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equivalence_reflects_normal_mono {D : Type u₂} [category.{v₁} D] [has_zero_morphisms D]
(F : C ⥤ D) [is_equivalence F] {X Y : C} {f : X ⟶ Y} (hf : normal_mono (F.map f)) :
normal_mono f | { Z := F.obj_preimage hf.Z,
g := full.preimage (hf.g ≫ (F.obj_obj_preimage_iso hf.Z).inv),
w := faithful.map_injective F $ by simp [reassoc_of hf.w],
is_limit := reflects_limit.reflects $
is_limit.of_cone_equiv (cones.postcompose_equivalence (comp_nat_iso F : _)) $
is_limit.of_iso_limit
(by exac... | def | category_theory.equivalence_reflects_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"
] | [] | If `F` is an equivalence and `F.map f` is a normal mono, then `f` is a normal mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_mono.regular_mono (f : X ⟶ Y) [I : normal_mono f] : regular_mono f | { left := I.g,
right := 0,
w := (by simpa using I.w),
..I } | instance | category_theory.normal_mono.regular_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"
] | [] | Every normal monomorphism is a regular monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_mono.lift' {W : C} (f : X ⟶ Y) [normal_mono f] (k : W ⟶ Y) (h : k ≫ normal_mono.g = 0) :
{l : W ⟶ X // l ≫ f = k} | kernel_fork.is_limit.lift' normal_mono.is_limit _ h | def | category_theory.normal_mono.lift' | 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"
] | [] | If `f` is a normal mono, then any map `k : W ⟶ Y` such that `k ≫ normal_mono.g = 0` induces
a morphism `l : W ⟶ X` such that `l ≫ f = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_of_is_pullback_snd_of_normal
{P Q R S : C} {f : P ⟶ Q} {g : P ⟶ R} {h : Q ⟶ S} {k : R ⟶ S}
[hn : normal_mono h] (comm : f ≫ h = g ≫ k) (t : is_limit (pullback_cone.mk _ _ comm)) :
normal_mono g | { Z := hn.Z,
g := k ≫ hn.g,
w := by rw [← reassoc_of comm, hn.w, has_zero_morphisms.comp_zero],
is_limit :=
begin
letI gr := regular_of_is_pullback_snd_of_regular comm t,
have q := (has_zero_morphisms.comp_zero k hn.Z).symm,
convert gr.is_limit,
dunfold kernel_fork.of_ι fork.of_ι,
congr, exa... | def | category_theory.normal_of_is_pullback_snd_of_normal | 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"
] | [
"comm",
"proof_irrel_heq"
] | The second leg of a pullback cone is a normal monomorphism if the right component is too.
See also `pullback.snd_of_mono` for the basic monomorphism version, and
`normal_of_is_pullback_fst_of_normal` for the flipped version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_of_is_pullback_fst_of_normal
{P Q R S : C} {f : P ⟶ Q} {g : P ⟶ R} {h : Q ⟶ S} {k : R ⟶ S}
[hn : normal_mono k] (comm : f ≫ h = g ≫ k) (t : is_limit (pullback_cone.mk _ _ comm)) :
normal_mono f | normal_of_is_pullback_snd_of_normal comm.symm (pullback_cone.flip_is_limit t) | def | category_theory.normal_of_is_pullback_fst_of_normal | 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"
] | [
"comm"
] | The first leg of a pullback cone is a normal monomorphism if the left component is too.
See also `pullback.fst_of_mono` for the basic monomorphism version, and
`normal_of_is_pullback_snd_of_normal` for the flipped version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_mono_category | (normal_mono_of_mono : ∀ {X Y : C} (f : X ⟶ Y) [mono f], normal_mono f) | class | category_theory.normal_mono_category | 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 mono category is a category in which every monomorphism is normal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_mono_of_mono [normal_mono_category C] (f : X ⟶ Y) [mono f] : normal_mono f | normal_mono_category.normal_mono_of_mono _ | def | category_theory.normal_mono_of_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"
] | [] | In a category in which every monomorphism is normal, we can express every monomorphism as
a kernel. This is not an instance because it would create an instance loop. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_mono_category_of_normal_mono_category [normal_mono_category C] :
regular_mono_category C | { regular_mono_of_mono := λ _ _ f _,
by { haveI := by exactI normal_mono_of_mono f, apply_instance } } | instance | category_theory.regular_mono_category_of_normal_mono_category | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normal_epi (f : X ⟶ Y) | (W : C)
(g : W ⟶ X)
(w : g ≫ f = 0)
(is_colimit : is_colimit (cokernel_cofork.of_π f w)) | class | category_theory.normal_epi | 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 epimorphism is a morphism which is the cokernel of some morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_reflects_normal_epi {D : Type u₂} [category.{v₁} D] [has_zero_morphisms D]
(F : C ⥤ D) [is_equivalence F] {X Y : C} {f : X ⟶ Y} (hf : normal_epi (F.map f)) :
normal_epi f | { W := F.obj_preimage hf.W,
g := full.preimage ((F.obj_obj_preimage_iso hf.W).hom ≫ hf.g),
w := faithful.map_injective F $ by simp [hf.w],
is_colimit := reflects_colimit.reflects $
is_colimit.of_cocone_equiv (cocones.precompose_equivalence (comp_nat_iso F).symm) $
is_colimit.of_iso_colimit
(by e... | def | category_theory.equivalence_reflects_normal_epi | 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"
] | [] | If `F` is an equivalence and `F.map f` is a normal epi, then `f` is a normal epi. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_epi.regular_epi (f : X ⟶ Y) [I : normal_epi f] : regular_epi f | { left := I.g,
right := 0,
w := (by simpa using I.w),
..I } | instance | category_theory.normal_epi.regular_epi | 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"
] | [] | Every normal epimorphism is a regular epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_epi.desc' {W : C} (f : X ⟶ Y) [normal_epi f] (k : X ⟶ W) (h : normal_epi.g ≫ k = 0) :
{l : Y ⟶ W // f ≫ l = k} | cokernel_cofork.is_colimit.desc' (normal_epi.is_colimit) _ h | def | category_theory.normal_epi.desc' | 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"
] | [] | If `f` is a normal epi, then every morphism `k : X ⟶ W` satisfying `normal_epi.g ≫ k = 0`
induces `l : Y ⟶ W` such that `f ≫ l = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_of_is_pushout_snd_of_normal {P Q R S : C} {f : P ⟶ Q} {g : P ⟶ R} {h : Q ⟶ S} {k : R ⟶ S}
[gn : normal_epi g] (comm : f ≫ h = g ≫ k) (t : is_colimit (pushout_cocone.mk _ _ comm)) :
normal_epi h | { W := gn.W,
g := gn.g ≫ f,
w := by rw [category.assoc, comm, reassoc_of gn.w, zero_comp],
is_colimit :=
begin
letI hn := regular_of_is_pushout_snd_of_regular comm t,
have q := (@zero_comp _ _ _ gn.W _ _ f).symm,
convert hn.is_colimit,
dunfold cokernel_cofork.of_π cofork.of_π,
congr, exact q... | def | category_theory.normal_of_is_pushout_snd_of_normal | 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"
] | [
"comm",
"proof_irrel_heq"
] | The second leg of a pushout cocone is a normal epimorphism if the right component is too.
