statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
vertex_group_isom_of_map {c d : C} (f : c ⟶ d) : (c ⟶ c) ≃* (d ⟶ d) | { to_fun := λ γ, inv f ≫ γ ≫ f,
inv_fun := λ δ, f ≫ δ ≫ inv f,
left_inv := λ γ, by simp_rw [category.assoc, comp_inv, category.comp_id,
←category.assoc, comp_inv, category.id_comp],
right_inv := λ δ, by simp_rw [category.assoc, inv_comp, ←category.assoc,
... | def | category_theory.groupoid.vertex_group_isom_of_map | category_theory.groupoid | src/category_theory/groupoid/vertex_group.lean | [
"category_theory.groupoid",
"category_theory.path_category",
"algebra.group.defs",
"algebra.hom.group",
"algebra.hom.equiv.basic",
"combinatorics.quiver.path"
] | [
"inv_fun"
] | An arrow in the groupoid defines, by conjugation, an isomorphism of groups between
its endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
vertex_group_isom_of_path {c d : C} (p : quiver.path c d) : (c ⟶ c) ≃* (d ⟶ d) | vertex_group_isom_of_map (compose_path p) | def | category_theory.groupoid.vertex_group_isom_of_path | category_theory.groupoid | src/category_theory/groupoid/vertex_group.lean | [
"category_theory.groupoid",
"category_theory.path_category",
"algebra.group.defs",
"algebra.hom.group",
"algebra.hom.equiv.basic",
"combinatorics.quiver.path"
] | [
"quiver.path"
] | A path in the groupoid defines an isomorphism between its endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.category_theory.functor.map_vertex_group {D : Type v} [groupoid D]
(φ : C ⥤ D) (c : C) : (c ⟶ c) →* (φ.obj c ⟶ φ.obj c) | { to_fun := φ.map,
map_one' := φ.map_id c,
map_mul' := φ.map_comp } | def | category_theory.functor.map_vertex_group | category_theory.groupoid | src/category_theory/groupoid/vertex_group.lean | [
"category_theory.groupoid",
"category_theory.path_category",
"algebra.group.defs",
"algebra.hom.group",
"algebra.hom.equiv.basic",
"combinatorics.quiver.path"
] | [] | A functor defines a morphism of vertex group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_idempotent_complete : Prop | (idempotents_split : ∀ (X : C) (p : X ⟶ X), p ≫ p = p →
∃ (Y : C) (i : Y ⟶ X) (e : X ⟶ Y), i ≫ e = 𝟙 Y ∧ e ≫ i = p) | class | category_theory.is_idempotent_complete | category_theory.idempotents | src/category_theory/idempotents/basic.lean | [
"category_theory.abelian.basic"
] | [] | A category is idempotent complete iff all idempotent endomorphisms `p`
split as a composition `p = e ≫ i` with `i ≫ e = 𝟙 _` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_idempotent_complete_iff_has_equalizer_of_id_and_idempotent :
is_idempotent_complete C ↔ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → has_equalizer (𝟙 X) p | begin
split,
{ introI,
intros X p hp,
rcases is_idempotent_complete.idempotents_split X p hp with ⟨Y, i, e, ⟨h₁, h₂⟩⟩,
exact ⟨nonempty.intro
{ cone := fork.of_ι i
(show i ≫ 𝟙 X = i ≫ p, by rw [comp_id, ← h₂, ← assoc, h₁, id_comp]),
is_limit := begin
apply fork.is_limit... | lemma | category_theory.idempotents.is_idempotent_complete_iff_has_equalizer_of_id_and_idempotent | category_theory.idempotents | src/category_theory/idempotents/basic.lean | [
"category_theory.abelian.basic"
] | [] | A category is idempotent complete iff for all idempotent endomorphisms,
the equalizer of the identity and this idempotent exists. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
idem_of_id_sub_idem [preadditive C]
{X : C} (p : X ⟶ X) (hp : p ≫ p = p) :
(𝟙 _ - p) ≫ (𝟙 _ - p) = (𝟙 _ - p) | by simp only [comp_sub, sub_comp, id_comp, comp_id, hp, sub_self, sub_zero] | lemma | category_theory.idempotents.idem_of_id_sub_idem | category_theory.idempotents | src/category_theory/idempotents/basic.lean | [
"category_theory.abelian.basic"
] | [] | In a preadditive category, when `p : X ⟶ X` is idempotent,
then `𝟙 X - p` is also idempotent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_idempotent_complete_iff_idempotents_have_kernels [preadditive C] :
is_idempotent_complete C ↔ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → has_kernel p | begin
rw is_idempotent_complete_iff_has_equalizer_of_id_and_idempotent,
split,
{ intros h X p hp,
haveI := h X (𝟙 _ - p) (idem_of_id_sub_idem p hp),
convert has_kernel_of_has_equalizer (𝟙 X) (𝟙 X - p),
rw [sub_sub_cancel], },
{ intros h X p hp,
haveI : has_kernel (𝟙 _ - p) := h X (𝟙 _ - p) ... | lemma | category_theory.idempotents.is_idempotent_complete_iff_idempotents_have_kernels | category_theory.idempotents | src/category_theory/idempotents/basic.lean | [
"category_theory.abelian.basic"
] | [] | A preadditive category is pseudoabelian iff all idempotent endomorphisms have a kernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_idempotent_complete_of_abelian (D : Type*) [category D] [abelian D] :
is_idempotent_complete D | by { rw is_idempotent_complete_iff_idempotents_have_kernels, intros, apply_instance, } | instance | category_theory.idempotents.is_idempotent_complete_of_abelian | category_theory.idempotents | src/category_theory/idempotents/basic.lean | [
"category_theory.abelian.basic"
] | [] | An abelian category is idempotent complete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
split_imp_of_iso {X X' : C} (φ : X ≅ X') (p : X ⟶ X) (p' : X' ⟶ X')
(hpp' : p ≫ φ.hom = φ.hom ≫ p')
(h : ∃ (Y : C) (i : Y ⟶ X) (e : X ⟶ Y), i ≫ e = 𝟙 Y ∧ e ≫ i = p) :
(∃ (Y' : C) (i' : Y' ⟶ X') (e' : X' ⟶ Y'), i' ≫ e' = 𝟙 Y' ∧ e' ≫ i' = p') | begin
rcases h with ⟨Y, i, e, ⟨h₁, h₂⟩⟩,
use [Y, i ≫ φ.hom, φ.inv ≫ e],
split,
{ slice_lhs 2 3 { rw φ.hom_inv_id, },
rw [id_comp, h₁], },
{ slice_lhs 2 3 { rw h₂, },
rw [hpp', ← assoc, φ.inv_hom_id, id_comp], }
end | lemma | category_theory.idempotents.split_imp_of_iso | category_theory.idempotents | src/category_theory/idempotents/basic.lean | [
"category_theory.abelian.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
split_iff_of_iso {X X' : C} (φ : X ≅ X') (p : X ⟶ X) (p' : X' ⟶ X')
(hpp' : p ≫ φ.hom = φ.hom ≫ p') :
(∃ (Y : C) (i : Y ⟶ X) (e : X ⟶ Y), i ≫ e = 𝟙 Y ∧ e ≫ i = p) ↔
(∃ (Y' : C) (i' : Y' ⟶ X') (e' : X' ⟶ Y'), i' ≫ e' = 𝟙 Y' ∧ e' ≫ i' = p') | begin
split,
{ exact split_imp_of_iso φ p p' hpp', },
{ apply split_imp_of_iso φ.symm p' p,
rw [← comp_id p, ← φ.hom_inv_id],
slice_rhs 2 3 { rw hpp', },
