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cokernel.of_iso_comp [has_cokernel f]
{Z : C} (l : Z ⟶ Y) (i : X ≅ Z) (h : i.hom ≫ l = f) :
is_colimit (cokernel_cofork.of_π (cokernel.π f) $
show l ≫ cokernel.π f = 0, by simp [i.eq_inv_comp.2 h]) | is_cokernel.of_iso_comp f l i h $ colimit.is_colimit _ | def | category_theory.limits.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 the cokernel of `f` is a cokernel of
`l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_cokernel.cokernel_iso {Z : C} (l : Y ⟶ Z) {s : cokernel_cofork f} (hs : is_colimit s)
(i : s.X ≅ Z) (h : cofork.π s ≫ i.hom = l) : is_colimit (cokernel_cofork.of_π l $
show f ≫ l = 0, by simp [←h]) | is_colimit.of_iso_colimit hs $ cocones.ext i $ λ j, by { cases j, { simp }, { exact h } } | def | category_theory.limits.is_cokernel.cokernel_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | If `s` is any colimit cokernel cocone over `f` and `i` is an isomorphism such that
`s.π ≫ i.hom = l`, then `l` is a cokernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cokernel.cokernel_iso [has_cokernel f]
{Z : C} (l : Y ⟶ Z) (i : cokernel f ≅ Z) (h : cokernel.π f ≫ i.hom = l) :
is_colimit (cokernel_cofork.of_π l $ by simp [←h]) | is_cokernel.cokernel_iso f l (colimit.is_colimit _) i h | def | category_theory.limits.cokernel.cokernel_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 `cokernel.π f ≫ i.hom = l`, then `l` is a cokernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_comparison [has_kernel f] [has_kernel (G.map f)] :
G.obj (kernel f) ⟶ kernel (G.map f) | kernel.lift _ (G.map (kernel.ι f)) (by simp only [←G.map_comp, kernel.condition, functor.map_zero]) | def | category_theory.limits.kernel_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The comparison morphism for the kernel of `f`.
This is an isomorphism iff `G` preserves the kernel of `f`; see
`category_theory/limits/preserves/shapes/kernels.lean` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_comparison_comp_ι [has_kernel f] [has_kernel (G.map f)] :
kernel_comparison f G ≫ kernel.ι (G.map f) = G.map (kernel.ι f) | kernel.lift_ι _ _ _ | lemma | category_theory.limits.kernel_comparison_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 | |
map_lift_kernel_comparison [has_kernel f] [has_kernel (G.map f)]
{Z : C} {h : Z ⟶ X} (w : h ≫ f = 0) :
G.map (kernel.lift _ h w) ≫ kernel_comparison f G =
kernel.lift _ (G.map h) (by simp only [←G.map_comp, w, functor.map_zero]) | by { ext, simp [← G.map_comp] } | lemma | category_theory.limits.map_lift_kernel_comparison | 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_comparison_comp_kernel_map {X' Y' : C} [has_kernel f]
[has_kernel (G.map f)] (g : X' ⟶ Y') [has_kernel g] [has_kernel (G.map g)] (p : X ⟶ X')
(q : Y ⟶ Y') (hpq : f ≫ q = p ≫ g) :
kernel_comparison f G ≫ (kernel.map (G.map f) (G.map g) (G.map p) (G.map q)
(by rw [←G.map_comp, hpq, G.map_comp])) =
G.ma... | kernel.lift_map _ _ (by rw [←G.map_comp, kernel.condition, G.map_zero]) _ _
(by rw [←G.map_comp, kernel.condition, G.map_zero]) _ _ _
(by simp only [←G.map_comp]; exact G.congr_map (kernel.lift_ι _ _ _).symm) _ | lemma | category_theory.limits.kernel_comparison_comp_kernel_map | 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_comparison [has_cokernel f] [has_cokernel (G.map f)] :
cokernel (G.map f) ⟶ G.obj (cokernel f) | cokernel.desc _ (G.map (coequalizer.π _ _))
(by simp only [←G.map_comp, cokernel.condition, functor.map_zero]) | def | category_theory.limits.cokernel_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | The comparison morphism for the cokernel of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π_comp_cokernel_comparison [has_cokernel f] [has_cokernel (G.map f)] :
cokernel.π (G.map f) ≫ cokernel_comparison f G = G.map (cokernel.π _) | cokernel.π_desc _ _ _ | lemma | category_theory.limits.π_comp_cokernel_comparison | 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_comparison_map_desc [has_cokernel f] [has_cokernel (G.map f)]
{Z : C} {h : Y ⟶ Z} (w : f ≫ h = 0) :
cokernel_comparison f G ≫ G.map (cokernel.desc _ h w) =
cokernel.desc _ (G.map h) (by simp only [←G.map_comp, w, functor.map_zero]) | by { ext, simp [← G.map_comp] } | lemma | category_theory.limits.cokernel_comparison_map_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_map_comp_cokernel_comparison {X' Y' : C} [has_cokernel f]
[has_cokernel (G.map f)] (g : X' ⟶ Y') [has_cokernel g] [has_cokernel (G.map g)]