See also `pushout.snd_of_epi` for the basic epimorphism version, and
`normal_of_is_pushout_fst_of_normal` for the flipped version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_of_is_pushout_fst_of_normal {P Q R S : C} {f : P ⟶ Q} {g : P ⟶ R} {h : Q ⟶ S} {k : R ⟶ S}
[hn : normal_epi f] (comm : f ≫ h = g ≫ k) (t : is_colimit (pushout_cocone.mk _ _ comm)) :
normal_epi k | normal_of_is_pushout_snd_of_normal comm.symm (pushout_cocone.flip_is_colimit t) | def | category_theory.normal_of_is_pushout_fst_of_normal | 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"
] | [
"comm"
] | The first leg of a pushout cocone is a normal epimorphism if the left component is too.
See also `pushout.fst_of_epi` for the basic epimorphism version, and
`normal_of_is_pushout_snd_of_normal` for the flipped version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_epi_of_normal_mono_unop {X Y : Cᵒᵖ} (f : X ⟶ Y) (m : normal_mono f.unop) :
normal_epi f | { W := op m.Z,
g := m.g.op,
w := congr_arg quiver.hom.op m.w,
is_colimit := cokernel_cofork.is_colimit.of_π _ _
(λ Z' g' w',
(kernel_fork.is_limit.lift' m.is_limit g'.unop (congr_arg quiver.hom.unop w')).1.op)
(λ Z' g' w',
congr_arg quiver.hom.op
(kernel_fork.is_limit.lift' m.is_limit ... | def | category_theory.normal_epi_of_normal_mono_unop | 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"
] | [
"quiver.hom.op",
"quiver.hom.unop",
"quiver.hom.unop_inj"
] | A normal mono becomes a normal epi in the opposite category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_mono_of_normal_epi_unop {X Y : Cᵒᵖ} (f : X ⟶ Y) (m : normal_epi f.unop) :
normal_mono f | { Z := op m.W,
g := m.g.op,
w := congr_arg quiver.hom.op m.w,
is_limit := kernel_fork.is_limit.of_ι _ _
(λ Z' g' w',
(cokernel_cofork.is_colimit.desc' m.is_colimit g'.unop (congr_arg quiver.hom.unop w')).1.op)
(λ Z' g' w',
congr_arg quiver.hom.op
(cokernel_cofork.is_colimit.desc' m.is_... | def | category_theory.normal_mono_of_normal_epi_unop | 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"
] | [
"quiver.hom.op",
"quiver.hom.unop",
"quiver.hom.unop_inj"
] | A normal epi becomes a normal mono in the opposite category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_epi_category | (normal_epi_of_epi : ∀ {X Y : C} (f : X ⟶ Y) [epi f], normal_epi f) | class | category_theory.normal_epi_category | 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 epi category is a category in which every epimorphism is normal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_epi_of_epi [normal_epi_category C] (f : X ⟶ Y) [epi f] : normal_epi f | normal_epi_category.normal_epi_of_epi _ | def | category_theory.normal_epi_of_epi | 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"
] | [] | In a category in which every epimorphism is normal, we can express every epimorphism as
a kernel. This is not an instance because it would create an instance loop. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_epi_category_of_normal_epi_category [normal_epi_category C] :
regular_epi_category C | { regular_epi_of_epi := λ _ _ f _,
by { haveI := by exactI normal_epi_of_epi f, apply_instance } } | instance | category_theory.regular_epi_category_of_normal_epi_category | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_of_mono {X Y Z : C} (a : X ⟶ Z) (b : Y ⟶ Z) [mono a] [mono b] :
has_limit (cospan a b) | let ⟨P, f, haf, i⟩ := normal_mono_of_mono a in
let ⟨Q, g, hbg, i'⟩ := normal_mono_of_mono b in
let ⟨a', ha'⟩ := kernel_fork.is_limit.lift' i (kernel.ι (prod.lift f g)) $
calc kernel.ι (prod.lift f g) ≫ f
= kernel.ι (prod.lift f g) ≫ prod.lift f g ≫ limits.prod.fst : by rw prod.lift_fst
... = (0 : kernel... | lemma | category_theory.normal_mono_category.pullback_of_mono | category_theory.limits.shapes.normal_mono | src/category_theory/limits/shapes/normal_mono/equalizers.lean | [
"category_theory.limits.shapes.normal_mono.basic",
"category_theory.limits.shapes.finite_products"
] | [] | The pullback of two monomorphisms exists. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
P {X Y : C} (f g : X ⟶ Y)
[mono (prod.lift (𝟙 X) f)] [mono (prod.lift (𝟙 X) g)] : C | pullback (prod.lift (𝟙 X) f) (prod.lift (𝟙 X) g) | abbreviation | category_theory.normal_mono_category.P | category_theory.limits.shapes.normal_mono | src/category_theory/limits/shapes/normal_mono/equalizers.lean | [
"category_theory.limits.shapes.normal_mono.basic",
"category_theory.limits.shapes.finite_products"
] | [] | The pullback of `(𝟙 X, f)` and `(𝟙 X, g)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit_parallel_pair {X Y : C} (f g : X ⟶ Y) : has_limit (parallel_pair f g) | have huv : (pullback.fst : P f g ⟶ X) = pullback.snd, from
calc (pullback.fst : P f g ⟶ X) = pullback.fst ≫ 𝟙 _ : eq.symm $ category.comp_id _
... = pullback.fst ≫ prod.lift (𝟙 X) f ≫ limits.prod.fst : by rw prod.lift_fst
... = pullback.snd ≫ prod.lift (𝟙 X) g ≫ limits.prod.fst : by rw pullback.condition_a... | lemma | category_theory.normal_mono_category.has_limit_parallel_pair | category_theory.limits.shapes.normal_mono | src/category_theory/limits/shapes/normal_mono/equalizers.lean | [
"category_theory.limits.shapes.normal_mono.basic",
"category_theory.limits.shapes.finite_products"
] | [] | The equalizer of `f` and `g` exists. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_equalizers : has_equalizers C | has_equalizers_of_has_limit_parallel_pair _ | instance | category_theory.normal_mono_category.has_equalizers | category_theory.limits.shapes.normal_mono | src/category_theory/limits/shapes/normal_mono/equalizers.lean | [
"category_theory.limits.shapes.normal_mono.basic",
"category_theory.limits.shapes.finite_products"
] | [] | A `normal_mono_category` category with finite products and kernels has all equalizers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_of_zero_cokernel {X Y : C} (f : X ⟶ Y) (Z : C)
(l : is_colimit (cokernel_cofork.of_π (0 : Y ⟶ Z) (show f ≫ 0 = 0, by simp))) : epi f | ⟨λ P u v huv,
begin
obtain ⟨W, w, hw, hl⟩ := normal_mono_of_mono (equalizer.ι u v),
obtain ⟨m, hm⟩ := equalizer.lift' f huv,
have hwf : f ≫ w = 0,
{ rw [←hm, category.assoc, hw, comp_zero] },
obtain ⟨n, hn⟩ := cokernel_cofork.is_colimit.desc' l _ hwf,
