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id_app (X : Top.{u}) (x : X) :
(𝟙 X : X → X) x = x | rfl | lemma | Top.id_app | topology.category.Top | src/topology/category/Top/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_app {X Y Z : Top.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
(f ≫ g : X → Z) x = g (f x) | rfl | lemma | Top.comp_app | topology.category.Top | src/topology/category/Top/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (X : Type u) [topological_space X] : Top | ⟨X⟩ | def | Top.of | 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"
] | Construct a bundled `Top` from the underlying type and the typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of (X : Type u) [topological_space X] : (of X : Type u) = X | rfl | lemma | Top.coe_of | topology.category.Top | src/topology/category/Top/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise",
"topology.continuous_function.basic"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete : Type u ⥤ Top.{u} | { obj := λ X, ⟨X, ⊥⟩,
map := λ X Y f, { to_fun := f, continuous_to_fun := continuous_bot } } | def | Top.discrete | topology.category.Top | src/topology/category/Top/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise",
"topology.continuous_function.basic"
] | [
"continuous_bot"
] | The discrete topology on any type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trivial : Type u ⥤ Top.{u} | { obj := λ X, ⟨X, ⊤⟩,
map := λ X Y f, { to_fun := f, continuous_to_fun := continuous_top } } | def | Top.trivial | topology.category.Top | src/topology/category/Top/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise",
"topology.continuous_function.basic"
] | [
"continuous_top"
] | The trivial topology on any type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_of_homeo {X Y : Top.{u}} (f : X ≃ₜ Y) : X ≅ Y | { hom := ⟨f⟩,
inv := ⟨f.symm⟩ } | def | Top.iso_of_homeo | topology.category.Top | src/topology/category/Top/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise",
"topology.continuous_function.basic"
] | [] | Any homeomorphisms induces an isomorphism in `Top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homeo_of_iso {X Y : Top.{u}} (f : X ≅ Y) : X ≃ₜ Y | { to_fun := f.hom,
inv_fun := f.inv,
left_inv := λ x, by simp,
right_inv := λ x, by simp,
continuous_to_fun := f.hom.continuous,
continuous_inv_fun := f.inv.continuous } | def | Top.homeo_of_iso | topology.category.Top | src/topology/category/Top/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise",
"topology.continuous_function.basic"
] | [
"inv_fun"
] | Any isomorphism in `Top` induces a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_iso_of_homeo {X Y : Top.{u}} (f : X ≃ₜ Y) : homeo_of_iso (iso_of_homeo f) = f | by { ext, refl } | lemma | Top.of_iso_of_homeo | topology.category.Top | src/topology/category/Top/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_homeo_of_iso {X Y : Top.{u}} (f : X ≅ Y) : iso_of_homeo (homeo_of_iso f) = f | by { ext, refl } | lemma | Top.of_homeo_of_iso | topology.category.Top | src/topology/category/Top/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_iff_comp_is_iso {X Y Z : Top} (f : X ⟶ Y) (g : Y ⟶ Z) [is_iso g] :
open_embedding (f ≫ g) ↔ open_embedding f | (Top.homeo_of_iso (as_iso g)).open_embedding.of_comp_iff f | lemma | Top.open_embedding_iff_comp_is_iso | topology.category.Top | src/topology/category/Top/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise",
"topology.continuous_function.basic"
] | [
"Top",
"Top.homeo_of_iso",
"open_embedding",
"open_embedding.of_comp_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_iff_is_iso_comp {X Y Z : Top} (f : X ⟶ Y) (g : Y ⟶ Z) [is_iso f] :
open_embedding (f ≫ g) ↔ open_embedding g | begin
split,
{ intro h,
convert h.comp (Top.homeo_of_iso (as_iso f).symm).open_embedding,
exact congr_arg _ (is_iso.inv_hom_id_assoc f g).symm },