(p : X ⟶ X') (q : Y ⟶ Y') (hpq : f ≫ q = p ≫ g) :
cokernel.map (G.map f) (G.map g) (G.map p) (G.map q) (by rw [←G.map_comp, hpq, G.map_comp])
≫ cokernel_compariso... | cokernel.map_desc _ _ (by rw [←G.map_comp, cokernel.condition, G.map_zero]) _ _
(by rw [←G.map_comp, cokernel.condition, G.map_zero]) _ _ _ _
(by simp only [←G.map_comp]; exact G.congr_map (cokernel.π_desc _ _ _)) | lemma | category_theory.limits.cokernel_map_comp_cokernel_comparison | 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_kernels : Prop | (has_limit : Π {X Y : C} (f : X ⟶ Y), has_kernel f . tactic.apply_instance) | class | category_theory.limits.has_kernels | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | `has_kernels` represents the existence of kernels for every morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_cokernels : Prop | (has_colimit : Π {X Y : C} (f : X ⟶ Y), has_cokernel f . tactic.apply_instance) | class | category_theory.limits.has_cokernels | category_theory.limits.shapes | src/category_theory/limits/shapes/kernels.lean | [
"category_theory.limits.preserves.shapes.zero"
] | [] | `has_cokernels` represents the existence of cokernels for every morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_kernels_of_has_equalizers [has_equalizers C] : has_kernels C | {} | instance | category_theory.limits.has_kernels_of_has_equalizers | 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_cokernels_of_has_coequalizers [has_coequalizers C] : has_cokernels C | {} | instance | category_theory.limits.has_cokernels_of_has_coequalizers | 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_kernel_pair | is_pullback a b f f | abbreviation | category_theory.is_kernel_pair | category_theory.limits.shapes | src/category_theory/limits/shapes/kernel_pair.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.regular_mono"
] | [] | `is_kernel_pair f a b` expresses that `(a, b)` is a kernel pair for `f`, i.e. `a ≫ f = b ≫ f`
and the square
R → X
↓ ↓
X → Y
is a pullback square.
This is just an abbreviation for `is_pullback a b f f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_of_mono [mono f] : is_kernel_pair f (𝟙 _) (𝟙 _) | ⟨⟨rfl⟩, ⟨pullback_cone.is_limit_mk_id_id _⟩⟩ | lemma | category_theory.is_kernel_pair.id_of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/kernel_pair.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.regular_mono"
] | [] | If `f` is a monomorphism, then `(𝟙 _, 𝟙 _)` is a kernel pair for `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift' {S : C} (k : is_kernel_pair f a b) (p q : S ⟶ X) (w : p ≫ f = q ≫ f) :
{ t : S ⟶ R // t ≫ a = p ∧ t ≫ b = q } | pullback_cone.is_limit.lift' k.is_limit _ _ w | def | category_theory.is_kernel_pair.lift' | category_theory.limits.shapes | src/category_theory/limits/shapes/kernel_pair.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.regular_mono"
] | [] | Given a pair of morphisms `p`, `q` to `X` which factor through `f`, they factor through any kernel
pair of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cancel_right {f₁ : X ⟶ Y} {f₂ : Y ⟶ Z} (comm : a ≫ f₁ = b ≫ f₁)
(big_k : is_kernel_pair (f₁ ≫ f₂) a b) :
is_kernel_pair f₁ a b | { w := comm,
is_limit' := ⟨pullback_cone.is_limit_aux' _ $ λ s,
begin
let s' : pullback_cone (f₁ ≫ f₂) (f₁ ≫ f₂) :=
pullback_cone.mk s.fst s.snd (s.condition_assoc _),
refine ⟨big_k.is_limit.lift s',
big_k.is_limit.fac _ walking_cospan.left,
big_k.is_limit.fac _ walking_cospan.... | lemma | category_theory.is_kernel_pair.cancel_right | category_theory.limits.shapes | src/category_theory/limits/shapes/kernel_pair.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.regular_mono"
] | [
"comm"
] | If `(a,b)` is a kernel pair for `f₁ ≫ f₂` and `a ≫ f₁ = b ≫ f₁`, then `(a,b)` is a kernel pair for
just `f₁`.
That is, to show that `(a,b)` is a kernel pair for `f₁` it suffices to only show the square
commutes, rather than to additionally show it's a pullback. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cancel_right_of_mono {f₁ : X ⟶ Y} {f₂ : Y ⟶ Z} [mono f₂]
(big_k : is_kernel_pair (f₁ ≫ f₂) a b) :
is_kernel_pair f₁ a b | cancel_right (begin rw [← cancel_mono f₂, assoc, assoc, big_k.w] end) big_k | lemma | category_theory.is_kernel_pair.cancel_right_of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/kernel_pair.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.regular_mono"
] | [] | If `(a,b)` is a kernel pair for `f₁ ≫ f₂` and `f₂` is mono, then `(a,b)` is a kernel pair for
just `f₁`.