rw [cofork.π_of_π, zero_comp] at hn,
haveI : is_iso (... | lemma | category_theory.normal_mono_category.epi_of_zero_cokernel | category_theory.limits.shapes.normal_mono | src/category_theory/limits/shapes/normal_mono/equalizers.lean | [
"category_theory.limits.shapes.normal_mono.basic",
"category_theory.limits.shapes.finite_products"
] | [] | If a zero morphism is a cokernel of `f`, then `f` is an epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_of_zero_cancel {X Y : C} (f : X ⟶ Y)
(hf : ∀ (Z : C) (g : Y ⟶ Z) (hgf : f ≫ g = 0), g = 0) : epi f | epi_of_zero_cokernel f 0 $ zero_cokernel_of_zero_cancel f hf | lemma | category_theory.normal_mono_category.epi_of_zero_cancel | category_theory.limits.shapes.normal_mono | src/category_theory/limits/shapes/normal_mono/equalizers.lean | [
"category_theory.limits.shapes.normal_mono.basic",
"category_theory.limits.shapes.finite_products"
] | [] | If `f ≫ g = 0` implies `g = 0` for all `g`, then `g` is a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pushout_of_epi {X Y Z : C} (a : X ⟶ Y) (b : X ⟶ Z) [epi a] [epi b] :
has_colimit (span a b) | let ⟨P, f, hfa, i⟩ := normal_epi_of_epi a in
let ⟨Q, g, hgb, i'⟩ := normal_epi_of_epi b in
let ⟨a', ha'⟩ := cokernel_cofork.is_colimit.desc' i (cokernel.π (coprod.desc f g)) $
calc f ≫ cokernel.π (coprod.desc f g)
= coprod.inl ≫ coprod.desc f g ≫ cokernel.π (coprod.desc f g) : by rw coprod.inl_desc_assoc
... ... | lemma | category_theory.normal_epi_category.pushout_of_epi | category_theory.limits.shapes.normal_mono | src/category_theory/limits/shapes/normal_mono/equalizers.lean | [
"category_theory.limits.shapes.normal_mono.basic",
"category_theory.limits.shapes.finite_products"
] | [] | The pushout of two epimorphisms exists. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Q {X Y : C} (f g : X ⟶ Y)
[epi (coprod.desc (𝟙 Y) f)] [epi (coprod.desc (𝟙 Y) g)] : C | pushout (coprod.desc (𝟙 Y) f) (coprod.desc (𝟙 Y) g) | abbreviation | category_theory.normal_epi_category.Q | category_theory.limits.shapes.normal_mono | src/category_theory/limits/shapes/normal_mono/equalizers.lean | [
"category_theory.limits.shapes.normal_mono.basic",
"category_theory.limits.shapes.finite_products"
] | [] | The pushout of `(𝟙 Y, f)` and `(𝟙 Y, g)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimit_parallel_pair {X Y : C} (f g : X ⟶ Y) : has_colimit (parallel_pair f g) | have huv : (pushout.inl : Y ⟶ Q f g) = pushout.inr, from
calc (pushout.inl : Y ⟶ Q f g) = 𝟙 _ ≫ pushout.inl : eq.symm $ category.id_comp _
... = (coprod.inl ≫ coprod.desc (𝟙 Y) f) ≫ pushout.inl : by rw coprod.inl_desc
... = (coprod.inl ≫ coprod.desc (𝟙 Y) g) ≫ pushout.inr :
by simp only [category.ass... | lemma | category_theory.normal_epi_category.has_colimit_parallel_pair | category_theory.limits.shapes.normal_mono | src/category_theory/limits/shapes/normal_mono/equalizers.lean | [
"category_theory.limits.shapes.normal_mono.basic",
"category_theory.limits.shapes.finite_products"
] | [] | The coequalizer of `f` and `g` exists. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coequalizers : has_coequalizers C | has_coequalizers_of_has_colimit_parallel_pair _ | instance | category_theory.normal_epi_category.has_coequalizers | category_theory.limits.shapes.normal_mono | src/category_theory/limits/shapes/normal_mono/equalizers.lean | [
"category_theory.limits.shapes.normal_mono.basic",
"category_theory.limits.shapes.finite_products"
] | [] | A `normal_epi_category` category with finite coproducts and cokernels has all coequalizers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_of_zero_kernel {X Y : C} (f : X ⟶ Y) (Z : C)
(l : is_limit (kernel_fork.of_ι (0 : Z ⟶ X) (show 0 ≫ f = 0, by simp))) : mono f | ⟨λ P u v huv,
begin
obtain ⟨W, w, hw, hl⟩ := normal_epi_of_epi (coequalizer.π u v),
obtain ⟨m, hm⟩ := coequalizer.desc' f huv,
have hwf : w ≫ f = 0,
{ rw [←hm, reassoc_of hw, zero_comp] },
obtain ⟨n, hn⟩ := kernel_fork.is_limit.lift' l _ hwf,
rw [fork.ι_of_ι, has_zero_morphisms.comp_zero] at hn,
haveI : ... | lemma | category_theory.normal_epi_category.mono_of_zero_kernel | category_theory.limits.shapes.normal_mono | src/category_theory/limits/shapes/normal_mono/equalizers.lean | [
"category_theory.limits.shapes.normal_mono.basic",
"category_theory.limits.shapes.finite_products"
] | [] | If a zero morphism is a kernel of `f`, then `f` is a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_of_cancel_zero {X Y : C} (f : X ⟶ Y)
(hf : ∀ (Z : C) (g : Z ⟶ X) (hgf : g ≫ f = 0), g = 0) : mono f | mono_of_zero_kernel f 0 $ zero_kernel_of_cancel_zero f hf | lemma | category_theory.normal_epi_category.mono_of_cancel_zero | category_theory.limits.shapes.normal_mono | src/category_theory/limits/shapes/normal_mono/equalizers.lean | [
"category_theory.limits.shapes.normal_mono.basic",
"category_theory.limits.shapes.finite_products"
] | [] | If `g ≫ f = 0` implies `g = 0` for all `g`, then `f` is a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear (R : Type w) [semiring R] (C : Type u) [category.{v} C] [preadditive C] | (hom_module : Π X Y : C, module R (X ⟶ Y) . tactic.apply_instance)
(smul_comp' : ∀ (X Y Z : C) (r : R) (f : X ⟶ Y) (g : Y ⟶ Z),
(r • f) ≫ g = r • (f ≫ g) . obviously)
(comp_smul' : ∀ (X Y Z : C) (f : X ⟶ Y) (r : R) (g : Y ⟶ Z),
f ≫ (r • g) = r • (f ≫ g) . obviously) | class | category_theory.linear | category_theory.linear | src/category_theory/linear/basic.lean | [
"category_theory.preadditive.basic",
"algebra.module.linear_map",
"algebra.invertible",
"algebra.algebra.basic"
] | [
"module",
"semiring"
] | A category is called `R`-linear if `P ⟶ Q` is an `R`-module such that composition is
`R`-linear in both variables. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preadditive_nat_linear : linear ℕ C | { smul_comp' := λ X Y Z r f g, (preadditive.right_comp X g).map_nsmul f r,
comp_smul' := λ X Y Z f r g, (preadditive.left_comp Z f).map_nsmul g r, } | instance | category_theory.linear.preadditive_nat_linear | category_theory.linear | src/category_theory/linear/basic.lean | [
"category_theory.preadditive.basic",
"algebra.module.linear_map",
"algebra.invertible",
"algebra.algebra.basic"
