statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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has_colimit_of_closed_under_colimits (h : closed_under_colimits_of_shape J P)
(F : J ⥤ full_subcategory P) [has_colimit (F ⋙ full_subcategory_inclusion P)] : has_colimit F | have creates_colimit F (full_subcategory_inclusion P),
from creates_colimit_full_subcategory_inclusion_of_closed h F,
by exactI has_colimit_of_created F (full_subcategory_inclusion P) | lemma | category_theory.limits.has_colimit_of_closed_under_colimits | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_shape_of_closed_under_colimits (h : closed_under_colimits_of_shape J P)
[has_colimits_of_shape J C] : has_colimits_of_shape J (full_subcategory P) | { has_colimit := λ F, has_colimit_of_closed_under_colimits h F } | lemma | category_theory.limits.has_colimits_of_shape_of_closed_under_colimits | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.lift_π_app (H : J ⥤ K ⥤ C) [has_limit H] (c : cone H) (j : J) (k : K) :
(limit.lift H c).app k ≫ (limit.π H j).app k = (c.π.app j).app k | congr_app (limit.lift_π c j) k | lemma | category_theory.limits.limit.lift_π_app | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit.ι_desc_app (H : J ⥤ K ⥤ C) [has_colimit H] (c : cocone H) (j : J) (k : K) :
(colimit.ι H j).app k ≫ (colimit.desc H c).app k = (c.ι.app j).app k | congr_app (colimit.ι_desc c j) k | lemma | category_theory.limits.colimit.ι_desc_app | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
evaluation_jointly_reflects_limits {F : J ⥤ K ⥤ C} (c : cone F)
(t : Π (k : K), is_limit (((evaluation K C).obj k).map_cone c)) : is_limit c | { lift := λ s,
{ app := λ k, (t k).lift ⟨s.X.obj k, whisker_right s.π ((evaluation K C).obj k)⟩,
naturality' := λ X Y f, (t Y).hom_ext $ λ j,
begin
rw [assoc, (t Y).fac _ j],
simpa using
((t X).fac_assoc ⟨s.X.obj X, whisker_right s.π ((evaluation K C).obj X)⟩ j _).symm,
end },
fac' :... | def | category_theory.limits.evaluation_jointly_reflects_limits | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [
"hom_ext",
"lift"
] | The evaluation functors jointly reflect limits: that is, to show a cone is a limit of `F`
it suffices to show that each evaluation cone is a limit. In other words, to prove a cone is
limiting you can show it's pointwise limiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
combine_cones (F : J ⥤ K ⥤ C) (c : Π (k : K), limit_cone (F.flip.obj k)) :
cone F | { X :=
{ obj := λ k, (c k).cone.X,
map := λ k₁ k₂ f, (c k₂).is_limit.lift ⟨_, (c k₁).cone.π ≫ F.flip.map f⟩,
map_id' := λ k, (c k).is_limit.hom_ext (λ j, by { dsimp, simp }),
map_comp' := λ k₁ k₂ k₃ f₁ f₂, (c k₃).is_limit.hom_ext (λ j, by simp) },
π :=
{ app := λ j, { app := λ k, (c k).cone.π.app j },... | def | category_theory.limits.combine_cones | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | Given a functor `F` and a collection of limit cones for each diagram `X ↦ F X k`, we can stitch
them together to give a cone for the diagram `F`.
`combined_is_limit` shows that the new cone is limiting, and `eval_combined` shows it is
(essentially) made up of the original cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
evaluate_combined_cones (F : J ⥤ K ⥤ C) (c : Π (k : K), limit_cone (F.flip.obj k)) (k : K) :
((evaluation K C).obj k).map_cone (combine_cones F c) ≅ (c k).cone | cones.ext (iso.refl _) (by tidy) | def | category_theory.limits.evaluate_combined_cones | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | The stitched together cones each project down to the original given cones (up to iso). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
combined_is_limit (F : J ⥤ K ⥤ C) (c : Π (k : K), limit_cone (F.flip.obj k)) :
is_limit (combine_cones F c) | evaluation_jointly_reflects_limits _
(λ k, (c k).is_limit.of_iso_limit (evaluate_combined_cones F c k).symm) | def | category_theory.limits.combined_is_limit | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | Stitching together limiting cones gives a limiting cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
evaluation_jointly_reflects_colimits {F : J ⥤ K ⥤ C} (c : cocone F)
(t : Π (k : K), is_colimit (((evaluation K C).obj k).map_cocone c)) : is_colimit c | { desc := λ s,
{ app := λ k, (t k).desc ⟨s.X.obj k, whisker_right s.ι ((evaluation K C).obj k)⟩,
naturality' := λ X Y f, (t X).hom_ext $ λ j,
begin
rw [(t X).fac_assoc _ j],
erw ← (c.ι.app j).naturality_assoc f,
erw (t Y).fac ⟨s.X.obj _, whisker_right s.ι _⟩ j,
dsimp,
simp,
e... | def | category_theory.limits.evaluation_jointly_reflects_colimits | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [
"hom_ext"
] | The evaluation functors jointly reflect colimits: that is, to show a cocone is a colimit of `F`
it suffices to show that each evaluation cocone is a colimit. In other words, to prove a cocone is
colimiting you can show it's pointwise colimiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
combine_cocones (F : J ⥤ K ⥤ C) (c : Π (k : K), colimit_cocone (F.flip.obj k)) :
cocone F | { X :=
{ obj := λ k, (c k).cocone.X,
map := λ k₁ k₂ f, (c k₁).is_colimit.desc ⟨_, F.flip.map f ≫ (c k₂).cocone.ι⟩,
map_id' := λ k, (c k).is_colimit.hom_ext (λ j, by { dsimp, simp }),
map_comp' := λ k₁ k₂ k₃ f₁ f₂, (c k₁).is_colimit.hom_ext (λ j, by simp) },
ι :=
{ app := λ j, { app := λ k, (c k).cocon... | def | category_theory.limits.combine_cocones | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | Given a functor `F` and a collection of colimit cocones for each diagram `X ↦ F X k`, we can stitch
them together to give a cocone for the diagram `F`.
