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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|---|---|---|---|---|---|---|---|---|---|---|
hom_ext {X Y : UniformSpace} {f g : X ⟶ Y} : (f : X → Y) = g → f = g | subtype.eq | lemma | UniformSpace.hom_ext | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"UniformSpace",
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_Top : has_forget₂ UniformSpace.{u} Top.{u} | { forget₂ :=
{ obj := λ X, Top.of X,
map := λ X Y f, { to_fun := f,
continuous_to_fun := uniform_continuous.continuous f.property }, }, } | instance | UniformSpace.has_forget_to_Top | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"Top.of",
"uniform_continuous.continuous"
] | The forgetful functor from uniform spaces to topological spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CpltSepUniformSpace | (α : Type u)
[is_uniform_space : uniform_space α]
[is_complete_space : complete_space α]
[is_separated : separated_space α] | structure | CpltSepUniformSpace | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"complete_space",
"separated_space",
"uniform_space"
] | A (bundled) complete separated uniform space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_UniformSpace (X : CpltSepUniformSpace) : UniformSpace | UniformSpace.of X | def | CpltSepUniformSpace.to_UniformSpace | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"CpltSepUniformSpace",
"UniformSpace",
"UniformSpace.of"
] | The function forgetting that a complete separated uniform spaces is complete and separated. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_space (X : CpltSepUniformSpace) : complete_space ((to_UniformSpace X).α) | CpltSepUniformSpace.is_complete_space X | instance | CpltSepUniformSpace.complete_space | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"CpltSepUniformSpace",
"complete_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
separated_space (X : CpltSepUniformSpace) : separated_space ((to_UniformSpace X).α) | CpltSepUniformSpace.is_separated X | instance | CpltSepUniformSpace.separated_space | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"CpltSepUniformSpace",
"separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (X : Type u) [uniform_space X] [complete_space X] [separated_space X] :
CpltSepUniformSpace | ⟨X⟩ | def | CpltSepUniformSpace.of | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"CpltSepUniformSpace",
"complete_space",
"separated_space",
"uniform_space"
] | Construct a bundled `UniformSpace` from the underlying type and the appropriate typeclasses. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of (X : Type u) [uniform_space X] [complete_space X] [separated_space X] :
(of X : Type u) = X | rfl | lemma | CpltSepUniformSpace.coe_of | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"complete_space",
"separated_space",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
category : large_category CpltSepUniformSpace | induced_category.category to_UniformSpace | instance | CpltSepUniformSpace.category | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"CpltSepUniformSpace"
] | The category instance on `CpltSepUniformSpace`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concrete_category : concrete_category CpltSepUniformSpace | induced_category.concrete_category to_UniformSpace | instance | CpltSepUniformSpace.concrete_category | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"CpltSepUniformSpace"
] | The concrete category instance on `CpltSepUniformSpace`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_forget_to_UniformSpace : has_forget₂ CpltSepUniformSpace UniformSpace | induced_category.has_forget₂ to_UniformSpace | instance | CpltSepUniformSpace.has_forget_to_UniformSpace | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"CpltSepUniformSpace",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
completion_functor : UniformSpace ⥤ CpltSepUniformSpace | { obj := λ X, CpltSepUniformSpace.of (completion X),
map := λ X Y f, ⟨completion.map f.1, completion.uniform_continuous_map⟩,
map_id' := λ X, subtype.eq completion.map_id,
map_comp' := λ X Y Z f g, subtype.eq (completion.map_comp g.property f.property).symm, }. | def | UniformSpace.completion_functor | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"CpltSepUniformSpace",
"CpltSepUniformSpace.of",
"UniformSpace"
] | The functor turning uniform spaces into complete separated uniform spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
completion_hom (X : UniformSpace) :
X ⟶ (forget₂ CpltSepUniformSpace UniformSpace).obj (completion_functor.obj X) | { val := (coe : X → completion X),
property := completion.uniform_continuous_coe X } | def | UniformSpace.completion_hom | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"CpltSepUniformSpace",
"UniformSpace"
] | The inclusion of a uniform space into its completion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
completion_hom_val (X : UniformSpace) (x) :
(completion_hom X) x = (x : completion X) | rfl | lemma | UniformSpace.completion_hom_val | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"UniformSpace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_hom {X : UniformSpace} {Y : CpltSepUniformSpace}
(f : X ⟶ (forget₂ CpltSepUniformSpace UniformSpace).obj Y) :
completion_functor.obj X ⟶ Y | { val := completion.extension f,
