statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
epi_pullback_of_epi_g [epi g] : epi (pullback.fst : pullback f g ⟶ X) | -- It will suffice to consider some morphism e : X ⟶ R such that
-- pullback.fst ≫ e = 0 and show that e = 0.
epi_of_cancel_zero _ $ λ R e h,
begin
-- Consider the morphism u := (e, 0) : X ⊞ Y ⟶ R.
let u := biprod.desc e (0 : Y ⟶ R),
-- The composite pullback f g ⟶ X ⊞ Y ⟶ R is zero by assumption.
have hu : pul... | instance | category_theory.abelian.epi_pullback_of_epi_g | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | In an abelian category, the pullback of an epimorphism is an epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_snd_of_is_limit [epi f] {s : pullback_cone f g} (hs : is_limit s) : epi s.snd | begin
convert epi_of_epi_fac (is_limit.cone_point_unique_up_to_iso_hom_comp (limit.is_limit _) hs _),
{ refl },
{ exact abelian.epi_pullback_of_epi_f _ _ }
end | lemma | category_theory.abelian.epi_snd_of_is_limit | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_fst_of_is_limit [epi g] {s : pullback_cone f g} (hs : is_limit s) : epi s.fst | begin
convert epi_of_epi_fac (is_limit.cone_point_unique_up_to_iso_hom_comp (limit.is_limit _) hs _),
{ refl },
{ exact abelian.epi_pullback_of_epi_g _ _ }
end | lemma | category_theory.abelian.epi_fst_of_is_limit | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_fst_of_factor_thru_epi_mono_factorization
(g₁ : Y ⟶ W) [epi g₁] (g₂ : W ⟶ Z) [mono g₂] (hg : g₁ ≫ g₂ = g) (f' : X ⟶ W) (hf : f' ≫ g₂ = f)
(t : pullback_cone f g) (ht : is_limit t) : epi t.fst | by apply epi_fst_of_is_limit _ _ (pullback_cone.is_limit_of_factors f g g₂ f' g₁ hf hg t ht) | lemma | category_theory.abelian.epi_fst_of_factor_thru_epi_mono_factorization | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | Suppose `f` and `g` are two morphisms with a common codomain and suppose we have written `g` as
an epimorphism followed by a monomorphism. If `f` factors through the mono part of this
factorization, then any pullback of `g` along `f` is an epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_pushout_of_mono_f [mono f] : mono (pushout.inr : Z ⟶ pushout f g) | mono_of_cancel_zero _ $ λ R e h,
begin
let u := biprod.lift (0 : R ⟶ Y) e,
have hu : u ≫ biproduct_to_pushout_is_cokernel.biproduct_to_pushout f g = 0 := by simpa,
have := mono_is_kernel_of_cokernel _
(biproduct_to_pushout_is_cokernel.is_colimit_biproduct_to_pushout f g),
obtain ⟨d, hd⟩ := kernel_fork.is_li... | instance | category_theory.abelian.mono_pushout_of_mono_f | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_pushout_of_mono_g [mono g] : mono (pushout.inl : Y ⟶ pushout f g) | mono_of_cancel_zero _ $ λ R e h,
begin
let u := biprod.lift e (0 : R ⟶ Z),
have hu : u ≫ biproduct_to_pushout_is_cokernel.biproduct_to_pushout f g = 0 := by simpa,
have := mono_is_kernel_of_cokernel _
(biproduct_to_pushout_is_cokernel.is_colimit_biproduct_to_pushout f g),
obtain ⟨d, hd⟩ := kernel_fork.is_li... | instance | category_theory.abelian.mono_pushout_of_mono_g | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_inr_of_is_colimit [mono f] {s : pushout_cocone f g} (hs : is_colimit s) : mono s.inr | begin
convert mono_of_mono_fac
(is_colimit.comp_cocone_point_unique_up_to_iso_hom hs (colimit.is_colimit _) _),
{ refl },
{ exact abelian.mono_pushout_of_mono_f _ _ }
end | lemma | category_theory.abelian.mono_inr_of_is_colimit | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_inl_of_is_colimit [mono g] {s : pushout_cocone f g} (hs : is_colimit s) : mono s.inl | begin
convert mono_of_mono_fac
(is_colimit.comp_cocone_point_unique_up_to_iso_hom hs (colimit.is_colimit _) _),
{ refl },
{ exact abelian.mono_pushout_of_mono_g _ _ }
end | lemma | category_theory.abelian.mono_inl_of_is_colimit | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_inl_of_factor_thru_epi_mono_factorization (f : X ⟶ Y) (g : X ⟶ Z)
(g₁ : X ⟶ W) [epi g₁] (g₂ : W ⟶ Z) [mono g₂] (hg : g₁ ≫ g₂ = g) (f' : W ⟶ Y) (hf : g₁ ≫ f' = f)
(t : pushout_cocone f g) (ht : is_colimit t) : mono t.inl | by apply mono_inl_of_is_colimit _ _ (pushout_cocone.is_colimit_of_factors _ _ _ _ _ hf hg t ht) | lemma | category_theory.abelian.mono_inl_of_factor_thru_epi_mono_factorization | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | Suppose `f` and `g` are two morphisms with a common domain and suppose we have written `g` as
an epimorphism followed by a monomorphism. If `f` factors through the epi part of this
factorization, then any pushout of `g` along `f` is a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abelian : abelian C | { has_finite_products := by apply_instance,
/- We need the `convert`s here because the instances we have are slightly different from the
instances we need: `has_kernels` depends on an instance of `has_zero_morphisms`. In the
case of `non_preadditive_abelian`, this instance is an explicit argument. However, in the... | def | category_theory.non_preadditive_abelian.abelian | category_theory.abelian | src/category_theory/abelian/basic.lean | [
"category_theory.limits.constructions.pullbacks",
"category_theory.preadditive.biproducts",
"category_theory.limits.shapes.images",
"category_theory.limits.constructions.limits_of_products_and_equalizers",
"category_theory.abelian.non_preadditive"
] | [] | Every non_preadditive_abelian category can be promoted to an abelian category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exact_iff_image_eq_kernel : exact f g ↔ image_subobject f = kernel_subobject g | begin
split,
{ intro h,
fapply subobject.eq_of_comm,
{ suffices : is_iso (image_to_kernel _ _ h.w),
{ exactI as_iso (image_to_kernel _ _ h.w), },
exact is_iso_of_mono_of_epi _, },
{ simp, }, },
{ apply exact_of_image_eq_kernel, },
end | theorem | category_theory.abelian.exact_iff_image_eq_kernel | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [
"image_to_kernel"
] | In an abelian category, a pair of morphisms `f : X ⟶ Y`, `g : Y ⟶ Z` is exact
iff `image_subobject f = kernel_subobject g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exact_iff : exact f g ↔ f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0 | begin
split,
{ intro h,
exact ⟨h.1, kernel_comp_cokernel f g h⟩ },
