Loogle!

Result

Found 24 declarations mentioning CategoryTheory.Limits.biproduct.map.

• CategoryTheory.Limits.biproduct.map 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (b : J) → f b ⟶ g b) : ⨁ f ⟶ ⨁ g
• CategoryTheory.Limits.biproduct.map_epi 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (j : J) → f j ⟶ g j) [∀ (j : J), CategoryTheory.Epi (p j)] : CategoryTheory.Epi (CategoryTheory.Limits.biproduct.map p)
• CategoryTheory.Limits.biproduct.map_mono 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (j : J) → f j ⟶ g j) [∀ (j : J), CategoryTheory.Mono (p j)] : CategoryTheory.Mono (CategoryTheory.Limits.biproduct.map p)
• CategoryTheory.Limits.biproduct.map_eq_map' 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (b : J) → f b ⟶ g b) : CategoryTheory.Limits.biproduct.map p = CategoryTheory.Limits.biproduct.map' p
• CategoryTheory.Limits.biproduct.mapIso_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (b : J) → f b ≅ g b) : (CategoryTheory.Limits.biproduct.mapIso p).hom = CategoryTheory.Limits.biproduct.map fun b => (p b).hom
• CategoryTheory.Limits.biproduct.mapIso_inv 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (b : J) → f b ≅ g b) : (CategoryTheory.Limits.biproduct.mapIso p).inv = CategoryTheory.Limits.biproduct.map fun b => (p b).inv
• CategoryTheory.Limits.biproduct.map_π 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (j : J) → f j ⟶ g j) (j : J) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map p) (CategoryTheory.Limits.biproduct.π g j) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f j) (p j)
• CategoryTheory.Limits.biproduct.ι_map 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (j : J) → f j ⟶ g j) (j : J) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.Limits.biproduct.map p) = CategoryTheory.CategoryStruct.comp (p j) (CategoryTheory.Limits.biproduct.ι g j)
• CategoryTheory.Limits.biproduct.lift_map 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] {P : C} (k : (j : J) → P ⟶ f j) (p : (j : J) → f j ⟶ g j) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.lift k) (CategoryTheory.Limits.biproduct.map p) = CategoryTheory.Limits.biproduct.lift fun j => CategoryTheory.CategoryStruct.comp (k j) (p j)
• CategoryTheory.Limits.biproduct.map_desc 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (j : J) → f j ⟶ g j) {P : C} (k : (j : J) → g j ⟶ P) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map p) (CategoryTheory.Limits.biproduct.desc k) = CategoryTheory.Limits.biproduct.desc fun j => CategoryTheory.CategoryStruct.comp (p j) (k j)
• CategoryTheory.Limits.biproduct.map_π_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (j : J) → f j ⟶ g j) (j : J) {Z : C} (h : g j ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map p) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π g j) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f j) (CategoryTheory.CategoryStruct.comp (p j) h)
• CategoryTheory.Limits.biproduct.ι_map_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (j : J) → f j ⟶ g j) (j : J) {Z : C} (h : ⨁ g ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι f j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map p) h) = CategoryTheory.CategoryStruct.comp (p j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι g j) h)
• CategoryTheory.Limits.biproduct.lift_map_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] {P : C} (k : (j : J) → P ⟶ f j) (p : (j : J) → f j ⟶ g j) {Z : C} (h : ⨁ g ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.lift k) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map p) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.lift fun j => CategoryTheory.CategoryStruct.comp (k j) (p j)) h
• CategoryTheory.Limits.biproduct.map_desc_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {f g : J → C} [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct g] (p : (j : J) → f j ⟶ g j) {P : C} (k : (j : J) → g j ⟶ P) {Z : C} (h : P ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map p) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.desc k) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.desc fun j => CategoryTheory.CategoryStruct.comp (p j) (k j)) h
• CategoryTheory.Limits.biproduct.toSubtype_fromSubtype 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] [DecidablePred p] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.toSubtype f p) (CategoryTheory.Limits.biproduct.fromSubtype f p) = CategoryTheory.Limits.biproduct.map fun j => if p j then CategoryTheory.CategoryStruct.id (f j) else 0
