statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
is_inverted_by.iff_of_iso (W : morphism_property C) {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) :
W.is_inverted_by F₁ ↔ W.is_inverted_by F₂ | begin
suffices : ∀ (X Y : C) (f : X ⟶ Y), is_iso (F₁.map f) ↔ is_iso (F₂.map f),
{ split,
exact λ h X Y f hf, by { rw ← this, exact h f hf, },
exact λ h X Y f hf, by { rw this, exact h f hf, }, },
intros X Y f,
exact (respects_iso.isomorphisms D).arrow_mk_iso_iff
(arrow.iso_mk (e.app X) (e.app Y) (b... | lemma | category_theory.morphism_property.is_inverted_by.iff_of_iso | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonal (P : morphism_property C) : morphism_property C | λ X Y f, P (pullback.diagonal f) | def | category_theory.morphism_property.diagonal | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | For `P : morphism_property C`, `P.diagonal` is a morphism property that holds for `f : X ⟶ Y`
whenever `P` holds for `X ⟶ Y xₓ Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonal_iff {X Y : C} {f : X ⟶ Y} : P.diagonal f ↔ P (pullback.diagonal f) | iff.rfl | lemma | category_theory.morphism_property.diagonal_iff | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso.diagonal (hP : P.respects_iso) : P.diagonal.respects_iso | begin
split,
{ introv H,
rwa [diagonal_iff, pullback.diagonal_comp, hP.cancel_left_is_iso, hP.cancel_left_is_iso,
← hP.cancel_right_is_iso _ _, ← pullback.condition, hP.cancel_left_is_iso],
apply_instance },
{ introv H,
delta diagonal,
rwa [pullback.diagonal_comp, hP.cancel_right_is_iso] }
e... | lemma | category_theory.morphism_property.respects_iso.diagonal | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_composition.diagonal
(hP : stable_under_composition P) (hP' : respects_iso P) (hP'' : stable_under_base_change P) :
P.diagonal.stable_under_composition | begin
introv X h₁ h₂,
rw [diagonal_iff, pullback.diagonal_comp],
apply hP, { assumption },
rw hP'.cancel_left_is_iso,
apply hP''.snd,
assumption
end | lemma | category_theory.morphism_property.stable_under_composition.diagonal | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change.diagonal
(hP : stable_under_base_change P) (hP' : respects_iso P) :
P.diagonal.stable_under_base_change | stable_under_base_change.mk hP'.diagonal
begin
introv h,
rw [diagonal_iff, diagonal_pullback_fst, hP'.cancel_left_is_iso, hP'.cancel_right_is_iso],
convert hP.base_change_map f _ _; simp; assumption
end | lemma | category_theory.morphism_property.stable_under_base_change.diagonal | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universally (P : morphism_property C) : morphism_property C | λ X Y f, ∀ ⦃X' Y' : C⦄ (i₁ : X' ⟶ X) (i₂ : Y' ⟶ Y) (f' : X' ⟶ Y')
(h : is_pullback f' i₁ i₂ f), P f' | def | category_theory.morphism_property.universally | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | `P.universally` holds for a morphism `f : X ⟶ Y` iff `P` holds for all `X ×[Y] Y' ⟶ Y'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
universally_respects_iso (P : morphism_property C) :
P.universally.respects_iso | begin
constructor,
{ intros X Y Z e f hf X' Z' i₁ i₂ f' H,
have : is_pullback (𝟙 _) (i₁ ≫ e.hom) i₁ e.inv := is_pullback.of_horiz_is_iso
⟨by rw [category.id_comp, category.assoc, e.hom_inv_id, category.comp_id]⟩,
replace this := this.paste_horiz H,
rw [iso.inv_hom_id_assoc, category.id_comp] at t... | lemma | category_theory.morphism_property.universally_respects_iso | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universally_stable_under_base_change (P : morphism_property C) :
P.universally.stable_under_base_change | λ X Y Y' S f g f' g' H h₁ Y'' X'' i₁ i₂ f'' H', h₁ _ _ _ (H'.paste_vert H.flip) | lemma | category_theory.morphism_property.universally_stable_under_base_change | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_composition.universally [has_pullbacks C]
