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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is_initial.to_eq_desc_cocone_morphism {F : J ⥤ C} {c : cocone F}
(hc : is_initial c) (s : cocone F) :
is_initial.to hc s = ((cocone.is_colimit_equiv_is_initial _).symm hc).desc_cocone_morphism s | by convert (is_colimit.desc_cocone_morphism_eq_is_initial_to _ s).symm | lemma | category_theory.limits.is_initial.to_eq_desc_cocone_morphism | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_colimit.of_preserves_cocone_initial {F : J ⥤ C} {F' : K ⥤ D} (G : cocone F ⥤ cocone F')
[preserves_colimit (functor.empty.{0} _) G] {c : cocone F} (hc : is_colimit c) :
is_colimit (G.obj c) | (cocone.is_colimit_equiv_is_initial _).symm $
(cocone.is_colimit_equiv_is_initial _ hc).is_initial_obj _ _ | def | category_theory.limits.is_colimit.of_preserves_cocone_initial | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | If `G : cocone F ⥤ cocone F'` preserves initial objects, it preserves colimit cocones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit.of_reflects_cocone_initial {F : J ⥤ C} {F' : K ⥤ D} (G : cocone F ⥤ cocone F')
[reflects_colimit (functor.empty.{0} _) G] {c : cocone F} (hc : is_colimit (G.obj c)) :
is_colimit c | (cocone.is_colimit_equiv_is_initial _).symm $
(cocone.is_colimit_equiv_is_initial _ hc).is_initial_of_obj _ _ | def | category_theory.limits.is_colimit.of_reflects_cocone_initial | category_theory.limits | src/category_theory/limits/cone_category.lean | [
"category_theory.adjunction.comma",
"category_theory.limits.preserves.shapes.terminal",
"category_theory.structured_arrow",
"category_theory.limits.shapes.equivalence"
] | [] | If `G : cocone F ⥤ cocone F'` reflects initial objects, it reflects colimit cocones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
wide_pullback_shape_connected (J : Type v₁) : is_connected (wide_pullback_shape J) | begin
apply is_connected.of_induct,
introv hp t,
cases j,
{ exact hp },
{ rwa t (wide_pullback_shape.hom.term j) }
end | instance | category_theory.wide_pullback_shape_connected | category_theory.limits | src/category_theory/limits/connected.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.is_connected",
"category_theory.limits.preserves.basic"
] | [
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
wide_pushout_shape_connected (J : Type v₁) : is_connected (wide_pushout_shape J) | begin
apply is_connected.of_induct,
introv hp t,
cases j,
{ exact hp },
{ rwa ← t (wide_pushout_shape.hom.init j) }
end | instance | category_theory.wide_pushout_shape_connected | category_theory.limits | src/category_theory/limits/connected.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.is_connected",
"category_theory.limits.preserves.basic"
] | [
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallel_pair_inhabited : inhabited walking_parallel_pair | ⟨walking_parallel_pair.one⟩ | instance | category_theory.parallel_pair_inhabited | category_theory.limits | src/category_theory/limits/connected.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.is_connected",
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
parallel_pair_connected : is_connected (walking_parallel_pair) | begin
apply is_connected.of_induct,
introv _ t,
cases j,
{ rwa t walking_parallel_pair_hom.left },
{ assumption }
end | instance | category_theory.parallel_pair_connected | category_theory.limits | src/category_theory/limits/connected.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.is_connected",
"category_theory.limits.preserves.basic"
] | [
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
γ₂ {K : J ⥤ C} (X : C) : K ⋙ prod.functor.obj X ⟶ K | { app := λ Y, limits.prod.snd } | def | category_theory.prod_preserves_connected_limits.γ₂ | category_theory.limits | src/category_theory/limits/connected.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.is_connected",
"category_theory.limits.preserves.basic"
] | [] | (Impl). The obvious natural transformation from (X × K -) to K. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
γ₁ {K : J ⥤ C} (X : C) : K ⋙ prod.functor.obj X ⟶ (functor.const J).obj X | { app := λ Y, limits.prod.fst } | def | category_theory.prod_preserves_connected_limits.γ₁ | category_theory.limits | src/category_theory/limits/connected.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.is_connected",
"category_theory.limits.preserves.basic"
] | [
"functor.const"
] | (Impl). The obvious natural transformation from (X × K -) to X | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_cone {X : C} {K : J ⥤ C} (s : cone (K ⋙ prod.functor.obj X)) : cone K | { X := s.X,
π := s.π ≫ γ₂ X } | def | category_theory.prod_preserves_connected_limits.forget_cone | category_theory.limits | src/category_theory/limits/connected.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.is_connected",
"category_theory.limits.preserves.basic"
] | [] | (Impl).
Given a cone for (X × K -), produce a cone for K using the natural transformation `γ₂` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_preserves_connected_limits [is_connected J] (X : C) :
preserves_limits_of_shape J (prod.functor.obj X) | { preserves_limit := λ K,
{ preserves := λ c l,
{ lift := λ s, prod.lift
(s.π.app (classical.arbitrary _) ≫ limits.prod.fst)
(l.lift (forget_cone s)),
fac' := λ s j,
begin
apply prod.hom_ext,
{ erw [assoc, lim_map_π, comp_id, limit.lift_π],
exact (nat_trans_fr... | def | category_theory.prod_preserves_connected_limits | category_theory.limits | src/category_theory/limits/connected.lean | [
"category_theory.limits.shapes.binary_products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.is_connected",
"category_theory.limits.preserves.basic"
] | [
"classical.arbitrary",
"is_connected",
"lift"
] | The functor `(X × -)` preserves any connected limit.