] | [
"map_nsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preadditive_int_linear : linear ℤ C | { smul_comp' := λ X Y Z r f g, (preadditive.right_comp X g).map_zsmul f r,
comp_smul' := λ X Y Z f r g, (preadditive.left_comp Z f).map_zsmul g r, } | instance | category_theory.linear.preadditive_int_linear | category_theory.linear | src/category_theory/linear/basic.lean | [
"category_theory.preadditive.basic",
"algebra.module.linear_map",
"algebra.invertible",
"algebra.algebra.basic"
] | [
"map_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_category : linear.{w v} R (induced_category C F) | { hom_module := λ X Y, @linear.hom_module R _ C _ _ _ (F X) (F Y),
smul_comp' := λ P Q R f f' g, smul_comp' _ _ _ _ _ _,
comp_smul' := λ P Q R f g g', comp_smul' _ _ _ _ _ _, } | instance | category_theory.linear.induced_category | category_theory.linear | src/category_theory/linear/basic.lean | [
"category_theory.preadditive.basic",
"algebra.module.linear_map",
"algebra.invertible",
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_subcategory (Z : C → Prop) : linear.{w v} R (full_subcategory Z) | { hom_module := λ X Y, @linear.hom_module R _ C _ _ _ X.obj Y.obj,
smul_comp' := λ P Q R f f' g, smul_comp' _ _ _ _ _ _,
comp_smul' := λ P Q R f g g', comp_smul' _ _ _ _ _ _, } | instance | category_theory.linear.full_subcategory | category_theory.linear | src/category_theory/linear/basic.lean | [
"category_theory.preadditive.basic",
"algebra.module.linear_map",
"algebra.invertible",
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_comp {X Y : C} (Z : C) (f : X ⟶ Y) : (Y ⟶ Z) →ₗ[R] (X ⟶ Z) | { to_fun := λ g, f ≫ g,
map_add' := by simp,
map_smul' := by simp, } | def | category_theory.linear.left_comp | category_theory.linear | src/category_theory/linear/basic.lean | [
"category_theory.preadditive.basic",
"algebra.module.linear_map",
"algebra.invertible",
"algebra.algebra.basic"
] | [] | Composition by a fixed left argument as an `R`-linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_comp (X : C) {Y Z : C} (g : Y ⟶ Z) : (X ⟶ Y) →ₗ[R] (X ⟶ Z) | { to_fun := λ f, f ≫ g,
map_add' := by simp,
map_smul' := by simp, } | def | category_theory.linear.right_comp | category_theory.linear | src/category_theory/linear/basic.lean | [
"category_theory.preadditive.basic",
"algebra.module.linear_map",
"algebra.invertible",
"algebra.algebra.basic"
] | [] | Composition by a fixed right argument as an `R`-linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_congr (k : Type*) {C : Type*} [category C] [semiring k]
[preadditive C] [linear k C] {X Y W Z : C} (f₁ : X ≅ Y) (f₂ : W ≅ Z) :
(X ⟶ W) ≃ₗ[k] (Y ⟶ Z) | { inv_fun := (left_comp k W f₁.hom).comp (right_comp k Y f₂.symm.hom),
left_inv := λ x, by simp only [iso.symm_hom, linear_map.to_fun_eq_coe, linear_map.coe_comp,
function.comp_app, left_comp_apply, right_comp_apply, category.assoc, iso.hom_inv_id,
category.comp_id, iso.hom_inv_id_assoc],
right_inv := λ x, ... | def | category_theory.linear.hom_congr | category_theory.linear | src/category_theory/linear/basic.lean | [
"category_theory.preadditive.basic",
"algebra.module.linear_map",
"algebra.invertible",
"algebra.algebra.basic"
] | [
"inv_fun",
"linear_map.coe_comp",
"linear_map.to_fun_eq_coe",
"semiring"
] | Given isomorphic objects `X ≅ Y, W ≅ Z` in a `k`-linear category, we have a `k`-linear
isomorphism between `Hom(X, W)` and `Hom(Y, Z).` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_congr_apply (k : Type*) {C : Type*} [category C] [semiring k]
[preadditive C] [linear k C] {X Y W Z : C} (f₁ : X ≅ Y) (f₂ : W ≅ Z) (f : X ⟶ W) :
hom_congr k f₁ f₂ f = (f₁.inv ≫ f) ≫ f₂.hom | rfl | lemma | category_theory.linear.hom_congr_apply | category_theory.linear | src/category_theory/linear/basic.lean | [
"category_theory.preadditive.basic",
"algebra.module.linear_map",
"algebra.invertible",
"algebra.algebra.basic"
] | [
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_congr_symm_apply (k : Type*) {C : Type*} [category C] [semiring k]
[preadditive C] [linear k C] {X Y W Z : C} (f₁ : X ≅ Y) (f₂ : W ≅ Z) (f : Y ⟶ Z) :
(hom_congr k f₁ f₂).symm f = f₁.hom ≫ f ≫ f₂.inv | rfl | lemma | category_theory.linear.hom_congr_symm_apply | category_theory.linear | src/category_theory/linear/basic.lean | [
"category_theory.preadditive.basic",
"algebra.module.linear_map",
"algebra.invertible",
"algebra.algebra.basic"
] | [
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (X Y Z : C) : (X ⟶ Y) →ₗ[S] ((Y ⟶ Z) →ₗ[S] (X ⟶ Z)) | { to_fun := λ f, left_comp S Z f,
map_add' := by { intros, ext, simp, },
map_smul' := by { intros, ext, simp, }, } | def | category_theory.linear.comp | category_theory.linear | src/category_theory/linear/basic.lean | [
"category_theory.preadditive.basic",
"algebra.module.linear_map",
"algebra.invertible",
"algebra.algebra.basic"
] | [] | Composition as a bilinear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_category_linear : linear R (C ⥤ D) | { hom_module := λ F G,
{ smul := λ r α,
{ app := λ X, r • α.app X,
naturality' := by { intros, rw [comp_smul, smul_comp, α.naturality] } },
one_smul := by { intros, ext, apply one_smul },
zero_smul := by { intros, ext, apply zero_smul },
smul_zero := by { intros, ext, apply smul_zero },
add_... | instance | category_theory.functor_category_linear | category_theory.linear | src/category_theory/linear/functor_category.lean | [
"category_theory.preadditive.functor_category",
"category_theory.linear.basic"
] | [
"add_smul",
"one_smul",
"smul_add",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_linear_map (X : C) : (F ⟶ G) →ₗ[R] (F.obj X ⟶ G.obj X) | { to_fun := λ α, α.app X,
map_add' := λ _ _, rfl,
map_smul' := λ _ _, rfl, } | def | category_theory.nat_trans.app_linear_map | category_theory.linear | src/category_theory/linear/functor_category.lean | [
"category_theory.preadditive.functor_category",
"category_theory.linear.basic"
] | [] | Application of a natural transformation at a fixed object,
as group homomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
app_smul (X : C) (r : R) (α : F ⟶ G) : (r • α).app X = r • α.app X | rfl | lemma | category_theory.nat_trans.app_smul | category_theory.linear | src/category_theory/linear/functor_category.lean | [