{ refine λ h, ⟨h.1, _⟩,
suffices hl : is_limit
(kernel_fork.of_ι (image_subobject f).arrow (image_subobject_arrow_comp_eq_zero h.1)),
{ have : image_to_kernel f g h.1 =
(is_limit.cone_point_unique_up_to_iso hl (limit.is_l... | theorem | category_theory.abelian.exact_iff | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [
"image_to_kernel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_iff' {cg : kernel_fork g} (hg : is_limit cg)
{cf : cokernel_cofork f} (hf : is_colimit cf) : exact f g ↔ f ≫ g = 0 ∧ cg.ι ≫ cf.π = 0 | begin
split,
{ intro h,
exact ⟨h.1, fork_ι_comp_cofork_π f g h cg cf⟩ },
{ rw exact_iff,
refine λ h, ⟨h.1, _⟩,
apply zero_of_epi_comp (is_limit.cone_point_unique_up_to_iso hg (limit.is_limit _)).hom,
apply zero_of_comp_mono
(is_colimit.cocone_point_unique_up_to_iso (colimit.is_colimit _) hf)... | theorem | category_theory.abelian.exact_iff' | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_tfae :
tfae [exact f g,
f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0,
image_subobject f = kernel_subobject g] | begin
tfae_have : 1 ↔ 2, { apply exact_iff },
tfae_have : 1 ↔ 3, { apply exact_iff_image_eq_kernel },
tfae_finish
end | theorem | category_theory.abelian.exact_tfae | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalence.exact_iff {D : Type u₁} [category.{v₁} D] [abelian D]
(F : C ⥤ D) [is_equivalence F] :
exact (F.map f) (F.map g) ↔ exact f g | begin
simp only [exact_iff, ← F.map_eq_zero_iff, F.map_comp, category.assoc,
← kernel_comparison_comp_ι g F, ← π_comp_cokernel_comparison f F],
rw [is_iso.comp_left_eq_zero (kernel_comparison g F), ← category.assoc,
is_iso.comp_right_eq_zero _ (cokernel_comparison f F)],
end | lemma | category_theory.abelian.is_equivalence.exact_iff | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_epi_comp_iff {W : C} (h : W ⟶ X) [epi h] : exact (h ≫ f) g ↔ exact f g | begin
refine ⟨λ hfg, _, λ h, exact_epi_comp h⟩,
let hc := is_cokernel_of_comp _ _ (colimit.is_colimit (parallel_pair (h ≫ f) 0))
(by rw [← cancel_epi h, ← category.assoc, cokernel_cofork.condition, comp_zero]) rfl,
refine (exact_iff' _ _ (limit.is_limit _) hc).2 ⟨_, ((exact_iff _ _).1 hfg).2⟩,
exact zero_of... | lemma | category_theory.abelian.exact_epi_comp_iff | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | The dual result is true even in non-abelian categories, see
`category_theory.exact_comp_mono_iff`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_image (h : exact f g) :
is_limit
(kernel_fork.of_ι (abelian.image.ι f) (image_ι_comp_eq_zero h.1) : kernel_fork g) | begin
rw exact_iff at h,
refine kernel_fork.is_limit.of_ι _ _ _ _ _,
{ refine λ W u hu, kernel.lift (cokernel.π f) u _,
rw [←kernel.lift_ι g u hu, category.assoc, h.2, has_zero_morphisms.comp_zero] },
tidy
end | def | category_theory.abelian.is_limit_image | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | If `(f, g)` is exact, then `abelian.image.ι f` is a kernel of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_image' (h : exact f g) :
is_limit (kernel_fork.of_ι (limits.image.ι f) (limits.image_ι_comp_eq_zero h.1)) | is_kernel.iso_kernel _ _ (is_limit_image f g h) (image_iso_image f).symm $ is_image.lift_fac _ _ | def | category_theory.abelian.is_limit_image' | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | If `(f, g)` is exact, then `image.ι f` is a kernel of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_coimage (h : exact f g) : is_colimit (cokernel_cofork.of_π (abelian.coimage.π g)
(abelian.comp_coimage_π_eq_zero h.1) : cokernel_cofork f) | begin
rw exact_iff at h,
refine cokernel_cofork.is_colimit.of_π _ _ _ _ _,
{ refine λ W u hu, cokernel.desc (kernel.ι g) u _,
rw [←cokernel.π_desc f u hu, ←category.assoc, h.2, has_zero_morphisms.zero_comp] },
tidy
end | def | category_theory.abelian.is_colimit_coimage | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | If `(f, g)` is exact, then `coimages.coimage.π g` is a cokernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_image (h : exact f g) : is_colimit
(cokernel_cofork.of_π (limits.factor_thru_image g) (comp_factor_thru_image_eq_zero h.1)) | is_cokernel.cokernel_iso _ _ (is_colimit_coimage f g h) (coimage_iso_image' g) $
(cancel_mono (limits.image.ι g)).1 $ by simp | def | category_theory.abelian.is_colimit_image | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | If `(f, g)` is exact, then `factor_thru_image g` is a cokernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exact_cokernel : exact f (cokernel.π f) | by { rw exact_iff, tidy } | lemma | category_theory.abelian.exact_cokernel | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel.desc.inv [epi g] (ex : exact f g) :
g ≫ inv (cokernel.desc _ _ ex.w) = cokernel.π _ | by simp | lemma | category_theory.abelian.cokernel.desc.inv | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel.lift.inv [mono f] (ex : exact f g) :
inv (kernel.lift _ _ ex.w) ≫ f = kernel.ι g | by simp | lemma | category_theory.abelian.kernel.lift.inv | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit_of_exact_of_epi [epi g] (h : exact f g) :
is_colimit (cokernel_cofork.of_π _ h.w) | is_colimit.of_iso_colimit (colimit.is_colimit _) $ cocones.ext
⟨cokernel.desc _ _ h.w, epi_desc g (cokernel.π f) ((exact_iff _ _).1 h).2,
(cancel_epi (cokernel.π f)).1 (by tidy), (cancel_epi g).1 (by tidy)⟩ (λ j, by cases j; simp) | def | category_theory.abelian.is_colimit_of_exact_of_epi | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | If `X ⟶ Y ⟶ Z ⟶ 0` is exact, then the second map is a cokernel of the first. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_of_exact_of_mono [mono f] (h : exact f g) :
is_limit (kernel_fork.of_ι _ h.w) | is_limit.of_iso_limit (limit.is_limit _) $ cones.ext
⟨mono_lift f (kernel.ι g) ((exact_iff _ _).1 h).2, kernel.lift _ _ h.w,
(cancel_mono (kernel.ι g)).1 (by tidy), (cancel_mono f).1 (by tidy)⟩ (λ j, by cases j; simp) | def | category_theory.abelian.is_limit_of_exact_of_mono | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | If `0 ⟶ X ⟶ Y ⟶ Z` is exact, then the first map is a kernel of the second. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exact_of_is_cokernel (w : f ≫ g = 0)