Note that this functor does not preserve the two most obvious disconnected limits - that is,
`(X × -)` does not preserve products or terminal object, eg `(X ⨯ A) ⨯ (X ⨯ B)` is not isomorphic to
`X ⨯ (A ⨯ B)` and `X ⨯ 1` is not isomorphic to `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
liftable_cone (K : J ⥤ C) (F : C ⥤ D) (c : cone (K ⋙ F)) | (lifted_cone : cone K)
(valid_lift : F.map_cone lifted_cone ≅ c) | structure | category_theory.liftable_cone | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | Define the lift of a cone: For a cone `c` for `K ⋙ F`, give a cone for `K`
which is a lift of `c`, i.e. the image of it under `F` is (iso) to `c`.
We will then use this as part of the definition of creation of limits:
every limit cone has a lift.
Note this definition is really only useful when `c` is a limit already. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
liftable_cocone (K : J ⥤ C) (F : C ⥤ D) (c : cocone (K ⋙ F)) | (lifted_cocone : cocone K)
(valid_lift : F.map_cocone lifted_cocone ≅ c) | structure | category_theory.liftable_cocone | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | Define the lift of a cocone: For a cocone `c` for `K ⋙ F`, give a cocone for
`K` which is a lift of `c`, i.e. the image of it under `F` is (iso) to `c`.
We will then use this as part of the definition of creation of colimits:
every limit cocone has a lift.
Note this definition is really only useful when `c` is a coli... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_limit (K : J ⥤ C) (F : C ⥤ D) extends reflects_limit K F | (lifts : Π c, is_limit c → liftable_cone K F c) | class | category_theory.creates_limit | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | Definition 3.3.1 of [Riehl].
We say that `F` creates limits of `K` if, given any limit cone `c` for `K ⋙ F`
(i.e. below) we can lift it to a cone "above", and further that `F` reflects
limits for `K`.
If `F` reflects isomorphisms, it suffices to show only that the lifted cone is
a limit - see `creates_limit_of_reflect... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_limits_of_shape (J : Type w) [category.{w'} J] (F : C ⥤ D) | (creates_limit : Π {K : J ⥤ C}, creates_limit K F . tactic.apply_instance) | class | category_theory.creates_limits_of_shape | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | `F` creates limits of shape `J` if `F` creates the limit of any diagram
`K : J ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_limits_of_size (F : C ⥤ D) | (creates_limits_of_shape : Π {J : Type w} [category.{w'} J],
creates_limits_of_shape J F . tactic.apply_instance) | class | category_theory.creates_limits_of_size | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | `F` creates limits if it creates limits of shape `J` for any `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_limits (F : C ⥤ D) | creates_limits_of_size.{v₂ v₂} F | abbreviation | category_theory.creates_limits | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | `F` creates small limits if it creates limits of shape `J` for any small `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_colimit (K : J ⥤ C) (F : C ⥤ D) extends reflects_colimit K F | (lifts : Π c, is_colimit c → liftable_cocone K F c) | class | category_theory.creates_colimit | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | Dual of definition 3.3.1 of [Riehl].
We say that `F` creates colimits of `K` if, given any limit cocone `c` for
`K ⋙ F` (i.e. below) we can lift it to a cocone "above", and further that `F`
reflects limits for `K`.
If `F` reflects isomorphisms, it suffices to show only that the lifted cocone is
a limit - see `creates_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_colimits_of_shape (J : Type w) [category.{w'} J] (F : C ⥤ D) | (creates_colimit : Π {K : J ⥤ C}, creates_colimit K F . tactic.apply_instance) | class | category_theory.creates_colimits_of_shape | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | `F` creates colimits of shape `J` if `F` creates the colimit of any diagram
`K : J ⥤ C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_colimits_of_size (F : C ⥤ D) | (creates_colimits_of_shape : Π {J : Type w} [category.{w'} J],
creates_colimits_of_shape J F . tactic.apply_instance) | class | category_theory.creates_colimits_of_size | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | `F` creates colimits if it creates colimits of shape `J` for any small `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_colimits (F : C ⥤ D) | creates_colimits_of_size.{v₂ v₂} F | abbreviation | category_theory.creates_colimits | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | `F` creates small colimits if it creates colimits of shape `J` for any small `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_limit {K : J ⥤ C} {F : C ⥤ D} [creates_limit K F] {c : cone (K ⋙ F)} (t : is_limit c) :
cone K | (creates_limit.lifts c t).lifted_cone | def | category_theory.lift_limit | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | `lift_limit t` is the cone for `K` given by lifting the limit `t` for `K ⋙ F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lifted_limit_maps_to_original {K : J ⥤ C} {F : C ⥤ D}
[creates_limit K F] {c : cone (K ⋙ F)} (t : is_limit c) :
F.map_cone (lift_limit t) ≅ c | (creates_limit.lifts c t).valid_lift | def | category_theory.lifted_limit_maps_to_original | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | The lifted cone has an image isomorphic to the original cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lifted_limit_is_limit {K : J ⥤ C} {F : C ⥤ D}
[creates_limit K F] {c : cone (K ⋙ F)} (t : is_limit c) :
