statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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ExactFunctor.forget_map {F G : C ⥤ₑ D} (α : F ⟶ G) :
(ExactFunctor.forget C D).map α = α | rfl | lemma | category_theory.ExactFunctor.forget_map | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
LeftExactFunctor.of (F : C ⥤ D) [preserves_finite_limits F] : C ⥤ₗ D | ⟨F, ⟨infer_instance⟩⟩ | def | category_theory.LeftExactFunctor.of | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | Turn a left exact functor into an object of the category `LeftExactFunctor C D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
RightExactFunctor.of (F : C ⥤ D) [preserves_finite_colimits F] : C ⥤ᵣ D | ⟨F, ⟨infer_instance⟩⟩ | def | category_theory.RightExactFunctor.of | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | Turn a right exact functor into an object of the category `RightExactFunctor C D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ExactFunctor.of (F : C ⥤ D) [preserves_finite_limits F] [preserves_finite_colimits F] :
C ⥤ₑ D | ⟨F, ⟨⟨infer_instance⟩, ⟨infer_instance⟩⟩⟩ | def | category_theory.ExactFunctor.of | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | Turn an exact functor into an object of the category `ExactFunctor C D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
LeftExactFunctor.of_fst (F : C ⥤ D) [preserves_finite_limits F] :
(LeftExactFunctor.of F).obj = F | rfl | lemma | category_theory.LeftExactFunctor.of_fst | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
RightExactFunctor.of_fst (F : C ⥤ D) [preserves_finite_colimits F] :
(RightExactFunctor.of F).obj = F | rfl | lemma | category_theory.RightExactFunctor.of_fst | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ExactFunctor.of_fst (F : C ⥤ D) [preserves_finite_limits F]
[preserves_finite_colimits F] : (ExactFunctor.of F).obj = F | rfl | lemma | category_theory.ExactFunctor.of_fst | 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_of (F : C ⥤ D) [preserves_finite_limits F] :
(LeftExactFunctor.forget C D).obj (LeftExactFunctor.of F) = F | rfl | lemma | category_theory.LeftExactFunctor.forget_obj_of | 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_of (F : C ⥤ D) [preserves_finite_colimits F] :
(RightExactFunctor.forget C D).obj (RightExactFunctor.of F) = F | rfl | lemma | category_theory.RightExactFunctor.forget_obj_of | 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_of (F : C ⥤ D) [preserves_finite_limits F]
[preserves_finite_colimits F] : (ExactFunctor.forget C D).obj (ExactFunctor.of F) = F | rfl | lemma | category_theory.ExactFunctor.forget_obj_of | category_theory.limits | src/category_theory/limits/exact_functor.lean | [
"category_theory.limits.preserves.finite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_cofiltered_limits_of_size : Prop | (has_limits_of_shape : Π (I : Type w) [category.{w'} I] [is_cofiltered I], has_limits_of_shape I C) | class | category_theory.limits.has_cofiltered_limits_of_size | category_theory.limits | src/category_theory/limits/filtered.lean | [
"category_theory.filtered",
"category_theory.limits.has_limits"
] | [] | Class for having all cofiltered limits of a given size. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_filtered_colimits_of_size : Prop | (has_colimits_of_shape : Π (I : Type w) [category.{w'} I] [is_filtered I],
has_colimits_of_shape I C) | class | category_theory.limits.has_filtered_colimits_of_size | category_theory.limits | src/category_theory/limits/filtered.lean | [
"category_theory.filtered",
"category_theory.limits.has_limits"
] | [] | Class for having all filtered colimits of a given size. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_shape_of_has_cofiltered_limits [has_cofiltered_limits_of_size.{w' w} C]
(I : Type w) [category.{w'} I] [is_cofiltered I] : has_limits_of_shape I C | has_cofiltered_limits_of_size.has_limits_of_shape _ | instance | category_theory.limits.has_limits_of_shape_of_has_cofiltered_limits | category_theory.limits | src/category_theory/limits/filtered.lean | [
"category_theory.filtered",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits_of_shape_of_has_filtered_colimits [has_filtered_colimits_of_size.{w' w} C]
(I : Type w) [category.{w'} I] [is_filtered I] : has_colimits_of_shape I C | has_filtered_colimits_of_size.has_colimits_of_shape _ | instance | category_theory.limits.has_colimits_of_shape_of_has_filtered_colimits | category_theory.limits | src/category_theory/limits/filtered.lean | [
"category_theory.filtered",
"category_theory.limits.has_limits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_limit_to_limit_colimit_injective :
function.injective (colimit_limit_to_limit_colimit F) | begin
classical,
casesI nonempty_fintype J,
-- Suppose we have two terms `x y` in the colimit (over `K`) of the limits (over `J`),
-- and that these have the same image under `colimit_limit_to_limit_colimit F`.
intros x y h,
-- These elements of the colimit have representatives somewhere:
obtain ⟨kx, x, r... | lemma | category_theory.limits.colimit_limit_to_limit_colimit_injective | category_theory.limits | src/category_theory/limits/filtered_colimit_commutes_finite_limit.lean | [
"category_theory.limits.colimit_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.basic"
] | [
"exists_prop_of_true",
"finset",
"finset.mem_image",
"finset.mem_univ",
"finset.univ",
"heq_iff_eq",
"nonempty_fintype"
] | This follows this proof from
* Borceux, Handbook of categorical algebra 1, Theorem 2.13.4 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_limit_to_limit_colimit_surjective :
function.surjective (colimit_limit_to_limit_colimit F) | begin
classical,
-- We begin with some element `x` in the limit (over J) over the colimits (over K),
intro x,
-- This consists of some coherent family of elements in the various colimits,
-- and so our first task is to pick representatives of these elements.