slice_rhs 1 2 { erw φ.inv_hom_id, },
simpa only [id_comp], },
end | lemma | category_theory.idempotents.split_iff_of_iso | category_theory.idempotents | src/category_theory/idempotents/basic.lean | [
"category_theory.abelian.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence.is_idempotent_complete {D : Type*} [category D] (ε : C ≌ D)
(h : is_idempotent_complete C) : is_idempotent_complete D | begin
refine ⟨_⟩,
intros X' p hp,
let φ := ε.counit_iso.symm.app X',
erw split_iff_of_iso φ p (φ.inv ≫ p ≫ φ.hom)
(by { slice_rhs 1 2 { rw φ.hom_inv_id, }, rw id_comp,}),
rcases is_idempotent_complete.idempotents_split (ε.inverse.obj X') (ε.inverse.map p)
(by rw [← ε.inverse.map_comp, hp]) with ⟨Y, i,... | lemma | category_theory.idempotents.equivalence.is_idempotent_complete | category_theory.idempotents | src/category_theory/idempotents/basic.lean | [
"category_theory.abelian.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_idempotent_complete_iff_of_equivalence {D : Type*} [category D] (ε : C ≌ D) :
is_idempotent_complete C ↔ is_idempotent_complete D | begin
split,
{ exact equivalence.is_idempotent_complete ε, },
{ exact equivalence.is_idempotent_complete ε.symm, },
end | lemma | category_theory.idempotents.is_idempotent_complete_iff_of_equivalence | category_theory.idempotents | src/category_theory/idempotents/basic.lean | [
"category_theory.abelian.basic"
] | [] | If `C` and `D` are equivalent categories, that `C` is idempotent complete iff `D` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_idempotent_complete_of_is_idempotent_complete_opposite
(h : is_idempotent_complete Cᵒᵖ) : is_idempotent_complete C | begin
refine ⟨_⟩,
intros X p hp,
rcases is_idempotent_complete.idempotents_split (op X) p.op
(by rw [← op_comp, hp]) with ⟨Y, i, e, ⟨h₁, h₂⟩⟩,
use [Y.unop, e.unop, i.unop],
split,
{ simpa only [← unop_comp, h₁], },
{ simpa only [← unop_comp, h₂], },
end | lemma | category_theory.idempotents.is_idempotent_complete_of_is_idempotent_complete_opposite | category_theory.idempotents | src/category_theory/idempotents/basic.lean | [
"category_theory.abelian.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_idempotent_complete_iff_opposite :
is_idempotent_complete Cᵒᵖ ↔ is_idempotent_complete C | begin
split,
{ exact is_idempotent_complete_of_is_idempotent_complete_opposite, },
{ intro h,
apply is_idempotent_complete_of_is_idempotent_complete_opposite,
rw is_idempotent_complete_iff_of_equivalence (op_op_equivalence C),
exact h, },
end | lemma | category_theory.idempotents.is_idempotent_complete_iff_opposite | category_theory.idempotents | src/category_theory/idempotents/basic.lean | [
"category_theory.abelian.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bicone [has_finite_biproducts C] {J : Type} [fintype J]
(F : J → karoubi C) : bicone F | { X :=
{ X := biproduct (λ j, (F j).X),
p := biproduct.map (λ j, (F j).p),
idem := begin
ext j,
simp only [biproduct.ι_map_assoc, biproduct.ι_map],
slice_lhs 1 2 { rw (F j).idem, },
end, },
π := λ j,
{ f := biproduct.map (λ j, (F j).p) ≫ bicone.π _ j,
comm := by simp only [as... | def | category_theory.idempotents.karoubi.biproducts.bicone | category_theory.idempotents | src/category_theory/idempotents/biproducts.lean | [
"category_theory.idempotents.karoubi"
] | [
"comm",
"fintype",
"hom_ext"
] | The `bicone` used in order to obtain the existence of
the biproduct of a functor `J ⥤ karoubi C` when the category `C` is additive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
karoubi_has_finite_biproducts [has_finite_biproducts C] :
has_finite_biproducts (karoubi C) | { out := λ n,
{ has_biproduct := λ F, begin
classical,
apply has_biproduct_of_total (biproducts.bicone F),
ext1, ext1,
simp only [id_eq, comp_id, biproducts.bicone_X_p, biproduct.ι_map],
rw [sum_hom, comp_sum, finset.sum_eq_single j], rotate,
{ intros j' h1 h2,
simp only [b... | lemma | category_theory.idempotents.karoubi.karoubi_has_finite_biproducts | category_theory.idempotents | src/category_theory/idempotents/biproducts.lean | [
"category_theory.idempotents.karoubi"
] | [
"finset.mem_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complement (P : karoubi C) : karoubi C | { X := P.X,
p := 𝟙 _ - P.p,
idem := idem_of_id_sub_idem P.p P.idem, } | def | category_theory.idempotents.karoubi.complement | category_theory.idempotents | src/category_theory/idempotents/biproducts.lean | [
"category_theory.idempotents.karoubi"
] | [] | `P.complement` is the formal direct factor of `P.X` given by the idempotent
endomorphism `𝟙 P.X - P.p` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
decomposition (P : karoubi C) : P ⊞ P.complement ≅ (to_karoubi _).obj P.X | { hom := biprod.desc P.decomp_id_i P.complement.decomp_id_i,
inv := biprod.lift P.decomp_id_p P.complement.decomp_id_p,
hom_inv_id' := begin
ext1,
{ simp only [← assoc, biprod.inl_desc, comp_id, biprod.lift_eq, comp_add,
← decomp_id, id_comp, add_right_eq_self],
convert zero_comp,
ext,
... | def | category_theory.idempotents.karoubi.decomposition | category_theory.idempotents | src/category_theory/idempotents/biproducts.lean | [
"category_theory.idempotents.karoubi"
] | [] | A formal direct factor `P : karoubi C` of an object `P.X : C` in a
preadditive category is actually a direct factor of the image `(to_karoubi C).obj P.X`
of `P.X` in the category `karoubi C` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
app_idem :
P.p.app X ≫ P.p.app X = P.p.app X | congr_app P.idem X | lemma | category_theory.idempotents.app_idem | category_theory.idempotents | src/category_theory/idempotents/functor_categories.lean | [
"category_theory.idempotents.karoubi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_p_comp : P.p.app X ≫ f.f.app X = f.f.app X | congr_app (p_comp f) X | lemma | category_theory.idempotents.app_p_comp | category_theory.idempotents | src/category_theory/idempotents/functor_categories.lean | [
"category_theory.idempotents.karoubi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_comp_p : f.f.app X ≫ Q.p.app X = f.f.app X | congr_app (comp_p f) X | lemma | category_theory.idempotents.app_comp_p | category_theory.idempotents | src/category_theory/idempotents/functor_categories.lean | [