{ exact λ h, h.comp (Top.homeo_of_iso (as_iso f)).open_embedding }
end | lemma | Top.open_embedding_iff_is_iso_comp | topology.category.Top | src/topology/category/Top/basic.lean | [
"category_theory.concrete_category.bundled_hom",
"category_theory.elementwise",
"topology.continuous_function.basic"
] | [
"Top",
"Top.homeo_of_iso",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_iff_surjective {X Y : Top.{u}} (f : X ⟶ Y) : epi f ↔ function.surjective f | begin
suffices : epi f ↔ epi ((forget Top).map f),
{ rw [this, category_theory.epi_iff_surjective], refl },
split,
{ introI, apply_instance },
{ apply functor.epi_of_epi_map }
end | lemma | Top.epi_iff_surjective | topology.category.Top | src/topology/category/Top/epi_mono.lean | [
"topology.category.Top.adjunctions"
] | [
"Top",
"category_theory.epi_iff_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_iff_injective {X Y : Top.{u}} (f : X ⟶ Y) : mono f ↔ function.injective f | begin
suffices : mono f ↔ mono ((forget Top).map f),
{ rw [this, category_theory.mono_iff_injective], refl },
split,
{ introI, apply_instance },
{ apply functor.mono_of_mono_map }
end | lemma | Top.mono_iff_injective | topology.category.Top | src/topology/category/Top/epi_mono.lean | [
"topology.category.Top.adjunctions"
] | [
"Top",
"category_theory.mono_iff_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opens_hom_has_coe_to_fun {U V : opens X} : has_coe_to_fun (U ⟶ V) (λ f, U → V) | ⟨λ f x, ⟨x, f.le x.2⟩⟩ | instance | topological_space.opens.opens_hom_has_coe_to_fun | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_le_left (U V : opens X) : U ⊓ V ⟶ U | inf_le_left.hom | def | topological_space.opens.inf_le_left | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"inf_le_left"
] | The inclusion `U ⊓ V ⟶ U` as a morphism in the category of open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_le_right (U V : opens X) : U ⊓ V ⟶ V | inf_le_right.hom | def | topological_space.opens.inf_le_right | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"inf_le_right"
] | The inclusion `U ⊓ V ⟶ V` as a morphism in the category of open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_supr {ι : Type*} (U : ι → opens X) (i : ι) : U i ⟶ supr U | (le_supr U i).hom | def | topological_space.opens.le_supr | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"le_supr",
"supr"
] | The inclusion `U i ⟶ supr U` as a morphism in the category of open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bot_le (U : opens X) : ⊥ ⟶ U | bot_le.hom | def | topological_space.opens.bot_le | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"bot_le"
] | The inclusion `⊥ ⟶ U` as a morphism in the category of open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_top (U : opens X) : U ⟶ ⊤ | le_top.hom | def | topological_space.opens.le_top | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"le_top"
] | The inclusion `U ⟶ ⊤` as a morphism in the category of open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_le_left_apply (U V : opens X) (x) :
(inf_le_left U V) x = ⟨x.1, (@_root_.inf_le_left _ _ U V : _ ≤ _) x.2⟩ | rfl | lemma | topological_space.opens.inf_le_left_apply | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_le_left_apply_mk (U V : opens X) (x) (m) :
(inf_le_left U V) ⟨x, m⟩ = ⟨x, (@_root_.inf_le_left _ _ U V : _ ≤ _) m⟩ | rfl | lemma | topological_space.opens.inf_le_left_apply_mk | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_supr_apply_mk {ι : Type*} (U : ι → opens X) (i : ι) (x) (m) :
(le_supr U i) ⟨x, m⟩ = ⟨x, (_root_.le_supr U i : _) m⟩ | rfl | lemma | topological_space.opens.le_supr_apply_mk | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"le_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Top (X : Top.{u}) : opens X ⥤ Top | { obj := λ U, ⟨U, infer_instance⟩,
map := λ U V i, ⟨λ x, ⟨x.1, i.le x.2⟩,
