statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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left_adjoint_lift_struct_equiv :
sq.lift_struct ≃ (sq.left_adjoint adj).lift_struct | { to_fun := λ l,
{ l := (adj.hom_equiv _ _).symm l.l,
fac_left' := by rw [← adj.hom_equiv_naturality_left_symm, l.fac_left],
fac_right' := by rw [← adj.hom_equiv_naturality_right_symm, l.fac_right], },
inv_fun := λ l,
{ l := (adj.hom_equiv _ _) l.l,
fac_left' := begin
rw [← adj.hom_equiv_natural... | def | category_theory.comm_sq.left_adjoint_lift_struct_equiv | category_theory.lifting_properties | src/category_theory/lifting_properties/adjunction.lean | [
"category_theory.lifting_properties.basic",
"category_theory.adjunction.basic"
] | [
"adj",
"inv_fun"
] | The liftings of a commutative are in bijection with the liftings of its (left)
adjoint square. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_adjoint_has_lift_iff :
has_lift (sq.left_adjoint adj) ↔ has_lift sq | begin
simp only [has_lift.iff],
exact equiv.nonempty_congr (sq.left_adjoint_lift_struct_equiv adj).symm,
end | lemma | category_theory.comm_sq.left_adjoint_has_lift_iff | category_theory.lifting_properties | src/category_theory/lifting_properties/adjunction.lean | [
"category_theory.lifting_properties.basic",
"category_theory.adjunction.basic"
] | [
"adj",
"equiv.nonempty_congr"
] | A (left) adjoint square has a lifting if and only if the original square has a lifting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_lifting_property_iff (adj : G ⊣ F) {A B : C} {X Y : D} (i : A ⟶ B) (p : X ⟶ Y) :
has_lifting_property (G.map i) p ↔ has_lifting_property i (F.map p) | begin
split; introI; constructor; intros f g sq,
{ rw ← sq.left_adjoint_has_lift_iff adj,
apply_instance, },
{ rw ← sq.right_adjoint_has_lift_iff adj,
apply_instance, },
end | lemma | category_theory.adjunction.has_lifting_property_iff | category_theory.lifting_properties | src/category_theory/lifting_properties/adjunction.lean | [
"category_theory.lifting_properties.basic",
"category_theory.adjunction.basic"
] | [
"adj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_lifting_property : Prop | (sq_has_lift : ∀ {f : A ⟶ X} {g : B ⟶ Y} (sq : comm_sq f i p g), sq.has_lift) | class | category_theory.has_lifting_property | category_theory.lifting_properties | src/category_theory/lifting_properties/basic.lean | [
"category_theory.comm_sq"
] | [] | `has_lifting_property i p` means that `i` has the left lifting
property with respect to `p`, or equivalently that `p` has
the right lifting property with respect to `i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sq_has_lift_of_has_lifting_property {f : A ⟶ X} {g : B ⟶ Y} (sq : comm_sq f i p g)
[hip : has_lifting_property i p] : sq.has_lift | by apply hip.sq_has_lift | instance | category_theory.sq_has_lift_of_has_lifting_property | category_theory.lifting_properties | src/category_theory/lifting_properties/basic.lean | [
"category_theory.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op (h : has_lifting_property i p) : has_lifting_property p.op i.op | ⟨λ f g sq, begin
simp only [comm_sq.has_lift.iff_unop, quiver.hom.unop_op],
apply_instance,
end⟩ | lemma | category_theory.has_lifting_property.op | category_theory.lifting_properties | src/category_theory/lifting_properties/basic.lean | [
"category_theory.comm_sq"
] | [
"quiver.hom.unop_op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop {A B X Y : Cᵒᵖ} {i : A ⟶ B} {p : X ⟶ Y}
(h : has_lifting_property i p) : has_lifting_property p.unop i.unop | ⟨λ f g sq, begin
rw comm_sq.has_lift.iff_op,
simp only [quiver.hom.op_unop],
apply_instance,
end⟩ | lemma | category_theory.has_lifting_property.unop | category_theory.lifting_properties | src/category_theory/lifting_properties/basic.lean | [
"category_theory.comm_sq"
] | [
"quiver.hom.op_unop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_op : has_lifting_property i p ↔ has_lifting_property p.op i.op | ⟨op, unop⟩ | lemma | category_theory.has_lifting_property.iff_op | category_theory.lifting_properties | src/category_theory/lifting_properties/basic.lean | [
"category_theory.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_unop {A B X Y : Cᵒᵖ} (i : A ⟶ B) (p : X ⟶ Y) :
has_lifting_property i p ↔ has_lifting_property p.unop i.unop | ⟨unop, op⟩ | lemma | category_theory.has_lifting_property.iff_unop | category_theory.lifting_properties | src/category_theory/lifting_properties/basic.lean | [
"category_theory.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_left_iso [is_iso i] : has_lifting_property i p | ⟨λ f g sq, comm_sq.has_lift.mk'
{ l := inv i ≫ f,
fac_left' := by simp only [is_iso.hom_inv_id_assoc],
fac_right' := by simp only [sq.w, assoc, is_iso.inv_hom_id_assoc], }⟩ | instance | category_theory.has_lifting_property.of_left_iso | category_theory.lifting_properties | src/category_theory/lifting_properties/basic.lean | [
"category_theory.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_right_iso [is_iso p] : has_lifting_property i p | ⟨λ f g sq, comm_sq.has_lift.mk'
{ l := g ≫ inv p,
fac_left' := by simp only [← sq.w_assoc, is_iso.hom_inv_id, comp_id],
fac_right' := by simp only [assoc, is_iso.inv_hom_id, comp_id], }⟩ | instance | category_theory.has_lifting_property.of_right_iso | category_theory.lifting_properties | src/category_theory/lifting_properties/basic.lean | [