"category_theory.idempotents.karoubi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_p_comm : P.p.app X ≫ f.f.app X = f.f.app X ≫ Q.p.app X | congr_app (p_comm f) X | lemma | category_theory.idempotents.app_p_comm | category_theory.idempotents | src/category_theory/idempotents/functor_categories.lean | [
"category_theory.idempotents.karoubi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_is_idempotent_complete [is_idempotent_complete C] :
is_idempotent_complete (J ⥤ C) | begin
refine ⟨_⟩,
intros F p hp,
have hC := (is_idempotent_complete_iff_has_equalizer_of_id_and_idempotent C).mp infer_instance,
haveI : ∀ (j : J), has_equalizer (𝟙 _) (p.app j) := λ j, hC _ _ (congr_app hp j),
/- We construct the direct factor `Y` associated to `p : F ⟶ F` by computing
the equalizer of ... | instance | category_theory.idempotents.functor_category_is_idempotent_complete | category_theory.idempotents | src/category_theory/idempotents/functor_categories.lean | [
"category_theory.idempotents.karoubi"
] | [
"functor.map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj (P : karoubi (J ⥤ C)) : J ⥤ karoubi C | { obj := λ j, ⟨P.X.obj j, P.p.app j, congr_app P.idem j⟩,
map := λ j j' φ,
{ f := P.p.app j ≫ P.X.map φ,
comm := begin
simp only [nat_trans.naturality, assoc],
have h := congr_app P.idem j,
rw [nat_trans.comp_app] at h,
slice_rhs 1 3 { erw [h, h], },
end }, } | def | category_theory.idempotents.karoubi_functor_category_embedding.obj | category_theory.idempotents | src/category_theory/idempotents/functor_categories.lean | [
"category_theory.idempotents.karoubi"
] | [
"comm"
] | On objects, the functor which sends a formal direct factor `P` of a
functor `F : J ⥤ C` to the functor `J ⥤ karoubi C` which sends `(j : J)` to
the corresponding direct factor of `F.obj j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map {P Q : karoubi (J ⥤ C)} (f : P ⟶ Q) : obj P ⟶ obj Q | { app := λ j, ⟨f.f.app j, congr_app f.comm j⟩, } | def | category_theory.idempotents.karoubi_functor_category_embedding.map | category_theory.idempotents | src/category_theory/idempotents/functor_categories.lean | [
"category_theory.idempotents.karoubi"
] | [] | Tautological action on maps of the functor `karoubi (J ⥤ C) ⥤ (J ⥤ karoubi C)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
karoubi_functor_category_embedding :
karoubi (J ⥤ C) ⥤ (J ⥤ karoubi C) | { obj := karoubi_functor_category_embedding.obj,
map := λ P Q, karoubi_functor_category_embedding.map, } | def | category_theory.idempotents.karoubi_functor_category_embedding | category_theory.idempotents | src/category_theory/idempotents/functor_categories.lean | [
"category_theory.idempotents.karoubi"
] | [] | The tautological fully faithful functor `karoubi (J ⥤ C) ⥤ (J ⥤ karoubi C)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_karoubi_comp_karoubi_functor_category_embedding :
(to_karoubi _) ⋙ karoubi_functor_category_embedding J C =
(whiskering_right J _ _).obj (to_karoubi C) | begin
apply functor.ext,
{ intros X Y f,
ext j,
dsimp [to_karoubi],
simp only [eq_to_hom_app, eq_to_hom_refl, id_comp],
erw [comp_id], },
{ intro X,
apply functor.ext,
{ intros j j' φ,
ext,
dsimp,
simpa only [comp_id, id_comp], },
{ intro j,
refl, }, }
end | lemma | category_theory.idempotents.to_karoubi_comp_karoubi_functor_category_embedding | category_theory.idempotents | src/category_theory/idempotents/functor_categories.lean | [
"category_theory.idempotents.karoubi"
] | [
"functor.ext"
] | The composition of `(J ⥤ C) ⥤ karoubi (J ⥤ C)` and `karoubi (J ⥤ C) ⥤ (J ⥤ karoubi C)`
equals the functor `(J ⥤ C) ⥤ (J ⥤ karoubi C)` given by the composition with
`to_karoubi C : C ⥤ karoubi C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_trans_eq {F G : karoubi C ⥤ D} (φ : F ⟶ G) (P : karoubi C) :
φ.app P = F.map (decomp_id_i P) ≫ φ.app P.X ≫ G.map (decomp_id_p P) | begin
rw [← φ.naturality, ← assoc, ← F.map_comp],
conv { to_lhs, rw [← id_comp (φ.app P), ← F.map_id], },
congr,
apply decomp_id,
end | lemma | category_theory.idempotents.nat_trans_eq | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [] | A natural transformation between functors `karoubi C ⥤ D` is determined
by its value on objects coming from `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj (F : C ⥤ karoubi D) : karoubi C ⥤ karoubi D | { obj := λ P, ⟨(F.obj P.X).X, (F.map P.p).f,
by simpa only [F.map_comp, hom_ext] using F.congr_map P.idem⟩,
map := λ P Q f, ⟨(F.map f.f).f,
by simpa only [F.map_comp, hom_ext] using F.congr_map f.comm⟩, } | def | category_theory.idempotents.functor_extension₁.obj | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [
"hom_ext"
] | The canonical extension of a functor `C ⥤ karoubi D` to a functor
`karoubi C ⥤ karoubi D` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map {F G : C ⥤ karoubi D} (φ : F ⟶ G) : obj F ⟶ obj G | { app := λ P,
{ f := (F.map P.p).f ≫ (φ.app P.X).f,
comm := begin
have h := φ.naturality P.p,
have h' := F.congr_map P.idem,
simp only [hom_ext, karoubi.comp_f, F.map_comp] at h h',
simp only [obj_obj_p, assoc, ← h],
slice_rhs 1 3 { rw [h', h'], },
end, },
naturality' := λ P Q ... | def | category_theory.idempotents.functor_extension₁.map | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [
"comm",
"hom_ext"
] | Extension of a natural transformation `φ` between functors
`C ⥤ karoubi D` to a natural transformation between the
extension of these functors to `karoubi C ⥤ karoubi D` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_extension₁ : (C ⥤ karoubi D) ⥤ (karoubi C ⥤ karoubi D) | { obj := functor_extension₁.obj,
map := λ F G, functor_extension₁.map,
map_id' := λ F, by { ext P, exact comp_p (F.map P.p), },
map_comp' := λ F G H φ φ', begin
ext P,
simp only [comp_f, functor_extension₁.map_app_f, nat_trans.comp_app, assoc],
have h := φ.naturality P.p,
have h' := F.congr_map P.... | def | category_theory.idempotents.functor_extension₁ | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [
"hom_ext"
] | The canonical functor `(C ⥤ karoubi D) ⥤ (karoubi C ⥤ karoubi D)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_extension₁_comp_whiskering_left_to_karoubi :
functor_extension₁ C D ⋙
(whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C) = 𝟭 _ | begin
