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kernel_fork.is_limit.of_ι {W : C} (g : W ⟶ X) (eq : g ≫ f = 0)
(lift : Π {W' : C} (g' : W' ⟶ X) (eq' : g' ≫ f = 0), W' ⟶ W)
(fac : ∀ {W' : C} (g' : W' ⟶ X) (eq' : g' ≫ f = 0), lift g' eq' ≫ g = g')
(uniq :
∀ {W' : C} (g' : W' ⟶ X) (eq' : g' ≫ f = 0) (m : W' ⟶ W) (w : m ≫ g = g'), m = lift g' eq') :
is_limit... | is_limit_aux _ (λ s, lift s.ι s.condition) (λ s, fac s.ι s.condition) (λ s, uniq s.ι s.condition) | def | category_theory.limits.kernel_fork.is_limit.of_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [
"lift"
] | This is a more convenient formulation to show that a `kernel_fork` constructed using
`kernel_fork.of_ι` is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_kernel_comp_mono {c : kernel_fork f} (i : is_limit c) {Z} (g : Y ⟶ Z) [hg : mono g]
{h : X ⟶ Z} (hh : h = f ≫ g) :
is_limit (kernel_fork.of_ι c.ι (by simp [hh]) : kernel_fork h) | fork.is_limit.mk' _ $ λ s,
let s' : kernel_fork f := fork.of_ι s.ι (by rw [←cancel_mono g]; simp [←hh, s.condition]) in
let l := kernel_fork.is_limit.lift' i s'.ι s'.condition in
⟨l.1, l.2, λ m hm, by apply fork.is_limit.hom_ext i; rw fork.ι_of_ι at hm; rw hm; exact l.2.symm⟩ | def | category_theory.limits.is_kernel_comp_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | Every kernel of `f` induces a kernel of `f ≫ g` if `g` is mono. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_kernel_comp_mono_lift {c : kernel_fork f} (i : is_limit c) {Z} (g : Y ⟶ Z) [hg : mono g]
{h : X ⟶ Z} (hh : h = f ≫ g) (s : kernel_fork h) :
(is_kernel_comp_mono i g hh).lift s
= i.lift (fork.of_ι s.ι (by { rw [←cancel_mono g, category.assoc, ←hh], simp })) | rfl | lemma | category_theory.limits.is_kernel_comp_mono_lift | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_kernel_of_comp {W : C} (g : Y ⟶ W) (h : X ⟶ W) {c : kernel_fork h} (i : is_limit c)
(hf : c.ι ≫ f = 0) (hfg : f ≫ g = h) : is_limit (kernel_fork.of_ι c.ι hf) | fork.is_limit.mk _
(λ s, i.lift (kernel_fork.of_ι s.ι (by simp [← hfg])))
(λ s, by simp only [kernel_fork.ι_of_ι, fork.is_limit.lift_ι])
(λ s m h, by { apply fork.is_limit.hom_ext i, simpa using h }) | def | category_theory.limits.is_kernel_of_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | Every kernel of `f ≫ g` is also a kernel of `f`, as long as `c.ι ≫ f` vanishes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel : C | equalizer f 0 | abbreviation | category_theory.limits.kernel | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The kernel of a morphism, expressed as the equalizer with the 0 morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.ι : kernel f ⟶ X | equalizer.ι f 0 | abbreviation | category_theory.limits.kernel.ι | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The map from `kernel f` into the source of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equalizer_as_kernel : equalizer.ι f 0 = kernel.ι f | rfl | lemma | category_theory.limits.equalizer_as_kernel | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel.condition : kernel.ι f ≫ f = 0 | kernel_fork.condition _ | lemma | category_theory.limits.kernel.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_is_kernel :
is_limit (fork.of_ι (kernel.ι f) ((kernel.condition f).trans (comp_zero.symm))) | is_limit.of_iso_limit (limit.is_limit _) (fork.ext (iso.refl _) (by tidy)) | def | category_theory.limits.kernel_is_kernel | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The kernel built from `kernel.ι f` is limiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.lift {W : C} (k : W ⟶ X) (h : k ≫ f = 0) : W ⟶ kernel f | (kernel_is_kernel f).lift (kernel_fork.of_ι k h) | abbreviation | category_theory.limits.kernel.lift | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [
"lift"
] | Given any morphism `k : W ⟶ X` satisfying `k ≫ f = 0`, `k` factors through `kernel.ι f`
via `kernel.lift : W ⟶ kernel f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.lift_ι {W : C} (k : W ⟶ X) (h : k ≫ f = 0) : kernel.lift f k h ≫ kernel.ι f = k | (kernel_is_kernel f).fac (kernel_fork.of_ι k h) walking_parallel_pair.zero | lemma | category_theory.limits.kernel.lift_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel.lift_zero {W : C} {h} : kernel.lift f (0 : W ⟶ X) h = 0 | by { ext, simp, } | lemma | category_theory.limits.kernel.lift_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel.lift_mono {W : C} (k : W ⟶ X) (h : k ≫ f = 0) [mono k] : mono (kernel.lift f k h) | ⟨λ Z g g' w,
begin
replace w := w =≫ kernel.ι f,
simp only [category.assoc, kernel.lift_ι] at w,
exact (cancel_mono k).1 w,
end⟩ | instance | category_theory.limits.kernel.lift_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel.lift' {W : C} (k : W ⟶ X) (h : k ≫ f = 0) : {l : W ⟶ kernel f // l ≫ kernel.ι f = k} | ⟨kernel.lift f k h, kernel.lift_ι _ _ _⟩ | def | category_theory.limits.kernel.lift' | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | Any morphism `k : W ⟶ X` satisfying `k ≫ f = 0` induces a morphism `l : W ⟶ kernel f` such that
