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preserves_limits (F : C ⥤ D) | preserves_limits_of_size.{v₂ v₂} F | abbreviation | category_theory.limits.preserves_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | We say that `F` preserves (small) limits if it sends small
limit cones over any diagram to limit cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_size (F : C ⥤ D) | (preserves_colimits_of_shape : Π {J : Type w} [category.{w'} J],
preserves_colimits_of_shape J F . tactic.apply_instance) | class | category_theory.limits.preserves_colimits_of_size | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | `preserves_colimits_of_size.{v u} F` means that `F` sends all colimit cocones over any
diagram `J ⥤ C` to colimit cocones, where `J : Type u` with `[category.{v} J]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits (F : C ⥤ D) | preserves_colimits_of_size.{v₂ v₂} F | abbreviation | category_theory.limits.preserves_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | We say that `F` preserves (small) limits if it sends small
limit cones over any diagram to limit cones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_of_preserves (F : C ⥤ D) {c : cone K} (t : is_limit c) [preserves_limit K F] :
is_limit (F.map_cone c) | preserves_limit.preserves t | def | category_theory.limits.is_limit_of_preserves | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A convenience function for `preserves_limit`, which takes the functor as an explicit argument to
guide typeclass resolution. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_of_preserves (F : C ⥤ D) {c : cocone K} (t : is_colimit c)
[preserves_colimit K F] :
is_colimit (F.map_cocone c) | preserves_colimit.preserves t | def | category_theory.limits.is_colimit_of_preserves | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A convenience function for `preserves_colimit`, which takes the functor as an explicit argument to
guide typeclass resolution. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limit_subsingleton (K : J ⥤ C) (F : C ⥤ D) :
subsingleton (preserves_limit K F) | by split; rintros ⟨a⟩ ⟨b⟩; congr | instance | category_theory.limits.preserves_limit_subsingleton | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_colimit_subsingleton (K : J ⥤ C) (F : C ⥤ D) :
subsingleton (preserves_colimit K F) | by split; rintros ⟨a⟩ ⟨b⟩; congr | instance | category_theory.limits.preserves_colimit_subsingleton | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_limits_of_shape_subsingleton (J : Type w) [category.{w'} J] (F : C ⥤ D) :
subsingleton (preserves_limits_of_shape J F) | by { split, intros, cases a, cases b, congr } | instance | category_theory.limits.preserves_limits_of_shape_subsingleton | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_colimits_of_shape_subsingleton (J : Type w) [category.{w'} J] (F : C ⥤ D) :
subsingleton (preserves_colimits_of_shape J F) | by { split, intros, cases a, cases b, congr } | instance | category_theory.limits.preserves_colimits_of_shape_subsingleton | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_limits_subsingleton (F : C ⥤ D) :
subsingleton (preserves_limits_of_size.{w' w} F) | by { split, intros, cases a, cases b, cc } | instance | category_theory.limits.preserves_limits_subsingleton | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_colimits_subsingleton (F : C ⥤ D) :
subsingleton (preserves_colimits_of_size.{w' w} F) | by { split, intros, cases a, cases b, cc } | instance | category_theory.limits.preserves_colimits_subsingleton | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_preserves_limits : preserves_limits_of_size.{w' w} (𝟭 C) | { preserves_limits_of_shape := λ J 𝒥,
{ preserves_limit := λ K, by exactI ⟨λ c h,
⟨λ s, h.lift ⟨s.X, λ j, s.π.app j, λ j j' f, s.π.naturality f⟩,
by cases K; rcases c with ⟨_, _, _⟩; intros s j; cases s; exact h.fac _ j,
by cases K; rcases c with ⟨_, _, _⟩; intros s m w; rcases s with ⟨_, _, _⟩;
exact h... | instance | category_theory.limits.id_preserves_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_preserves_colimits : preserves_colimits_of_size.{w' w} (𝟭 C) | { preserves_colimits_of_shape := λ J 𝒥,
{ preserves_colimit := λ K, by exactI ⟨λ c h,
⟨λ s, h.desc ⟨s.X, λ j, s.ι.app j, λ j j' f, s.ι.naturality f⟩,
by cases K; rcases c with ⟨_, _, _⟩; intros s j; cases s; exact h.fac _ j,
by cases K; rcases c with ⟨_, _, _⟩; intros s m w; rcases s with ⟨_, _, _⟩;
exa... | instance | category_theory.limits.id_preserves_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_preserves_limit [preserves_limit K F] [preserves_limit (K ⋙ F) G] :
preserves_limit K (F ⋙ G) | ⟨λ c h, preserves_limit.preserves (preserves_limit.preserves h)⟩ | instance | category_theory.limits.comp_preserves_limit | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_preserves_limits_of_shape
[preserves_limits_of_shape J F] [preserves_limits_of_shape J G] :
preserves_limits_of_shape J (F ⋙ G) | {} | instance | category_theory.limits.comp_preserves_limits_of_shape | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_preserves_limits
[preserves_limits_of_size.{w' w} F] [preserves_limits_of_size.{w' w} G] :
preserves_limits_of_size.{w' w} (F ⋙ G) | {} | instance | category_theory.limits.comp_preserves_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_preserves_colimit [preserves_colimit K F] [preserves_colimit (K ⋙ F) G] :
preserves_colimit K (F ⋙ G) | ⟨λ c h, preserves_colimit.preserves (preserves_colimit.preserves h)⟩ | instance | category_theory.limits.comp_preserves_colimit | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_preserves_colimits_of_shape
