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mono_of_is_limit_parallel_family [nonempty J] {c : cone (parallel_family f)}
(i : is_limit c) :
mono (trident.ι c) | { right_cancellation := λ Z h k w, trident.is_limit.hom_ext i w } | lemma | category_theory.limits.mono_of_is_limit_parallel_family | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | The wide equalizer morphism in any limit cone is a monomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_wide_coequalizer | has_colimit (parallel_family f) | abbreviation | category_theory.limits.has_wide_coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | `has_wide_coequalizer f g` represents a particular choice of colimiting cocone
for the parallel family of morphisms `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_coequalizer : C | colimit (parallel_family f) | abbreviation | category_theory.limits.wide_coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | If a wide coequalizer of `f`, we can access an arbitrary choice of such by
saying `wide_coequalizer f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_coequalizer.π : Y ⟶ wide_coequalizer f | colimit.ι (parallel_family f) one | abbreviation | category_theory.limits.wide_coequalizer.π | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | If a wide_coequalizer of `f` exists, we can access the corresponding projection by
saying `wide_coequalizer.π f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_coequalizer.cotrident : cotrident f | colimit.cocone (parallel_family f) | abbreviation | category_theory.limits.wide_coequalizer.cotrident | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | An arbitrary choice of coequalizer cocone for a parallel family `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_coequalizer.cotrident_π :
(wide_coequalizer.cotrident f).π = wide_coequalizer.π f | rfl | lemma | category_theory.limits.wide_coequalizer.cotrident_π | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wide_coequalizer.cotrident_ι_app_one :
(wide_coequalizer.cotrident f).ι.app one = wide_coequalizer.π f | rfl | lemma | category_theory.limits.wide_coequalizer.cotrident_ι_app_one | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wide_coequalizer.condition (j₁ j₂ : J) :
f j₁ ≫ wide_coequalizer.π f = f j₂ ≫ wide_coequalizer.π f | cotrident.condition j₁ j₂ $ colimit.cocone $ parallel_family f | lemma | category_theory.limits.wide_coequalizer.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wide_coequalizer_is_wide_coequalizer [nonempty J] :
is_colimit (cotrident.of_π (wide_coequalizer.π f) (wide_coequalizer.condition f)) | is_colimit.of_iso_colimit (colimit.is_colimit _) (cotrident.ext (iso.refl _) (by tidy)) | def | category_theory.limits.wide_coequalizer_is_wide_coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | The cotrident built from `wide_coequalizer.π f` is colimiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_coequalizer.desc [nonempty J] {W : C} (k : Y ⟶ W)
(h : ∀ j₁ j₂, f j₁ ≫ k = f j₂ ≫ k) :
wide_coequalizer f ⟶ W | colimit.desc (parallel_family f) (cotrident.of_π k h) | abbreviation | category_theory.limits.wide_coequalizer.desc | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | Any morphism `k : Y ⟶ W` satisfying `∀ j₁ j₂, f j₁ ≫ k = f j₂ ≫ k` factors through the
wide coequalizer of `f` via `wide_coequalizer.desc : wide_coequalizer f ⟶ W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_coequalizer.π_desc [nonempty J] {W : C} (k : Y ⟶ W) (h : ∀ j₁ j₂, f j₁ ≫ k = f j₂ ≫ k) :
wide_coequalizer.π f ≫ wide_coequalizer.desc k h = k | colimit.ι_desc _ _ | lemma | category_theory.limits.wide_coequalizer.π_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wide_coequalizer.desc' [nonempty J] {W : C} (k : Y ⟶ W) (h : ∀ j₁ j₂, f j₁ ≫ k = f j₂ ≫ k) :
{l : wide_coequalizer f ⟶ W // wide_coequalizer.π f ≫ l = k} | ⟨wide_coequalizer.desc k h, wide_coequalizer.π_desc _ _⟩ | def | category_theory.limits.wide_coequalizer.desc' | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | Any morphism `k : Y ⟶ W` satisfying `∀ j₁ j₂, f j₁ ≫ k = f j₂ ≫ k` induces a morphism
`l : wide_coequalizer f ⟶ W` satisfying `wide_coequalizer.π ≫ g = l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_coequalizer.hom_ext [nonempty J] {W : C} {k l : wide_coequalizer f ⟶ W}
(h : wide_coequalizer.π f ≫ k = wide_coequalizer.π f ≫ l) : k = l | cotrident.is_colimit.hom_ext (colimit.is_colimit _) h | lemma | category_theory.limits.wide_coequalizer.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | Two maps from a wide coequalizer are equal if they are equal when composed with the wide