{P : morphism_property C} (hP : P.stable_under_composition) :
P.universally.stable_under_composition | begin
intros X Y Z f g hf hg X' Z' i₁ i₂ f' H,
have := pullback.lift_fst _ _ (H.w.trans (category.assoc _ _ _).symm),
rw ← this at H ⊢,
apply hP _ _ _ (hg _ _ _ $ is_pullback.of_has_pullback _ _),
exact hf _ _ _ (H.of_right (pullback.lift_snd _ _ _) (is_pullback.of_has_pullback i₂ g))
end | lemma | category_theory.morphism_property.stable_under_composition.universally | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universally_le (P : morphism_property C) :
P.universally ≤ P | begin
intros X Y f hf,
exact hf (𝟙 _) (𝟙 _) _ (is_pullback.of_vert_is_iso ⟨by rw [category.comp_id, category.id_comp]⟩)
end | lemma | category_theory.morphism_property.universally_le | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change.universally_eq
{P : morphism_property C} (hP : P.stable_under_base_change) :
P.universally = P | P.universally_le.antisymm $ λ X Y f hf X' Y' i₁ i₂ f' H, hP H.flip hf | lemma | category_theory.morphism_property.stable_under_base_change.universally_eq | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universally_mono : monotone (universally : morphism_property C → morphism_property C) | λ P₁ P₂ h X Y f h₁ X' Y' i₁ i₂ f' H, h _ _ _ (h₁ _ _ _ H) | lemma | category_theory.morphism_property.universally_mono | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective : morphism_property C | λ X Y f, injective f | def | category_theory.morphism_property.injective | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | Injectiveness (in a concrete category) as a `morphism_property` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
surjective : morphism_property C | λ X Y f, surjective f | def | category_theory.morphism_property.surjective | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | Surjectiveness (in a concrete category) as a `morphism_property` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bijective : morphism_property C | λ X Y f, bijective f | def | category_theory.morphism_property.bijective | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | Bijectiveness (in a concrete category) as a `morphism_property` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bijective_eq_sup : morphism_property.bijective C =
morphism_property.injective C ⊓ morphism_property.surjective C | rfl | lemma | category_theory.morphism_property.bijective_eq_sup | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_stable_under_composition :
(morphism_property.injective C).stable_under_composition | λ X Y Z f g hf hg, by { delta morphism_property.injective, rw coe_comp, exact hg.comp hf } | lemma | category_theory.morphism_property.injective_stable_under_composition | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_stable_under_composition :
(morphism_property.surjective C).stable_under_composition | λ X Y Z f g hf hg, by { delta morphism_property.surjective, rw coe_comp, exact hg.comp hf } | lemma | category_theory.morphism_property.surjective_stable_under_composition | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bijective_stable_under_composition :
(morphism_property.bijective C).stable_under_composition | λ X Y Z f g hf hg, by { delta morphism_property.bijective, rw coe_comp, exact hg.comp hf } | lemma | category_theory.morphism_property.bijective_stable_under_composition | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_respects_iso :
(morphism_property.injective C).respects_iso | (injective_stable_under_composition C).respects_iso
(λ X Y e, ((forget C).map_iso e).to_equiv.injective) | lemma | category_theory.morphism_property.injective_respects_iso | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_respects_iso :
(morphism_property.surjective C).respects_iso | (surjective_stable_under_composition C).respects_iso