`l ≫ kernel.ι f = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.map {X' Y' : C} (f' : X' ⟶ Y') [has_kernel f']
(p : X ⟶ X') (q : Y ⟶ Y') (w : f ≫ q = p ≫ f') : kernel f ⟶ kernel f' | kernel.lift f' (kernel.ι f ≫ p) (by simp [←w]) | abbreviation | category_theory.limits.kernel.map | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | A commuting square induces a morphism of kernels. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.lift_map {X Y Z X' Y' Z' : C}
(f : X ⟶ Y) (g : Y ⟶ Z) [has_kernel g] (w : f ≫ g = 0)
(f' : X' ⟶ Y') (g' : Y' ⟶ Z') [has_kernel g'] (w' : f' ≫ g' = 0)
(p : X ⟶ X') (q : Y ⟶ Y') (r : Z ⟶ Z') (h₁ : f ≫ q = p ≫ f') (h₂ : g ≫ r = q ≫ g') :
kernel.lift g f w ≫ kernel.map g g' q r h₂ = p ≫ kernel.lift g' f' w' | by { ext, simp [h₁], } | lemma | category_theory.limits.kernel.lift_map | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | Given a commutative diagram
X --f--> Y --g--> Z
| | |
| | |
v v v
X' -f'-> Y' -g'-> Z'
with horizontal arrows composing to zero,
then we obtain a commutative square
X ---> kernel g
| |
| | kernel.map
| |
v v
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.map_iso {X' Y' : C} (f' : X' ⟶ Y') [has_kernel f']
(p : X ≅ X') (q : Y ≅ Y') (w : f ≫ q.hom = p.hom ≫ f') : kernel f ≅ kernel f' | { hom := kernel.map f f' p.hom q.hom w,
inv := kernel.map f' f p.inv q.inv (by { refine (cancel_mono q.hom).1 _, simp [w], }), } | def | category_theory.limits.kernel.map_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | A commuting square of isomorphisms induces an isomorphism of kernels. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.ι_zero_is_iso : is_iso (kernel.ι (0 : X ⟶ Y)) | equalizer.ι_of_self _ | instance | category_theory.limits.kernel.ι_zero_is_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | Every kernel of the zero morphism is an isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_of_epi_kernel [epi (kernel.ι f)] : f = 0 | (cancel_epi (kernel.ι f)).1 (by simp) | lemma | category_theory.limits.eq_zero_of_epi_kernel | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_zero_iso_source : kernel (0 : X ⟶ Y) ≅ X | equalizer.iso_source_of_self 0 | def | category_theory.limits.kernel_zero_iso_source | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The kernel of a zero morphism is isomorphic to the source. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_zero_iso_source_hom :
kernel_zero_iso_source.hom = kernel.ι (0 : X ⟶ Y) | rfl | lemma | category_theory.limits.kernel_zero_iso_source_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_zero_iso_source_inv :
kernel_zero_iso_source.inv = kernel.lift (0 : X ⟶ Y) (𝟙 X) (by simp) | by { ext, simp [kernel_zero_iso_source], } | lemma | category_theory.limits.kernel_zero_iso_source_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_iso_of_eq {f g : X ⟶ Y} [has_kernel f] [has_kernel g] (h : f = g) :
kernel f ≅ kernel g | has_limit.iso_of_nat_iso (by simp[h]) | def | category_theory.limits.kernel_iso_of_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | If two morphisms are known to be equal, then their kernels are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_iso_of_eq_refl {h : f = f} : kernel_iso_of_eq h = iso.refl (kernel f) | by { ext, simp [kernel_iso_of_eq], } | lemma | category_theory.limits.kernel_iso_of_eq_refl | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_iso_of_eq_hom_comp_ι {f g : X ⟶ Y} [has_kernel f] [has_kernel g] (h : f = g) :
(kernel_iso_of_eq h).hom ≫ kernel.ι _ = kernel.ι _ | by { unfreezingI { induction h, simp } } | lemma | category_theory.limits.kernel_iso_of_eq_hom_comp_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_iso_of_eq_inv_comp_ι {f g : X ⟶ Y} [has_kernel f] [has_kernel g] (h : f = g) :
(kernel_iso_of_eq h).inv ≫ kernel.ι _ = kernel.ι _ | by { unfreezingI { induction h, simp } } | lemma | category_theory.limits.kernel_iso_of_eq_inv_comp_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_kernel_iso_of_eq_hom {Z} {f g : X ⟶ Y} [has_kernel f] [has_kernel g]
(h : f = g) (e : Z ⟶ X) (he) :
kernel.lift _ e he ≫ (kernel_iso_of_eq h).hom = kernel.lift _ e (by simp [← h, he]) | by { unfreezingI { induction h, simp } } | lemma | category_theory.limits.lift_comp_kernel_iso_of_eq_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_kernel_iso_of_eq_inv {Z} {f g : X ⟶ Y} [has_kernel f] [has_kernel g]
(h : f = g) (e : Z ⟶ X) (he) :
kernel.lift _ e he ≫ (kernel_iso_of_eq h).inv = kernel.lift _ e (by simp [h, he]) | by { unfreezingI { induction h, simp } } | lemma | category_theory.limits.lift_comp_kernel_iso_of_eq_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_iso_of_eq_trans {f g h : X ⟶ Y} [has_kernel f] [has_kernel g] [has_kernel h]
(w₁ : f = g) (w₂ : g = h) :