property := completion.uniform_continuous_extension } | def | UniformSpace.extension_hom | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"CpltSepUniformSpace",
"UniformSpace"
] | The mate of a morphism from a `UniformSpace` to a `CpltSepUniformSpace`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extension_hom_val {X : UniformSpace} {Y : CpltSepUniformSpace}
(f : X ⟶ (forget₂ _ _).obj Y) (x) :
(extension_hom f) x = completion.extension f x | rfl. | lemma | UniformSpace.extension_hom_val | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"CpltSepUniformSpace",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_comp_coe {X : UniformSpace} {Y : CpltSepUniformSpace}
(f : to_UniformSpace (CpltSepUniformSpace.of (completion X)) ⟶ to_UniformSpace Y) :
extension_hom (completion_hom X ≫ f) = f | by { apply subtype.eq, funext x, exact congr_fun (completion.extension_comp_coe f.property) x } | lemma | UniformSpace.extension_comp_coe | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"CpltSepUniformSpace",
"CpltSepUniformSpace.of",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adj : completion_functor ⊣ forget₂ CpltSepUniformSpace UniformSpace | adjunction.mk_of_hom_equiv
{ hom_equiv := λ X Y,
{ to_fun := λ f, completion_hom X ≫ f,
inv_fun := λ f, extension_hom f,
left_inv := λ f, by { dsimp, erw extension_comp_coe },
right_inv := λ f,
begin
apply subtype.eq, funext x, cases f,
exact @completion.extension_coe _ _ _ _ _ (CpltSepUni... | def | UniformSpace.adj | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"CpltSepUniformSpace",
"CpltSepUniformSpace.separated_space",
"UniformSpace",
"adj",
"hom_ext",
"inv_fun"
] | The completion functor is left adjoint to the forgetful functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CompHaus | (to_Top : Top)
[is_compact : compact_space to_Top]
[is_hausdorff : t2_space to_Top] | structure | CompHaus | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"Top",
"compact_space",
"is_compact",
"t2_space"
] | The type of Compact Hausdorff topological spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category : category CompHaus | induced_category.category to_Top | instance | CompHaus.category | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"CompHaus"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category : concrete_category CompHaus | induced_category.concrete_category _ | instance | CompHaus.concrete_category | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"CompHaus"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_Top {X : CompHaus} : (X.to_Top : Type*) = X | rfl | lemma | CompHaus.coe_to_Top | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"CompHaus"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of : CompHaus | { to_Top := Top.of X,
is_compact := ‹_›,
is_hausdorff := ‹_› } | def | CompHaus.of | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"CompHaus",
"Top.of",
"is_compact"
] | A constructor for objects of the category `CompHaus`,
taking a type, and bundling the compact Hausdorff topology
found by typeclass inference. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of : (CompHaus.of X : Type _) = X | rfl | lemma | CompHaus.coe_of | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"CompHaus.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map {X Y : CompHaus.{u}} (f : X ⟶ Y) : is_closed_map f | λ C hC, (hC.is_compact.image f.continuous).is_closed | lemma | CompHaus.is_closed_map | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"is_closed",
"is_closed_map"
] | Any continuous function on compact Hausdorff spaces is a closed map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_bijective {X Y : CompHaus.{u}} (f : X ⟶ Y) (bij : function.bijective f) :
is_iso f | begin
let E := equiv.of_bijective _ bij,
have hE : continuous E.symm,
{ rw continuous_iff_is_closed,
intros S hS,
rw ← E.image_eq_preimage,
exact is_closed_map f S hS },
refine ⟨⟨⟨E.symm, hE⟩, _, _⟩⟩,
{ ext x,
apply E.symm_apply_apply },
{ ext x,
apply E.apply_symm_apply }
end | lemma | CompHaus.is_iso_of_bijective | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"continuous",
"continuous_iff_is_closed",
"equiv.of_bijective",
"is_closed_map"
] | Any continuous bijection of compact Hausdorff spaces is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_of_bijective {X Y : CompHaus.{u}} (f : X ⟶ Y) (bij : function.bijective f) : X ≅ Y | by letI := is_iso_of_bijective _ bij; exact as_iso f | def | CompHaus.iso_of_bijective | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [] | Any continuous bijection of compact Hausdorff spaces induces an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CompHaus_to_Top : CompHaus.{u} ⥤ Top.{u} | induced_functor _ | def | CompHaus_to_Top | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [] | The fully faithful embedding of `CompHaus` in `Top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CompHaus.forget_reflects_isomorphisms : reflects_isomorphisms (forget CompHaus.{u}) | ⟨by introsI A B f hf; exact CompHaus.is_iso_of_bijective _ ((is_iso_iff_bijective f).mp hf)⟩ | instance | CompHaus.forget_reflects_isomorphisms | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"CompHaus.is_iso_of_bijective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
