Loogle!

Result

Found 11 declarations mentioning CategoryTheory.Subgroupoid.map.

• CategoryTheory.Subgroupoid.map 📋 Mathlib.CategoryTheory.Groupoid.Subgroupoid
  {C : Type u} [CategoryTheory.Groupoid C] {D : Type u_1} [CategoryTheory.Groupoid D] (φ : CategoryTheory.Functor C D) (hφ : Function.Injective φ.obj) (S : CategoryTheory.Subgroupoid C) : CategoryTheory.Subgroupoid D
• CategoryTheory.Subgroupoid.im.eq_1 📋 Mathlib.CategoryTheory.Groupoid.Subgroupoid
  {C : Type u} [CategoryTheory.Groupoid C] {D : Type u_1} [CategoryTheory.Groupoid D] (φ : CategoryTheory.Functor C D) (hφ : Function.Injective φ.obj) : CategoryTheory.Subgroupoid.im φ hφ = CategoryTheory.Subgroupoid.map φ hφ ⊤
• CategoryTheory.Subgroupoid.isNormal_map 📋 Mathlib.CategoryTheory.Groupoid.Subgroupoid
  {C : Type u} [CategoryTheory.Groupoid C] (S : CategoryTheory.Subgroupoid C) {D : Type u_1} [CategoryTheory.Groupoid D] (φ : CategoryTheory.Functor C D) (hφ : Function.Injective φ.obj) (hφ' : CategoryTheory.Subgroupoid.im φ hφ = ⊤) (Sn : S.IsNormal) : (CategoryTheory.Subgroupoid.map φ hφ S).IsNormal
• CategoryTheory.Subgroupoid.le_comap_map 📋 Mathlib.CategoryTheory.Groupoid.Subgroupoid
  {C : Type u} [CategoryTheory.Groupoid C] {D : Type u_1} [CategoryTheory.Groupoid D] (φ : CategoryTheory.Functor C D) (hφ : Function.Injective φ.obj) (S : CategoryTheory.Subgroupoid C) : S ≤ CategoryTheory.Subgroupoid.comap φ (CategoryTheory.Subgroupoid.map φ hφ S)
• CategoryTheory.Subgroupoid.map_comap_le 📋 Mathlib.CategoryTheory.Groupoid.Subgroupoid
  {C : Type u} [CategoryTheory.Groupoid C] {D : Type u_1} [CategoryTheory.Groupoid D] (φ : CategoryTheory.Functor C D) (hφ : Function.Injective φ.obj) (T : CategoryTheory.Subgroupoid D) : CategoryTheory.Subgroupoid.map φ hφ (CategoryTheory.Subgroupoid.comap φ T) ≤ T
• CategoryTheory.Subgroupoid.galoisConnection_map_comap 📋 Mathlib.CategoryTheory.Groupoid.Subgroupoid
  {C : Type u} [CategoryTheory.Groupoid C] {D : Type u_1} [CategoryTheory.Groupoid D] (φ : CategoryTheory.Functor C D) (hφ : Function.Injective φ.obj) : GaloisConnection (CategoryTheory.Subgroupoid.map φ hφ) (CategoryTheory.Subgroupoid.comap φ)
• CategoryTheory.Subgroupoid.map_objs_eq 📋 Mathlib.CategoryTheory.Groupoid.Subgroupoid
  {C : Type u} [CategoryTheory.Groupoid C] (S : CategoryTheory.Subgroupoid C) {D : Type u_1} [CategoryTheory.Groupoid D] (φ : CategoryTheory.Functor C D) (hφ : Function.Injective φ.obj) : (CategoryTheory.Subgroupoid.map φ hφ S).objs = φ.obj '' S.objs
• CategoryTheory.Subgroupoid.mem_map_objs_iff 📋 Mathlib.CategoryTheory.Groupoid.Subgroupoid
  {C : Type u} [CategoryTheory.Groupoid C] (S : CategoryTheory.Subgroupoid C) {D : Type u_1} [CategoryTheory.Groupoid D] (φ : CategoryTheory.Functor C D) (hφ : Function.Injective φ.obj) (d : D) : d ∈ (CategoryTheory.Subgroupoid.map φ hφ S).objs ↔ ∃ c ∈ S.objs, φ.obj c = d
• CategoryTheory.Subgroupoid.map_le_iff_le_comap 📋 Mathlib.CategoryTheory.Groupoid.Subgroupoid
  {C : Type u} [CategoryTheory.Groupoid C] {D : Type u_1} [CategoryTheory.Groupoid D] (φ : CategoryTheory.Functor C D) (hφ : Function.Injective φ.obj) (S : CategoryTheory.Subgroupoid C) (T : CategoryTheory.Subgroupoid D) : CategoryTheory.Subgroupoid.map φ hφ S ≤ T ↔ S ≤ CategoryTheory.Subgroupoid.comap φ T
• CategoryTheory.Subgroupoid.map_mono 📋 Mathlib.CategoryTheory.Groupoid.Subgroupoid
  {C : Type u} [CategoryTheory.Groupoid C] {D : Type u_1} [CategoryTheory.Groupoid D] (φ : CategoryTheory.Functor C D) (hφ : Function.Injective φ.obj) (S T : CategoryTheory.Subgroupoid C) : S ≤ T → CategoryTheory.Subgroupoid.map φ hφ S ≤ CategoryTheory.Subgroupoid.map φ hφ T
• CategoryTheory.Subgroupoid.mem_map_iff 📋 Mathlib.CategoryTheory.Groupoid.Subgroupoid
  {C : Type u} [CategoryTheory.Groupoid C] {D : Type u_1} [CategoryTheory.Groupoid D] (φ : CategoryTheory.Functor C D) (hφ : Function.Injective φ.obj) (S : CategoryTheory.Subgroupoid C) {c d : D} (f : c ⟶ d) : f ∈ (CategoryTheory.Subgroupoid.map φ hφ S).arrows c d ↔ ∃ a b g, ∃ (ha : φ.obj a = c) (hb : φ.obj b = d) (_ : g ∈ S.arrows a b), f = CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom ⋯) (CategoryTheory.CategoryStruct.comp (φ.map g) (CategoryTheory.eqToHom hb))

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