coequalizer map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_coequalizer.π_epi [nonempty J] : epi (wide_coequalizer.π f) | { left_cancellation := λ Z h k w, wide_coequalizer.hom_ext w } | instance | category_theory.limits.wide_coequalizer.π_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | A wide coequalizer morphism is an epimorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_of_is_colimit_parallel_family [nonempty J] {c : cocone (parallel_family f)}
(i : is_colimit c) :
epi (c.ι.app one) | { left_cancellation := λ Z h k w, cotrident.is_colimit.hom_ext i w } | lemma | category_theory.limits.epi_of_is_colimit_parallel_family | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | The wide coequalizer morphism in any colimit cocone is an epimorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_wide_equalizers | Π J, has_limits_of_shape (walking_parallel_family.{w} J) C | abbreviation | category_theory.limits.has_wide_equalizers | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | `has_wide_equalizers` represents a choice of wide equalizer for every family of morphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_wide_coequalizers | Π J, has_colimits_of_shape (walking_parallel_family.{w} J) C | abbreviation | category_theory.limits.has_wide_coequalizers | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | `has_wide_coequalizers` represents a choice of wide coequalizer for every family of morphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_wide_equalizers_of_has_limit_parallel_family
[Π {J : Type w} {X Y : C} {f : J → (X ⟶ Y)}, has_limit (parallel_family f)] :
has_wide_equalizers.{w} C | λ J, { has_limit := λ F, has_limit_of_iso (diagram_iso_parallel_family F).symm } | lemma | category_theory.limits.has_wide_equalizers_of_has_limit_parallel_family | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | If `C` has all limits of diagrams `parallel_family f`, then it has all wide equalizers | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_wide_coequalizers_of_has_colimit_parallel_family
[Π {J : Type w} {X Y : C} {f : J → (X ⟶ Y)}, has_colimit (parallel_family f)] :
has_wide_coequalizers.{w} C | λ J, { has_colimit := λ F, has_colimit_of_iso (diagram_iso_parallel_family F) } | lemma | category_theory.limits.has_wide_coequalizers_of_has_colimit_parallel_family | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | If `C` has all colimits of diagrams `parallel_family f`, then it has all wide coequalizers | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_equalizers_of_has_wide_equalizers [has_wide_equalizers.{w} C] : has_equalizers C | has_limits_of_shape_of_equivalence.{w} walking_parallel_family_equiv_walking_parallel_pair | instance | category_theory.limits.has_equalizers_of_has_wide_equalizers | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coequalizers_of_has_wide_coequalizers [has_wide_coequalizers.{w} C] :
has_coequalizers C | has_colimits_of_shape_of_equivalence.{w} walking_parallel_family_equiv_walking_parallel_pair | instance | category_theory.limits.has_coequalizers_of_has_wide_coequalizers | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_equalizers.lean | [
"category_theory.limits.has_limits",
"category_theory.limits.shapes.equalizers"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wide_pullback_shape | option J | def | category_theory.limits.wide_pullback_shape | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | A wide pullback shape for any type `J` can be written simply as `option J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pushout_shape | option J | def | category_theory.limits.wide_pushout_shape | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | A wide pushout shape for any type `J` can be written simply as `option J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom : wide_pullback_shape J → wide_pullback_shape J → Type w
| id : Π X, hom X X
| term : Π (j : J), hom (some j) none | inductive | category_theory.limits.wide_pullback_shape.hom | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The type of arrows for the shape indexing a wide pullback. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
struct : category_struct (wide_pullback_shape J) | { hom := hom,
id := λ j, hom.id j,
comp := λ j₁ j₂ j₃ f g,
begin
cases f,
exact g,
cases g,
apply hom.term _