StoneCech_obj (X : Top) : CompHaus | CompHaus.of (stone_cech X) | def | StoneCech_obj | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"CompHaus",
"CompHaus.of",
"Top",
"stone_cech"
] | (Implementation) The object part of the compactification functor from topological spaces to
compact Hausdorff spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stone_cech_equivalence (X : Top.{u}) (Y : CompHaus.{u}) :
(StoneCech_obj X ⟶ Y) ≃ (X ⟶ CompHaus_to_Top.obj Y) | { to_fun := λ f,
{ to_fun := f ∘ stone_cech_unit,
continuous_to_fun := f.2.comp (@continuous_stone_cech_unit X _) },
inv_fun := λ f,
{ to_fun := stone_cech_extend f.2,
continuous_to_fun := continuous_stone_cech_extend f.2 },
left_inv :=
begin
rintro ⟨f : stone_cech X ⟶ Y, hf : continuous f⟩,
e... | def | stone_cech_equivalence | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"StoneCech_obj",
"continuous",
"continuous.ext_on",
"continuous_stone_cech_extend",
"continuous_stone_cech_unit",
"dense_range_stone_cech_unit",
"inv_fun",
"stone_cech",
"stone_cech_extend",
"stone_cech_extend_extends",
"stone_cech_unit"
] | (Implementation) The bijection of homsets to establish the reflective adjunction of compact
Hausdorff spaces in topological spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Top_to_CompHaus : Top.{u} ⥤ CompHaus.{u} | adjunction.left_adjoint_of_equiv stone_cech_equivalence.{u} (λ _ _ _ _ _, rfl) | def | Top_to_CompHaus | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [] | The Stone-Cech compactification functor from topological spaces to compact Hausdorff spaces,
left adjoint to the inclusion functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Top_to_CompHaus_obj (X : Top) : ↥(Top_to_CompHaus.obj X) = stone_cech X | rfl | lemma | Top_to_CompHaus_obj | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"Top",
"stone_cech"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CompHaus_to_Top.reflective : reflective CompHaus_to_Top | { to_is_right_adjoint := ⟨Top_to_CompHaus, adjunction.adjunction_of_equiv_left _ _⟩ } | instance | CompHaus_to_Top.reflective | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"CompHaus_to_Top"
] | The category of compact Hausdorff spaces is reflective in the category of topological spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CompHaus_to_Top.creates_limits : creates_limits CompHaus_to_Top | monadic_creates_limits _ | instance | CompHaus_to_Top.creates_limits | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"CompHaus_to_Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CompHaus.has_limits : limits.has_limits CompHaus | has_limits_of_has_limits_creates_limits CompHaus_to_Top | instance | CompHaus.has_limits | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"CompHaus",
"CompHaus_to_Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CompHaus.has_colimits : limits.has_colimits CompHaus | has_colimits_of_reflective CompHaus_to_Top | instance | CompHaus.has_colimits | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"CompHaus",
"CompHaus_to_Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_cone {J : Type v} [small_category J] (F : J ⥤ CompHaus.{max v u}) :
limits.cone F | { X :=
{ to_Top := (Top.limit_cone (F ⋙ CompHaus_to_Top)).X,
is_compact := begin
show compact_space ↥{u : Π j, (F.obj j) | ∀ {i j : J} (f : i ⟶ j), (F.map f) (u i) = u j},
rw ← is_compact_iff_compact_space,
apply is_closed.is_compact,
have : {u : Π j, F.obj j | ∀ {i j : J} (f : i ⟶ j), F.m... | def | CompHaus.limit_cone | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"CompHaus_to_Top",
"Top.limit_cone",
"compact_space",
"continuous_apply",
"continuous_map.continuous",
"is_closed.is_compact",
"is_closed_Inter",
"is_closed_eq",
"is_compact",
"is_compact_iff_compact_space",
"set.mem_Inter",
"t2_space"
] | An explicit limit cone for a functor `F : J ⥤ CompHaus`, defined in terms of
`Top.limit_cone`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_is_limit {J : Type v} [small_category J] (F : J ⥤ CompHaus.{max v u}) :
limits.is_limit (limit_cone F) | { lift := λ S,
(Top.limit_cone_is_limit (F ⋙ CompHaus_to_Top)).lift (CompHaus_to_Top.map_cone S),
uniq' := λ S m h, (Top.limit_cone_is_limit _).uniq (CompHaus_to_Top.map_cone S) _ h } | def | CompHaus.limit_cone_is_limit | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"CompHaus_to_Top",
"Top.limit_cone_is_limit",
"lift"
] | The limit cone `CompHaus.limit_cone F` is indeed a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_iff_surjective {X Y : CompHaus.{u}} (f : X ⟶ Y) : epi f ↔ function.surjective f | begin
split,
{ contrapose!,
rintros ⟨y, hy⟩ hf,
let C := set.range f,
have hC : is_closed C := (is_compact_range f.continuous).is_closed,
let D := {y},
have hD : is_closed D := is_closed_singleton,
have hCD : disjoint C D,
{ rw set.disjoint_singleton_right, rintro ⟨y', hy'⟩, exact hy y' ... | lemma | CompHaus.epi_iff_surjective | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"CompHaus",
"category_theory.epi_iff_surjective",
"compact_space",
"continuous_map.coe_mk",
"disjoint",
"exists_continuous_zero_one_of_closed",
"is_closed",
"is_closed_singleton",
"is_compact_range",
"normal_of_compact_t2",
"normal_space",
"pi.one_apply",
"set.Icc",
"set.disjoint_singleton... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_iff_injective {X Y : CompHaus.{u}} (f : X ⟶ Y) : mono f ↔ function.injective f | begin
split,