kernel_iso_of_eq w₁ ≪≫ kernel_iso_of_eq w₂ = kernel_iso_of_eq (w₁.trans w₂) | by { unfreezingI { induction w₁, induction w₂, }, ext, simp [kernel_iso_of_eq], } | lemma | category_theory.limits.kernel_iso_of_eq_trans | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_not_epi_of_nonzero (w : f ≠ 0) : ¬epi (kernel.ι f) | λ I, by exactI w (eq_zero_of_epi_kernel f) | lemma | category_theory.limits.kernel_not_epi_of_nonzero | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_not_iso_of_nonzero (w : f ≠ 0) : (is_iso (kernel.ι f)) → false | λ I, kernel_not_epi_of_nonzero w $ by { resetI, apply_instance } | lemma | category_theory.limits.kernel_not_iso_of_nonzero | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_kernel_comp_mono {X Y Z : C} (f : X ⟶ Y) [has_kernel f] (g : Y ⟶ Z) [mono g] :
has_kernel (f ≫ g) | ⟨⟨{ cone := _, is_limit := is_kernel_comp_mono (limit.is_limit _) g rfl }⟩⟩ | instance | category_theory.limits.has_kernel_comp_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_comp_mono {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [has_kernel f] [mono g] :
kernel (f ≫ g) ≅ kernel f | { hom := kernel.lift _ (kernel.ι _) (by { rw [←cancel_mono g], simp, }),
inv := kernel.lift _ (kernel.ι _) (by simp), } | def | category_theory.limits.kernel_comp_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | When `g` is a monomorphism, the kernel of `f ≫ g` is isomorphic to the kernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_kernel_iso_comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [is_iso f] [has_kernel g] :
has_kernel (f ≫ g) | { exists_limit :=
⟨{ cone := kernel_fork.of_ι (kernel.ι g ≫ inv f) (by simp),
is_limit := is_limit_aux _ (λ s, kernel.lift _ (s.ι ≫ f) (by tidy)) (by tidy)
(λ s m w, by { simp_rw [←w], ext, simp, }), }⟩ } | instance | category_theory.limits.has_kernel_iso_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel_is_iso_comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [is_iso f] [has_kernel g] :
kernel (f ≫ g) ≅ kernel g | { hom := kernel.lift _ (kernel.ι _ ≫ f) (by simp),
inv := kernel.lift _ (kernel.ι _ ≫ inv f) (by simp), } | def | category_theory.limits.kernel_is_iso_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | When `f` is an isomorphism, the kernel of `f ≫ g` is isomorphic to the kernel of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.zero_kernel_fork : kernel_fork f | { X := 0,
π := { app := λ j, 0 }} | def | category_theory.limits.kernel.zero_kernel_fork | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The morphism from the zero object determines a cone on a kernel diagram | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.is_limit_cone_zero_cone [mono f] : is_limit (kernel.zero_kernel_fork f) | fork.is_limit.mk _ (λ s, 0)
(λ s, by { erw zero_comp,
convert (zero_of_comp_mono f _).symm,
exact kernel_fork.condition _ })
(λ _ _ _, zero_of_to_zero _) | def | category_theory.limits.kernel.is_limit_cone_zero_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The map from the zero object is a kernel of a monomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.of_mono [has_kernel f] [mono f] : kernel f ≅ 0 | functor.map_iso (cones.forget _) $ is_limit.unique_up_to_iso
(limit.is_limit (parallel_pair f 0)) (kernel.is_limit_cone_zero_cone f) | def | category_theory.limits.kernel.of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The kernel of a monomorphism is isomorphic to the zero object | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.ι_of_mono [has_kernel f] [mono f] : kernel.ι f = 0 | zero_of_source_iso_zero _ (kernel.of_mono f) | lemma | category_theory.limits.kernel.ι_of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The kernel morphism of a monomorphism is a zero morphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_kernel_of_cancel_zero {X Y : C} (f : X ⟶ Y)
(hf : ∀ (Z : C) (g : Z ⟶ X) (hgf : g ≫ f = 0), g = 0) :
is_limit (kernel_fork.of_ι (0 : 0 ⟶ X) (show 0 ≫ f = 0, by simp)) | fork.is_limit.mk _ (λ s, 0)
(λ s, by rw [hf _ _ (kernel_fork.condition s), zero_comp])
(λ s m h, by ext) | def | category_theory.limits.zero_kernel_of_cancel_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | If `g ≫ f = 0` implies `g = 0` for all `g`, then `0 : 0 ⟶ X` is a kernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_kernel.of_comp_iso {Z : C} (l : X ⟶ Z) (i : Z ≅ Y) (h : l ≫ i.hom = f)
{s : kernel_fork f} (hs : is_limit s) : is_limit (kernel_fork.of_ι (fork.ι s) $
show fork.ι s ≫ l = 0, by simp [←i.comp_inv_eq.2 h.symm]) | fork.is_limit.mk _
(λ s, hs.lift $ kernel_fork.of_ι (fork.ι s) $ by simp [←h])
(λ s, by simp)
(λ s m h, by { apply fork.is_limit.hom_ext hs, simpa using h }) | def | category_theory.limits.is_kernel.of_comp_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | If `i` is an isomorphism such that `l ≫ i.hom = f`, then any kernel of `f` is a kernel of `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.of_comp_iso [has_kernel f]