is_limit (lift_limit t) | reflects_limit.reflects (is_limit.of_iso_limit t (lifted_limit_maps_to_original t).symm) | def | category_theory.lifted_limit_is_limit | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | The lifted cone is a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit_of_created (K : J ⥤ C) (F : C ⥤ D)
[has_limit (K ⋙ F)] [creates_limit K F] : has_limit K | has_limit.mk { cone := lift_limit (limit.is_limit (K ⋙ F)),
is_limit := lifted_limit_is_limit _ } | lemma | category_theory.has_limit_of_created | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `F` creates the limit of `K` and `K ⋙ F` has a limit, then `K` has a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_shape_of_has_limits_of_shape_creates_limits_of_shape (F : C ⥤ D)
[has_limits_of_shape J D] [creates_limits_of_shape J F] : has_limits_of_shape J C | ⟨λ G, has_limit_of_created G F⟩ | lemma | category_theory.has_limits_of_shape_of_has_limits_of_shape_creates_limits_of_shape | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `F` creates limits of shape `J`, and `D` has limits of shape `J`, then
`C` has limits of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_has_limits_creates_limits (F : C ⥤ D) [has_limits_of_size.{w w'} D]
[creates_limits_of_size.{w w'} F] : has_limits_of_size.{w w'} C | ⟨λ J I, by exactI has_limits_of_shape_of_has_limits_of_shape_creates_limits_of_shape F⟩ | lemma | category_theory.has_limits_of_has_limits_creates_limits | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `F` creates limits, and `D` has all limits, then `C` has all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_colimit {K : J ⥤ C} {F : C ⥤ D} [creates_colimit K F] {c : cocone (K ⋙ F)}
(t : is_colimit c) :
cocone K | (creates_colimit.lifts c t).lifted_cocone | def | category_theory.lift_colimit | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | `lift_colimit t` is the cocone for `K` given by lifting the colimit `t` for `K ⋙ F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lifted_colimit_maps_to_original {K : J ⥤ C} {F : C ⥤ D}
[creates_colimit K F] {c : cocone (K ⋙ F)} (t : is_colimit c) :
F.map_cocone (lift_colimit t) ≅ c | (creates_colimit.lifts c t).valid_lift | def | category_theory.lifted_colimit_maps_to_original | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | The lifted cocone has an image isomorphic to the original cocone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lifted_colimit_is_colimit {K : J ⥤ C} {F : C ⥤ D}
[creates_colimit K F] {c : cocone (K ⋙ F)} (t : is_colimit c) :
is_colimit (lift_colimit t) | reflects_colimit.reflects (is_colimit.of_iso_colimit t (lifted_colimit_maps_to_original t).symm) | def | category_theory.lifted_colimit_is_colimit | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | The lifted cocone is a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimit_of_created (K : J ⥤ C) (F : C ⥤ D)
[has_colimit (K ⋙ F)] [creates_colimit K F] : has_colimit K | has_colimit.mk { cocone := lift_colimit (colimit.is_colimit (K ⋙ F)),
is_colimit := lifted_colimit_is_colimit _ } | lemma | category_theory.has_colimit_of_created | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `F` creates the limit of `K` and `K ⋙ F` has a limit, then `K` has a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_of_shape_of_has_colimits_of_shape_creates_colimits_of_shape (F : C ⥤ D)
[has_colimits_of_shape J D] [creates_colimits_of_shape J F] : has_colimits_of_shape J C | ⟨λ G, has_colimit_of_created G F⟩ | lemma | category_theory.has_colimits_of_shape_of_has_colimits_of_shape_creates_colimits_of_shape | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `F` creates colimits of shape `J`, and `D` has colimits of shape `J`, then
`C` has colimits of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimits_of_has_colimits_creates_colimits (F : C ⥤ D) [has_colimits_of_size.{w w'} D]
[creates_colimits_of_size.{w w'} F] : has_colimits_of_size.{w w'} C | ⟨λ J I, by exactI has_colimits_of_shape_of_has_colimits_of_shape_creates_colimits_of_shape F⟩ | lemma | category_theory.has_colimits_of_has_colimits_creates_colimits | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `F` creates colimits, and `D` has all colimits, then `C` has all colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflects_limits_of_shape_of_creates_limits_of_shape (F : C ⥤ D)
[creates_limits_of_shape J F] : reflects_limits_of_shape J F | {} | instance | category_theory.reflects_limits_of_shape_of_creates_limits_of_shape | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_limits_of_creates_limits (F : C ⥤ D)
[creates_limits_of_size.{w w'} F] : reflects_limits_of_size.{w w'} F | {} | instance | category_theory.reflects_limits_of_creates_limits | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_colimits_of_shape_of_creates_colimits_of_shape (F : C ⥤ D)
[creates_colimits_of_shape J F] : reflects_colimits_of_shape J F | {} | instance | category_theory.reflects_colimits_of_shape_of_creates_colimits_of_shape | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflects_colimits_of_creates_colimits (F : C ⥤ D)
[creates_colimits_of_size.{w w'} F] : reflects_colimits_of_size.{w w'} F | {} | instance | category_theory.reflects_colimits_of_creates_colimits | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lifts_to_limit (K : J ⥤ C) (F : C ⥤ D) (c : cone (K ⋙ F)) (t : is_limit c)
extends liftable_cone K F c | (makes_limit : is_limit lifted_cone) | structure | category_theory.lifts_to_limit | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | A helper to show a functor creates limits. In particular, if we can show
that for any limit cone `c` for `K ⋙ F`, there is a lift of it which is
a limit and `F` reflects isomorphisms, then `F` creates limits.