"category_theory.preadditive.functor_category",
"category_theory.linear.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor.linear {C D : Type*} [category C] [category D]
[preadditive C] [preadditive D] [linear R C] [linear R D] (F : C ⥤ D) [F.additive] : Prop | (map_smul' : Π {X Y : C} {f : X ⟶ Y} {r : R}, F.map (r • f) = r • F.map f . obviously) | class | category_theory.functor.linear | category_theory.linear | src/category_theory/linear/linear_functor.lean | [
"category_theory.preadditive.additive_functor",
"category_theory.linear.basic"
] | [] | An additive functor `F` is `R`-linear provided `F.map` is an `R`-module morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_smul {X Y : C} (r : R) (f : X ⟶ Y) : F.map (r • f) = r • F.map f | functor.linear.map_smul' | lemma | category_theory.functor.map_smul | category_theory.linear | src/category_theory/linear/linear_functor.lean | [
"category_theory.preadditive.additive_functor",
"category_theory.linear.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_linear_map {X Y : C} : (X ⟶ Y) →ₗ[R] (F.obj X ⟶ F.obj Y) | { map_smul' := λ r f, F.map_smul r f,
..F.map_add_hom } | def | category_theory.functor.map_linear_map | category_theory.linear | src/category_theory/linear/linear_functor.lean | [
"category_theory.preadditive.additive_functor",
"category_theory.linear.basic"
] | [] | `F.map_linear_map` is an `R`-linear map whose underlying function is `F.map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_map_linear_map {X Y : C} : ⇑(F.map_linear_map R : (X ⟶ Y) →ₗ[R] _) = @map C _ D _ F X Y | rfl | lemma | category_theory.functor.coe_map_linear_map | category_theory.linear | src/category_theory/linear/linear_functor.lean | [
"category_theory.preadditive.additive_functor",
"category_theory.linear.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_functor_linear : functor.linear R (induced_functor F) | {} | instance | category_theory.functor.induced_functor_linear | category_theory.linear | src/category_theory/linear/linear_functor.lean | [
"category_theory.preadditive.additive_functor",
"category_theory.linear.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
full_subcategory_inclusion_linear
{C : Type*} [category C] [preadditive C] [category_theory.linear R C] (Z : C → Prop) :
(full_subcategory_inclusion Z).linear R | {} | instance | category_theory.functor.full_subcategory_inclusion_linear | category_theory.linear | src/category_theory/linear/linear_functor.lean | [
"category_theory.preadditive.additive_functor",
"category_theory.linear.basic"
] | [
"category_theory.linear"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_linear : F.linear ℕ | { map_smul' := λ X Y f r, F.map_add_hom.map_nsmul f r, } | instance | category_theory.functor.nat_linear | category_theory.linear | src/category_theory/linear/linear_functor.lean | [
"category_theory.preadditive.additive_functor",
"category_theory.linear.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_linear : F.linear ℤ | { map_smul' := λ X Y f r, (F.map_add_hom : (X ⟶ Y) →+ (F.obj X ⟶ F.obj Y)).map_zsmul f r, } | instance | category_theory.functor.int_linear | category_theory.linear | src/category_theory/linear/linear_functor.lean | [
"category_theory.preadditive.additive_functor",
"category_theory.linear.basic"
] | [
"map_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rat_linear : F.linear ℚ | { map_smul' := λ X Y f r, F.map_add_hom.to_rat_linear_map.map_smul r f, } | instance | category_theory.functor.rat_linear | category_theory.linear | src/category_theory/linear/linear_functor.lean | [
"category_theory.preadditive.additive_functor",
"category_theory.linear.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse_linear (e : C ≌ D) [e.functor.additive] [e.functor.linear R] :
e.inverse.linear R | { map_smul' := λ X Y r f, by { apply e.functor.map_injective, simp, }, } | instance | category_theory.equivalence.inverse_linear | category_theory.linear | src/category_theory/linear/linear_functor.lean | [
"category_theory.preadditive.additive_functor",
"category_theory.linear.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_yoneda : C ⥤ Cᵒᵖ ⥤ Module R | { obj := λ X,
{ obj := λ Y, Module.of R (unop Y ⟶ X),
map := λ Y Y' f, linear.left_comp R _ f.unop,
map_comp' := λ _ _ _ f g, linear_map.ext $ λ _, category.assoc _ _ _,
map_id' := λ Y, linear_map.ext $ λ _, category.id_comp _ },
map := λ X X' f,
{ app := λ Y, linear.right_comp R _ f,
naturality' ... | def | category_theory.linear_yoneda | category_theory.linear | src/category_theory/linear/yoneda.lean | [
"algebra.category.Module.basic",
"category_theory.linear.basic",
"category_theory.preadditive.yoneda.basic"
] | [
"Module",
"Module.coe_comp",
"Module.id_apply",
"Module.of",
"linear_map.ext"
] | The Yoneda embedding for `R`-linear categories `C`,
sending an object `X : C` to the `Module R`-valued presheaf on `C`,
with value on `Y : Cᵒᵖ` given by `Module.of R (unop Y ⟶ X)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_coyoneda : Cᵒᵖ ⥤ C ⥤ Module R | { obj := λ Y,
{ obj := λ X, Module.of R (unop Y ⟶ X),
map := λ Y Y', linear.right_comp _ _,
map_id' := λ Y, linear_map.ext $ λ _, category.comp_id _,
map_comp' := λ _ _ _ f g, linear_map.ext $ λ _, eq.symm (category.assoc _ _ _) },
map := λ Y Y' f,
{ app := λ X, linear.left_comp _ _ f.unop,
natura... | def | category_theory.linear_coyoneda | category_theory.linear | src/category_theory/linear/yoneda.lean | [
"algebra.category.Module.basic",
"category_theory.linear.basic",
"category_theory.preadditive.yoneda.basic"
] | [
"Module",
"Module.coe_comp",
"Module.id_apply",
"Module.of",
"linear_map.ext"
] | The Yoneda embedding for `R`-linear categories `C`,
sending an object `Y : Cᵒᵖ` to the `Module R`-valued copresheaf on `C`,
with value on `X : C` given by `Module.of R (unop Y ⟶ X)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_yoneda_obj_additive (X : C) : ((linear_yoneda R C).obj X).additive | {} | instance | category_theory.linear_yoneda_obj_additive | category_theory.linear | src/category_theory/linear/yoneda.lean | [
"algebra.category.Module.basic",