(h : is_colimit (cokernel_cofork.of_π _ w)) : exact f g | begin
refine (exact_iff _ _).2 ⟨w, _⟩,
have := h.fac (cokernel_cofork.of_π _ (cokernel.condition f)) walking_parallel_pair.one,
simp only [cofork.of_π_ι_app] at this,
rw [← this, ← category.assoc, kernel.condition, zero_comp]
end | lemma | category_theory.abelian.exact_of_is_cokernel | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_of_is_kernel (w : f ≫ g = 0)
(h : is_limit (kernel_fork.of_ι _ w)) : exact f g | begin
refine (exact_iff _ _).2 ⟨w, _⟩,
have := h.fac (kernel_fork.of_ι _ (kernel.condition g)) walking_parallel_pair.zero,
simp only [fork.of_ι_π_app] at this,
rw [← this, category.assoc, cokernel.condition, comp_zero]
end | lemma | category_theory.abelian.exact_of_is_kernel | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_iff_exact_image_ι : exact f g ↔ exact (abelian.image.ι f) g | by conv_lhs { rw ← abelian.image.fac f }; apply exact_epi_comp_iff | lemma | category_theory.abelian.exact_iff_exact_image_ι | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact_iff_exact_coimage_π : exact f g ↔ exact f (coimage.π g) | by conv_lhs { rw ← abelian.coimage.fac g}; apply exact_comp_mono_iff | lemma | category_theory.abelian.exact_iff_exact_coimage_π | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tfae_mono : tfae [mono f, kernel.ι f = 0, exact (0 : Z ⟶ X) f] | begin
tfae_have : 3 → 2,
{ exact kernel_ι_eq_zero_of_exact_zero_left Z },
tfae_have : 1 → 3,
{ introsI, exact exact_zero_left_of_mono Z },
tfae_have : 2 → 1,
{ exact mono_of_kernel_ι_eq_zero _ },
tfae_finish
end | lemma | category_theory.abelian.tfae_mono | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_iff_kernel_ι_eq_zero : mono f ↔ kernel.ι f = 0 | (tfae_mono X f).out 0 1 | lemma | category_theory.abelian.mono_iff_kernel_ι_eq_zero | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tfae_epi : tfae [epi f, cokernel.π f = 0, exact f (0 : Y ⟶ Z)] | begin
tfae_have : 3 → 2,
{ rw exact_iff,
rintro ⟨-, h⟩,
exact zero_of_epi_comp _ h },
tfae_have : 1 → 3,
{ rw exact_iff,
introI,
exact ⟨by simp, by simp [cokernel.π_of_epi]⟩ },
tfae_have : 2 → 1,
{ exact epi_of_cokernel_π_eq_zero _ },
tfae_finish
end | lemma | category_theory.abelian.tfae_epi | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_iff_cokernel_π_eq_zero : epi f ↔ cokernel.π f = 0 | (tfae_epi X f).out 0 1 | lemma | category_theory.abelian.epi_iff_cokernel_π_eq_zero | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact.op (h : exact f g) : exact g.op f.op | begin
rw exact_iff,
refine ⟨by simp [← op_comp, h.w], quiver.hom.unop_inj _⟩,
simp only [unop_comp, cokernel.π_op, eq_to_hom_refl, kernel.ι_op, category.id_comp,
category.assoc, kernel_comp_cokernel_assoc _ _ h, zero_comp, comp_zero, unop_zero],
end | lemma | category_theory.abelian.exact.op | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [
"quiver.hom.unop_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact.op_iff : exact g.op f.op ↔ exact f g | ⟨λ e, begin
rw ← is_equivalence.exact_iff _ _ (op_op_equivalence C).inverse,
exact exact.op _ _ e
end, exact.op _ _⟩ | lemma | category_theory.abelian.exact.op_iff | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact.unop {X Y Z : Cᵒᵖ} (g : X ⟶ Y) (f : Y ⟶ Z) (h : exact g f) : exact f.unop g.unop | begin
rw [← f.op_unop, ← g.op_unop] at h,
rwa ← exact.op_iff,
end | lemma | category_theory.abelian.exact.unop | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exact.unop_iff {X Y Z : Cᵒᵖ} (g : X ⟶ Y) (f : Y ⟶ Z) : exact f.unop g.unop ↔ exact g f | ⟨λ e, by rwa [← f.op_unop, ← g.op_unop, ← exact.op_iff] at e, λ e, @@exact.unop _ _ g f e⟩ | lemma | category_theory.abelian.exact.unop_iff | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_exact_sequences_of_preserves_zero_morphisms_of_faithful [faithful F] :
reflects_exact_sequences F | { reflects := λ X Y Z f g hfg,
begin
rw [abelian.exact_iff, ← F.map_comp, F.map_eq_zero_iff] at hfg,
refine (abelian.exact_iff _ _).2 ⟨hfg.1, F.zero_of_map_zero _ _⟩,
obtain ⟨k, hk⟩ := kernel.lift' (F.map g) (F.map (kernel.ι g))
(by simp only [← F.map_comp, kernel.condition, category_theory.functor.... | instance | category_theory.functor.reflects_exact_sequences_of_preserves_zero_morphisms_of_faithful | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [
"category_theory.functor.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_exact {X Y Z : A} (f : X ⟶ Y) (g : Y ⟶ Z) (e1 : exact f g) :
exact (L.map f) (L.map g) | begin
let hcoker := is_colimit_of_has_cokernel_of_preserves_colimit L f,
let hker := is_limit_of_has_kernel_of_preserves_limit L g,
refine (exact_iff' _ _ hker hcoker).2 ⟨by simp [← L.map_comp, e1.1], _⟩,
rw [fork.ι_of_ι, cofork.π_of_π, ← L.map_comp, kernel_comp_cokernel _ _ e1, L.map_zero]
end | lemma | category_theory.functor.map_exact | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | A functor preserving finite limits and finite colimits preserves exactness. The converse
result is also true, see `functor.preserves_finite_limits_of_map_exact` and
`functor.preserves_finite_colimits_of_map_exact`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_zero_morphisms_of_map_exact : L.preserves_zero_morphisms | begin
replace h := (h (exact_of_zero (𝟙 0) (𝟙 0))).w,
rw [L.map_id, category.comp_id] at h,
exact preserves_zero_morphisms_of_map_zero_object (id_zero_equiv_iso_zero _ h),
end | lemma | category_theory.functor.preserves_zero_morphisms_of_map_exact | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | A functor which preserves exactness preserves zero morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_monomorphisms_of_map_exact : L.preserves_monomorphisms | { preserves := λ X Y f hf,
begin
letI := preserves_zero_morphisms_of_map_exact L h,
apply ((tfae_mono (L.obj 0) (L.map f)).out 2 0).mp,
rw ←L.map_zero,
exact h (((tfae_mono 0 f).out 0 2).mp hf)
end } | lemma | category_theory.functor.preserves_monomorphisms_of_map_exact | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | A functor which preserves exactness preserves monomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_epimorphisms_of_map_exact : L.preserves_epimorphisms | { preserves := λ X Y f hf,
begin
letI := preserves_zero_morphisms_of_map_exact L h,
