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limit.post_π (j : J) :
limit.post F G ≫ limit.π (F ⋙ G) j = G.map (limit.π F j) | by { erw is_limit.fac, refl } | lemma | category_theory.limits.limit.post_π | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.lift_post (c : cone F) :
G.map (limit.lift F c) ≫ limit.post F G = limit.lift (F ⋙ G) (G.map_cone c) | by { ext, rw [assoc, limit.post_π, ←G.map_comp, limit.lift_π, limit.lift_π], refl } | lemma | category_theory.limits.limit.lift_post | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.post_post
{E : Type u''} [category.{v''} E] (H : D ⥤ E) [has_limit ((F ⋙ G) ⋙ H)] :
/- H G (limit F) ⟶ H (limit (F ⋙ G)) ⟶ limit ((F ⋙ G) ⋙ H) equals -/
/- H G (limit F) ⟶ limit (F ⋙ (G ⋙ H)) -/
H.map (limit.post F G) ≫ limit.post (F ⋙ G) H = limit.post F (G ⋙ H) | by ext; erw [assoc, limit.post_π, ←H.map_comp, limit.post_π, limit.post_π]; refl | lemma | category_theory.limits.limit.post_post | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.pre_post {D : Type u'} [category.{v'} D]
(E : K ⥤ J) (F : J ⥤ C) (G : C ⥤ D)
[has_limit F] [has_limit (E ⋙ F)] [has_limit (F ⋙ G)] [has_limit ((E ⋙ F) ⋙ G)] :
/- G (limit F) ⟶ G (limit (E ⋙ F)) ⟶ limit ((E ⋙ F) ⋙ G) vs -/
/- G (limit F) ⟶ limit F ⋙ G ⟶ limit (E ⋙ (F ⋙ G)) or -/
G.map (limit.pre F E) ≫ limit... | by ext; erw [assoc, limit.post_π, ←G.map_comp, limit.pre_π, assoc, limit.pre_π, limit.post_π]; refl | lemma | category_theory.limits.limit.pre_post | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limit_equivalence_comp (e : K ≌ J) [has_limit F] : has_limit (e.functor ⋙ F) | has_limit.mk { cone := cone.whisker e.functor (limit.cone F),
is_limit := is_limit.whisker_equivalence (limit.is_limit F) e, } | instance | category_theory.limits.has_limit_equivalence_comp | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limit_of_equivalence_comp (e : K ≌ J) [has_limit (e.functor ⋙ F)] : has_limit F | begin
haveI : has_limit (e.inverse ⋙ e.functor ⋙ F) := limits.has_limit_equivalence_comp e.symm,
apply has_limit_of_iso (e.inv_fun_id_assoc F),
end | lemma | category_theory.limits.has_limit_of_equivalence_comp | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | If a `E ⋙ F` has a limit, and `E` is an equivalence, we can construct a limit of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lim : (J ⥤ C) ⥤ C | { obj := λ F, limit F,
map := λ F G α, lim_map α,
map_id' := λ F, by { ext, erw [lim_map_π, category.id_comp, category.comp_id] },
map_comp' := λ F G H α β,
by ext; erw [assoc, is_limit.fac, is_limit.fac, ←assoc, is_limit.fac, assoc]; refl } | def | category_theory.limits.lim | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"lim"
] | `limit F` is functorial in `F`, when `C` has all limits of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lim_map_eq_lim_map : lim.map α = lim_map α | rfl | lemma | category_theory.limits.lim_map_eq_lim_map | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.map_pre [has_limits_of_shape K C] (E : K ⥤ J) :
lim.map α ≫ limit.pre G E = limit.pre F E ≫ lim.map (whisker_left E α) | by { ext, simp } | lemma | category_theory.limits.limit.map_pre | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.map_pre' [has_limits_of_shape K C]
(F : J ⥤ C) {E₁ E₂ : K ⥤ J} (α : E₁ ⟶ E₂) :
limit.pre F E₂ = limit.pre F E₁ ≫ lim.map (whisker_right α F) | by ext1; simp [← category.assoc] | lemma | category_theory.limits.limit.map_pre' | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.id_pre (F : J ⥤ C) :
limit.pre F (𝟭 _) = lim.map (functor.left_unitor F).inv | by tidy | lemma | category_theory.limits.limit.id_pre | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.map_post {D : Type u'} [category.{v'} D] [has_limits_of_shape J D] (H : C ⥤ D) :
/- H (limit F) ⟶ H (limit G) ⟶ limit (G ⋙ H) vs
H (limit F) ⟶ limit (F ⋙ H) ⟶ limit (G ⋙ H) -/
H.map (lim_map α) ≫ limit.post G H = limit.post F H ≫ lim_map (whisker_right α H) | begin
ext,
simp only [whisker_right_app, lim_map_π, assoc, limit.post_π_assoc, limit.post_π, ← H.map_comp],
end | lemma | category_theory.limits.limit.map_post | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lim_yoneda : lim ⋙ yoneda ⋙ (whiskering_right _ _ _).obj ulift_functor.{u₁}
≅ category_theory.cones J C | nat_iso.of_components (λ F, nat_iso.of_components (λ W, limit.hom_iso F (unop W)) (by tidy))
(by tidy) | def | category_theory.limits.lim_yoneda | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"category_theory.cones",
"lim"
] | The isomorphism between
morphisms from `W` to the cone point of the limit cone for `F`
and cones over `F` with cone point `W`