The converse of `comp_of_mono`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_of_mono {f₁ : X ⟶ Y} {f₂ : Y ⟶ Z} [mono f₂] (small_k : is_kernel_pair f₁ a b) :
is_kernel_pair (f₁ ≫ f₂) a b | { w := by rw [small_k.w_assoc],
is_limit' := ⟨pullback_cone.is_limit_aux' _ $ λ s,
begin
refine ⟨_, _, _, _⟩,
apply (pullback_cone.is_limit.lift' small_k.is_limit s.fst s.snd _).1,
rw [← cancel_mono f₂, assoc, s.condition, assoc],
apply (pullback_cone.is_limit.lift' small_k.is_limit s.fst s.snd _).2... | lemma | category_theory.is_kernel_pair.comp_of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/kernel_pair.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.regular_mono"
] | [] | If `(a,b)` is a kernel pair for `f₁` and `f₂` is mono, then `(a,b)` is a kernel pair for `f₁ ≫ f₂`.
The converse of `cancel_right_of_mono`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_coequalizer (k : is_kernel_pair f a b) [r : regular_epi f] :
is_colimit (cofork.of_π f k.w) | begin
let t := k.is_limit.lift (pullback_cone.mk _ _ r.w),
have ht : t ≫ a = r.left := k.is_limit.fac _ walking_cospan.left,
have kt : t ≫ b = r.right := k.is_limit.fac _ walking_cospan.right,
apply cofork.is_colimit.mk _ _ _ _,
{ intro s,
apply (cofork.is_colimit.desc' r.is_colimit s.π _).1,
rw [← ht... | def | category_theory.is_kernel_pair.to_coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/kernel_pair.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.regular_mono"
] | [] | If `(a,b)` is the kernel pair of `f`, and `f` is a coequalizer morphism for some parallel pair, then
`f` is a coequalizer morphism of `a` and `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback {X Y Z A : C} {g : Y ⟶ Z} {a₁ a₂ : A ⟶ Y}
(h : is_kernel_pair g a₁ a₂) (f : X ⟶ Z) [has_pullback f g] [has_pullback f (a₁ ≫ g)] :
is_kernel_pair (pullback.fst : pullback f g ⟶ X)
(pullback.map f _ f _ (𝟙 X) a₁ (𝟙 Z) (by simp) $ category.comp_id _)
(pullback.map _ _ _ _ (𝟙 X) a₂ (𝟙 Z) (by ... | begin
refine ⟨⟨_⟩, ⟨_⟩⟩,
{ rw [pullback.lift_fst, pullback.lift_fst] },
{ fapply pullback_cone.is_limit_aux',
intro s,
refine ⟨pullback.lift (s.fst ≫ pullback.fst)
(h.lift' (s.fst ≫ pullback.snd) (s.snd ≫ pullback.snd) _).1 _, _, _, _⟩,
{ simp_rw [category.assoc, ← pullback.condition, ← category... | lemma | category_theory.is_kernel_pair.pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/kernel_pair.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.regular_mono"
] | [] | If `a₁ a₂ : A ⟶ Y` is a kernel pair for `g : Y ⟶ Z`, then `a₁ ×[Z] X` and `a₂ ×[Z] X`
(`A ×[Z] X ⟶ Y ×[Z] X`) is a kernel pair for `Y ×[Z] X ⟶ X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono_of_is_iso_fst (h : is_kernel_pair f a b) [is_iso a] :
mono f | begin
obtain ⟨l, h₁, h₂⟩ := limits.pullback_cone.is_limit.lift' h.is_limit (𝟙 _) (𝟙 _) (by simp [h.w]),
rw [is_pullback.cone_fst, ← is_iso.eq_comp_inv, category.id_comp] at h₁,
rw [h₁, is_iso.inv_comp_eq, category.comp_id] at h₂,
constructor,
intros Z g₁ g₂ e,
obtain ⟨l', rfl, rfl⟩ := limits.pullback_cone... | lemma | category_theory.is_kernel_pair.mono_of_is_iso_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/kernel_pair.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.regular_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_of_mono (h : is_kernel_pair f a b) [mono f] :
is_iso a | begin
rw ← show _ = a, from (category.comp_id _).symm.trans ((is_kernel_pair.id_of_mono f)
.is_limit.cone_point_unique_up_to_iso_inv_comp h.is_limit walking_cospan.left),
apply_instance,
end | lemma | category_theory.is_kernel_pair.is_iso_of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/kernel_pair.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.regular_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_is_iso_of_mono [is_iso a] [mono f] : is_kernel_pair f a a | begin
delta is_kernel_pair,
convert_to is_pullback a (a ≫ 𝟙 X) (𝟙 X ≫ f) f,
{ rw category.comp_id }, { rw category.id_comp },
exact (is_pullback.of_horiz_is_iso ⟨rfl⟩).paste_vert (is_kernel_pair.id_of_mono f)
end | lemma | category_theory.is_kernel_pair.of_is_iso_of_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/kernel_pair.lean | [
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.comm_sq",
"category_theory.limits.shapes.regular_mono"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_multicospan {L R : Type w} (fst snd : R → L) : Type w
| left : L → walking_multicospan
| right : R → walking_multicospan | inductive | category_theory.limits.walking_multicospan | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The type underlying the multiequalizer diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_multispan {L R : Type w} (fst snd : L → R) : Type w
| left : L → walking_multispan
| right : R → walking_multispan | inductive | category_theory.limits.walking_multispan | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The type underlying the multiecoqualizer diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom : Π (a b : walking_multicospan fst snd), Type w
| id (A) : hom A A
| fst (b) : hom (left (fst b)) (right b)