apply ((tfae_epi (L.obj 0) (L.map f)).out 2 0).mp,
rw ←L.map_zero,
exact h (((tfae_epi 0 f).out 0 2).mp hf)
end } | lemma | category_theory.functor.preserves_epimorphisms_of_map_exact | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | A functor which preserves exactness preserves epimorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_kernels_of_map_exact (X Y : A) (f : X ⟶ Y) :
preserves_limit (parallel_pair f 0) L | { preserves := λ c ic,
begin
letI := preserves_zero_morphisms_of_map_exact L h,
letI := preserves_monomorphisms_of_map_exact L h,
letI := mono_of_is_limit_fork ic,
have hf := (is_limit_map_cone_fork_equiv' L (kernel_fork.condition c)).symm
(is_limit_of_exact_of_mono (L.map (fork.ι c)) (L.map f)
... | def | category_theory.functor.preserves_kernels_of_map_exact | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | A functor which preserves exactness preserves kernels. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_cokernels_of_map_exact (X Y : A) (f : X ⟶ Y) :
preserves_colimit (parallel_pair f 0) L | { preserves := λ c ic,
begin
letI := preserves_zero_morphisms_of_map_exact L h,
letI := preserves_epimorphisms_of_map_exact L h,
letI := epi_of_is_colimit_cofork ic,
have hf := (is_colimit_map_cocone_cofork_equiv' L (cokernel_cofork.condition c)).symm
(is_colimit_of_exact_of_epi (L.map f) (L.map... | def | category_theory.functor.preserves_cokernels_of_map_exact | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | A functor which preserves exactness preserves zero cokernels. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_limits_of_map_exact : preserves_finite_limits L | begin
letI := preserves_zero_morphisms_of_map_exact L h,
letI := preserves_kernels_of_map_exact L h,
apply preserves_finite_limits_of_preserves_kernels,
end | def | category_theory.functor.preserves_finite_limits_of_map_exact | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | A functor which preserves exactness is left exact, i.e. preserves finite limits.
This is part of the inverse implication to `functor.map_exact`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_colimits_of_map_exact : preserves_finite_colimits L | begin
letI := preserves_zero_morphisms_of_map_exact L h,
letI := preserves_cokernels_of_map_exact L h,
apply preserves_finite_colimits_of_preserves_cokernels,
end | def | category_theory.functor.preserves_finite_colimits_of_map_exact | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | A functor which preserves exactness is right exact, i.e. preserves finite colimits.
This is part of the inverse implication to `functor.map_exact`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_limits_of_preserves_monos_and_cokernels
[preserves_zero_morphisms L] [preserves_monomorphisms L]
[∀ {X Y} (f : X ⟶ Y), preserves_colimit (parallel_pair f 0) L] : preserves_finite_limits L | begin
apply preserves_finite_limits_of_map_exact,
intros X Y Z f g h,
rw [← abelian.coimage.fac g, L.map_comp, exact_comp_mono_iff],
exact exact_of_is_cokernel _ _ _
(is_colimit_cofork_map_of_is_colimit' L _ (is_colimit_coimage f g h))
end | def | category_theory.functor.preserves_finite_limits_of_preserves_monos_and_cokernels | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | A functor preserving zero morphisms, monos, and cokernels preserves finite limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_finite_colimits_of_preserves_epis_and_kernels
[preserves_zero_morphisms L] [preserves_epimorphisms L]
[∀ {X Y} (f : X ⟶ Y), preserves_limit (parallel_pair f 0) L] : preserves_finite_colimits L | begin
apply preserves_finite_colimits_of_map_exact,
intros X Y Z f g h,
rw [← abelian.image.fac f, L.map_comp, exact_epi_comp_iff],
exact exact_of_is_kernel _ _ _ (is_limit_fork_map_of_is_limit' L _ (is_limit_image f g h))
end | def | category_theory.functor.preserves_finite_colimits_of_preserves_epis_and_kernels | category_theory.abelian | src/category_theory/abelian/exact.lean | [
"category_theory.abelian.opposite",
"category_theory.limits.preserves.shapes.zero",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.preadditive.left_exact",
"category_theory.adjunction.limits",
"algebra.homology.exact",
"tactic.tfae"
] | [] | A functor preserving zero morphisms, epis, and kernels preserves finite colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Ext (n : ℕ) : Cᵒᵖ ⥤ C ⥤ Module R | functor.flip
{ obj := λ Y, (((linear_yoneda R C).obj Y).right_op.left_derived n).left_op,
map := λ Y Y' f, (nat_trans.left_derived ((linear_yoneda R C).map f).right_op n).left_op,
map_id' := begin
intros X,
ext Y : 2,
dsimp only [nat_trans.id_app, nat_trans.left_op_app,
nat_trans.right_op_app, fun... | def | Ext | category_theory.abelian | src/category_theory/abelian/ext.lean | [
"algebra.category.Module.abelian",
"category_theory.functor.left_derived",
"category_theory.linear.yoneda",
"category_theory.abelian.opposite",
"category_theory.abelian.projective"
] | [
"Module",
"map_comp",
"map_id"
] | `Ext R C n` is defined by deriving in the first argument of `(X, Y) ↦ Module.of R (unop X ⟶ Y)`
(which is the second argument of `linear_yoneda`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Ext_succ_of_projective (X Y : C) [projective X] (n : ℕ) :
((Ext R C (n+1)).obj (opposite.op X)).obj Y ≅ 0 | let E := (((linear_yoneda R C).obj Y).right_op.left_derived_obj_projective_succ n X).unop.symm in
E ≪≫
{ hom := 0,
inv := 0,
hom_inv_id' := begin
let Z : (Module R)ᵒᵖ := 0,
rw [← (0 : 0 ⟶ Z.unop).unop_op, ← (0 : Z.unop ⟶ 0).unop_op,
← unop_id, ← unop_comp],
congr' 1,
dsimp,
dec_trivial,
... | def | Ext_succ_of_projective | category_theory.abelian | src/category_theory/abelian/ext.lean | [
"algebra.category.Module.abelian",
"category_theory.functor.left_derived",
"category_theory.linear.yoneda",
"category_theory.abelian.opposite",
"category_theory.abelian.projective"
] | [
"Ext",
"Module",
"opposite.op"
] | If `X : C` is projective and `n : ℕ`, then `Ext^(n + 1) X Y ≅ 0` for any `Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coimage_obj_iso : (abelian.coimage α).obj X ≅ abelian.coimage (α.app X) | preserves_cokernel.iso ((evaluation C D).obj X) _ ≪≫
cokernel.map_iso _ _ (preserves_kernel.iso ((evaluation C D).obj X) _) (iso.refl _)