[preserves_colimits_of_shape J F] [preserves_colimits_of_shape J G] :
preserves_colimits_of_shape J (F ⋙ G) | {} | instance | category_theory.limits.comp_preserves_colimits_of_shape | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_preserves_colimits
[preserves_colimits_of_size.{w' w} F] [preserves_colimits_of_size.{w' w} G] :
preserves_colimits_of_size.{w' w} (F ⋙ G) | {} | instance | category_theory.limits.comp_preserves_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_limit_of_preserves_limit_cone {F : C ⥤ D} {t : cone K}
(h : is_limit t) (hF : is_limit (F.map_cone t)) : preserves_limit K F | ⟨λ t' h', is_limit.of_iso_limit hF (functor.map_iso _ (is_limit.unique_up_to_iso h h'))⟩ | def | category_theory.limits.preserves_limit_of_preserves_limit_cone | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | If F preserves one limit cone for the diagram K,
then it preserves any limit cone for K. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limit_of_iso_diagram {K₁ K₂ : J ⥤ C} (F : C ⥤ D) (h : K₁ ≅ K₂)
[preserves_limit K₁ F] : preserves_limit K₂ F | { preserves := λ c t,
begin
apply is_limit.postcompose_inv_equiv (iso_whisker_right h F : _) _ _,
have := (is_limit.postcompose_inv_equiv h c).symm t,
apply is_limit.of_iso_limit (is_limit_of_preserves F this),
refine cones.ext (iso.refl _) (λ j, by tidy),
end } | def | category_theory.limits.preserves_limit_of_iso_diagram | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer preservation of limits along a natural isomorphism in the diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limit_of_nat_iso (K : J ⥤ C) {F G : C ⥤ D} (h : F ≅ G) [preserves_limit K F] :
preserves_limit K G | { preserves := λ c t, is_limit.map_cone_equiv h (preserves_limit.preserves t) } | def | category_theory.limits.preserves_limit_of_nat_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer preservation of a limit along a natural isomorphism in the functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_shape_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [preserves_limits_of_shape J F] :
preserves_limits_of_shape J G | { preserves_limit := λ K, preserves_limit_of_nat_iso K h } | def | category_theory.limits.preserves_limits_of_shape_of_nat_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer preservation of limits of shape along a natural isomorphism in the functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [preserves_limits_of_size.{w w'} F] :
preserves_limits_of_size.{w w'} G | { preserves_limits_of_shape := λ J 𝒥₁, by exactI preserves_limits_of_shape_of_nat_iso h } | def | category_theory.limits.preserves_limits_of_nat_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer preservation of limits along a natural isomorphism in the functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_shape_of_equiv {J' : Type w₂} [category.{w₂'} J'] (e : J ≌ J')
(F : C ⥤ D) [preserves_limits_of_shape J F] :
preserves_limits_of_shape J' F | { preserves_limit := λ K,
{ preserves := λ c t,
begin
let equ := e.inv_fun_id_assoc (K ⋙ F),
have := (is_limit_of_preserves F (t.whisker_equivalence e)).whisker_equivalence e.symm,
apply ((is_limit.postcompose_hom_equiv equ _).symm this).of_iso_limit,
refine cones.ext (iso.refl _) (λ j, _)... | def | category_theory.limits.preserves_limits_of_shape_of_equiv | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer preservation of limits along a equivalence in the shape. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_size_shrink (F : C ⥤ D)
[preserves_limits_of_size.{(max w w₂) (max w' w₂')} F] : preserves_limits_of_size.{w w'} F | ⟨λ J hJ, by exactI preserves_limits_of_shape_of_equiv
(ulift_hom_ulift_category.equiv.{w₂ w₂'} J).symm F⟩ | def | category_theory.limits.preserves_limits_of_size_shrink | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | `preserves_limits_of_size_shrink.{w w'} F` tries to obtain `preserves_limits_of_size.{w w'} F`
from some other `preserves_limits_of_size F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_smallest_limits_of_preserves_limits
(F : C ⥤ D) [preserves_limits_of_size.{v₃ u₃} F] : preserves_limits_of_size.{0 0} F | preserves_limits_of_size_shrink F | def | category_theory.limits.preserves_smallest_limits_of_preserves_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Preserving limits at any universe level implies preserving limits in universe `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimit_of_preserves_colimit_cocone {F : C ⥤ D} {t : cocone K}
(h : is_colimit t) (hF : is_colimit (F.map_cocone t)) : preserves_colimit K F | ⟨λ t' h', is_colimit.of_iso_colimit hF (functor.map_iso _ (is_colimit.unique_up_to_iso h h'))⟩ | def | category_theory.limits.preserves_colimit_of_preserves_colimit_cocone | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | If F preserves one colimit cocone for the diagram K,
then it preserves any colimit cocone for K. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimit_of_iso_diagram {K₁ K₂ : J ⥤ C} (F : C ⥤ D) (h : K₁ ≅ K₂)
[preserves_colimit K₁ F] : preserves_colimit K₂ F | { preserves := λ c t,
begin
apply is_colimit.precompose_hom_equiv (iso_whisker_right h F : _) _ _,
have := (is_colimit.precompose_hom_equiv h c).symm t,
apply is_colimit.of_iso_colimit (is_colimit_of_preserves F this),
refine cocones.ext (iso.refl _) (λ j, by tidy),