Usually, `F` creating limits says that _any_ lift of `c` is a limit, but
here we only need to show that our par... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lifts_to_colimit (K : J ⥤ C) (F : C ⥤ D) (c : cocone (K ⋙ F)) (t : is_colimit c)
extends liftable_cocone K F c | (makes_colimit : is_colimit lifted_cocone) | structure | category_theory.lifts_to_colimit | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | A helper to show a functor creates colimits. In particular, if we can show
that for any limit cocone `c` for `K ⋙ F`, there is a lift of it which is
a limit and `F` reflects isomorphisms, then `F` creates colimits.
Usually, `F` creating colimits says that _any_ lift of `c` is a colimit, but
here we only need to show th... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_limit_of_reflects_iso {K : J ⥤ C} {F : C ⥤ D} [reflects_isomorphisms F]
(h : Π c t, lifts_to_limit K F c t) :
creates_limit K F | { lifts := λ c t, (h c t).to_liftable_cone,
to_reflects_limit :=
{ reflects := λ (d : cone K) (hd : is_limit (F.map_cone d)),
begin
let d' : cone K := (h (F.map_cone d) hd).to_liftable_cone.lifted_cone,
let i : F.map_cone d' ≅ F.map_cone d := (h (F.map_cone d) hd).to_liftable_cone.valid_lift,
... | def | category_theory.creates_limit_of_reflects_iso | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `F` reflects isomorphisms and we can lift any limit cone to a limit cone,
then `F` creates limits.
In particular here we don't need to assume that F reflects limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_limit_of_fully_faithful_of_lift' {K : J ⥤ C} {F : C ⥤ D} [full F] [faithful F]
{l : cone (K ⋙ F)} (hl : is_limit l) (c : cone K) (i : F.map_cone c ≅ l) : creates_limit K F | creates_limit_of_reflects_iso (λ c' t,
{ lifted_cone := c,
valid_lift := i ≪≫ is_limit.unique_up_to_iso hl t,
makes_limit := is_limit.of_faithful F (is_limit.of_iso_limit hl i.symm) _
(λ s, F.image_preimage _) }) | def | category_theory.creates_limit_of_fully_faithful_of_lift' | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_limit_of_fully_faithful_of_lift {K : J ⥤ C} {F : C ⥤ D}
[full F] [faithful F] [has_limit (K ⋙ F)]
(c : cone K) (i : F.map_cone c ≅ limit.cone (K ⋙ F)) : creates_limit K F | creates_limit_of_fully_faithful_of_lift' (limit.is_limit _) c i | def | category_theory.creates_limit_of_fully_faithful_of_lift | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_limit_of_fully_faithful_of_iso' {K : J ⥤ C} {F : C ⥤ D} [full F] [faithful F]
{l : cone (K ⋙ F)} (hl : is_limit l) (X : C) (i : F.obj X ≅ l.X) : creates_limit K F | creates_limit_of_fully_faithful_of_lift' hl
({ X := X,
π :=
{ app := λ j, F.preimage (i.hom ≫ l.π.app j),
naturality' := λ Y Z f, F.map_injective $ by { dsimp, simpa using (l.w f).symm } } })
(cones.ext i (λ j, by simp only [functor.image_preimage, functor.map_cone_π_app])) | def | category_theory.creates_limit_of_fully_faithful_of_iso' | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_limit_of_fully_faithful_of_iso {K : J ⥤ C} {F : C ⥤ D}
[full F] [faithful F] [has_limit (K ⋙ F)]
(X : C) (i : F.obj X ≅ limit (K ⋙ F)) : creates_limit K F | creates_limit_of_fully_faithful_of_iso' (limit.is_limit _) X i | def | category_theory.creates_limit_of_fully_faithful_of_iso | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_limit_of_creates_limit_and_has_limit (K : J ⥤ C) (F : C ⥤ D)
[creates_limit K F] [has_limit (K ⋙ F)] :
preserves_limit K F | { preserves := λ c t, is_limit.of_iso_limit (limit.is_limit _)
((lifted_limit_maps_to_original (limit.is_limit _)).symm ≪≫
((cones.functoriality K F).map_iso
((lifted_limit_is_limit (limit.is_limit _)).unique_up_to_iso t))) } | instance | category_theory.preserves_limit_of_creates_limit_and_has_limit | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | `F` preserves the limit of `K` if it creates the limit and `K ⋙ F` has the limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limit_of_shape_of_creates_limits_of_shape_and_has_limits_of_shape (F : C ⥤ D)
[creates_limits_of_shape J F] [has_limits_of_shape J D] :
preserves_limits_of_shape J F | {} | instance | category_theory.preserves_limit_of_shape_of_creates_limits_of_shape_and_has_limits_of_shape | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | `F` preserves the limit of shape `J` if it creates these limits and `D` has them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_limits_of_creates_limits_and_has_limits (F : C ⥤ D)
[creates_limits_of_size.{w w'} F]
[has_limits_of_size.{w w'} D] :
preserves_limits_of_size.{w w'} F | {} | instance | category_theory.preserves_limits_of_creates_limits_and_has_limits | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | `F` preserves limits if it creates limits and `D` has limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_colimit_of_reflects_iso {K : J ⥤ C} {F : C ⥤ D} [reflects_isomorphisms F]
(h : Π c t, lifts_to_colimit K F c t) :
creates_colimit K F | { lifts := λ c t, (h c t).to_liftable_cocone,
to_reflects_colimit :=
{ reflects := λ (d : cocone K) (hd : is_colimit (F.map_cocone d)),
begin
let d' : cocone K := (h (F.map_cocone d) hd).to_liftable_cocone.lifted_cocone,
let i : F.map_cocone d' ≅ F.map_cocone d :=
(h (F.map_cocone d) hd).to_... | def | category_theory.creates_colimit_of_reflects_iso | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `F` reflects isomorphisms and we can lift any colimit cocone to a colimit cocone,
then `F` creates colimits.
