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|---|---|---|---|---|---|---|---|---|---|---|
iff_is_split_mono_eq_zero {X Y : C} (f : X ⟶ Y) [is_split_mono f] : is_zero X ↔ f = 0 | begin
rw iff_id_eq_zero,
split,
{ intro h, rw [←category.id_comp f, h, zero_comp], },
{ intro h, rw [←is_split_mono.id f], simp [h], },
end | lemma | category_theory.limits.is_zero.iff_is_split_mono_eq_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_is_split_epi_eq_zero {X Y : C} (f : X ⟶ Y) [is_split_epi f] : is_zero Y ↔ f = 0 | begin
rw iff_id_eq_zero,
split,
{ intro h, rw [←category.comp_id f, h, comp_zero], },
{ intro h, rw [←is_split_epi.id f], simp [h], },
end | lemma | category_theory.limits.is_zero.iff_is_split_epi_eq_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mono {X Y : C} (f : X ⟶ Y) [mono f] (i : is_zero Y) : is_zero X | begin
unfreezingI { have hf := i.eq_zero_of_tgt f, subst hf, },
exact is_zero.of_mono_zero X Y,
end | lemma | category_theory.limits.is_zero.of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_epi {X Y : C} (f : X ⟶ Y) [epi f] (i : is_zero X) : is_zero Y | begin
unfreezingI { have hf := i.eq_zero_of_src f, subst hf, },
exact is_zero.of_epi_zero X Y,
end | lemma | category_theory.limits.is_zero.of_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_zero.has_zero_morphisms {O : C} (hO : is_zero O) : has_zero_morphisms C | { has_zero := λ X Y,
{ zero := hO.from X ≫ hO.to Y },
zero_comp' := λ X Y Z f, by { rw category.assoc, congr, apply hO.eq_of_src, },
comp_zero' := λ X Y Z f, by { rw ←category.assoc, congr, apply hO.eq_of_tgt, }} | def | category_theory.limits.is_zero.has_zero_morphisms | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | A category with a zero object has zero morphisms.
It is rarely a good idea to use this. Many categories that have a zero object have zero
morphisms for some other reason, for example from additivity. Library code that uses
`zero_morphisms_of_zero_object` will then be incompatible with these categories beca... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_morphisms_of_zero_object : has_zero_morphisms C | { has_zero := λ X Y,
{ zero := (default : X ⟶ 0) ≫ default },
zero_comp' := λ X Y Z f, by { dunfold has_zero.zero, rw category.assoc, congr, },
comp_zero' := λ X Y Z f, by { dunfold has_zero.zero, rw ←category.assoc, congr, }} | def | category_theory.limits.has_zero_object.zero_morphisms_of_zero_object | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | A category with a zero object has zero morphisms.
It is rarely a good idea to use this. Many categories that have a zero object have zero
morphisms for some other reason, for example from additivity. Library code that uses
`zero_morphisms_of_zero_object` will then be incompatible with these categories beca... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_iso_is_initial_hom {X : C} (t : is_initial X) :
(zero_iso_is_initial t).hom = 0 | by ext | lemma | category_theory.limits.has_zero_object.zero_iso_is_initial_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_iso_is_initial_inv {X : C} (t : is_initial X) :
(zero_iso_is_initial t).inv = 0 | by ext | lemma | category_theory.limits.has_zero_object.zero_iso_is_initial_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_iso_is_terminal_hom {X : C} (t : is_terminal X) :
(zero_iso_is_terminal t).hom = 0 | by ext | lemma | category_theory.limits.has_zero_object.zero_iso_is_terminal_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_iso_is_terminal_inv {X : C} (t : is_terminal X) :