end } | def | category_theory.limits.preserves_colimit_of_iso_diagram | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer preservation of colimits along a natural isomorphism in the shape. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimit_of_nat_iso (K : J ⥤ C) {F G : C ⥤ D} (h : F ≅ G) [preserves_colimit K F] :
preserves_colimit K G | { preserves := λ c t, is_colimit.map_cocone_equiv h (preserves_colimit.preserves t) } | def | category_theory.limits.preserves_colimit_of_nat_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer preservation of a colimit along a natural isomorphism in the functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_shape_of_nat_iso {F G : C ⥤ D} (h : F ≅ G)
[preserves_colimits_of_shape J F] : preserves_colimits_of_shape J G | { preserves_colimit := λ K, preserves_colimit_of_nat_iso K h } | def | category_theory.limits.preserves_colimits_of_shape_of_nat_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer preservation of colimits of shape along a natural isomorphism in the functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [preserves_colimits_of_size.{w w'} F] :
preserves_colimits_of_size.{w w'} G | { preserves_colimits_of_shape := λ J 𝒥₁, by exactI preserves_colimits_of_shape_of_nat_iso h } | def | category_theory.limits.preserves_colimits_of_nat_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer preservation of colimits along a natural isomorphism in the functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_shape_of_equiv {J' : Type w₂} [category.{w₂'} J'] (e : J ≌ J')
(F : C ⥤ D) [preserves_colimits_of_shape J F] :
preserves_colimits_of_shape J' F | { preserves_colimit := λ K,
{ preserves := λ c t,
begin
let equ := e.inv_fun_id_assoc (K ⋙ F),
have := (is_colimit_of_preserves F (t.whisker_equivalence e)).whisker_equivalence e.symm,
apply ((is_colimit.precompose_inv_equiv equ _).symm this).of_iso_colimit,
refine cocones.ext (iso.refl _)... | def | category_theory.limits.preserves_colimits_of_shape_of_equiv | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer preservation of colimits along a equivalence in the shape. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_size_shrink (F : C ⥤ D)
[preserves_colimits_of_size.{(max w w₂) (max w' w₂')} F] : preserves_colimits_of_size.{w w'} F | ⟨λ J hJ, by exactI preserves_colimits_of_shape_of_equiv
(ulift_hom_ulift_category.equiv.{w₂ w₂'} J).symm F⟩ | def | category_theory.limits.preserves_colimits_of_size_shrink | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | `preserves_colimits_of_size_shrink.{w w'} F` tries to obtain `preserves_colimits_of_size.{w w'} F`
from some other `preserves_colimits_of_size F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_smallest_colimits_of_preserves_colimits
(F : C ⥤ D) [preserves_colimits_of_size.{v₃ u₃} F] : preserves_colimits_of_size.{0 0} F | preserves_colimits_of_size_shrink F | def | category_theory.limits.preserves_smallest_colimits_of_preserves_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Preserving colimits at any universe implies preserving colimits at universe `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_limit (K : J ⥤ C) (F : C ⥤ D) | (reflects : Π {c : cone K}, is_limit (F.map_cone c) → is_limit c) | class | category_theory.limits.reflects_limit | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A functor `F : C ⥤ D` reflects limits for `K : J ⥤ C` if
whenever the image of a cone over `K` under `F` is a limit cone in `D`,
the cone was already a limit cone in `C`.
Note that we do not assume a priori that `D` actually has any limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_colimit (K : J ⥤ C) (F : C ⥤ D) | (reflects : Π {c : cocone K}, is_colimit (F.map_cocone c) → is_colimit c) | class | category_theory.limits.reflects_colimit | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A functor `F : C ⥤ D` reflects colimits for `K : J ⥤ C` if
whenever the image of a cocone over `K` under `F` is a colimit cocone in `D`,
the cocone was already a colimit cocone in `C`.
Note that we do not assume a priori that `D` actually has any colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_limits_of_shape (J : Type w) [category.{w'} J] (F : C ⥤ D) | (reflects_limit : Π {K : J ⥤ C}, reflects_limit K F . tactic.apply_instance) | class | category_theory.limits.reflects_limits_of_shape | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A functor `F : C ⥤ D` reflects limits of shape `J` if
whenever the image of a cone over some `K : J ⥤ C` under `F` is a limit cone in `D`,
the cone was already a limit cone in `C`.
Note that we do not assume a priori that `D` actually has any limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_colimits_of_shape (J : Type w) [category.{w'} J] (F : C ⥤ D) | (reflects_colimit : Π {K : J ⥤ C}, reflects_colimit K F . tactic.apply_instance) | class | category_theory.limits.reflects_colimits_of_shape | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A functor `F : C ⥤ D` reflects colimits of shape `J` if
whenever the image of a cocone over some `K : J ⥤ C` under `F` is a colimit cocone in `D`,
the cocone was already a colimit cocone in `C`.
Note that we do not assume a priori that `D` actually has any colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_limits_of_size (F : C ⥤ D) | (reflects_limits_of_shape : Π {J : Type w} [category.{w'} J],
reflects_limits_of_shape J F . tactic.apply_instance) | class | category_theory.limits.reflects_limits_of_size | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A functor `F : C ⥤ D` reflects limits if
whenever the image of a cone over some `K : J ⥤ C` under `F` is a limit cone in `D`,
the cone was already a limit cone in `C`.