(λ X Y e, ((forget C).map_iso e).to_equiv.surjective) | lemma | category_theory.morphism_property.surjective_respects_iso | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bijective_respects_iso :
(morphism_property.bijective C).respects_iso | (bijective_stable_under_composition C).respects_iso
(λ X Y e, ((forget C).map_iso e).to_equiv.bijective) | lemma | category_theory.morphism_property.bijective_respects_iso | category_theory | src/category_theory/morphism_property.lean | [
"category_theory.limits.shapes.diagonal",
"category_theory.arrow",
"category_theory.limits.shapes.comm_sq",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app {F G : C ⥤ D} (α : F ≅ G) (X : C) : F.obj X ≅ G.obj X | { hom := α.hom.app X,
inv := α.inv.app X,
hom_inv_id' := begin rw [← comp_app, iso.hom_inv_id], refl end,
inv_hom_id' := begin rw [← comp_app, iso.inv_hom_id], refl end } | def | category_theory.iso.app | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | The application of a natural isomorphism to an object. We put this definition in a different
namespace, so that we can use `α.app` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_inv_id_app {F G : C ⥤ D} (α : F ≅ G) (X : C) :
α.hom.app X ≫ α.inv.app X = 𝟙 (F.obj X) | congr_fun (congr_arg nat_trans.app α.hom_inv_id) X | lemma | category_theory.iso.hom_inv_id_app | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_hom_id_app {F G : C ⥤ D} (α : F ≅ G) (X : C) :
α.inv.app X ≫ α.hom.app X = 𝟙 (G.obj X) | congr_fun (congr_arg nat_trans.app α.inv_hom_id) X | lemma | category_theory.iso.inv_hom_id_app | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_app {F G H : C ⥤ D} (α : F ≅ G) (β : G ≅ H) (X : C) :
(α ≪≫ β).app X = α.app X ≪≫ β.app X | rfl | lemma | category_theory.nat_iso.trans_app | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_hom {F G : C ⥤ D} (α : F ≅ G) (X : C) : (α.app X).hom = α.hom.app X | rfl | lemma | category_theory.nat_iso.app_hom | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_inv {F G : C ⥤ D} (α : F ≅ G) (X : C) : (α.app X).inv = α.inv.app X | rfl | lemma | category_theory.nat_iso.app_inv | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_app_is_iso (α : F ≅ G) (X : C) : is_iso (α.hom.app X) | ⟨⟨α.inv.app X,
⟨by rw [←comp_app, iso.hom_inv_id, ←id_app], by rw [←comp_app, iso.inv_hom_id, ←id_app]⟩⟩⟩ | instance | category_theory.nat_iso.hom_app_is_iso | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_app_is_iso (α : F ≅ G) (X : C) : is_iso (α.inv.app X) | ⟨⟨α.hom.app X,
⟨by rw [←comp_app, iso.inv_hom_id, ←id_app], by rw [←comp_app, iso.hom_inv_id, ←id_app]⟩⟩⟩ | instance | category_theory.nat_iso.inv_app_is_iso | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_nat_iso_hom_left {X : C} {Z : D} (g g' : G.obj X ⟶ Z) :
α.hom.app X ≫ g = α.hom.app X ≫ g' ↔ g = g' | by simp only [cancel_epi] | lemma | category_theory.nat_iso.cancel_nat_iso_hom_left | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_nat_iso_inv_left {X : C} {Z : D} (g g' : F.obj X ⟶ Z) :
α.inv.app X ≫ g = α.inv.app X ≫ g' ↔ g = g' | by simp only [cancel_epi] | lemma | category_theory.nat_iso.cancel_nat_iso_inv_left | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_nat_iso_hom_right {X : D} {Y : C} (f f' : X ⟶ F.obj Y) :
f ≫ α.hom.app Y = f' ≫ α.hom.app Y ↔ f = f' | by simp only [cancel_mono] | lemma | category_theory.nat_iso.cancel_nat_iso_hom_right | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_nat_iso_inv_right {X : D} {Y : C} (f f' : X ⟶ G.obj Y) :
f ≫ α.inv.app Y = f' ≫ α.inv.app Y ↔ f = f' | by simp only [cancel_mono] | lemma | category_theory.nat_iso.cancel_nat_iso_inv_right | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_nat_iso_hom_right_assoc {W X X' : D} {Y : C}
(f : W ⟶ X) (g : X ⟶ F.obj Y) (f' : W ⟶ X') (g' : X' ⟶ F.obj Y) :
f ≫ g ≫ α.hom.app Y = f' ≫ g' ≫ α.hom.app Y ↔ f ≫ g = f' ≫ g' | by simp only [←category.assoc, cancel_mono] | lemma | category_theory.nat_iso.cancel_nat_iso_hom_right_assoc | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_nat_iso_inv_right_assoc {W X X' : D} {Y : C}