(embedding.continuous_iff embedding_subtype_coe).2 continuous_induced_dom⟩ } | def | topological_space.opens.to_Top | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"Top",
"embedding.continuous_iff",
"embedding_subtype_coe"
] | The functor from open sets in `X` to `Top`,
realising each open set as a topological space itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Top_map (X : Top.{u}) {U V : opens X} {f : U ⟶ V} {x} {h} :
((to_Top X).map f) ⟨x, h⟩ = ⟨x, f.le h⟩ | rfl | lemma | topological_space.opens.to_Top_map | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion {X : Top.{u}} (U : opens X) : (to_Top X).obj U ⟶ X | { to_fun := _,
continuous_to_fun := continuous_subtype_coe } | def | topological_space.opens.inclusion | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"continuous_subtype_coe"
] | The inclusion map from an open subset to the whole space, as a morphism in `Top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_embedding {X : Top.{u}} (U : opens X) : open_embedding (inclusion U) | is_open.open_embedding_subtype_coe U.2 | lemma | topological_space.opens.open_embedding | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"is_open.open_embedding_subtype_coe",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_top_iso (X : Top.{u}) : (to_Top X).obj ⊤ ≅ X | { hom := inclusion ⊤,
inv := ⟨λ x, ⟨x, trivial⟩, continuous_def.2 $ λ U ⟨S, hS, hSU⟩, hSU ▸ hS⟩ } | def | topological_space.opens.inclusion_top_iso | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | The inclusion of the top open subset (i.e. the whole space) is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (f : X ⟶ Y) : opens Y ⥤ opens X | { obj := λ U, ⟨ f ⁻¹' U, U.is_open.preimage f.continuous ⟩,
map := λ U V i, ⟨ ⟨ λ x h, i.le h ⟩ ⟩ }. | def | topological_space.opens.map | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | `opens.map f` gives the functor from open sets in Y to open set in X,
given by taking preimages under f. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_coe (f : X ⟶ Y) (U : opens Y) :
↑((map f).obj U) = f ⁻¹' U | rfl | lemma | topological_space.opens.map_coe | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_obj (f : X ⟶ Y) (U) (p) :
(map f).obj ⟨U, p⟩ = ⟨f ⁻¹' U, p.preimage f.continuous⟩ | rfl | lemma | topological_space.opens.map_obj | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id_obj (U : opens X) : (map (𝟙 X)).obj U = U | let ⟨_,_⟩ := U in rfl | lemma | topological_space.opens.map_id_obj | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id_obj' (U) (p) : (map (𝟙 X)).obj ⟨U, p⟩ = ⟨U, p⟩ | rfl | lemma | topological_space.opens.map_id_obj' | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id_obj_unop (U : (opens X)ᵒᵖ) : (map (𝟙 X)).obj (unop U) = unop U | let ⟨_,_⟩ := U.unop in rfl | lemma | topological_space.opens.map_id_obj_unop | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_map_id_obj (U : (opens X)ᵒᵖ) : (map (𝟙 X)).op.obj U = U | by simp | lemma | topological_space.opens.op_map_id_obj | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_map_top (f : X ⟶ Y) (U : opens X) : U ⟶ (map f).obj ⊤ | le_top U | def | topological_space.opens.le_map_top | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"le_top"
] | The inclusion `U ⟶ (map f).obj ⊤` as a morphism in the category of open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_comp_obj (f : X ⟶ Y) (g : Y ⟶ Z) (U) :
(map (f ≫ g)).obj U = (map f).obj ((map g).obj U) | rfl | lemma | topological_space.opens.map_comp_obj | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp_obj' (f : X ⟶ Y) (g : Y ⟶ Z) (U) (p) :
(map (f ≫ g)).obj ⟨U, p⟩ = (map f).obj ((map g).obj ⟨U, p⟩) | rfl | lemma | topological_space.opens.map_comp_obj' | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp_map (f : X ⟶ Y) (g : Y ⟶ Z) {U V} (i : U ⟶ V) :