"category_theory.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_comp_left [has_lifting_property i p] [has_lifting_property i' p] :
has_lifting_property (i ≫ i') p | ⟨λ f g sq, begin
have fac := sq.w,
rw assoc at fac,
exact comm_sq.has_lift.mk'
{ l := (comm_sq.mk (comm_sq.mk fac).fac_right).lift,
fac_left' := by simp only [assoc, comm_sq.fac_left],
fac_right' := by simp only [comm_sq.fac_right], },
end⟩ | instance | category_theory.has_lifting_property.of_comp_left | category_theory.lifting_properties | src/category_theory/lifting_properties/basic.lean | [
"category_theory.comm_sq"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_comp_right [has_lifting_property i p] [has_lifting_property i p'] :
has_lifting_property i (p ≫ p') | ⟨λ f g sq, begin
have fac := sq.w,
rw ← assoc at fac,
let sq₂ := (comm_sq.mk ((comm_sq.mk fac).fac_left.symm)).lift,
exact comm_sq.has_lift.mk'
{ l := (comm_sq.mk ((comm_sq.mk fac).fac_left.symm)).lift,
fac_left' := by simp only [comm_sq.fac_left],
fac_right' := by simp only [comm_sq.fac_right_a... | instance | category_theory.has_lifting_property.of_comp_right | category_theory.lifting_properties | src/category_theory/lifting_properties/basic.lean | [
"category_theory.comm_sq"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_arrow_iso_left {A B A' B' X Y : C} {i : A ⟶ B} {i' : A' ⟶ B'}
(e : arrow.mk i ≅ arrow.mk i') (p : X ⟶ Y)
[hip : has_lifting_property i p] : has_lifting_property i' p | by { rw arrow.iso_w' e, apply_instance, } | lemma | category_theory.has_lifting_property.of_arrow_iso_left | category_theory.lifting_properties | src/category_theory/lifting_properties/basic.lean | [
"category_theory.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_arrow_iso_right {A B X Y X' Y' : C} (i : A ⟶ B) {p : X ⟶ Y} {p' : X' ⟶ Y'}
(e : arrow.mk p ≅ arrow.mk p')
[hip : has_lifting_property i p] : has_lifting_property i p' | by { rw arrow.iso_w' e, apply_instance, } | lemma | category_theory.has_lifting_property.of_arrow_iso_right | category_theory.lifting_properties | src/category_theory/lifting_properties/basic.lean | [
"category_theory.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_of_arrow_iso_left {A B A' B' X Y : C} {i : A ⟶ B} {i' : A' ⟶ B'}
(e : arrow.mk i ≅ arrow.mk i') (p : X ⟶ Y) :
has_lifting_property i p ↔ has_lifting_property i' p | by { split; introI, exacts [of_arrow_iso_left e p, of_arrow_iso_left e.symm p], } | lemma | category_theory.has_lifting_property.iff_of_arrow_iso_left | category_theory.lifting_properties | src/category_theory/lifting_properties/basic.lean | [
"category_theory.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_of_arrow_iso_right {A B X Y X' Y' : C} (i : A ⟶ B) {p : X ⟶ Y} {p' : X' ⟶ Y'}
(e : arrow.mk p ≅ arrow.mk p') :
has_lifting_property i p ↔ has_lifting_property i p' | by { split; introI, exacts [of_arrow_iso_right i e, of_arrow_iso_right i e.symm], } | lemma | category_theory.has_lifting_property.iff_of_arrow_iso_right | category_theory.lifting_properties | src/category_theory/lifting_properties/basic.lean | [
"category_theory.comm_sq"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bicone
| left : bicone
| right : bicone
| diagram (val : J) : bicone | inductive | category_theory.bicone | category_theory.limits | src/category_theory/limits/bicones.lean | [
"category_theory.limits.cones",
"category_theory.fin_category"
] | [] | Given a category `J`, construct a walking `bicone J` by adjoining two elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fin_bicone [fintype J] : fintype (bicone J) | { elems := [bicone.left, bicone.right].to_finset ∪ finset.image bicone.diagram (fintype.elems J),
complete := λ j, by { cases j; simp, exact fintype.complete j, }, } | instance | category_theory.fin_bicone | category_theory.limits | src/category_theory/limits/bicones.lean | [
"category_theory.limits.cones",
"category_theory.fin_category"
] | [
"finset.image",
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bicone_hom : bicone J → bicone J → Type (max u₁ v₁)
| left_id : bicone_hom bicone.left bicone.left
| right_id : bicone_hom bicone.right bicone.right
| left (j : J) : bicone_hom bicone.left (bicone.diagram j)
| right (j : J) : bicone_hom bicone.right (bicone.diagram j)
| diagram {j k : J} (f : j ⟶ k) : bicone_hom (bico... | inductive | category_theory.bicone_hom | category_theory.limits | src/category_theory/limits/bicones.lean | [
"category_theory.limits.cones",
"category_theory.fin_category"
] | [] | The homs for a walking `bicone J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bicone_hom.decidable_eq {j k : bicone J} : decidable_eq (bicone_hom J j k) | λ f g, by { cases f; cases g; simp; apply_instance } | instance | category_theory.bicone_hom.decidable_eq | category_theory.limits | src/category_theory/limits/bicones.lean | [
"category_theory.limits.cones",
"category_theory.fin_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bicone_category_struct : category_struct (bicone J) | { hom := bicone_hom J,
id := λ j, bicone.cases_on j
bicone_hom.left_id bicone_hom.right_id (λ k, bicone_hom.diagram (𝟙 k)),
comp := λ X Y Z f g, by
{ cases f, exact g, exact g,
cases g, exact bicone_hom.left g_k,
cases g, exact bicone_hom.right g_k,