{Z : C} (l : X ⟶ Z) (i : Z ≅ Y) (h : l ≫ i.hom = f) :
is_limit (kernel_fork.of_ι (kernel.ι f) $
show kernel.ι f ≫ l = 0, by simp [←i.comp_inv_eq.2 h.symm]) | is_kernel.of_comp_iso f l i h $ limit.is_limit _ | def | category_theory.limits.kernel.of_comp_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | If `i` is an isomorphism such that `l ≫ i.hom = f`, then the kernel of `f` is a kernel of `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_kernel.iso_kernel {Z : C} (l : Z ⟶ X) {s : kernel_fork f} (hs : is_limit s)
(i : Z ≅ s.X) (h : i.hom ≫ fork.ι s = l) : is_limit (kernel_fork.of_ι l $
show l ≫ f = 0, by simp [←h]) | is_limit.of_iso_limit hs $ cones.ext i.symm $ λ j,
by { cases j, { exact (iso.eq_inv_comp i).2 h }, { simp } } | def | category_theory.limits.is_kernel.iso_kernel | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | If `s` is any limit kernel cone over `f` and if `i` is an isomorphism such that
`i.hom ≫ s.ι = l`, then `l` is a kernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.iso_kernel [has_kernel f]
{Z : C} (l : Z ⟶ X) (i : Z ≅ kernel f) (h : i.hom ≫ kernel.ι f = l) :
is_limit (kernel_fork.of_ι l $ by simp [←h]) | is_kernel.iso_kernel f l (limit.is_limit _) i h | def | category_theory.limits.kernel.iso_kernel | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | If `i` is an isomorphism such that `i.hom ≫ kernel.ι f = l`, then `l` is a kernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.ι_of_zero : is_iso (kernel.ι (0 : X ⟶ Y)) | equalizer.ι_of_self _ | lemma | category_theory.limits.kernel.ι_of_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The kernel morphism of a zero morphism is an isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_cofork | cofork f 0 | abbreviation | category_theory.limits.cokernel_cofork | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | A cokernel cofork is just a cofork where the second morphism is a zero morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_cofork.condition (s : cokernel_cofork f) : f ≫ s.π = 0 | by rw [cofork.condition, zero_comp] | lemma | category_theory.limits.cokernel_cofork.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_cofork.π_eq_zero (s : cokernel_cofork f) : s.ι.app zero = 0 | by simp [cofork.app_zero_eq_comp_π_right] | lemma | category_theory.limits.cokernel_cofork.π_eq_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_cofork.of_π {Z : C} (π : Y ⟶ Z) (w : f ≫ π = 0) : cokernel_cofork f | cofork.of_π π $ by rw [w, zero_comp] | abbreviation | category_theory.limits.cokernel_cofork.of_π | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | A morphism `π` satisfying `f ≫ π = 0` determines a cokernel cofork on `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_cofork.π_of_π {X Y P : C} (f : X ⟶ Y) (π : Y ⟶ P) (w : f ≫ π = 0) :
cofork.π (cokernel_cofork.of_π π w) = π | rfl | lemma | category_theory.limits.cokernel_cofork.π_of_π | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_of_π (s : cofork f 0) : s ≅ cofork.of_π (cofork.π s) (cofork.condition s) | cocones.ext (iso.refl _) $ λ j, by cases j; tidy | def | category_theory.limits.iso_of_π | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | Every cokernel cofork `s` is isomorphic (actually, equal) to `cofork.of_π (cofork.π s) _`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_π_congr {P : C} {π π' : Y ⟶ P} {w : f ≫ π = 0} (h : π = π') :
cokernel_cofork.of_π π w ≅ cokernel_cofork.of_π π' (by rw [←h, w]) | cocones.ext (iso.refl _) $ λ j, by cases j; tidy | def | category_theory.limits.of_π_congr | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | If `π = π'`, then `cokernel_cofork.of_π π _` and `cokernel_cofork.of_π π' _` are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_cofork.is_colimit.desc' {s : cokernel_cofork f} (hs : is_colimit s) {W : C} (k : Y ⟶ W)
(h : f ≫ k = 0) : {l : s.X ⟶ W // cofork.π s ≫ l = k} | ⟨hs.desc $ cokernel_cofork.of_π _ h, hs.fac _ _⟩ | def | category_theory.limits.cokernel_cofork.is_colimit.desc' | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | If `s` is a colimit cokernel cofork, then every `k : Y ⟶ W` satisfying `f ≫ k = 0` induces
`l : s.X ⟶ W` such that `cofork.π s ≫ l = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_aux (t : cokernel_cofork f)
(desc : Π (s : cokernel_cofork f), t.X ⟶ s.X)
(fac : ∀ (s : cokernel_cofork f), t.π ≫ desc s = s.π)
(uniq : ∀ (s : cokernel_cofork f) (m : t.X ⟶ s.X) (w : t.π ≫ m = s.π), m = desc s) :
is_colimit t | { desc := desc,
fac' := λ s j, by { cases j, { simp, }, { exact fac s, }, },
uniq' := λ s m w, uniq s m (w limits.walking_parallel_pair.one), } | def | category_theory.limits.is_colimit_aux | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | This is a slightly more convenient method to verify that a cokernel cofork is a colimit cocone.