`combined_is_colimit` shows that the new cocone is colimiting, and `eval_combined` shows it is
(essentially) made up of the original cocones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
evaluate_combined_cocones
(F : J ⥤ K ⥤ C) (c : Π (k : K), colimit_cocone (F.flip.obj k)) (k : K) :
((evaluation K C).obj k).map_cocone (combine_cocones F c) ≅ (c k).cocone | cocones.ext (iso.refl _) (by tidy) | def | category_theory.limits.evaluate_combined_cocones | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | The stitched together cocones each project down to the original given cocones (up to iso). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
combined_is_colimit (F : J ⥤ K ⥤ C) (c : Π (k : K), colimit_cocone (F.flip.obj k)) :
is_colimit (combine_cocones F c) | evaluation_jointly_reflects_colimits _
(λ k, (c k).is_colimit.of_iso_colimit (evaluate_combined_cocones F c k).symm) | def | category_theory.limits.combined_is_colimit | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | Stitching together colimiting cocones gives a colimiting cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
functor_category_has_limits_of_shape
[has_limits_of_shape J C] : has_limits_of_shape J (K ⥤ C) | { has_limit := λ F, has_limit.mk
{ cone := combine_cones F (λ k, get_limit_cone _),
is_limit := combined_is_limit _ _ } } | instance | category_theory.limits.functor_category_has_limits_of_shape | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_has_colimits_of_shape
[has_colimits_of_shape J C] : has_colimits_of_shape J (K ⥤ C) | { has_colimit := λ F, has_colimit.mk
{ cocone := combine_cocones _ (λ k, get_colimit_cocone _),
is_colimit := combined_is_colimit _ _ } } | instance | category_theory.limits.functor_category_has_colimits_of_shape | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_has_limits_of_size [has_limits_of_size.{v₁ u₁} C] :
has_limits_of_size.{v₁ u₁} (K ⥤ C) | ⟨infer_instance⟩ | instance | category_theory.limits.functor_category_has_limits_of_size | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_category_has_colimits_of_size [has_colimits_of_size.{v₁ u₁} C] :
has_colimits_of_size.{v₁ u₁} (K ⥤ C) | ⟨infer_instance⟩ | instance | category_theory.limits.functor_category_has_colimits_of_size | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
evaluation_preserves_limits_of_shape [has_limits_of_shape J C] (k : K) :
preserves_limits_of_shape J ((evaluation K C).obj k) | { preserves_limit :=
λ F, preserves_limit_of_preserves_limit_cone (combined_is_limit _ _) $
is_limit.of_iso_limit (limit.is_limit _)
(evaluate_combined_cones F _ k).symm } | instance | category_theory.limits.evaluation_preserves_limits_of_shape | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_obj_iso_limit_comp_evaluation [has_limits_of_shape J C] (F : J ⥤ K ⥤ C) (k : K) :
(limit F).obj k ≅ limit (F ⋙ ((evaluation K C).obj k)) | preserves_limit_iso ((evaluation K C).obj k) F | def | category_theory.limits.limit_obj_iso_limit_comp_evaluation | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | If `F : J ⥤ K ⥤ C` is a functor into a functor category which has a limit,
then the evaluation of that limit at `k` is the limit of the evaluations of `F.obj j` at `k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_obj_iso_limit_comp_evaluation_hom_π
[has_limits_of_shape J C] (F : J ⥤ (K ⥤ C)) (j : J) (k : K) :
(limit_obj_iso_limit_comp_evaluation F k).hom ≫ limit.π (F ⋙ ((evaluation K C).obj k)) j =
(limit.π F j).app k | begin
dsimp [limit_obj_iso_limit_comp_evaluation],
simp,
end | lemma | category_theory.limits.limit_obj_iso_limit_comp_evaluation_hom_π | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_obj_iso_limit_comp_evaluation_inv_π_app
[has_limits_of_shape J C] (F : J ⥤ (K ⥤ C)) (j : J) (k : K):
(limit_obj_iso_limit_comp_evaluation F k).inv ≫ (limit.π F j).app k =
limit.π (F ⋙ ((evaluation K C).obj k)) j | begin
dsimp [limit_obj_iso_limit_comp_evaluation],
rw iso.inv_comp_eq,
simp,
end | lemma | category_theory.limits.limit_obj_iso_limit_comp_evaluation_inv_π_app | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_map_limit_obj_iso_limit_comp_evaluation_hom
[has_limits_of_shape J C] {i j : K} (F : J ⥤ K ⥤ C) (f : i ⟶ j) :
(limit F).map f ≫ (limit_obj_iso_limit_comp_evaluation _ _).hom =
(limit_obj_iso_limit_comp_evaluation _ _).hom ≫
lim_map (whisker_left _ ((evaluation _ _).map f)) | by { ext, dsimp, simp } | lemma | category_theory.limits.limit_map_limit_obj_iso_limit_comp_evaluation_hom | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_obj_iso_limit_comp_evaluation_inv_limit_map
[has_limits_of_shape J C] {i j : K} (F : J ⥤ K ⥤ C) (f : i ⟶ j) :
(limit_obj_iso_limit_comp_evaluation _ _).inv ≫ (limit F).map f =