have z := λ j, jointly_surjective'.{v v} (limit.π... | lemma | category_theory.limits.colimit_limit_to_limit_colimit_surjective | category_theory.limits | src/category_theory/limits/filtered_colimit_commutes_finite_limit.lean | [
"category_theory.limits.colimit_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.basic"
] | [
"finset",
"finset.mem_bUnion",
"finset.mem_image",
"finset.mem_insert",
"finset.mem_singleton",
"finset.mem_univ",
"heq_iff_eq"
] | This follows this proof from
* Borceux, Handbook of categorical algebra 1, Theorem 2.13.4
although with different names. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_limit_to_limit_colimit_is_iso :
is_iso (colimit_limit_to_limit_colimit F) | (is_iso_iff_bijective _).mpr
⟨colimit_limit_to_limit_colimit_injective F, colimit_limit_to_limit_colimit_surjective F⟩ | instance | category_theory.limits.colimit_limit_to_limit_colimit_is_iso | category_theory.limits | src/category_theory/limits/filtered_colimit_commutes_finite_limit.lean | [
"category_theory.limits.colimit_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_limit_to_limit_colimit_cone_iso (F : J ⥤ K ⥤ Type v) :
is_iso (colimit_limit_to_limit_colimit_cone F) | begin
haveI : is_iso (colimit_limit_to_limit_colimit_cone F).hom,
{ dsimp only [colimit_limit_to_limit_colimit_cone], apply_instance },
apply cones.cone_iso_of_hom_iso,
end | instance | category_theory.limits.colimit_limit_to_limit_colimit_cone_iso | category_theory.limits | src/category_theory/limits/filtered_colimit_commutes_finite_limit.lean | [
"category_theory.limits.colimit_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filtered_colim_preserves_finite_limits_of_types :
preserves_finite_limits (colim : (K ⥤ Type v) ⥤ _) | begin
apply preserves_finite_limits_of_preserves_finite_limits_of_size.{v},
intros J _ _, resetI, constructor,
intro F, constructor,
intros c hc,
apply is_limit.of_iso_limit (limit.is_limit _),
symmetry, transitivity (colim.map_cone (limit.cone F)),
exact functor.map_iso _ (hc.unique_up_to_iso (limit.is_l... | instance | category_theory.limits.filtered_colim_preserves_finite_limits_of_types | category_theory.limits | src/category_theory/limits/filtered_colimit_commutes_finite_limit.lean | [
"category_theory.limits.colimit_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filtered_colim_preserves_finite_limits :
preserves_limits_of_shape J (colim : (K ⥤ C) ⥤ _) | begin
haveI : preserves_limits_of_shape J ((colim : (K ⥤ C) ⥤ _) ⋙ forget C) :=
preserves_limits_of_shape_of_nat_iso (preserves_colimit_nat_iso _).symm,
exactI preserves_limits_of_shape_of_reflects_of_preserves _ (forget C)
end | instance | category_theory.limits.filtered_colim_preserves_finite_limits | category_theory.limits | src/category_theory/limits/filtered_colimit_commutes_finite_limit.lean | [
"category_theory.limits.colimit_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_limit_iso (F : J ⥤ K ⥤ C) :
colimit (limit F) ≅ limit (colimit F.flip) | (is_limit_of_preserves colim (limit.is_limit _)).cone_point_unique_up_to_iso (limit.is_limit _) ≪≫
(has_limit.iso_of_nat_iso (colimit_flip_iso_comp_colim _).symm) | def | category_theory.limits.colimit_limit_iso | category_theory.limits | src/category_theory/limits/filtered_colimit_commutes_finite_limit.lean | [
"category_theory.limits.colimit_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.basic"
] | [] | A curried version of the fact that filtered colimits commute with finite limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_colimit_limit_iso_limit_π (F : J ⥤ K ⥤ C) (a) (b) :
colimit.ι (limit F) a ≫ (colimit_limit_iso F).hom ≫ limit.π (colimit F.flip) b =
(limit.π F b).app a ≫ (colimit.ι F.flip a).app b | begin
dsimp [colimit_limit_iso],
simp only [functor.map_cone_π_app, iso.symm_hom,
limits.limit.cone_point_unique_up_to_iso_hom_comp_assoc, limits.limit.cone_π,
limits.colimit.ι_map_assoc, limits.colimit_flip_iso_comp_colim_inv_app, assoc,
limits.has_limit.iso_of_nat_iso_hom_π],
congr' 1,
simp only [... | lemma | category_theory.limits.ι_colimit_limit_iso_limit_π | category_theory.limits | src/category_theory/limits/filtered_colimit_commutes_finite_limit.lean | [
"category_theory.limits.colimit_limit",
"category_theory.limits.preserves.functor_category",
"category_theory.limits.preserves.finite",
"category_theory.limits.shapes.finite_limits",
"category_theory.limits.preserves.filtered",
"category_theory.concrete_category.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
final (F : C ⥤ D) : Prop | (out (d : D) : is_connected (structured_arrow d F)) | class | category_theory.functor.final | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"is_connected"
] | A functor `F : C ⥤ D` is final if for every `d : D`, the comma category of morphisms `d ⟶ F.obj c`
is connected.
See <https://stacks.math.columbia.edu/tag/04E6> | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial (F : C ⥤ D) : Prop | (out (d : D) : is_connected (costructured_arrow F d)) | class | category_theory.functor.initial | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"is_connected"
] | A functor `F : C ⥤ D` is initial if for every `d : D`, the comma category of morphisms
`F.obj c ⟶ d` is connected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