{ introsI hf x₁ x₂ h,
let g₁ : of punit ⟶ X := ⟨λ _, x₁, continuous_const⟩,
let g₂ : of punit ⟶ X := ⟨λ _, x₂, continuous_const⟩,
have : g₁ ≫ f = g₂ ≫ f, by { ext, exact h },
rw cancel_mono at this,
apply_fun (λ e, e punit.star) at this,
exact this },
{ rw ← category_theory.mo... | lemma | CompHaus.mono_iff_injective | topology.category.CompHaus | src/topology/category/CompHaus/basic.lean | [
"category_theory.adjunction.reflective",
"topology.stone_cech",
"category_theory.monad.limits",
"topology.urysohns_lemma",
"topology.category.Top.limits.basic"
] | [
"CompHaus",
"category_theory.mono_iff_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
projective_ultrafilter (X : Type*) :
projective (of $ ultrafilter X) | { factors := λ Y Z f g hg,
begin
rw epi_iff_surjective at hg,
obtain ⟨g', hg'⟩ := hg.has_right_inverse,
let t : X → Y := g' ∘ f ∘ (pure : X → ultrafilter X),
let h : ultrafilter X → Y := ultrafilter.extend t,
have hh : continuous h := continuous_ultrafilter_extend _,
use ⟨h, hh⟩,
apply fai... | instance | CompHaus.projective_ultrafilter | topology.category.CompHaus | src/topology/category/CompHaus/projective.lean | [
"topology.category.CompHaus.basic",
"topology.stone_cech",
"category_theory.preadditive.projective"
] | [
"CompHaus",
"continuous",
"continuous_map.coe_mk",
"continuous_ultrafilter_extend",
"ultrafilter",
"ultrafilter.extend",
"ultrafilter_extend_extends"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
projective_presentation (X : CompHaus) : projective_presentation X | { P := of $ ultrafilter X,
f := ⟨_, continuous_ultrafilter_extend id⟩,
projective := CompHaus.projective_ultrafilter X,
epi := concrete_category.epi_of_surjective _ $
λ x, ⟨(pure x : ultrafilter X), congr_fun (ultrafilter_extend_extends (𝟙 X)) x⟩ } | def | CompHaus.projective_presentation | topology.category.CompHaus | src/topology/category/CompHaus/projective.lean | [
"topology.category.CompHaus.basic",
"topology.stone_cech",
"category_theory.preadditive.projective"
] | [
"CompHaus",
"CompHaus.projective_ultrafilter",
"continuous_ultrafilter_extend",
"ultrafilter",
"ultrafilter_extend_extends"
] | For any compact Hausdorff space `X`,
the natural map `ultrafilter X → X` is a projective presentation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fintype_diagram : discrete_quotient X ⥤ Fintype | { obj := λ S, by haveI := fintype.of_finite S; exact Fintype.of S,
map := λ S T f, discrete_quotient.of_le f.le } | def | Profinite.fintype_diagram | topology.category.Profinite | src/topology/category/Profinite/as_limit.lean | [
"topology.category.Profinite.basic",
"topology.discrete_quotient"
] | [
"Fintype",
"Fintype.of",
"discrete_quotient",
"discrete_quotient.of_le",
"fintype.of_finite"
] | The functor `discrete_quotient X ⥤ Fintype` whose limit is isomorphic to `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram : discrete_quotient X ⥤ Profinite | X.fintype_diagram ⋙ Fintype.to_Profinite | abbreviation | Profinite.diagram | topology.category.Profinite | src/topology/category/Profinite/as_limit.lean | [
"topology.category.Profinite.basic",
"topology.discrete_quotient"
] | [
"Fintype.to_Profinite",
"Profinite",
"discrete_quotient"
] | An abbreviation for `X.fintype_diagram ⋙ Fintype_to_Profinite`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_limit_cone : category_theory.limits.cone X.diagram | { X := X,
π := { app := λ S, ⟨S.proj, S.proj_is_locally_constant.continuous⟩ } } | def | Profinite.as_limit_cone | topology.category.Profinite | src/topology/category/Profinite/as_limit.lean | [
"topology.category.Profinite.basic",
"topology.discrete_quotient"
] | [
"category_theory.limits.cone"
] | A cone over `X.diagram` whose cone point is `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_as_limit_cone_lift :
is_iso ((limit_cone_is_limit X.diagram).lift X.as_limit_cone) | is_iso_of_bijective _
begin
refine ⟨λ a b h, _, λ a, _⟩,
{ refine discrete_quotient.eq_of_forall_proj_eq (λ S, _),
apply_fun (λ f : (limit_cone X.diagram).X, f.val S) at h,
exact h },
{ obtain ⟨b, hb⟩ := discrete_quotient.exists_of_compat
(λ S, a.val S) (λ _ _ h, a.prop (hom_of_le h)),
refine ⟨b... | instance | Profinite.is_iso_as_limit_cone_lift | topology.category.Profinite | src/topology/category/Profinite/as_limit.lean | [
"topology.category.Profinite.basic",
"topology.discrete_quotient"
] | [
"discrete_quotient.eq_of_forall_proj_eq",
"discrete_quotient.exists_of_compat",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_as_limit_cone_lift : X ≅ (limit_cone X.diagram).X | as_iso $ (limit_cone_is_limit _).lift X.as_limit_cone | def | Profinite.iso_as_limit_cone_lift | topology.category.Profinite | src/topology/category/Profinite/as_limit.lean | [
"topology.category.Profinite.basic",
"topology.discrete_quotient"
] | [
"lift"
] | The isomorphism between `X` and the explicit limit of `X.diagram`,
induced by lifting `X.as_limit_cone`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_limit_cone_iso : X.as_limit_cone ≅ limit_cone _ | limits.cones.ext (iso_as_limit_cone_lift _) (λ _, rfl) | def | Profinite.as_limit_cone_iso | topology.category.Profinite | src/topology/category/Profinite/as_limit.lean | [
"topology.category.Profinite.basic",
"topology.discrete_quotient"
] | [] | The isomorphism of cones `X.as_limit_cone` and `Profinite.limit_cone X.diagram`.