cases g, exact bicone_hom.diagram (f_f ≫ g_f) }... | instance | category_theory.bicone_category_struct | category_theory.limits | src/category_theory/limits/bicones.lean | [
"category_theory.limits.cones",
"category_theory.fin_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bicone_category : category (bicone J) | { id_comp' := λ X Y f, by { cases f; simp },
comp_id' := λ X Y f, by { cases f; simp },
assoc' := λ W X Y Z f g h, by { cases f; cases g; cases h; simp } } | instance | category_theory.bicone_category | category_theory.limits | src/category_theory/limits/bicones.lean | [
"category_theory.limits.cones",
"category_theory.fin_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bicone_mk {C : Type u₁} [category.{v₁} C]
{F : J ⥤ C} (c₁ c₂ : cone F) : bicone J ⥤ C | { obj := λ X, bicone.cases_on X c₁.X c₂.X (λ j, F.obj j),
map := λ X Y f, by
{ cases f, exact (𝟙 _), exact (𝟙 _),
exact c₁.π.app f_1,
exact c₂.π.app f_1,
exact F.map f_f, },
map_id' := λ X, by { cases X; simp },
map_comp' := λ X Y Z f g, by
{ cases f,
exact (category.id_comp _).symm,
exa... | def | category_theory.bicone_mk | category_theory.limits | src/category_theory/limits/bicones.lean | [
"category_theory.limits.cones",
"category_theory.fin_category"
] | [] | Given a diagram `F : J ⥤ C` and two `cone F`s, we can join them into a diagram `bicone J ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fin_bicone_hom [fin_category J] (j k : bicone J) : fintype (j ⟶ k) | begin
cases j; cases k,
exact { elems := {bicone_hom.left_id}, complete := λ f, by { cases f, simp } },
exact { elems := ∅, complete := λ f, by { cases f } },
exact { elems := {bicone_hom.left k}, complete := λ f, by { cases f, simp } },
exact { elems := ∅, complete := λ f, by { cases f } },
exact { elems :... | instance | category_theory.fin_bicone_hom | category_theory.limits | src/category_theory/limits/bicones.lean | [
"category_theory.limits.cones",
"category_theory.fin_category"
] | [
"finset.image",
"finset.mem_image",
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bicone_small_category : small_category (bicone J) | category_theory.bicone_category J | instance | category_theory.bicone_small_category | category_theory.limits | src/category_theory/limits/bicones.lean | [
"category_theory.limits.cones",
"category_theory.fin_category"
] | [
"category_theory.bicone_category"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bicone_fin_category [fin_category J] : fin_category (bicone J) | {} | instance | category_theory.bicone_fin_category | category_theory.limits | src/category_theory/limits/bicones.lean | [
"category_theory.limits.cones",
"category_theory.fin_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id_left_eq_curry_map {j : J} {k k' : K} {f : k ⟶ k'} :
F.map ((𝟙 j, f) : (j, k) ⟶ (j, k')) = ((curry.obj F).obj j).map f | rfl | lemma | category_theory.limits.map_id_left_eq_curry_map | category_theory.limits | src/category_theory/limits/colimit_limit.lean | [
"category_theory.limits.types",
"category_theory.functor.currying",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id_right_eq_curry_swap_map {j j' : J} {f : j ⟶ j'} {k : K} :
F.map ((f, 𝟙 k) : (j, k) ⟶ (j', k)) = ((curry.obj (swap K J ⋙ F)).obj k).map f | rfl | lemma | category_theory.limits.map_id_right_eq_curry_swap_map | category_theory.limits | src/category_theory/limits/colimit_limit.lean | [
"category_theory.limits.types",
"category_theory.functor.currying",
"category_theory.limits.functor_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_limit_to_limit_colimit :
colimit ((curry.obj (swap K J ⋙ F)) ⋙ lim) ⟶ limit ((curry.obj F) ⋙ colim) | limit.lift ((curry.obj F) ⋙ colim)
{ X := _,
π :=
{ app := λ j, colimit.desc ((curry.obj (swap K J ⋙ F)) ⋙ lim)
{ X := _,
ι :=
{ app := λ k,
limit.π ((curry.obj (swap K J ⋙ F)).obj k) j ≫ colimit.ι ((curry.obj F).obj j) k,
naturality' :=
begin
dsimp,
int... | def | category_theory.limits.colimit_limit_to_limit_colimit | category_theory.limits | src/category_theory/limits/colimit_limit.lean | [
"category_theory.limits.types",
"category_theory.functor.currying",
"category_theory.limits.functor_category"
] | [
"lim"
] | The universal morphism
$\colim_k \lim_j F(j,k) → \lim_j \colim_k F(j, k)$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_colimit_limit_to_limit_colimit_π (j) (k) :
colimit.ι _ k ≫ colimit_limit_to_limit_colimit F ≫ limit.π _ j =
limit.π ((curry.obj (swap K J ⋙ F)).obj k) j ≫ colimit.ι ((curry.obj F).obj j) k | by { dsimp [colimit_limit_to_limit_colimit], simp, } | lemma | category_theory.limits.ι_colimit_limit_to_limit_colimit_π | category_theory.limits | src/category_theory/limits/colimit_limit.lean | [
"category_theory.limits.types",
"category_theory.functor.currying",
"category_theory.limits.functor_category"
] | [] | Since `colimit_limit_to_limit_colimit` is a morphism from a colimit to a limit,
this lemma characterises it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_colimit_limit_to_limit_colimit_π_apply (F : J × K ⥤ Type v) (j) (k) (f) :
limit.π ((curry.obj F) ⋙ colim) j
(colimit_limit_to_limit_colimit F (colimit.ι ((curry.obj (swap K J ⋙ F)) ⋙ lim) k f)) =
colimit.ι ((curry.obj F).obj j) k (limit.π ((curry.obj (swap K J ⋙ F)).obj k) j f) | by { dsimp [colimit_limit_to_limit_colimit], simp, } | lemma | category_theory.limits.ι_colimit_limit_to_limit_colimit_π_apply | category_theory.limits | src/category_theory/limits/colimit_limit.lean | [