| snd (b) : hom (left (snd b)) (right b) | inductive | category_theory.limits.walking_multicospan.hom | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | Morphisms for `walking_multicospan`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom.comp : Π {A B C : walking_multicospan fst snd} (f : hom A B) (g : hom B C),
hom A C | | _ _ _ (hom.id X) f := f
| _ _ _ (hom.fst b) (hom.id X) := hom.fst b
| _ _ _ (hom.snd b) (hom.id X) := hom.snd b | def | category_theory.limits.walking_multicospan.hom.comp | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | Composition of morphisms for `walking_multicospan`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom : Π (a b : walking_multispan fst snd), Type v
| id (A) : hom A A
| fst (a) : hom (left a) (right (fst a))
| snd (a) : hom (left a) (right (snd a)) | inductive | category_theory.limits.walking_multispan.hom | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | Morphisms for `walking_multispan`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom.comp : Π {A B C : walking_multispan fst snd} (f : hom A B) (g : hom B C),
hom A C | | _ _ _ (hom.id X) f := f
| _ _ _ (hom.fst a) (hom.id X) := hom.fst a
| _ _ _ (hom.snd a) (hom.id X) := hom.snd a | def | category_theory.limits.walking_multispan.hom.comp | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | Composition of morphisms for `walking_multispan`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multicospan_index (C : Type u) [category.{v} C] | (L R : Type w)
(fst_to snd_to : R → L)
(left : L → C)
(right : R → C)
(fst : Π b, left (fst_to b) ⟶ right b)
(snd : Π b, left (snd_to b) ⟶ right b) | structure | category_theory.limits.multicospan_index | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | This is a structure encapsulating the data necessary to define a `multicospan`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multispan_index (C : Type u) [category.{v} C] | (L R : Type w)
(fst_from snd_from : L → R)
(left : L → C)
(right : R → C)
(fst : Π a, left a ⟶ right (fst_from a))
(snd : Π a, left a ⟶ right (snd_from a)) | structure | category_theory.limits.multispan_index | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | This is a structure encapsulating the data necessary to define a `multispan`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multicospan : walking_multicospan I.fst_to I.snd_to ⥤ C | { obj := λ x,
match x with
| walking_multicospan.left a := I.left a
| walking_multicospan.right b := I.right b
end,
map := λ x y f,
match x, y, f with
| _, _, walking_multicospan.hom.id x := 𝟙 _
| _, _, walking_multicospan.hom.fst b := I.fst _
| _, _, walking_multicospan.hom.snd b := I.snd _
end,
... | def | category_theory.limits.multicospan_index.multicospan | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The multicospan associated to `I : multicospan_index`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multicospan_obj_left (a) :
I.multicospan.obj (walking_multicospan.left a) = I.left a | rfl | lemma | category_theory.limits.multicospan_index.multicospan_obj_left | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multicospan_obj_right (b) :
I.multicospan.obj (walking_multicospan.right b) = I.right b | rfl | lemma | category_theory.limits.multicospan_index.multicospan_obj_right | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multicospan_map_fst (b) :
I.multicospan.map (walking_multicospan.hom.fst b) = I.fst b | rfl | lemma | category_theory.limits.multicospan_index.multicospan_map_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multicospan_map_snd (b) :
I.multicospan.map (walking_multicospan.hom.snd b) = I.snd b | rfl | lemma | category_theory.limits.multicospan_index.multicospan_map_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_pi_map : ∏ I.left ⟶ ∏ I.right | pi.lift (λ b, pi.π I.left (I.fst_to b) ≫ I.fst b) | def | category_theory.limits.multicospan_index.fst_pi_map | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The induced map `∏ I.left ⟶ ∏ I.right` via `I.fst`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_pi_map : ∏ I.left ⟶ ∏ I.right | pi.lift (λ b, pi.π I.left (I.snd_to b) ≫ I.snd b) | def | category_theory.limits.multicospan_index.snd_pi_map | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The induced map `∏ I.left ⟶ ∏ I.right` via `I.snd`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_pi_map_π (b) : I.fst_pi_map ≫ pi.π I.right b = pi.π I.left _ ≫ I.fst b | by simp [fst_pi_map] | lemma | category_theory.limits.multicospan_index.fst_pi_map_π | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_pi_map_π (b) : I.snd_pi_map ≫ pi.π I.right b = pi.π I.left _ ≫ I.snd b | by simp [snd_pi_map] | lemma | category_theory.limits.multicospan_index.snd_pi_map_π | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallel_pair_diagram | parallel_pair I.fst_pi_map I.snd_pi_map | def | category_theory.limits.multicospan_index.parallel_pair_diagram | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | Taking the multiequalizer over the multicospan index is equivalent to taking the equalizer over