The underlying isomorphism is defeq to `X.iso_as_limit_cone_lift`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_limit : category_theory.limits.is_limit X.as_limit_cone | limits.is_limit.of_iso_limit (limit_cone_is_limit _) X.as_limit_cone_iso.symm | def | Profinite.as_limit | topology.category.Profinite | src/topology/category/Profinite/as_limit.lean | [
"topology.category.Profinite.basic",
"topology.discrete_quotient"
] | [
"category_theory.limits.is_limit"
] | `X.as_limit_cone` is indeed a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lim : limits.limit_cone X.diagram | ⟨X.as_limit_cone, X.as_limit⟩ | def | Profinite.lim | topology.category.Profinite | src/topology/category/Profinite/as_limit.lean | [
"topology.category.Profinite.basic",
"topology.discrete_quotient"
] | [
"lim"
] | A bundled version of `X.as_limit_cone` and `X.as_limit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Profinite | (to_CompHaus : CompHaus)
[is_totally_disconnected : totally_disconnected_space to_CompHaus] | structure | Profinite | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"CompHaus",
"is_totally_disconnected",
"totally_disconnected_space"
] | The type of profinite topological spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (X : Type*) [topological_space X] [compact_space X] [t2_space X]
[totally_disconnected_space X] : Profinite | ⟨⟨⟨X⟩⟩⟩ | def | Profinite.of | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite",
"compact_space",
"t2_space",
"topological_space",
"totally_disconnected_space"
] | Construct a term of `Profinite` from a type endowed with the structure of a
compact, Hausdorff and totally disconnected topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
category : category Profinite | induced_category.category to_CompHaus | instance | Profinite.category | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concrete_category : concrete_category Profinite | induced_category.concrete_category _ | instance | Profinite.concrete_category | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget₂ : has_forget₂ Profinite Top | induced_category.has_forget₂ _ | instance | Profinite.has_forget₂ | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite",
"Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_CompHaus {X : Profinite} : (X.to_CompHaus : Type*) = X | rfl | lemma | Profinite.coe_to_CompHaus | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_id (X : Profinite) : (𝟙 X : X → X) = id | rfl | lemma | Profinite.coe_id | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp {X Y Z : Profinite} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g : X → Z) = g ∘ f | rfl | lemma | Profinite.coe_comp | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Profinite_to_CompHaus : Profinite ⥤ CompHaus | induced_functor _ | def | Profinite_to_CompHaus | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"CompHaus",
"Profinite"
] | The fully faithful embedding of `Profinite` in `CompHaus`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Profinite.to_Top : Profinite ⥤ Top | forget₂ _ _ | def | Profinite.to_Top | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite",
"Top"
] | The fully faithful embedding of `Profinite` in `Top`. This is definitionally the same as the
obvious composite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Profinite.to_CompHaus_to_Top :
Profinite_to_CompHaus ⋙ CompHaus_to_Top = Profinite.to_Top | rfl | lemma | Profinite.to_CompHaus_to_Top | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"CompHaus_to_Top",
"Profinite.to_Top",
"Profinite_to_CompHaus"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CompHaus.to_Profinite_obj (X : CompHaus.{u}) : Profinite.{u} | { to_CompHaus :=
{ to_Top := Top.of (connected_components X),
is_compact := quotient.compact_space,
is_hausdorff := connected_components.t2 },
is_totally_disconnected := connected_components.totally_disconnected_space } | def | CompHaus.to_Profinite_obj | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Top.of",
"connected_components",
"connected_components.t2",
"connected_components.totally_disconnected_space",
"is_compact",
"is_totally_disconnected",
"quotient.compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Profinite.to_CompHaus_equivalence (X : CompHaus.{u}) (Y : Profinite.{u}) :
(CompHaus.to_Profinite_obj X ⟶ Y) ≃ (X ⟶ Profinite_to_CompHaus.obj Y) | { to_fun := λ f, f.comp ⟨quotient.mk', continuous_quotient_mk⟩,
inv_fun := λ g,
{ to_fun := continuous.connected_components_lift g.2,
continuous_to_fun := continuous.connected_components_lift_continuous g.2},
left_inv := λ f, continuous_map.ext $ connected_components.surjective_coe.forall.2 $ λ a, rfl,
... | def | Profinite.to_CompHaus_equivalence | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"CompHaus.to_Profinite_obj",
"continuous.connected_components_lift",
"continuous.connected_components_lift_continuous",
"continuous_map.ext",
"inv_fun"
] | (Implementation) The bijection of homsets to establish the reflective adjunction of Profinite
spaces in compact Hausdorff spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CompHaus.to_Profinite : CompHaus ⥤ Profinite | adjunction.left_adjoint_of_equiv Profinite.to_CompHaus_equivalence (λ _ _ _ _ _, rfl) | def | CompHaus.to_Profinite | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"CompHaus",
"Profinite",
"Profinite.to_CompHaus_equivalence"
] | The connected_components functor from compact Hausdorff spaces to profinite spaces,
left adjoint to the inclusion functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CompHaus.to_Profinite_obj' (X : CompHaus) :
↥(CompHaus.to_Profinite.obj X) = connected_components X | rfl | lemma | CompHaus.to_Profinite_obj' | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"CompHaus",
"connected_components"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Fintype.bot_topology (A : Fintype) : topological_space A | ⊥ | def | Fintype.bot_topology | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Fintype",
"topological_space"
] | Finite types are given the discrete topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Fintype.discrete_topology (A : Fintype) : discrete_topology A | ⟨rfl⟩ | lemma | Fintype.discrete_topology | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Fintype",
"discrete_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Fintype.to_Profinite : Fintype ⥤ Profinite | { obj := λ A, Profinite.of A,
map := λ _ _ f, ⟨f⟩ } | def | Fintype.to_Profinite | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Fintype",
"Profinite",
"Profinite.of"
] | The natural functor from `Fintype` to `Profinite`, endowing a finite type with the
discrete topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone {J : Type u} [small_category J] (F : J ⥤ Profinite.{u}) :
limits.cone F | { X :=
{ to_CompHaus := (CompHaus.limit_cone.{u u} (F ⋙ Profinite_to_CompHaus)).X,
is_totally_disconnected :=
begin
change totally_disconnected_space ↥{u : Π (j : J), (F.obj j) | _},
exact subtype.totally_disconnected_space,
end },
π := { app := (CompHaus.limit_cone.{u u} (F ⋙ Profinite_to_C... | def | Profinite.limit_cone | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite_to_CompHaus",
"is_totally_disconnected",
"subtype.totally_disconnected_space",
"totally_disconnected_space"
] | An explicit limit cone for a functor `F : J ⥤ Profinite`, defined in terms of
`Top.limit_cone`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_is_limit {J : Type u} [small_category J] (F : J ⥤ Profinite.{u}) :