is natural in `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_lim_adj : (const J : C ⥤ (J ⥤ C)) ⊣ lim | { hom_equiv := λ c g,
{ to_fun := λ f, limit.lift _ ⟨c, f⟩,
inv_fun := λ f, { app := λ j, f ≫ limit.π _ _ , naturality' := by tidy },
left_inv := λ _, nat_trans.ext _ _ $ funext $ λ j, limit.lift_π _ _,
right_inv := λ α, limit.hom_ext $ λ j, limit.lift_π _ _ },
unit := { app := λ c, limit.lift _ ⟨_, 𝟙 ... | def | category_theory.limits.const_lim_adj | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"inv_fun",
"lim"
] | The constant functor and limit functor are adjoint to each other | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lim_map_mono' {F G : J ⥤ C} [has_limits_of_shape J C] (α : F ⟶ G)
[mono α] : mono (lim_map α) | (lim : (J ⥤ C) ⥤ C).map_mono α | instance | category_theory.limits.lim_map_mono' | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"lim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lim_map_mono {F G : J ⥤ C} [has_limit F] [has_limit G] (α : F ⟶ G)
[∀ j, mono (α.app j)] : mono (lim_map α) | ⟨λ Z u v h, limit.hom_ext $ λ j, (cancel_mono (α.app j)).1 $ by simpa using h =≫ limit.π _ j⟩ | instance | category_theory.limits.lim_map_mono | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits_of_shape_of_equivalence {J' : Type u₂} [category.{v₂} J']
(e : J ≌ J') [has_limits_of_shape J C] : has_limits_of_shape J' C | by { constructor, intro F, apply has_limit_of_equivalence_comp e, apply_instance } | lemma | category_theory.limits.has_limits_of_shape_of_equivalence | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | We can transport limits of shape `J` along an equivalence `J ≌ J'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_size_shrink [has_limits_of_size.{(max v₁ v₂) (max u₁ u₂)} C] :
has_limits_of_size.{v₁ u₁} C | ⟨λ J hJ, by exactI has_limits_of_shape_of_equivalence
(ulift_hom_ulift_category.equiv.{v₂ u₂} J).symm⟩ | lemma | category_theory.limits.has_limits_of_size_shrink | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | `has_limits_of_size_shrink.{v u} C` tries to obtain `has_limits_of_size.{v u} C`
from some other `has_limits_of_size C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_smallest_limits_of_has_limits [has_limits C] :
has_limits_of_size.{0 0} C | has_limits_of_size_shrink.{0 0} C | instance | category_theory.limits.has_smallest_limits_of_has_limits | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_cocone (F : J ⥤ C) | (cocone : cocone F)
(is_colimit : is_colimit cocone) | structure | category_theory.limits.colimit_cocone | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | `colimit_cocone F` contains a cocone over `F` together with the information that it is a
colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimit (F : J ⥤ C) : Prop | mk' :: (exists_colimit : nonempty (colimit_cocone F)) | class | category_theory.limits.has_colimit | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"mk'"
] | `has_colimit F` represents the mere existence of a colimit for `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimit.mk {F : J ⥤ C} (d : colimit_cocone F) : has_colimit F | ⟨nonempty.intro d⟩ | lemma | category_theory.limits.has_colimit.mk | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
get_colimit_cocone (F : J ⥤ C) [has_colimit F] : colimit_cocone F | classical.choice $ has_colimit.exists_colimit | def | category_theory.limits.get_colimit_cocone | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | Use the axiom of choice to extract explicit `colimit_cocone F` from `has_colimit F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_of_shape : Prop | (has_colimit : Π F : J ⥤ C, has_colimit F . tactic.apply_instance) | class | category_theory.limits.has_colimits_of_shape | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | `C` has colimits of shape `J` if there exists a colimit for every functor `F : J ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_of_size (C : Type u) [category.{v} C] : Prop | (has_colimits_of_shape :
Π (J : Type u₁) [𝒥 : category.{v₁} J], has_colimits_of_shape J C . tactic.apply_instance) | class | category_theory.limits.has_colimits_of_size | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | `C` has all colimits of size `v₁ u₁` (`has_colimits_of_size.{v₁ u₁} C`)
if it has colimits of every shape `J : Type u₁` with `[category.{v₁} J]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits (C : Type u) [category.{v} C] : Prop | has_colimits_of_size.{v v} C | abbreviation | category_theory.limits.has_colimits | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | `C` has all (small) colimits if it has colimits of every shape that is as big as its hom-sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits.has_colimits_of_shape {C : Type u} [category.{v} C] [has_colimits C]