refine functor.ext _ _,
{ intro F,
refine functor.ext _ _,
{ intro X,
ext,
{ dsimp,
rw [id_comp, comp_id, F.map_id, id_eq], },
{ refl, }, },
{ intros X Y f,
ext,
dsimp,
simp only [comp_id, eq_to_hom_f, eq_to_hom_refl, comp_p, functor_extension₁.obj_obj_p... | lemma | category_theory.idempotents.functor_extension₁_comp_whiskering_left_to_karoubi | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [
"functor.ext",
"functor.map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_extension₁_comp_whiskering_left_to_karoubi_iso :
functor_extension₁ C D ⋙
(whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C) ≅ 𝟭 _ | eq_to_iso (functor_extension₁_comp_whiskering_left_to_karoubi C D) | def | category_theory.idempotents.functor_extension₁_comp_whiskering_left_to_karoubi_iso | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [] | The natural isomorphism expressing that functors `karoubi C ⥤ karoubi D` obtained
using `functor_extension₁` actually extends the original functors `C ⥤ karoubi D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
karoubi_universal₁.counit_iso :
(whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C) ⋙
functor_extension₁ C D ≅ 𝟭 _ | nat_iso.of_components (λ G,
{ hom :=
{ app := λ P,
{ f := (G.map (decomp_id_p P)).f,
comm := by simpa only [hom_ext, G.map_comp, G.map_id] using G.congr_map
(show P.decomp_id_p = (to_karoubi C).map P.p ≫ P.decomp_id_p ≫ 𝟙 _, by simp), },
naturality' := λ P Q f,
by simpa only... | def | category_theory.idempotents.karoubi_universal₁.counit_iso | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [
"comm",
"hom_ext"
] | The counit isomorphism of the equivalence `(C ⥤ karoubi D) ≌ (karoubi C ⥤ karoubi D)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
karoubi_universal₁ : (C ⥤ karoubi D) ≌ (karoubi C ⥤ karoubi D) | { functor := functor_extension₁ C D,
inverse := (whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C),
unit_iso := (functor_extension₁_comp_whiskering_left_to_karoubi_iso C D).symm,
counit_iso := karoubi_universal₁.counit_iso C D,
functor_unit_iso_comp' := λ F, begin
ext P,
dsimp [functor_exten... | def | category_theory.idempotents.karoubi_universal₁ | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [
"hom_ext"
] | The equivalence of categories `(C ⥤ karoubi D) ≌ (karoubi C ⥤ karoubi D)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_extension₁_comp (F : C ⥤ karoubi D) (G : D ⥤ karoubi E) :
(functor_extension₁ C E).obj (F ⋙ (functor_extension₁ D E).obj G) =
(functor_extension₁ C D).obj F ⋙ (functor_extension₁ D E).obj G | functor.ext (by tidy) (λ X Y f, by { dsimp, simpa only [id_comp, comp_id], }) | lemma | category_theory.idempotents.functor_extension₁_comp | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [
"functor.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_extension₂ : (C ⥤ D) ⥤ (karoubi C ⥤ karoubi D) | (whiskering_right C D (karoubi D)).obj (to_karoubi D) ⋙ functor_extension₁ C D | def | category_theory.idempotents.functor_extension₂ | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [] | The canonical functor `(C ⥤ D) ⥤ (karoubi C ⥤ karoubi D)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_extension₂_comp_whiskering_left_to_karoubi :
functor_extension₂ C D ⋙ (whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C) =
(whiskering_right C D (karoubi D)).obj (to_karoubi D) | by simp only [functor_extension₂, functor.assoc,
functor_extension₁_comp_whiskering_left_to_karoubi, functor.comp_id] | lemma | category_theory.idempotents.functor_extension₂_comp_whiskering_left_to_karoubi | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_extension₂_comp_whiskering_left_to_karoubi_iso :
functor_extension₂ C D ⋙ (whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C) ≅
(whiskering_right C D (karoubi D)).obj (to_karoubi D) | eq_to_iso (functor_extension₂_comp_whiskering_left_to_karoubi C D) | def | category_theory.idempotents.functor_extension₂_comp_whiskering_left_to_karoubi_iso | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [] | The natural isomorphism expressing that functors `karoubi C ⥤ karoubi D` obtained
using `functor_extension₂` actually extends the original functors `C ⥤ D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
karoubi_universal₂ : (C ⥤ D) ≌ (karoubi C ⥤ karoubi D) | (equivalence.congr_right (to_karoubi D).as_equivalence).trans
(karoubi_universal₁ C D) | def | category_theory.idempotents.karoubi_universal₂ | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [] | The equivalence of categories `(C ⥤ D) ≌ (karoubi C ⥤ karoubi D)` when `D`
is idempotent complete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
karoubi_universal₂_functor_eq :
(karoubi_universal₂ C D).functor = functor_extension₂ C D | rfl | lemma | category_theory.idempotents.karoubi_universal₂_functor_eq | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_extension : (C ⥤ D) ⥤ (karoubi C ⥤ D) | functor_extension₂ C D ⋙ (whiskering_right (karoubi C) (karoubi D) D).obj
(to_karoubi_is_equivalence D).inverse | def | category_theory.idempotents.functor_extension | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [] | The extension of functors functor `(C ⥤ D) ⥤ (karoubi C ⥤ D)`
when `D` is idempotent compltete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
karoubi_universal : (C ⥤ D) ≌ (karoubi C ⥤ D) | (karoubi_universal₂ C D).trans (equivalence.congr_right (to_karoubi D).as_equivalence.symm) | def | category_theory.idempotents.karoubi_universal | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [] | The equivalence `(C ⥤ D) ≌ (karoubi C ⥤ D)` when `D` is idempotent complete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
karoubi_universal_functor_eq :
(karoubi_universal C D).functor = functor_extension C D | rfl | lemma | category_theory.idempotents.karoubi_universal_functor_eq | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
whiskering_left_obj_preimage_app {F G : karoubi C ⥤ D}
(τ : to_karoubi _ ⋙ F ⟶ to_karoubi _ ⋙ G) (P : karoubi C) :
(((whiskering_left _ _ _).obj (to_karoubi _)).preimage τ).app P =
F.map P.decomp_id_i ≫ τ.app P.X ≫ G.map P.decomp_id_p | begin
rw nat_trans_eq,
congr' 2,
exact congr_app (((whiskering_left _ _ _).obj (to_karoubi _)).image_preimage τ) P.X,