limits.is_limit (limit_cone F) | { lift := λ S, (CompHaus.limit_cone_is_limit.{u u} (F ⋙ Profinite_to_CompHaus)).lift
(Profinite_to_CompHaus.map_cone S),
uniq' := λ S m h,
(CompHaus.limit_cone_is_limit.{u u} _).uniq (Profinite_to_CompHaus.map_cone S) _ h } | def | Profinite.limit_cone_is_limit | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite_to_CompHaus",
"lift"
] | The limit cone `Profinite.limit_cone F` is indeed a limit cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Profinite_adj_to_CompHaus : CompHaus.to_Profinite ⊣ Profinite_to_CompHaus | adjunction.adjunction_of_equiv_left _ _ | def | Profinite.to_Profinite_adj_to_CompHaus | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"CompHaus.to_Profinite",
"Profinite_to_CompHaus"
] | The adjunction between CompHaus.to_Profinite and Profinite.to_CompHaus | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_CompHaus.reflective : reflective Profinite_to_CompHaus | { to_is_right_adjoint := ⟨CompHaus.to_Profinite, Profinite.to_Profinite_adj_to_CompHaus⟩ } | instance | Profinite.to_CompHaus.reflective | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite_to_CompHaus"
] | The category of profinite sets is reflective in the category of compact hausdroff spaces | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_CompHaus.creates_limits : creates_limits Profinite_to_CompHaus | monadic_creates_limits _ | instance | Profinite.to_CompHaus.creates_limits | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite_to_CompHaus"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Top.reflective : reflective Profinite.to_Top | reflective.comp Profinite_to_CompHaus CompHaus_to_Top | instance | Profinite.to_Top.reflective | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"CompHaus_to_Top",
"Profinite.to_Top",
"Profinite_to_CompHaus"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Top.creates_limits : creates_limits Profinite.to_Top | monadic_creates_limits _ | instance | Profinite.to_Top.creates_limits | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite.to_Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_limits : limits.has_limits Profinite | has_limits_of_has_limits_creates_limits Profinite.to_Top | instance | Profinite.has_limits | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite",
"Profinite.to_Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_colimits : limits.has_colimits Profinite | has_colimits_of_reflective Profinite_to_CompHaus | instance | Profinite.has_colimits | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite",
"Profinite_to_CompHaus"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_limits : limits.preserves_limits (forget Profinite) | by apply limits.comp_preserves_limits Profinite.to_Top (forget Top) | instance | Profinite.forget_preserves_limits | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite",
"Profinite.to_Top",
"Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map : is_closed_map f | CompHaus.is_closed_map _ | lemma | Profinite.is_closed_map | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"CompHaus.is_closed_map",
"is_closed_map"
] | Any morphism of profinite spaces is a closed map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_iso_of_bijective (bij : function.bijective f) : is_iso f | begin
haveI := CompHaus.is_iso_of_bijective (Profinite_to_CompHaus.map f) bij,
exact is_iso_of_fully_faithful Profinite_to_CompHaus _
end | lemma | Profinite.is_iso_of_bijective | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"CompHaus.is_iso_of_bijective",
"Profinite_to_CompHaus"
] | Any continuous bijection of profinite spaces induces an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_of_bijective (bij : function.bijective f) : X ≅ Y | by letI := Profinite.is_iso_of_bijective f bij; exact as_iso f | def | Profinite.iso_of_bijective | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite.is_iso_of_bijective"
] | Any continuous bijection of profinite spaces induces an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_reflects_isomorphisms : reflects_isomorphisms (forget Profinite) | ⟨by introsI A B f hf; exact Profinite.is_iso_of_bijective _ ((is_iso_iff_bijective f).mp hf)⟩ | instance | Profinite.forget_reflects_isomorphisms | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite",
"Profinite.is_iso_of_bijective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_of_homeo (f : X ≃ₜ Y) : X ≅ Y | { hom := ⟨f, f.continuous⟩,
inv := ⟨f.symm, f.symm.continuous⟩,
hom_inv_id' := by { ext x, exact f.symm_apply_apply x },
inv_hom_id' := by { ext x, exact f.apply_symm_apply x } } | def | Profinite.iso_of_homeo | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [] | Construct an isomorphism from a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homeo_of_iso (f : X ≅ Y) : X ≃ₜ Y | { to_fun := f.hom,
inv_fun := f.inv,
left_inv := λ x, by { change (f.hom ≫ f.inv) x = x, rw [iso.hom_inv_id, coe_id, id.def] },
right_inv := λ x, by { change (f.inv ≫ f.hom) x = x, rw [iso.inv_hom_id, coe_id, id.def] },
continuous_to_fun := f.hom.continuous,
continuous_inv_fun := f.inv.continuous } | def | Profinite.homeo_of_iso | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"inv_fun"
] | Construct a homeomorphism from an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_equiv_homeo : (X ≅ Y) ≃ (X ≃ₜ Y) | { to_fun := homeo_of_iso,
inv_fun := iso_of_homeo,
left_inv := λ f, by { ext, refl },
right_inv := λ f, by { ext, refl } } | def | Profinite.iso_equiv_homeo | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"inv_fun"
] | The equivalence between isomorphisms in `Profinite` and homeomorphisms
of topological spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
epi_iff_surjective {X Y : Profinite.{u}} (f : X ⟶ Y) : epi f ↔ function.surjective f | begin
split,
{ contrapose!,
rintros ⟨y, hy⟩ hf, resetI,
let C := set.range f,
have hC : is_closed C := (is_compact_range f.continuous).is_closed,
let U := Cᶜ,
have hyU : y ∈ U,
{ refine set.mem_compl _, rintro ⟨y', hy'⟩, exact hy y' hy' },
have hUy : U ∈ 𝓝 y := hC.compl_mem_nhds hyU,
... | lemma | Profinite.epi_iff_surjective | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"Profinite",
"category_theory.epi_iff_surjective",
"is_closed",
"is_compact_range",
"locally_constant.continuous",
"locally_constant.of_clopen",
"set.mem_compl",
"set.mem_compl_iff",
"set.mem_range_self",
"set.range",
"top_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_iff_injective {X Y : Profinite.{u}} (f : X ⟶ Y) : mono f ↔ function.injective f | begin
split,
{ intro h,
haveI : limits.preserves_limits Profinite_to_CompHaus := infer_instance,
haveI : mono (Profinite_to_CompHaus.map f) := infer_instance,
rwa ← CompHaus.mono_iff_injective },
{ rw ← category_theory.mono_iff_injective,
apply (forget Profinite).mono_of_mono_map }
end | lemma | Profinite.mono_iff_injective | topology.category.Profinite | src/topology/category/Profinite/basic.lean | [
"topology.category.CompHaus.basic",
"topology.connected",
"topology.subset_properties",
"topology.locally_constant.basic",
"category_theory.adjunction.reflective",
"category_theory.monad.limits",
"category_theory.Fintype"
] | [
"CompHaus.mono_iff_injective",
"Profinite",
"Profinite_to_CompHaus",
"category_theory.mono_iff_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_clopen_of_cofiltered {U : set C.X} (hU : is_clopen U) :
∃ (j : J) (V : set (F.obj j)) (hV : is_clopen V), U = C.π.app j ⁻¹' V | begin
-- First, we have the topological basis of the cofiltered limit obtained by pulling back
-- clopen sets from the factors in the limit. By continuity, all such sets are again clopen.