It only asks for a proof of facts that carry any mathematical content | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_cofork.is_colimit.of_π {Z : C} (g : Y ⟶ Z) (eq : f ≫ g = 0)
(desc : Π {Z' : C} (g' : Y ⟶ Z') (eq' : f ≫ g' = 0), Z ⟶ Z')
(fac : ∀ {Z' : C} (g' : Y ⟶ Z') (eq' : f ≫ g' = 0), g ≫ desc g' eq' = g')
(uniq :
∀ {Z' : C} (g' : Y ⟶ Z') (eq' : f ≫ g' = 0) (m : Z ⟶ Z') (w : g ≫ m = g'), m = desc g' eq') :
is... | is_colimit_aux _ (λ s, desc s.π s.condition) (λ s, fac s.π s.condition) (λ s, uniq s.π s.condition) | def | category_theory.limits.cokernel_cofork.is_colimit.of_π | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | This is a more convenient formulation to show that a `cokernel_cofork` constructed using
`cokernel_cofork.of_π` is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_cokernel_epi_comp {c : cokernel_cofork f} (i : is_colimit c) {W} (g : W ⟶ X) [hg : epi g]
{h : W ⟶ Y} (hh : h = g ≫ f) :
is_colimit (cokernel_cofork.of_π c.π (by rw [hh]; simp) : cokernel_cofork h) | cofork.is_colimit.mk' _ $ λ s,
let s' : cokernel_cofork f := cofork.of_π s.π
(by { apply hg.left_cancellation, rw [←category.assoc, ←hh, s.condition], simp }) in
let l := cokernel_cofork.is_colimit.desc' i s'.π s'.condition in
⟨l.1, l.2,
λ m hm, by apply cofork.is_colimit.hom_ext i; rw cofork.π_of_π at hm... | def | category_theory.limits.is_cokernel_epi_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | Every cokernel of `f` induces a cokernel of `g ≫ f` if `g` is epi. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_cokernel_epi_comp_desc {c : cokernel_cofork f} (i : is_colimit c) {W}
(g : W ⟶ X) [hg : epi g] {h : W ⟶ Y} (hh : h = g ≫ f) (s : cokernel_cofork h) :
(is_cokernel_epi_comp i g hh).desc s
= i.desc (cofork.of_π s.π (by { rw [←cancel_epi g, ←category.assoc, ←hh], simp })) | rfl | lemma | category_theory.limits.is_cokernel_epi_comp_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cokernel_of_comp {W : C} (g : W ⟶ X) (h : W ⟶ Y) {c : cokernel_cofork h} (i : is_colimit c)
(hf : f ≫ c.π = 0) (hfg : g ≫ f = h) : is_colimit (cokernel_cofork.of_π c.π hf) | cofork.is_colimit.mk _
(λ s, i.desc (cokernel_cofork.of_π s.π (by simp [← hfg])))
(λ s, by simp only [cokernel_cofork.π_of_π, cofork.is_colimit.π_desc])
(λ s m h, by { apply cofork.is_colimit.hom_ext i, simpa using h }) | def | category_theory.limits.is_cokernel_of_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | Every cokernel of `g ≫ f` is also a cokernel of `f`, as long as `f ≫ c.π` vanishes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel : C | coequalizer f 0 | abbreviation | category_theory.limits.cokernel | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The cokernel of a morphism, expressed as the coequalizer with the 0 morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel.π : Y ⟶ cokernel f | coequalizer.π f 0 | abbreviation | category_theory.limits.cokernel.π | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The map from the target of `f` to `cokernel f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coequalizer_as_cokernel : coequalizer.π f 0 = cokernel.π f | rfl | lemma | category_theory.limits.coequalizer_as_cokernel | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel.condition : f ≫ cokernel.π f = 0 | cokernel_cofork.condition _ | lemma | category_theory.limits.cokernel.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_is_cokernel :
is_colimit (cofork.of_π (cokernel.π f) ((cokernel.condition f).trans (zero_comp.symm))) | is_colimit.of_iso_colimit (colimit.is_colimit _) (cofork.ext (iso.refl _) (by tidy)) | def | category_theory.limits.cokernel_is_cokernel | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The cokernel built from `cokernel.π f` is colimiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel.desc {W : C} (k : Y ⟶ W) (h : f ≫ k = 0) : cokernel f ⟶ W | (cokernel_is_cokernel f).desc (cokernel_cofork.of_π k h) | abbreviation | category_theory.limits.cokernel.desc | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | Given any morphism `k : Y ⟶ W` such that `f ≫ k = 0`, `k` factors through `cokernel.π f`
via `cokernel.desc : cokernel f ⟶ W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel.π_desc {W : C} (k : Y ⟶ W) (h : f ≫ k = 0) :