(f : W ⟶ X) (g : X ⟶ G.obj Y) (f' : W ⟶ X') (g' : X' ⟶ G.obj Y) :
f ≫ g ≫ α.inv.app Y = f' ≫ g' ≫ α.inv.app Y ↔ f ≫ g = f' ≫ g' | by simp only [←category.assoc, cancel_mono] | lemma | category_theory.nat_iso.cancel_nat_iso_inv_right_assoc | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_inv_app {F G : C ⥤ D} (e : F ≅ G) (X : C) :
inv (e.inv.app X) = e.hom.app X | by { ext, simp } | lemma | category_theory.nat_iso.inv_inv_app | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
naturality_1 (α : F ≅ G) (f : X ⟶ Y) :
α.inv.app X ≫ F.map f ≫ α.hom.app Y = G.map f | by simp | lemma | category_theory.nat_iso.naturality_1 | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
naturality_2 (α : F ≅ G) (f : X ⟶ Y) :
α.hom.app X ≫ G.map f ≫ α.inv.app Y = F.map f | by simp | lemma | category_theory.nat_iso.naturality_2 | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
naturality_1' (α : F ⟶ G) (f : X ⟶ Y) [is_iso (α.app X)] :
inv (α.app X) ≫ F.map f ≫ α.app Y = G.map f | by simp | lemma | category_theory.nat_iso.naturality_1' | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
naturality_2' (α : F ⟶ G) (f : X ⟶ Y) [is_iso (α.app Y)] :
α.app X ≫ G.map f ≫ inv (α.app Y) = F.map f | by rw [←category.assoc, ←naturality, category.assoc, is_iso.hom_inv_id, category.comp_id] | lemma | category_theory.nat_iso.naturality_2' | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_app_of_is_iso (α : F ⟶ G) [is_iso α] (X) : is_iso (α.app X) | ⟨⟨(inv α).app X,
⟨congr_fun (congr_arg nat_trans.app (is_iso.hom_inv_id α)) X,
congr_fun (congr_arg nat_trans.app (is_iso.inv_hom_id α)) X⟩⟩⟩ | instance | category_theory.nat_iso.is_iso_app_of_is_iso | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | The components of a natural isomorphism are isomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_inv_app (α : F ⟶ G) [is_iso α] (X) : (inv α).app X = inv (α.app X) | by { ext, rw ←nat_trans.comp_app, simp, } | lemma | category_theory.nat_iso.is_iso_inv_app | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_map_inv_app (F : C ⥤ D ⥤ E) {X Y : C} (e : X ≅ Y) (Z : D) :
inv ((F.map e.inv).app Z) = (F.map e.hom).app Z | by { ext, simp, } | lemma | category_theory.nat_iso.inv_map_inv_app | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_components (app : ∀ X : C, F.obj X ≅ G.obj X)
(naturality : ∀ {X Y : C} (f : X ⟶ Y), F.map f ≫ (app Y).hom = (app X).hom ≫ G.map f) :
F ≅ G | { hom := { app := λ X, (app X).hom },
inv :=
{ app := λ X, (app X).inv,
naturality' := λ X Y f,
begin
have h := congr_arg (λ f, (app X).inv ≫ (f ≫ (app Y).inv)) (naturality f).symm,
simp only [iso.inv_hom_id_assoc, iso.hom_inv_id, assoc, comp_id, cancel_mono] at h,
exact h
end }, } | def | category_theory.nat_iso.of_components | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | Construct a natural isomorphism between functors by giving object level isomorphisms,
and checking naturality only in the forward direction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_components.app (app' : ∀ X : C, F.obj X ≅ G.obj X) (naturality) (X) :
(of_components app' naturality).app X = app' X | by tidy | lemma | category_theory.nat_iso.of_components.app | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_of_is_iso_app (α : F ⟶ G) [∀ X : C, is_iso (α.app X)] : is_iso α | ⟨(is_iso.of_iso (of_components (λ X, as_iso (α.app X)) (by tidy))).1⟩ | lemma | category_theory.nat_iso.is_iso_of_is_iso_app | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hcomp {F G : C ⥤ D} {H I : D ⥤ E} (α : F ≅ G) (β : H ≅ I) : F ⋙ H ≅ G ⋙ I | begin
refine ⟨α.hom ◫ β.hom, α.inv ◫ β.inv, _, _⟩,
{ ext, rw [←nat_trans.exchange], simp, refl },
ext, rw [←nat_trans.exchange], simp, refl