begin
dsimp,
simp only [category.comp_id],
exact (kernel_comparison_comp_ι _ ((evaluation C D).obj X)).symm,
end | def | category_theory.abelian.functor_category.coimage_obj_iso | category_theory.abelian | src/category_theory/abelian/functor_category.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.functor_category",
"category_theory.limits.shapes.functor_category",
"category_theory.limits.preserves.shapes.kernels"
] | [] | The abelian coimage in a functor category can be calculated componentwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_obj_iso : (abelian.image α).obj X ≅ abelian.image (α.app X) | preserves_kernel.iso ((evaluation C D).obj X) _ ≪≫
kernel.map_iso _ _ (iso.refl _) (preserves_cokernel.iso ((evaluation C D).obj X) _)
begin
apply (cancel_mono (preserves_cokernel.iso ((evaluation C D).obj X) α).inv).1,
simp only [category.assoc, iso.hom_inv_id],
dsimp,
simp only [category.id_comp, ... | def | category_theory.abelian.functor_category.image_obj_iso | category_theory.abelian | src/category_theory/abelian/functor_category.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.functor_category",
"category_theory.limits.shapes.functor_category",
"category_theory.limits.preserves.shapes.kernels"
] | [] | The abelian image in a functor category can be calculated componentwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coimage_image_comparison_app :
coimage_image_comparison (α.app X) =
(coimage_obj_iso α X).inv ≫ (coimage_image_comparison α).app X ≫ (image_obj_iso α X).hom | begin
ext,
dsimp,
simp only [category.comp_id, category.id_comp, category.assoc,
coimage_image_factorisation, limits.cokernel.π_desc_assoc, limits.kernel.lift_ι],
simp only [←evaluation_obj_map C D X],
erw kernel_comparison_comp_ι _ ((evaluation C D).obj X),
erw π_comp_cokernel_comparison_assoc _ ((eval... | lemma | category_theory.abelian.functor_category.coimage_image_comparison_app | category_theory.abelian | src/category_theory/abelian/functor_category.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.functor_category",
"category_theory.limits.shapes.functor_category",
"category_theory.limits.preserves.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coimage_image_comparison_app' :
(coimage_image_comparison α).app X =
(coimage_obj_iso α X).hom ≫ coimage_image_comparison (α.app X) ≫ (image_obj_iso α X).inv | by simp only [coimage_image_comparison_app, iso.hom_inv_id_assoc, iso.hom_inv_id, category.assoc,
category.comp_id] | lemma | category_theory.abelian.functor_category.coimage_image_comparison_app' | category_theory.abelian | src/category_theory/abelian/functor_category.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.functor_category",
"category_theory.limits.shapes.functor_category",
"category_theory.limits.preserves.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_is_iso_coimage_image_comparison :
is_iso (abelian.coimage_image_comparison α) | begin
haveI : ∀ X : C, is_iso ((abelian.coimage_image_comparison α).app X),
{ intros, rw coimage_image_comparison_app', apply_instance, },
apply nat_iso.is_iso_of_is_iso_app,
end | instance | category_theory.abelian.functor_category.functor_category_is_iso_coimage_image_comparison | category_theory.abelian | src/category_theory/abelian/functor_category.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.functor_category",
"category_theory.limits.shapes.functor_category",
"category_theory.limits.preserves.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_abelian : abelian (C ⥤ D) | abelian.of_coimage_image_comparison_is_iso | instance | category_theory.abelian.functor_category_abelian | category_theory.abelian | src/category_theory/abelian/functor_category.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.functor_category",
"category_theory.limits.shapes.functor_category",
"category_theory.limits.preserves.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_abelian' : abelian (C ⥤ D) | abelian.functor_category_abelian.{u u+1 u u} | instance | category_theory.abelian.functor_category_abelian' | category_theory.abelian | src/category_theory/abelian/functor_category.lean | [
"category_theory.abelian.basic",
"category_theory.preadditive.functor_category",
"category_theory.limits.shapes.functor_category",
"category_theory.limits.preserves.shapes.kernels"
] | [] | A variant with specialized universes for a common case. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_injective_coseparator [has_limits C] [enough_injectives C] (G : C)
(hG : is_separator G) : ∃ G : C, injective G ∧ is_coseparator G | begin
haveI : well_powered C := well_powered_of_is_detector G hG.is_detector,
haveI : has_products_of_shape (subobject (op G)) C := has_products_of_shape_of_small _ _,
let T : C := injective.under (pi_obj (λ P : subobject (op G), unop P)),
refine ⟨T, infer_instance, (preadditive.is_coseparator_iff _).2 (λ X Y f... | theorem | category_theory.abelian.has_injective_coseparator | category_theory.abelian | src/category_theory/abelian/generator.lean | [
"category_theory.abelian.subobject",
"category_theory.limits.essentially_small",
"category_theory.preadditive.injective",
"category_theory.preadditive.generator",
"category_theory.abelian.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_projective_separator [has_colimits C] [enough_projectives C] (G : C)
(hG : is_coseparator G) : ∃ G : C, projective G ∧ is_separator G | begin
obtain ⟨T, hT₁, hT₂⟩ := has_injective_coseparator (op G) ((is_separator_op_iff _).2 hG),
exactI ⟨unop T, infer_instance, (is_separator_unop_iff _).2 hT₂⟩
end | theorem | category_theory.abelian.has_projective_separator | category_theory.abelian | src/category_theory/abelian/generator.lean | [
"category_theory.abelian.subobject",
"category_theory.limits.essentially_small",
"category_theory.preadditive.injective",
"category_theory.preadditive.generator",
"category_theory.abelian.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homology_c : A | cokernel (kernel.lift g f w) | abbreviation | category_theory.abelian.homology_c | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [] | The cokernel of `kernel.lift g f w`. This is isomorphic to `homology f g w`.