cokernel.π f ≫ cokernel.desc f k h = k | (cokernel_is_cokernel f).fac (cokernel_cofork.of_π k h) walking_parallel_pair.one | lemma | category_theory.limits.cokernel.π_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel.desc_zero {W : C} {h} : cokernel.desc f (0 : Y ⟶ W) h = 0 | by { ext, simp, } | lemma | category_theory.limits.cokernel.desc_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel.desc_epi
{W : C} (k : Y ⟶ W) (h : f ≫ k = 0) [epi k] : epi (cokernel.desc f k h) | ⟨λ Z g g' w,
begin
replace w := cokernel.π f ≫= w,
simp only [cokernel.π_desc_assoc] at w,
exact (cancel_epi k).1 w,
end⟩ | instance | category_theory.limits.cokernel.desc_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel.desc' {W : C} (k : Y ⟶ W) (h : f ≫ k = 0) :
{l : cokernel f ⟶ W // cokernel.π f ≫ l = k} | ⟨cokernel.desc f k h, cokernel.π_desc _ _ _⟩ | def | category_theory.limits.cokernel.desc' | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | Any morphism `k : Y ⟶ W` satisfying `f ≫ k = 0` induces `l : cokernel f ⟶ W` such that
`cokernel.π f ≫ l = k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel.map {X' Y' : C} (f' : X' ⟶ Y') [has_cokernel f']
(p : X ⟶ X') (q : Y ⟶ Y') (w : f ≫ q = p ≫ f') : cokernel f ⟶ cokernel f' | cokernel.desc f (q ≫ cokernel.π f') (by simp [reassoc_of w]) | abbreviation | category_theory.limits.cokernel.map | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | A commuting square induces a morphism of cokernels. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel.map_desc {X Y Z X' Y' Z' : C}
(f : X ⟶ Y) [has_cokernel f] (g : Y ⟶ Z) (w : f ≫ g = 0)
(f' : X' ⟶ Y') [has_cokernel f'] (g' : Y' ⟶ Z') (w' : f' ≫ g' = 0)
(p : X ⟶ X') (q : Y ⟶ Y') (r : Z ⟶ Z') (h₁ : f ≫ q = p ≫ f') (h₂ : g ≫ r = q ≫ g') :
cokernel.map f f' p q h₁ ≫ cokernel.desc f' g' w' = cokernel.des... | by { ext, simp [h₂], } | lemma | category_theory.limits.cokernel.map_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | Given a commutative diagram
X --f--> Y --g--> Z
| | |
| | |
v v v
X' -f'-> Y' -g'-> Z'
with horizontal arrows composing to zero,
then we obtain a commutative square
cokernel f ---> Z
| |
| cokernel.map |
| |
v ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel.map_iso {X' Y' : C} (f' : X' ⟶ Y') [has_cokernel f']
(p : X ≅ X') (q : Y ≅ Y') (w : f ≫ q.hom = p.hom ≫ f') : cokernel f ≅ cokernel f' | { hom := cokernel.map f f' p.hom q.hom w,
inv := cokernel.map f' f p.inv q.inv (by { refine (cancel_mono q.hom).1 _, simp [w], }), } | def | category_theory.limits.cokernel.map_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | A commuting square of isomorphisms induces an isomorphism of cokernels. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel.π_zero_is_iso :
is_iso (cokernel.π (0 : X ⟶ Y)) | coequalizer.π_of_self _ | instance | category_theory.limits.cokernel.π_zero_is_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The cokernel of the zero morphism is an isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_of_mono_cokernel [mono (cokernel.π f)] : f = 0 | (cancel_mono (cokernel.π f)).1 (by simp) | lemma | category_theory.limits.eq_zero_of_mono_cokernel | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_zero_iso_target : cokernel (0 : X ⟶ Y) ≅ Y | coequalizer.iso_target_of_self 0 | def | category_theory.limits.cokernel_zero_iso_target | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The cokernel of a zero morphism is isomorphic to the target. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_zero_iso_target_hom :
cokernel_zero_iso_target.hom = cokernel.desc (0 : X ⟶ Y) (𝟙 Y) (by simp) | by { ext, simp [cokernel_zero_iso_target], } | lemma | category_theory.limits.cokernel_zero_iso_target_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_zero_iso_target_inv :
cokernel_zero_iso_target.inv = cokernel.π (0 : X ⟶ Y) | rfl | lemma | category_theory.limits.cokernel_zero_iso_target_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_iso_of_eq {f g : X ⟶ Y} [has_cokernel f] [has_cokernel g] (h : f = g) :
cokernel f ≅ cokernel g | has_colimit.iso_of_nat_iso (by simp[h]) | def | category_theory.limits.cokernel_iso_of_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | If two morphisms are known to be equal, then their cokernels are isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel_iso_of_eq_refl {h : f = f} : cokernel_iso_of_eq h = iso.refl (cokernel f) | by { ext, simp [cokernel_iso_of_eq], } | lemma | category_theory.limits.cokernel_iso_of_eq_refl | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
π_comp_cokernel_iso_of_eq_hom {f g : X ⟶ Y} [has_cokernel f] [has_cokernel g] (h : f = g) :