(zero_iso_is_terminal t).inv = 0 | by ext | lemma | category_theory.limits.has_zero_object.zero_iso_is_terminal_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_iso_initial_hom [has_initial C] : zero_iso_initial.hom = (0 : 0 ⟶ ⊥_ C) | by ext | lemma | category_theory.limits.has_zero_object.zero_iso_initial_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_iso_initial_inv [has_initial C] : zero_iso_initial.inv = (0 : ⊥_ C ⟶ 0) | by ext | lemma | category_theory.limits.has_zero_object.zero_iso_initial_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_iso_terminal_hom [has_terminal C] : zero_iso_terminal.hom = (0 : 0 ⟶ ⊤_ C) | by ext | lemma | category_theory.limits.has_zero_object.zero_iso_terminal_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_iso_terminal_inv [has_terminal C] : zero_iso_terminal.inv = (0 : ⊤_ C ⟶ 0) | by ext | lemma | category_theory.limits.has_zero_object.zero_iso_terminal_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_zero.map [has_zero_object D] [has_zero_morphisms D] {F : C ⥤ D} (hF : is_zero F)
{X Y : C} (f : X ⟶ Y) : F.map f = 0 | (hF.obj _).eq_of_src _ _ | lemma | category_theory.limits.is_zero.map | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.category_theory.functor.zero_obj [has_zero_object D]
(X : C) : is_zero ((0 : C ⥤ D).obj X) | (is_zero_zero _).obj _ | lemma | category_theory.functor.zero_obj | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.category_theory.zero_map [has_zero_object D] [has_zero_morphisms D]
{X Y : C} (f : X ⟶ Y) : (0 : C ⥤ D).map f = 0 | (is_zero_zero _).map _ | lemma | category_theory.zero_map | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_zero : 𝟙 (0 : C) = (0 : 0 ⟶ 0) | by ext | lemma | category_theory.limits.id_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_of_to_zero {X : C} (f : X ⟶ 0) : f = 0 | by ext | lemma | category_theory.limits.zero_of_to_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_of_target_iso_zero {X Y : C} (f : X ⟶ Y) (i : Y ≅ 0) : f = 0 | begin
have h : f = f ≫ i.hom ≫ 𝟙 0 ≫ i.inv := by simp only [iso.hom_inv_id, id_comp, comp_id],
simpa using h,
end | lemma | category_theory.limits.zero_of_target_iso_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_of_from_zero {X : C} (f : 0 ⟶ X) : f = 0 | by ext | lemma | category_theory.limits.zero_of_from_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | An arrow starting at the zero object is zero | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_of_source_iso_zero {X Y : C} (f : X ⟶ Y) (i : X ≅ 0) : f = 0 | begin
have h : f = i.hom ≫ 𝟙 0 ≫ i.inv ≫ f := by simp only [iso.hom_inv_id_assoc, id_comp, comp_id],
simpa using h,
end | lemma | category_theory.limits.zero_of_source_iso_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_of_source_iso_zero' {X Y : C} (f : X ⟶ Y) (i : is_isomorphic X 0) : f = 0 | zero_of_source_iso_zero f (nonempty.some i) | lemma | category_theory.limits.zero_of_source_iso_zero' | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [
"nonempty.some"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_of_target_iso_zero' {X Y : C} (f : X ⟶ Y) (i : is_isomorphic Y 0) : f = 0 | zero_of_target_iso_zero f (nonempty.some i) | lemma | category_theory.limits.zero_of_target_iso_zero' | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [
"nonempty.some"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_of_source_iso_zero {X Y : C} (f : X ⟶ Y) (i : X ≅ 0) : mono f | ⟨λ Z g h w, by rw [zero_of_target_iso_zero g i, zero_of_target_iso_zero h i]⟩ | lemma | category_theory.limits.mono_of_source_iso_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_of_target_iso_zero {X Y : C} (f : X ⟶ Y) (i : Y ≅ 0) : epi f | ⟨λ Z g h w, by rw [zero_of_source_iso_zero g i, zero_of_source_iso_zero h i]⟩ | lemma | category_theory.limits.epi_of_target_iso_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_zero_equiv_iso_zero (X : C) : (𝟙 X = 0) ≃ (X ≅ 0) | { to_fun := λ h, { hom := 0, inv := 0, },
inv_fun := λ i, zero_of_target_iso_zero (𝟙 X) i,
left_inv := by tidy,
right_inv := by tidy, } | def | category_theory.limits.id_zero_equiv_iso_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [
"inv_fun"
] | An object `X` has `𝟙 X = 0` if and only if it is isomorphic to the zero object.