"category_theory.limits.types",
"category_theory.functor.currying",
"category_theory.limits.functor_category"
] | [
"lim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_limit_to_limit_colimit_cone (G : J ⥤ K ⥤ C) [has_limit G] :
colim.map_cone (limit.cone G) ⟶ limit.cone (G ⋙ colim) | { hom := colim.map (limit_iso_swap_comp_lim G).hom ≫
colimit_limit_to_limit_colimit (uncurry.obj G : _) ≫
lim.map (whisker_right (currying.unit_iso.app G).inv colim),
w' := λ j,
begin
ext1 k,
simp only [limit_obj_iso_limit_comp_evaluation_hom_π_assoc, iso.app_inv,
ι_colimit_limit_to_limit_coli... | def | category_theory.limits.colimit_limit_to_limit_colimit_cone | category_theory.limits | src/category_theory/limits/colimit_limit.lean | [
"category_theory.limits.types",
"category_theory.functor.currying",
"category_theory.limits.functor_category"
] | [] | The map `colimit_limit_to_limit_colimit` realized as a map of cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_auxiliary_cone (c₁ : cone (F ⋙ fst L R)) :
cone ((F ⋙ snd L R) ⋙ R) | (cones.postcompose (whisker_left F (comma.nat_trans L R) : _)).obj (L.map_cone c₁) | def | category_theory.comma.limit_auxiliary_cone | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | (Implementation). An auxiliary cone which is useful in order to construct limits
in the comma category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_of_preserves [preserves_limit (F ⋙ snd L R) R]
(c₁ : cone (F ⋙ fst L R)) {c₂ : cone (F ⋙ snd L R)} (t₂ : is_limit c₂) :
cone F | { X :=
{ left := c₁.X,
right := c₂.X,
hom := (is_limit_of_preserves R t₂).lift (limit_auxiliary_cone _ c₁) },
π :=
{ app := λ j,
{ left := c₁.π.app j,
right := c₂.π.app j,
w' := ((is_limit_of_preserves R t₂).fac (limit_auxiliary_cone F c₁) j).symm },
naturality' := λ j₁ j₂ t, by ext; d... | def | category_theory.comma.cone_of_preserves | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [
"lift"
] | If `R` preserves the appropriate limit, then given a cone for `F ⋙ fst L R : J ⥤ L` and a
limit cone for `F ⋙ snd L R : J ⥤ R` we can build a cone for `F` which will turn out to be a limit
cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_of_preserves_is_limit [preserves_limit (F ⋙ snd L R) R]
{c₁ : cone (F ⋙ fst L R)} (t₁ : is_limit c₁)
{c₂ : cone (F ⋙ snd L R)} (t₂ : is_limit c₂) :
is_limit (cone_of_preserves F c₁ t₂) | { lift := λ s,
{ left := t₁.lift ((fst L R).map_cone s),
right := t₂.lift ((snd L R).map_cone s),
w' := (is_limit_of_preserves R t₂).hom_ext $ λ j,
begin
rw [cone_of_preserves_X_hom, assoc, assoc, (is_limit_of_preserves R t₂).fac,
limit_auxiliary_cone_π_app, ←L.map_comp_assoc, t₁.fac, R.map_... | def | category_theory.comma.cone_of_preserves_is_limit | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [
"hom_ext",
"lift"
] | Provided that `R` preserves the appropriate limit, then the cone in `cone_of_preserves` is a
limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_auxiliary_cocone (c₂ : cocone (F ⋙ snd L R)) :
cocone ((F ⋙ fst L R) ⋙ L) | (cocones.precompose (whisker_left F (comma.nat_trans L R) : _)).obj (R.map_cocone c₂) | def | category_theory.comma.colimit_auxiliary_cocone | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | (Implementation). An auxiliary cocone which is useful in order to construct colimits
in the comma category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_of_preserves [preserves_colimit (F ⋙ fst L R) L]
{c₁ : cocone (F ⋙ fst L R)} (t₁ : is_colimit c₁) (c₂ : cocone (F ⋙ snd L R)) :
cocone F | { X :=
{ left := c₁.X,
right := c₂.X,
hom := (is_colimit_of_preserves L t₁).desc (colimit_auxiliary_cocone _ c₂) },
ι :=
{ app := λ j,
{ left := c₁.ι.app j,
right := c₂.ι.app j,
w' := ((is_colimit_of_preserves L t₁).fac (colimit_auxiliary_cocone _ c₂) j) },
naturality' := λ j₁ j₂ t, by... | def | category_theory.comma.cocone_of_preserves | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | If `L` preserves the appropriate colimit, then given a colimit cocone for `F ⋙ fst L R : J ⥤ L` and
a cocone for `F ⋙ snd L R : J ⥤ R` we can build a cocone for `F` which will turn out to be a
colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocone_of_preserves_is_colimit [preserves_colimit (F ⋙ fst L R) L]
{c₁ : cocone (F ⋙ fst L R)} (t₁ : is_colimit c₁)
{c₂ : cocone (F ⋙ snd L R)} (t₂ : is_colimit c₂) :
is_colimit (cocone_of_preserves F t₁ c₂) | { desc := λ s,
{ left := t₁.desc ((fst L R).map_cocone s),
right := t₂.desc ((snd L R).map_cocone s),
w' := (is_colimit_of_preserves L t₁).hom_ext $ λ j,
begin
rw [cocone_of_preserves_X_hom, (is_colimit_of_preserves L t₁).fac_assoc,
colimit_auxiliary_cocone_ι_app, assoc, ←R.map_comp, t₂.fac,... | def | category_theory.comma.cocone_of_preserves_is_colimit | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [
"hom_ext"
] | Provided that `L` preserves the appropriate colimit, then the cocone in `cocone_of_preserves` is
a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit (F : J ⥤ comma L R)
[has_limit (F ⋙ fst L R)] [has_limit (F ⋙ snd L R)]
[preserves_limit (F ⋙ snd L R) R] :
has_limit F | has_limit.mk ⟨_, cone_of_preserves_is_limit _ (limit.is_limit _) (limit.is_limit _)⟩ | instance | category_theory.comma.has_limit | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits_of_shape