end | lemma | category_theory.idempotents.whiskering_left_obj_preimage_app | category_theory.idempotents | src/category_theory/idempotents/functor_extension.lean | [
"category_theory.idempotents.karoubi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
p_comp_d : P.p.f n ≫ f.f.f n = f.f.f n | homological_complex.congr_hom (p_comp f) n | lemma | category_theory.idempotents.karoubi.homological_complex.p_comp_d | category_theory.idempotents | src/category_theory/idempotents/homological_complex.lean | [
"algebra.homology.additive",
"category_theory.idempotents.karoubi"
] | [
"homological_complex.congr_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_p_d : f.f.f n ≫ Q.p.f n = f.f.f n | homological_complex.congr_hom (comp_p f) n | lemma | category_theory.idempotents.karoubi.homological_complex.comp_p_d | category_theory.idempotents | src/category_theory/idempotents/homological_complex.lean | [
"algebra.homology.additive",
"category_theory.idempotents.karoubi"
] | [
"homological_complex.congr_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
p_comm_f : P.p.f n ≫ f.f.f n = f.f.f n ≫ Q.p.f n | homological_complex.congr_hom (p_comm f) n | lemma | category_theory.idempotents.karoubi.homological_complex.p_comm_f | category_theory.idempotents | src/category_theory/idempotents/homological_complex.lean | [
"algebra.homology.additive",
"category_theory.idempotents.karoubi"
] | [
"homological_complex.congr_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
p_idem : P.p.f n ≫ P.p.f n = P.p.f n | homological_complex.congr_hom P.idem n | lemma | category_theory.idempotents.karoubi.homological_complex.p_idem | category_theory.idempotents | src/category_theory/idempotents/homological_complex.lean | [
"algebra.homology.additive",
"category_theory.idempotents.karoubi"
] | [
"homological_complex.congr_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
obj (P : karoubi (homological_complex C c)) : homological_complex (karoubi C) c | { X := λ n, ⟨P.X.X n, P.p.f n, by simpa only [homological_complex.comp_f]
using homological_complex.congr_hom P.idem n⟩,
d := λ i j,
{ f := P.p.f i ≫ P.X.d i j,
comm := by tidy, },
shape' := λ i j hij, by simp only [hom_eq_zero_iff,
P.X.shape i j hij, limits.comp_zero], } | def | category_theory.idempotents.karoubi_homological_complex_equivalence.functor.obj | category_theory.idempotents | src/category_theory/idempotents/homological_complex.lean | [
"algebra.homology.additive",
"category_theory.idempotents.karoubi"
] | [
"comm",
"homological_complex",
"homological_complex.comp_f",
"homological_complex.congr_hom"
] | The functor `karoubi (homological_complex C c) ⥤ homological_complex (karoubi C) c`,
on objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map {P Q : karoubi (homological_complex C c)} (f : P ⟶ Q) : obj P ⟶ obj Q | { f:= λ n,
{ f:= f.f.f n,
comm := by simp, }, } | def | category_theory.idempotents.karoubi_homological_complex_equivalence.functor.map | category_theory.idempotents | src/category_theory/idempotents/homological_complex.lean | [
"algebra.homology.additive",
"category_theory.idempotents.karoubi"
] | [
"comm",
"homological_complex"
] | The functor `karoubi (homological_complex C c) ⥤ homological_complex (karoubi C) c`,
on morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor : karoubi (homological_complex C c) ⥤ homological_complex (karoubi C) c | { obj := functor.obj,
map := λ P Q f, functor.map f, } | def | category_theory.idempotents.karoubi_homological_complex_equivalence.functor | category_theory.idempotents | src/category_theory/idempotents/homological_complex.lean | [
"algebra.homology.additive",
"category_theory.idempotents.karoubi"
] | [
"homological_complex"
] | The functor `karoubi (homological_complex C c) ⥤ homological_complex (karoubi C) c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
obj (K : homological_complex (karoubi C) c) : karoubi (homological_complex C c) | { X :=
{ X := λ n, (K.X n).X,
d := λ i j, (K.d i j).f,
shape' := λ i j hij, hom_eq_zero_iff.mp (K.shape i j hij),
d_comp_d' := λ i j k hij hjk, by { simpa only [comp_f]
using hom_eq_zero_iff.mp (K.d_comp_d i j k), }, },
p :=
{ f := λ n, (K.X n).p,
comm' := by simp, },
idem := by tidy, ... | def | category_theory.idempotents.karoubi_homological_complex_equivalence.inverse.obj | category_theory.idempotents | src/category_theory/idempotents/homological_complex.lean | [
"algebra.homology.additive",
"category_theory.idempotents.karoubi"
] | [
"homological_complex"
] | The functor `homological_complex (karoubi C) c ⥤ karoubi (homological_complex C c)`,
on objects | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map {K L : homological_complex (karoubi C) c} (f : K ⟶ L) : obj K ⟶ obj L | { f:=
{ f := λ n, (f.f n).f,
comm' := λ i j hij, by simpa only [comp_f]
using hom_ext.mp (f.comm' i j hij), },
comm := by tidy, } | def | category_theory.idempotents.karoubi_homological_complex_equivalence.inverse.map | category_theory.idempotents | src/category_theory/idempotents/homological_complex.lean | [
"algebra.homology.additive",
"category_theory.idempotents.karoubi"
] | [
"comm",
"homological_complex"
] | The functor `homological_complex (karoubi C) c ⥤ karoubi (homological_complex C c)`,
on morphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse :
homological_complex (karoubi C) c ⥤ karoubi (homological_complex C c) | { obj := inverse.obj,
map := λ K L f, inverse.map f, } | def | category_theory.idempotents.karoubi_homological_complex_equivalence.inverse | category_theory.idempotents | src/category_theory/idempotents/homological_complex.lean | [
"algebra.homology.additive",
"category_theory.idempotents.karoubi"
] | [
"homological_complex"
] | The functor `homological_complex (karoubi C) c ⥤ karoubi (homological_complex C c)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
counit_iso : inverse ⋙ functor ≅ 𝟭 (homological_complex (karoubi C) c) | eq_to_iso (functor.ext (λ P, homological_complex.ext (by tidy) (by tidy)) (by tidy)) | def | category_theory.idempotents.karoubi_homological_complex_equivalence.counit_iso | category_theory.idempotents | src/category_theory/idempotents/homological_complex.lean | [
