Loogle!

Result

Found 14 declarations mentioning TopologicalSpace.OpenNhds.map.

• TopologicalSpace.OpenNhds.map 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X Y : TopCat} (f : X ⟶ Y) (x : ↑X) : CategoryTheory.Functor (TopologicalSpace.OpenNhds ((CategoryTheory.ConcreteCategory.hom f) x)) (TopologicalSpace.OpenNhds x)
• IsOpenMap.adjunctionNhds 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X Y : TopCat} {f : X ⟶ Y} (h : IsOpenMap ⇑(CategoryTheory.ConcreteCategory.hom f)) (x : ↑X) : h.functorNhds x ⊣ TopologicalSpace.OpenNhds.map f x
• Topology.IsInducing.adjunctionNhds 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X Y : TopCat} {f : X ⟶ Y} (h : Topology.IsInducing ⇑(CategoryTheory.ConcreteCategory.hom f)) (x : ↑X) : TopologicalSpace.OpenNhds.map f x ⊣ h.functorNhds x
• TopologicalSpace.OpenNhds.map_id_obj 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X : TopCat} (x : ↑X) (U : TopologicalSpace.OpenNhds ((CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.id X)) x)) : (TopologicalSpace.OpenNhds.map (CategoryTheory.CategoryStruct.id X) x).obj U = U
• TopologicalSpace.OpenNhds.map_id_obj_unop 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X : TopCat} (x : ↑X) (U : (TopologicalSpace.OpenNhds x)ᵒᵖ) : (TopologicalSpace.OpenNhds.map (CategoryTheory.CategoryStruct.id X) x).obj (Opposite.unop U) = Opposite.unop U
• TopologicalSpace.OpenNhds.map_id_obj' 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X : TopCat} (x : ↑X) (U : Set ↑X) (p : IsOpen U) (q : (CategoryTheory.ConcreteCategory.hom (CategoryTheory.CategoryStruct.id X)) x ∈ { carrier := U, is_open' := p }) : (TopologicalSpace.OpenNhds.map (CategoryTheory.CategoryStruct.id X) x).obj { obj := { carrier := U, is_open' := p }, property := q } = { obj := { carrier := U, is_open' := p }, property := q }
• TopologicalSpace.OpenNhds.op_map_id_obj 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X : TopCat} (x : ↑X) (U : (TopologicalSpace.OpenNhds x)ᵒᵖ) : (TopologicalSpace.OpenNhds.map (CategoryTheory.CategoryStruct.id X) x).op.obj U = U
• TopologicalSpace.OpenNhds.inclusionMapIso 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X Y : TopCat} (f : X ⟶ Y) (x : ↑X) : (TopologicalSpace.OpenNhds.inclusion ((CategoryTheory.ConcreteCategory.hom f) x)).comp (TopologicalSpace.Opens.map f) ≅ (TopologicalSpace.OpenNhds.map f x).comp (TopologicalSpace.OpenNhds.inclusion x)
• TopologicalSpace.OpenNhds.map_obj 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X Y : TopCat} (f : X ⟶ Y) (x : ↑X) (U : TopologicalSpace.Opens ↑Y) (q : (CategoryTheory.ConcreteCategory.hom f) x ∈ U) : (TopologicalSpace.OpenNhds.map f x).obj { obj := U, property := q } = { obj := (TopologicalSpace.Opens.map f).obj U, property := q }
• TopologicalSpace.OpenNhds.inclusionMapIso_inv 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X Y : TopCat} (f : X ⟶ Y) (x : ↑X) : (TopologicalSpace.OpenNhds.inclusionMapIso f x).inv = CategoryTheory.CategoryStruct.id ((TopologicalSpace.OpenNhds.map f x).comp (TopologicalSpace.OpenNhds.inclusion x))
• TopologicalSpace.OpenNhds.inclusionMapIso_hom 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X Y : TopCat} (f : X ⟶ Y) (x : ↑X) : (TopologicalSpace.OpenNhds.inclusionMapIso f x).hom = CategoryTheory.CategoryStruct.id ((TopologicalSpace.OpenNhds.inclusion ((CategoryTheory.ConcreteCategory.hom f) x)).comp (TopologicalSpace.Opens.map f))
• TopologicalSpace.OpenNhds.inclusionMapIso_inv_app 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X Y : TopCat} (f : X ⟶ Y) (x : ↑X) (X✝ : TopologicalSpace.OpenNhds ((CategoryTheory.ConcreteCategory.hom f) x)) : (TopologicalSpace.OpenNhds.inclusionMapIso f x).inv.app X✝ = CategoryTheory.CategoryStruct.id ((TopologicalSpace.OpenNhds.inclusion x).obj ((TopologicalSpace.OpenNhds.map f x).obj X✝))
• TopologicalSpace.OpenNhds.inclusionMapIso_hom_app 📋 Mathlib.Topology.Category.TopCat.OpenNhds
  {X Y : TopCat} (f : X ⟶ Y) (x : ↑X) (X✝ : TopologicalSpace.OpenNhds ((CategoryTheory.ConcreteCategory.hom f) x)) : (TopologicalSpace.OpenNhds.inclusionMapIso f x).hom.app X✝ = CategoryTheory.CategoryStruct.id ((TopologicalSpace.Opens.map f).obj ((TopologicalSpace.OpenNhds.inclusion ((CategoryTheory.ConcreteCategory.hom f) x)).obj X✝))
• TopCat.Presheaf.stalkPushforward.eq_1 📋 Mathlib.Topology.Sheaves.Stalks
  (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimits C] {X Y : TopCat} (f : X ⟶ Y) (F : TopCat.Presheaf C X) (x : ↑X) : TopCat.Presheaf.stalkPushforward C f F x = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colim.map (CategoryTheory.whiskerRight (CategoryTheory.NatTrans.op (TopologicalSpace.OpenNhds.inclusionMapIso f x).inv) F)) (CategoryTheory.Limits.colimit.pre (((CategoryTheory.whiskeringLeft (TopologicalSpace.OpenNhds x)ᵒᵖ (TopologicalSpace.Opens ↑X)ᵒᵖ C).obj (TopologicalSpace.OpenNhds.inclusion x).op).obj F) (TopologicalSpace.OpenNhds.map f x).op)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 19971e9 serving mathlib revision 40fea08