(map (f ≫ g)).map i = (map f).map ((map g).map i) | rfl | lemma | topological_space.opens.map_comp_map | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp_obj_unop (f : X ⟶ Y) (g : Y ⟶ Z) (U) :
(map (f ≫ g)).obj (unop U) = (map f).obj ((map g).obj (unop U)) | rfl | lemma | topological_space.opens.map_comp_obj_unop | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_map_comp_obj (f : X ⟶ Y) (g : Y ⟶ Z) (U) :
(map (f ≫ g)).op.obj U = (map f).op.obj ((map g).op.obj U) | rfl | lemma | topological_space.opens.op_map_comp_obj | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_supr (f : X ⟶ Y) {ι : Type*} (U : ι → opens Y) :
(map f).obj (supr U) = supr ((map f).obj ∘ U) | begin
ext1, rw [supr_def, supr_def, map_obj],
dsimp, rw set.preimage_Union, refl,
end | lemma | topological_space.opens.map_supr | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"map_supr",
"set.preimage_Union",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : map (𝟙 X) ≅ 𝟭 (opens X) | { hom := { app := λ U, eq_to_hom (map_id_obj U) },
inv := { app := λ U, eq_to_hom (map_id_obj U).symm } } | def | topological_space.opens.map_id | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"map_id"
] | The functor `opens X ⥤ opens X` given by taking preimages under the identity function
is naturally isomorphic to the identity functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_id_eq : map (𝟙 X) = 𝟭 (opens X) | by { unfold map, congr, ext, refl, ext } | lemma | topological_space.opens.map_id_eq | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp (f : X ⟶ Y) (g : Y ⟶ Z) : map (f ≫ g) ≅ map g ⋙ map f | { hom := { app := λ U, eq_to_hom (map_comp_obj f g U) },
inv := { app := λ U, eq_to_hom (map_comp_obj f g U).symm } } | def | topological_space.opens.map_comp | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"map_comp"
] | The natural isomorphism between taking preimages under `f ≫ g`, and the composite
of taking preimages under `g`, then preimages under `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_comp_eq (f : X ⟶ Y) (g : Y ⟶ Z) : map (f ≫ g) = map g ⋙ map f | rfl | lemma | topological_space.opens.map_comp_eq | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_iso (f g : X ⟶ Y) (h : f = g) : map f ≅ map g | nat_iso.of_components (λ U, eq_to_iso (congr_fun (congr_arg functor.obj (congr_arg map h)) U) )
(by obviously) | def | topological_space.opens.map_iso | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq (f g : X ⟶ Y) (h : f = g) : map f = map g | by { unfold map, congr, ext, rw h, rw h, assumption' } | lemma | topological_space.opens.map_eq | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_iso_refl (f : X ⟶ Y) (h) : map_iso f f h = iso.refl (map _) | rfl | lemma | topological_space.opens.map_iso_refl | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_iso_hom_app (f g : X ⟶ Y) (h : f = g) (U : opens Y) :
(map_iso f g h).hom.app U = eq_to_hom (congr_fun (congr_arg functor.obj (congr_arg map h)) U) | rfl | lemma | topological_space.opens.map_iso_hom_app | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_iso_inv_app (f g : X ⟶ Y) (h : f = g) (U : opens Y) :
(map_iso f g h).inv.app U =
eq_to_hom (congr_fun (congr_arg functor.obj (congr_arg map h.symm)) U) | rfl | lemma | topological_space.opens.map_iso_inv_app | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map_iso {X Y : Top.{u}} (H : X ≅ Y) : opens Y ≌ opens X | { functor := map H.hom,
inverse := map H.inv,
unit_iso := nat_iso.of_components (λ U, eq_to_iso (by simp [map, set.preimage_preimage]))
(by { intros _ _ _, simp }),
counit_iso := nat_iso.of_components (λ U, eq_to_iso (by simp [map, set.preimage_preimage]))
(by { intros _ _ _, simp }) } | def | topological_space.opens.map_map_iso | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"set.preimage_preimage"
] | A homeomorphism of spaces gives an equivalence of categories of open sets.