Because `X ≅ 0` contains data (even if a subsingleton), we express this `↔` as an `≃`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_zero_equiv_iso_zero_apply_hom (X : C) (h : 𝟙 X = 0) :
((id_zero_equiv_iso_zero X) h).hom = 0 | rfl | lemma | category_theory.limits.id_zero_equiv_iso_zero_apply_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_zero_equiv_iso_zero_apply_inv (X : C) (h : 𝟙 X = 0) :
((id_zero_equiv_iso_zero X) h).inv = 0 | rfl | lemma | category_theory.limits.id_zero_equiv_iso_zero_apply_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_zero_of_mono_zero {X Y : C} (h : mono (0 : X ⟶ Y)) : X ≅ 0 | { hom := 0,
inv := 0,
hom_inv_id' := (cancel_mono (0 : X ⟶ Y)).mp (by simp) } | def | category_theory.limits.iso_zero_of_mono_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | If `0 : X ⟶ Y` is an monomorphism, then `X ≅ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_zero_of_epi_zero {X Y : C} (h : epi (0 : X ⟶ Y)) : Y ≅ 0 | { hom := 0,
inv := 0,
hom_inv_id' := (cancel_epi (0 : X ⟶ Y)).mp (by simp) } | def | category_theory.limits.iso_zero_of_epi_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | If `0 : X ⟶ Y` is an epimorphism, then `Y ≅ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_zero_of_mono_eq_zero {X Y : C} {f : X ⟶ Y} [mono f] (h : f = 0) : X ≅ 0 | by { unfreezingI { subst h, }, apply iso_zero_of_mono_zero ‹_›, } | def | category_theory.limits.iso_zero_of_mono_eq_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | If a monomorphism out of `X` is zero, then `X ≅ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_zero_of_epi_eq_zero {X Y : C} {f : X ⟶ Y} [epi f] (h : f = 0) : Y ≅ 0 | by { unfreezingI { subst h, }, apply iso_zero_of_epi_zero ‹_›, } | def | category_theory.limits.iso_zero_of_epi_eq_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | If an epimorphism in to `Y` is zero, then `Y ≅ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_of_is_isomorphic_zero {X : C} (P : is_isomorphic X 0) : X ≅ 0 | { hom := 0,
inv := 0,
hom_inv_id' :=
begin
casesI P,
rw ←P.hom_inv_id,
rw ←category.id_comp P.inv,
simp,
end,
inv_hom_id' := by simp, } | def | category_theory.limits.iso_of_is_isomorphic_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | If an object `X` is isomorphic to 0, there's no need to use choice to construct
an explicit isomorphism: the zero morphism suffices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_zero_equiv (X Y : C) : is_iso (0 : X ⟶ Y) ≃ (𝟙 X = 0 ∧ 𝟙 Y = 0) | { to_fun := by { introsI i, rw ←is_iso.hom_inv_id (0 : X ⟶ Y),
rw ←is_iso.inv_hom_id (0 : X ⟶ Y), simp },
inv_fun := λ h, ⟨⟨(0 : Y ⟶ X), by tidy⟩⟩,
left_inv := by tidy,
right_inv := by tidy, } | def | category_theory.limits.is_iso_zero_equiv | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [
"inv_fun"
] | A zero morphism `0 : X ⟶ Y` is an isomorphism if and only if
the identities on both `X` and `Y` are zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_zero_self_equiv (X : C) : is_iso (0 : X ⟶ X) ≃ (𝟙 X = 0) | by simpa using is_iso_zero_equiv X X | def | category_theory.limits.is_iso_zero_self_equiv | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | A zero morphism `0 : X ⟶ X` is an isomorphism if and only if
the identity on `X` is zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_zero_equiv_iso_zero (X Y : C) : is_iso (0 : X ⟶ Y) ≃ (X ≅ 0) × (Y ≅ 0) | begin
-- This is lame, because `prod` can't cope with `Prop`, so we can't use `equiv.prod_congr`.