(J : Type v) [category.{v} J] :
has_colimits_of_shape J C | has_colimits_of_size.has_colimits_of_shape J | lemma | category_theory.limits.has_colimits.has_colimits_of_shape | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit_of_has_colimits_of_shape
{J : Type u₁} [category.{v₁} J] [H : has_colimits_of_shape J C] (F : J ⥤ C) : has_colimit F | has_colimits_of_shape.has_colimit F | instance | category_theory.limits.has_colimit_of_has_colimits_of_shape | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_shape_of_has_colimits_of_size {J : Type u₁} [category.{v₁} J]
[H : has_colimits_of_size.{v₁ u₁} C] : has_colimits_of_shape J C | has_colimits_of_size.has_colimits_of_shape J | instance | category_theory.limits.has_colimits_of_shape_of_has_colimits_of_size | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.cocone (F : J ⥤ C) [has_colimit F] : cocone F | (get_colimit_cocone F).cocone | def | category_theory.limits.colimit.cocone | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | An arbitrary choice of colimit cocone of a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit (F : J ⥤ C) [has_colimit F] | (colimit.cocone F).X | def | category_theory.limits.colimit | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | An arbitrary choice of colimit object of a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit.ι (F : J ⥤ C) [has_colimit F] (j : J) : F.obj j ⟶ colimit F | (colimit.cocone F).ι.app j | def | category_theory.limits.colimit.ι | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | The coprojection from a value of the functor to the colimit object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit.cocone_ι {F : J ⥤ C} [has_colimit F] (j : J) :
(colimit.cocone F).ι.app j = colimit.ι _ j | rfl | lemma | category_theory.limits.colimit.cocone_ι | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.cocone_X {F : J ⥤ C} [has_colimit F] :
(colimit.cocone F).X = colimit F | rfl | lemma | category_theory.limits.colimit.cocone_X | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.w (F : J ⥤ C) [has_colimit F] {j j' : J} (f : j ⟶ j') :
F.map f ≫ colimit.ι F j' = colimit.ι F j | (colimit.cocone F).w f | lemma | category_theory.limits.colimit.w | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.is_colimit (F : J ⥤ C) [has_colimit F] : is_colimit (colimit.cocone F) | (get_colimit_cocone F).is_colimit | def | category_theory.limits.colimit.is_colimit | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | Evidence that the arbitrary choice of cocone is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit.desc (F : J ⥤ C) [has_colimit F] (c : cocone F) : colimit F ⟶ c.X | (colimit.is_colimit F).desc c | def | category_theory.limits.colimit.desc | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | The morphism from the colimit object to the cone point of any other cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit.is_colimit_desc {F : J ⥤ C} [has_colimit F] (c : cocone F) :
(colimit.is_colimit F).desc c = colimit.desc F c | rfl | lemma | category_theory.limits.colimit.is_colimit_desc | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.ι_desc {F : J ⥤ C} [has_colimit F] (c : cocone F) (j : J) :
colimit.ι F j ≫ colimit.desc F c = c.ι.app j | is_colimit.fac _ c j | lemma | category_theory.limits.colimit.ι_desc | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | We have lots of lemmas describing how to simplify `colimit.ι F j ≫ _`,
and combined with `colimit.ext` we rely on these lemmas for many calculations.
However, since `category.assoc` is a `@[simp]` lemma, often expressions are
right associated, and it's hard to apply these lemmas about `colimit.ι`.
We thus use `reasso... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colim_map {F G : J ⥤ C} [has_colimit F] [has_colimit G] (α : F ⟶ G) : colimit F ⟶ colimit G | is_colimit.map (colimit.is_colimit F) _ α | def | category_theory.limits.colim_map | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | Functoriality of colimits.
Usually this morphism should be accessed through `colim.map`,
but may be needed separately when you have specified colimits for the source and target functors,
but not necessarily for all functors of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_colim_map {F G : J ⥤ C} [has_colimit F] [has_colimit G] (α : F ⟶ G) (j : J) :
colimit.ι F j ≫ colim_map α = α.app j ≫ colimit.ι G j | colimit.ι_desc _ j | lemma | category_theory.limits.ι_colim_map | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.cocone_morphism {F : J ⥤ C} [has_colimit F] (c : cocone F) :