See `homology_iso_cokernel_lift`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_k : A | kernel (cokernel.desc f g w) | abbreviation | category_theory.abelian.homology_k | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [] | The kernel of `cokernel.desc f g w`. This is isomorphic to `homology f g w`.
See `homology_iso_kernel_desc`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_c_to_k : homology_c f g w ⟶ homology_k f g w | cokernel.desc _ (kernel.lift _ (kernel.ι _ ≫ cokernel.π _) (by simp)) begin
apply limits.equalizer.hom_ext,
simp,
end | abbreviation | category_theory.abelian.homology_c_to_k | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [] | The canonical map from `homology_c` to `homology_k`.
This is an isomorphism, and it is used in obtaining the API for `homology f g w`
in the bottom of this file. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_iso_kernel_desc : homology f g w ≅ kernel (cokernel.desc f g w) | homology_iso_cokernel_lift _ _ _ ≪≫ as_iso (category_theory.abelian.homology_c_to_k _ _ _) | def | homology_iso_kernel_desc | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"category_theory.abelian.homology_c_to_k",
"homology",
"homology_iso_cokernel_lift"
] | The homology associated to `f` and `g` is isomorphic to a kernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π' : kernel g ⟶ homology f g w | cokernel.π _ ≫ (homology_iso_cokernel_lift _ _ _).inv | def | homology.π' | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"homology",
"homology_iso_cokernel_lift"
] | The canonical map from the kernel of `g` to the homology of `f` and `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι : homology f g w ⟶ cokernel f | (homology_iso_kernel_desc _ _ _).hom ≫ kernel.ι _ | def | homology.ι | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"homology",
"homology_iso_kernel_desc"
] | The canonical map from the homology of `f` and `g` to the cokernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc' {W : A} (e : kernel g ⟶ W) (he : kernel.lift g f w ≫ e = 0) :
homology f g w ⟶ W | (homology_iso_cokernel_lift _ _ _).hom ≫ cokernel.desc _ e he | def | homology.desc' | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"homology",
"homology_iso_cokernel_lift"
] | Obtain a morphism from the homology, given a morphism from the kernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift {W : A} (e : W ⟶ cokernel f) (he : e ≫ cokernel.desc f g w = 0) :
W ⟶ homology f g w | kernel.lift _ e he ≫ (homology_iso_kernel_desc _ _ _).inv | def | homology.lift | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"homology",
"homology_iso_kernel_desc",
"lift"
] | Obtain a moprhism to the homology, given a morphism to the kernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π'_desc' {W : A} (e : kernel g ⟶ W) (he : kernel.lift g f w ≫ e = 0) :
π' f g w ≫ desc' f g w e he = e | by { dsimp [π', desc'], simp } | lemma | homology.π'_desc' | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_ι {W : A} (e : W ⟶ cokernel f) (he : e ≫ cokernel.desc f g w = 0) :
lift f g w e he ≫ ι _ _ _ = e | by { dsimp [ι, lift], simp } | lemma | homology.lift_ι | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
condition_π' : kernel.lift g f w ≫ π' f g w = 0 | by { dsimp [π'], simp } | lemma | homology.condition_π' | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
condition_ι : ι f g w ≫ cokernel.desc f g w = 0 | by { dsimp [ι], simp } | lemma | homology.condition_ι | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_from_ext {W : A} (a b : homology f g w ⟶ W)
(h : π' f g w ≫ a = π' f g w ≫ b) : a = b | begin
dsimp [π'] at h,
apply_fun (λ e, (homology_iso_cokernel_lift f g w).inv ≫ e),
swap,
{ intros i j hh,
apply_fun (λ e, (homology_iso_cokernel_lift f g w).hom ≫ e) at hh,
simpa using hh },
simp only [category.assoc] at h,
exact coequalizer.hom_ext h,
end | lemma | homology.hom_from_ext | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"homology",
"homology_iso_cokernel_lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_to_ext {W : A} (a b : W ⟶ homology f g w)
(h : a ≫ ι f g w = b ≫ ι f g w) : a = b | begin
dsimp [ι] at h,
apply_fun (λ e, e ≫ (homology_iso_kernel_desc f g w).hom),
swap,
{ intros i j hh,
apply_fun (λ e, e ≫ (homology_iso_kernel_desc f g w).inv) at hh,
simpa using hh },
simp only [← category.assoc] at h,
exact equalizer.hom_ext h,
end | lemma | homology.hom_to_ext | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"homology",
"homology_iso_kernel_desc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
π'_ι : π' f g w ≫ ι f g w = kernel.ι _ ≫ cokernel.π _ | by { dsimp [π', ι, homology_iso_kernel_desc], simp } | lemma | homology.π'_ι | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"homology_iso_kernel_desc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
π'_eq_π : (kernel_subobject_iso _).hom ≫ π' f g w = π _ _ _ | begin
dsimp [π', homology_iso_cokernel_lift],
simp only [← category.assoc],
rw iso.comp_inv_eq,
dsimp [π, homology_iso_cokernel_image_to_kernel'],
simp,
end | lemma | homology.π'_eq_π | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"homology_iso_cokernel_image_to_kernel'",
"homology_iso_cokernel_lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
π'_map (α β h) :
π' _ _ _ ≫ map w w' α β h = kernel.map _ _ α.right β.right (by simp [h,β.w.symm]) ≫ π' _ _ _ | begin
apply_fun (λ e, (kernel_subobject_iso _).hom ≫ e),
swap,
{ intros i j hh,
apply_fun (λ e, (kernel_subobject_iso _).inv ≫ e) at hh,
simpa using hh },
dsimp [map],
simp only [π'_eq_π_assoc],
dsimp [π],
simp only [cokernel.π_desc],
rw [← iso.inv_comp_eq, ← category.assoc],
have : (limits.ke... | lemma | homology.π'_map | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"homology_iso_cokernel_image_to_kernel'",