refine (is_iso_zero_equiv X Y).trans _,
symmetry,
fsplit,
{ rintros ⟨eX, eY⟩, fsplit,
exact (id_zero_equiv_iso_zero X).symm eX,
exact (id_zero_equiv_iso_zero Y).symm eY, },
{ rintros ⟨hX, hY⟩, fsplit,
... | def | category_theory.limits.is_iso_zero_equiv_iso_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | A zero morphism `0 : X ⟶ Y` is an isomorphism if and only if
`X` and `Y` are isomorphic to the zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_source_target_iso_zero {X Y : C} (f : X ⟶ Y) (i : X ≅ 0) (j : Y ≅ 0) : is_iso f | begin
rw zero_of_source_iso_zero f i,
exact (is_iso_zero_equiv_iso_zero _ _).inv_fun ⟨i, j⟩,
end | lemma | category_theory.limits.is_iso_of_source_target_iso_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_zero_self_equiv_iso_zero (X : C) : is_iso (0 : X ⟶ X) ≃ (X ≅ 0) | (is_iso_zero_equiv_iso_zero X X).trans subsingleton_prod_self_equiv | def | category_theory.limits.is_iso_zero_self_equiv_iso_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [
"subsingleton_prod_self_equiv"
] | A zero morphism `0 : X ⟶ X` is an isomorphism if and only if
`X` is isomorphic to the zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_zero_object_of_has_initial_object
[has_zero_morphisms C] [has_initial C] : has_zero_object C | begin
refine ⟨⟨⊥_ C, λ X, ⟨⟨⟨0⟩, by tidy⟩⟩, λ X, ⟨⟨⟨0⟩, λ f, _⟩⟩⟩⟩,
calc
f = f ≫ 𝟙 _ : (category.comp_id _).symm
... = f ≫ 0 : by congr
... = 0 : has_zero_morphisms.comp_zero _ _
end | lemma | category_theory.limits.has_zero_object_of_has_initial_object | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | If there are zero morphisms, any initial object is a zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_zero_object_of_has_terminal_object
[has_zero_morphisms C] [has_terminal C] : has_zero_object C | begin
refine ⟨⟨⊤_ C, λ X, ⟨⟨⟨0⟩, λ f, _⟩⟩, λ X, ⟨⟨⟨0⟩, by tidy⟩⟩⟩⟩,
calc
f = 𝟙 _ ≫ f : (category.id_comp _).symm
... = 0 ≫ f : by congr
... = 0 : zero_comp
end | lemma | category_theory.limits.has_zero_object_of_has_terminal_object | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | If there are zero morphisms, any terminal object is a zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_ι_comp_eq_zero {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} [has_image f]
[epi (factor_thru_image f)] (h : f ≫ g = 0) : image.ι f ≫ g = 0 | zero_of_epi_comp (factor_thru_image f) $ by simp [h] | lemma | category_theory.limits.image_ι_comp_eq_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_factor_thru_image_eq_zero {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} [has_image g]
(h : f ≫ g = 0) : f ≫ factor_thru_image g = 0 | zero_of_comp_mono (image.ι g) $ by simp [h] | lemma | category_theory.limits.comp_factor_thru_image_eq_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_factorisation_zero (X Y : C) : mono_factorisation (0 : X ⟶ Y) | { I := 0, m := 0, e := 0, } | def | category_theory.limits.mono_factorisation_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | The zero morphism has a `mono_factorisation` through the zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_factorisation_zero (X Y : C) : image_factorisation (0 : X ⟶ Y) | { F := mono_factorisation_zero X Y,
is_image := { lift := λ F', 0 } } | def | category_theory.limits.image_factorisation_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [
"lift"