end } | instance | category_theory.limits.wide_pullback_shape.struct | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom.inhabited : inhabited (hom none none) | ⟨hom.id (none : wide_pullback_shape J)⟩ | instance | category_theory.limits.wide_pullback_shape.hom.inhabited | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_hom : quiver.is_thin (wide_pullback_shape J) | λ _ _, ⟨by tidy⟩ | instance | category_theory.limits.wide_pullback_shape.subsingleton_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [
"quiver.is_thin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
category : small_category (wide_pullback_shape J) | thin_category | instance | category_theory.limits.wide_pullback_shape.category | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_id (X : wide_pullback_shape J) : hom.id X = 𝟙 X | rfl | lemma | category_theory.limits.wide_pullback_shape.hom_id | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wide_cospan (B : C) (objs : J → C) (arrows : Π (j : J), objs j ⟶ B) :
wide_pullback_shape J ⥤ C | { obj := λ j, option.cases_on j B objs,
map := λ X Y f,
begin
cases f with _ j,
{ apply (𝟙 _) },
{ exact arrows j }
end,
map_comp' := λ _ _ _ f g,
begin
cases f,
{ simpa },
cases g,
simp
end } | def | category_theory.limits.wide_pullback_shape.wide_cospan | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | Construct a functor out of the wide pullback shape given a J-indexed collection of arrows to a
fixed object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram_iso_wide_cospan (F : wide_pullback_shape J ⥤ C) :
F ≅ wide_cospan (F.obj none) (λ j, F.obj (some j)) (λ j, F.map (hom.term j)) | nat_iso.of_components (λ j, eq_to_iso $ by tidy) $ by tidy | def | category_theory.limits.wide_pullback_shape.diagram_iso_wide_cospan | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | Every diagram is naturally isomorphic (actually, equal) to a `wide_cospan` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_cone {F : wide_pullback_shape J ⥤ C} {X : C}
(f : X ⟶ F.obj none) (π : Π j, X ⟶ F.obj (some j))
(w : ∀ j, π j ≫ F.map (hom.term j) = f) : cone F | { X := X,
π :=
{ app := λ j, match j with
| none := f
| (some j) := π j
end,
naturality' := λ j j' f, by { cases j; cases j'; cases f; unfold_aux; dsimp; simp [w], }, } } | def | category_theory.limits.wide_pullback_shape.mk_cone | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | Construct a cone over a wide cospan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equivalence_of_equiv (J' : Type w') (h : J ≃ J') :
wide_pullback_shape J ≌ wide_pullback_shape J' | { functor := wide_cospan none (λ j, some (h j)) (λ j, hom.term (h j)),
inverse := wide_cospan none (λ j, some (h.inv_fun j)) (λ j, hom.term (h.inv_fun j)),
unit_iso := nat_iso.of_components (λ j, by cases j; simp)
(λ j k f, by { simp only [eq_iff_true_of_subsingleton]}),
counit_iso := nat_iso.of_components (λ... | def | category_theory.limits.wide_pullback_shape.equivalence_of_equiv | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [
"eq_iff_true_of_subsingleton"
] | Wide pullback diagrams of equivalent index types are equivlent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift_equivalence :
ulift_hom.{w'} (ulift.{w'} (wide_pullback_shape J)) ≌ wide_pullback_shape (ulift J) | (ulift_hom_ulift_category.equiv.{w' w' w w} (wide_pullback_shape J)).symm.trans
(equivalence_of_equiv _ (equiv.ulift.{w' w}.symm : J ≃ ulift.{w'} J)) | def | category_theory.limits.wide_pullback_shape.ulift_equivalence | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | Lifting universe and morphism levels preserves wide pullback diagrams. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom : wide_pushout_shape J → wide_pushout_shape J → Type w
| id : Π X, hom X X
| init : Π (j : J), hom none (some j) | inductive | category_theory.limits.wide_pushout_shape.hom | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The type of arrows for the shape indexing a wide psuhout. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
struct : category_struct (wide_pushout_shape J) | { hom := hom,
id := λ j, hom.id j,
comp := λ j₁ j₂ j₃ f g,
begin
cases f,
exact g,
cases g,
apply hom.init _
end } | instance | category_theory.limits.wide_pushout_shape.struct | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom.inhabited : inhabited (hom none none) | ⟨hom.id (none : wide_pushout_shape J)⟩ | instance | category_theory.limits.wide_pushout_shape.hom.inhabited | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_hom : quiver.is_thin (wide_pushout_shape J) | λ _ _, ⟨by tidy⟩ | instance | category_theory.limits.wide_pushout_shape.subsingleton_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [
"quiver.is_thin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
category : small_category (wide_pushout_shape J) | thin_category | instance | category_theory.limits.wide_pushout_shape.category | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_id (X : wide_pushout_shape J) : hom.id X = 𝟙 X | rfl | lemma | category_theory.limits.wide_pushout_shape.hom_id | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wide_span (B : C) (objs : J → C) (arrows : Π (j : J), B ⟶ objs j) : wide_pushout_shape J ⥤ C | { obj := λ j, option.cases_on j B objs,
map := λ X Y f,
begin
cases f with _ j,
{ apply (𝟙 _) },
{ exact arrows j }
end,
map_comp' := by { rintros (_|_) (_|_) (_|_) (_|_) (_|_); simpa <|> simp } } | def | category_theory.limits.wide_pushout_shape.wide_span | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | Construct a functor out of the wide pushout shape given a J-indexed collection of arrows from a
fixed object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram_iso_wide_span (F : wide_pushout_shape J ⥤ C) :
F ≅ wide_span (F.obj none) (λ j, F.obj (some j)) (λ j, F.map (hom.init j)) | nat_iso.of_components (λ j, eq_to_iso $ by tidy) $ by tidy | def | category_theory.limits.wide_pushout_shape.diagram_iso_wide_span | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | Every diagram is naturally isomorphic (actually, equal) to a `wide_span` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_cocone {F : wide_pushout_shape J ⥤ C} {X : C}
(f : F.obj none ⟶ X) (ι : Π j, F.obj (some j) ⟶ X)
(w : ∀ j, F.map (hom.init j) ≫ ι j = f) : cocone F | { X := X,
ι :=
{ app := λ j, match j with
| none := f
| (some j) := ι j
end,
naturality' := λ j j' f, by { cases j; cases j'; cases f; unfold_aux; dsimp; simp [w], }, } } | def | category_theory.limits.wide_pushout_shape.mk_cocone | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | Construct a cocone over a wide span. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_wide_pullbacks : Prop | Π (J : Type w), has_limits_of_shape (wide_pullback_shape J) C | abbreviation | category_theory.limits.has_wide_pullbacks | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | `has_wide_pullbacks` represents a choice of wide pullback for every collection of morphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_wide_pushouts : Prop | Π (J : Type w), has_colimits_of_shape (wide_pushout_shape J) C | abbreviation | category_theory.limits.has_wide_pushouts | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | `has_wide_pushouts` represents a choice of wide pushout for every collection of morphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_wide_pullback (B : C) (objs : J → C)
(arrows : Π (j : J), objs j ⟶ B) : Prop | has_limit (wide_pullback_shape.wide_cospan B objs arrows) | abbreviation | category_theory.limits.has_wide_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | `has_wide_pullback B objs arrows` means that `wide_cospan B objs arrows` has a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_wide_pushout (B : C) (objs : J → C)
(arrows : Π (j : J), B ⟶ objs j) : Prop | has_colimit (wide_pushout_shape.wide_span B objs arrows) | abbreviation | category_theory.limits.has_wide_pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | `has_wide_pushout B objs arrows` means that `wide_span B objs arrows` has a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pullback (B : C) (objs : J → C) (arrows : Π (j : J), objs j ⟶ B)
[has_wide_pullback B objs arrows] : C | limit (wide_pullback_shape.wide_cospan B objs arrows) | abbreviation | category_theory.limits.wide_pullback | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | A choice of wide pullback. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pushout (B : C) (objs : J → C) (arrows : Π (j : J), B ⟶ objs j)
[has_wide_pushout B objs arrows] : C | colimit (wide_pushout_shape.wide_span B objs arrows) | abbreviation | category_theory.limits.wide_pushout | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | A choice of wide pushout. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π (j : J) : wide_pullback _ _ arrows ⟶ objs j | limit.π (wide_pullback_shape.wide_cospan _ _ _) (option.some j) | abbreviation | category_theory.limits.wide_pullback.π | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The `j`-th projection from the pullback. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