lim_map (whisker_left _ ((evaluation _ _).map f)) ≫
(limit_obj_iso_limit_comp_evaluation _ _).inv | by rw [iso.inv_comp_eq, ← category.assoc, iso.eq_comp_inv,
limit_map_limit_obj_iso_limit_comp_evaluation_hom] | lemma | category_theory.limits.limit_obj_iso_limit_comp_evaluation_inv_limit_map | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_obj_ext {H : J ⥤ K ⥤ C} [has_limits_of_shape J C]
{k : K} {W : C} {f g : W ⟶ (limit H).obj k}
(w : ∀ j, f ≫ (limits.limit.π H j).app k = g ≫ (limits.limit.π H j).app k) : f = g | begin
apply (cancel_mono (limit_obj_iso_limit_comp_evaluation H k).hom).1,
ext,
simpa using w j,
end | lemma | category_theory.limits.limit_obj_ext | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
evaluation_preserves_colimits_of_shape [has_colimits_of_shape J C] (k : K) :
preserves_colimits_of_shape J ((evaluation K C).obj k) | { preserves_colimit :=
λ F, preserves_colimit_of_preserves_colimit_cocone (combined_is_colimit _ _) $
is_colimit.of_iso_colimit (colimit.is_colimit _)
(evaluate_combined_cocones F _ k).symm } | instance | category_theory.limits.evaluation_preserves_colimits_of_shape | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_obj_iso_colimit_comp_evaluation [has_colimits_of_shape J C] (F : J ⥤ K ⥤ C) (k : K) :
(colimit F).obj k ≅ colimit (F ⋙ ((evaluation K C).obj k)) | preserves_colimit_iso ((evaluation K C).obj k) F | def | category_theory.limits.colimit_obj_iso_colimit_comp_evaluation | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | If `F : J ⥤ K ⥤ C` is a functor into a functor category which has a colimit,
then the evaluation of that colimit at `k` is the colimit of the evaluations of `F.obj j` at `k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_obj_iso_colimit_comp_evaluation_ι_inv
[has_colimits_of_shape J C] (F : J ⥤ (K ⥤ C)) (j : J) (k : K) :
colimit.ι (F ⋙ ((evaluation K C).obj k)) j ≫ (colimit_obj_iso_colimit_comp_evaluation F k).inv =
(colimit.ι F j).app k | begin
dsimp [colimit_obj_iso_colimit_comp_evaluation],
simp,
end | lemma | category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_ι_inv | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_obj_iso_colimit_comp_evaluation_ι_app_hom
[has_colimits_of_shape J C] (F : J ⥤ (K ⥤ C)) (j : J) (k : K) :
(colimit.ι F j).app k ≫ (colimit_obj_iso_colimit_comp_evaluation F k).hom =
colimit.ι (F ⋙ ((evaluation K C).obj k)) j | begin
dsimp [colimit_obj_iso_colimit_comp_evaluation],
rw ←iso.eq_comp_inv,
simp,
end | lemma | category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_ι_app_hom | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_obj_iso_colimit_comp_evaluation_inv_colimit_map
[has_colimits_of_shape J C] (F : J ⥤ K ⥤ C) {i j : K} (f : i ⟶ j) :
(colimit_obj_iso_colimit_comp_evaluation _ _).inv ≫ (colimit F).map f =
colim_map (whisker_left _ ((evaluation _ _).map f)) ≫
(colimit_obj_iso_colimit_comp_evaluation _ _).inv | by { ext, dsimp, simp } | lemma | category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_inv_colimit_map | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_map_colimit_obj_iso_colimit_comp_evaluation_hom
[has_colimits_of_shape J C] (F : J ⥤ K ⥤ C) {i j : K} (f : i ⟶ j) :
(colimit F).map f ≫ (colimit_obj_iso_colimit_comp_evaluation _ _).hom =
(colimit_obj_iso_colimit_comp_evaluation _ _).hom ≫
colim_map (whisker_left _ ((evaluation _ _).map f)) | by rw [← iso.inv_comp_eq, ← category.assoc, ← iso.eq_comp_inv,
colimit_obj_iso_colimit_comp_evaluation_inv_colimit_map] | lemma | category_theory.limits.colimit_map_colimit_obj_iso_colimit_comp_evaluation_hom | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_obj_ext {H : J ⥤ K ⥤ C} [has_colimits_of_shape J C]
{k : K} {W : C} {f g : (colimit H).obj k ⟶ W}
(w : ∀ j, (colimit.ι H j).app k ≫ f = (colimit.ι H j).app k ≫ g) : f = g | begin
apply (cancel_epi (colimit_obj_iso_colimit_comp_evaluation H k).inv).1,
ext,
simpa using w j,
end | lemma | category_theory.limits.colimit_obj_ext | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
evaluation_preserves_limits [has_limits C] (k : K) :
preserves_limits ((evaluation K C).obj k) | { preserves_limits_of_shape := λ J 𝒥, by resetI; apply_instance } | instance | category_theory.limits.evaluation_preserves_limits | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_limit_of_evaluation (F : D ⥤ K ⥤ C) (G : J ⥤ D)
(H : Π (k : K), preserves_limit G (F ⋙ (evaluation K C).obj k : D ⥤ C)) :
preserves_limit G F | ⟨λ c hc,
begin
apply evaluation_jointly_reflects_limits,
intro X,
haveI := H X,
change is_limit ((F ⋙ (evaluation K C).obj X).map_cone c),
exact preserves_limit.preserves hc,