(colimit.cocone F) ⟶ c | (colimit.is_colimit F).desc_cocone_morphism c | def | category_theory.limits.colimit.cocone_morphism | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | The cocone morphism from the arbitrary choice of colimit cocone to any cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit.cocone_morphism_hom {F : J ⥤ C} [has_colimit F] (c : cocone F) :
(colimit.cocone_morphism c).hom = colimit.desc F c | rfl | lemma | category_theory.limits.colimit.cocone_morphism_hom | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.ι_cocone_morphism {F : J ⥤ C} [has_colimit F] (c : cocone F) (j : J) :
colimit.ι F j ≫ (colimit.cocone_morphism c).hom = c.ι.app j | by simp | lemma | category_theory.limits.colimit.ι_cocone_morphism | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.comp_cocone_point_unique_up_to_iso_hom {F : J ⥤ C} [has_colimit F]
{c : cocone F} (hc : is_colimit c) (j : J) :
colimit.ι F j ≫ (is_colimit.cocone_point_unique_up_to_iso (colimit.is_colimit _) hc).hom =
c.ι.app j | is_colimit.comp_cocone_point_unique_up_to_iso_hom _ _ _ | lemma | category_theory.limits.colimit.comp_cocone_point_unique_up_to_iso_hom | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.comp_cocone_point_unique_up_to_iso_inv {F : J ⥤ C} [has_colimit F]
{c : cocone F} (hc : is_colimit c) (j : J) :
colimit.ι F j ≫ (is_colimit.cocone_point_unique_up_to_iso hc (colimit.is_colimit _)).inv =
c.ι.app j | is_colimit.comp_cocone_point_unique_up_to_iso_inv _ _ _ | lemma | category_theory.limits.colimit.comp_cocone_point_unique_up_to_iso_inv | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.exists_unique {F : J ⥤ C} [has_colimit F] (t : cocone F) :
∃! (d : colimit F ⟶ t.X), ∀ j, colimit.ι F j ≫ d = t.ι.app j | (colimit.is_colimit F).exists_unique _ | lemma | category_theory.limits.colimit.exists_unique | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.iso_colimit_cocone {F : J ⥤ C} [has_colimit F] (t : colimit_cocone F) :
colimit F ≅ t.cocone.X | is_colimit.cocone_point_unique_up_to_iso (colimit.is_colimit F) t.is_colimit | def | category_theory.limits.colimit.iso_colimit_cocone | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | Given any other colimit cocone for `F`, the chosen `colimit F` is isomorphic to the cocone point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit.iso_colimit_cocone_ι_hom
{F : J ⥤ C} [has_colimit F] (t : colimit_cocone F) (j : J) :
colimit.ι F j ≫ (colimit.iso_colimit_cocone t).hom = t.cocone.ι.app j | by { dsimp [colimit.iso_colimit_cocone, is_colimit.cocone_point_unique_up_to_iso], tidy, } | lemma | category_theory.limits.colimit.iso_colimit_cocone_ι_hom | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.iso_colimit_cocone_ι_inv
{F : J ⥤ C} [has_colimit F] (t : colimit_cocone F) (j : J) :
t.cocone.ι.app j ≫ (colimit.iso_colimit_cocone t).inv = colimit.ι F j | by { dsimp [colimit.iso_colimit_cocone, is_colimit.cocone_point_unique_up_to_iso], tidy, } | lemma | category_theory.limits.colimit.iso_colimit_cocone_ι_inv | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.hom_ext {F : J ⥤ C} [has_colimit F] {X : C} {f f' : colimit F ⟶ X}
(w : ∀ j, colimit.ι F j ≫ f = colimit.ι F j ≫ f') : f = f' | (colimit.is_colimit F).hom_ext w | lemma | category_theory.limits.colimit.hom_ext | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.desc_cocone {F : J ⥤ C} [has_colimit F] :
colimit.desc F (colimit.cocone F) = 𝟙 (colimit F) | (colimit.is_colimit _).desc_self | lemma | category_theory.limits.colimit.desc_cocone | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.hom_iso (F : J ⥤ C) [has_colimit F] (W : C) :
ulift.{u₁} (colimit F ⟶ W : Type v) ≅ (F.cocones.obj W) | (colimit.is_colimit F).hom_iso W | def | category_theory.limits.colimit.hom_iso | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | The isomorphism (in `Type`) between
morphisms from the colimit object to a specified object `W`,
and cocones with cone point `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit.hom_iso_hom (F : J ⥤ C) [has_colimit F] {W : C} (f : ulift (colimit F ⟶ W)) :
(colimit.hom_iso F W).hom f = (colimit.cocone F).ι ≫ (const J).map f.down | (colimit.is_colimit F).hom_iso_hom f | lemma | category_theory.limits.colimit.hom_iso_hom | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.hom_iso' (F : J ⥤ C) [has_colimit F] (W : C) :
ulift.{u₁} ((colimit F ⟶ W) : Type v) ≅