[has_limits_of_shape J A] [has_limits_of_shape J B] [preserves_limits_of_shape J R] :
has_limits_of_shape J (comma L R) | {} | instance | category_theory.comma.has_limits_of_shape | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits [has_limits A] [has_limits B] [preserves_limits R] :
has_limits (comma L R) | ⟨infer_instance⟩ | instance | category_theory.comma.has_limits | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit (F : J ⥤ comma L R)
[has_colimit (F ⋙ fst L R)] [has_colimit (F ⋙ snd L R)]
[preserves_colimit (F ⋙ fst L R) L] :
has_colimit F | has_colimit.mk ⟨_, cocone_of_preserves_is_colimit _ (colimit.is_colimit _) (colimit.is_colimit _)⟩ | instance | category_theory.comma.has_colimit | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_shape
[has_colimits_of_shape J A] [has_colimits_of_shape J B] [preserves_colimits_of_shape J L] :
has_colimits_of_shape J (comma L R) | {} | instance | category_theory.comma.has_colimits_of_shape | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits [has_colimits A] [has_colimits B] [preserves_colimits L] :
has_colimits (comma L R) | ⟨infer_instance⟩ | instance | category_theory.comma.has_colimits | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limit (F : J ⥤ arrow T)
[i₁ : has_limit (F ⋙ left_func)] [i₂ : has_limit (F ⋙ right_func)] :
has_limit F | @@comma.has_limit _ _ _ _ _ i₁ i₂ _ | instance | category_theory.arrow.has_limit | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits_of_shape [has_limits_of_shape J T] : has_limits_of_shape J (arrow T) | {} | instance | category_theory.arrow.has_limits_of_shape | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits [has_limits T] : has_limits (arrow T) | ⟨infer_instance⟩ | instance | category_theory.arrow.has_limits | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit (F : J ⥤ arrow T)
[i₁ : has_colimit (F ⋙ left_func)] [i₂ : has_colimit (F ⋙ right_func)] :
has_colimit F | @@comma.has_colimit _ _ _ _ _ i₁ i₂ _ | instance | category_theory.arrow.has_colimit | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_shape [has_colimits_of_shape J T] : has_colimits_of_shape J (arrow T) | {} | instance | category_theory.arrow.has_colimits_of_shape | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits [has_colimits T] : has_colimits (arrow T) | ⟨infer_instance⟩ | instance | category_theory.arrow.has_colimits | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limit [i₁ : has_limit (F ⋙ proj X G)] [i₂ : preserves_limit (F ⋙ proj X G) G] :
has_limit F | @@comma.has_limit _ _ _ _ _ _ i₁ i₂ | instance | category_theory.structured_arrow.has_limit | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits_of_shape [has_limits_of_shape J A] [preserves_limits_of_shape J G] :
has_limits_of_shape J (structured_arrow X G) | {} | instance | category_theory.structured_arrow.has_limits_of_shape | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits [has_limits A] [preserves_limits G] :
has_limits (structured_arrow X G) | ⟨infer_instance⟩ | instance | category_theory.structured_arrow.has_limits | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_limit [i : preserves_limit (F ⋙ proj X G) G] :
creates_limit F (proj X G) | creates_limit_of_reflects_iso $ λ c t,
{ lifted_cone := @@comma.cone_of_preserves _ _ _ _ _ i punit_cone t,
makes_limit := comma.cone_of_preserves_is_limit _ punit_cone_is_limit _,
valid_lift := cones.ext (iso.refl _) $ λ j, (id_comp _).symm } | instance | category_theory.structured_arrow.creates_limit | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_limits_of_shape [preserves_limits_of_shape J G] :
creates_limits_of_shape J (proj X G) | {} | instance | category_theory.structured_arrow.creates_limits_of_shape | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_limits [preserves_limits G] :
creates_limits (proj X G : _) | ⟨⟩ | instance | category_theory.structured_arrow.creates_limits | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_right_of_mono [has_pullbacks A] [preserves_limits_of_shape walking_cospan G]
{Y Z : structured_arrow X G} (f : Y ⟶ Z) [mono f] : mono f.right | show mono ((proj X G).map f), from infer_instance | instance | category_theory.structured_arrow.mono_right_of_mono | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_iff_mono_right [has_pullbacks A] [preserves_limits_of_shape walking_cospan G]
{Y Z : structured_arrow X G} (f : Y ⟶ Z) : mono f ↔ mono f.right | ⟨λ h, by exactI infer_instance, λ h, by exactI mono_of_mono_right f⟩ | lemma | category_theory.structured_arrow.mono_iff_mono_right | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit [i₁ : has_colimit (F ⋙ proj G X)] [i₂ : preserves_colimit (F ⋙ proj G X) G] :
has_colimit F | @@comma.has_colimit _ _ _ _ _ i₁ _ i₂ | instance | category_theory.costructured_arrow.has_colimit | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_shape [has_colimits_of_shape J A] [preserves_colimits_of_shape J G] :
has_colimits_of_shape J (costructured_arrow G X) | {} | instance | category_theory.costructured_arrow.has_colimits_of_shape | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits [has_colimits A] [preserves_colimits G] :
has_colimits (costructured_arrow G X) | ⟨infer_instance⟩ | instance | category_theory.costructured_arrow.has_colimits | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_colimit [i : preserves_colimit (F ⋙ proj G X) G] :