] | The factorisation through the zero object is an image factorisation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_image_zero {X Y : C} : has_image (0 : X ⟶ Y) | has_image.mk $ image_factorisation_zero _ _ | instance | category_theory.limits.has_image_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_zero {X Y : C} : image (0 : X ⟶ Y) ≅ 0 | is_image.iso_ext (image.is_image (0 : X ⟶ Y)) (image_factorisation_zero X Y).is_image | def | category_theory.limits.image_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | The image of a zero morphism is the zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_zero' {X Y : C} {f : X ⟶ Y} (h : f = 0) [has_image f] : image f ≅ 0 | image.eq_to_iso h ≪≫ image_zero | def | category_theory.limits.image_zero' | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | The image of a morphism which is equal to zero is the zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.ι_zero {X Y : C} [has_image (0 : X ⟶ Y)] : image.ι (0 : X ⟶ Y) = 0 | begin
rw ←image.lift_fac (mono_factorisation_zero X Y),
simp,
end | lemma | category_theory.limits.image.ι_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image.ι_zero' [has_equalizers C] {X Y : C} {f : X ⟶ Y} (h : f = 0) [has_image f] :
image.ι f = 0 | by { rw image.eq_fac h, simp } | lemma | category_theory.limits.image.ι_zero' | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | If we know `f = 0`,
it requires a little work to conclude `image.ι f = 0`,
because `f = g` only implies `image f ≅ image g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_mono_sigma_ι {β : Type u'} [has_zero_morphisms C] (f : β → C)
[has_colimit (discrete.functor f)] (b : β) : is_split_mono (sigma.ι f b) | is_split_mono.mk'
{ retraction := sigma.desc $ pi.single b (𝟙 _) } | instance | category_theory.limits.is_split_mono_sigma_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_epi_pi_π {β : Type u'} [has_zero_morphisms C] (f : β → C)
[has_limit (discrete.functor f)] (b : β) : is_split_epi (pi.π f b) | is_split_epi.mk'
{ section_ := pi.lift $ pi.single b (𝟙 _) } | instance | category_theory.limits.is_split_epi_pi_π | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | In the presence of zero morphisms, projections into a product are (split) epimorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_mono_coprod_inl [has_zero_morphisms C] {X Y : C} [has_colimit (pair X Y)] :
is_split_mono (coprod.inl : X ⟶ X ⨿ Y) | is_split_mono.mk'
{ retraction := coprod.desc (𝟙 X) 0, } | instance | category_theory.limits.is_split_mono_coprod_inl | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_mono_coprod_inr [has_zero_morphisms C] {X Y : C} [has_colimit (pair X Y)] :
is_split_mono (coprod.inr : Y ⟶ X ⨿ Y) | is_split_mono.mk'
{ retraction := coprod.desc 0 (𝟙 Y), } | instance | category_theory.limits.is_split_mono_coprod_inr | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_epi_prod_fst [has_zero_morphisms C] {X Y : C} [has_limit (pair X Y)] :
is_split_epi (prod.fst : X ⨯ Y ⟶ X) | is_split_epi.mk'
{ section_ := prod.lift (𝟙 X) 0, } | instance | category_theory.limits.is_split_epi_prod_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | In the presence of zero morphisms, projections into a product are (split) epimorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_split_epi_prod_snd [has_zero_morphisms C] {X Y : C} [has_limit (pair X Y)] :
is_split_epi (prod.snd : X ⨯ Y ⟶ Y) | is_split_epi.mk'
{ section_ := prod.lift 0 (𝟙 Y), } | instance | category_theory.limits.is_split_epi_prod_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | In the presence of zero morphisms, projections into a product are (split) epimorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_zero (X : C) : Prop | (unique_to : ∀ Y, nonempty (unique (X ⟶ Y)))
(unique_from : ∀ Y, nonempty (unique (Y ⟶ X))) | structure | category_theory.limits.is_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [
"unique"
] | An object `X` in a category is a *zero object* if for every object `Y`
there is a unique morphism `to : X → Y` and a unique morphism `from : Y → X`.