"homology_iso_cokernel_lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_desc'_lift_left (α β h) : map w w' α β h =
homology.desc' _ _ _ (homology.lift _ _ _ (kernel.ι _ ≫ β.left ≫ cokernel.π _) (by simp))
(by { ext, simp only [←h, category.assoc, zero_comp, lift_ι, kernel.lift_ι_assoc],
erw ← reassoc_of α.w, simp } ) | begin
apply homology.hom_from_ext,
simp only [π'_map, π'_desc'],
dsimp [π', lift],
rw iso.eq_comp_inv,
dsimp [homology_iso_kernel_desc],
ext,
simp [h],
end | lemma | homology.map_eq_desc'_lift_left | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"homology.desc'",
"homology.hom_from_ext",
"homology.lift",
"homology_iso_kernel_desc",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_lift_desc'_left (α β h) : map w w' α β h =
homology.lift _ _ _ (homology.desc' _ _ _ (kernel.ι _ ≫ β.left ≫ cokernel.π _)
(by { simp only [kernel.lift_ι_assoc, ← h], erw ← reassoc_of α.w, simp }))
(by { ext, simp }) | by { rw map_eq_desc'_lift_left, ext, simp } | lemma | homology.map_eq_lift_desc'_left | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"homology.desc'",
"homology.lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_desc'_lift_right (α β h) : map w w' α β h =
homology.desc' _ _ _ (homology.lift _ _ _ (kernel.ι _ ≫ α.right ≫ cokernel.π _) (by simp [h]))
(by { ext, simp only [category.assoc, zero_comp, lift_ι, kernel.lift_ι_assoc],
erw ← reassoc_of α.w, simp } ) | by { rw map_eq_desc'_lift_left, ext, simp [h] } | lemma | homology.map_eq_desc'_lift_right | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"homology.desc'",
"homology.lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_lift_desc'_right (α β h) : map w w' α β h =
homology.lift _ _ _ (homology.desc' _ _ _ (kernel.ι _ ≫ α.right ≫ cokernel.π _)
(by { simp only [kernel.lift_ι_assoc], erw ← reassoc_of α.w, simp }))
(by { ext, simp [h] }) | by { rw map_eq_desc'_lift_right, ext, simp } | lemma | homology.map_eq_lift_desc'_right | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"homology.desc'",
"homology.lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ι (α β h) :
map w w' α β h ≫ ι f' g' w' = ι f g w ≫ cokernel.map f f' α.left β.left (by simp [h, β.w.symm]) | begin
rw [map_eq_lift_desc'_left, lift_ι],
ext,
simp only [← category.assoc],
rw [π'_ι, π'_desc', category.assoc, category.assoc, cokernel.π_desc],
end | lemma | homology.map_ι | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homology_iso (C : homological_complex A c) (j : ι) :
F.obj (C.homology j) ≅ ((F.map_homological_complex _).obj C).homology j | (preserves_cokernel.iso _ _).trans (cokernel.map_iso _ _ ((F.map_iso (image_subobject_iso _)).trans
((preserves_image.iso _ _).symm.trans (image_subobject_iso _).symm))
((F.map_iso (kernel_subobject_iso _)).trans ((preserves_kernel.iso _ _).trans
(kernel_subobject_iso _).symm))
begin
dsimp,
ext,
sim... | def | category_theory.functor.homology_iso | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"homological_complex",
"homology",
"image_to_kernel_arrow"
] | When `F` is an exact additive functor, `F(Hᵢ(X)) ≅ Hᵢ(F(X))` for `X` a complex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homology_functor_iso (i : ι) :
homology_functor A c i ⋙ F ≅ F.map_homological_complex c ⋙ homology_functor B c i | nat_iso.of_components (λ X, homology_iso F X i)
begin
intros X Y f,
dsimp,
rw [←iso.inv_comp_eq, ←category.assoc, ←iso.eq_comp_inv],
refine coequalizer.hom_ext _,
dsimp [homology_iso],
simp only [homology.map, ←category.assoc, cokernel.π_desc],
simp only [category.assoc, cokernel_comparison_map_desc, coke... | def | category_theory.functor.homology_functor_iso | category_theory.abelian | src/category_theory/abelian/homology.lean | [
"algebra.homology.additive",
"category_theory.abelian.pseudoelements",
"category_theory.limits.preserves.shapes.kernels",
"category_theory.limits.preserves.shapes.images"
] | [
"homological_complex.d_from",
"homological_complex.hom.next",
"homological_complex.hom.sq_from_left",
"homological_complex.hom.sq_from_right",
"homology.map",
"homology_functor"
] | If `F` is an exact additive functor, then `F` commutes with `Hᵢ` (up to natural isomorphism). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image : C | kernel (cokernel.π f) | abbreviation | category_theory.abelian.image | category_theory.abelian | src/category_theory/abelian/images.lean | [
"category_theory.limits.shapes.kernels"
] | [] | The kernel of the cokernel of `f` is called the (abelian) image of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.ι : abelian.image f ⟶ Q | kernel.ι (cokernel.π f) | abbreviation | category_theory.abelian.image.ι | category_theory.abelian | src/category_theory/abelian/images.lean | [
"category_theory.limits.shapes.kernels"
] | [] | The inclusion of the image into the codomain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
factor_thru_image : P ⟶ abelian.image f | kernel.lift (cokernel.π f) f $ cokernel.condition f | abbreviation | category_theory.abelian.factor_thru_image | category_theory.abelian | src/category_theory/abelian/images.lean | [
"category_theory.limits.shapes.kernels"
] | [] | There is a canonical epimorphism `p : P ⟶ image f` for every `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image.fac :
abelian.factor_thru_image f ≫ image.ι f = f | kernel.lift_ι _ _ _ | lemma | category_theory.abelian.image.fac | category_theory.abelian | src/category_theory/abelian/images.lean | [
"category_theory.limits.shapes.kernels"
] | [] | `f` factors through its image via the canonical morphism `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_factor_thru_image [mono f] : mono (abelian.factor_thru_image f) | mono_of_mono_fac $ image.fac f | instance | category_theory.abelian.mono_factor_thru_image | category_theory.abelian | src/category_theory/abelian/images.lean | [