creates_colimit F (proj G X) | creates_colimit_of_reflects_iso $ λ c t,
{ lifted_cocone := @@comma.cocone_of_preserves _ _ _ _ _ i t punit_cocone,
makes_colimit := comma.cocone_of_preserves_is_colimit _ _ punit_cocone_is_colimit,
valid_lift := cocones.ext (iso.refl _) $ λ j, comp_id _ } | instance | category_theory.costructured_arrow.creates_colimit | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_colimits_of_shape [preserves_colimits_of_shape J G] :
creates_colimits_of_shape J (proj G X) | {} | instance | category_theory.costructured_arrow.creates_colimits_of_shape | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_colimits [preserves_colimits G] :
creates_colimits (proj G X : _) | ⟨⟩ | instance | category_theory.costructured_arrow.creates_colimits | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_left_of_epi [has_pushouts A] [preserves_colimits_of_shape walking_span G]
{Y Z : costructured_arrow G X} (f : Y ⟶ Z) [epi f] : epi f.left | show epi ((proj G X).map f), from infer_instance | instance | category_theory.costructured_arrow.epi_left_of_epi | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_iff_epi_left [has_pushouts A] [preserves_colimits_of_shape walking_span G]
{Y Z : costructured_arrow G X} (f : Y ⟶ Z) : epi f ↔ epi f.left | ⟨λ h, by exactI infer_instance, λ h, by exactI epi_of_epi_left f⟩ | lemma | category_theory.costructured_arrow.epi_iff_epi_left | category_theory.limits | src/category_theory/limits/comma.lean | [
"category_theory.arrow",
"category_theory.limits.constructions.epi_mono",
"category_theory.limits.creates",
"category_theory.limits.unit",
"category_theory.structured_arrow"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.to_product_injective_of_is_limit {D : cone F} (hD : is_limit D) :
function.injective (λ (x : D.X) (j : J), D.π.app j x) | begin
let E := (forget C).map_cone D,
let hE : is_limit E := is_limit_of_preserves _ hD,
let G := types.limit_cone.{w v} (F ⋙ forget C),
let hG := types.limit_cone_is_limit.{w v} (F ⋙ forget C),
let T : E.X ≅ G.X := hE.cone_point_unique_up_to_iso hG,
change function.injective (T.hom ≫ (λ x j, G.π.app j x)),... | lemma | category_theory.limits.concrete.to_product_injective_of_is_limit | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.is_limit_ext {D : cone F} (hD : is_limit D) (x y : D.X) :
(∀ j, D.π.app j x = D.π.app j y) → x = y | λ h, concrete.to_product_injective_of_is_limit _ hD (funext h) | lemma | category_theory.limits.concrete.is_limit_ext | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.limit_ext [has_limit F] (x y : limit F) :
(∀ j, limit.π F j x = limit.π F j y) → x = y | concrete.is_limit_ext F (limit.is_limit _) _ _ | lemma | category_theory.limits.concrete.limit_ext | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.wide_pullback_ext {B : C} {ι : Type w} {X : ι → C} (f : Π j : ι, X j ⟶ B)
[has_wide_pullback B X f] [preserves_limit (wide_cospan B X f) (forget C)]
(x y : wide_pullback B X f) (h₀ : base f x = base f y)
(h : ∀ j, π f j x = π f j y) : x = y | begin
apply concrete.limit_ext,
rintro (_|j),
{ exact h₀ },
{ apply h }
end | lemma | category_theory.limits.concrete.wide_pullback_ext | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.wide_pullback_ext' {B : C} {ι : Type w} [nonempty ι]
{X : ι → C} (f : Π j : ι, X j ⟶ B) [has_wide_pullback.{w} B X f]
[preserves_limit (wide_cospan B X f) (forget C)]
(x y : wide_pullback B X f) (h : ∀ j, π f j x = π f j y) : x = y | begin
apply concrete.wide_pullback_ext _ _ _ _ h,
inhabit ι,
simp only [← π_arrow f (arbitrary _), comp_apply, h],
end | lemma | category_theory.limits.concrete.wide_pullback_ext' | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.multiequalizer_ext {I : multicospan_index.{w} C} [has_multiequalizer I]
[preserves_limit I.multicospan (forget C)] (x y : multiequalizer I)
(h : ∀ (t : I.L), multiequalizer.ι I t x = multiequalizer.ι I t y) : x = y | begin
apply concrete.limit_ext,
rintros (a|b),
{ apply h },
{ rw [← limit.w I.multicospan (walking_multicospan.hom.fst b),
comp_apply, comp_apply, h] }
end | lemma | category_theory.limits.concrete.multiequalizer_ext | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.multiequalizer_equiv_aux (I : multicospan_index C) :
(I.multicospan ⋙ (forget C)).sections ≃
{ x : Π (i : I.L), I.left i // ∀ (i : I.R), I.fst i (x _) = I.snd i (x _) } | { to_fun := λ x, ⟨λ i, x.1 (walking_multicospan.left _), λ i, begin
have a := x.2 (walking_multicospan.hom.fst i),
have b := x.2 (walking_multicospan.hom.snd i),
rw ← b at a,
exact a,
end⟩,
inv_fun := λ x,
{ val := λ j,
match j with
| walking_multicospan.left a := x.1 _
| walking_multi... | def | category_theory.limits.concrete.multiequalizer_equiv_aux | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [
"inv_fun"
] | An auxiliary equivalence to be used in `multiequalizer_equiv` below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concrete.multiequalizer_equiv (I : multicospan_index.{w} C) [has_multiequalizer I]
[preserves_limit I.multicospan (forget C)] : (multiequalizer I : C) ≃
{ x : Π (i : I.L), I.left i // ∀ (i : I.R), I.fst i (x _) = I.snd i (x _) } | let h1 := (limit.is_limit I.multicospan),
h2 := (is_limit_of_preserves (forget C) h1),
E := h2.cone_point_unique_up_to_iso (types.limit_cone_is_limit _) in