have hB := Top.is_topological_basis_cofiltered_limit.{u}
(F ⋙ Profinite.to_Top)
(Profinite.to_Top.map_cone C)
(is_... | theorem | Profinite.exists_clopen_of_cofiltered | topology.category.Profinite | src/topology/category/Profinite/cofiltered_limit.lean | [
"topology.category.Profinite.basic",
"topology.locally_constant.basic",
"topology.discrete_quotient",
"topology.category.Top.limits.cofiltered",
"topology.category.Top.limits.konig"
] | [
"Profinite.coe_comp",
"Profinite.to_Top",
"continuity",
"is_clopen",
"is_clopen_bUnion_finset",
"is_clopen_univ",
"is_topological_basis_clopen",
"set.mem_Union",
"set.preimage_Union",
"set.preimage_comp",
"topological_space.is_topological_basis"
] | If `X` is a cofiltered limit of profinite sets, then any clopen subset of `X` arises from
a clopen set in one of the terms in the limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_locally_constant_fin_two (f : locally_constant C.X (fin 2)) :
∃ (j : J) (g : locally_constant (F.obj j) (fin 2)), f = g.comap (C.π.app _) | begin
let U := f ⁻¹' {0},
have hU : is_clopen U := f.is_locally_constant.is_clopen_fiber _,
obtain ⟨j,V,hV,h⟩ := exists_clopen_of_cofiltered C hC hU,
use [j, locally_constant.of_clopen hV],
apply locally_constant.locally_constant_eq_of_fiber_zero_eq,
rw locally_constant.coe_comap _ _ (C.π.app j).continuous,... | lemma | Profinite.exists_locally_constant_fin_two | topology.category.Profinite | src/topology/category/Profinite/cofiltered_limit.lean | [
"topology.category.Profinite.basic",
"topology.locally_constant.basic",
"topology.discrete_quotient",
"topology.category.Top.limits.cofiltered",
"topology.category.Top.limits.konig"
] | [
"continuous",
"is_clopen",
"locally_constant",
"locally_constant.coe_comap",
"locally_constant.locally_constant_eq_of_fiber_zero_eq",
"locally_constant.of_clopen",
"locally_constant.of_clopen_fiber_zero",
"set.preimage_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_locally_constant_finite_aux {α : Type*} [finite α] (f : locally_constant C.X α) :
∃ (j : J) (g : locally_constant (F.obj j) (α → fin 2)),
f.map (λ a b, if a = b then (0 : fin 2) else 1) = g.comap (C.π.app _) | begin
casesI nonempty_fintype α,
let ι : α → α → fin 2 := λ x y, if x = y then 0 else 1,
let ff := (f.map ι).flip,
have hff := λ (a : α), exists_locally_constant_fin_two _ hC (ff a),
choose j g h using hff,
let G : finset J := finset.univ.image j,
obtain ⟨j0,hj0⟩ := is_cofiltered.inf_objs_exists G,
have... | theorem | Profinite.exists_locally_constant_finite_aux | topology.category.Profinite | src/topology/category/Profinite/cofiltered_limit.lean | [
"topology.category.Profinite.basic",
"topology.locally_constant.basic",
"topology.discrete_quotient",
"topology.category.Top.limits.cofiltered",
"topology.category.Top.limits.konig"
] | [
"continuity",
"finite",
"finset",
"locally_constant",
"locally_constant.coe_comap",
"locally_constant.comap",
"locally_constant.flip",
"locally_constant.unflip",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_locally_constant_finite_nonempty {α : Type*} [finite α] [nonempty α]
(f : locally_constant C.X α) :
∃ (j : J) (g : locally_constant (F.obj j) α), f = g.comap (C.π.app _) | begin
inhabit α,
obtain ⟨j,gg,h⟩ := exists_locally_constant_finite_aux _ hC f,
let ι : α → α → fin 2 := λ a b, if a = b then 0 else 1,
let σ : (α → fin 2) → α := λ f, if h : ∃ (a : α), ι a = f then h.some else arbitrary _,
refine ⟨j, gg.map σ, _⟩,
ext,
rw locally_constant.coe_comap _ _ (C.π.app j).continu... | theorem | Profinite.exists_locally_constant_finite_nonempty | topology.category.Profinite | src/topology/category/Profinite/cofiltered_limit.lean | [
"topology.category.Profinite.basic",
"topology.locally_constant.basic",
"topology.discrete_quotient",
"topology.category.Top.limits.cofiltered",
"topology.category.Top.limits.konig"
] | [
"bot_ne_top",
"continuous",
"finite",
"locally_constant",
"locally_constant.coe_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_locally_constant {α : Type*} (f : locally_constant C.X α) :
∃ (j : J) (g : locally_constant (F.obj j) α), f = g.comap (C.π.app _) | begin
let S := f.discrete_quotient,
let ff : S → α := f.lift,
casesI is_empty_or_nonempty S,
{ suffices : ∃ j, is_empty (F.obj j),
{ refine this.imp (λ j hj, _),
refine ⟨⟨hj.elim, λ A, _⟩, _⟩,
{ convert is_open_empty,
exact @set.eq_empty_of_is_empty _ hj _ },
{ ext x,