end⟩ | def | category_theory.limits.preserves_limit_of_evaluation | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | `F : D ⥤ K ⥤ C` preserves the limit of some `G : J ⥤ D` if it does for each `k : K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_shape_of_evaluation (F : D ⥤ K ⥤ C) (J : Type*) [category J]
(H : Π (k : K), preserves_limits_of_shape J (F ⋙ (evaluation K C).obj k)) :
preserves_limits_of_shape J F | ⟨λ G, preserves_limit_of_evaluation F G (λ k, preserves_limits_of_shape.preserves_limit)⟩ | def | category_theory.limits.preserves_limits_of_shape_of_evaluation | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | `F : D ⥤ K ⥤ C` preserves limits of shape `J` if it does for each `k : K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_evaluation (F : D ⥤ K ⥤ C)
(H : Π (k : K), preserves_limits_of_size.{w' w} (F ⋙ (evaluation K C).obj k)) :
preserves_limits_of_size.{w' w} F | ⟨λ L hL, by exactI preserves_limits_of_shape_of_evaluation
F L (λ k, preserves_limits_of_size.preserves_limits_of_shape)⟩ | def | category_theory.limits.preserves_limits_of_evaluation | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | `F : D ⥤ K ⥤ C` preserves all limits if it does for each `k : K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_const : preserves_limits_of_size.{w' w} (const D : C ⥤ _) | preserves_limits_of_evaluation _ $ λ X, preserves_limits_of_nat_iso $ iso.symm $
const_comp_evaluation_obj _ _ | instance | category_theory.limits.preserves_limits_const | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | The constant functor `C ⥤ (D ⥤ C)` preserves limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
evaluation_preserves_colimits [has_colimits C] (k : K) :
preserves_colimits ((evaluation K C).obj k) | { preserves_colimits_of_shape := λ J 𝒥, by resetI; apply_instance } | instance | category_theory.limits.evaluation_preserves_colimits | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_colimit_of_evaluation (F : D ⥤ K ⥤ C) (G : J ⥤ D)
(H : Π (k), preserves_colimit G (F ⋙ (evaluation K C).obj k)) : preserves_colimit G F | ⟨λ c hc,
begin
apply evaluation_jointly_reflects_colimits,
intro X,
haveI := H X,
change is_colimit ((F ⋙ (evaluation K C).obj X).map_cocone c),
exact preserves_colimit.preserves hc,
end⟩ | def | category_theory.limits.preserves_colimit_of_evaluation | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | `F : D ⥤ K ⥤ C` preserves the colimit of some `G : J ⥤ D` if it does for each `k : K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_shape_of_evaluation (F : D ⥤ K ⥤ C) (J : Type*) [category J]
(H : Π (k : K), preserves_colimits_of_shape J (F ⋙ (evaluation K C).obj k)) :
preserves_colimits_of_shape J F | ⟨λ G, preserves_colimit_of_evaluation F G (λ k, preserves_colimits_of_shape.preserves_colimit)⟩ | def | category_theory.limits.preserves_colimits_of_shape_of_evaluation | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | `F : D ⥤ K ⥤ C` preserves all colimits of shape `J` if it does for each `k : K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_evaluation (F : D ⥤ K ⥤ C)
(H : Π (k : K), preserves_colimits_of_size.{w' w} (F ⋙ (evaluation K C).obj k)) :
preserves_colimits_of_size.{w' w} F | ⟨λ L hL, by exactI preserves_colimits_of_shape_of_evaluation
F L (λ k, preserves_colimits_of_size.preserves_colimits_of_shape)⟩ | def | category_theory.limits.preserves_colimits_of_evaluation | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | `F : D ⥤ K ⥤ C` preserves all colimits if it does for each `k : K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_const : preserves_colimits_of_size.{w' w} (const D : C ⥤ _) | preserves_colimits_of_evaluation _ $ λ X, preserves_colimits_of_nat_iso $ iso.symm $
const_comp_evaluation_obj _ _ | instance | category_theory.limits.preserves_colimits_const | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | The constant functor `C ⥤ (D ⥤ C)` preserves colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_iso_flip_comp_lim [has_limits_of_shape J C] (F : J ⥤ K ⥤ C) :
limit F ≅ F.flip ⋙ lim | nat_iso.of_components (limit_obj_iso_limit_comp_evaluation F) $ by tidy | def | category_theory.limits.limit_iso_flip_comp_lim | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [
"lim"
] | The limit of a diagram `F : J ⥤ K ⥤ C` is isomorphic to the functor given by
the individual limits on objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_flip_iso_comp_lim [has_limits_of_shape J C] (F : K ⥤ J ⥤ C) :
limit F.flip ≅ F ⋙ lim | nat_iso.of_components (λ k,
limit_obj_iso_limit_comp_evaluation F.flip k ≪≫
has_limit.iso_of_nat_iso (flip_comp_evaluation _ _)) $ by tidy | def | category_theory.limits.limit_flip_iso_comp_lim | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [
"lim"