equiv.trans E.to_equiv (concrete.multiequalizer_equiv_aux I) | def | category_theory.limits.concrete.multiequalizer_equiv | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [
"equiv.trans"
] | The equivalence between the noncomputable multiequalizer and
and the concrete multiequalizer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concrete.multiequalizer_equiv_apply (I : multicospan_index.{w} C) [has_multiequalizer I]
[preserves_limit I.multicospan (forget C)] (x : multiequalizer I) (i : I.L) :
((concrete.multiequalizer_equiv I) x : Π (i : I.L), I.left i) i = multiequalizer.ι I i x | rfl | lemma | category_theory.limits.concrete.multiequalizer_equiv_apply | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cokernel_funext {C : Type*} [category C] [has_zero_morphisms C] [concrete_category C]
{M N K : C} {f : M ⟶ N} [has_cokernel f] {g h : cokernel f ⟶ K}
(w : ∀ (n : N), g (cokernel.π f n) = h (cokernel.π f n)) : g = h | begin
apply coequalizer.hom_ext,
apply concrete_category.hom_ext _ _,
simpa using w,
end | lemma | category_theory.limits.cokernel_funext | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.from_union_surjective_of_is_colimit {D : cocone F} (hD : is_colimit D) :
let ff : (Σ (j : J), F.obj j) → D.X | λ a, D.ι.app a.1 a.2 in function.surjective ff :=
begin
intro ff,
let E := (forget C).map_cocone D,
let hE : is_colimit E := is_colimit_of_preserves _ hD,
let G := types.colimit_cocone.{v v} (F ⋙ forget C),
let hG := types.colimit_cocone_is_colimit.{v v} (F ⋙ forget C),
let T : E ≅ G := hE.unique_up_to_iso ... | lemma | category_theory.limits.concrete.from_union_surjective_of_is_colimit | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.is_colimit_exists_rep {D : cocone F} (hD : is_colimit D) (x : D.X) :
∃ (j : J) (y : F.obj j), D.ι.app j y = x | begin
obtain ⟨a, rfl⟩ := concrete.from_union_surjective_of_is_colimit F hD x,
exact ⟨a.1, a.2, rfl⟩,
end | lemma | category_theory.limits.concrete.is_colimit_exists_rep | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.colimit_exists_rep [has_colimit F] (x : colimit F) :
∃ (j : J) (y : F.obj j), colimit.ι F j y = x | concrete.is_colimit_exists_rep F (colimit.is_colimit _) x | lemma | category_theory.limits.concrete.colimit_exists_rep | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.is_colimit_rep_eq_of_exists {D : cocone F} {i j : J} (hD : is_colimit D)
(x : F.obj i) (y : F.obj j) (h : ∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y) :
D.ι.app i x = D.ι.app j y | begin
let E := (forget C).map_cocone D,
let hE : is_colimit E := is_colimit_of_preserves _ hD,
let G := types.colimit_cocone.{v v} (F ⋙ forget C),
let hG := types.colimit_cocone_is_colimit.{v v} (F ⋙ forget C),
let T : E ≅ G := hE.unique_up_to_iso hG,
let TX : E.X ≅ G.X := (cocones.forget _).map_iso T,
ap... | lemma | category_theory.limits.concrete.is_colimit_rep_eq_of_exists | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.colimit_rep_eq_of_exists [has_colimit F] {i j : J}
(x : F.obj i) (y : F.obj j) (h : ∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y) :
colimit.ι F i x = colimit.ι F j y | concrete.is_colimit_rep_eq_of_exists F (colimit.is_colimit _) x y h | lemma | category_theory.limits.concrete.colimit_rep_eq_of_exists | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.is_colimit_exists_of_rep_eq {D : cocone F} {i j : J} (hD : is_colimit D)
(x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :
∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y | begin
let E := (forget C).map_cocone D,
let hE : is_colimit E := is_colimit_of_preserves _ hD,
let G := types.colimit_cocone.{v v} (F ⋙ forget C),
let hG := types.colimit_cocone_is_colimit.{v v} (F ⋙ forget C),
let T : E ≅ G := hE.unique_up_to_iso hG,
let TX : E.X ≅ G.X := (cocones.forget _).map_iso T,
ap... | lemma | category_theory.limits.concrete.is_colimit_exists_of_rep_eq | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.is_colimit_rep_eq_iff_exists {D : cocone F} {i j : J}
(hD : is_colimit D) (x : F.obj i) (y : F.obj j) :
D.ι.app i x = D.ι.app j y ↔ ∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y | ⟨concrete.is_colimit_exists_of_rep_eq _ hD _ _, concrete.is_colimit_rep_eq_of_exists _ hD _ _⟩ | theorem | category_theory.limits.concrete.is_colimit_rep_eq_iff_exists | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.colimit_exists_of_rep_eq [has_colimit F] {i j : J}
(x : F.obj i) (y : F.obj j) (h : colimit.ι F _ x = colimit.ι F _ y) :
∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y | concrete.is_colimit_exists_of_rep_eq F (colimit.is_colimit _) x y h | lemma | category_theory.limits.concrete.colimit_exists_of_rep_eq | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.colimit_rep_eq_iff_exists [has_colimit F] {i j : J}
(x : F.obj i) (y : F.obj j) :
colimit.ι F i x = colimit.ι F j y ↔ ∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y | ⟨concrete.colimit_exists_of_rep_eq _ _ _, concrete.colimit_rep_eq_of_exists _ _ _⟩ | theorem | category_theory.limits.concrete.colimit_rep_eq_iff_exists | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.wide_pushout_exists_rep {B : C} {α : Type*} {X : α → C} (f : Π j : α, B ⟶ X j)
[has_wide_pushout.{v} B X f] [preserves_colimit (wide_span B X f) (forget C)]
(x : wide_pushout B X f) : (∃ y : B, head f y = x) ∨ (∃ (i : α) (y : X i), ι f i y = x) | begin
obtain ⟨_ | j, y, rfl⟩ := concrete.colimit_exists_rep _ x,