end | def | category_theory.nat_iso.hcomp | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | Horizontal composition of natural isomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_map_iff {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) {X Y : C} (f : X ⟶ Y) :
is_iso (F₁.map f) ↔ is_iso (F₂.map f) | begin
revert F₁ F₂,
suffices : ∀ {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) (hf : is_iso (F₁.map f)), is_iso (F₂.map f),
{ exact λ F₁ F₂ e, ⟨this e, this e.symm⟩, },
introsI F₁ F₂ e hf,
refine is_iso.mk ⟨e.inv.app Y ≫ inv (F₁.map f) ≫ e.hom.app X, _, _⟩,
{ simp only [nat_trans.naturality_assoc, is_iso.hom_inv_id_assoc, ... | lemma | category_theory.nat_iso.is_iso_map_iff | category_theory | src/category_theory/natural_isomorphism.lean | [
"category_theory.functor.category",
"category_theory.isomorphism"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans (F G : C ⥤ D) : Type (max u₁ v₂) | (app : Π X : C, F.obj X ⟶ G.obj X)
(naturality' : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), F.map f ≫ app Y = app X ≫ G.map f . obviously) | structure | category_theory.nat_trans | category_theory | src/category_theory/natural_transformation.lean | [
"category_theory.functor.basic"
] | [] | `nat_trans F G` represents a natural transformation between functors `F` and `G`.
The field `app` provides the components of the natural transformation.
Naturality is expressed by `α.naturality_lemma`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_app {F G : C ⥤ D} {α β : nat_trans F G} (h : α = β) (X : C) : α.app X = β.app X | congr_fun (congr_arg nat_trans.app h) X | lemma | category_theory.congr_app | category_theory | src/category_theory/natural_transformation.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id (F : C ⥤ D) : nat_trans F F | { app := λ X, 𝟙 (F.obj X) } | def | category_theory.nat_trans.id | category_theory | src/category_theory/natural_transformation.lean | [
"category_theory.functor.basic"
] | [] | `nat_trans.id F` is the identity natural transformation on a functor `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_app' (F : C ⥤ D) (X : C) : (nat_trans.id F).app X = 𝟙 (F.obj X) | rfl | lemma | category_theory.nat_trans.id_app' | category_theory | src/category_theory/natural_transformation.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vcomp (α : nat_trans F G) (β : nat_trans G H) : nat_trans F H | { app := λ X, (α.app X) ≫ (β.app X) } | def | category_theory.nat_trans.vcomp | category_theory | src/category_theory/natural_transformation.lean | [
"category_theory.functor.basic"
] | [] | `vcomp α β` is the vertical compositions of natural transformations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vcomp_app (α : nat_trans F G) (β : nat_trans G H) (X : C) :
(vcomp α β).app X = (α.app X) ≫ (β.app X) | rfl | lemma | category_theory.nat_trans.vcomp_app | category_theory | src/category_theory/natural_transformation.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noetherian_object (X : C) : Prop | (subobject_gt_well_founded : well_founded ((>) : subobject X → subobject X → Prop)) | class | category_theory.noetherian_object | category_theory | src/category_theory/noetherian.lean | [
"category_theory.subobject.lattice",
"category_theory.essentially_small",
"category_theory.simple"
] | [] | A noetherian object is an object
which does not have infinite increasing sequences of subobjects.
See https://stacks.math.columbia.edu/tag/0FCG | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
artinian_object (X : C) : Prop | (subobject_lt_well_founded [] : well_founded ((<) : subobject X → subobject X → Prop)) | class | category_theory.artinian_object | category_theory | src/category_theory/noetherian.lean | [
"category_theory.subobject.lattice",
"category_theory.essentially_small",
"category_theory.simple"
] | [] | An artinian object is an object
which does not have infinite decreasing sequences of subobjects.