base : wide_pullback _ _ arrows ⟶ B | limit.π (wide_pullback_shape.wide_cospan _ _ _) option.none | abbreviation | category_theory.limits.wide_pullback.base | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The unique map to the base from the pullback. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π_arrow (j : J) : π arrows j ≫ arrows _ = base arrows | by apply limit.w (wide_pullback_shape.wide_cospan _ _ _) (wide_pullback_shape.hom.term j) | lemma | category_theory.limits.wide_pullback.π_arrow | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift {X : C} (f : X ⟶ B) (fs : Π (j : J), X ⟶ objs j)
(w : ∀ j, fs j ≫ arrows j = f) : X ⟶ wide_pullback _ _ arrows | limit.lift (wide_pullback_shape.wide_cospan _ _ _)
(wide_pullback_shape.mk_cone f fs $ by exact w) | abbreviation | category_theory.limits.wide_pullback.lift | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [
"lift"
] | Lift a collection of morphisms to a morphism to the pullback. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_π (j : J) : lift f fs w ≫ π arrows j = fs _ | by { simp, refl } | lemma | category_theory.limits.wide_pullback.lift_π | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_base : lift f fs w ≫ base arrows = f | by { simp, refl } | lemma | category_theory.limits.wide_pullback.lift_base | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_lift_of_comp_eq (g : X ⟶ wide_pullback _ _ arrows) :
(∀ j : J, g ≫ π arrows j = fs j) → g ≫ base arrows = f → g = lift f fs w | begin
intros h1 h2,
apply (limit.is_limit (wide_pullback_shape.wide_cospan B objs arrows)).uniq
(wide_pullback_shape.mk_cone f fs $ by exact w),
rintro (_|_),
{ apply h2 },
{ apply h1 }
end | lemma | category_theory.limits.wide_pullback.eq_lift_of_comp_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_eq_lift (g : X ⟶ wide_pullback _ _ arrows) :
g = lift (g ≫ base arrows) (λ j, g ≫ π arrows j) (by tidy) | begin
apply eq_lift_of_comp_eq,
tidy,
end | lemma | category_theory.limits.wide_pullback.hom_eq_lift | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext (g1 g2 : X ⟶ wide_pullback _ _ arrows) :
(∀ j : J, g1 ≫ π arrows j = g2 ≫ π arrows j) →
g1 ≫ base arrows = g2 ≫ base arrows → g1 = g2 | begin
intros h1 h2,
apply limit.hom_ext,
rintros (_|_),
{ apply h2 },
{ apply h1 },
end | lemma | category_theory.limits.wide_pullback.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι (j : J) : objs j ⟶ wide_pushout _ _ arrows | colimit.ι (wide_pushout_shape.wide_span _ _ _) (option.some j) | abbreviation | category_theory.limits.wide_pushout.ι | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The `j`-th inclusion to the pushout. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
head : B ⟶ wide_pushout B objs arrows | colimit.ι (wide_pushout_shape.wide_span _ _ _) option.none | abbreviation | category_theory.limits.wide_pushout.head | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The unique map from the head to the pushout. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arrow_ι (j : J) : arrows j ≫ ι arrows j = head arrows | by apply colimit.w (wide_pushout_shape.wide_span _ _ _) (wide_pushout_shape.hom.init j) | lemma | category_theory.limits.wide_pushout.arrow_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
desc {X : C} (f : B ⟶ X) (fs : Π (j : J), objs j ⟶ X)
(w : ∀ j, arrows j ≫ fs j = f) : wide_pushout _ _ arrows ⟶ X | colimit.desc (wide_pushout_shape.wide_span B objs arrows)
(wide_pushout_shape.mk_cocone f fs $ by exact w) | abbreviation | category_theory.limits.wide_pushout.desc | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | Descend a collection of morphisms to a morphism from the pushout. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_desc (j : J) : ι arrows j ≫ desc f fs w = fs _ | by { simp, refl } | lemma | category_theory.limits.wide_pushout.ι_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
head_desc : head arrows ≫ desc f fs w = f | by { simp, refl } | lemma | category_theory.limits.wide_pushout.head_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_desc_of_comp_eq (g : wide_pushout _ _ arrows ⟶ X) :
(∀ j : J, ι arrows j ≫ g = fs j) → head arrows ≫ g = f → g = desc f fs w | begin
intros h1 h2,
apply (colimit.is_colimit (wide_pushout_shape.wide_span B objs arrows)).uniq
(wide_pushout_shape.mk_cocone f fs $ by exact w),
rintro (_|_),
{ apply h2 },
{ apply h1 }
end | lemma | category_theory.limits.wide_pushout.eq_desc_of_comp_eq | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_eq_desc (g : wide_pushout _ _ arrows ⟶ X) :