{ use y },
{ right,
use [j,y] }
end | lemma | category_theory.limits.concrete.wide_pushout_exists_rep | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete.wide_pushout_exists_rep' {B : C} {α : Type*} [nonempty α] {X : α → C}
(f : Π j : α, B ⟶ X j) [has_wide_pushout.{v} B X f]
[preserves_colimit (wide_span B X f) (forget C)] (x : wide_pushout B X f) :
∃ (i : α) (y : X i), ι f i y = x | begin
rcases concrete.wide_pushout_exists_rep f x with ⟨y, rfl⟩ | ⟨i, y, rfl⟩,
{ inhabit α,
use [arbitrary _, f _ y],
simp only [← arrow_ι _ (arbitrary α), comp_apply] },
{ use [i,y] }
end | lemma | category_theory.limits.concrete.wide_pushout_exists_rep' | category_theory.limits | src/category_theory/limits/concrete_category.lean | [
"category_theory.limits.preserves.basic",
"category_theory.limits.types",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.multiequalizer",
"category_theory.concrete_category.basic",
"category_theory.limits.shapes.kernels",
"tactic.apply_fun"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cones : Cᵒᵖ ⥤ Type (max u₁ v₃) | (const J).op ⋙ yoneda.obj F | def | category_theory.functor.cones | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | `F.cones` is the functor assigning to an object `X` the type of
natural transformations from the constant functor with value `X` to `F`.
An object representing this functor is a limit of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocones : C ⥤ Type (max u₁ v₃) | const J ⋙ coyoneda.obj (op F) | def | category_theory.functor.cocones | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | `F.cocones` is the functor assigning to an object `X` the type of
natural transformations from `F` to the constant functor with value `X`.
An object corepresenting this functor is a colimit of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cones : (J ⥤ C) ⥤ (Cᵒᵖ ⥤ Type (max u₁ v₃)) | { obj := functor.cones,
map := λ F G f, whisker_left (const J).op (yoneda.map f) } | def | category_theory.cones | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Functorially associated to each functor `J ⥤ C`, we have the `C`-presheaf consisting of
cones with a given cone point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocones : (J ⥤ C)ᵒᵖ ⥤ (C ⥤ Type (max u₁ v₃)) | { obj := λ F, functor.cocones (unop F),
map := λ F G f, whisker_left (const J) (coyoneda.map f) } | def | category_theory.cocones | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | Contravariantly associated to each functor `J ⥤ C`, we have the `C`-copresheaf consisting of
cocones with a given cocone point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone (F : J ⥤ C) | (X : C)
(π : (const J).obj X ⟶ F) | structure | category_theory.limits.cone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | A `c : cone F` is:
* an object `c.X` and
* a natural transformation `c.π : c.X ⟶ F` from the constant `c.X` functor to `F`.
`cone F` is equivalent, via `cone.equiv` below, to `Σ X, F.cones.obj X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited_cone (F : discrete punit ⥤ C) : inhabited (cone F) | ⟨{ X := F.obj ⟨⟨⟩⟩,
π :=
{ app := λ ⟨⟨⟩⟩, 𝟙 _, }, }⟩ | instance | category_theory.limits.inhabited_cone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cone.w {F : J ⥤ C} (c : cone F) {j j' : J} (f : j ⟶ j') :
c.π.app j ≫ F.map f = c.π.app j' | by { rw ← c.π.naturality f, apply id_comp } | lemma | category_theory.limits.cone.w | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone (F : J ⥤ C) | (X : C)
(ι : F ⟶ (const J).obj X) | structure | category_theory.limits.cocone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | A `c : cocone F` is
* an object `c.X` and
* a natural transformation `c.ι : F ⟶ c.X` from `F` to the constant `c.X` functor.
`cocone F` is equivalent, via `cone.equiv` below, to `Σ X, F.cocones.obj X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited_cocone (F : discrete punit ⥤ C) : inhabited (cocone F) | ⟨{ X := F.obj ⟨⟨⟩⟩,
ι :=
{ app := λ ⟨⟨⟩⟩, 𝟙 _, }, }⟩ | instance | category_theory.limits.inhabited_cocone | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocone.w {F : J ⥤ C} (c : cocone F) {j j' : J} (f : j ⟶ j') :
F.map f ≫ c.ι.app j' = c.ι.app j | by { rw c.ι.naturality f, apply comp_id } | lemma | category_theory.limits.cocone.w | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv (F : J ⥤ C) : cone F ≅ Σ X, F.cones.obj X | { hom := λ c, ⟨op c.X, c.π⟩,
inv := λ c, { X := c.1.unop, π := c.2 },
hom_inv_id' := by { ext1, cases x, refl },
inv_hom_id' := by { ext1, cases x, refl } } | def | category_theory.limits.cone.equiv | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [
"equiv"
] | The isomorphism between a cone on `F` and an element of the functor `F.cones`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extensions (c : cone F) :
yoneda.obj c.X ⋙ ulift_functor.{u₁} ⟶ F.cones | { app := λ X f, (const J).map f.down ≫ c.π } | def | category_theory.limits.cone.extensions | category_theory.limits | src/category_theory/limits/cones.lean | [
"category_theory.functor.const",
"category_theory.discrete_category",
"category_theory.yoneda",
"category_theory.functor.reflects_isomorphisms"
] | [] | A map to the vertex of a cone naturally induces a cone by composition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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