{ p : Π j, F.obj j ⟶ W // ∀ {j j'} (f : j ⟶ j'), F.map f ≫ p j' = p j } | (colimit.is_colimit F).hom_iso' W | def | category_theory.limits.colimit.hom_iso' | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | The isomorphism (in `Type`) between
morphisms from the colimit object to a specified object `W`,
and an explicit componentwise description of cocones with cone point `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit.desc_extend (F : J ⥤ C) [has_colimit F] (c : cocone F) {X : C} (f : c.X ⟶ X) :
colimit.desc F (c.extend f) = colimit.desc F c ≫ f | begin
ext1, rw [←category.assoc], simp
end | lemma | category_theory.limits.colimit.desc_extend | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit_of_iso {F G : J ⥤ C} [has_colimit F] (α : G ≅ F) : has_colimit G | has_colimit.mk
{ cocone := (cocones.precompose α.hom).obj (colimit.cocone F),
is_colimit :=
{ desc := λ s, colimit.desc F ((cocones.precompose α.inv).obj s),
fac' := λ s j,
begin
rw [cocones.precompose_obj_ι, nat_trans.comp_app, colimit.cocone_ι],
rw [category.assoc, colimit.ι_desc, ←nat_iso.app... | lemma | category_theory.limits.has_colimit_of_iso | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit.of_cocones_iso {K : Type u₁} [category.{v₂} K] (F : J ⥤ C)
(G : K ⥤ C) (h : F.cocones ≅ G.cocones)
[has_colimit F] : has_colimit G | has_colimit.mk ⟨_, is_colimit.of_nat_iso (is_colimit.nat_iso (colimit.is_colimit F) ≪≫ h)⟩ | lemma | category_theory.limits.has_colimit.of_cocones_iso | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | If a functor `G` has the same collection of cocones as a functor `F`
which has a colimit, then `G` also has a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimit.iso_of_nat_iso {F G : J ⥤ C} [has_colimit F] [has_colimit G] (w : F ≅ G) :
colimit F ≅ colimit G | is_colimit.cocone_points_iso_of_nat_iso (colimit.is_colimit F) (colimit.is_colimit G) w | def | category_theory.limits.has_colimit.iso_of_nat_iso | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | The colimits of `F : J ⥤ C` and `G : J ⥤ C` are isomorphic,
if the functors are naturally isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimit.iso_of_nat_iso_ι_hom {F G : J ⥤ C} [has_colimit F] [has_colimit G]
(w : F ≅ G) (j : J) :
colimit.ι F j ≫ (has_colimit.iso_of_nat_iso w).hom = w.hom.app j ≫ colimit.ι G j | is_colimit.comp_cocone_points_iso_of_nat_iso_hom _ _ _ _ | lemma | category_theory.limits.has_colimit.iso_of_nat_iso_ι_hom | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit.iso_of_nat_iso_ι_inv {F G : J ⥤ C} [has_colimit F] [has_colimit G]
(w : F ≅ G) (j : J) :
colimit.ι G j ≫ (has_colimit.iso_of_nat_iso w).inv = w.inv.app j ≫ colimit.ι F j | is_colimit.comp_cocone_points_iso_of_nat_iso_inv _ _ _ _ | lemma | category_theory.limits.has_colimit.iso_of_nat_iso_ι_inv | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit.iso_of_nat_iso_hom_desc {F G : J ⥤ C} [has_colimit F] [has_colimit G]
(t : cocone G) (w : F ≅ G) :
(has_colimit.iso_of_nat_iso w).hom ≫ colimit.desc G t =
colimit.desc F ((cocones.precompose w.hom).obj _) | is_colimit.cocone_points_iso_of_nat_iso_hom_desc _ _ _ | lemma | category_theory.limits.has_colimit.iso_of_nat_iso_hom_desc | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit.iso_of_nat_iso_inv_desc {F G : J ⥤ C} [has_colimit F] [has_colimit G]
(t : cocone F) (w : F ≅ G) :
(has_colimit.iso_of_nat_iso w).inv ≫ colimit.desc F t =
colimit.desc G ((cocones.precompose w.inv).obj _) | is_colimit.cocone_points_iso_of_nat_iso_inv_desc _ _ _ | lemma | category_theory.limits.has_colimit.iso_of_nat_iso_inv_desc | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit.iso_of_equivalence {F : J ⥤ C} [has_colimit F] {G : K ⥤ C} [has_colimit G]
(e : J ≌ K) (w : e.functor ⋙ G ≅ F) : colimit F ≅ colimit G | is_colimit.cocone_points_iso_of_equivalence (colimit.is_colimit F) (colimit.is_colimit G) e w | def | category_theory.limits.has_colimit.iso_of_equivalence | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | The colimits of `F : J ⥤ C` and `G : K ⥤ C` are isomorphic,
if there is an equivalence `e : J ≌ K` making the triangle commute up to natural isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimit.iso_of_equivalence_hom_π {F : J ⥤ C} [has_colimit F] {G : K ⥤ C} [has_colimit G]
(e : J ≌ K) (w : e.functor ⋙ G ≅ F) (j : J) :
colimit.ι F j ≫ (has_colimit.iso_of_equivalence e w).hom =
F.map (e.unit.app j) ≫ w.inv.app _ ≫ colimit.ι G _ | begin
simp [has_colimit.iso_of_equivalence, is_colimit.cocone_points_iso_of_equivalence_inv],
dsimp,
simp,
end | lemma | category_theory.limits.has_colimit.iso_of_equivalence_hom_π | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit.iso_of_equivalence_inv_π {F : J ⥤ C} [has_colimit F] {G : K ⥤ C} [has_colimit G]