] | A variant of `limit_iso_flip_comp_lim` where the arguemnts of `F` are flipped. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_iso_swap_comp_lim [has_limits_of_shape J C] (G : J ⥤ K ⥤ C) :
limit G ≅ curry.obj (swap K J ⋙ uncurry.obj G) ⋙ lim | limit_iso_flip_comp_lim G ≪≫ iso_whisker_right (flip_iso_curry_swap_uncurry _) _ | def | category_theory.limits.limit_iso_swap_comp_lim | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [
"lim"
] | For a functor `G : J ⥤ K ⥤ C`, its limit `K ⥤ C` is given by `(G' : K ⥤ J ⥤ C) ⋙ lim`.
Note that this does not require `K` to be small. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_iso_flip_comp_colim [has_colimits_of_shape J C] (F : J ⥤ K ⥤ C) :
colimit F ≅ F.flip ⋙ colim | nat_iso.of_components (colimit_obj_iso_colimit_comp_evaluation F) $ by tidy | def | category_theory.limits.colimit_iso_flip_comp_colim | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | The colimit of a diagram `F : J ⥤ K ⥤ C` is isomorphic to the functor given by
the individual colimits on objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_flip_iso_comp_colim [has_colimits_of_shape J C] (F : K ⥤ J ⥤ C) :
colimit F.flip ≅ F ⋙ colim | nat_iso.of_components (λ k,
colimit_obj_iso_colimit_comp_evaluation _ _ ≪≫
has_colimit.iso_of_nat_iso (flip_comp_evaluation _ _)) $ by tidy | def | category_theory.limits.colimit_flip_iso_comp_colim | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | A variant of `colimit_iso_flip_comp_colim` where the arguemnts of `F` are flipped. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_iso_swap_comp_colim [has_colimits_of_shape J C] (G : J ⥤ K ⥤ C) :
colimit G ≅ curry.obj (swap K J ⋙ uncurry.obj G) ⋙ colim | colimit_iso_flip_comp_colim G ≪≫ iso_whisker_right (flip_iso_curry_swap_uncurry _) _ | def | category_theory.limits.colimit_iso_swap_comp_colim | category_theory.limits | src/category_theory/limits/functor_category.lean | [
"category_theory.limits.preserves.limits"
] | [] | For a functor `G : J ⥤ K ⥤ C`, its colimit `K ⥤ C` is given by `(G' : K ⥤ J ⥤ C) ⋙ colim`.
Note that this does not require `K` to be small. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone (F : J ⥤ C) | (cone : cone F)
(is_limit : is_limit cone) | structure | category_theory.limits.limit_cone | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | `limit_cone F` contains a cone over `F` together with the information that it is a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit (F : J ⥤ C) : Prop | mk' :: (exists_limit : nonempty (limit_cone F)) | class | category_theory.limits.has_limit | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"mk'"
] | `has_limit F` represents the mere existence of a limit for `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit.mk {F : J ⥤ C} (d : limit_cone F) : has_limit F | ⟨nonempty.intro d⟩ | lemma | category_theory.limits.has_limit.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_limit_cone (F : J ⥤ C) [has_limit F] : limit_cone F | classical.choice $ has_limit.exists_limit | def | category_theory.limits.get_limit_cone | 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 `limit_cone F` from `has_limit F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_shape : Prop | (has_limit : Π F : J ⥤ C, has_limit F . tactic.apply_instance) | class | category_theory.limits.has_limits_of_shape | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | `C` has limits of shape `J` if there exists a limit for every functor `F : J ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_size (C : Type u) [category.{v} C] : Prop | (has_limits_of_shape :
Π (J : Type u₁) [𝒥 : category.{v₁} J], has_limits_of_shape J C . tactic.apply_instance) | class | category_theory.limits.has_limits_of_size | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | `C` has all limits of size `v₁ u₁` (`has_limits_of_size.{v₁ u₁} C`)
if it has limits of every shape `J : Type u₁` with `[category.{v₁} J]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits (C : Type u) [category.{v} C] : Prop | has_limits_of_size.{v v} C | abbreviation | category_theory.limits.has_limits | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | `C` has all (small) limits if it has limits of every shape that is as big as its hom-sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits.has_limits_of_shape {C : Type u} [category.{v} C] [has_limits C]
(J : Type v) [category.{v} J] :
has_limits_of_shape J C | has_limits_of_size.has_limits_of_shape J | lemma | category_theory.limits.has_limits.has_limits_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_limit_of_has_limits_of_shape
{J : Type u₁} [category.{v₁} J] [H : has_limits_of_shape J C] (F : J ⥤ C) : has_limit F | has_limits_of_shape.has_limit F | instance | category_theory.limits.has_limit_of_has_limits_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_limits_of_shape_of_has_limits