See https://stacks.math.columbia.edu/tag/0FCF | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
noetherian extends essentially_small C | (noetherian_object : ∀ (X : C), noetherian_object X) | class | category_theory.noetherian | category_theory | src/category_theory/noetherian.lean | [
"category_theory.subobject.lattice",
"category_theory.essentially_small",
"category_theory.simple"
] | [] | A category is noetherian if it is essentially small and all objects are noetherian. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
artinian extends essentially_small C | (artinian_object : ∀ (X : C), artinian_object X) | class | category_theory.artinian | category_theory | src/category_theory/noetherian.lean | [
"category_theory.subobject.lattice",
"category_theory.essentially_small",
"category_theory.simple"
] | [] | A category is artinian if it is essentially small and all objects are artinian. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_simple_subobject {X : C} [artinian_object X] (h : ¬ is_zero X) :
∃ (Y : subobject X), simple (Y : C) | begin
haveI : nontrivial (subobject X) := nontrivial_of_not_is_zero h,
haveI := is_atomic_of_order_bot_well_founded_lt (artinian_object.subobject_lt_well_founded X),
have := is_atomic.eq_bot_or_exists_atom_le (⊤ : subobject X),
obtain ⟨Y, s⟩ := (is_atomic.eq_bot_or_exists_atom_le (⊤ : subobject X)).resolve_left... | lemma | category_theory.exists_simple_subobject | category_theory | src/category_theory/noetherian.lean | [
"category_theory.subobject.lattice",
"category_theory.essentially_small",
"category_theory.simple"
] | [
"is_atomic_of_order_bot_well_founded_lt",
"nontrivial",
"top_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simple_subobject {X : C} [artinian_object X] (h : ¬ is_zero X) : C | (exists_simple_subobject h).some | def | category_theory.simple_subobject | category_theory | src/category_theory/noetherian.lean | [
"category_theory.subobject.lattice",
"category_theory.essentially_small",
"category_theory.simple"
] | [] | Choose an arbitrary simple subobject of a non-zero artinian object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simple_subobject_arrow {X : C} [artinian_object X] (h : ¬ is_zero X) :
simple_subobject h ⟶ X | (exists_simple_subobject h).some.arrow | def | category_theory.simple_subobject_arrow | category_theory | src/category_theory/noetherian.lean | [
"category_theory.subobject.lattice",
"category_theory.essentially_small",
"category_theory.simple"
] | [] | The monomorphism from the arbitrary simple subobject of a non-zero artinian object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quiver.hom.op_inj {X Y : C} :
function.injective (quiver.hom.op : (X ⟶ Y) → (op Y ⟶ op X)) | λ _ _ H, congr_arg quiver.hom.unop H | lemma | quiver.hom.op_inj | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.op",
"quiver.hom.unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quiver.hom.unop_inj {X Y : Cᵒᵖ} :
function.injective (quiver.hom.unop : (X ⟶ Y) → (unop Y ⟶ unop X)) | λ _ _ H, congr_arg quiver.hom.op H | lemma | quiver.hom.unop_inj | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.op",
"quiver.hom.unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quiver.hom.unop_op {X Y : C} (f : X ⟶ Y) : f.op.unop = f | rfl | lemma | quiver.hom.unop_op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quiver.hom.op_unop {X Y : Cᵒᵖ} (f : X ⟶ Y) : f.unop.op = f | rfl | lemma | quiver.hom.op_unop | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
category.opposite : category.{v₁} Cᵒᵖ | { comp := λ _ _ _ f g, (g.unop ≫ f.unop).op,
id := λ X, (𝟙 (unop X)).op } | instance | category_theory.category.opposite | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | The opposite category.