g = desc (head arrows ≫ g) (λ j, ι arrows j ≫ g) (λ j, by { rw ← category.assoc, simp }) | begin
apply eq_desc_of_comp_eq,
tidy,
end | lemma | category_theory.limits.wide_pushout.hom_eq_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext (g1 g2 : wide_pushout _ _ arrows ⟶ X) :
(∀ j : J, ι arrows j ≫ g1 = ι arrows j ≫ g2) →
head arrows ≫ g1 = head arrows ≫ g2 → g1 = g2 | begin
intros h1 h2,
apply colimit.hom_ext,
rintros (_|_),
{ apply h2 },
{ apply h1 },
end | lemma | category_theory.limits.wide_pushout.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wide_pullback_shape_op_map : Π (X Y : wide_pullback_shape J),
(X ⟶ Y) → ((op X : (wide_pushout_shape J)ᵒᵖ) ⟶ (op Y : (wide_pushout_shape J)ᵒᵖ)) | | _ _ (wide_pullback_shape.hom.id X) := quiver.hom.op (wide_pushout_shape.hom.id _)
| _ _ (wide_pullback_shape.hom.term j) := quiver.hom.op (wide_pushout_shape.hom.init _) | def | category_theory.limits.wide_pullback_shape_op_map | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [
"quiver.hom.op"
] | The action on morphisms of the obvious functor
`wide_pullback_shape_op : wide_pullback_shape J ⥤ (wide_pushout_shape J)ᵒᵖ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pullback_shape_op : wide_pullback_shape J ⥤ (wide_pushout_shape J)ᵒᵖ | { obj := λ X, op X,
map := wide_pullback_shape_op_map J, } | def | category_theory.limits.wide_pullback_shape_op | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The obvious functor `wide_pullback_shape J ⥤ (wide_pushout_shape J)ᵒᵖ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pushout_shape_op_map : Π (X Y : wide_pushout_shape J),
(X ⟶ Y) → ((op X : (wide_pullback_shape J)ᵒᵖ) ⟶ (op Y : (wide_pullback_shape J)ᵒᵖ)) | | _ _ (wide_pushout_shape.hom.id X) := quiver.hom.op (wide_pullback_shape.hom.id _)
| _ _ (wide_pushout_shape.hom.init j) := quiver.hom.op (wide_pullback_shape.hom.term _) | def | category_theory.limits.wide_pushout_shape_op_map | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [
"quiver.hom.op"
] | The action on morphisms of the obvious functor
`wide_pushout_shape_op : `wide_pushout_shape J ⥤ (wide_pullback_shape J)ᵒᵖ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pushout_shape_op : wide_pushout_shape J ⥤ (wide_pullback_shape J)ᵒᵖ | { obj := λ X, op X,
map := wide_pushout_shape_op_map J, } | def | category_theory.limits.wide_pushout_shape_op | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The obvious functor `wide_pushout_shape J ⥤ (wide_pullback_shape J)ᵒᵖ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pullback_shape_unop : (wide_pullback_shape J)ᵒᵖ ⥤ wide_pushout_shape J | (wide_pullback_shape_op J).left_op | def | category_theory.limits.wide_pullback_shape_unop | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The obvious functor `(wide_pullback_shape J)ᵒᵖ ⥤ wide_pushout_shape J` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pushout_shape_unop : (wide_pushout_shape J)ᵒᵖ ⥤ wide_pullback_shape J | (wide_pushout_shape_op J).left_op | def | category_theory.limits.wide_pushout_shape_unop | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The obvious functor `(wide_pushout_shape J)ᵒᵖ ⥤ wide_pullback_shape J` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pushout_shape_op_unop : wide_pushout_shape_unop J ⋙ wide_pullback_shape_op J ≅ 𝟭 _ | nat_iso.of_components (λ X, iso.refl _) (λ X Y f, dec_trivial) | def | category_theory.limits.wide_pushout_shape_op_unop | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The inverse of the unit isomorphism of the equivalence
`wide_pushout_shape_op_equiv : (wide_pushout_shape J)ᵒᵖ ≌ wide_pullback_shape J` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pushout_shape_unop_op : wide_pushout_shape_op J ⋙ wide_pullback_shape_unop J ≅ 𝟭 _ | nat_iso.of_components (λ X, iso.refl _) (λ X Y f, dec_trivial) | def | category_theory.limits.wide_pushout_shape_unop_op | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The counit isomorphism of the equivalence
`wide_pullback_shape_op_equiv : (wide_pullback_shape J)ᵒᵖ ≌ wide_pushout_shape J` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pullback_shape_op_unop : wide_pullback_shape_unop J ⋙ wide_pushout_shape_op J ≅ 𝟭 _ | nat_iso.of_components (λ X, iso.refl _) (λ X Y f, dec_trivial) | def | category_theory.limits.wide_pullback_shape_op_unop | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The inverse of the unit isomorphism of the equivalence