final_op_of_initial (F : C ⥤ D) [initial F] : final F.op | { out := λ d, is_connected_of_equivalent (costructured_arrow_op_equivalence F (unop d)) } | instance | category_theory.functor.final_op_of_initial | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial_op_of_final (F : C ⥤ D) [final F] : initial F.op | { out := λ d, is_connected_of_equivalent (structured_arrow_op_equivalence F (unop d)) } | instance | category_theory.functor.initial_op_of_final | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
final_of_initial_op (F : C ⥤ D) [initial F.op] : final F | { out := λ d, @is_connected_of_is_connected_op _ _
(is_connected_of_equivalent (structured_arrow_op_equivalence F d).symm) } | lemma | category_theory.functor.final_of_initial_op | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial_of_final_op (F : C ⥤ D) [final F.op] : initial F | { out := λ d, @is_connected_of_is_connected_op _ _
(is_connected_of_equivalent (costructured_arrow_op_equivalence F d).symm) } | lemma | category_theory.functor.initial_of_final_op | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
final_of_adjunction {L : C ⥤ D} {R : D ⥤ C} (adj : L ⊣ R) : final R | { out := λ c,
let u : structured_arrow c R := structured_arrow.mk (adj.unit.app c) in
@zigzag_is_connected _ _ ⟨u⟩ $ λ f g, relation.refl_trans_gen.trans
(relation.refl_trans_gen.single (show zag f u, from
or.inr ⟨structured_arrow.hom_mk ((adj.hom_equiv c f.right).symm f.hom) (by simp)⟩))
(relation.re... | lemma | category_theory.functor.final_of_adjunction | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"adj",
"relation.refl_trans_gen.single",
"relation.refl_trans_gen.trans"
] | If a functor `R : D ⥤ C` is a right adjoint, it is final. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial_of_adjunction {L : C ⥤ D} {R : D ⥤ C} (adj : L ⊣ R) : initial L | { out := λ d,
let u : costructured_arrow L d := costructured_arrow.mk (adj.counit.app d) in
@zigzag_is_connected _ _ ⟨u⟩ $ λ f g, relation.refl_trans_gen.trans
(relation.refl_trans_gen.single (show zag f u, from
or.inl ⟨costructured_arrow.hom_mk (adj.hom_equiv f.left d f.hom) (by simp)⟩))
(relation.re... | lemma | category_theory.functor.initial_of_adjunction | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"adj",
"relation.refl_trans_gen.single",
"relation.refl_trans_gen.trans"
] | If a functor `L : C ⥤ D` is a left adjoint, it is initial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
final_of_is_right_adjoint (F : C ⥤ D) [h : is_right_adjoint F] : final F | final_of_adjunction h.adj | instance | category_theory.functor.final_of_is_right_adjoint | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial_of_is_left_adjoint (F : C ⥤ D) [h : is_left_adjoint F] : initial F | initial_of_adjunction h.adj | instance | category_theory.functor.initial_of_is_left_adjoint | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (d : D) : C | (classical.arbitrary (structured_arrow d F)).right | def | category_theory.functor.final.lift | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"classical.arbitrary",
"lift"
] | When `F : C ⥤ D` is cofinal, we denote by `lift F d` an arbitrary choice of object in `C` such that
there exists a morphism `d ⟶ F.obj (lift F d)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_to_lift (d : D) : d ⟶ F.obj (lift F d) | (classical.arbitrary (structured_arrow d F)).hom | def | category_theory.functor.final.hom_to_lift | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"classical.arbitrary",
"lift"
] | When `F : C ⥤ D` is cofinal, we denote by `hom_to_lift` an arbitrary choice of morphism
`d ⟶ F.obj (lift F d)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induction {d : D} (Z : Π (X : C) (k : d ⟶ F.obj X), Sort*)
(h₁ : Π X₁ X₂ (k₁ : d ⟶ F.obj X₁) (k₂ : d ⟶ F.obj X₂) (f : X₁ ⟶ X₂),
(k₁ ≫ F.map f = k₂) → Z X₁ k₁ → Z X₂ k₂)
(h₂ : Π X₁ X₂ (k₁ : d ⟶ F.obj X₁) (k₂ : d ⟶ F.obj X₂) (f : X₁ ⟶ X₂),
(k₁ ≫ F.map f = k₂) → Z X₂ k₂ → Z X₁ k₁)
{X₀ : C} {k₀ : d ⟶ F.obj X₀... | begin
apply nonempty.some,
apply @is_preconnected_induction _ _ _
(λ (Y : structured_arrow d F), Z Y.right Y.hom) _ _ (structured_arrow.mk k₀) z,
{ intros j₁ j₂ f a, fapply h₁ _ _ _ _ f.right _ a, convert f.w.symm, dsimp, simp, },
{ intros j₁ j₂ f a, fapply h₂ _ _ _ _ f.right _ a, convert f.w.symm, dsimp, s... | def | category_theory.functor.final.induction | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"lift",
"nonempty.some"
] | We provide an induction principle for reasoning about `lift` and `hom_to_lift`.
We want to perform some construction (usually just a proof) about
the particular choices `lift F d` and `hom_to_lift F d`,
it suffices to perform that construction for some other pair of choices
(denoted `X₀ : C` and `k₀ : d ⟶ F.obj X₀` bel... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_cocone : cocone (F ⋙ G) ⥤ cocone G | { obj := λ c,
{ X := c.X,
ι :=
{ app := λ X, G.map (hom_to_lift F X) ≫ c.ι.app (lift F X),
naturality' := λ X Y f,
begin
dsimp, simp,
-- This would be true if we'd chosen `lift F X` to be `lift F Y`
-- and `hom_to_lift F X` to be `f ≫ hom_to_lift F Y`.
apply inducti... | def | category_theory.functor.final.extend_cocone | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"lift"
] | Given a cocone over `F ⋙ G`, we can construct a `cocone G` with the same cocone point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_comp_aux (s : cocone (F ⋙ G)) (j : C) :
G.map (hom_to_lift F (F.obj j)) ≫ s.ι.app (lift F (F.obj j)) =
s.ι.app j | begin
-- This point is that this would be true if we took `lift (F.obj j)` to just be `j`
-- and `hom_to_lift (F.obj j)` to be `𝟙 (F.obj j)`.