• CategoryTheory.Limits.biproduct.toSubtype_fromSubtype_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) [CategoryTheory.Limits.HasBiproduct f] (p : J → Prop) [CategoryTheory.Limits.HasBiproduct (Subtype.restrict p f)] [DecidablePred p] {Z : C} (h : ⨁ f ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.toSubtype f p) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.fromSubtype f p) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map fun j => if p j then CategoryTheory.CategoryStruct.id (f j) else 0) h
• CategoryTheory.Limits.biproduct.map_matrix 📋 Mathlib.CategoryTheory.Preadditive.Biproducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {J K : Type} [Finite J] [CategoryTheory.Limits.HasFiniteBiproducts C] [Finite K] {f g : J → C} {h : K → C} (m : (k : J) → f k ⟶ g k) (n : (j : J) → (k : K) → g j ⟶ h k) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map m) (CategoryTheory.Limits.biproduct.matrix n) = CategoryTheory.Limits.biproduct.matrix fun j k => CategoryTheory.CategoryStruct.comp (m j) (n j k)
• CategoryTheory.Limits.biproduct.matrix_map 📋 Mathlib.CategoryTheory.Preadditive.Biproducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {J K : Type} [Finite J] [CategoryTheory.Limits.HasFiniteBiproducts C] [Finite K] {f : J → C} {g h : K → C} (m : (j : J) → (k : K) → f j ⟶ g k) (n : (k : K) → g k ⟶ h k) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.matrix m) (CategoryTheory.Limits.biproduct.map n) = CategoryTheory.Limits.biproduct.matrix fun j k => CategoryTheory.CategoryStruct.comp (m j k) (n k)
• CategoryTheory.Limits.biproduct.map_matrix_assoc 📋 Mathlib.CategoryTheory.Preadditive.Biproducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {J K : Type} [Finite J] [CategoryTheory.Limits.HasFiniteBiproducts C] [Finite K] {f g : J → C} {h : K → C} (m : (k : J) → f k ⟶ g k) (n : (j : J) → (k : K) → g j ⟶ h k) {Z : C} (h✝ : ⨁ h ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map m) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.matrix n) h✝) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.matrix fun j k => CategoryTheory.CategoryStruct.comp (m j) (n j k)) h✝
• CategoryTheory.Limits.biproduct.matrix_map_assoc 📋 Mathlib.CategoryTheory.Preadditive.Biproducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {J K : Type} [Finite J] [CategoryTheory.Limits.HasFiniteBiproducts C] [Finite K] {f : J → C} {g h : K → C} (m : (j : J) → (k : K) → f j ⟶ g k) (n : (k : K) → g k ⟶ h k) {Z : C} (h✝ : ⨁ h ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.matrix m) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map n) h✝) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.matrix fun j k => CategoryTheory.CategoryStruct.comp (m j k) (n k)) h✝
• CategoryTheory.Limits.biproduct.map_eq 📋 Mathlib.CategoryTheory.Preadditive.Biproducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {J : Type} [Fintype J] [CategoryTheory.Limits.HasFiniteBiproducts C] {f g : J → C} {h : (j : J) → f j ⟶ g j} : CategoryTheory.Limits.biproduct.map h = ∑ j, CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.π f j) (CategoryTheory.CategoryStruct.comp (h j) (CategoryTheory.Limits.biproduct.ι g j))
• CategoryTheory.Idempotents.Karoubi.Biproducts.bicone_pt_p 📋 Mathlib.CategoryTheory.Idempotents.Biproducts
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteBiproducts C] {J : Type} [Finite J] (F : J → CategoryTheory.Idempotents.Karoubi C) : (CategoryTheory.Idempotents.Karoubi.Biproducts.bicone F).pt.p = CategoryTheory.Limits.biproduct.map fun j => (F j).p
• CategoryTheory.Idempotents.Karoubi.Biproducts.bicone_ι_f 📋 Mathlib.CategoryTheory.Idempotents.Biproducts
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteBiproducts C] {J : Type} [Finite J] (F : J → CategoryTheory.Idempotents.Karoubi C) (j : J) : ((CategoryTheory.Idempotents.Karoubi.Biproducts.bicone F).ι j).f = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.ι (fun j => (F j).X) j) (CategoryTheory.Limits.biproduct.map fun j => (F j).p)
• CategoryTheory.Idempotents.Karoubi.Biproducts.bicone_π_f 📋 Mathlib.CategoryTheory.Idempotents.Biproducts
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteBiproducts C] {J : Type} [Finite J] (F : J → CategoryTheory.Idempotents.Karoubi C) (j : J) : ((CategoryTheory.Idempotents.Karoubi.Biproducts.bicone F).π j).f = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biproduct.map fun j => (F j).p) ((CategoryTheory.Limits.biproduct.bicone fun j => (F j).X).π j)

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