(e : J ≌ K) (w : e.functor ⋙ G ≅ F) (k : K) :
colimit.ι G k ≫ (has_colimit.iso_of_equivalence e w).inv =
G.map (e.counit_inv.app k) ≫ w.hom.app (e.inverse.obj k) ≫ colimit.ι F (e.inverse.obj k) | begin
simp [has_colimit.iso_of_equivalence, is_colimit.cocone_points_iso_of_equivalence_inv],
dsimp,
simp,
end | lemma | category_theory.limits.has_colimit.iso_of_equivalence_inv_π | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.pre : colimit (E ⋙ F) ⟶ colimit F | colimit.desc (E ⋙ F) ((colimit.cocone F).whisker E) | def | category_theory.limits.colimit.pre | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | The canonical morphism from the colimit of `E ⋙ F` to the colimit of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit.ι_pre (k : K) :
colimit.ι (E ⋙ F) k ≫ colimit.pre F E = colimit.ι F (E.obj k) | by { erw is_colimit.fac, refl, } | lemma | category_theory.limits.colimit.ι_pre | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.pre_desc (c : cocone F) :
colimit.pre F E ≫ colimit.desc F c = colimit.desc (E ⋙ F) (c.whisker E) | by ext; rw [←assoc, colimit.ι_pre]; simp | lemma | category_theory.limits.colimit.pre_desc | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.pre_pre : colimit.pre (E ⋙ F) D ≫ colimit.pre F E = colimit.pre F (D ⋙ E) | begin
ext j,
rw [←assoc, colimit.ι_pre, colimit.ι_pre],
letI : has_colimit ((D ⋙ E) ⋙ F) := show has_colimit (D ⋙ E ⋙ F), by apply_instance,
exact (colimit.ι_pre F (D ⋙ E) j).symm
end | lemma | category_theory.limits.colimit.pre_pre | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.pre_eq (s : colimit_cocone (E ⋙ F)) (t : colimit_cocone F) :
colimit.pre F E =
(colimit.iso_colimit_cocone s).hom ≫ s.is_colimit.desc ((t.cocone).whisker E) ≫
(colimit.iso_colimit_cocone t).inv | by tidy | lemma | category_theory.limits.colimit.pre_eq | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.post : colimit (F ⋙ G) ⟶ G.obj (colimit F) | colimit.desc (F ⋙ G) (G.map_cocone (colimit.cocone F)) | def | category_theory.limits.colimit.post | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | The canonical morphism from `G` applied to the colimit of `F ⋙ G`
to `G` applied to the colimit of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit.ι_post (j : J) :
colimit.ι (F ⋙ G) j ≫ colimit.post F G = G.map (colimit.ι F j) | by { erw is_colimit.fac, refl, } | lemma | category_theory.limits.colimit.ι_post | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.post_desc (c : cocone F) :
colimit.post F G ≫ G.map (colimit.desc F c) = colimit.desc (F ⋙ G) (G.map_cocone c) | by { ext, rw [←assoc, colimit.ι_post, ←G.map_comp, colimit.ι_desc, colimit.ι_desc], refl } | lemma | category_theory.limits.colimit.post_desc | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.post_post
{E : Type u''} [category.{v''} E] (H : D ⥤ E) [has_colimit ((F ⋙ G) ⋙ H)] :
/- H G (colimit F) ⟶ H (colimit (F ⋙ G)) ⟶ colimit ((F ⋙ G) ⋙ H) equals -/
/- H G (colimit F) ⟶ colimit (F ⋙ (G ⋙ H)) -/
colimit.post (F ⋙ G) H ≫ H.map (colimit.post F G) = colimit.post F (G ⋙ H) | begin
ext,
rw [←assoc, colimit.ι_post, ←H.map_comp, colimit.ι_post],
exact (colimit.ι_post F (G ⋙ H) j).symm
end | lemma | category_theory.limits.colimit.post_post | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.pre_post {D : Type u'} [category.{v'} D]
(E : K ⥤ J) (F : J ⥤ C) (G : C ⥤ D)
[has_colimit F] [has_colimit (E ⋙ F)] [has_colimit (F ⋙ G)] [H : has_colimit ((E ⋙ F) ⋙ G)] :
/- G (colimit F) ⟶ G (colimit (E ⋙ F)) ⟶ colimit ((E ⋙ F) ⋙ G) vs -/
/- G (colimit F) ⟶ colimit F ⋙ G ⟶ colimit (E ⋙ (F ⋙ G)) or -/
col... | begin
ext,
rw [←assoc, colimit.ι_post, ←G.map_comp, colimit.ι_pre, ←assoc],
letI : has_colimit (E ⋙ F ⋙ G) := show has_colimit ((E ⋙ F) ⋙ G), by apply_instance,
erw [colimit.ι_pre (F ⋙ G) E j, colimit.ι_post]
end | lemma | category_theory.limits.colimit.pre_post | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit_equivalence_comp (e : K ≌ J) [has_colimit F] : has_colimit (e.functor ⋙ F) | has_colimit.mk { cocone := cocone.whisker e.functor (colimit.cocone F),
is_colimit := is_colimit.whisker_equivalence (colimit.is_colimit F) e, } | instance | category_theory.limits.has_colimit_equivalence_comp | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimit_of_equivalence_comp (e : K ≌ J) [has_colimit (e.functor ⋙ F)] : has_colimit F | begin
haveI : has_colimit (e.inverse ⋙ e.functor ⋙ F) := limits.has_colimit_equivalence_comp e.symm,
apply has_colimit_of_iso (e.inv_fun_id_assoc F).symm,