"algebra.homology.additive",
"category_theory.idempotents.karoubi"
] | [
"functor.ext",
"homological_complex",
"homological_complex.ext"
] | The counit isomorphism of the equivalence
`karoubi (homological_complex C c) ≌ homological_complex (karoubi C) c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_iso : 𝟭 (karoubi (homological_complex C c)) ≅ functor ⋙ inverse | { hom :=
{ app := λ P,
{ f :=
{ f := λ n, P.p.f n,
comm' := λ i j hij, begin
dsimp,
simp only [homological_complex.hom.comm, homological_complex.hom.comm_assoc,
homological_complex.p_idem],
end },
comm := by { ext n, dsimp, simp only [homological_complex... | def | category_theory.idempotents.karoubi_homological_complex_equivalence.unit_iso | category_theory.idempotents | src/category_theory/idempotents/homological_complex.lean | [
"algebra.homology.additive",
"category_theory.idempotents.karoubi"
] | [
"comm",
"homological_complex",
"homological_complex.comp_f",
"homological_complex.hom.comm"
] | The unit isomorphism of the equivalence
`karoubi (homological_complex C c) ≌ homological_complex (karoubi C) c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
karoubi_homological_complex_equivalence :
karoubi (homological_complex C c) ≌ homological_complex (karoubi C) c | { functor := karoubi_homological_complex_equivalence.functor,
inverse := karoubi_homological_complex_equivalence.inverse,
unit_iso := karoubi_homological_complex_equivalence.unit_iso,
counit_iso := karoubi_homological_complex_equivalence.counit_iso, } | def | category_theory.idempotents.karoubi_homological_complex_equivalence | category_theory.idempotents | src/category_theory/idempotents/homological_complex.lean | [
"algebra.homology.additive",
"category_theory.idempotents.karoubi"
] | [
"homological_complex"
] | The equivalence `karoubi (homological_complex C c) ≌ homological_complex (karoubi C) c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
karoubi_chain_complex_equivalence :
karoubi (chain_complex C α) ≌ chain_complex (karoubi C) α | karoubi_homological_complex_equivalence C (complex_shape.down α) | def | category_theory.idempotents.karoubi_chain_complex_equivalence | category_theory.idempotents | src/category_theory/idempotents/homological_complex.lean | [
"algebra.homology.additive",
"category_theory.idempotents.karoubi"
] | [
"chain_complex",
"complex_shape.down"
] | The equivalence `karoubi (chain_complex C α) ≌ chain_complex (karoubi C) α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
karoubi_cochain_complex_equivalence :
karoubi (cochain_complex C α) ≌ cochain_complex (karoubi C) α | karoubi_homological_complex_equivalence C (complex_shape.up α) | def | category_theory.idempotents.karoubi_cochain_complex_equivalence | category_theory.idempotents | src/category_theory/idempotents/homological_complex.lean | [
"algebra.homology.additive",
"category_theory.idempotents.karoubi"
] | [
"cochain_complex",
"complex_shape.up"
] | The equivalence `karoubi (cochain_complex C α) ≌ cochain_complex (karoubi C) α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
karoubi | (X : C) (p : X ⟶ X) (idem : p ≫ p = p) | structure | category_theory.idempotents.karoubi | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | In a preadditive category `C`, when an object `X` decomposes as `X ≅ P ⨿ Q`, one may
consider `P` as a direct factor of `X` and up to unique isomorphism, it is determined by the
obvious idempotent `X ⟶ P ⟶ X` which is the projection onto `P` with kernel `Q`. More generally,
one may define a formal direct factor of an o... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {P Q : karoubi C} (h_X : P.X = Q.X)
(h_p : P.p ≫ eq_to_hom h_X = eq_to_hom h_X ≫ Q.p) : P = Q | begin
cases P,
cases Q,
dsimp at h_X h_p,
subst h_X,
simpa only [true_and, eq_self_iff_true, id_comp, eq_to_hom_refl,
heq_iff_eq, comp_id] using h_p,
end | lemma | category_theory.idempotents.karoubi.ext | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [
"heq_iff_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom (P Q : karoubi C) | (f : P.X ⟶ Q.X) (comm : f = P.p ≫ f ≫ Q.p) | structure | category_theory.idempotents.karoubi.hom | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [
"comm"
] | A morphism `P ⟶ Q` in the category `karoubi C` is a morphism in the underlying category
`C` which satisfies a relation, which in the preadditive case, expresses that it induces a
map between the corresponding "formal direct factors" and that it vanishes on the complement
formal direct factor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_ext {P Q : karoubi C} {f g : hom P Q} : f = g ↔ f.f = g.f | begin
split,
{ intro h, rw h, },
{ ext, }
end | lemma | category_theory.idempotents.karoubi.hom_ext | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
p_comp {P Q : karoubi C} (f : hom P Q) : P.p ≫ f.f = f.f | by rw [f.comm, ← assoc, P.idem] | lemma | category_theory.idempotents.karoubi.p_comp | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_p {P Q : karoubi C} (f : hom P Q) : f.f ≫ Q.p = f.f | by rw [f.comm, assoc, assoc, Q.idem] | lemma | category_theory.idempotents.karoubi.comp_p | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
p_comm {P Q : karoubi C} (f : hom P Q) : P.p ≫ f.f = f.f ≫ Q.p | by rw [p_comp, comp_p] | lemma | category_theory.idempotents.karoubi.p_comm | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_proof {P Q R : karoubi C} (g : hom Q R) (f : hom P Q) :
f.f ≫ g.f = P.p ≫ (f.f ≫ g.f) ≫ R.p | by rw [assoc, comp_p, ← assoc, p_comp] | lemma | category_theory.idempotents.karoubi.comp_proof | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_f {P Q R : karoubi C} (f : P ⟶ Q) (g : Q ⟶ R) :
(f ≫ g).f = f.f ≫ g.f | by refl | lemma | category_theory.idempotents.karoubi.comp_f | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_eq {P : karoubi C} : 𝟙 P = ⟨P.p, by repeat { rw P.idem, }⟩ | by refl | lemma | category_theory.idempotents.karoubi.id_eq | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe : has_coe_t C (karoubi C) | ⟨λ X, ⟨X, 𝟙 X, by rw comp_id⟩⟩ | instance | category_theory.idempotents.karoubi.coe | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | It is possible to coerce an object of `C` into an object of `karoubi C`.