In particular here we don't need to assume that F reflects colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_colimit_of_fully_faithful_of_lift' {K : J ⥤ C} {F : C ⥤ D} [full F] [faithful F]
{l : cocone (K ⋙ F)} (hl : is_colimit l) (c : cocone K) (i : F.map_cocone c ≅ l) :
creates_colimit K F | creates_colimit_of_reflects_iso (λ c' t,
{ lifted_cocone := c,
valid_lift := i ≪≫ is_colimit.unique_up_to_iso hl t,
makes_colimit := is_colimit.of_faithful F (is_colimit.of_iso_colimit hl i.symm) _
(λ s, F.image_preimage _) }) | def | category_theory.creates_colimit_of_fully_faithful_of_lift' | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_colimit_of_fully_faithful_of_lift {K : J ⥤ C} {F : C ⥤ D}
[full F] [faithful F] [has_colimit (K ⋙ F)]
(c : cocone K) (i : F.map_cocone c ≅ colimit.cocone (K ⋙ F)) : creates_colimit K F | creates_colimit_of_fully_faithful_of_lift' (colimit.is_colimit _) c i | def | category_theory.creates_colimit_of_fully_faithful_of_lift | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_colimit_of_fully_faithful_of_iso' {K : J ⥤ C} {F : C ⥤ D} [full F] [faithful F]
{l : cocone (K ⋙ F)} (hl : is_colimit l) (X : C) (i : F.obj X ≅ l.X) : creates_colimit K F | creates_colimit_of_fully_faithful_of_lift' hl
({ X := X,
ι :=
{ app := λ j, F.preimage (l.ι.app j ≫ i.inv),
naturality' := λ Y Z f, F.map_injective $
by { dsimp, simpa [← cancel_mono i.hom] using (l.w f) } } })
(cocones.ext i (λ j, by simp)) | def | category_theory.creates_colimit_of_fully_faithful_of_iso' | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_colimit_of_fully_faithful_of_iso {K : J ⥤ C} {F : C ⥤ D}
[full F] [faithful F] [has_colimit (K ⋙ F)]
(X : C) (i : F.obj X ≅ colimit (K ⋙ F)) : creates_colimit K F | creates_colimit_of_fully_faithful_of_iso' (colimit.is_colimit _) X i | def | category_theory.creates_colimit_of_fully_faithful_of_iso | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preserves_colimit_of_creates_colimit_and_has_colimit (K : J ⥤ C) (F : C ⥤ D)
[creates_colimit K F] [has_colimit (K ⋙ F)] :
preserves_colimit K F | { preserves := λ c t, is_colimit.of_iso_colimit (colimit.is_colimit _)
((lifted_colimit_maps_to_original (colimit.is_colimit _)).symm ≪≫
((cocones.functoriality K F).map_iso
((lifted_colimit_is_colimit (colimit.is_colimit _)).unique_up_to_iso t))) } | instance | category_theory.preserves_colimit_of_creates_colimit_and_has_colimit | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | `F` preserves the colimit of `K` if it creates the colimit and `K ⋙ F` has the colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimit_of_shape_of_creates_colimits_of_shape_and_has_colimits_of_shape
(F : C ⥤ D) [creates_colimits_of_shape J F] [has_colimits_of_shape J D] :
preserves_colimits_of_shape J F | {} | instance | category_theory.preserves_colimit_of_shape_of_creates_colimits_of_shape_and_has_colimits_of_shape | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | `F` preserves the colimit of shape `J` if it creates these colimits and `D` has them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preserves_colimits_of_creates_colimits_and_has_colimits (F : C ⥤ D)
[creates_colimits_of_size.{w w'} F] [has_colimits_of_size.{w w'} D] :
preserves_colimits_of_size.{w w'} F | {} | instance | category_theory.preserves_colimits_of_creates_colimits_and_has_colimits | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | `F` preserves limits if it creates limits and `D` has limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_limit_of_iso_diagram {K₁ K₂ : J ⥤ C} (F : C ⥤ D) (h : K₁ ≅ K₂)
[creates_limit K₁ F] : creates_limit K₂ F | { lifts := λ c t,
let t' := (is_limit.postcompose_inv_equiv (iso_whisker_right h F : _) c).symm t in
{ lifted_cone := (cones.postcompose h.hom).obj (lift_limit t'),
valid_lift :=
F.map_cone_postcompose ≪≫
(cones.postcompose (iso_whisker_right h F).hom).map_iso
(lifted_limit_maps_to_o... | def | category_theory.creates_limit_of_iso_diagram | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | Transfer creation of limits along a natural isomorphism in the diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_limit_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [creates_limit K F] :
creates_limit K G | { lifts := λ c t,
{ lifted_cone :=
lift_limit ((is_limit.postcompose_inv_equiv (iso_whisker_left K h : _) c).symm t),
valid_lift :=
begin
refine (is_limit.map_cone_equiv h _).unique_up_to_iso t,
apply is_limit.of_iso_limit _ ((lifted_limit_maps_to_original _).symm),
apply (is_limit.pos... | def | category_theory.creates_limit_of_nat_iso | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `F` creates the limit of `K` and `F ≅ G`, then `G` creates the limit of `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_limits_of_shape_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [creates_limits_of_shape J F] :