exact... | theorem | Profinite.exists_locally_constant | topology.category.Profinite | src/topology/category/Profinite/cofiltered_limit.lean | [
"topology.category.Profinite.basic",
"topology.locally_constant.basic",
"topology.discrete_quotient",
"topology.category.Top.limits.cofiltered",
"topology.category.Top.limits.konig"
] | [
"Profinite.to_Top",
"Top.nonempty_limit_cone_of_compact_t2_cofiltered_system",
"compact_space",
"continuous",
"is_empty",
"is_empty_or_nonempty",
"is_open_empty",
"locally_constant",
"locally_constant.coe_comap",
"not_forall",
"not_nonempty_iff",
"set.eq_empty_of_is_empty",
"t2_space"
] | Any locally constant function from a cofiltered limit of profinite sets factors through
one of the components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
projective_ultrafilter (X : Type u) :
projective (of $ ultrafilter X) | { factors := λ Y Z f g hg,
begin
rw epi_iff_surjective at hg,
obtain ⟨g', hg'⟩ := hg.has_right_inverse,
let t : X → Y := g' ∘ f ∘ (pure : X → ultrafilter X),
let h : ultrafilter X → Y := ultrafilter.extend t,
have hh : continuous h := continuous_ultrafilter_extend _,
use ⟨h, hh⟩,
apply fai... | instance | Profinite.projective_ultrafilter | topology.category.Profinite | src/topology/category/Profinite/projective.lean | [
"topology.category.Profinite.basic",
"topology.stone_cech",
"category_theory.preadditive.projective"
] | [
"Profinite",
"continuous",
"continuous_map.coe_mk",
"continuous_ultrafilter_extend",
"ultrafilter",
"ultrafilter.extend",
"ultrafilter_extend_extends"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
projective_presentation (X : Profinite.{u}) : projective_presentation X | { P := of $ ultrafilter X,
f := ⟨_, continuous_ultrafilter_extend id⟩,
projective := Profinite.projective_ultrafilter X,
epi := concrete_category.epi_of_surjective _ $
λ x, ⟨(pure x : ultrafilter X), congr_fun (ultrafilter_extend_extends (𝟙 X)) x⟩ } | def | Profinite.projective_presentation | topology.category.Profinite | src/topology/category/Profinite/projective.lean | [
"topology.category.Profinite.basic",
"topology.stone_cech",
"category_theory.preadditive.projective"
] | [
"Profinite.projective_ultrafilter",
"continuous_ultrafilter_extend",
"ultrafilter",
"ultrafilter_extend_extends"
] | For any profinite `X`, the natural map `ultrafilter X → X` is a projective presentation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adj₁ : discrete ⊣ forget Top.{u} | adjunction.mk_of_unit_counit
{ unit := { app := λ X, id },
counit := { app := λ X, ⟨id, continuous_bot⟩ } } | def | Top.adj₁ | topology.category.Top | src/topology/category/Top/adjunctions.lean | [
"topology.category.Top.basic",
"category_theory.adjunction.basic"
] | [] | Equipping a type with the discrete topology is left adjoint to the forgetful functor
`Top ⥤ Type`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adj₂ : forget Top.{u} ⊣ trivial | adjunction.mk_of_unit_counit
{ unit := { app := λ X, ⟨id, continuous_top⟩ },
counit := { app := λ X, id } } | def | Top.adj₂ | topology.category.Top | src/topology/category/Top/adjunctions.lean | [
"topology.category.Top.basic",
"category_theory.adjunction.basic"
] | [] | Equipping a type with the trivial topology is right adjoint to the forgetful functor
`Top ⥤ Type`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Top : Type (u+1) | bundled topological_space | def | Top | topology.category.Top | src/topology/category/Top/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise",
"topology.continuous_function.basic"
] | [
"topological_space"
] | The category of topological spaces and continuous maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bundled_hom : bundled_hom @continuous_map | ⟨@continuous_map.to_fun, @continuous_map.id, @continuous_map.comp, @continuous_map.coe_injective⟩ | instance | Top.bundled_hom | topology.category.Top | src/topology/category/Top/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise",
"topology.continuous_function.basic"
] | [
"continuous_map",
"continuous_map.comp",
"continuous_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_space_unbundled (x : Top) : topological_space x | x.str | instance | Top.topological_space_unbundled | topology.category.Top | src/topology/category/Top/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise",
"topology.continuous_function.basic"
] | [
"Top",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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