See <https://stacks.math.columbia.edu/tag/001M>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_comp {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} :
(f ≫ g).op = g.op ≫ f.op | rfl | lemma | category_theory.op_comp | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_id {X : C} : (𝟙 X).op = 𝟙 (op X) | rfl | lemma | category_theory.op_id | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_comp {X Y Z : Cᵒᵖ} {f : X ⟶ Y} {g : Y ⟶ Z} :
(f ≫ g).unop = g.unop ≫ f.unop | rfl | lemma | category_theory.unop_comp | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_id {X : Cᵒᵖ} : (𝟙 X).unop = 𝟙 (unop X) | rfl | lemma | category_theory.unop_id | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_id_op {X : C} : (𝟙 (op X)).unop = 𝟙 X | rfl | lemma | category_theory.unop_id_op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_id_unop {X : Cᵒᵖ} : (𝟙 (unop X)).op = 𝟙 X | rfl | lemma | category_theory.op_id_unop | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_op : (Cᵒᵖ)ᵒᵖ ⥤ C | { obj := λ X, unop (unop X),
map := λ X Y f, f.unop.unop } | def | category_theory.op_op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | The functor from the double-opposite of a category to the underlying category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_unop : C ⥤ Cᵒᵖᵒᵖ | { obj := λ X, op (op X),
map := λ X Y f, f.op.op } | def | category_theory.unop_unop | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | The functor from a category to its double-opposite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_op_equivalence : Cᵒᵖᵒᵖ ≌ C | { functor := op_op C,
inverse := unop_unop C,
unit_iso := iso.refl (𝟭 Cᵒᵖᵒᵖ),
counit_iso := iso.refl (unop_unop C ⋙ op_op C) } | def | category_theory.op_op_equivalence | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | The double opposite category is equivalent to the original. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_op {X Y : C} (f : X ⟶ Y) [is_iso f] : is_iso f.op | ⟨⟨(inv f).op,
⟨quiver.hom.unop_inj (by tidy), quiver.hom.unop_inj (by tidy)⟩⟩⟩ | instance | category_theory.is_iso_op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.unop_inj"
] | If `f` is an isomorphism, so is `f.op` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_op {X Y : C} (f : X ⟶ Y) [is_iso f.op] : is_iso f | ⟨⟨(inv (f.op)).unop,
⟨quiver.hom.op_inj (by simp), quiver.hom.op_inj (by simp)⟩⟩⟩ | lemma | category_theory.is_iso_of_op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.op_inj"
] | If `f.op` is an isomorphism `f` must be too.
(This cannot be an instance as it would immediately loop!) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_op_iff {X Y : C} (f : X ⟶ Y) : is_iso f.op ↔ is_iso f | ⟨λ hf, by exactI is_iso_of_op _, λ hf, by exactI infer_instance⟩ | lemma | category_theory.is_iso_op_iff | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_unop_iff {X Y : Cᵒᵖ} (f : X ⟶ Y) : is_iso f.unop ↔ is_iso f | by rw [← is_iso_op_iff f.unop, quiver.hom.op_unop] | lemma | category_theory.is_iso_unop_iff | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.op_unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_unop {X Y : Cᵒᵖ} (f : X ⟶ Y) [is_iso f] : is_iso f.unop | (is_iso_unop_iff _).2 infer_instance | instance | category_theory.is_iso_unop | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_inv {X Y : C} (f : X ⟶ Y) [is_iso f] : (inv f).op = inv f.op | by { ext, rw [← op_comp, is_iso.inv_hom_id, op_id] } | lemma | category_theory.op_inv | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_inv {X Y : Cᵒᵖ} (f : X ⟶ Y) [is_iso f] : (inv f).unop = inv f.unop | by { ext, rw [← unop_comp, is_iso.inv_hom_id, unop_id] } | lemma | category_theory.unop_inv | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op (F : C ⥤ D) : Cᵒᵖ ⥤ Dᵒᵖ | { obj := λ X, op (F.obj (unop X)),
map := λ X Y f, (F.map f.unop).op } | def | category_theory.functor.op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | The opposite of a functor, i.e. considering a functor `F : C ⥤ D` as a functor `Cᵒᵖ ⥤ Dᵒᵖ`.
In informal mathematics no distinction is made between these. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop (F : Cᵒᵖ ⥤ Dᵒᵖ) : C ⥤ D | { obj := λ X, unop (F.obj (op X)),
map := λ X Y f, (F.map f.op).unop } | def | category_theory.functor.unop | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | Given a functor `F : Cᵒᵖ ⥤ Dᵒᵖ` we can take the "unopposite" functor `F : C ⥤ D`.