{J : Type u₁} [category.{v₁} J] [H : has_limits_of_size.{v₁ u₁} C] : has_limits_of_shape J C | has_limits_of_size.has_limits_of_shape J | instance | category_theory.limits.has_limits_of_shape_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 | |
limit.cone (F : J ⥤ C) [has_limit F] : cone F | (get_limit_cone F).cone | def | category_theory.limits.limit.cone | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | An arbitrary choice of limit cone for a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit (F : J ⥤ C) [has_limit F] | (limit.cone F).X | def | category_theory.limits.limit | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | An arbitrary choice of limit object of a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit.π (F : J ⥤ C) [has_limit F] (j : J) : limit F ⟶ F.obj j | (limit.cone F).π.app j | def | category_theory.limits.limit.π | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | The projection from the limit object to a value of the functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit.cone_X {F : J ⥤ C} [has_limit F] :
(limit.cone F).X = limit F | rfl | lemma | category_theory.limits.limit.cone_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 | |
limit.cone_π {F : J ⥤ C} [has_limit F] :
(limit.cone F).π.app = limit.π _ | rfl | lemma | category_theory.limits.limit.cone_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.w (F : J ⥤ C) [has_limit F] {j j' : J} (f : j ⟶ j') :
limit.π F j ≫ F.map f = limit.π F j' | (limit.cone F).w f | lemma | category_theory.limits.limit.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 | |
limit.is_limit (F : J ⥤ C) [has_limit F] : is_limit (limit.cone F) | (get_limit_cone F).is_limit | def | category_theory.limits.limit.is_limit | 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 cone provied by `limit.cone F` is a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit.lift (F : J ⥤ C) [has_limit F] (c : cone F) : c.X ⟶ limit F | (limit.is_limit F).lift c | def | category_theory.limits.limit.lift | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"lift"
] | The morphism from the cone point of any other cone to the limit object. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit.is_limit_lift {F : J ⥤ C} [has_limit F] (c : cone F) :
(limit.is_limit F).lift c = limit.lift F c | rfl | lemma | category_theory.limits.limit.is_limit_lift | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit.lift_π {F : J ⥤ C} [has_limit F] (c : cone F) (j : J) :
limit.lift F c ≫ limit.π F j = c.π.app j | is_limit.fac _ c j | lemma | category_theory.limits.limit.lift_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 | |
lim_map {F G : J ⥤ C} [has_limit F] [has_limit G] (α : F ⟶ G) : limit F ⟶ limit G | is_limit.map _ (limit.is_limit G) α | def | category_theory.limits.lim_map | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | Functoriality of limits.
Usually this morphism should be accessed through `lim.map`,
but may be needed separately when you have specified limits for the source and target functors,
but not necessarily for all functors of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lim_map_π {F G : J ⥤ C} [has_limit F] [has_limit G] (α : F ⟶ G) (j : J) :
lim_map α ≫ limit.π G j = limit.π F j ≫ α.app j | limit.lift_π _ j | lemma | category_theory.limits.lim_map_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.cone_morphism {F : J ⥤ C} [has_limit F] (c : cone F) :
c ⟶ limit.cone F | (limit.is_limit F).lift_cone_morphism c | def | category_theory.limits.limit.cone_morphism | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | The cone morphism from any cone to the arbitrary choice of limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit.cone_morphism_hom {F : J ⥤ C} [has_limit F] (c : cone F) :
(limit.cone_morphism c).hom = limit.lift F c | rfl | lemma | category_theory.limits.limit.cone_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 | |
limit.cone_morphism_π {F : J ⥤ C} [has_limit F] (c : cone F) (j : J) :
(limit.cone_morphism c).hom ≫ limit.π F j = c.π.app j | by simp | lemma | category_theory.limits.limit.cone_morphism_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.cone_point_unique_up_to_iso_hom_comp {F : J ⥤ C} [has_limit F]
{c : cone F} (hc : is_limit c) (j : J) :
(is_limit.cone_point_unique_up_to_iso hc (limit.is_limit _)).hom ≫ limit.π F j = c.π.app j | is_limit.cone_point_unique_up_to_iso_hom_comp _ _ _ | lemma | category_theory.limits.limit.cone_point_unique_up_to_iso_hom_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 | |
limit.cone_point_unique_up_to_iso_inv_comp {F : J ⥤ C} [has_limit F]
{c : cone F} (hc : is_limit c) (j : J) :