end | lemma | category_theory.limits.has_colimit_of_equivalence_comp | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | If a `E ⋙ F` has a colimit, and `E` is an equivalence, we can construct a colimit of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colim : (J ⥤ C) ⥤ C | { obj := λ F, colimit F,
map := λ F G α, colim_map α,
map_id' := λ F, by { ext, erw [ι_colim_map, id_comp, comp_id] },
map_comp' := λ F G H α β,
by { ext, erw [←assoc, is_colimit.fac, is_colimit.fac, assoc, is_colimit.fac, ←assoc], refl } } | def | category_theory.limits.colim | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | `colimit F` is functorial in `F`, when `C` has all colimits of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit.ι_map (j : J) :
colimit.ι F j ≫ colim.map α = α.app j ≫ colimit.ι G j | by apply is_colimit.fac | lemma | category_theory.limits.colimit.ι_map | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.map_desc (c : cocone G) :
colim.map α ≫ colimit.desc G c = colimit.desc F ((cocones.precompose α).obj c) | by ext; rw [←assoc, colimit.ι_map, assoc, colimit.ι_desc, colimit.ι_desc]; refl | lemma | category_theory.limits.colimit.map_desc | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.pre_map [has_colimits_of_shape K C] (E : K ⥤ J) :
colimit.pre F E ≫ colim.map α = colim.map (whisker_left E α) ≫ colimit.pre G E | by ext; rw [←assoc, colimit.ι_pre, colimit.ι_map, ←assoc, colimit.ι_map, assoc, colimit.ι_pre]; refl | lemma | category_theory.limits.colimit.pre_map | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.pre_map' [has_colimits_of_shape K C]
(F : J ⥤ C) {E₁ E₂ : K ⥤ J} (α : E₁ ⟶ E₂) :
colimit.pre F E₁ = colim.map (whisker_right α F) ≫ colimit.pre F E₂ | by ext1; simp [← category.assoc] | lemma | category_theory.limits.colimit.pre_map' | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.pre_id (F : J ⥤ C) :
colimit.pre F (𝟭 _) = colim.map (functor.left_unitor F).hom | by tidy | lemma | category_theory.limits.colimit.pre_id | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.map_post {D : Type u'} [category.{v'} D] [has_colimits_of_shape J D] (H : C ⥤ D) :
/- H (colimit F) ⟶ H (colimit G) ⟶ colimit (G ⋙ H) vs
H (colimit F) ⟶ colimit (F ⋙ H) ⟶ colimit (G ⋙ H) -/
colimit.post F H ≫ H.map (colim.map α) = colim.map (whisker_right α H) ≫ colimit.post G H | begin
ext,
rw [←assoc, colimit.ι_post, ←H.map_comp, colimit.ι_map, H.map_comp],
rw [←assoc, colimit.ι_map, assoc, colimit.ι_post],
refl
end | lemma | category_theory.limits.colimit.map_post | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colim_coyoneda : colim.op ⋙ coyoneda ⋙ (whiskering_right _ _ _).obj ulift_functor.{u₁} ≅
category_theory.cocones J C | nat_iso.of_components (λ F, nat_iso.of_components (colimit.hom_iso (unop F)) (by tidy))
(by tidy) | def | category_theory.limits.colim_coyoneda | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"category_theory.cocones"
] | The isomorphism between
morphisms from the cone point of the colimit cocone for `F` to `W`
and cocones over `F` with cone point `W`
is natural in `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colim_const_adj : (colim : (J ⥤ C) ⥤ C) ⊣ const J | { hom_equiv := λ f c,
{ to_fun := λ g, { app := λ _, colimit.ι _ _ ≫ g, naturality' := by tidy },
inv_fun := λ g, colimit.desc _ ⟨_, g⟩,
left_inv := λ _, colimit.hom_ext $ λ j, colimit.ι_desc _ _,
right_inv := λ _, nat_trans.ext _ _ $ funext $ λ j, colimit.ι_desc _ _ },
unit := { app := λ g, { app := co... | def | category_theory.limits.colim_const_adj | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"inv_fun"
] | The colimit functor and constant functor are adjoint to each other | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colim_map_epi' {F G : J ⥤ C} [has_colimits_of_shape J C] (α : F ⟶ G) [epi α] :
epi (colim_map α) | (colim : (J ⥤ C) ⥤ C).map_epi α | instance | category_theory.limits.colim_map_epi' | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colim_map_epi {F G : J ⥤ C} [has_colimit F] [has_colimit G] (α : F ⟶ G)
[∀ j, epi (α.app j)] : epi (colim_map α) | ⟨λ Z u v h, colimit.hom_ext $ λ j, (cancel_epi (α.app j)).1 $ by simpa using colimit.ι _ j ≫= h⟩ | instance | category_theory.limits.colim_map_epi | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_shape_of_equivalence {J' : Type u₂} [category.{v₂} J']
(e : J ≌ J') [has_colimits_of_shape J C] : has_colimits_of_shape J' C | by { constructor, intro F, apply has_colimit_of_equivalence_comp e, apply_instance } | lemma | category_theory.limits.has_colimits_of_shape_of_equivalence | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | We can transport colimits of shape `J` along an equivalence `J ≌ J'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_of_size_shrink [has_colimits_of_size.{(max v₁ v₂) (max u₁ u₂)} C] :