Note that we do not assume a priori that `D` actually has any limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_limits (F : C ⥤ D) | reflects_limits_of_size.{v₂ v₂} F | abbreviation | category_theory.limits.reflects_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A functor `F : C ⥤ D` reflects (small) limits if
whenever the image of a cone over some `K : J ⥤ C` under `F` is a limit cone in `D`,
the cone was already a limit cone in `C`.
Note that we do not assume a priori that `D` actually has any limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_colimits_of_size (F : C ⥤ D) | (reflects_colimits_of_shape : Π {J : Type w} [category.{w'} J],
reflects_colimits_of_shape J F . tactic.apply_instance) | class | category_theory.limits.reflects_colimits_of_size | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A functor `F : C ⥤ D` reflects colimits if
whenever the image of a cocone over some `K : J ⥤ C` under `F` is a colimit cocone in `D`,
the cocone was already a colimit cocone in `C`.
Note that we do not assume a priori that `D` actually has any colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_colimits (F : C ⥤ D) | reflects_colimits_of_size.{v₂ v₂} F | abbreviation | category_theory.limits.reflects_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A functor `F : C ⥤ D` reflects (small) colimits if
whenever the image of a cocone over some `K : J ⥤ C` under `F` is a colimit cocone in `D`,
the cocone was already a colimit cocone in `C`.
Note that we do not assume a priori that `D` actually has any colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_of_reflects (F : C ⥤ D) {c : cone K} (t : is_limit (F.map_cone c))
[reflects_limit K F] : is_limit c | reflects_limit.reflects t | def | category_theory.limits.is_limit_of_reflects | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A convenience function for `reflects_limit`, which takes the functor as an explicit argument to
guide typeclass resolution. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_of_reflects (F : C ⥤ D) {c : cocone K} (t : is_colimit (F.map_cocone c))
[reflects_colimit K F] :
is_colimit c | reflects_colimit.reflects t | def | category_theory.limits.is_colimit_of_reflects | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A convenience function for `reflects_colimit`, which takes the functor as an explicit argument to
guide typeclass resolution. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_limit_subsingleton (K : J ⥤ C) (F : C ⥤ D) : subsingleton (reflects_limit K F) | by split; rintros ⟨a⟩ ⟨b⟩; congr | instance | category_theory.limits.reflects_limit_subsingleton | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_colimit_subsingleton (K : J ⥤ C) (F : C ⥤ D) :
subsingleton (reflects_colimit K F) | by split; rintros ⟨a⟩ ⟨b⟩; congr | instance | category_theory.limits.reflects_colimit_subsingleton | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_limits_of_shape_subsingleton (J : Type w) [category.{w'} J] (F : C ⥤ D) :
subsingleton (reflects_limits_of_shape J F) | by { split, intros, cases a, cases b, congr } | instance | category_theory.limits.reflects_limits_of_shape_subsingleton | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_colimits_of_shape_subsingleton (J : Type w) [category.{w'} J] (F : C ⥤ D) :
subsingleton (reflects_colimits_of_shape J F) | by { split, intros, cases a, cases b, congr } | instance | category_theory.limits.reflects_colimits_of_shape_subsingleton | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_limits_subsingleton (F : C ⥤ D) :
subsingleton (reflects_limits_of_size.{w' w} F) | by { split, intros, cases a, cases b, cc } | instance | category_theory.limits.reflects_limits_subsingleton | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_colimits_subsingleton (F : C ⥤ D) :
subsingleton (reflects_colimits_of_size.{w' w} F) | by { split, intros, cases a, cases b, cc } | instance | category_theory.limits.reflects_colimits_subsingleton | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_limit_of_reflects_limits_of_shape (K : J ⥤ C) (F : C ⥤ D)
[H : reflects_limits_of_shape J F] : reflects_limit K F | reflects_limits_of_shape.reflects_limit | instance | category_theory.limits.reflects_limit_of_reflects_limits_of_shape | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_colimit_of_reflects_colimits_of_shape (K : J ⥤ C) (F : C ⥤ D)
[H : reflects_colimits_of_shape J F] : reflects_colimit K F | reflects_colimits_of_shape.reflects_colimit | instance | category_theory.limits.reflects_colimit_of_reflects_colimits_of_shape | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_limits_of_shape_of_reflects_limits (J : Type w) [category.{w'} J] (F : C ⥤ D)
[H : reflects_limits_of_size.{w' w} F] : reflects_limits_of_shape J F | reflects_limits_of_size.reflects_limits_of_shape | instance | category_theory.limits.reflects_limits_of_shape_of_reflects_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_colimits_of_shape_of_reflects_colimits (J : Type w) [category.{w'} J]
(F : C ⥤ D) [H : reflects_colimits_of_size.{w' w} F] : reflects_colimits_of_shape J F | reflects_colimits_of_size.reflects_colimits_of_shape | instance | category_theory.limits.reflects_colimits_of_shape_of_reflects_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_reflects_limits : reflects_limits_of_size.{w w'} (𝟭 C) | { reflects_limits_of_shape := λ J 𝒥,
{ reflects_limit := λ K, by exactI ⟨λ c h,
⟨λ s, h.lift ⟨s.X, λ j, s.π.app j, λ j j' f, s.π.naturality f⟩,