cokernel.π _ ≫ (cokernel_iso_of_eq h).hom = cokernel.π _ | by { unfreezingI { induction h, simp } } | lemma | category_theory.limits.π_comp_cokernel_iso_of_eq_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
π_comp_cokernel_iso_of_eq_inv {f g : X ⟶ Y} [has_cokernel f] [has_cokernel g] (h : f = g) :
cokernel.π _ ≫ (cokernel_iso_of_eq h).inv = cokernel.π _ | by { unfreezingI { induction h, simp } } | lemma | category_theory.limits.π_comp_cokernel_iso_of_eq_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_iso_of_eq_hom_comp_desc {Z} {f g : X ⟶ Y} [has_cokernel f] [has_cokernel g]
(h : f = g) (e : Y ⟶ Z) (he) :
(cokernel_iso_of_eq h).hom ≫ cokernel.desc _ e he = cokernel.desc _ e (by simp [h, he]) | by { unfreezingI { induction h, simp } } | lemma | category_theory.limits.cokernel_iso_of_eq_hom_comp_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_iso_of_eq_inv_comp_desc {Z} {f g : X ⟶ Y} [has_cokernel f] [has_cokernel g]
(h : f = g) (e : Y ⟶ Z) (he) :
(cokernel_iso_of_eq h).inv ≫ cokernel.desc _ e he = cokernel.desc _ e (by simp [← h, he]) | by { unfreezingI { induction h, simp } } | lemma | category_theory.limits.cokernel_iso_of_eq_inv_comp_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_iso_of_eq_trans {f g h : X ⟶ Y} [has_cokernel f] [has_cokernel g] [has_cokernel h]
(w₁ : f = g) (w₂ : g = h) :
cokernel_iso_of_eq w₁ ≪≫ cokernel_iso_of_eq w₂ = cokernel_iso_of_eq (w₁.trans w₂) | by { unfreezingI { induction w₁, induction w₂, }, ext, simp [cokernel_iso_of_eq], } | lemma | category_theory.limits.cokernel_iso_of_eq_trans | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_not_mono_of_nonzero (w : f ≠ 0) : ¬mono (cokernel.π f) | λ I, by exactI w (eq_zero_of_mono_cokernel f) | lemma | category_theory.limits.cokernel_not_mono_of_nonzero | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_not_iso_of_nonzero (w : f ≠ 0) : (is_iso (cokernel.π f)) → false | λ I, cokernel_not_mono_of_nonzero w $ by { resetI, apply_instance } | lemma | category_theory.limits.cokernel_not_iso_of_nonzero | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_cokernel_comp_iso {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [has_cokernel f] [is_iso g] :
has_cokernel (f ≫ g) | { exists_colimit :=
⟨{ cocone := cokernel_cofork.of_π (inv g ≫ cokernel.π f) (by simp),
is_colimit := is_colimit_aux _ (λ s, cokernel.desc _ (g ≫ s.π)
(by { rw [←category.assoc, cokernel_cofork.condition], }))
(by tidy)
(λ s m w, by { simp_rw [←w], ext, simp, }), }⟩ } | instance | category_theory.limits.has_cokernel_comp_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_comp_is_iso {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [has_cokernel f] [is_iso g] :
cokernel (f ≫ g) ≅ cokernel f | { hom := cokernel.desc _ (inv g ≫ cokernel.π f) (by simp),
inv := cokernel.desc _ (g ≫ cokernel.π (f ≫ g)) (by rw [←category.assoc, cokernel.condition]), } | def | category_theory.limits.cokernel_comp_is_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | When `g` is an isomorphism, the cokernel of `f ≫ g` is isomorphic to the cokernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_cokernel_epi_comp {X Y : C} (f : X ⟶ Y) [has_cokernel f] {W} (g : W ⟶ X) [epi g] :
has_cokernel (g ≫ f) | ⟨⟨{ cocone := _, is_colimit := is_cokernel_epi_comp (colimit.is_colimit _) g rfl }⟩⟩ | instance | category_theory.limits.has_cokernel_epi_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_epi_comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [epi f] [has_cokernel g] :
cokernel (f ≫ g) ≅ cokernel g | { hom := cokernel.desc _ (cokernel.π g) (by simp),
inv := cokernel.desc _ (cokernel.π (f ≫ g)) (by { rw [←cancel_epi f, ←category.assoc], simp, }), } | def | category_theory.limits.cokernel_epi_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | When `f` is an epimorphism, the cokernel of `f ≫ g` is isomorphic to the cokernel of `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel.zero_cokernel_cofork : cokernel_cofork f | { X := 0,
ι := { app := λ j, 0 } } | def | category_theory.limits.cokernel.zero_cokernel_cofork | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The morphism to the zero object determines a cocone on a cokernel diagram | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel.is_colimit_cocone_zero_cocone [epi f] :
is_colimit (cokernel.zero_cokernel_cofork f) | cofork.is_colimit.mk _ (λ s, 0)