In informal mathematics no distinction is made between these. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_unop_iso (F : C ⥤ D) : F.op.unop ≅ F | nat_iso.of_components (λ X, iso.refl _) (by tidy) | def | category_theory.functor.op_unop_iso | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | The isomorphism between `F.op.unop` and `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_op_iso (F : Cᵒᵖ ⥤ Dᵒᵖ) : F.unop.op ≅ F | nat_iso.of_components (λ X, iso.refl _) (by tidy) | def | category_theory.functor.unop_op_iso | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | The isomorphism between `F.unop.op` and `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_hom : (C ⥤ D)ᵒᵖ ⥤ (Cᵒᵖ ⥤ Dᵒᵖ) | { obj := λ F, (unop F).op,
map := λ F G α,
{ app := λ X, (α.unop.app (unop X)).op,
naturality' := λ X Y f, quiver.hom.unop_inj (α.unop.naturality f.unop).symm } } | def | category_theory.functor.op_hom | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.unop_inj"
] | Taking the opposite of a functor is functorial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_inv : (Cᵒᵖ ⥤ Dᵒᵖ) ⥤ (C ⥤ D)ᵒᵖ | { obj := λ F, op F.unop,
map := λ F G α, quiver.hom.op
{ app := λ X, (α.app (op X)).unop,
naturality' := λ X Y f, quiver.hom.op_inj $ (α.naturality f.op).symm } } | def | category_theory.functor.op_inv | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.op",
"quiver.hom.op_inj"
] | Take the "unopposite" of a functor is functorial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_op (F : C ⥤ Dᵒᵖ) : Cᵒᵖ ⥤ D | { obj := λ X, unop (F.obj (unop X)),
map := λ X Y f, (F.map f.unop).unop } | def | category_theory.functor.left_op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | Another variant of the opposite of functor, turning a functor `C ⥤ Dᵒᵖ` into a functor `Cᵒᵖ ⥤ D`.
In informal mathematics no distinction is made. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_op (F : Cᵒᵖ ⥤ D) : C ⥤ Dᵒᵖ | { obj := λ X, op (F.obj (op X)),
map := λ X Y f, (F.map f.op).op } | def | category_theory.functor.right_op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | Another variant of the opposite of functor, turning a functor `Cᵒᵖ ⥤ D` into a functor `C ⥤ Dᵒᵖ`.
In informal mathematics no distinction is made. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_op_faithful {F : Cᵒᵖ ⥤ D} [faithful F] : faithful F.right_op | { map_injective' := λ X Y f g h, quiver.hom.op_inj (map_injective F (quiver.hom.op_inj h)) } | instance | category_theory.functor.right_op_faithful | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.op_inj"
] | If F is faithful then the right_op of F is also faithful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_op_faithful {F : C ⥤ Dᵒᵖ} [faithful F] : faithful F.left_op | { map_injective' := λ X Y f g h, quiver.hom.unop_inj (map_injective F (quiver.hom.unop_inj h)) } | instance | category_theory.functor.left_op_faithful | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.unop_inj"
] | If F is faithful then the left_op of F is also faithful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_op_right_op_iso (F : C ⥤ Dᵒᵖ) : F.left_op.right_op ≅ F | nat_iso.of_components (λ X, iso.refl _) (by tidy) | def | category_theory.functor.left_op_right_op_iso | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | The isomorphism between `F.left_op.right_op` and `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_op_left_op_iso (F : Cᵒᵖ ⥤ D) : F.right_op.left_op ≅ F | nat_iso.of_components (λ X, iso.refl _) (by tidy) | def | category_theory.functor.right_op_left_op_iso | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | The isomorphism between `F.right_op.left_op` and `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_op_left_op_eq (F : Cᵒᵖ ⥤ D) : F.right_op.left_op = F | by { cases F, refl, } | lemma | category_theory.functor.right_op_left_op_eq | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | Whenever possible, it is advisable to use the isomorphism `right_op_left_op_iso`
instead of this equality of functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op (α : F ⟶ G) : G.op ⟶ F.op | { app := λ X, (α.app (unop X)).op,
naturality' := λ X Y f, quiver.hom.unop_inj (by simp) } | def | category_theory.nat_trans.op | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.unop_inj"
] | The opposite of a natural transformation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_id (F : C ⥤ D) : nat_trans.op (𝟙 F) = 𝟙 (F.op) | rfl | lemma | category_theory.nat_trans.op_id | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop {F G : Cᵒᵖ ⥤ Dᵒᵖ} (α : F ⟶ G) : G.unop ⟶ F.unop | { app := λ X, (α.app (op X)).unop,
naturality' := λ X Y f, quiver.hom.op_inj (by simp) } | def | category_theory.nat_trans.unop | category_theory | src/category_theory/opposites.lean | [
"category_theory.equivalence"
] | [
"quiver.hom.op_inj"
] | The "unopposite" of a natural transformation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.