This is a characteristic predicate for `has_zero_object`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
«to» (h : is_zero X) (Y : C) : X ⟶ Y | @default (X ⟶ Y) $ @unique.inhabited _ $ (h.unique_to Y).some | def | category_theory.limits.is_zero.«to» | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | If `h : is_zero X`, then `h.to Y` is a choice of unique morphism `X → Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_to (h : is_zero X) (f : X ⟶ Y) : f = h.to Y | @unique.eq_default _ (id _) _ | lemma | category_theory.limits.is_zero.eq_to | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [
"unique.eq_default"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_eq (h : is_zero X) (f : X ⟶ Y) : h.to Y = f | (h.eq_to f).symm | lemma | category_theory.limits.is_zero.to_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
«from» (h : is_zero X) (Y : C) : Y ⟶ X | @default (Y ⟶ X) $ @unique.inhabited _ $ (h.unique_from Y).some | def | category_theory.limits.is_zero.«from» | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | If `h : is_zero X`, then `h.from Y` is a choice of unique morphism `Y → X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_from (h : is_zero X) (f : Y ⟶ X) : f = h.from Y | @unique.eq_default _ (id _) _ | lemma | category_theory.limits.is_zero.eq_from | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [
"unique.eq_default"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_eq (h : is_zero X) (f : Y ⟶ X) : h.from Y = f | (h.eq_from f).symm | lemma | category_theory.limits.is_zero.from_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_src (hX : is_zero X) (f g : X ⟶ Y) : f = g | (hX.eq_to f).trans (hX.eq_to g).symm | lemma | category_theory.limits.is_zero.eq_of_src | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_tgt (hX : is_zero X) (f g : Y ⟶ X) : f = g | (hX.eq_from f).trans (hX.eq_from g).symm | lemma | category_theory.limits.is_zero.eq_of_tgt | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso (hX : is_zero X) (hY : is_zero Y) : X ≅ Y | { hom := hX.to Y,
inv := hX.from Y,
hom_inv_id' := hX.eq_of_src _ _,
inv_hom_id' := hY.eq_of_src _ _, } | def | category_theory.limits.is_zero.iso | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [
"iso"
] | Any two zero objects are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_initial (hX : is_zero X) : is_initial X | @is_initial.of_unique _ _ X $ λ Y, (hX.unique_to Y).some | def | category_theory.limits.is_zero.is_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | A zero object is in particular initial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_terminal (hX : is_zero X) : is_terminal X | @is_terminal.of_unique _ _ X $ λ Y, (hX.unique_from Y).some | def | category_theory.limits.is_zero.is_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | A zero object is in particular terminal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_is_initial (hX : is_zero X) (hY : is_initial Y) : X ≅ Y | hX.is_initial.unique_up_to_iso hY | def | category_theory.limits.is_zero.iso_is_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | The (unique) isomorphism between any initial object and the zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_is_terminal (hX : is_zero X) (hY : is_terminal Y) : X ≅ Y | hX.is_terminal.unique_up_to_iso hY | def | category_theory.limits.is_zero.iso_is_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | The (unique) isomorphism between any terminal object and the zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_iso (hY : is_zero Y) (e : X ≅ Y) : is_zero X | begin
refine ⟨λ Z, ⟨⟨⟨e.hom ≫ hY.to Z⟩, λ f, _⟩⟩, λ Z, ⟨⟨⟨hY.from Z ≫ e.inv⟩, λ f, _⟩⟩⟩,
{ rw ← cancel_epi e.inv, apply hY.eq_of_src, },
{ rw ← cancel_mono e.hom, apply hY.eq_of_tgt, },
end | lemma | category_theory.limits.is_zero.of_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op (h : is_zero X) : is_zero (opposite.op X) | ⟨λ Y, ⟨⟨⟨(h.from (opposite.unop Y)).op⟩, λ f, quiver.hom.unop_inj (h.eq_of_tgt _ _)⟩⟩,
λ Y, ⟨⟨⟨(h.to (opposite.unop Y)).op⟩, λ f, quiver.hom.unop_inj (h.eq_of_src _ _)⟩⟩⟩ | lemma | category_theory.limits.is_zero.op | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [
"opposite.op",