the two morphsims `∏ I.left ⇉ ∏ I.right`. This is the diagram of the latter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multispan : walking_multispan I.fst_from I.snd_from ⥤ C | { obj := λ x,
match x with
| walking_multispan.left a := I.left a
| walking_multispan.right b := I.right b
end,
map := λ x y f,
match x, y, f with
| _, _, walking_multispan.hom.id x := 𝟙 _
| _, _, walking_multispan.hom.fst b := I.fst _
| _, _, walking_multispan.hom.snd b := I.snd _
end,
map_id' :... | def | category_theory.limits.multispan_index.multispan | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The multispan associated to `I : multispan_index`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multispan_obj_left (a) :
I.multispan.obj (walking_multispan.left a) = I.left a | rfl | lemma | category_theory.limits.multispan_index.multispan_obj_left | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multispan_obj_right (b) :
I.multispan.obj (walking_multispan.right b) = I.right b | rfl | lemma | category_theory.limits.multispan_index.multispan_obj_right | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multispan_map_fst (a) :
I.multispan.map (walking_multispan.hom.fst a) = I.fst a | rfl | lemma | category_theory.limits.multispan_index.multispan_map_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multispan_map_snd (a) :
I.multispan.map (walking_multispan.hom.snd a) = I.snd a | rfl | lemma | category_theory.limits.multispan_index.multispan_map_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_sigma_map : ∐ I.left ⟶ ∐ I.right | sigma.desc (λ b, I.fst b ≫ sigma.ι _ (I.fst_from b)) | def | category_theory.limits.multispan_index.fst_sigma_map | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The induced map `∐ I.left ⟶ ∐ I.right` via `I.fst`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_sigma_map : ∐ I.left ⟶ ∐ I.right | sigma.desc (λ b, I.snd b ≫ sigma.ι _ (I.snd_from b)) | def | category_theory.limits.multispan_index.snd_sigma_map | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The induced map `∐ I.left ⟶ ∐ I.right` via `I.snd`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_fst_sigma_map (b) : sigma.ι I.left b ≫ I.fst_sigma_map = I.fst b ≫ sigma.ι I.right _ | by simp [fst_sigma_map] | lemma | category_theory.limits.multispan_index.ι_fst_sigma_map | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_snd_sigma_map (b) : sigma.ι I.left b ≫ I.snd_sigma_map = I.snd b ≫ sigma.ι I.right _ | by simp [snd_sigma_map] | lemma | category_theory.limits.multispan_index.ι_snd_sigma_map | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallel_pair_diagram | parallel_pair I.fst_sigma_map I.snd_sigma_map | abbreviation | category_theory.limits.multispan_index.parallel_pair_diagram | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | Taking the multicoequalizer over the multispan index is equivalent to taking the coequalizer over
the two morphsims `∐ I.left ⇉ ∐ I.right`. This is the diagram of the latter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multifork (I : multicospan_index C) | cone I.multicospan | abbreviation | category_theory.limits.multifork | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | A multifork is a cone over a multicospan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multicofork (I : multispan_index C) | cocone I.multispan | abbreviation | category_theory.limits.multicofork | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | A multicofork is a cocone over a multispan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι (a : I.L) : K.X ⟶ I.left a | K.π.app (walking_multicospan.left _) | def | category_theory.limits.multifork.ι | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The maps from the cone point of a multifork to the objects on the left. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
app_left_eq_ι (a) : K.π.app (walking_multicospan.left a) = K.ι a | rfl | lemma | category_theory.limits.multifork.app_left_eq_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_right_eq_ι_comp_fst (b) :
K.π.app (walking_multicospan.right b) = K.ι (I.fst_to b) ≫ I.fst b | by { rw ← K.w (walking_multicospan.hom.fst b), refl } | lemma | category_theory.limits.multifork.app_right_eq_ι_comp_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
app_right_eq_ι_comp_snd (b) :
K.π.app (walking_multicospan.right b) = K.ι (I.snd_to b) ≫ I.snd b | by { rw ← K.w (walking_multicospan.hom.snd b), refl } | lemma | category_theory.limits.multifork.app_right_eq_ι_comp_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_comp_ι (K₁ K₂ : multifork I) (f : K₁ ⟶ K₂) (j : I.L) :
f.hom ≫ K₂.ι j = K₁.ι j | f.w (walking_multicospan.left j) | lemma | category_theory.limits.multifork.hom_comp_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_ι (I : multicospan_index C) (P : C) (ι : Π a, P ⟶ I.left a)
(w : ∀ b, ι (I.fst_to b) ≫ I.fst b = ι (I.snd_to b) ≫ I.snd b) :
multifork I | { X := P,
π :=
{ app := λ x,
match x with
| walking_multicospan.left a := ι _
| walking_multicospan.right b := ι (I.fst_to b) ≫ I.fst b