creates_limits_of_shape J G | { creates_limit := λ K, creates_limit_of_nat_iso h } | def | category_theory.creates_limits_of_shape_of_nat_iso | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `F` creates limits of shape `J` and `F ≅ G`, then `G` creates limits of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_limits_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [creates_limits_of_size.{w w'} F] :
creates_limits_of_size.{w w'} G | { creates_limits_of_shape := λ J 𝒥₁, by exactI creates_limits_of_shape_of_nat_iso h } | def | category_theory.creates_limits_of_nat_iso | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `F` creates limits and `F ≅ G`, then `G` creates limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_colimit_of_iso_diagram {K₁ K₂ : J ⥤ C} (F : C ⥤ D) (h : K₁ ≅ K₂)
[creates_colimit K₁ F] : creates_colimit K₂ F | { lifts := λ c t,
let t' := (is_colimit.precompose_hom_equiv (iso_whisker_right h F : _) c).symm t in
{ lifted_cocone := (cocones.precompose h.inv).obj (lift_colimit t'),
valid_lift :=
F.map_cocone_precompose ≪≫
(cocones.precompose (iso_whisker_right h F).inv).map_iso
(lifted_colimit... | def | category_theory.creates_colimit_of_iso_diagram | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | Transfer creation of colimits along a natural isomorphism in the diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_colimit_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [creates_colimit K F] :
creates_colimit K G | { lifts := λ c t,
{ lifted_cocone :=
lift_colimit ((is_colimit.precompose_hom_equiv (iso_whisker_left K h : _) c).symm t),
valid_lift :=
begin
refine (is_colimit.map_cocone_equiv h _).unique_up_to_iso t,
apply is_colimit.of_iso_colimit _ ((lifted_colimit_maps_to_original _).symm),
appl... | def | category_theory.creates_colimit_of_nat_iso | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `F` creates the colimit of `K` and `F ≅ G`, then `G` creates the colimit of `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_colimits_of_shape_of_nat_iso {F G : C ⥤ D} (h : F ≅ G)
[creates_colimits_of_shape J F] : creates_colimits_of_shape J G | { creates_colimit := λ K, creates_colimit_of_nat_iso h } | def | category_theory.creates_colimits_of_shape_of_nat_iso | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `F` creates colimits of shape `J` and `F ≅ G`, then `G` creates colimits of shape `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_colimits_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [creates_colimits_of_size.{w w'} F] :
creates_colimits_of_size.{w w'} G | { creates_colimits_of_shape := λ J 𝒥₁, by exactI creates_colimits_of_shape_of_nat_iso h } | def | category_theory.creates_colimits_of_nat_iso | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If `F` creates colimits and `F ≅ G`, then `G` creates colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lifts_to_limit_of_creates (K : J ⥤ C) (F : C ⥤ D)
[creates_limit K F] (c : cone (K ⋙ F)) (t : is_limit c) :
lifts_to_limit K F c t | { lifted_cone := lift_limit t,
valid_lift := lifted_limit_maps_to_original t,
makes_limit := lifted_limit_is_limit t } | def | category_theory.lifts_to_limit_of_creates | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If F creates the limit of K, any cone lifts to a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lifts_to_colimit_of_creates (K : J ⥤ C) (F : C ⥤ D)
[creates_colimit K F] (c : cocone (K ⋙ F)) (t : is_colimit c) :
lifts_to_colimit K F c t | { lifted_cocone := lift_colimit t,
valid_lift := lifted_colimit_maps_to_original t,
makes_colimit := lifted_colimit_is_colimit t } | def | category_theory.lifts_to_colimit_of_creates | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | If F creates the colimit of K, any cocone lifts to a colimit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_lifts_cone (c : cone (K ⋙ 𝟭 C)) : liftable_cone K (𝟭 C) c | { lifted_cone :=
{ X := c.X,
π := c.π ≫ K.right_unitor.hom },
valid_lift := cones.ext (iso.refl _) (by tidy) } | def | category_theory.id_lifts_cone | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | Any cone lifts through the identity functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_creates_limits : creates_limits_of_size.{w w'} (𝟭 C) | { creates_limits_of_shape := λ J 𝒥, by exactI
{ creates_limit := λ F, { lifts := λ c t, id_lifts_cone c } } } | instance | category_theory.id_creates_limits | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | The identity functor creates all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_lifts_cocone (c : cocone (K ⋙ 𝟭 C)) : liftable_cocone K (𝟭 C) c | { lifted_cocone :=
{ X := c.X,
ι := K.right_unitor.inv ≫ c.ι },
valid_lift := cocones.ext (iso.refl _) (by tidy) } | def | category_theory.id_lifts_cocone | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | Any cocone lifts through the identity functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_creates_colimits : creates_colimits_of_size.{w w'} (𝟭 C) | { creates_colimits_of_shape := λ J 𝒥, by exactI