TODO: define `order_iso.equivalence`, use it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map.functor {X Y : Top} {f : X ⟶ Y} (hf : is_open_map f) :
opens X ⥤ opens Y | { obj := λ U, ⟨f '' U, hf U U.2⟩,
map := λ U V h, ⟨⟨set.image_subset _ h.down.down⟩⟩ } | def | is_open_map.functor | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"Top",
"is_open_map"
] | An open map `f : X ⟶ Y` induces a functor `opens X ⥤ opens Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map.adjunction {X Y : Top} {f : X ⟶ Y} (hf : is_open_map f) :
adjunction hf.functor (topological_space.opens.map f) | adjunction.mk_of_unit_counit
{ unit := { app := λ U, hom_of_le $ λ x hxU, ⟨x, hxU, rfl⟩ },
counit := { app := λ V, hom_of_le $ λ y ⟨x, hfxV, hxy⟩, hxy ▸ hfxV } } | def | is_open_map.adjunction | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"Top",
"is_open_map",
"topological_space.opens.map"
] | An open map `f : X ⟶ Y` induces an adjunction between `opens X` and `opens Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map.functor_full_of_mono {X Y : Top} {f : X ⟶ Y} (hf : is_open_map f)
[H : mono f] : full hf.functor | { preimage := λ U V i, hom_of_le (λ x hx, by
{ obtain ⟨y, hy, eq⟩ := i.le ⟨x, hx, rfl⟩, exact (Top.mono_iff_injective f).mp H eq ▸ hy }) } | instance | is_open_map.functor_full_of_mono | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"Top",
"Top.mono_iff_injective",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map.functor_faithful {X Y : Top} {f : X ⟶ Y} (hf : is_open_map f) :
faithful hf.functor | {} | instance | is_open_map.functor_faithful | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"Top",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_obj_top {X : Top} (U : opens X) :
U.open_embedding.is_open_map.functor.obj ⊤ = U | by { ext1, exact set.image_univ.trans subtype.range_coe } | lemma | topological_space.opens.open_embedding_obj_top | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"Top",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_map_eq_top {X : Top} (U : opens X) :
(opens.map U.inclusion).obj U = ⊤ | by { ext1, exact subtype.coe_preimage_self _ } | lemma | topological_space.opens.inclusion_map_eq_top | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"Top",
"subtype.coe_preimage_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjunction_counit_app_self {X : Top} (U : opens X) :
U.open_embedding.is_open_map.adjunction.counit.app U = eq_to_hom (by simp) | by ext | lemma | topological_space.opens.adjunction_counit_app_self | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_top_functor (X : Top) :
(@opens.open_embedding X ⊤).is_open_map.functor =
map (inclusion_top_iso X).inv | begin
apply functor.hext, intro, abstract obj_eq { ext,
exact ⟨ λ ⟨⟨_,_⟩,h,rfl⟩, h, λ h, ⟨⟨x,trivial⟩,h,rfl⟩ ⟩ },
intros, apply subsingleton.helim, congr' 1,
iterate 2 {apply inclusion_top_functor.obj_eq},
end | lemma | topological_space.opens.inclusion_top_functor | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"Top",
"is_open_map.functor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_obj_map_obj {X Y : Top} {f : X ⟶ Y} (hf : is_open_map f) (U : opens Y) :
hf.functor.obj ((opens.map f).obj U) = hf.functor.obj ⊤ ⊓ U | begin
ext, split,
{ rintros ⟨x, hx, rfl⟩, exact ⟨⟨x, trivial, rfl⟩, hx⟩ },
{ rintros ⟨⟨x, -, rfl⟩, hx⟩, exact ⟨x, hx, rfl⟩ }
end | lemma | topological_space.opens.functor_obj_map_obj | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"Top",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_map_eq_inf {X : Top} (U V : opens X) :
U.open_embedding.is_open_map.functor.obj ((opens.map U.inclusion).obj V) = V ⊓ U | by { ext1, refine set.image_preimage_eq_inter_range.trans _, simpa } | lemma | topological_space.opens.functor_map_eq_inf | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_functor_eq' {X U : Top} (f : U ⟶ X) (hf : _root_.open_embedding f) (V) :
((opens.map f).obj $ hf.is_open_map.functor.obj V) = V | opens.ext $ set.preimage_image_eq _ hf.inj | lemma | topological_space.opens.map_functor_eq' | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"Top",
"set.preimage_image_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_functor_eq {X : Top} {U : opens X} (V : opens U) :