end,
naturality' := begin
rintros (_|_) (_|_) (_|_|_),
any_goals { symmetry, dsimp, rw category.id_comp, apply category.comp_id },
{ dsimp, rw... | def | category_theory.limits.multifork.of_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | Construct a multifork using a collection `ι` of morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
condition (b) :
K.ι (I.fst_to b) ≫ I.fst b = K.ι (I.snd_to b) ≫ I.snd b | by rw [←app_right_eq_ι_comp_fst, ←app_right_eq_ι_comp_snd] | lemma | category_theory.limits.multifork.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit.mk
(lift : Π (E : multifork I), E.X ⟶ K.X)
(fac : ∀ (E : multifork I) (i : I.L), lift E ≫ K.ι i = E.ι i)
(uniq : ∀ (E : multifork I) (m : E.X ⟶ K.X),
(∀ i : I.L, m ≫ K.ι i = E.ι i) → m = lift E) : is_limit K | { lift := lift,
fac' := begin
rintros E (a|b),
{ apply fac },
{ rw [← E.w (walking_multicospan.hom.fst b), ← K.w (walking_multicospan.hom.fst b),
← category.assoc],
congr' 1,
apply fac }
end,
uniq' := begin
rintros E m hm,
apply uniq,
intros i,
apply hm,
end } | def | category_theory.limits.multifork.is_limit.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [
"lift"
] | This definition provides a convenient way to show that a multifork is a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_condition : pi.lift K.ι ≫ I.fst_pi_map = pi.lift K.ι ≫ I.snd_pi_map | by { ext, discrete_cases, simp, } | lemma | category_theory.limits.multifork.pi_condition | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_pi_fork (K : multifork I) : fork I.fst_pi_map I.snd_pi_map | { X := K.X,
π :=
{ app := λ x,
match x with
| walking_parallel_pair.zero := pi.lift K.ι
| walking_parallel_pair.one := pi.lift K.ι ≫ I.fst_pi_map
end,
naturality' :=
begin
rintros (_|_) (_|_) (_|_|_),
any_goals { symmetry, dsimp, rw category.id_comp, apply category.comp_id },
... | def | category_theory.limits.multifork.to_pi_fork | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | Given a multifork, we may obtain a fork over `∏ I.left ⇉ ∏ I.right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_pi_fork_π_app_zero : K.to_pi_fork.ι = pi.lift K.ι | rfl | lemma | category_theory.limits.multifork.to_pi_fork_π_app_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_pi_fork_π_app_one :
K.to_pi_fork.π.app walking_parallel_pair.one = pi.lift K.ι ≫ I.fst_pi_map | rfl | lemma | category_theory.limits.multifork.to_pi_fork_π_app_one | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_pi_fork (c : fork I.fst_pi_map I.snd_pi_map) : multifork I | { X := c.X,
π :=
{ app := λ x,
match x with
| walking_multicospan.left a := c.ι ≫ pi.π _ _
| walking_multicospan.right b := c.ι ≫ I.fst_pi_map ≫ pi.π _ _
end,
naturality' :=
begin
rintros (_|_) (_|_) (_|_|_),
any_goals { symmetry, dsimp, rw category.id_comp, apply category.comp_i... | def | category_theory.limits.multifork.of_pi_fork | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | Given a fork over `∏ I.left ⇉ ∏ I.right`, we may obtain a multifork. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_pi_fork_π_app_left (c : fork I.fst_pi_map I.snd_pi_map) (a) :
(of_pi_fork I c).ι a = c.ι ≫ pi.π _ _ | rfl | lemma | category_theory.limits.multifork.of_pi_fork_π_app_left | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_pi_fork_π_app_right (c : fork I.fst_pi_map I.snd_pi_map) (a) :
(of_pi_fork I c).π.app (walking_multicospan.right a) = c.ι ≫ I.fst_pi_map ≫ pi.π _ _ | rfl | lemma | category_theory.limits.multifork.of_pi_fork_π_app_right | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_pi_fork_functor : multifork I ⥤ fork I.fst_pi_map I.snd_pi_map | { obj := multifork.to_pi_fork,
map := λ K₁ K₂ f,
{ hom := f.hom,
w' := begin
rintro (_|_),
{ ext, dsimp, simp },
{ ext,
simp only [multifork.to_pi_fork_π_app_one, multifork.pi_condition, category.assoc],
dsimp [snd_pi_map],
simp },
end } } | def | category_theory.limits.multicospan_index.to_pi_fork_functor | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | `multifork.to_pi_fork` is functorial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_pi_fork_functor : fork I.fst_pi_map I.snd_pi_map ⥤ multifork I | { obj := multifork.of_pi_fork I, map := λ K₁ K₂ f, { hom := f.hom, w' := by rintros (_|_); simp } } | def | category_theory.limits.multicospan_index.of_pi_fork_functor | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | `multifork.of_pi_fork` is functorial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multifork_equiv_pi_fork : multifork I ≌ fork I.fst_pi_map I.snd_pi_map | { functor := to_pi_fork_functor I,
inverse := of_pi_fork_functor I,
unit_iso := nat_iso.of_components (λ K, cones.ext (iso.refl _)
(by { rintros (_|_); dsimp; simp[←fork.app_one_eq_ι_comp_left, -fork.app_one_eq_ι_comp_left] }))
(λ K₁ K₂ f, by { ext, simp }),
counit_iso := nat_iso.of_components (λ K, fork.... | def | category_theory.limits.multicospan_index.multifork_equiv_pi_fork | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The category of multiforks is equivalent to the category of forks over `∏ I.left ⇉ ∏ I.right`.