apply induction F (λ X k, G.map k ≫ s.ι.app X = (s.ι.app j : _)),
{ intros j₁ j₂ k₁ k₂ f w h, rw ←w, rw ← s.w f at h, simpa using h, },
{ intros j₁ j₂ k₁ k₂ f w h, rw ... | lemma | category_theory.functor.final.colimit_cocone_comp_aux | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocones_equiv : cocone (F ⋙ G) ≌ cocone G | { functor := extend_cocone,
inverse := cocones.whiskering F,
unit_iso := nat_iso.of_components (λ c, cocones.ext (iso.refl _) (by tidy)) (by tidy),
counit_iso := nat_iso.of_components (λ c, cocones.ext (iso.refl _) (by tidy)) (by tidy), }. | def | category_theory.functor.final.cocones_equiv | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | If `F` is cofinal,
the category of cocones on `F ⋙ G` is equivalent to the category of cocones on `G`,
for any `G : D ⥤ E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_whisker_equiv (t : cocone G) : is_colimit (t.whisker F) ≃ is_colimit t | is_colimit.of_cocone_equiv (cocones_equiv F G).symm | def | category_theory.functor.final.is_colimit_whisker_equiv | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | When `F : C ⥤ D` is cofinal, and `t : cocone G` for some `G : D ⥤ E`,
`t.whisker F` is a colimit cocone exactly when `t` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_colimit_extend_cocone_equiv (t : cocone (F ⋙ G)) :
is_colimit (extend_cocone.obj t) ≃ is_colimit t | is_colimit.of_cocone_equiv (cocones_equiv F G) | def | category_theory.functor.final.is_colimit_extend_cocone_equiv | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | When `F` is cofinal, and `t : cocone (F ⋙ G)`,
`extend_cocone.obj t` is a colimit coconne exactly when `t` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_comp (t : colimit_cocone G) :
colimit_cocone (F ⋙ G) | { cocone := _,
is_colimit := (is_colimit_whisker_equiv F _).symm (t.is_colimit) } | def | category_theory.functor.final.colimit_cocone_comp | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | Given a colimit cocone over `G : D ⥤ E` we can construct a colimit cocone over `F ⋙ G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_has_colimit [has_colimit G] :
has_colimit (F ⋙ G) | has_colimit.mk (colimit_cocone_comp F (get_colimit_cocone G)) | instance | category_theory.functor.final.comp_has_colimit | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_pre_is_iso_aux {t : cocone G} (P : is_colimit t) :
((is_colimit_whisker_equiv F _).symm P).desc (t.whisker F) = 𝟙 t.X | begin
dsimp [is_colimit_whisker_equiv],
apply P.hom_ext,
intro j,
dsimp, simp,
end | lemma | category_theory.functor.final.colimit_pre_is_iso_aux | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_pre_is_iso [has_colimit G] :
is_iso (colimit.pre G F) | begin
rw colimit.pre_eq (colimit_cocone_comp F (get_colimit_cocone G)) (get_colimit_cocone G),
erw colimit_pre_is_iso_aux,
dsimp,
apply_instance,
end | instance | category_theory.functor.final.colimit_pre_is_iso | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_iso [has_colimit G] : colimit (F ⋙ G) ≅ colimit G | as_iso (colimit.pre G F) | def | category_theory.functor.final.colimit_iso | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | When `F : C ⥤ D` is cofinal, and `G : D ⥤ E` has a colimit, then `F ⋙ G` has a colimit also and
`colimit (F ⋙ G) ≅ colimit G`
https://stacks.math.columbia.edu/tag/04E7 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_of_comp (t : colimit_cocone (F ⋙ G)) :
colimit_cocone G | { cocone := extend_cocone.obj t.cocone,
is_colimit := (is_colimit_extend_cocone_equiv F _).symm (t.is_colimit), } | def | category_theory.functor.final.colimit_cocone_of_comp | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | Given a colimit cocone over `F ⋙ G` we can construct a colimit cocone over `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_colimit_of_comp [has_colimit (F ⋙ G)] :
has_colimit G | has_colimit.mk (colimit_cocone_of_comp F (get_colimit_cocone (F ⋙ G))) | lemma | category_theory.functor.final.has_colimit_of_comp | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | When `F` is cofinal, and `F ⋙ G` has a colimit, then `G` has a colimit also.
We can't make this an instance, because `F` is not determined by the goal.
(Even if this weren't a problem, it would cause a loop with `comp_has_colimit`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_iso' [has_colimit (F ⋙ G)] : colimit (F ⋙ G) ≅ colimit G | as_iso (colimit.pre G F) | def | category_theory.functor.final.colimit_iso' | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | When `F` is cofinal, and `F ⋙ G` has a colimit, then `G` has a colimit also and
`colimit (F ⋙ G) ≅ colimit G`
https://stacks.math.columbia.edu/tag/04E7 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_comp_coyoneda_iso (d : D) [is_iso (colimit.pre (coyoneda.obj (op d)) F)] :
colimit (F ⋙ coyoneda.obj (op d)) ≅ punit | as_iso (colimit.pre (coyoneda.obj (op d)) F) ≪≫ coyoneda.colimit_coyoneda_iso (op d) | def | category_theory.functor.final.colimit_comp_coyoneda_iso | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | If the universal morphism `colimit (F ⋙ coyoneda.obj (op d)) ⟶ colimit (coyoneda.obj (op d))`
is an isomorphism (as it always is when `F` is cofinal),
then `colimit (F ⋙ coyoneda.obj (op d)) ≅ punit`
(simply because `colimit (coyoneda.obj (op d)) ≅ punit`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zigzag_of_eqv_gen_quot_rel {F : C ⥤ D} {d : D} {f₁ f₂ : Σ X, d ⟶ F.obj X}
(t : eqv_gen (types.quot.rel.{v v} (F ⋙ coyoneda.obj (op d))) f₁ f₂) :
zigzag (structured_arrow.mk f₁.2) (structured_arrow.mk f₂.2) | begin
induction t,
case eqv_gen.rel : x y r
{ obtain ⟨f, w⟩ := r,
fconstructor,
swap 2, fconstructor,
left, fsplit,
exact structured_arrow.hom_mk f (by tidy), },
case eqv_gen.refl
{ fconstructor, },
case eqv_gen.symm : x y h ih
{ apply zigzag_symmetric,
exact ih, },
case eqv_gen.tran... | lemma | category_theory.functor.final.zigzag_of_eqv_gen_quot_rel | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"ih",
"relation.refl_trans_gen.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofinal_of_colimit_comp_coyoneda_iso_punit
(I : Π d, colimit (F ⋙ coyoneda.obj (op d)) ≅ punit) : final F | ⟨λ d, begin
haveI : nonempty (structured_arrow d F),
{ have := (I d).inv punit.star,
obtain ⟨j, y, rfl⟩ := limits.types.jointly_surjective'.{v v} this,
exact ⟨structured_arrow.mk y⟩, },
apply zigzag_is_connected,
rintros ⟨⟨⟨⟩⟩,X₁,f₁⟩ ⟨⟨⟨⟩⟩,X₂,f₂⟩,
dsimp at *,
let y₁ := colimit.ι (F ⋙ coyoneda.obj (o... | lemma | category_theory.functor.final.cofinal_of_colimit_comp_coyoneda_iso_punit | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | If `colimit (F ⋙ coyoneda.obj (op d)) ≅ punit` for all `d : D`, then `F` is cofinal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift (d : D) : C | (classical.arbitrary (costructured_arrow F d)).left | def | category_theory.functor.initial.lift | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"classical.arbitrary",
"lift"
] | When `F : C ⥤ D` is initial, we denote by `lift F d` an arbitrary choice of object in `C` such that
there exists a morphism `F.obj (lift F d) ⟶ d`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_to_lift (d : D) : F.obj (lift F d) ⟶ d | (classical.arbitrary (costructured_arrow F d)).hom | def | category_theory.functor.initial.hom_to_lift | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"classical.arbitrary",
"lift"
] | When `F : C ⥤ D` is initial, we denote by `hom_to_lift` an arbitrary choice of morphism
`F.obj (lift F d) ⟶ d`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induction {d : D} (Z : Π (X : C) (k : F.obj X ⟶ d), Sort*)
(h₁ : Π X₁ X₂ (k₁ : F.obj X₁ ⟶ d) (k₂ : F.obj X₂ ⟶ d) (f : X₁ ⟶ X₂),
(F.map f ≫ k₂ = k₁) → Z X₁ k₁ → Z X₂ k₂)
(h₂ : Π X₁ X₂ (k₁ : F.obj X₁ ⟶ d) (k₂ : F.obj X₂ ⟶ d) (f : X₁ ⟶ X₂),
(F.map f ≫ k₂ = k₁) → Z X₂ k₂ → Z X₁ k₁)
{X₀ : C} {k₀ : F.obj X₀ ⟶ d... | begin
apply nonempty.some,
apply @is_preconnected_induction _ _ _
(λ Y : costructured_arrow F d, Z Y.left Y.hom) _ _ (costructured_arrow.mk k₀) z,
{ intros j₁ j₂ f a, fapply h₁ _ _ _ _ f.left _ a, convert f.w, dsimp, simp, },
{ intros j₁ j₂ f a, fapply h₂ _ _ _ _ f.left _ a, convert f.w, dsimp, simp, },
end | def | category_theory.functor.initial.induction | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"lift",
"nonempty.some"
] | We provide an induction principle for reasoning about `lift` and `hom_to_lift`.