((opens.map U.inclusion).obj $ U.open_embedding.is_open_map.functor.obj V) = V | topological_space.opens.map_functor_eq' _ U.open_embedding V | lemma | topological_space.opens.map_functor_eq | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"Top",
"topological_space.opens.map_functor_eq'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjunction_counit_map_functor {X : Top} {U : opens X} (V : opens U) :
U.open_embedding.is_open_map.adjunction.counit.app (U.open_embedding.is_open_map.functor.obj V)
= eq_to_hom (by { conv_rhs { rw ← V.map_functor_eq }, refl }) | by ext | lemma | topological_space.opens.adjunction_counit_map_functor | topology.category.Top | src/topology/category/Top/opens.lean | [
"category_theory.category.preorder",
"category_theory.eq_to_hom",
"topology.category.Top.epi_mono",
"topology.sets.opens"
] | [
"Top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_nhds (x : X) | full_subcategory (λ (U : opens X), x ∈ U) | def | topological_space.open_nhds | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [] | The type of open neighbourhoods of a point `x` in a (bundled) topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_nhds_category (x : X) : category.{u} (open_nhds x) | by {unfold open_nhds, apply_instance} | instance | topological_space.open_nhds.open_nhds_category | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opens_nhds_hom_has_coe_to_fun {x : X} {U V : open_nhds x} :
has_coe_to_fun (U ⟶ V) (λ _, U.1 → V.1) | ⟨λ f x, ⟨x, f.le x.2⟩⟩ | instance | topological_space.open_nhds.opens_nhds_hom_has_coe_to_fun | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_le_left {x : X} (U V : open_nhds x) : U ⊓ V ⟶ U | hom_of_le inf_le_left | def | topological_space.open_nhds.inf_le_left | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [
"inf_le_left"
] | The inclusion `U ⊓ V ⟶ U` as a morphism in the category of open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_le_right {x : X} (U V : open_nhds x) : U ⊓ V ⟶ V | hom_of_le inf_le_right | def | topological_space.open_nhds.inf_le_right | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [
"inf_le_right"
] | The inclusion `U ⊓ V ⟶ V` as a morphism in the category of open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion (x : X) : open_nhds x ⥤ opens X | full_subcategory_inclusion _ | def | topological_space.open_nhds.inclusion | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [] | The inclusion functor from open neighbourhoods of `x`
to open sets in the ambient topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion_obj (x : X) (U) (p) : (inclusion x).obj ⟨U,p⟩ = U | rfl | lemma | topological_space.open_nhds.inclusion_obj | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding {x : X} (U : open_nhds x) : open_embedding (U.1.inclusion) | U.1.open_embedding | lemma | topological_space.open_nhds.open_embedding | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (x : X) : open_nhds (f x) ⥤ open_nhds x | { obj := λ U, ⟨(opens.map f).obj U.1, U.2⟩,
map := λ U V i, (opens.map f).map i } | def | topological_space.open_nhds.map | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [] | The preimage functor from neighborhoods of `f x` to neighborhoods of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_obj (x : X) (U) (q) : (map f x).obj ⟨U, q⟩ = ⟨(opens.map f).obj U, by tidy⟩ | rfl | lemma | topological_space.open_nhds.map_obj | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id_obj (x : X) (U) : (map (𝟙 X) x).obj U = U | by tidy | lemma | topological_space.open_nhds.map_id_obj | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id_obj' (x : X) (U) (p) (q) : (map (𝟙 X) x).obj ⟨⟨U, p⟩, q⟩ = ⟨⟨U, p⟩, q⟩ | rfl | lemma | topological_space.open_nhds.map_id_obj' | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id_obj_unop (x : X) (U : (open_nhds x)ᵒᵖ) : (map (𝟙 X) x).obj (unop U) = unop U | by simp | lemma | topological_space.open_nhds.map_id_obj_unop | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_map_id_obj (x : X) (U : (open_nhds x)ᵒᵖ) : (map (𝟙 X) x).op.obj U = U | by simp | lemma | topological_space.open_nhds.op_map_id_obj | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_map_iso (x : X) : inclusion (f x) ⋙ opens.map f ≅ map f x ⋙ inclusion x | nat_iso.of_components