by cases K; rcases c with ⟨_, _, _⟩; intros s j; cases s; exact h.fac _ j,
by cases K; rcases c with ⟨_, _, _⟩; intros s m w; rcases s with ⟨_, _, _⟩;
exact h.u... | instance | category_theory.limits.id_reflects_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_reflects_colimits : reflects_colimits_of_size.{w w'} (𝟭 C) | { reflects_colimits_of_shape := λ J 𝒥,
{ reflects_colimit := λ K, by exactI ⟨λ c h,
⟨λ s, h.desc ⟨s.X, λ j, s.ι.app j, λ j j' f, s.ι.naturality f⟩,
by cases K; rcases c with ⟨_, _, _⟩; intros s j; cases s; exact h.fac _ j,
by cases K; rcases c with ⟨_, _, _⟩; intros s m w; rcases s with ⟨_, _, _⟩;
exact... | instance | category_theory.limits.id_reflects_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_reflects_limit [reflects_limit K F] [reflects_limit (K ⋙ F) G] :
reflects_limit K (F ⋙ G) | ⟨λ c h, reflects_limit.reflects (reflects_limit.reflects h)⟩ | instance | category_theory.limits.comp_reflects_limit | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_reflects_limits_of_shape
[reflects_limits_of_shape J F] [reflects_limits_of_shape J G] :
reflects_limits_of_shape J (F ⋙ G) | {} | instance | category_theory.limits.comp_reflects_limits_of_shape | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_reflects_limits
[reflects_limits_of_size.{w' w} F] [reflects_limits_of_size.{w' w} G] :
reflects_limits_of_size.{w' w} (F ⋙ G) | {} | instance | category_theory.limits.comp_reflects_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_reflects_colimit [reflects_colimit K F] [reflects_colimit (K ⋙ F) G] :
reflects_colimit K (F ⋙ G) | ⟨λ c h, reflects_colimit.reflects (reflects_colimit.reflects h)⟩ | instance | category_theory.limits.comp_reflects_colimit | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_reflects_colimits_of_shape
[reflects_colimits_of_shape J F] [reflects_colimits_of_shape J G] :
reflects_colimits_of_shape J (F ⋙ G) | {} | instance | category_theory.limits.comp_reflects_colimits_of_shape | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_reflects_colimits
[reflects_colimits_of_size.{w' w} F] [reflects_colimits_of_size.{w' w} G] :
reflects_colimits_of_size.{w' w} (F ⋙ G) | {} | instance | category_theory.limits.comp_reflects_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_limit_of_reflects_of_preserves [preserves_limit K (F ⋙ G)]
[reflects_limit (K ⋙ F) G] : preserves_limit K F | ⟨λ c h,
begin
apply is_limit_of_reflects G,
apply is_limit_of_preserves (F ⋙ G) h,
end⟩ | def | category_theory.limits.preserves_limit_of_reflects_of_preserves | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | If `F ⋙ G` preserves limits for `K`, and `G` reflects limits for `K ⋙ F`,
then `F` preserves limits for `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_shape_of_reflects_of_preserves [preserves_limits_of_shape J (F ⋙ G)]
[reflects_limits_of_shape J G] : preserves_limits_of_shape J F | { preserves_limit := λ K, preserves_limit_of_reflects_of_preserves F G } | def | category_theory.limits.preserves_limits_of_shape_of_reflects_of_preserves | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | If `F ⋙ G` preserves limits of shape `J` and `G` reflects limits of shape `J`, then `F` preserves
limits of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_reflects_of_preserves
[preserves_limits_of_size.{w' w} (F ⋙ G)] [reflects_limits_of_size.{w' w} G] :
preserves_limits_of_size.{w' w} F | { preserves_limits_of_shape := λ J 𝒥₁,
by exactI preserves_limits_of_shape_of_reflects_of_preserves F G } | def | category_theory.limits.preserves_limits_of_reflects_of_preserves | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | If `F ⋙ G` preserves limits and `G` reflects limits, then `F` preserves limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_limit_of_iso_diagram {K₁ K₂ : J ⥤ C} (F : C ⥤ D) (h : K₁ ≅ K₂)
[reflects_limit K₁ F] : reflects_limit K₂ F | { reflects := λ c t,
begin
apply is_limit.postcompose_inv_equiv h c (is_limit_of_reflects F _),
apply ((is_limit.postcompose_inv_equiv (iso_whisker_right h F : _) _).symm t).of_iso_limit _,
exact cones.ext (iso.refl _) (by tidy),
end } | def | category_theory.limits.reflects_limit_of_iso_diagram | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer reflection of limits along a natural isomorphism in the diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_limit_of_nat_iso (K : J ⥤ C) {F G : C ⥤ D} (h : F ≅ G) [reflects_limit K F] :
reflects_limit K G | { reflects := λ c t, reflects_limit.reflects (is_limit.map_cone_equiv h.symm t) } | def | category_theory.limits.reflects_limit_of_nat_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer reflection of a limit along a natural isomorphism in the functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_limits_of_shape_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [reflects_limits_of_shape J F] :
reflects_limits_of_shape J G | { reflects_limit := λ K, reflects_limit_of_nat_iso K h } | def | category_theory.limits.reflects_limits_of_shape_of_nat_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer reflection of limits of shape along a natural isomorphism in the functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_limits_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [reflects_limits_of_size.{w' w} F] :