It then follows from `category_theory.is_limit_of_preserves_cone_terminal` (or `reflects`) that it
preserves and reflects limit cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π (b : I.R) : I.right b ⟶ K.X | K.ι.app (walking_multispan.right _) | def | category_theory.limits.multicofork.π | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The maps to the cocone point of a multicofork from the objects on the right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π_eq_app_right (b) : K.ι.app (walking_multispan.right _) = K.π b | rfl | lemma | category_theory.limits.multicofork.π_eq_app_right | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_app_right (a) :
K.ι.app (walking_multispan.left a) = I.fst a ≫ K.π _ | by { rw ← K.w (walking_multispan.hom.fst a), refl } | lemma | category_theory.limits.multicofork.fst_app_right | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_app_right (a) :
K.ι.app (walking_multispan.left a) = I.snd a ≫ K.π _ | by { rw ← K.w (walking_multispan.hom.snd a), refl } | lemma | category_theory.limits.multicofork.snd_app_right | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_π (I : multispan_index C) (P : C) (π : Π b, I.right b ⟶ P)
(w : ∀ a, I.fst a ≫ π (I.fst_from a) = I.snd a ≫ π (I.snd_from a)) :
multicofork I | { X := P,
ι :=
{ app := λ x,
match x with
| walking_multispan.left a := I.fst a ≫ π _
| walking_multispan.right b := π _
end,
naturality' := begin
rintros (_|_) (_|_) (_|_|_),
any_goals { dsimp, rw category.comp_id, apply category.id_comp },
{ dsimp, rw category.comp_id, refl }... | def | category_theory.limits.multicofork.of_π | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | Construct a multicofork using a collection `π` of morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
condition (a) : I.fst a ≫ K.π (I.fst_from a) = I.snd a ≫ K.π (I.snd_from a) | by rw [←K.snd_app_right, ←K.fst_app_right] | lemma | category_theory.limits.multicofork.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit.mk
(desc : Π (E : multicofork I), K.X ⟶ E.X)
(fac : ∀ (E : multicofork I) (i : I.R), K.π i ≫ desc E = E.π i)
(uniq : ∀ (E : multicofork I) (m : K.X ⟶ E.X),
(∀ i : I.R, K.π i ≫ m = E.π i) → m = desc E) : is_colimit K | { desc := desc,
fac' := begin
rintros S (a|b),
{ rw [← K.w (walking_multispan.hom.fst a), ← S.w (walking_multispan.hom.fst a),
category.assoc],
congr' 1,
apply fac },
{ apply fac },
end,
uniq' := begin
intros S m hm,
apply uniq,
intros i,
apply hm
end } | def | category_theory.limits.multicofork.is_colimit.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | This definition provides a convenient way to show that a multicofork is a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_condition :
I.fst_sigma_map ≫ sigma.desc K.π = I.snd_sigma_map ≫ sigma.desc K.π | by { ext, discrete_cases, simp, } | lemma | category_theory.limits.multicofork.sigma_condition | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_sigma_cofork (K : multicofork I) : cofork I.fst_sigma_map I.snd_sigma_map | { X := K.X,
ι :=
{ app := λ x,
match x with
| walking_parallel_pair.zero := I.fst_sigma_map ≫ sigma.desc K.π
| walking_parallel_pair.one := sigma.desc K.π
end,
naturality' :=
begin
rintros (_|_) (_|_) (_|_|_),
any_goals { dsimp, rw category.comp_id, apply category.id_comp },
... | def | category_theory.limits.multicofork.to_sigma_cofork | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | Given a multicofork, we may obtain a cofork over `∐ I.left ⇉ ∐ I.right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_sigma_cofork_π : K.to_sigma_cofork.π = sigma.desc K.π | rfl | lemma | category_theory.limits.multicofork.to_sigma_cofork_π | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_sigma_cofork (c : cofork I.fst_sigma_map I.snd_sigma_map) : multicofork I | { X := c.X,
ι :=
{ app := λ x,
match x with
| walking_multispan.left a := (sigma.ι I.left a : _) ≫ I.fst_sigma_map ≫ c.π
| walking_multispan.right b := (sigma.ι I.right b : _) ≫ c.π
end,
naturality' :=
begin
rintros (_|_) (_|_) (_|_|_),
any_goals { dsimp, rw category.comp_id, app... | def | category_theory.limits.multicofork.of_sigma_cofork | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | Given a cofork over `∐ I.left ⇉ ∐ I.right`, we may obtain a multicofork. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_sigma_cofork_ι_app_left (c : cofork I.fst_sigma_map I.snd_sigma_map) (a) :
(of_sigma_cofork I c).ι.app (walking_multispan.left a) =
(sigma.ι I.left a : _) ≫ I.fst_sigma_map ≫ c.π | rfl | lemma | category_theory.limits.multicofork.of_sigma_cofork_ι_app_left | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_sigma_cofork_ι_app_right (c : cofork I.fst_sigma_map I.snd_sigma_map) (b) :