See also the functor `to_karoubi`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_X (X : C) : (X : karoubi C).X = X | by refl | lemma | category_theory.idempotents.karoubi.coe_X | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_p (X : C) : (X : karoubi C).p = 𝟙 X | by refl | lemma | category_theory.idempotents.karoubi.coe_p | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_to_hom_f {P Q : karoubi C} (h : P = Q) :
karoubi.hom.f (eq_to_hom h) = P.p ≫ eq_to_hom (congr_arg karoubi.X h) | by { subst h, simp only [eq_to_hom_refl, karoubi.id_eq, comp_id], } | lemma | category_theory.idempotents.karoubi.eq_to_hom_f | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_karoubi : C ⥤ karoubi C | { obj := λ X, ⟨X, 𝟙 X, by rw comp_id⟩,
map := λ X Y f, ⟨f, by simp only [comp_id, id_comp]⟩ } | def | category_theory.idempotents.to_karoubi | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | The obvious fully faithful functor `to_karoubi` sends an object `X : C` to the obvious
formal direct factor of `X` given by `𝟙 X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_eq_zero_iff [preadditive C] {P Q : karoubi C} {f : hom P Q} : f = 0 ↔ f.f = 0 | hom_ext | lemma | category_theory.idempotents.karoubi.hom_eq_zero_iff | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_hom [preadditive C] (P Q : karoubi C) : add_monoid_hom (P ⟶ Q) (P.X ⟶ Q.X) | { to_fun := λ f, f.f,
map_zero' := rfl,
map_add' := λ f g, rfl } | def | category_theory.idempotents.karoubi.inclusion_hom | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [
"add_monoid_hom"
] | The map sending `f : P ⟶ Q` to `f.f : P.X ⟶ Q.X` is additive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_hom [preadditive C] {P Q : karoubi C} {α : Type*} (s : finset α) (f : α → (P ⟶ Q)) :
(∑ x in s, f x).f = ∑ x in s, (f x).f | add_monoid_hom.map_sum (inclusion_hom P Q) f s | lemma | category_theory.idempotents.karoubi.sum_hom | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_karoubi_is_equivalence [is_idempotent_complete C] :
is_equivalence (to_karoubi C) | equivalence.of_fully_faithfully_ess_surj (to_karoubi C) | def | category_theory.idempotents.to_karoubi_is_equivalence | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | If `C` is idempotent complete, the functor `to_karoubi : C ⥤ karoubi C` is an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_karoubi_equivalence [is_idempotent_complete C] : C ≌ karoubi C | by { haveI := to_karoubi_is_equivalence C, exact functor.as_equivalence (to_karoubi C), } | def | category_theory.idempotents.to_karoubi_equivalence | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | The equivalence `C ≅ karoubi C` when `C` is idempotent complete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_karoubi_equivalence_functor_additive
[preadditive C] [is_idempotent_complete C] :
(to_karoubi_equivalence C).functor.additive | (infer_instance : (to_karoubi C).additive) | instance | category_theory.idempotents.to_karoubi_equivalence_functor_additive | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decomp_id_i (P : karoubi C) : P ⟶ P.X | ⟨P.p, by erw [coe_p, comp_id, P.idem]⟩ | def | category_theory.idempotents.karoubi.decomp_id_i | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | The split mono which appears in the factorisation `decomp_id P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
decomp_id_p (P : karoubi C) : (P.X : karoubi C) ⟶ P | ⟨P.p, by erw [coe_p, id_comp, P.idem]⟩ | def | category_theory.idempotents.karoubi.decomp_id_p | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | The split epi which appears in the factorisation `decomp_id P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
decomp_id (P : karoubi C) :
𝟙 P = (decomp_id_i P) ≫ (decomp_id_p P) | by { ext, simp only [comp_f, id_eq, P.idem, decomp_id_i, decomp_id_p], } | lemma | category_theory.idempotents.karoubi.decomp_id | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | The formal direct factor of `P.X` given by the idempotent `P.p` in the category `C`
is actually a direct factor in the category `karoubi C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
decomp_p (P : karoubi C) :
(to_karoubi C).map P.p = (decomp_id_p P) ≫ (decomp_id_i P) | by { ext, simp only [comp_f, decomp_id_p_f, decomp_id_i_f, P.idem, to_karoubi_map_f], } | lemma | category_theory.idempotents.karoubi.decomp_p | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decomp_id_i_to_karoubi (X : C) : decomp_id_i ((to_karoubi C).obj X) = 𝟙 _ | by { ext, refl, } | lemma | category_theory.idempotents.karoubi.decomp_id_i_to_karoubi | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decomp_id_p_to_karoubi (X : C) : decomp_id_p ((to_karoubi C).obj X) = 𝟙 _ | by { ext, refl, } | lemma | category_theory.idempotents.karoubi.decomp_id_p_to_karoubi | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decomp_id_i_naturality {P Q : karoubi C} (f : P ⟶ Q) : f ≫ decomp_id_i _ =
decomp_id_i _ ≫ ⟨f.f, by erw [comp_id, id_comp]⟩ | by { ext, simp only [comp_f, decomp_id_i_f, karoubi.comp_p, karoubi.p_comp], } | lemma | category_theory.idempotents.karoubi.decomp_id_i_naturality | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decomp_id_p_naturality {P Q : karoubi C} (f : P ⟶ Q) : decomp_id_p P ≫ f =
(⟨f.f, by erw [comp_id, id_comp]⟩ : (P.X : karoubi C) ⟶ Q.X) ≫ decomp_id_p Q | by { ext, simp only [comp_f, decomp_id_p_f, karoubi.comp_p, karoubi.p_comp], } | lemma | category_theory.idempotents.karoubi.decomp_id_p_naturality | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_hom [preadditive C] {P Q : karoubi C} (n : ℤ) (f : P ⟶ Q) :
(n • f).f = n • f.f | map_zsmul (inclusion_hom P Q) n f | lemma | category_theory.idempotents.karoubi.zsmul_hom | category_theory.idempotents | src/category_theory/idempotents/karoubi.lean | [
"category_theory.idempotents.basic",