We want to perform some construction (usually just a proof) about
the particular choices `lift F d` and `hom_to_lift F d`,
it suffices to perform that construction for some other pair of choices
(denoted `X₀ : C` and `k₀ : F.obj X₀ ⟶ d` bel... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_cone : cone (F ⋙ G) ⥤ cone G | { obj := λ c,
{ X := c.X,
π :=
{ app := λ d, c.π.app (lift F d) ≫ G.map (hom_to_lift F d),
naturality' := λ X Y f,
begin
dsimp, simp,
-- This would be true if we'd chosen `lift F Y` to be `lift F X`
-- and `hom_to_lift F Y` to be `hom_to_lift F X ≫ f`.
apply inducti... | def | category_theory.functor.initial.extend_cone | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"lift"
] | Given a cone over `F ⋙ G`, we can construct a `cone G` with the same cocone point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_comp_aux (s : cone (F ⋙ G)) (j : C) :
s.π.app (lift F (F.obj j)) ≫ G.map (hom_to_lift F (F.obj j)) =
s.π.app j | begin
-- This point is that this would be true if we took `lift (F.obj j)` to just be `j`
-- and `hom_to_lift (F.obj j)` to be `𝟙 (F.obj j)`.
apply induction F (λ X k, s.π.app X ≫ G.map k = (s.π.app j : _)),
{ intros j₁ j₂ k₁ k₂ f w h, rw ←s.w f, rw ←w at h, simpa using h, },
{ intros j₁ j₂ k₁ k₂ f w h, rw ←... | lemma | category_theory.functor.initial.limit_cone_comp_aux | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cones_equiv : cone (F ⋙ G) ≌ cone G | { functor := extend_cone,
inverse := cones.whiskering F,
unit_iso := nat_iso.of_components (λ c, cones.ext (iso.refl _) (by tidy)) (by tidy),
counit_iso := nat_iso.of_components (λ c, cones.ext (iso.refl _) (by tidy)) (by tidy), }. | def | category_theory.functor.initial.cones_equiv | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | If `F` is initial,
the category of cones on `F ⋙ G` is equivalent to the category of cones on `G`,
for any `G : D ⥤ E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_whisker_equiv (t : cone G) : is_limit (t.whisker F) ≃ is_limit t | is_limit.of_cone_equiv (cones_equiv F G).symm | def | category_theory.functor.initial.is_limit_whisker_equiv | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | When `F : C ⥤ D` is initial, and `t : cone G` for some `G : D ⥤ E`,
`t.whisker F` is a limit cone exactly when `t` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_limit_extend_cone_equiv (t : cone (F ⋙ G)) :
is_limit (extend_cone.obj t) ≃ is_limit t | is_limit.of_cone_equiv (cones_equiv F G) | def | category_theory.functor.initial.is_limit_extend_cone_equiv | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | When `F` is initial, and `t : cone (F ⋙ G)`,
`extend_cone.obj t` is a limit cone exactly when `t` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_comp (t : limit_cone G) :
limit_cone (F ⋙ G) | { cone := _,
is_limit := (is_limit_whisker_equiv F _).symm (t.is_limit) } | def | category_theory.functor.initial.limit_cone_comp | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | Given a limit cone over `G : D ⥤ E` we can construct a limit cone over `F ⋙ G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_has_limit [has_limit G] :
has_limit (F ⋙ G) | has_limit.mk (limit_cone_comp F (get_limit_cone G)) | instance | category_theory.functor.initial.comp_has_limit | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_pre_is_iso_aux {t : cone G} (P : is_limit t) :
((is_limit_whisker_equiv F _).symm P).lift (t.whisker F) = 𝟙 t.X | begin
dsimp [is_limit_whisker_equiv],
apply P.hom_ext,
intro j,
simp,
end | lemma | category_theory.functor.initial.limit_pre_is_iso_aux | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_pre_is_iso [has_limit G] :
is_iso (limit.pre G F) | begin
rw limit.pre_eq (limit_cone_comp F (get_limit_cone G)) (get_limit_cone G),
erw limit_pre_is_iso_aux,
dsimp,
apply_instance,
end | instance | category_theory.functor.initial.limit_pre_is_iso | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_iso [has_limit G] : limit (F ⋙ G) ≅ limit G | (as_iso (limit.pre G F)).symm | def | category_theory.functor.initial.limit_iso | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | When `F : C ⥤ D` is initial, and `G : D ⥤ E` has a limit, then `F ⋙ G` has a limit also and
`limit (F ⋙ G) ≅ limit G`
https://stacks.math.columbia.edu/tag/04E7 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_of_comp (t : limit_cone (F ⋙ G)) :
limit_cone G | { cone := extend_cone.obj t.cone,
is_limit := (is_limit_extend_cone_equiv F _).symm (t.is_limit), } | def | category_theory.functor.initial.limit_cone_of_comp | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | Given a limit cone over `F ⋙ G` we can construct a limit cone over `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit_of_comp [has_limit (F ⋙ G)] :
has_limit G | has_limit.mk (limit_cone_of_comp F (get_limit_cone (F ⋙ G))) | lemma | category_theory.functor.initial.has_limit_of_comp | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | When `F` is initial, and `F ⋙ G` has a limit, then `G` has a limit also.