reflects_limits_of_size.{w' w} G | { reflects_limits_of_shape := λ J 𝒥₁, by exactI reflects_limits_of_shape_of_nat_iso h } | def | category_theory.limits.reflects_limits_of_nat_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer reflection of limits along a natural isomorphism in the functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_limits_of_shape_of_equiv {J' : Type w₂} [category.{w₂'} J'] (e : J ≌ J')
(F : C ⥤ D) [reflects_limits_of_shape J F] :
reflects_limits_of_shape J' F | { reflects_limit := λ K,
{ reflects := λ c t,
begin
apply is_limit.of_whisker_equivalence e,
apply is_limit_of_reflects F,
apply is_limit.of_iso_limit _ (functor.map_cone_whisker _).symm,
exact is_limit.whisker_equivalence t _,
end } } | def | category_theory.limits.reflects_limits_of_shape_of_equiv | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer reflection of limits along a equivalence in the shape. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_limits_of_size_shrink (F : C ⥤ D)
[reflects_limits_of_size.{(max w w₂) (max w' w₂')} F] : reflects_limits_of_size.{w w'} F | ⟨λ J hJ, by exactI reflects_limits_of_shape_of_equiv
(ulift_hom_ulift_category.equiv.{w₂ w₂'} J).symm F⟩ | def | category_theory.limits.reflects_limits_of_size_shrink | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | `reflects_limits_of_size_shrink.{w w'} F` tries to obtain `reflects_limits_of_size.{w w'} F`
from some other `reflects_limits_of_size F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_smallest_limits_of_reflects_limits
(F : C ⥤ D) [reflects_limits_of_size.{v₃ u₃} F] : reflects_limits_of_size.{0 0} F | reflects_limits_of_size_shrink F | def | category_theory.limits.reflects_smallest_limits_of_reflects_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Reflecting limits at any universe implies reflecting limits at universe `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_limit_of_reflects_isomorphisms (F : J ⥤ C) (G : C ⥤ D)
[reflects_isomorphisms G] [has_limit F] [preserves_limit F G] :
reflects_limit F G | { reflects := λ c t,
begin
apply is_limit.of_point_iso (limit.is_limit F),
change is_iso ((cones.forget _).map ((limit.is_limit F).lift_cone_morphism c)),
apply (cones.forget F).map_is_iso _,
apply is_iso_of_reflects_iso _ (cones.functoriality F G),
refine t.hom_is_iso (is_limit_of_preserves G (li... | def | category_theory.limits.reflects_limit_of_reflects_isomorphisms | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | If the limit of `F` exists and `G` preserves it, then if `G` reflects isomorphisms then it
reflects the limit of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_limits_of_shape_of_reflects_isomorphisms {G : C ⥤ D}
[reflects_isomorphisms G] [has_limits_of_shape J C] [preserves_limits_of_shape J G] :
reflects_limits_of_shape J G | { reflects_limit := λ F, reflects_limit_of_reflects_isomorphisms F G } | def | category_theory.limits.reflects_limits_of_shape_of_reflects_isomorphisms | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | If `C` has limits of shape `J` and `G` preserves them, then if `G` reflects isomorphisms then it
reflects limits of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_limits_of_reflects_isomorphisms {G : C ⥤ D}
[reflects_isomorphisms G] [has_limits_of_size.{w' w} C] [preserves_limits_of_size.{w' w} G] :
reflects_limits_of_size.{w' w} G | { reflects_limits_of_shape := λ J 𝒥₁,
by exactI reflects_limits_of_shape_of_reflects_isomorphisms } | def | category_theory.limits.reflects_limits_of_reflects_isomorphisms | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | If `C` has limits and `G` preserves limits, then if `G` reflects isomorphisms then it reflects
limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimit_of_reflects_of_preserves [preserves_colimit K (F ⋙ G)]
[reflects_colimit (K ⋙ F) G] : preserves_colimit K F | ⟨λ c h,
begin
apply is_colimit_of_reflects G,
apply is_colimit_of_preserves (F ⋙ G) h,
end⟩ | def | category_theory.limits.preserves_colimit_of_reflects_of_preserves | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | If `F ⋙ G` preserves colimits for `K`, and `G` reflects colimits for `K ⋙ F`,
then `F` preserves colimits for `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_shape_of_reflects_of_preserves [preserves_colimits_of_shape J (F ⋙ G)]
[reflects_colimits_of_shape J G] : preserves_colimits_of_shape J F | { preserves_colimit := λ K, preserves_colimit_of_reflects_of_preserves F G } | def | category_theory.limits.preserves_colimits_of_shape_of_reflects_of_preserves | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | If `F ⋙ G` preserves colimits of shape `J` and `G` reflects colimits of shape `J`, then `F`
preserves colimits of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_reflects_of_preserves [preserves_colimits_of_size.{w' w} (F ⋙ G)]
[reflects_colimits_of_size.{w' w} G] : preserves_colimits_of_size.{w' w} F | { preserves_colimits_of_shape := λ J 𝒥₁,
by exactI preserves_colimits_of_shape_of_reflects_of_preserves F G } | def | category_theory.limits.preserves_colimits_of_reflects_of_preserves | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | If `F ⋙ G` preserves colimits and `G` reflects colimits, then `F` preserves colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_colimit_of_iso_diagram {K₁ K₂ : J ⥤ C} (F : C ⥤ D) (h : K₁ ≅ K₂)
[reflects_colimit K₁ F] : reflects_colimit K₂ F | { reflects := λ c t,
begin
apply is_colimit.precompose_hom_equiv h c (is_colimit_of_reflects F _),
apply ((is_colimit.precompose_hom_equiv (iso_whisker_right h F : _) _).symm t).of_iso_colimit _,
exact cocones.ext (iso.refl _) (by tidy),