"category_theory.linear.basic",
"category_theory.preadditive.yoneda.basic"
] | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_coyoneda_obj_additive (Y : Cᵒᵖ) : ((linear_coyoneda R C).obj Y).additive | {} | instance | category_theory.linear_coyoneda_obj_additive | category_theory.linear | src/category_theory/linear/yoneda.lean | [
"algebra.category.Module.basic",
"category_theory.linear.basic",
"category_theory.preadditive.yoneda.basic"
] | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whiskering_linear_yoneda : linear_yoneda R C ⋙
(whiskering_right _ _ _).obj (forget (Module.{v} R)) = yoneda | rfl | lemma | category_theory.whiskering_linear_yoneda | category_theory.linear | src/category_theory/linear/yoneda.lean | [
"algebra.category.Module.basic",
"category_theory.linear.basic",
"category_theory.preadditive.yoneda.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whiskering_linear_yoneda₂ : linear_yoneda R C ⋙
(whiskering_right _ _ _).obj (forget₂ (Module.{v} R) AddCommGroup.{v}) = preadditive_yoneda | rfl | lemma | category_theory.whiskering_linear_yoneda₂ | category_theory.linear | src/category_theory/linear/yoneda.lean | [
"algebra.category.Module.basic",
"category_theory.linear.basic",
"category_theory.preadditive.yoneda.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whiskering_linear_coyoneda : linear_coyoneda R C ⋙
(whiskering_right _ _ _).obj (forget (Module.{v} R)) = coyoneda | rfl | lemma | category_theory.whiskering_linear_coyoneda | category_theory.linear | src/category_theory/linear/yoneda.lean | [
"algebra.category.Module.basic",
"category_theory.linear.basic",
"category_theory.preadditive.yoneda.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whiskering_linear_coyoneda₂ : linear_coyoneda R C ⋙
(whiskering_right _ _ _).obj (forget₂ (Module.{v} R) AddCommGroup.{v}) = preadditive_coyoneda | rfl | lemma | category_theory.whiskering_linear_coyoneda₂ | category_theory.linear | src/category_theory/linear/yoneda.lean | [
"algebra.category.Module.basic",
"category_theory.linear.basic",
"category_theory.preadditive.yoneda.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_yoneda_full : full (linear_yoneda R C) | let yoneda_full : full (linear_yoneda R C ⋙
(whiskering_right _ _ _).obj (forget (Module.{v} R))) := yoneda.yoneda_full in
by exactI full.of_comp_faithful (linear_yoneda R C)
(((whiskering_right _ _ _)).obj (forget (Module.{v} R))) | instance | category_theory.linear_yoneda_full | category_theory.linear | src/category_theory/linear/yoneda.lean | [
"algebra.category.Module.basic",
"category_theory.linear.basic",
"category_theory.preadditive.yoneda.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_coyoneda_full : full (linear_coyoneda R C) | let coyoneda_full : full (linear_coyoneda R C ⋙
(whiskering_right _ _ _).obj (forget (Module.{v} R))) := coyoneda.coyoneda_full in
by exactI full.of_comp_faithful (linear_coyoneda R C)
(((whiskering_right _ _ _)).obj (forget (Module.{v} R))) | instance | category_theory.linear_coyoneda_full | category_theory.linear | src/category_theory/linear/yoneda.lean | [
"algebra.category.Module.basic",
"category_theory.linear.basic",
"category_theory.preadditive.yoneda.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_yoneda_faithful : faithful (linear_yoneda R C) | faithful.of_comp_eq (whiskering_linear_yoneda R C) | instance | category_theory.linear_yoneda_faithful | category_theory.linear | src/category_theory/linear/yoneda.lean | [
"algebra.category.Module.basic",
"category_theory.linear.basic",
"category_theory.preadditive.yoneda.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_coyoneda_faithful : faithful (linear_coyoneda R C) | faithful.of_comp_eq (whiskering_linear_coyoneda R C) | instance | category_theory.linear_coyoneda_faithful | category_theory.linear | src/category_theory/linear/yoneda.lean | [
"algebra.category.Module.basic",
"category_theory.linear.basic",
"category_theory.preadditive.yoneda.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
loc_quiver (W : morphism_property C) | (obj : C) | structure | category_theory.localization.construction.loc_quiver | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [] | If `W : morphism_property C`, `loc_quiver W` is a quiver with the same objects
as `C`, and whose morphisms are those in `C` and placeholders for formal
inverses of the morphisms in `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_paths (X : C) : paths (loc_quiver W) | ⟨X⟩ | def | category_theory.localization.construction.ι_paths | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [] | The object in the path category of `loc_quiver W` attached to an object in
the category `C` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ψ₁ {X Y : C} (f : X ⟶ Y) : ι_paths W X ⟶ ι_paths W Y | paths.of.map (sum.inl f) | def | category_theory.localization.construction.ψ₁ | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [] | The morphism in the path category associated to a morphism in the original category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ψ₂ {X Y : C} (w : X ⟶ Y) (hw : W w) : ι_paths W Y ⟶ ι_paths W X | paths.of.map (sum.inr ⟨w, hw⟩) | def | category_theory.localization.construction.ψ₂ | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [] | The morphism in the path category corresponding to a formal inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
relations : hom_rel (paths (loc_quiver W))
| id (X : C) :
relations (ψ₁ W (𝟙 X)) (𝟙 _)
| comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) :
relations (ψ₁ W (f ≫ g)) (ψ₁ W f ≫ ψ₁ W g)
| Winv₁ {X Y : C} (w : X ⟶ Y) (hw : W w) :
relations (ψ₁ W w ≫ ψ₂ W w hw) (𝟙 _)
| Winv₂ {X Y : C} (w : X ⟶ Y) (hw : W w) :
relations (... | inductive | category_theory.localization.construction.relations | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [
"hom_rel"
] | The relations by which we take the quotient in order to get the localized category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization | category_theory.quotient (localization.construction.relations W) | def | category_theory.morphism_property.localization | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [
"category_theory.quotient",
"localization"
] | The localized category obtained by formally inverting the morphisms
in `W : morphism_property C` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Q : C ⥤ W.localization | { obj := λ X, (quotient.functor _).obj (paths.of.obj ⟨X⟩),
map := λ X Y f, (quotient.functor _).map (ψ₁ W f),
map_id' := λ X, quotient.sound _ (relations.id X),
map_comp' := λ X Z Y f g, quotient.sound _ (relations.comp f g), } | def | category_theory.morphism_property.Q | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [] | The obvious functor `C ⥤ W.localization` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Wiso {X Y : C} (w : X ⟶ Y) (hw : W w) : iso (W.Q.obj X) (W.Q.obj Y) | { hom := W.Q.map w,
inv := (quotient.functor _).map (paths.of.map (sum.inr ⟨w, hw⟩)),
hom_inv_id' := quotient.sound _ (relations.Winv₁ w hw),
inv_hom_id' := quotient.sound _ (relations.Winv₂ w hw), } | def | category_theory.localization.construction.Wiso | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [
"iso"
] | The isomorphism in `W.localization` associated to a morphism `w` in W | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Winv {X Y : C} (w : X ⟶ Y) (hw : W w) | (Wiso w hw).inv | abbreviation | category_theory.localization.construction.Winv | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [] | The formal inverse in `W.localization` of a morphism `w` in `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.category_theory.morphism_property.Q_inverts : W.is_inverted_by W.Q | λ X Y w hw, is_iso.of_iso (localization.construction.Wiso w hw) | lemma | category_theory.morphism_property.Q_inverts | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_to_path_category : paths (loc_quiver W) ⥤ D | Quiv.lift
{ obj := λ X, G.obj X.obj,
map := λ X Y, begin
rintro (f|⟨g, hg⟩),
{ exact G.map f, },
{ haveI := hG g hg,
exact inv (G.map g), },
end, } | def | category_theory.localization.construction.lift_to_path_category | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [] | The lifting of a functor to the path category of `loc_quiver W` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : W.localization ⥤ D | quotient.lift (relations W) (lift_to_path_category G hG)
begin
rintro ⟨X⟩ ⟨Y⟩ f₁ f₂ r,
rcases r,
tidy,
end | def | category_theory.localization.construction.lift | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [
"lift"
] | The lifting of a functor `C ⥤ D` inverting `W` as a functor `W.localization ⥤ D` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fac : W.Q ⋙ lift G hG = G | functor.ext (λ X, rfl)
begin
intros X Y f,
simp only [functor.comp_map, eq_to_hom_refl, comp_id, id_comp],
dsimp [lift, lift_to_path_category, morphism_property.Q],
rw compose_path_to_path,
end | lemma | category_theory.localization.construction.fac | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [
"functor.ext",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniq (G₁ G₂ : W.localization ⥤ D) (h : W.Q ⋙ G₁ = W.Q ⋙ G₂) :
G₁ = G₂ | begin
suffices h' : quotient.functor _ ⋙ G₁ = quotient.functor _ ⋙ G₂,
{ refine functor.ext _ _,
{ rintro ⟨⟨X⟩⟩,
apply functor.congr_obj h, },
{ rintros ⟨⟨X⟩⟩ ⟨⟨Y⟩⟩ ⟨f⟩,
apply functor.congr_hom h', }, },
{ refine paths.ext_functor _ _,
{ ext X,
cases X,
apply functor.congr_obj ... | lemma | category_theory.localization.construction.uniq | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [
"functor.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj_equiv : C ≃ W.localization | { to_fun := W.Q.obj,
inv_fun := λ X, X.as.obj,
left_inv := λ X, rfl,
right_inv := by { rintro ⟨⟨X⟩⟩, refl, }, } | def | category_theory.localization.construction.obj_equiv | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [
"inv_fun"
] | The canonical bijection between objects in a category and its
localization with respect to a morphism_property `W` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
morphism_property_is_top
(P : morphism_property W.localization)
(hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f))
(hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (Winv w hw))
(hP₃ : P.stable_under_composition) : P = ⊤ | begin
ext X Y f,
split,
{ intro hf,
simp only [pi.top_apply], },
{ intro hf, clear hf,
let G : _ ⥤ W.localization := quotient.functor _,
suffices : ∀ (X₁ X₂ : C) (p : localization.construction.ι_paths W X₁ ⟶
localization.construction.ι_paths W X₂), P (G.map p),
{ rcases X with ⟨⟨X⟩⟩,
... | lemma | category_theory.localization.construction.morphism_property_is_top | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [
"functor.map_id",
"pi.top_apply",
"quiver.hom.to_path"
] | A `morphism_property` in `W.localization` is satisfied by all
morphisms in the localized category if it contains the image of the
morphisms in the original category, the inverses of the morphisms
in `W` and if it is stable under composition | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
morphism_property_is_top'
(P : morphism_property W.localization)
(hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f))
(hP₂ : ∀ ⦃X Y : W.localization⦄ (e : X ≅ Y) (he : P e.hom), P e.inv)
(hP₃ : P.stable_under_composition) : P = ⊤ | morphism_property_is_top P hP₁ (λ X Y w hw, hP₂ _ (by exact hP₁ w)) hP₃ | lemma | category_theory.localization.construction.morphism_property_is_top' | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [] | A `morphism_property` in `W.localization` is satisfied by all
morphisms in the localized category if it contains the image of the
morphisms in the original category, if is stable under composition
and if the property is stable by passing to inverses. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
app (X : W.localization) : F₁.obj X ⟶ F₂.obj X | eq_to_hom (congr_arg F₁.obj ((obj_equiv W).right_inv X).symm) ≫
τ.app ((obj_equiv W).inv_fun X) ≫ eq_to_hom (congr_arg F₂.obj ((obj_equiv W).right_inv X)) | def | category_theory.localization.construction.nat_trans_extension.app | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [
"inv_fun"
] | If `F₁` and `F₂` are functors `W.localization ⥤ D` and if we have
`τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂`, we shall define a natural transformation `F₁ ⟶ F₂`.