(is_limit.cone_point_unique_up_to_iso (limit.is_limit _) hc).inv ≫ limit.π F j = c.π.app j | is_limit.cone_point_unique_up_to_iso_inv_comp _ _ _ | lemma | category_theory.limits.limit.cone_point_unique_up_to_iso_inv_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 | |
limit.exists_unique {F : J ⥤ C} [has_limit F] (t : cone F) :
∃! (l : t.X ⟶ limit F), ∀ j, l ≫ limit.π F j = t.π.app j | (limit.is_limit F).exists_unique _ | lemma | category_theory.limits.limit.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 | |
limit.iso_limit_cone {F : J ⥤ C} [has_limit F] (t : limit_cone F) :
limit F ≅ t.cone.X | is_limit.cone_point_unique_up_to_iso (limit.is_limit F) t.is_limit | def | category_theory.limits.limit.iso_limit_cone | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | Given any other limit cone for `F`, the chosen `limit F` is isomorphic to the cone point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit.iso_limit_cone_hom_π
{F : J ⥤ C} [has_limit F] (t : limit_cone F) (j : J) :
(limit.iso_limit_cone t).hom ≫ t.cone.π.app j = limit.π F j | by { dsimp [limit.iso_limit_cone, is_limit.cone_point_unique_up_to_iso], tidy, } | lemma | category_theory.limits.limit.iso_limit_cone_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 | |
limit.iso_limit_cone_inv_π
{F : J ⥤ C} [has_limit F] (t : limit_cone F) (j : J) :
(limit.iso_limit_cone t).inv ≫ limit.π F j = t.cone.π.app j | by { dsimp [limit.iso_limit_cone, is_limit.cone_point_unique_up_to_iso], tidy, } | lemma | category_theory.limits.limit.iso_limit_cone_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 | |
limit.hom_ext {F : J ⥤ C} [has_limit F] {X : C} {f f' : X ⟶ limit F}
(w : ∀ j, f ≫ limit.π F j = f' ≫ limit.π F j) : f = f' | (limit.is_limit F).hom_ext w | lemma | category_theory.limits.limit.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 | |
limit.lift_map {F G : J ⥤ C} [has_limit F] [has_limit G] (c : cone F) (α : F ⟶ G) :
limit.lift F c ≫ lim_map α = limit.lift G ((cones.postcompose α).obj c) | by { ext, rw [assoc, lim_map_π, limit.lift_π_assoc, limit.lift_π], refl } | lemma | category_theory.limits.limit.lift_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.lift_cone {F : J ⥤ C} [has_limit F] :
limit.lift F (limit.cone F) = 𝟙 (limit F) | (limit.is_limit _).lift_self | lemma | category_theory.limits.limit.lift_cone | 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.hom_iso (F : J ⥤ C) [has_limit F] (W : C) :
ulift.{u₁} (W ⟶ limit F : Type v) ≅ (F.cones.obj (op W)) | (limit.is_limit F).hom_iso W | def | category_theory.limits.limit.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 a specified object `W` to the limit object,
and cones with cone point `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit.hom_iso_hom (F : J ⥤ C) [has_limit F] {W : C} (f : ulift (W ⟶ limit F)) :
(limit.hom_iso F W).hom f = (const J).map f.down ≫ (limit.cone F).π | (limit.is_limit F).hom_iso_hom f | lemma | category_theory.limits.limit.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 | |
limit.hom_iso' (F : J ⥤ C) [has_limit F] (W : C) :
ulift.{u₁} ((W ⟶ limit F) : Type v) ≅
{ p : Π j, W ⟶ F.obj j // ∀ {j j' : J} (f : j ⟶ j'), p j ≫ F.map f = p j' } | (limit.is_limit F).hom_iso' W | def | category_theory.limits.limit.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 a specified object `W` to the limit object,
and an explicit componentwise description of cones with cone point `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit.lift_extend {F : J ⥤ C} [has_limit F] (c : cone F) {X : C} (f : X ⟶ c.X) :
limit.lift F (c.extend f) = f ≫ limit.lift F c | by obviously | lemma | category_theory.limits.limit.lift_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_limit_of_iso {F G : J ⥤ C} [has_limit F] (α : F ≅ G) : has_limit G | has_limit.mk
{ cone := (cones.postcompose α.hom).obj (limit.cone F),
is_limit :=
{ lift := λ s, limit.lift F ((cones.postcompose α.inv).obj s),
fac' := λ s j,
begin
rw [cones.postcompose_obj_π, nat_trans.comp_app, limit.cone_π, ←category.assoc, limit.lift_π],
simp
end,
uniq' := λ s m w,
... | lemma | category_theory.limits.has_limit_of_iso | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [
"lift"
] | If a functor `F` has a limit, so does any naturally isomorphic functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit.of_cones_iso {J K : Type u₁} [category.{v₁} J] [category.{v₂} K] (F : J ⥤ C)
(G : K ⥤ C) (h : F.cones ≅ G.cones) [has_limit F] : has_limit G | has_limit.mk ⟨_, is_limit.of_nat_iso ((is_limit.nat_iso (limit.is_limit F)) ≪≫ h)⟩ | lemma | category_theory.limits.has_limit.of_cones_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_limit.iso_of_nat_iso {F G : J ⥤ C} [has_limit F] [has_limit G] (w : F ≅ G) :
