Loogle!

Result

Found 10 declarations mentioning CategoryTheory.Limits.image.map.

• CategoryTheory.Limits.im_map 📋 Mathlib.CategoryTheory.Limits.Shapes.Images
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasImages C] [CategoryTheory.Limits.HasImageMaps C] {X✝ Y✝ : CategoryTheory.Arrow C} (st : X✝ ⟶ Y✝) : CategoryTheory.Limits.im.map st = CategoryTheory.Limits.image.map st
• CategoryTheory.Limits.image.map 📋 Mathlib.CategoryTheory.Limits.Shapes.Images
  {C : Type u} [CategoryTheory.Category.{v, u} C] {f g : CategoryTheory.Arrow C} [CategoryTheory.Limits.HasImage f.hom] [CategoryTheory.Limits.HasImage g.hom] (sq : f ⟶ g) [CategoryTheory.Limits.HasImageMap sq] : CategoryTheory.Limits.image f.hom ⟶ CategoryTheory.Limits.image g.hom
• CategoryTheory.Limits.image.map_id 📋 Mathlib.CategoryTheory.Limits.Shapes.Images
  {C : Type u} [CategoryTheory.Category.{v, u} C] (f : CategoryTheory.Arrow C) [CategoryTheory.Limits.HasImage f.hom] [CategoryTheory.Limits.HasImageMap (CategoryTheory.CategoryStruct.id f)] : CategoryTheory.Limits.image.map (CategoryTheory.CategoryStruct.id f) = CategoryTheory.CategoryStruct.id (CategoryTheory.Limits.image f.hom)
• CategoryTheory.Limits.image.map_homMk'_ι 📋 Mathlib.CategoryTheory.Limits.Shapes.Images
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y P Q : C} {k : X ⟶ Y} [CategoryTheory.Limits.HasImage k] {l : P ⟶ Q} [CategoryTheory.Limits.HasImage l] {m : X ⟶ P} {n : Y ⟶ Q} (w : CategoryTheory.CategoryStruct.comp m l = CategoryTheory.CategoryStruct.comp k n) [CategoryTheory.Limits.HasImageMap (CategoryTheory.Arrow.homMk' m n w)] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.image.map (CategoryTheory.Arrow.homMk' m n w)) (CategoryTheory.Limits.image.ι l) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.image.ι k) n
• CategoryTheory.Limits.image.map_comp 📋 Mathlib.CategoryTheory.Limits.Shapes.Images
  {C : Type u} [CategoryTheory.Category.{v, u} C] {f g : CategoryTheory.Arrow C} [CategoryTheory.Limits.HasImage f.hom] [CategoryTheory.Limits.HasImage g.hom] (sq : f ⟶ g) [CategoryTheory.Limits.HasImageMap sq] {h : CategoryTheory.Arrow C} [CategoryTheory.Limits.HasImage h.hom] (sq' : g ⟶ h) [CategoryTheory.Limits.HasImageMap sq'] [CategoryTheory.Limits.HasImageMap (CategoryTheory.CategoryStruct.comp sq sq')] : CategoryTheory.Limits.image.map (CategoryTheory.CategoryStruct.comp sq sq') = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.image.map sq) (CategoryTheory.Limits.image.map sq')
• CategoryTheory.Limits.image.factor_map 📋 Mathlib.CategoryTheory.Limits.Shapes.Images
  {C : Type u} [CategoryTheory.Category.{v, u} C] {f g : CategoryTheory.Arrow C} [CategoryTheory.Limits.HasImage f.hom] [CategoryTheory.Limits.HasImage g.hom] (sq : f ⟶ g) [CategoryTheory.Limits.HasImageMap sq] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.factorThruImage f.hom) (CategoryTheory.Limits.image.map sq) = CategoryTheory.CategoryStruct.comp sq.left (CategoryTheory.Limits.factorThruImage g.hom)
• CategoryTheory.Limits.image.map_ι 📋 Mathlib.CategoryTheory.Limits.Shapes.Images
  {C : Type u} [CategoryTheory.Category.{v, u} C] {f g : CategoryTheory.Arrow C} [CategoryTheory.Limits.HasImage f.hom] [CategoryTheory.Limits.HasImage g.hom] (sq : f ⟶ g) [CategoryTheory.Limits.HasImageMap sq] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.image.map sq) (CategoryTheory.Limits.image.ι g.hom) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.image.ι f.hom) sq.right
• CategoryTheory.Limits.imageSubobjectMap.eq_1 📋 Mathlib.CategoryTheory.Subobject.Limits
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y Z : C} {f : W ⟶ X} [CategoryTheory.Limits.HasImage f] {g : Y ⟶ Z} [CategoryTheory.Limits.HasImage g] (sq : CategoryTheory.Arrow.mk f ⟶ CategoryTheory.Arrow.mk g) [CategoryTheory.Limits.HasImageMap sq] : CategoryTheory.Limits.imageSubobjectMap sq = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.imageSubobjectIso f).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.image.map sq) (CategoryTheory.Limits.imageSubobjectIso g).inv)
• CategoryTheory.Limits.imageSubobjectIso_comp_image_map 📋 Mathlib.CategoryTheory.Subobject.Limits
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y Z : C} {f : W ⟶ X} [CategoryTheory.Limits.HasImage f] {g : Y ⟶ Z} [CategoryTheory.Limits.HasImage g] (sq : CategoryTheory.Arrow.mk f ⟶ CategoryTheory.Arrow.mk g) [CategoryTheory.Limits.HasImageMap sq] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.imageSubobjectIso (CategoryTheory.Arrow.mk f).hom).hom (CategoryTheory.Limits.image.map sq) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.imageSubobjectMap sq) (CategoryTheory.Limits.imageSubobjectIso g).hom
• CategoryTheory.Limits.image_map_comp_imageSubobjectIso_inv 📋 Mathlib.CategoryTheory.Subobject.Limits
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y Z : C} {f : W ⟶ X} [CategoryTheory.Limits.HasImage f] {g : Y ⟶ Z} [CategoryTheory.Limits.HasImage g] (sq : CategoryTheory.Arrow.mk f ⟶ CategoryTheory.Arrow.mk g) [CategoryTheory.Limits.HasImageMap sq] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.image.map sq) (CategoryTheory.Limits.imageSubobjectIso (CategoryTheory.Arrow.mk g).hom).inv = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.imageSubobjectIso f).inv (CategoryTheory.Limits.imageSubobjectMap sq)

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