This is the `app` field of this natural transformation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
app_eq (X : C) : (app τ) (W.Q.obj X) = τ.app X | by simpa only [app, eq_to_hom_refl, comp_id, id_comp] | lemma | category_theory.localization.construction.nat_trans_extension.app_eq | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans_extension {F₁ F₂ : W.localization ⥤ D} (τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂) :
F₁ ⟶ F₂ | { app := nat_trans_extension.app τ,
naturality' := λ X Y f, begin
have h := morphism_property_is_top'
(morphism_property.naturality_property (nat_trans_extension.app τ)) _
(morphism_property.naturality_property.is_stable_under_inverse _)
(morphism_property.naturality_property.is_stable_under_com... | def | category_theory.localization.construction.nat_trans_extension | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [
"pi.top_apply"
] | If `F₁` and `F₂` are functors `W.localization ⥤ D`, a natural transformation `F₁ ⟶ F₂`
can be obtained from a natural transformation `W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans_extension_hcomp {F G : W.localization ⥤ D} (τ : W.Q ⋙ F ⟶ W.Q ⋙ G) :
(𝟙 W.Q) ◫ nat_trans_extension τ = τ | begin
ext X,
simp only [nat_trans.hcomp_app, nat_trans.id_app, G.map_id, comp_id,
nat_trans_extension_app, nat_trans_extension.app_eq],
end | lemma | category_theory.localization.construction.nat_trans_extension_hcomp | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_trans_hcomp_injective {F G : W.localization ⥤ D} {τ₁ τ₂ : F ⟶ G}
(h : 𝟙 W.Q ◫ τ₁ = 𝟙 W.Q ◫ τ₂) : τ₁ = τ₂ | begin
ext X,
have eq := (obj_equiv W).right_inv X,
simp only [obj_equiv] at eq,
rw [← eq, ← nat_trans.id_hcomp_app, ← nat_trans.id_hcomp_app, h],
end | lemma | category_theory.localization.construction.nat_trans_hcomp_injective | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor : (W.localization ⥤ D) ⥤ (W.functors_inverting D) | full_subcategory.lift _ ((whiskering_left _ _ D).obj W.Q)
(λ F, morphism_property.is_inverted_by.of_comp W W.Q W.Q_inverts _) | def | category_theory.localization.construction.whiskering_left_equivalence.functor | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [] | The functor `(W.localization ⥤ D) ⥤ (W.functors_inverting D)` induced by the
composition with `W.Q : C ⥤ W.localization`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse : (W.functors_inverting D) ⥤ (W.localization ⥤ D) | { obj := λ G, lift G.obj G.property,
map := λ G₁ G₂ τ, nat_trans_extension (eq_to_hom (by rw fac) ≫ τ ≫ eq_to_hom (by rw fac)),
map_id' := λ G, nat_trans_hcomp_injective begin
rw nat_trans_extension_hcomp,
ext X,
simpa only [nat_trans.comp_app, eq_to_hom_app, eq_to_hom_refl, comp_id, id_comp,
nat_... | def | category_theory.localization.construction.whiskering_left_equivalence.inverse | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [
"functor.map_id",
"lift"
] | The function `(W.functors_inverting D) ⥤ (W.localization ⥤ D)` induced by
`construction.lift`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_iso : 𝟭 (W.localization ⥤ D) ≅ functor W D ⋙ inverse W D | eq_to_iso
begin
refine functor.ext (λ G, _) (λ G₁ G₂ τ, _),
{ apply uniq,
dsimp [functor],
rw fac, },
{ apply nat_trans_hcomp_injective,
ext X,
simp only [functor.id_map, nat_trans.hcomp_app, comp_id, functor.comp_map,
inverse_map, nat_trans.comp_app, eq_to_hom_app, eq_to_hom_refl, nat_trans... | def | category_theory.localization.construction.whiskering_left_equivalence.unit_iso | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [
"functor.ext"
] | The unit isomorphism of the equivalence of categories `whiskering_left_equivalence W D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
counit_iso : inverse W D ⋙ functor W D ≅ 𝟭 (W.functors_inverting D) | eq_to_iso
begin
refine functor.ext _ _,
{ rintro ⟨G, hG⟩,
ext1,
apply fac, },
{ rintros ⟨G₁, hG₁⟩ ⟨G₂, hG₂⟩ f,
ext X,
apply nat_trans_extension.app_eq, },
end | def | category_theory.localization.construction.whiskering_left_equivalence.counit_iso | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [
"functor.ext"
] | The counit isomorphism of the equivalence of categories `whiskering_left_equivalence W D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
whiskering_left_equivalence : (W.localization ⥤ D) ≌ W.functors_inverting D | { functor := whiskering_left_equivalence.functor W D,
inverse := whiskering_left_equivalence.inverse W D,
unit_iso := whiskering_left_equivalence.unit_iso W D,
counit_iso := whiskering_left_equivalence.counit_iso W D,
functor_unit_iso_comp' := λ F, begin
ext X,
simpa only [eq_to_hom_app, whiskering_left... | def | category_theory.localization.construction.whiskering_left_equivalence | category_theory.localization | src/category_theory/localization/construction.lean | [
"category_theory.morphism_property",
"category_theory.category.Quiv"
] | [] | The equivalence of categories `(W.localization ⥤ D) ≌ (W.functors_inverting D)`
induced by the composition with `W.Q : C ⥤ W.localization`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_universal_property_fixed_target.op {E : Type*} [category E]
(h : strict_universal_property_fixed_target L W Eᵒᵖ):
strict_universal_property_fixed_target L.op W.op E | { inverts := h.inverts.op,
lift := λ F hF, (h.lift F.right_op hF.right_op).left_op,
fac := λ F hF, begin
convert congr_arg functor.left_op (h.fac F.right_op hF.right_op),
exact F.right_op_left_op_eq.symm,
end,
uniq := λ F₁ F₂ eq, begin
suffices : F₁.right_op = F₂.right_op,
{ rw [← F₁.right_op_le... | def | category_theory.localization.strict_universal_property_fixed_target.op | category_theory.localization | src/category_theory/localization/opposite.lean | [
"category_theory.localization.predicate"
] | [
"lift"
] | If `L : C ⥤ D` satisfies the universal property of the localisation
for `W : morphism_property C`, then `L.op` also does. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization_op : W.Q.op.is_localization W.op | functor.is_localization.mk' W.Q.op W.op
(strict_universal_property_fixed_target_Q W _).op
(strict_universal_property_fixed_target_Q W _).op | instance | category_theory.localization.is_localization_op | category_theory.localization | src/category_theory/localization/opposite.lean | [
"category_theory.localization.predicate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization.op [h : L.is_localization W] : L.op.is_localization W.op | is_localization.of_equivalence_target W.Q.op W.op L.op
(localization.equivalence_from_model L W).op
(nat_iso.op (localization.Q_comp_equivalence_from_model_functor_iso L W).symm) | instance | category_theory.functor.is_localization.op | category_theory.localization | src/category_theory/localization/opposite.lean | [
"category_theory.localization.predicate"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization : Prop | (inverts : W.is_inverted_by L)
(nonempty_is_equivalence : nonempty (is_equivalence (localization.construction.lift L inverts))) | class | category_theory.functor.is_localization | category_theory.localization | src/category_theory/localization/predicate.lean | [
"category_theory.localization.construction"
] | [
"is_localization"
] | The predicate expressing that, up to equivalence, a functor `L : C ⥤ D`
identifies the category `D` with the localized category of `C` with respect
to `W : morphism_property C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Q_is_localization : W.Q.is_localization W | { inverts := W.Q_inverts,
nonempty_is_equivalence := begin
suffices : localization.construction.lift W.Q W.Q_inverts = 𝟭 _,
{ apply nonempty.intro, rw this, apply_instance, },
apply localization.construction.uniq,
simpa only [localization.construction.fac],
end, } | instance | category_theory.functor.Q_is_localization | category_theory.localization | src/category_theory/localization/predicate.lean | [
"category_theory.localization.construction"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_universal_property_fixed_target | (inverts : W.is_inverted_by L)
(lift : Π (F : C ⥤ E) (hF : W.is_inverted_by F), D ⥤ E)
(fac : Π (F : C ⥤ E) (hF : W.is_inverted_by F), L ⋙ lift F hF = F)
(uniq : Π (F₁ F₂ : D ⥤ E) (h : L ⋙ F₁ = L ⋙ F₂), F₁ = F₂) | structure | category_theory.localization.strict_universal_property_fixed_target | category_theory.localization | src/category_theory/localization/predicate.lean | [
"category_theory.localization.construction"
] | [
"lift"
] | This universal property states that a functor `L : C ⥤ D` inverts morphisms
in `W` and the all functors `D ⥤ E` (for a fixed category `E`) uniquely factors
through `L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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