"category_theory.preadditive.additive_functor",
"category_theory.equivalence"
] | [
"map_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse : karoubi (karoubi C) ⥤ karoubi C | { obj := λ P, ⟨P.X.X, P.p.f, by simpa only [hom_ext] using P.idem⟩,
map := λ P Q f, ⟨f.f.f, by simpa only [hom_ext] using f.comm⟩, } | def | category_theory.idempotents.karoubi_karoubi.inverse | category_theory.idempotents | src/category_theory/idempotents/karoubi_karoubi.lean | [
"category_theory.idempotents.karoubi"
] | [
"hom_ext"
] | The canonical functor `karoubi (karoubi C) ⥤ karoubi C` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_iso : 𝟭 (karoubi C) ≅ to_karoubi (karoubi C) ⋙ inverse C | eq_to_iso (functor.ext (by tidy) (by tidy)) | def | category_theory.idempotents.karoubi_karoubi.unit_iso | category_theory.idempotents | src/category_theory/idempotents/karoubi_karoubi.lean | [
"category_theory.idempotents.karoubi"
] | [
"functor.ext"
] | The unit isomorphism of the equivalence | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
counit_iso : inverse C ⋙ to_karoubi (karoubi C) ≅ 𝟭 (karoubi (karoubi C)) | { hom :=
{ app := λ P,
{ f :=
{ f := P.p.1,
comm := begin
have h := P.idem,
simp only [hom_ext, comp_f] at h,
erw [← assoc, h, comp_p],
end, },
comm := begin
have h := P.idem,
simp only [hom_ext, comp_f] at h ⊢,
erw [h, h],
en... | def | category_theory.idempotents.karoubi_karoubi.counit_iso | category_theory.idempotents | src/category_theory/idempotents/karoubi_karoubi.lean | [
"category_theory.idempotents.karoubi"
] | [
"comm",
"hom_ext"
] | The counit isomorphism of the equivalence | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence : karoubi C ≌ karoubi (karoubi C) | { functor := to_karoubi (karoubi C),
inverse := karoubi_karoubi.inverse C,
unit_iso := karoubi_karoubi.unit_iso C,
counit_iso := karoubi_karoubi.counit_iso C, } | def | category_theory.idempotents.karoubi_karoubi.equivalence | category_theory.idempotents | src/category_theory/idempotents/karoubi_karoubi.lean | [
"category_theory.idempotents.karoubi"
] | [] | The equivalence `karoubi C ≌ karoubi (karoubi C)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence.additive_functor [preadditive C] :
functor.additive (equivalence C).functor | by { dsimp, apply_instance, } | instance | category_theory.idempotents.karoubi_karoubi.equivalence.additive_functor | category_theory.idempotents | src/category_theory/idempotents/karoubi_karoubi.lean | [
"category_theory.idempotents.karoubi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equivalence.additive_inverse [preadditive C] :
functor.additive (equivalence C).inverse | by { dsimp, apply_instance, } | instance | category_theory.idempotents.karoubi_karoubi.equivalence.additive_inverse | category_theory.idempotents | src/category_theory/idempotents/karoubi_karoubi.lean | [
"category_theory.idempotents.karoubi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_adjoint :
comm_sq (adj.hom_equiv _ _ u) i (F.map p) (adj.hom_equiv _ _ v) | ⟨begin
simp only [adjunction.hom_equiv_unit, assoc, ← F.map_comp, sq.w],
rw [F.map_comp, adjunction.unit_naturality_assoc],
end⟩ | lemma | category_theory.comm_sq.right_adjoint | category_theory.lifting_properties | src/category_theory/lifting_properties/adjunction.lean | [
"category_theory.lifting_properties.basic",
"category_theory.adjunction.basic"
] | [] | When we have an adjunction `G ⊣ F`, any commutative square where the left
map is of the form `G.map i` and the right map is `p` has an "adjoint" commutative
square whose left map is `i` and whose right map is `F.map p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_adjoint_lift_struct_equiv :
sq.lift_struct ≃ (sq.right_adjoint adj).lift_struct | { to_fun := λ l,
{ l := adj.hom_equiv _ _ l.l,
fac_left' := by rw [← adj.hom_equiv_naturality_left, l.fac_left],
fac_right' := by rw [← adjunction.hom_equiv_naturality_right, l.fac_right], },
inv_fun := λ l,
{ l := (adj.hom_equiv _ _).symm l.l,
fac_left' := begin
rw [← adjunction.hom_equiv_natur... | def | category_theory.comm_sq.right_adjoint_lift_struct_equiv | category_theory.lifting_properties | src/category_theory/lifting_properties/adjunction.lean | [
"category_theory.lifting_properties.basic",
"category_theory.adjunction.basic"
] | [
"adj",
"inv_fun"
] | The liftings of a commutative are in bijection with the liftings of its (right)
adjoint square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_adjoint_has_lift_iff :
has_lift (sq.right_adjoint adj) ↔ has_lift sq | begin
simp only [has_lift.iff],
exact equiv.nonempty_congr (sq.right_adjoint_lift_struct_equiv adj).symm,
end | lemma | category_theory.comm_sq.right_adjoint_has_lift_iff | category_theory.lifting_properties | src/category_theory/lifting_properties/adjunction.lean | [
"category_theory.lifting_properties.basic",
"category_theory.adjunction.basic"
] | [
"adj",
"equiv.nonempty_congr"
] | A square has a lifting if and only if its (right) adjoint square has a lifting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_adjoint :
comm_sq ((adj.hom_equiv _ _).symm u) (G.map i) p
((adj.hom_equiv _ _).symm v) | ⟨begin
simp only [adjunction.hom_equiv_counit, assoc,
← G.map_comp_assoc, ← sq.w],
rw [G.map_comp, assoc, adjunction.counit_naturality],
end⟩ | lemma | category_theory.comm_sq.left_adjoint | category_theory.lifting_properties | src/category_theory/lifting_properties/adjunction.lean | [
"category_theory.lifting_properties.basic",
"category_theory.adjunction.basic"
] | [] | When we have an adjunction `G ⊣ F`, any commutative square where the left
map is of the form `i` and the right map is `F.map p` has an "adjoint" commutative
square whose left map is `G.map i` and whose right map is `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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