has_colimits_of_size.{v₁ u₁} C | ⟨λ J hJ, by exactI has_colimits_of_shape_of_equivalence
(ulift_hom_ulift_category.equiv.{v₂ u₂} J).symm⟩ | lemma | category_theory.limits.has_colimits_of_size_shrink | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | `has_colimits_of_size_shrink.{v u} C` tries to obtain `has_colimits_of_size.{v u} C`
from some other `has_colimits_of_size C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_smallest_colimits_of_has_colimits [has_colimits C] :
has_colimits_of_size.{0 0} C | has_colimits_of_size_shrink.{0 0} C | instance | category_theory.limits.has_smallest_colimits_of_has_colimits | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit.op {t : cone F} (P : is_limit t) : is_colimit t.op | { desc := λ s, (P.lift s.unop).op,
fac' := λ s j, congr_arg quiver.hom.op (P.fac s.unop (unop j)),
uniq' := λ s m w,
begin
rw ← P.uniq s.unop m.unop,
{ refl, },
{ dsimp, intro j, rw ← w, refl, }
end } | def | category_theory.limits.is_limit.op | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"quiver.hom.op"
] | If `t : cone F` is a limit cone, then `t.op : cocone F.op` is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit.op {t : cocone F} (P : is_colimit t) : is_limit t.op | { lift := λ s, (P.desc s.unop).op,
fac' := λ s j, congr_arg quiver.hom.op (P.fac s.unop (unop j)),
uniq' := λ s m w,
begin
rw ← P.uniq s.unop m.unop,
{ refl, },
{ dsimp, intro j, rw ← w, refl, }
end } | def | category_theory.limits.is_colimit.op | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"lift",
"quiver.hom.op"
] | If `t : cocone F` is a colimit cocone, then `t.op : cone F.op` is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit.unop {t : cone F.op} (P : is_limit t) : is_colimit t.unop | { desc := λ s, (P.lift s.op).unop,
fac' := λ s j, congr_arg quiver.hom.unop (P.fac s.op (op j)),
uniq' := λ s m w,
begin
rw ← P.uniq s.op m.op,
{ refl, },
{ dsimp, intro j, rw ← w, refl, }
end } | def | category_theory.limits.is_limit.unop | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"quiver.hom.unop"
] | If `t : cone F.op` is a limit cone, then `t.unop : cocone F` is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit.unop {t : cocone F.op} (P : is_colimit t) : is_limit t.unop | { lift := λ s, (P.desc s.op).unop,
fac' := λ s j, congr_arg quiver.hom.unop (P.fac s.op (op j)),
uniq' := λ s m w,
begin
rw ← P.uniq s.op m.op,
{ refl, },
{ dsimp, intro j, rw ← w, refl, }
end } | def | category_theory.limits.is_colimit.unop | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"lift",
"quiver.hom.unop"
] | If `t : cocone F.op` is a colimit cocone, then `t.unop : cone F.` is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_equiv_is_colimit_op {t : cone F} : is_limit t ≃ is_colimit t.op | equiv_of_subsingleton_of_subsingleton
is_limit.op (λ P, P.unop.of_iso_limit (cones.ext (iso.refl _) (by tidy))) | def | category_theory.limits.is_limit_equiv_is_colimit_op | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"equiv_of_subsingleton_of_subsingleton"
] | `t : cone F` is a limit cone if and only is `t.op : cocone F.op` is a colimit cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_equiv_is_limit_op {t : cocone F} : is_colimit t ≃ is_limit t.op | equiv_of_subsingleton_of_subsingleton
is_colimit.op (λ P, P.unop.of_iso_colimit (cocones.ext (iso.refl _) (by tidy))) | def | category_theory.limits.is_colimit_equiv_is_limit_op | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"equiv_of_subsingleton_of_subsingleton"
] | `t : cocone F` is a colimit cocone if and only is `t.op : cone F.op` is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit (t : cone F) | (lift : Π (s : cone F), s.X ⟶ t.X)
(fac' : ∀ (s : cone F) (j : J), lift s ≫ t.π.app j = s.π.app j . obviously)
(uniq' : ∀ (s : cone F) (m : s.X ⟶ t.X) (w : ∀ j : J, m ≫ t.π.app j = s.π.app j),
m = lift s . obviously) | structure | category_theory.limits.is_limit | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [
"lift"
] | A cone `t` on `F` is a limit cone if each cone on `F` admits a unique
cone morphism to `t`.
See <https://stacks.math.columbia.edu/tag/002E>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton {t : cone F} : subsingleton (is_limit t) | ⟨by intros P Q; cases P; cases Q; congr; ext; solve_by_elim⟩ | instance | category_theory.limits.is_limit.subsingleton | category_theory.limits | src/category_theory/limits/is_limit.lean | [
"category_theory.adjunction.basic",
"category_theory.limits.cones"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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