`wide_pullback_shape_op_equiv : (wide_pullback_shape J)ᵒᵖ ≌ wide_pushout_shape J` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pullback_shape_unop_op : wide_pullback_shape_op J ⋙ wide_pushout_shape_unop J ≅ 𝟭 _ | nat_iso.of_components (λ X, iso.refl _) (λ X Y f, dec_trivial) | def | category_theory.limits.wide_pullback_shape_unop_op | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The counit isomorphism of the equivalence
`wide_pushout_shape_op_equiv : (wide_pushout_shape J)ᵒᵖ ≌ wide_pullback_shape J` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pushout_shape_op_equiv : (wide_pushout_shape J)ᵒᵖ ≌ wide_pullback_shape J | { functor := wide_pushout_shape_unop J,
inverse := wide_pullback_shape_op J,
unit_iso := (wide_pushout_shape_op_unop J).symm,
counit_iso := wide_pullback_shape_unop_op J, } | def | category_theory.limits.wide_pushout_shape_op_equiv | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The duality equivalence `(wide_pushout_shape J)ᵒᵖ ≌ wide_pullback_shape J` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pullback_shape_op_equiv : (wide_pullback_shape J)ᵒᵖ ≌ wide_pushout_shape J | { functor := wide_pullback_shape_unop J,
inverse := wide_pushout_shape_op J,
unit_iso := (wide_pullback_shape_op_unop J).symm,
counit_iso := wide_pushout_shape_unop_op J, } | def | category_theory.limits.wide_pullback_shape_op_equiv | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | The duality equivalence `(wide_pullback_shape J)ᵒᵖ ≌ wide_pushout_shape J` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_wide_pullbacks_shrink [has_wide_pullbacks.{max w w'} C] : has_wide_pullbacks.{w} C | λ J, has_limits_of_shape_of_equivalence
(wide_pullback_shape.equivalence_of_equiv _ equiv.ulift.{w'}) | lemma | category_theory.limits.has_wide_pullbacks_shrink | category_theory.limits.shapes | src/category_theory/limits/shapes/wide_pullbacks.lean | [
"category_theory.limits.has_limits",
"category_theory.thin"
] | [] | If a category has wide pullbacks on a higher universe level it also has wide pullbacks
on a lower universe level. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_zero_morphisms | [has_zero : Π X Y : C, has_zero (X ⟶ Y)]
(comp_zero' : ∀ {X Y : C} (f : X ⟶ Y) (Z : C), f ≫ (0 : Y ⟶ Z) = (0 : X ⟶ Z) . obviously)
(zero_comp' : ∀ (X : C) {Y Z : C} (f : Y ⟶ Z), (0 : X ⟶ Y) ≫ f = (0 : X ⟶ Z) . obviously) | class | category_theory.limits.has_zero_morphisms | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | A category "has zero morphisms" if there is a designated "zero morphism" in each morphism space,
and compositions of zero morphisms with anything give the zero morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_zero [has_zero_morphisms C] {X Y : C} {f : X ⟶ Y} {Z : C} :
f ≫ (0 : Y ⟶ Z) = (0 : X ⟶ Z) | has_zero_morphisms.comp_zero f Z | lemma | category_theory.limits.comp_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_comp [has_zero_morphisms C] {X : C} {Y Z : C} {f : Y ⟶ Z} :
(0 : X ⟶ Y) ≫ f = (0 : X ⟶ Z) | has_zero_morphisms.zero_comp X f | lemma | category_theory.limits.zero_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zero_morphisms_pempty : has_zero_morphisms (discrete pempty) | { has_zero := by tidy } | instance | category_theory.limits.has_zero_morphisms_pempty | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [
"pempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zero_morphisms_punit : has_zero_morphisms (discrete punit) | { has_zero := by tidy } | instance | category_theory.limits.has_zero_morphisms_punit | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_aux (I J : has_zero_morphisms C)
(w : ∀ X Y : C, (@has_zero_morphisms.has_zero _ _ I X Y).zero =
(@has_zero_morphisms.has_zero _ _ J X Y).zero) : I = J | begin
casesI I, casesI J,
congr,
{ ext X Y,
exact w X Y },
{ apply proof_irrel_heq, },
{ apply proof_irrel_heq, }
end | lemma | category_theory.limits.has_zero_morphisms.ext_aux | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [
"proof_irrel_heq"
] | This lemma will be immediately superseded by `ext`, below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext (I J : has_zero_morphisms C) : I = J | begin
apply ext_aux,
intros X Y,
rw ←@has_zero_morphisms.comp_zero _ _ I X X (@has_zero_morphisms.has_zero _ _ J X X).zero,
rw @has_zero_morphisms.zero_comp _ _ J,
end | lemma | category_theory.limits.has_zero_morphisms.ext | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | If you're tempted to use this lemma "in the wild", you should probably
carefully consider whether you've made a mistake in allowing two
instances of `has_zero_morphisms` to exist at all.