We can't make this an instance, because `F` is not determined by the goal.
(Even if this weren't a problem, it would cause a loop with `comp_has_limit`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_iso' [has_limit (F ⋙ G)] : limit (F ⋙ G) ≅ limit G | (as_iso (limit.pre G F)).symm | def | category_theory.functor.initial.limit_iso' | category_theory.limits | src/category_theory/limits/final.lean | [
"category_theory.punit",
"category_theory.structured_arrow",
"category_theory.is_connected",
"category_theory.limits.yoneda",
"category_theory.limits.types"
] | [] | When `F` is initial, and `F ⋙ G` has a limit, then `G` has a limit also and
`limit (F ⋙ G) ≅ limit G`
https://stacks.math.columbia.edu/tag/04E7 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram_of_cones | (obj : Π j : J, cone (F.obj j))
(map : Π {j j' : J} (f : j ⟶ j'), (cones.postcompose (F.map f)).obj (obj j) ⟶ obj j')
(id : ∀ j : J, (map (𝟙 j)).hom = 𝟙 _ . obviously)
(comp : ∀ {j₁ j₂ j₃ : J} (f : j₁ ⟶ j₂) (g : j₂ ⟶ j₃),
(map (f ≫ g)).hom = (map f).hom ≫ (map g).hom . obviously) | structure | category_theory.limits.diagram_of_cones | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_of_cones.cone_points (D : diagram_of_cones F) :
J ⥤ C | { obj := λ j, (D.obj j).X,
map := λ j j' f, (D.map f).hom,
map_id' := λ j, D.id j,
map_comp' := λ j₁ j₂ j₃ f g, D.comp f g, } | def | category_theory.limits.diagram_of_cones.cone_points | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [] | Extract the functor `J ⥤ C` consisting of the cone points and the maps between them,
from a `diagram_of_cones`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_of_cone_uncurry
{D : diagram_of_cones F} (Q : Π j, is_limit (D.obj j))
(c : cone (uncurry.obj F)) :
cone (D.cone_points) | { X := c.X,
π :=
{ app := λ j, (Q j).lift
{ X := c.X,
π :=
{ app := λ k, c.π.app (j, k),
naturality' := λ k k' f,
begin
dsimp, simp only [category.id_comp],
have := @nat_trans.naturality _ _ _ _ _ _ c.π (j, k) (j, k') (𝟙 j, f),
dsimp at this,
... | def | category_theory.limits.cone_of_cone_uncurry | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [
"hom_ext",
"lift"
] | Given a diagram `D` of limit cones over the `F.obj j`, and a cone over `uncurry.obj F`,
we can construct a cone over the diagram consisting of the cone points from `D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cone_of_cone_uncurry_is_limit
{D : diagram_of_cones F} (Q : Π j, is_limit (D.obj j))
{c : cone (uncurry.obj F)} (P : is_limit c) :
is_limit (cone_of_cone_uncurry Q c) | { lift := λ s, P.lift
{ X := s.X,
π :=
{ app := λ p, s.π.app p.1 ≫ (D.obj p.1).π.app p.2,
naturality' := λ p p' f,
begin
dsimp, simp only [category.id_comp, category.assoc],
rcases p with ⟨j, k⟩,
rcases p' with ⟨j', k'⟩,
rcases f with ⟨fj, fk⟩,
dsimp,
... | def | category_theory.limits.cone_of_cone_uncurry_is_limit | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [
"hom_ext",
"lift"
] | `cone_of_cone_uncurry Q c` is a limit cone when `c` is a limit cone.` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram_of_cones.mk_of_has_limits : diagram_of_cones F | { obj := λ j, limit.cone (F.obj j),
map := λ j j' f, { hom := lim.map (F.map f), }, } | def | category_theory.limits.diagram_of_cones.mk_of_has_limits | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [] | Given a functor `F : J ⥤ K ⥤ C`, with all needed limits,
we can construct a diagram consisting of the limit cone over each functor `F.obj j`,
and the universal cone morphisms between these. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram_of_cones_inhabited : inhabited (diagram_of_cones F) | ⟨diagram_of_cones.mk_of_has_limits F⟩ | instance | category_theory.limits.diagram_of_cones_inhabited | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_of_cones.mk_of_has_limits_cone_points :
(diagram_of_cones.mk_of_has_limits F).cone_points = (F ⋙ lim) | rfl | lemma | category_theory.limits.diagram_of_cones.mk_of_has_limits_cone_points | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [
"lim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_uncurry_iso_limit_comp_lim : limit (uncurry.obj F) ≅ limit (F ⋙ lim) | begin
let c := limit.cone (uncurry.obj F),
let P : is_limit c := limit.is_limit _,
let G := diagram_of_cones.mk_of_has_limits F,
let Q : Π j, is_limit (G.obj j) := λ j, limit.is_limit _,
have Q' := cone_of_cone_uncurry_is_limit Q P,
have Q'' := (limit.is_limit (F ⋙ lim)),
exact is_limit.cone_point_unique_... | def | category_theory.limits.limit_uncurry_iso_limit_comp_lim | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [
"lim"
] | The Fubini theorem for a functor `F : J ⥤ K ⥤ C`,
showing that the limit of `uncurry.obj F` can be computed as
the limit of the limits of the functors `F.obj j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_uncurry_iso_limit_comp_lim_hom_π_π {j} {k} :
(limit_uncurry_iso_limit_comp_lim F).hom ≫ limit.π _ j ≫ limit.π _ k = limit.π _ (j, k) | begin
dsimp [limit_uncurry_iso_limit_comp_lim, is_limit.cone_point_unique_up_to_iso,
