symm_id (X : D) : (adj.hom_equiv _ X).symm (𝟙 _) = adj.counit.app X | by simp | lemma | category_theory.adjunction.hom_equiv_symm_id | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_equiv_naturality_left_symm (f : X' ⟶ X) (g : X ⟶ G.obj Y) :
(adj.hom_equiv X' Y).symm (f ≫ g) = F.map f ≫ (adj.hom_equiv X Y).symm g | by rw [hom_equiv_counit, F.map_comp, assoc, adj.hom_equiv_counit.symm] | lemma | category_theory.adjunction.hom_equiv_naturality_left_symm | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_equiv_naturality_left (f : X' ⟶ X) (g : F.obj X ⟶ Y) :
(adj.hom_equiv X' Y) (F.map f ≫ g) = f ≫ (adj.hom_equiv X Y) g | by rw [← equiv.eq_symm_apply]; simp [-hom_equiv_unit] | lemma | category_theory.adjunction.hom_equiv_naturality_left | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"equiv.eq_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_equiv_naturality_right (f : F.obj X ⟶ Y) (g : Y ⟶ Y') :
(adj.hom_equiv X Y') (f ≫ g) = (adj.hom_equiv X Y) f ≫ G.map g | by rw [hom_equiv_unit, G.map_comp, ← assoc, ←hom_equiv_unit] | lemma | category_theory.adjunction.hom_equiv_naturality_right | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_equiv_naturality_right_symm (f : X ⟶ G.obj Y) (g : Y ⟶ Y') :
(adj.hom_equiv X Y').symm (f ≫ G.map g) = (adj.hom_equiv X Y).symm f ≫ g | by rw [equiv.symm_apply_eq]; simp [-hom_equiv_counit] | lemma | category_theory.adjunction.hom_equiv_naturality_right_symm | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"equiv.symm_apply_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_triangle :
(whisker_right adj.unit F) ≫ (whisker_left F adj.counit) = nat_trans.id _ | begin
ext, dsimp,
erw [← adj.hom_equiv_counit, equiv.symm_apply_eq, adj.hom_equiv_unit],
simp
end | lemma | category_theory.adjunction.left_triangle | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"equiv.symm_apply_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_triangle :
(whisker_left G adj.unit) ≫ (whisker_right adj.counit G) = nat_trans.id _ | begin
ext, dsimp,
erw [← adj.hom_equiv_unit, ← equiv.eq_symm_apply, adj.hom_equiv_counit],
simp
end | lemma | category_theory.adjunction.right_triangle | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [
"equiv.eq_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_triangle_components :
F.map (adj.unit.app X) ≫ adj.counit.app (F.obj X) = 𝟙 (F.obj X) | congr_arg (λ (t : nat_trans _ (𝟭 C ⋙ F)), t.app X) adj.left_triangle | lemma | category_theory.adjunction.left_triangle_components | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_triangle_components {Y : D} :
adj.unit.app (G.obj Y) ≫ G.map (adj.counit.app Y) = 𝟙 (G.obj Y) | congr_arg (λ (t : nat_trans _ (G ⋙ 𝟭 C)), t.app Y) adj.right_triangle | lemma | category_theory.adjunction.right_triangle_components | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
counit_naturality {X Y : D} (f : X ⟶ Y) :
F.map (G.map f) ≫ (adj.counit).app Y = (adj.counit).app X ≫ f | adj.counit.naturality f | lemma | category_theory.adjunction.counit_naturality | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_naturality {X Y : C} (f : X ⟶ Y) :
(adj.unit).app X ≫ G.map (F.map f) = f ≫ (adj.unit).app Y | (adj.unit.naturality f).symm | lemma | category_theory.adjunction.unit_naturality | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_equiv_apply_eq {A : C} {B : D} (f : F.obj A ⟶ B) (g : A ⟶ G.obj B) :
adj.hom_equiv A B f = g ↔ f = (adj.hom_equiv A B).symm g | ⟨λ h, by {cases h, simp}, λ h, by {cases h, simp}⟩ | lemma | category_theory.adjunction.hom_equiv_apply_eq | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_hom_equiv_apply {A : C} {B : D} (f : F.obj A ⟶ B) (g : A ⟶ G.obj B) :
g = adj.hom_equiv A B f ↔ (adj.hom_equiv A B).symm g = f | ⟨λ h, by {cases h, simp}, λ h, by {cases h, simp}⟩ | lemma | category_theory.adjunction.eq_hom_equiv_apply | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
core_hom_equiv (F : C ⥤ D) (G : D ⥤ C) | (hom_equiv : Π (X Y), (F.obj X ⟶ Y) ≃ (X ⟶ G.obj Y))
(hom_equiv_naturality_left_symm' : Π {X' X Y} (f : X' ⟶ X) (g : X ⟶ G.obj Y),
(hom_equiv X' Y).symm (f ≫ g) = F.map f ≫ (hom_equiv X Y).symm g . obviously)
(hom_equiv_naturality_right' : Π {X Y Y'} (f : F.obj X ⟶ Y) (g : Y ⟶ Y'),
(hom_equiv X Y') (f ≫ g) = (hom_e... | structure | category_theory.adjunction.core_hom_equiv | category_theory.adjunction | src/category_theory/adjunction/basic.lean | [
"category_theory.equivalence"
] | [] | This is an auxiliary data structure useful for constructing adjunctions.
See `adjunction.mk_of_hom_equiv`.
This structure won't typically be used anywhere else. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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