{ creates_colimit := λ F, { lifts := λ c t, id_lifts_cocone c } } } | instance | category_theory.id_creates_colimits | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | The identity functor creates all colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited_liftable_cone (c : cone (K ⋙ 𝟭 C)) :
inhabited (liftable_cone K (𝟭 C) c) | ⟨id_lifts_cone c⟩ | instance | category_theory.inhabited_liftable_cone | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | Satisfy the inhabited linter | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited_liftable_cocone (c : cocone (K ⋙ 𝟭 C)) :
inhabited (liftable_cocone K (𝟭 C) c) | ⟨id_lifts_cocone c⟩ | instance | category_theory.inhabited_liftable_cocone | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inhabited_lifts_to_limit (K : J ⥤ C) (F : C ⥤ D)
[creates_limit K F] (c : cone (K ⋙ F)) (t : is_limit c) :
inhabited (lifts_to_limit _ _ _ t) | ⟨lifts_to_limit_of_creates K F c t⟩ | instance | category_theory.inhabited_lifts_to_limit | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | Satisfy the inhabited linter | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited_lifts_to_colimit (K : J ⥤ C) (F : C ⥤ D)
[creates_colimit K F] (c : cocone (K ⋙ F)) (t : is_colimit c) :
inhabited (lifts_to_colimit _ _ _ t) | ⟨lifts_to_colimit_of_creates K F c t⟩ | instance | category_theory.inhabited_lifts_to_colimit | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_creates_limit [creates_limit K F] [creates_limit (K ⋙ F) G] :
creates_limit K (F ⋙ G) | { lifts := λ c t,
{ lifted_cone := lift_limit (lifted_limit_is_limit t),
valid_lift := (cones.functoriality (K ⋙ F) G).map_iso
(lifted_limit_maps_to_original (lifted_limit_is_limit t)) ≪≫
(lifted_limit_maps_to_original t) } } | instance | category_theory.comp_creates_limit | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_creates_limits_of_shape [creates_limits_of_shape J F] [creates_limits_of_shape J G] :
creates_limits_of_shape J (F ⋙ G) | { creates_limit := infer_instance } | instance | category_theory.comp_creates_limits_of_shape | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_creates_limits [creates_limits_of_size.{w w'} F] [creates_limits_of_size.{w w'} G] :
creates_limits_of_size.{w w'} (F ⋙ G) | { creates_limits_of_shape := infer_instance } | instance | category_theory.comp_creates_limits | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_creates_colimit [creates_colimit K F] [creates_colimit (K ⋙ F) G] :
creates_colimit K (F ⋙ G) | { lifts := λ c t,
{ lifted_cocone := lift_colimit (lifted_colimit_is_colimit t),
valid_lift := (cocones.functoriality (K ⋙ F) G).map_iso
(lifted_colimit_maps_to_original (lifted_colimit_is_colimit t)) ≪≫
(lifted_colimit_maps_to_original t) } } | instance | category_theory.comp_creates_colimit | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_creates_colimits_of_shape
[creates_colimits_of_shape J F] [creates_colimits_of_shape J G] :
creates_colimits_of_shape J (F ⋙ G) | { creates_colimit := infer_instance } | instance | category_theory.comp_creates_colimits_of_shape | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_creates_colimits [creates_colimits_of_size.{w w'} F]
[creates_colimits_of_size.{w w'} G] : creates_colimits_of_size.{w w'} (F ⋙ G) | { creates_colimits_of_shape := infer_instance } | instance | category_theory.comp_creates_colimits | category_theory.limits | src/category_theory/limits/creates.lean | [
"category_theory.limits.preserves.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits_of_shape_of_essentially_small [essentially_small.{w₁} J]
[has_limits_of_size.{w₁ w₁} C] : has_limits_of_shape J C | has_limits_of_shape_of_equivalence $ equivalence.symm $ equiv_small_model.{w₁} J | lemma | category_theory.limits.has_limits_of_shape_of_essentially_small | category_theory.limits | src/category_theory/limits/essentially_small.lean | [
"category_theory.limits.shapes.products",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_shape_of_essentially_small [essentially_small.{w₁} J]
[has_colimits_of_size.{w₁ w₁} C] : has_colimits_of_shape J C | has_colimits_of_shape_of_equivalence $ equivalence.symm $ equiv_small_model.{w₁} J | lemma | category_theory.limits.has_colimits_of_shape_of_essentially_small | category_theory.limits | src/category_theory/limits/essentially_small.lean | [
"category_theory.limits.shapes.products",
"category_theory.essentially_small"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_products_of_shape_of_small (β : Type w₂) [small.{w₁} β] [has_products.{w₁} C] :
has_products_of_shape β C | has_limits_of_shape_of_equivalence $ discrete.equivalence $ equiv.symm $ equiv_shrink β | lemma | category_theory.limits.has_products_of_shape_of_small | category_theory.limits | src/category_theory/limits/essentially_small.lean | [