is_limit.unique_up_to_iso],
simp,
end | lemma | category_theory.limits.limit_uncurry_iso_limit_comp_lim_hom_π_π | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_uncurry_iso_limit_comp_lim_inv_π {j} {k} :
(limit_uncurry_iso_limit_comp_lim F).inv ≫ limit.π _ (j, k) = limit.π _ j ≫ limit.π _ k | begin
rw [←cancel_epi (limit_uncurry_iso_limit_comp_lim F).hom],
simp,
end | lemma | category_theory.limits.limit_uncurry_iso_limit_comp_lim_inv_π | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_flip_comp_lim_iso_limit_comp_lim : limit (F.flip ⋙ lim) ≅ limit (F ⋙ lim) | (limit_uncurry_iso_limit_comp_lim _).symm ≪≫
has_limit.iso_of_nat_iso (uncurry_obj_flip _) ≪≫
(has_limit.iso_of_equivalence (prod.braiding _ _)
(nat_iso.of_components (λ _, by refl) (by tidy))) ≪≫
limit_uncurry_iso_limit_comp_lim _ | def | category_theory.limits.limit_flip_comp_lim_iso_limit_comp_lim | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [
"lim"
] | The limit of `F.flip ⋙ lim` is isomorphic to the limit of `F ⋙ lim`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_flip_comp_lim_iso_limit_comp_lim_hom_π_π (j) (k) :
(limit_flip_comp_lim_iso_limit_comp_lim F).hom ≫ limit.π _ j ≫ limit.π _ k =
limit.π _ k ≫ limit.π _ j | by { dsimp [limit_flip_comp_lim_iso_limit_comp_lim], simp, dsimp, simp, } | lemma | category_theory.limits.limit_flip_comp_lim_iso_limit_comp_lim_hom_π_π | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_flip_comp_lim_iso_limit_comp_lim_inv_π_π (k) (j) :
(limit_flip_comp_lim_iso_limit_comp_lim F).inv ≫ limit.π _ k ≫ limit.π _ j =
limit.π _ j ≫ limit.π _ k | by { dsimp [limit_flip_comp_lim_iso_limit_comp_lim], simp, dsimp, simp, dsimp, simp, } | lemma | category_theory.limits.limit_flip_comp_lim_iso_limit_comp_lim_inv_π_π | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_iso_limit_curry_comp_lim : limit G ≅ limit ((curry.obj G) ⋙ lim) | begin
have i : G ≅ uncurry.obj ((@curry J _ K _ C _).obj G) := currying.symm.unit_iso.app G,
haveI : limits.has_limit (uncurry.obj ((@curry J _ K _ C _).obj G)) :=
has_limit_of_iso i,
transitivity limit (uncurry.obj ((@curry J _ K _ C _).obj G)),
apply has_limit.iso_of_nat_iso i,
exact limit_uncurry_iso_l... | def | category_theory.limits.limit_iso_limit_curry_comp_lim | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [
"lim"
] | The Fubini theorem for a functor `G : J × K ⥤ C`,
showing that the limit of `G` can be computed as
the limit of the limits of the functors `G.obj (j, _)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_iso_limit_curry_comp_lim_hom_π_π {j} {k} :
(limit_iso_limit_curry_comp_lim G).hom ≫ limit.π _ j ≫ limit.π _ k = limit.π _ (j, k) | by simp [limit_iso_limit_curry_comp_lim, is_limit.cone_point_unique_up_to_iso,
is_limit.unique_up_to_iso] | lemma | category_theory.limits.limit_iso_limit_curry_comp_lim_hom_π_π | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_iso_limit_curry_comp_lim_inv_π {j} {k} :
(limit_iso_limit_curry_comp_lim G).inv ≫ limit.π _ (j, k) = limit.π _ j ≫ limit.π _ k | begin
rw [←cancel_epi (limit_iso_limit_curry_comp_lim G).hom],
simp,
end | lemma | category_theory.limits.limit_iso_limit_curry_comp_lim_inv_π | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_curry_swap_comp_lim_iso_limit_curry_comp_lim :
limit ((curry.obj (swap K J ⋙ G)) ⋙ lim) ≅ limit ((curry.obj G) ⋙ lim) | calc
limit ((curry.obj (swap K J ⋙ G)) ⋙ lim)
≅ limit (swap K J ⋙ G) : (limit_iso_limit_curry_comp_lim _).symm
... ≅ limit G : has_limit.iso_of_equivalence (braiding K J) (iso.refl _)
... ≅ limit ((curry.obj G) ⋙ lim) : limit_iso_limit_curry_comp_lim _ | def | category_theory.limits.limit_curry_swap_comp_lim_iso_limit_curry_comp_lim | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [
"lim"
] | A variant of the Fubini theorem for a functor `G : J × K ⥤ C`,
showing that $\lim_k \lim_j G(j,k) ≅ \lim_j \lim_k G(j,k)$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_curry_swap_comp_lim_iso_limit_curry_comp_lim_hom_π_π {j} {k} :
(limit_curry_swap_comp_lim_iso_limit_curry_comp_lim G).hom ≫ limit.π _ j ≫ limit.π _ k =
limit.π _ k ≫ limit.π _ j | begin
dsimp [limit_curry_swap_comp_lim_iso_limit_curry_comp_lim],
simp only [iso.refl_hom, braiding_counit_iso_hom_app, limits.has_limit.iso_of_equivalence_hom_π,
iso.refl_inv, limit_iso_limit_curry_comp_lim_hom_π_π, eq_to_iso_refl, category.assoc],
erw [nat_trans.id_app], -- Why can't `simp` do this`?
dsim... | lemma | category_theory.limits.limit_curry_swap_comp_lim_iso_limit_curry_comp_lim_hom_π_π | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_curry_swap_comp_lim_iso_limit_curry_comp_lim_inv_π_π {j} {k} :
(limit_curry_swap_comp_lim_iso_limit_curry_comp_lim G).inv ≫ limit.π _ k ≫ limit.π _ j =
limit.π _ j ≫ limit.π _ k | begin
dsimp [limit_curry_swap_comp_lim_iso_limit_curry_comp_lim],
simp only [iso.refl_hom, braiding_counit_iso_hom_app, limits.has_limit.iso_of_equivalence_inv_π,
iso.refl_inv, limit_iso_limit_curry_comp_lim_hom_π_π, eq_to_iso_refl, category.assoc],
erw [nat_trans.id_app], -- Why can't `simp` do this`?