See, particularly, the note on `zero_morphisms_of_zero_object` below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_zero_morphisms_opposite [has_zero_morphisms C] :
has_zero_morphisms Cᵒᵖ | { has_zero := λ X Y, ⟨(0 : unop Y ⟶ unop X).op⟩,
comp_zero' := λ X Y f Z, congr_arg quiver.hom.op (has_zero_morphisms.zero_comp (unop Z) f.unop),
zero_comp' := λ X Y Z f, congr_arg quiver.hom.op (has_zero_morphisms.comp_zero f.unop (unop X)), } | instance | category_theory.limits.has_zero_morphisms_opposite | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [
"quiver.hom.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_of_comp_mono {X Y Z : C} {f : X ⟶ Y} (g : Y ⟶ Z) [mono g] (h : f ≫ g = 0) : f = 0 | by { rw [←zero_comp, cancel_mono] at h, exact h } | lemma | category_theory.limits.zero_of_comp_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_of_epi_comp {X Y Z : C} (f : X ⟶ Y) {g : Y ⟶ Z} [epi f] (h : f ≫ g = 0) : g = 0 | by { rw [←comp_zero, cancel_epi] at h, exact h } | lemma | category_theory.limits.zero_of_epi_comp | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_image_eq_zero {X Y : C} {f : X ⟶ Y} [has_image f] (w : image.ι f = 0) : f = 0 | by rw [←image.fac f, w, has_zero_morphisms.comp_zero] | lemma | category_theory.limits.eq_zero_of_image_eq_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonzero_image_of_nonzero {X Y : C} {f : X ⟶ Y} [has_image f] (w : f ≠ 0) : image.ι f ≠ 0 | λ h, w (eq_zero_of_image_eq_zero h) | lemma | category_theory.limits.nonzero_image_of_nonzero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_app (F G : C ⥤ D) (j : C) : (0 : F ⟶ G).app j = 0 | rfl | lemma | category_theory.limits.zero_app | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_src {X Y : C} (o : is_zero X) (f : X ⟶ Y) : f = 0 | o.eq_of_src _ _ | lemma | category_theory.limits.is_zero.eq_zero_of_src | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_tgt {X Y : C} (o : is_zero Y) (f : X ⟶ Y) : f = 0 | o.eq_of_tgt _ _ | lemma | category_theory.limits.is_zero.eq_zero_of_tgt | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_id_eq_zero (X : C) : is_zero X ↔ (𝟙 X = 0) | ⟨λ h, h.eq_of_src _ _,
λ h, ⟨
λ Y, ⟨⟨⟨0⟩, λ f, by { rw [←id_comp f, ←id_comp default, h, zero_comp, zero_comp], }⟩⟩,
λ Y, ⟨⟨⟨0⟩, λ f, by { rw [←comp_id f, ←comp_id default, h, comp_zero, comp_zero], }⟩⟩⟩⟩ | lemma | category_theory.limits.is_zero.iff_id_eq_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mono_zero (X Y : C) [mono (0 : X ⟶ Y)] : is_zero X | (iff_id_eq_zero X).mpr ((cancel_mono (0 : X ⟶ Y)).1 (by simp)) | lemma | category_theory.limits.is_zero.of_mono_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_epi_zero (X Y : C) [epi (0 : X ⟶ Y)] : is_zero Y | (iff_id_eq_zero Y).mpr ((cancel_epi (0 : X ⟶ Y)).1 (by simp)) | lemma | category_theory.limits.is_zero.of_epi_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mono_eq_zero {X Y : C} (f : X ⟶ Y) [mono f] (h : f = 0) : is_zero X | by { unfreezingI { subst h, }, apply of_mono_zero X Y, } | lemma | category_theory.limits.is_zero.of_mono_eq_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_epi_eq_zero {X Y : C} (f : X ⟶ Y) [epi f] (h : f = 0) : is_zero Y | by { unfreezingI { subst h, }, apply of_epi_zero X Y, } | lemma | category_theory.limits.is_zero.of_epi_eq_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/zero_morphisms.lean | [
"data.pi.algebra",
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.images",
"category_theory.isomorphism_classes",
"category_theory.limits.shapes.zero_objects"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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