end } | def | category_theory.limits.reflects_colimit_of_iso_diagram | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer reflection of colimits along a natural isomorphism in the diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_colimit_of_nat_iso (K : J ⥤ C) {F G : C ⥤ D} (h : F ≅ G) [reflects_colimit K F] :
reflects_colimit K G | { reflects := λ c t, reflects_colimit.reflects (is_colimit.map_cocone_equiv h.symm t) } | def | category_theory.limits.reflects_colimit_of_nat_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer reflection of a colimit along a natural isomorphism in the functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_colimits_of_shape_of_nat_iso {F G : C ⥤ D} (h : F ≅ G)
[reflects_colimits_of_shape J F] : reflects_colimits_of_shape J G | { reflects_colimit := λ K, reflects_colimit_of_nat_iso K h } | def | category_theory.limits.reflects_colimits_of_shape_of_nat_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer reflection of colimits of shape along a natural isomorphism in the functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_colimits_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [reflects_colimits_of_size.{w w'} F] :
reflects_colimits_of_size.{w w'} G | { reflects_colimits_of_shape := λ J 𝒥₁, by exactI reflects_colimits_of_shape_of_nat_iso h } | def | category_theory.limits.reflects_colimits_of_nat_iso | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer reflection of colimits along a natural isomorphism in the functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_colimits_of_shape_of_equiv {J' : Type w₂} [category.{w₂'} J'] (e : J ≌ J')
(F : C ⥤ D) [reflects_colimits_of_shape J F] :
reflects_colimits_of_shape J' F | { reflects_colimit := λ K,
{ reflects := λ c t,
begin
apply is_colimit.of_whisker_equivalence e,
apply is_colimit_of_reflects F,
apply is_colimit.of_iso_colimit _ (functor.map_cocone_whisker _).symm,
exact is_colimit.whisker_equivalence t _,
end } } | def | category_theory.limits.reflects_colimits_of_shape_of_equiv | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Transfer reflection of colimits along a equivalence in the shape. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_colimits_of_size_shrink (F : C ⥤ D)
[reflects_colimits_of_size.{(max w w₂) (max w' w₂')} F] : reflects_colimits_of_size.{w w'} F | ⟨λ J hJ, by exactI reflects_colimits_of_shape_of_equiv
(ulift_hom_ulift_category.equiv.{w₂ w₂'} J).symm F⟩ | def | category_theory.limits.reflects_colimits_of_size_shrink | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | `reflects_colimits_of_size_shrink.{w w'} F` tries to obtain `reflects_colimits_of_size.{w w'} F`
from some other `reflects_colimits_of_size F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_smallest_colimits_of_reflects_colimits
(F : C ⥤ D) [reflects_colimits_of_size.{v₃ u₃} F] : reflects_colimits_of_size.{0 0} F | reflects_colimits_of_size_shrink F | def | category_theory.limits.reflects_smallest_colimits_of_reflects_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | Reflecting colimits at any universe implies reflecting colimits at universe `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_colimit_of_reflects_isomorphisms (F : J ⥤ C) (G : C ⥤ D)
[reflects_isomorphisms G] [has_colimit F] [preserves_colimit F G] :
reflects_colimit F G | { reflects := λ c t,
begin
apply is_colimit.of_point_iso (colimit.is_colimit F),
change is_iso ((cocones.forget _).map ((colimit.is_colimit F).desc_cocone_morphism c)),
apply (cocones.forget F).map_is_iso _,
apply is_iso_of_reflects_iso _ (cocones.functoriality F G),
refine (is_colimit_of_preserve... | def | category_theory.limits.reflects_colimit_of_reflects_isomorphisms | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | If the colimit of `F` exists and `G` preserves it, then if `G` reflects isomorphisms then it
reflects the colimit of `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_colimits_of_shape_of_reflects_isomorphisms {G : C ⥤ D}
[reflects_isomorphisms G] [has_colimits_of_shape J C] [preserves_colimits_of_shape J G] :
reflects_colimits_of_shape J G | { reflects_colimit := λ F, reflects_colimit_of_reflects_isomorphisms F G } | def | category_theory.limits.reflects_colimits_of_shape_of_reflects_isomorphisms | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | If `C` has colimits of shape `J` and `G` preserves them, then if `G` reflects isomorphisms then it
reflects colimits of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_colimits_of_reflects_isomorphisms {G : C ⥤ D}
[reflects_isomorphisms G] [has_colimits_of_size.{w' w} C] [preserves_colimits_of_size.{w' w} G] :
reflects_colimits_of_size.{w' w} G | { reflects_colimits_of_shape := λ J 𝒥₁,
by exactI reflects_colimits_of_shape_of_reflects_isomorphisms } | def | category_theory.limits.reflects_colimits_of_reflects_isomorphisms | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | If `C` has colimits and `G` preserves colimits, then if `G` reflects isomorphisms then it reflects
colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fully_faithful_reflects_limits [full F] [faithful F] : reflects_limits_of_size.{w w'} F | { reflects_limits_of_shape := λ J 𝒥₁, by exactI
{ reflects_limit := λ K,
{ reflects := λ c t,
is_limit.mk_cone_morphism (λ s, (cones.functoriality K F).preimage (t.lift_cone_morphism _)) $
begin
apply (λ s m, (cones.functoriality K F).map_injective _),
rw [functor.image_preimage],