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coimage : C | cokernel (kernel.ι f) | abbreviation | category_theory.abelian.coimage | category_theory.abelian | src/category_theory/abelian/images.lean | [
"category_theory.limits.shapes.kernels"
] | [] | The cokernel of the kernel of `f` is called the (abelian) coimage of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coimage.π : P ⟶ abelian.coimage f | cokernel.π (kernel.ι f) | abbreviation | category_theory.abelian.coimage.π | category_theory.abelian | src/category_theory/abelian/images.lean | [
"category_theory.limits.shapes.kernels"
] | [] | The projection onto the coimage. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
factor_thru_coimage : abelian.coimage f ⟶ Q | cokernel.desc (kernel.ι f) f $ kernel.condition f | abbreviation | category_theory.abelian.factor_thru_coimage | category_theory.abelian | src/category_theory/abelian/images.lean | [
"category_theory.limits.shapes.kernels"
] | [] | There is a canonical monomorphism `i : coimage f ⟶ Q`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coimage.fac : coimage.π f ≫ abelian.factor_thru_coimage f = f | cokernel.π_desc _ _ _ | lemma | category_theory.abelian.coimage.fac | category_theory.abelian | src/category_theory/abelian/images.lean | [
"category_theory.limits.shapes.kernels"
] | [] | `f` factors through its coimage via the canonical morphism `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_factor_thru_coimage [epi f] : epi (abelian.factor_thru_coimage f) | epi_of_epi_fac $ coimage.fac f | instance | category_theory.abelian.epi_factor_thru_coimage | category_theory.abelian | src/category_theory/abelian/images.lean | [
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coimage_image_comparison : abelian.coimage f ⟶ abelian.image f | cokernel.desc (kernel.ι f) (kernel.lift (cokernel.π f) f (by simp)) $ (by { ext, simp, }) | def | category_theory.abelian.coimage_image_comparison | category_theory.abelian | src/category_theory/abelian/images.lean | [
"category_theory.limits.shapes.kernels"
] | [] | The canonical map from the abelian coimage to the abelian image.
In any abelian category this is an isomorphism.
Conversely, any additive category with kernels and cokernels and
in which this is always an isomorphism, is abelian.
See <https://stacks.math.columbia.edu/tag/0107> | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coimage_image_comparison' : abelian.coimage f ⟶ abelian.image f | kernel.lift (cokernel.π f) (cokernel.desc (kernel.ι f) f (by simp)) (by { ext, simp, }) | def | category_theory.abelian.coimage_image_comparison' | category_theory.abelian | src/category_theory/abelian/images.lean | [
"category_theory.limits.shapes.kernels"
] | [] | An alternative formulation of the canonical map from the abelian coimage to the abelian image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coimage_image_comparison_eq_coimage_image_comparison' :
coimage_image_comparison f = coimage_image_comparison' f | by { ext, simp [coimage_image_comparison, coimage_image_comparison'], } | lemma | category_theory.abelian.coimage_image_comparison_eq_coimage_image_comparison' | category_theory.abelian | src/category_theory/abelian/images.lean | [
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coimage_image_factorisation :
coimage.π f ≫ coimage_image_comparison f ≫ image.ι f = f | by simp [coimage_image_comparison] | lemma | category_theory.abelian.coimage_image_factorisation | category_theory.abelian | src/category_theory/abelian/images.lean | [
"category_theory.limits.shapes.kernels"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_finite_colimits_preadditive_yoneda_obj_of_injective (J : C)
[hP : injective J] : preserves_finite_colimits (preadditive_yoneda_obj J) | begin
letI := (injective_iff_preserves_epimorphisms_preadditive_yoneda_obj' J).mp hP,
apply functor.preserves_finite_colimits_of_preserves_epis_and_kernels,
end | def | category_theory.preserves_finite_colimits_preadditive_yoneda_obj_of_injective | category_theory.abelian | src/category_theory/abelian/injective.lean | [
"category_theory.abelian.exact",
"category_theory.preadditive.injective",
"category_theory.preadditive.yoneda.limits",
"category_theory.preadditive.yoneda.injective"
] | [] | The preadditive Yoneda functor on `J` preserves colimits if `J` is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
injective_of_preserves_finite_colimits_preadditive_yoneda_obj (J : C)
[hP : preserves_finite_colimits (preadditive_yoneda_obj J)] : injective J | begin
rw injective_iff_preserves_epimorphisms_preadditive_yoneda_obj',
apply_instance
end | lemma | category_theory.injective_of_preserves_finite_colimits_preadditive_yoneda_obj | category_theory.abelian | src/category_theory/abelian/injective.lean | [
"category_theory.abelian.exact",
"category_theory.preadditive.injective",
"category_theory.preadditive.yoneda.limits",
"category_theory.preadditive.yoneda.injective"
] | [] | An object is injective if its preadditive Yoneda functor preserves finite colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
desc_f_zero {Y Z : C} (f : Z ⟶ Y) (I : InjectiveResolution Y) (J : InjectiveResolution Z) :
J.cocomplex.X 0 ⟶ I.cocomplex.X 0 | factor_thru (f ≫ I.ι.f 0) (J.ι.f 0) | def | category_theory.InjectiveResolution.desc_f_zero | category_theory.abelian | src/category_theory/abelian/injective_resolution.lean | [
"algebra.homology.quasi_iso",
"category_theory.preadditive.injective_resolution",
"algebra.homology.homotopy_category"
] | [] | Auxiliary construction for `desc`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.