(of_sigma_cofork I c).ι.app (walking_multispan.right b) = (sigma.ι I.right b : _) ≫ c.π | rfl | lemma | category_theory.limits.multicofork.of_sigma_cofork_ι_app_right | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_sigma_cofork_functor : multicofork I ⥤ cofork I.fst_sigma_map I.snd_sigma_map | { obj := multicofork.to_sigma_cofork, map := λ K₁ K₂ f, { hom := f.hom } } | def | category_theory.limits.multispan_index.to_sigma_cofork_functor | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | `multicofork.to_sigma_cofork` is functorial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_sigma_cofork_functor : cofork I.fst_sigma_map I.snd_sigma_map ⥤ multicofork I | { obj := multicofork.of_sigma_cofork I,
map := λ K₁ K₂ f, { hom := f.hom, w' := by rintros (_|_); simp } } | def | category_theory.limits.multispan_index.of_sigma_cofork_functor | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | `multicofork.of_sigma_cofork` is functorial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multicofork_equiv_sigma_cofork : multicofork I ≌ cofork I.fst_sigma_map I.snd_sigma_map | { functor := to_sigma_cofork_functor I,
inverse := of_sigma_cofork_functor I,
unit_iso := nat_iso.of_components (λ K, cocones.ext (iso.refl _)
(by { rintros (_|_); dsimp; simp }))
(λ K₁ K₂ f, by { ext, simp }),
counit_iso := nat_iso.of_components (λ K, cofork.ext (iso.refl _)
(by { ext ⟨j⟩, dsimp,... | def | category_theory.limits.multispan_index.multicofork_equiv_sigma_cofork | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The category of multicoforks is equivalent to the category of coforks over `∐ I.left ⇉ ∐ I.right`.
It then follows from `category_theory.is_colimit_of_preserves_cocone_initial` (or `reflects`) that
it preserves and reflects colimit cocones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_multiequalizer (I : multicospan_index C) | has_limit I.multicospan | abbreviation | category_theory.limits.has_multiequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | For `I : multicospan_index C`, we say that it has a multiequalizer if the associated
multicospan has a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multiequalizer (I : multicospan_index C) [has_multiequalizer I] : C | limit I.multicospan | abbreviation | category_theory.limits.multiequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The multiequalizer of `I : multicospan_index C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_multicoequalizer (I : multispan_index C) | has_colimit I.multispan | abbreviation | category_theory.limits.has_multicoequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | For `I : multispan_index C`, we say that it has a multicoequalizer if
the associated multicospan has a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multicoequalizer (I : multispan_index C) [has_multicoequalizer I] : C | colimit I.multispan | abbreviation | category_theory.limits.multicoequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The multiecoqualizer of `I : multispan_index C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι (a : I.L) : multiequalizer I ⟶ I.left a | limit.π _ (walking_multicospan.left a) | abbreviation | category_theory.limits.multiequalizer.ι | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The canonical map from the multiequalizer to the objects on the left. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multifork : multifork I | limit.cone _ | abbreviation | category_theory.limits.multiequalizer.multifork | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The multifork associated to the multiequalizer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multifork_ι (a) :
(multiequalizer.multifork I).ι a = multiequalizer.ι I a | rfl | lemma | category_theory.limits.multiequalizer.multifork_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multifork_π_app_left (a) :
(multiequalizer.multifork I).π.app (walking_multicospan.left a) =
multiequalizer.ι I a | rfl | lemma | category_theory.limits.multiequalizer.multifork_π_app_left | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
condition (b) :
multiequalizer.ι I (I.fst_to b) ≫ I.fst b =
multiequalizer.ι I (I.snd_to b) ≫ I.snd b | multifork.condition _ _ | lemma | category_theory.limits.multiequalizer.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (W : C) (k : Π a, W ⟶ I.left a)
(h : ∀ b, k (I.fst_to b) ≫ I.fst b = k (I.snd_to b) ≫ I.snd b) :
W ⟶ multiequalizer I | limit.lift _ (multifork.of_ι I _ k h) | abbreviation | category_theory.limits.multiequalizer.lift | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [
"lift"
] | Construct a morphism to the multiequalizer from its universal property. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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