(λ s, by { erw zero_comp,
convert (zero_of_epi_comp f _).symm,
exact cokernel_cofork.condition _ })
(λ _ _ _, zero_of_from_zero _) | def | category_theory.limits.cokernel.is_colimit_cocone_zero_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The morphism to the zero object is a cokernel of an epimorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel.of_epi [has_cokernel f] [epi f] : cokernel f ≅ 0 | functor.map_iso (cocones.forget _) $ is_colimit.unique_up_to_iso
(colimit.is_colimit (parallel_pair f 0)) (cokernel.is_colimit_cocone_zero_cocone f) | def | category_theory.limits.cokernel.of_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The cokernel of an epimorphism is isomorphic to the zero object | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel.π_of_epi [has_cokernel f] [epi f] : cokernel.π f = 0 | zero_of_target_iso_zero _ (cokernel.of_epi f) | lemma | category_theory.limits.cokernel.π_of_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The cokernel morphism of an epimorphism is a zero morphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_factorisation.kernel_ι_comp [has_kernel f] (F : mono_factorisation f) :
kernel.ι f ≫ F.e = 0 | by rw [← cancel_mono F.m, zero_comp, category.assoc, F.fac, kernel.condition] | lemma | category_theory.limits.mono_factorisation.kernel_ι_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_image_ι {X Y : C} (f : X ⟶ Y)
[has_image f] [has_cokernel (image.ι f)] [has_cokernel f] [epi (factor_thru_image f)] :
cokernel (image.ι f) ≅ cokernel f | { hom := cokernel.desc _ (cokernel.π f)
begin
have w := cokernel.condition f,
conv at w { to_lhs, congr, rw ←image.fac f, },
rw [←has_zero_morphisms.comp_zero (limits.factor_thru_image f), category.assoc, cancel_epi]
at w,
exact w,
end,
inv := cokernel.desc _ (cokernel.π _)
begin
conv ... | def | category_theory.limits.cokernel_image_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The cokernel of the image inclusion of a morphism `f` is isomorphic to the cokernel of `f`.
(This result requires that the factorisation through the image is an epimorphism.
This holds in any category with equalizers.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel.π_of_zero :
is_iso (cokernel.π (0 : X ⟶ Y)) | coequalizer.π_of_self _ | lemma | category_theory.limits.cokernel.π_of_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The cokernel of a zero morphism is an isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel.of_cokernel_of_epi [has_cokernel f]
[has_kernel (cokernel.π f)] [epi f] : is_iso (kernel.ι (cokernel.π f)) | equalizer.ι_of_eq $ cokernel.π_of_epi f | instance | category_theory.limits.kernel.of_cokernel_of_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The kernel of the cokernel of an epimorphism is an isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel.of_kernel_of_mono [has_kernel f]
[has_cokernel (kernel.ι f)] [mono f] : is_iso (cokernel.π (kernel.ι f)) | coequalizer.π_of_eq $ kernel.ι_of_mono f | instance | category_theory.limits.cokernel.of_kernel_of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The cokernel of the kernel of a monomorphism is an isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_cokernel_of_zero_cancel {X Y : C} (f : X ⟶ Y)
(hf : ∀ (Z : C) (g : Y ⟶ Z) (hgf : f ≫ g = 0), g = 0) :
is_colimit (cokernel_cofork.of_π (0 : Y ⟶ 0) (show f ≫ 0 = 0, by simp)) | cofork.is_colimit.mk _ (λ s, 0)
(λ s, by rw [hf _ _ (cokernel_cofork.condition s), comp_zero])
(λ s m h, by ext) | def | category_theory.limits.zero_cokernel_of_zero_cancel | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | If `f ≫ g = 0` implies `g = 0` for all `g`, then `0 : Y ⟶ 0` is a cokernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_cokernel.of_iso_comp {Z : C} (l : Z ⟶ Y) (i : X ≅ Z) (h : i.hom ≫ l = f)
{s : cokernel_cofork f} (hs : is_colimit s) : is_colimit (cokernel_cofork.of_π (cofork.π s) $
show l ≫ cofork.π s = 0, by simp [i.eq_inv_comp.2 h]) | cofork.is_colimit.mk _
(λ s, hs.desc $ cokernel_cofork.of_π (cofork.π s) $ by simp [←h])
(λ s, by simp)
(λ s m h, by { apply cofork.is_colimit.hom_ext hs, simpa using h }) | def | category_theory.limits.is_cokernel.of_iso_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | If `i` is an isomorphism such that `i.hom ≫ l = f`, then any cokernel of `f` is a cokernel of
`l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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