"opposite.unop",
"quiver.hom.unop_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop {X : Cᵒᵖ} (h : is_zero X) : is_zero (opposite.unop X) | ⟨λ Y, ⟨⟨⟨(h.from (opposite.op Y)).unop⟩, λ f, quiver.hom.op_inj (h.eq_of_tgt _ _)⟩⟩,
λ Y, ⟨⟨⟨(h.to (opposite.op Y)).unop⟩, λ f, quiver.hom.op_inj (h.eq_of_src _ _)⟩⟩⟩ | lemma | category_theory.limits.is_zero.unop | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [
"opposite.op",
"opposite.unop",
"quiver.hom.op_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso.is_zero_iff {X Y : C} (e : X ≅ Y) :
is_zero X ↔ is_zero Y | ⟨λ h, h.of_iso e.symm, λ h, h.of_iso e⟩ | lemma | category_theory.iso.is_zero_iff | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor.is_zero (F : C ⥤ D) (hF : ∀ X, is_zero (F.obj X)) :
is_zero F | begin
split; intros G; refine ⟨⟨⟨_⟩, _⟩⟩,
{ refine { app := λ X, (hF _).to _, naturality' := _ },
intros, exact (hF _).eq_of_src _ _ },
{ intro f, ext, apply (hF _).eq_of_src _ _ },
{ refine { app := λ X, (hF _).from _, naturality' := _ },
intros, exact (hF _).eq_of_tgt _ _ },
{ intro f, ext, apply (h... | lemma | category_theory.functor.is_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zero_object : Prop | (zero : ∃ X : C, is_zero X) | class | category_theory.limits.has_zero_object | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | A category "has a zero object" if it has an object which is both initial and terminal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_zero_object_punit : has_zero_object (discrete punit) | { zero := ⟨⟨⟨⟩⟩, by tidy, by tidy⟩, } | instance | category_theory.limits.has_zero_object_punit | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zero_object.has_zero : has_zero C | { zero := has_zero_object.zero.some } | def | category_theory.limits.has_zero_object.has_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | Construct a `has_zero C` for a category with a zero object.
This can not be a global instance as it will trigger for every `has_zero C` typeclass search. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_zero_zero : is_zero (0 : C) | has_zero_object.zero.some_spec | lemma | category_theory.limits.is_zero_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zero_object_op : has_zero_object Cᵒᵖ | ⟨⟨opposite.op 0, is_zero.op (is_zero_zero C)⟩⟩ | instance | category_theory.limits.has_zero_object_op | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zero_object_unop [has_zero_object Cᵒᵖ] : has_zero_object C | ⟨⟨opposite.unop 0, is_zero.unop (is_zero_zero Cᵒᵖ)⟩⟩ | lemma | category_theory.limits.has_zero_object_unop | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_zero.has_zero_object {X : C} (hX : is_zero X) : has_zero_object C | ⟨⟨X, hX⟩⟩ | lemma | category_theory.limits.is_zero.has_zero_object | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_zero.iso_zero [has_zero_object C] {X : C} (hX : is_zero X) : X ≅ 0 | hX.iso (is_zero_zero C) | def | category_theory.limits.is_zero.iso_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | Every zero object is isomorphic to *the* zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_zero.obj [has_zero_object D] {F : C ⥤ D} (hF : is_zero F) (X : C) :
is_zero (F.obj X) | begin
let G : C ⥤ D := (category_theory.functor.const C).obj 0,
have hG : is_zero G := functor.is_zero _ (λ X, is_zero_zero _),
let e : F ≅ G := hF.iso hG,
exact (is_zero_zero _).of_iso (e.app X),
end | lemma | category_theory.limits.is_zero.obj | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [
"category_theory.functor.const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_to (X : C) : unique (0 ⟶ X) | ((is_zero_zero C).unique_to X).some | def | category_theory.limits.has_zero_object.unique_to | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [
"unique"
] | There is a unique morphism from the zero object to any object `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_from (X : C) : unique (X ⟶ 0) | ((is_zero_zero C).unique_from X).some | def | category_theory.limits.has_zero_object.unique_from | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [
"unique"