dsim... | lemma | category_theory.limits.limit_curry_swap_comp_lim_iso_limit_curry_comp_lim_inv_π_π | category_theory.limits | src/category_theory/limits/fubini.lean | [
"category_theory.limits.has_limits",
"category_theory.products.basic",
"category_theory.functor.currying"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_under_limits_of_shape {C : Type u} [category.{v} C] (J : Type w) [category.{w'} J]
(P : C → Prop) : Prop | ∀ ⦃F : J ⥤ C⦄ ⦃c : cone F⦄ (hc : is_limit c), (∀ j, P (F.obj j)) → P c.X | def | category_theory.limits.closed_under_limits_of_shape | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | We say that a property is closed under limits of shape `J` if whenever all objects in a
`J`-shaped diagram have the property, any limit of this diagram also has the property. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_under_colimits_of_shape {C : Type u} [category.{v} C] (J : Type w) [category.{w'} J]
(P : C → Prop) : Prop | ∀ ⦃F : J ⥤ C⦄ ⦃c : cocone F⦄ (hc : is_colimit c), (∀ j, P (F.obj j)) → P c.X | def | category_theory.limits.closed_under_colimits_of_shape | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | We say that a property is closed under colimits of shape `J` if whenever all objects in a
`J`-shaped diagram have the property, any colimit of this diagram also has the property. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_under_limits_of_shape.limit (h : closed_under_limits_of_shape J P) {F : J ⥤ C}
[has_limit F] : (∀ j, P (F.obj j)) → P (limit F) | h (limit.is_limit _) | lemma | category_theory.limits.closed_under_limits_of_shape.limit | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_under_colimits_of_shape.colimit (h : closed_under_colimits_of_shape J P) {F : J ⥤ C}
[has_colimit F] : (∀ j, P (F.obj j)) → P (colimit F) | h (colimit.is_colimit _) | lemma | category_theory.limits.closed_under_colimits_of_shape.colimit | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_limit_full_subcategory_inclusion' (F : J ⥤ full_subcategory P)
{c : cone (F ⋙ full_subcategory_inclusion P)} (hc : is_limit c) (h : P c.X) :
creates_limit F (full_subcategory_inclusion P) | creates_limit_of_fully_faithful_of_iso' hc ⟨_, h⟩ (iso.refl _) | def | category_theory.limits.creates_limit_full_subcategory_inclusion' | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | If a `J`-shaped diagram in `full_subcategory P` has a limit cone in `C` whose cone point lives
in the full subcategory, then this defines a limit in the full subcategory. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_limit_full_subcategory_inclusion (F : J ⥤ full_subcategory P)
[has_limit (F ⋙ full_subcategory_inclusion P)]
(h : P (limit (F ⋙ full_subcategory_inclusion P))) :
creates_limit F (full_subcategory_inclusion P) | creates_limit_full_subcategory_inclusion' F (limit.is_limit _) h | def | category_theory.limits.creates_limit_full_subcategory_inclusion | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | If a `J`-shaped diagram in `full_subcategory P` has a limit in `C` whose cone point lives in the
full subcategory, then this defines a limit in the full subcategory. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_colimit_full_subcategory_inclusion' (F : J ⥤ full_subcategory P)
{c : cocone (F ⋙ full_subcategory_inclusion P)} (hc : is_colimit c) (h : P c.X) :
creates_colimit F (full_subcategory_inclusion P) | creates_colimit_of_fully_faithful_of_iso' hc ⟨_, h⟩ (iso.refl _) | def | category_theory.limits.creates_colimit_full_subcategory_inclusion' | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | If a `J`-shaped diagram in `full_subcategory P` has a colimit cocone in `C` whose cocone point
lives in the full subcategory, then this defines a colimit in the full subcategory. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_colimit_full_subcategory_inclusion (F : J ⥤ full_subcategory P)
[has_colimit (F ⋙ full_subcategory_inclusion P)]
(h : P (colimit (F ⋙ full_subcategory_inclusion P))) :
creates_colimit F (full_subcategory_inclusion P) | creates_colimit_full_subcategory_inclusion' F (colimit.is_colimit _) h | def | category_theory.limits.creates_colimit_full_subcategory_inclusion | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | If a `J`-shaped diagram in `full_subcategory P` has a colimit in `C` whose cocone point lives in
the full subcategory, then this defines a colimit in the full subcategory. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_limit_full_subcategory_inclusion_of_closed (h : closed_under_limits_of_shape J P)
(F : J ⥤ full_subcategory P) [has_limit (F ⋙ full_subcategory_inclusion P)] :
creates_limit F (full_subcategory_inclusion P) | creates_limit_full_subcategory_inclusion F (h.limit (λ j, (F.obj j).property)) | def | category_theory.limits.creates_limit_full_subcategory_inclusion_of_closed | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | If `P` is closed under limits of shape `J`, then the inclusion creates such limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_limits_of_shape_full_subcategory_inclusion (h : closed_under_limits_of_shape J P)
[has_limits_of_shape J C] : creates_limits_of_shape J (full_subcategory_inclusion P) | { creates_limit := λ F, creates_limit_full_subcategory_inclusion_of_closed h F } | def | category_theory.limits.creates_limits_of_shape_full_subcategory_inclusion | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | If `P` is closed under limits of shape `J`, then the inclusion creates such limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limit_of_closed_under_limits (h : closed_under_limits_of_shape J P)
(F : J ⥤ full_subcategory P) [has_limit (F ⋙ full_subcategory_inclusion P)] : has_limit F | have creates_limit F (full_subcategory_inclusion P),
from creates_limit_full_subcategory_inclusion_of_closed h F,
by exactI has_limit_of_created F (full_subcategory_inclusion P) | lemma | category_theory.limits.has_limit_of_closed_under_limits | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits_of_shape_of_closed_under_limits (h : closed_under_limits_of_shape J P)
[has_limits_of_shape J C] : has_limits_of_shape J (full_subcategory P) | { has_limit := λ F, has_limit_of_closed_under_limits h F } | lemma | category_theory.limits.has_limits_of_shape_of_closed_under_limits | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
creates_colimit_full_subcategory_inclusion_of_closed (h : closed_under_colimits_of_shape J P)
(F : J ⥤ full_subcategory P) [has_colimit (F ⋙ full_subcategory_inclusion P)] :
creates_colimit F (full_subcategory_inclusion P) | creates_colimit_full_subcategory_inclusion F (h.colimit (λ j, (F.obj j).property)) | def | category_theory.limits.creates_colimit_full_subcategory_inclusion_of_closed | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | If `P` is closed under colimits of shape `J`, then the inclusion creates such colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
creates_colimits_of_shape_full_subcategory_inclusion
(h : closed_under_colimits_of_shape J P) [has_colimits_of_shape J C] :
creates_colimits_of_shape J (full_subcategory_inclusion P) | { creates_colimit := λ F, creates_colimit_full_subcategory_inclusion_of_closed h F } | def | category_theory.limits.creates_colimits_of_shape_full_subcategory_inclusion | category_theory.limits | src/category_theory/limits/full_subcategory.lean | [
"category_theory.limits.creates"
] | [] | If `P` is closed under colimits of shape `J`, then the inclusion creates such colimits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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