limit F ≅ limit G | is_limit.cone_points_iso_of_nat_iso (limit.is_limit F) (limit.is_limit G) w | def | category_theory.limits.has_limit.iso_of_nat_iso | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | The limits of `F : J ⥤ C` and `G : J ⥤ C` are isomorphic,
if the functors are naturally isomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit.iso_of_nat_iso_hom_π {F G : J ⥤ C} [has_limit F] [has_limit G]
(w : F ≅ G) (j : J) :
(has_limit.iso_of_nat_iso w).hom ≫ limit.π G j = limit.π F j ≫ w.hom.app j | is_limit.cone_points_iso_of_nat_iso_hom_comp _ _ _ _ | lemma | category_theory.limits.has_limit.iso_of_nat_iso_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_limit.iso_of_nat_iso_inv_π {F G : J ⥤ C} [has_limit F] [has_limit G]
(w : F ≅ G) (j : J) :
(has_limit.iso_of_nat_iso w).inv ≫ limit.π F j = limit.π G j ≫ w.inv.app j | is_limit.cone_points_iso_of_nat_iso_inv_comp _ _ _ _ | lemma | category_theory.limits.has_limit.iso_of_nat_iso_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 | |
has_limit.lift_iso_of_nat_iso_hom {F G : J ⥤ C} [has_limit F] [has_limit G] (t : cone F)
(w : F ≅ G) :
limit.lift F t ≫ (has_limit.iso_of_nat_iso w).hom =
limit.lift G ((cones.postcompose w.hom).obj _) | is_limit.lift_comp_cone_points_iso_of_nat_iso_hom _ _ _ | lemma | category_theory.limits.has_limit.lift_iso_of_nat_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 | |
has_limit.lift_iso_of_nat_iso_inv {F G : J ⥤ C} [has_limit F] [has_limit G] (t : cone G)
(w : F ≅ G) :
limit.lift G t ≫ (has_limit.iso_of_nat_iso w).inv =
limit.lift F ((cones.postcompose w.inv).obj _) | is_limit.lift_comp_cone_points_iso_of_nat_iso_inv _ _ _ | lemma | category_theory.limits.has_limit.lift_iso_of_nat_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 | |
has_limit.iso_of_equivalence {F : J ⥤ C} [has_limit F] {G : K ⥤ C} [has_limit G]
(e : J ≌ K) (w : e.functor ⋙ G ≅ F) : limit F ≅ limit G | is_limit.cone_points_iso_of_equivalence (limit.is_limit F) (limit.is_limit G) e w | def | category_theory.limits.has_limit.iso_of_equivalence | category_theory.limits | src/category_theory/limits/has_limits.lean | [
"category_theory.limits.is_limit",
"category_theory.category.ulift"
] | [] | The limits 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_limit.iso_of_equivalence_hom_π {F : J ⥤ C} [has_limit F] {G : K ⥤ C} [has_limit G]
(e : J ≌ K) (w : e.functor ⋙ G ≅ F) (k : K) :
(has_limit.iso_of_equivalence e w).hom ≫ limit.π G k =
limit.π F (e.inverse.obj k) ≫ w.inv.app (e.inverse.obj k) ≫ G.map (e.counit.app k) | begin
simp only [has_limit.iso_of_equivalence, is_limit.cone_points_iso_of_equivalence_hom],
dsimp,
simp,
end | lemma | category_theory.limits.has_limit.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_limit.iso_of_equivalence_inv_π {F : J ⥤ C} [has_limit F] {G : K ⥤ C} [has_limit G]
(e : J ≌ K) (w : e.functor ⋙ G ≅ F) (j : J) :
(has_limit.iso_of_equivalence e w).inv ≫ limit.π F j =
limit.π G (e.functor.obj j) ≫ w.hom.app j | begin
simp only [has_limit.iso_of_equivalence, is_limit.cone_points_iso_of_equivalence_hom],
dsimp,
simp,
end | lemma | category_theory.limits.has_limit.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 | |
limit.pre : limit F ⟶ limit (E ⋙ F) | limit.lift (E ⋙ F) ((limit.cone F).whisker E) | def | category_theory.limits.limit.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 limit of `F` to the limit of `E ⋙ F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit.pre_π (k : K) :
limit.pre F E ≫ limit.π (E ⋙ F) k = limit.π F (E.obj k) | by { erw is_limit.fac, refl } | lemma | category_theory.limits.limit.pre_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_pre (c : cone F) :
limit.lift F c ≫ limit.pre F E = limit.lift (E ⋙ F) (c.whisker E) | by ext; simp | lemma | category_theory.limits.limit.lift_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.pre_pre : limit.pre F E ≫ limit.pre (E ⋙ F) D = limit.pre F (D ⋙ E) | by ext j; erw [assoc, limit.pre_π, limit.pre_π, limit.pre_π]; refl | lemma | category_theory.limits.limit.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 | |
limit.pre_eq (s : limit_cone (E ⋙ F)) (t : limit_cone F) :
limit.pre F E =
(limit.iso_limit_cone t).hom ≫ s.is_limit.lift ((t.cone).whisker E) ≫
(limit.iso_limit_cone s).inv | by tidy | lemma | category_theory.limits.limit.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 | |
limit.post : G.obj (limit F) ⟶ limit (F ⋙ G) | limit.lift (F ⋙ G) (G.map_cone (limit.cone F)) | def | category_theory.limits.limit.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 limit of `F` to the limit of `F ⋙ G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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