] | There is a unique morphism from any object `X` to the zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_zero_ext {X : C} (f g : X ⟶ 0) : f = g | (is_zero_zero C).eq_of_tgt _ _ | lemma | category_theory.limits.has_zero_object.to_zero_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_zero_ext {X : C} (f g : 0 ⟶ X) : f = g | (is_zero_zero C).eq_of_src _ _ | lemma | category_theory.limits.has_zero_object.from_zero_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_to_zero_is_iso (f : (0 : C) ⟶ 0) :
is_iso f | by convert (show is_iso (𝟙 (0 : C)), by apply_instance) | instance | category_theory.limits.has_zero_object.zero_to_zero_is_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_is_initial : is_initial (0 : C) | (is_zero_zero C).is_initial | def | category_theory.limits.has_zero_object.zero_is_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | A zero object is in particular initial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_is_terminal : is_terminal (0 : C) | (is_zero_zero C).is_terminal | def | category_theory.limits.has_zero_object.zero_is_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | A zero object is in particular terminal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_initial : has_initial C | has_initial_of_unique 0 | instance | category_theory.limits.has_zero_object.has_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | A zero object is in particular initial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_terminal : has_terminal C | has_terminal_of_unique 0 | instance | category_theory.limits.has_zero_object.has_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | A zero object is in particular terminal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_iso_is_initial {X : C} (t : is_initial X) : 0 ≅ X | zero_is_initial.unique_up_to_iso t | def | category_theory.limits.has_zero_object.zero_iso_is_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | The (unique) isomorphism between any initial object and the zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_iso_is_terminal {X : C} (t : is_terminal X) : 0 ≅ X | zero_is_terminal.unique_up_to_iso t | def | category_theory.limits.has_zero_object.zero_iso_is_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | The (unique) isomorphism between any terminal object and the zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_iso_initial [has_initial C] : 0 ≅ ⊥_ C | zero_is_initial.unique_up_to_iso initial_is_initial | def | category_theory.limits.has_zero_object.zero_iso_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | The (unique) isomorphism between the chosen initial object and the chosen zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_iso_terminal [has_terminal C] : 0 ≅ ⊤_ C | zero_is_terminal.unique_up_to_iso terminal_is_terminal | def | category_theory.limits.has_zero_object.zero_iso_terminal | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | The (unique) isomorphism between the chosen terminal object and the chosen zero object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_initial : initial_mono_class C | initial_mono_class.of_is_initial zero_is_initial (λ X, category_theory.mono _) | instance | category_theory.limits.has_zero_object.has_strict_initial | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [
"category_theory.mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor.is_zero_iff [has_zero_object D] (F : C ⥤ D) :
is_zero F ↔ ∀ X, is_zero (F.obj X) | ⟨λ hF X, hF.obj X, functor.is_zero _⟩ | lemma | category_theory.functor.is_zero_iff | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_objects.lean | [
"category_theory.limits.shapes.terminal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normal_mono (f : X ⟶ Y) | (Z : C)
(g : Y ⟶ Z)
(w : f ≫ g = 0)
(is_limit : is_limit (kernel_fork.of_ι f w)) | class | category_theory.normal_mono | category_theory.limits.shapes.normal_mono | src/category_theory/limits/shapes/normal_mono/basic.lean | [
"category_theory.limits.shapes.regular_mono",
"category_theory.limits.shapes.kernels",
"category_theory.limits.preserves.basic"
] | [] | A normal monomorphism is a morphism which is the kernel of some morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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