(λ U, begin split, exact 𝟙 _, exact 𝟙 _ end)
(by tidy) | def | topological_space.open_nhds.inclusion_map_iso | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [] | `opens.map f` and `open_nhds.map f` form a commuting square (up to natural isomorphism)
with the inclusion functors into `opens X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion_map_iso_hom (x : X) : (inclusion_map_iso f x).hom = 𝟙 _ | rfl | lemma | topological_space.open_nhds.inclusion_map_iso_hom | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion_map_iso_inv (x : X) : (inclusion_map_iso f x).inv = 𝟙 _ | rfl | lemma | topological_space.open_nhds.inclusion_map_iso_inv | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_nhds (h : is_open_map f) (x : X) :
open_nhds x ⥤ open_nhds (f x) | { obj := λ U, ⟨h.functor.obj U.1, ⟨x, U.2, rfl⟩⟩,
map := λ U V i, h.functor.map i } | def | is_open_map.functor_nhds | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [
"is_open_map"
] | An open map `f : X ⟶ Y` induces a functor `open_nhds x ⥤ open_nhds (f x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjunction_nhds (h : is_open_map f) (x : X) :
is_open_map.functor_nhds h x ⊣ open_nhds.map f x | adjunction.mk_of_unit_counit
{ unit := { app := λ U, hom_of_le $ λ x hxU, ⟨x, hxU, rfl⟩ },
counit := { app := λ V, hom_of_le $ λ y ⟨x, hfxV, hxy⟩, hxy ▸ hfxV } } | def | is_open_map.adjunction_nhds | topology.category.Top | src/topology/category/Top/open_nhds.lean | [
"topology.category.Top.opens"
] | [
"is_open_map",
"is_open_map.functor_nhds"
] | An open map `f : X ⟶ Y` induces an adjunction between `open_nhds x` and `open_nhds (f x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone (F : J ⥤ Top.{max v u}) : cone F | { X := Top.of {u : Π j : J, F.obj j | ∀ {i j : J} (f : i ⟶ j), F.map f (u i) = u j},
π :=
{ app := λ j,
{ to_fun := λ u, u.val j,
continuous_to_fun := show continuous ((λ u : Π j : J, F.obj j, u j) ∘ subtype.val),
by continuity } } } | def | Top.limit_cone | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [
"Top.of",
"continuity",
"continuous"
] | A choice of limit cone for a functor `F : J ⥤ Top`.
Generally you should just use `limit.cone F`, unless you need the actual definition
(which is in terms of `types.limit_cone`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_infi (F : J ⥤ Top.{max v u}) : cone F | { X := ⟨(types.limit_cone (F ⋙ forget)).X, ⨅j,
(F.obj j).str.induced ((types.limit_cone (F ⋙ forget)).π.app j)⟩,
π :=
{ app := λ j, ⟨(types.limit_cone (F ⋙ forget)).π.app j,
continuous_iff_le_induced.mpr (infi_le _ _)⟩,
naturality' := λ j j' f, continuous_map.coe_injective
((types... | def | Top.limit_cone_infi | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [
"continuous_map.coe_injective",
"infi_le"
] | A choice of limit cone for a functor `F : J ⥤ Top` whose topology is defined as an
infimum of topologies infimum.
Generally you should just use `limit.cone F`, unless you need the actual definition
(which is in terms of `types.limit_cone`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_is_limit (F : J ⥤ Top.{max v u}) : is_limit (limit_cone F) | { lift := λ S, { to_fun := λ x, ⟨λ j, S.π.app _ x, λ i j f, by { dsimp, erw ← S.w f, refl }⟩ },
uniq' := λ S m h, by { ext : 3, simpa [← h] } } | def | Top.limit_cone_is_limit | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [
"lift"
] | The chosen cone `Top.limit_cone F` for a functor `F : J ⥤ Top` is a limit cone.
Generally you should just use `limit.is_limit F`, unless you need the actual definition
(which is in terms of `types.limit_cone_is_limit`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_infi_is_limit (F : J ⥤ Top.{max v u}) : is_limit (limit_cone_infi F) | by { refine is_limit.of_faithful forget (types.limit_cone_is_limit _) (λ s, ⟨_, _⟩) (λ s, rfl),
exact continuous_iff_coinduced_le.mpr (le_infi $ λ j,
coinduced_le_iff_le_induced.mp $ (continuous_iff_coinduced_le.mp (s.π.app j).continuous :
_) ) } | def | Top.limit_cone_infi_is_limit | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [
"continuous",
"le_infi"
] | The chosen cone `Top.limit_cone_infi F` for a functor `F : J ⥤ Top` is a limit cone.