... | def | category_theory.limits.fully_faithful_reflects_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A fully faithful functor reflects limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fully_faithful_reflects_colimits [full F] [faithful F] : reflects_colimits_of_size.{w w'} F | { reflects_colimits_of_shape := λ J 𝒥₁, by exactI
{ reflects_colimit := λ K,
{ reflects := λ c t,
is_colimit.mk_cocone_morphism
(λ s, (cocones.functoriality K F).preimage (t.desc_cocone_morphism _)) $
begin
apply (λ s m, (cocones.functoriality K F).map_injective _),
rw [functo... | def | category_theory.limits.fully_faithful_reflects_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/basic.lean | [
"category_theory.limits.has_limits"
] | [] | A fully faithful functor reflects colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_filtered_colimits (F : C ⥤ D) : Type (max u₁ u₂ (v+1)) | (preserves_filtered_colimits : Π (J : Type v) [small_category J] [is_filtered J],
preserves_colimits_of_shape J F) | class | category_theory.limits.preserves_filtered_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/filtered.lean | [
"category_theory.limits.preserves.basic",
"category_theory.filtered"
] | [] | A functor is said to preserve filtered colimits, if it preserves all colimits of shape `J`, where
`J` is a filtered category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits.preserves_filtered_colimits (F : C ⥤ D) [preserves_colimits F] :
preserves_filtered_colimits F | { preserves_filtered_colimits := infer_instance } | instance | category_theory.limits.preserves_colimits.preserves_filtered_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/filtered.lean | [
"category_theory.limits.preserves.basic",
"category_theory.filtered"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_preserves_filtered_colimits (F : C ⥤ D) (G : D ⥤ E)
[preserves_filtered_colimits F] [preserves_filtered_colimits G] :
preserves_filtered_colimits (F ⋙ G) | { preserves_filtered_colimits := λ J _ _, by exactI infer_instance } | instance | category_theory.limits.comp_preserves_filtered_colimits | category_theory.limits.preserves | src/category_theory/limits/preserves/filtered.lean | [
"category_theory.limits.preserves.basic",
"category_theory.filtered"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_cofiltered_limits (F : C ⥤ D) : Type (max u₁ u₂ (v+1)) | (preserves_cofiltered_limits : Π (J : Type v) [small_category J] [is_cofiltered J],
preserves_limits_of_shape J F) | class | category_theory.limits.preserves_cofiltered_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/filtered.lean | [
"category_theory.limits.preserves.basic",
"category_theory.filtered"
] | [] | A functor is said to preserve cofiltered limits, if it preserves all limits of shape `J`, where
`J` is a cofiltered category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits.preserves_cofiltered_limits (F : C ⥤ D) [preserves_limits F] :
preserves_cofiltered_limits F | { preserves_cofiltered_limits := infer_instance } | instance | category_theory.limits.preserves_limits.preserves_cofiltered_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/filtered.lean | [
"category_theory.limits.preserves.basic",
"category_theory.filtered"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_preserves_cofiltered_limits (F : C ⥤ D) (G : D ⥤ E)
[preserves_cofiltered_limits F] [preserves_cofiltered_limits G] :
preserves_cofiltered_limits (F ⋙ G) | { preserves_cofiltered_limits := λ J _ _, by exactI infer_instance } | instance | category_theory.limits.comp_preserves_cofiltered_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/filtered.lean | [
"category_theory.limits.preserves.basic",
"category_theory.filtered"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_finite_limits (F : C ⥤ D) | (preserves_finite_limits : Π (J : Type) [small_category J] [fin_category J],
preserves_limits_of_shape J F . tactic.apply_instance) | class | category_theory.limits.preserves_finite_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [] | A functor is said to preserve finite limits, if it preserves all limits of shape `J`,
where `J : Type` is a finite category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_shape_of_preserves_finite_limits (F : C ⥤ D)
[preserves_finite_limits F] (J : Type w) [small_category J] [fin_category J] :
preserves_limits_of_shape J F | by apply preserves_limits_of_shape_of_equiv (fin_category.equiv_as_type J) | instance | category_theory.limits.preserves_limits_of_shape_of_preserves_finite_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [] | Preserving finite limits also implies preserving limits over finite shapes in higher universes,
though through a noncomputable instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_size.preserves_finite_limits (F : C ⥤ D)
[preserves_limits_of_size.{w w₂} F] : preserves_finite_limits F | ⟨λ J sJ fJ,
begin
haveI := preserves_smallest_limits_of_preserves_limits F,
exact preserves_limits_of_shape_of_equiv (fin_category.equiv_as_type J) F,
end⟩ | def | category_theory.limits.preserves_limits_of_size.preserves_finite_limits | category_theory.limits.preserves | src/category_theory/limits/preserves/finite.lean | [
"category_theory.limits.preserves.basic",
"category_theory.fin_category"
] | [] | If we preserve limits of some arbitrary size, then we preserve all finite limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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