"category_theory.limits.shapes.products",
"category_theory.essentially_small"
] | [
"equiv.symm",
"equiv_shrink"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coproducts_of_shape_of_small (β : Type w₂) [small.{w₁} β] [has_coproducts.{w₁} C] :
has_coproducts_of_shape β C | has_colimits_of_shape_of_equivalence $ discrete.equivalence $ equiv.symm $ equiv_shrink β | lemma | category_theory.limits.has_coproducts_of_shape_of_small | category_theory.limits | src/category_theory/limits/essentially_small.lean | [
"category_theory.limits.shapes.products",
"category_theory.essentially_small"
] | [
"equiv.symm",
"equiv_shrink"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
LeftExactFunctor | full_subcategory (λ F : C ⥤ D, nonempty (preserves_finite_limits F)) | def | category_theory.LeftExactFunctor | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | Bundled left-exact functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
LeftExactFunctor.forget : (C ⥤ₗ D) ⥤ (C ⥤ D) | full_subcategory_inclusion _ | def | category_theory.LeftExactFunctor.forget | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | A left exact functor is in particular a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
RightExactFunctor | full_subcategory (λ F : C ⥤ D, nonempty (preserves_finite_colimits F)) | def | category_theory.RightExactFunctor | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | Bundled right-exact functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
RightExactFunctor.forget : (C ⥤ᵣ D) ⥤ (C ⥤ D) | full_subcategory_inclusion _ | def | category_theory.RightExactFunctor.forget | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | A right exact functor is in particular a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ExactFunctor | full_subcategory
(λ F : C ⥤ D, nonempty (preserves_finite_limits F) ∧ nonempty (preserves_finite_colimits F)) | def | category_theory.ExactFunctor | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | Bundled exact functors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ExactFunctor.forget : (C ⥤ₑ D) ⥤ (C ⥤ D) | full_subcategory_inclusion _ | def | category_theory.ExactFunctor.forget | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | An exact functor is in particular a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
LeftExactFunctor.of_exact : (C ⥤ₑ D) ⥤ (C ⥤ₗ D) | full_subcategory.map (λ X, and.left) | def | category_theory.LeftExactFunctor.of_exact | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | Turn an exact functor into a left exact functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
RightExactFunctor.of_exact : (C ⥤ₑ D) ⥤ (C ⥤ᵣ D) | full_subcategory.map (λ X, and.right) | def | category_theory.RightExactFunctor.of_exact | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | Turn an exact functor into a left exact functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
LeftExactFunctor.of_exact_obj (F : C ⥤ₑ D) :
(LeftExactFunctor.of_exact C D).obj F = ⟨F.1, F.2.1⟩ | rfl | lemma | category_theory.LeftExactFunctor.of_exact_obj | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
RightExactFunctor.of_exact_obj (F : C ⥤ₑ D) :
(RightExactFunctor.of_exact C D).obj F = ⟨F.1, F.2.2⟩ | rfl | lemma | category_theory.RightExactFunctor.of_exact_obj | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
LeftExactFunctor.of_exact_map {F G : C ⥤ₑ D} (α : F ⟶ G) :
(LeftExactFunctor.of_exact C D).map α = α | rfl | lemma | category_theory.LeftExactFunctor.of_exact_map | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
RightExactFunctor.of_exact_map {F G : C ⥤ₑ D} (α : F ⟶ G) :
(RightExactFunctor.of_exact C D).map α = α | rfl | lemma | category_theory.RightExactFunctor.of_exact_map | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
LeftExactFunctor.forget_obj (F : C ⥤ₗ D) :
(LeftExactFunctor.forget C D).obj F = F.1 | rfl | lemma | category_theory.LeftExactFunctor.forget_obj | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
RightExactFunctor.forget_obj (F : C ⥤ᵣ D) :
(RightExactFunctor.forget C D).obj F = F.1 | rfl | lemma | category_theory.RightExactFunctor.forget_obj | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ExactFunctor.forget_obj (F : C ⥤ₑ D) :
(ExactFunctor.forget C D).obj F = F.1 | rfl | lemma | category_theory.ExactFunctor.forget_obj | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
LeftExactFunctor.forget_map {F G : C ⥤ₗ D} (α : F ⟶ G) :
(LeftExactFunctor.forget C D).map α = α | rfl | lemma | category_theory.LeftExactFunctor.forget_map | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
RightExactFunctor.forget_map {F G : C ⥤ᵣ D} (α : F ⟶ G) :
(RightExactFunctor.forget C D).map α = α | rfl | lemma | category_theory.RightExactFunctor.forget_map | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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