Generally you should just use `limit.is_limit F`, unless you need the actual definition
(which is in terms of `types.limit_cone_is_limit`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Top_has_limits_of_size : has_limits_of_size.{v} Top.{max v u} | { has_limits_of_shape := λ J 𝒥, by exactI
{ has_limit := λ F, has_limit.mk { cone := limit_cone F, is_limit := limit_cone_is_limit F } } } | instance | Top.Top_has_limits_of_size | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Top_has_limits : has_limits Top.{u} | Top.Top_has_limits_of_size.{u u} | instance | Top.Top_has_limits | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget : Top.{max v u} ⥤ Type (max v u)) | { preserves_limits_of_shape := λ J 𝒥,
{ preserves_limit := λ F,
by exactI preserves_limit_of_preserves_limit_cone
(limit_cone_is_limit F) (types.limit_cone_is_limit (F ⋙ forget)) } } | instance | Top.forget_preserves_limits_of_size | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_limits : preserves_limits (forget : Top.{u} ⥤ Type u) | Top.forget_preserves_limits_of_size.{u u} | instance | Top.forget_preserves_limits | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_cocone (F : J ⥤ Top.{max v u}) : cocone F | { X := ⟨(types.colimit_cocone (F ⋙ forget)).X, ⨆ j,
(F.obj j).str.coinduced ((types.colimit_cocone (F ⋙ forget)).ι.app j)⟩,
ι :=
{ app := λ j, ⟨(types.colimit_cocone (F ⋙ forget)).ι.app j,
continuous_iff_coinduced_le.mpr (le_supr _ j)⟩,
naturality' := λ j j' f, continuous_map.coe_inject... | def | Top.colimit_cocone | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [
"continuous_map.coe_injective",
"le_supr"
] | A choice of colimit cocone for a functor `F : J ⥤ Top`.
Generally you should just use `colimit.coone F`, unless you need the actual definition
(which is in terms of `types.colimit_cocone`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_is_colimit (F : J ⥤ Top.{max v u}) : is_colimit (colimit_cocone F) | by { refine is_colimit.of_faithful forget (types.colimit_cocone_is_colimit _) (λ s, ⟨_, _⟩)
(λ s, rfl),
exact continuous_iff_le_induced.mpr (supr_le $ λ j,
coinduced_le_iff_le_induced.mp $ (continuous_iff_coinduced_le.mp (s.ι.app j).continuous :
_) ) } | def | Top.colimit_cocone_is_colimit | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [
"continuous",
"supr_le"
] | The chosen cocone `Top.colimit_cocone F` for a functor `F : J ⥤ Top` is a colimit cocone.
Generally you should just use `colimit.is_colimit F`, unless you need the actual definition
(which is in terms of `types.colimit_cocone_is_colimit`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Top_has_colimits_of_size : has_colimits_of_size.{v} Top.{max v u} | { has_colimits_of_shape := λ J 𝒥, by exactI
{ has_colimit := λ F, has_colimit.mk { cocone := colimit_cocone F, is_colimit :=
colimit_cocone_is_colimit F } } } | instance | Top.Top_has_colimits_of_size | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Top_has_colimits : has_colimits Top.{u} | Top.Top_has_colimits_of_size.{u u} | instance | Top.Top_has_colimits | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_colimits_of_size :
preserves_colimits_of_size.{v v} (forget : Top.{max v u} ⥤ Type (max v u)) | { preserves_colimits_of_shape := λ J 𝒥,
{ preserves_colimit := λ F,
by exactI preserves_colimit_of_preserves_colimit_cocone
(colimit_cocone_is_colimit F) (types.colimit_cocone_is_colimit (F ⋙ forget)) } } | instance | Top.forget_preserves_colimits_of_size | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_colimits : preserves_colimits (forget : Top.{u} ⥤ Type u) | Top.forget_preserves_colimits_of_size.{u u} | instance | Top.forget_preserves_colimits | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_terminal_punit : is_terminal (Top.of punit.{u+1}) | begin
haveI : ∀ X, unique (X ⟶ Top.of punit.{u+1}) :=
λ X, ⟨⟨⟨λ x, punit.star, by continuity⟩⟩, λ f, by ext⟩,
exact limits.is_terminal.of_unique _,
end | def | Top.is_terminal_punit | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [
"Top.of",
"unique"
] | The terminal object of `Top` is `punit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
terminal_iso_punit : ⊤_ Top.{u} ≅ Top.of punit | terminal_is_terminal.unique_up_to_iso is_terminal_punit | def | Top.terminal_iso_punit | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [
"Top.of"
] | The terminal object of `Top` is `punit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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