Loogle!

Result

Found 8 declarations mentioning CategoryTheory.Limits.end_.map.

• CategoryTheory.Limits.end_.map_id 📋 Mathlib.CategoryTheory.Limits.Shapes.End
  {J : Type u} [CategoryTheory.Category.{v, u} J] {C : Type u'} [CategoryTheory.Category.{v', u'} C] {F : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} [CategoryTheory.Limits.HasEnd F] : CategoryTheory.Limits.end_.map (CategoryTheory.CategoryStruct.id F) = CategoryTheory.CategoryStruct.id (CategoryTheory.Limits.end_ F)
• CategoryTheory.Limits.end_.map 📋 Mathlib.CategoryTheory.Limits.Shapes.End
  {J : Type u} [CategoryTheory.Category.{v, u} J] {C : Type u'} [CategoryTheory.Category.{v', u'} C] {F : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} [CategoryTheory.Limits.HasEnd F] {F' : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} [CategoryTheory.Limits.HasEnd F'] (f : F ⟶ F') : CategoryTheory.Limits.end_ F ⟶ CategoryTheory.Limits.end_ F'
• CategoryTheory.Limits.endFunctor_map 📋 Mathlib.CategoryTheory.Limits.Shapes.End
  (J : Type u) [CategoryTheory.Category.{v, u} J] (C : Type u') [CategoryTheory.Category.{v', u'} C] [∀ (F : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)), CategoryTheory.Limits.HasEnd F] {X✝ Y✝ : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} (f : X✝ ⟶ Y✝) : (CategoryTheory.Limits.endFunctor J C).map f = CategoryTheory.Limits.end_.map f
• CategoryTheory.Limits.end_.map_comp 📋 Mathlib.CategoryTheory.Limits.Shapes.End
  {J : Type u} [CategoryTheory.Category.{v, u} J] {C : Type u'} [CategoryTheory.Category.{v', u'} C] {F : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} [CategoryTheory.Limits.HasEnd F] {F' : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} [CategoryTheory.Limits.HasEnd F'] (f : F ⟶ F') {F'' : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} [CategoryTheory.Limits.HasEnd F''] (g : F' ⟶ F'') : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.end_.map f) (CategoryTheory.Limits.end_.map g) = CategoryTheory.Limits.end_.map (CategoryTheory.CategoryStruct.comp f g)
• CategoryTheory.Limits.end_.map_comp_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.End
  {J : Type u} [CategoryTheory.Category.{v, u} J] {C : Type u'} [CategoryTheory.Category.{v', u'} C] {F : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} [CategoryTheory.Limits.HasEnd F] {F' : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} [CategoryTheory.Limits.HasEnd F'] (f : F ⟶ F') {F'' : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} [CategoryTheory.Limits.HasEnd F''] (g : F' ⟶ F'') {Z : C} (h : CategoryTheory.Limits.end_ F'' ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.end_.map f) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.end_.map g) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.end_.map (CategoryTheory.CategoryStruct.comp f g)) h
• CategoryTheory.Limits.end_.map.eq_1 📋 Mathlib.CategoryTheory.Limits.Shapes.End
  {J : Type u} [CategoryTheory.Category.{v, u} J] {C : Type u'} [CategoryTheory.Category.{v', u'} C] {F : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} [CategoryTheory.Limits.HasEnd F] {F' : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} [CategoryTheory.Limits.HasEnd F'] (f : F ⟶ F') : CategoryTheory.Limits.end_.map f = CategoryTheory.Limits.end_.lift (fun x => CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.end_.π F x) ((f.app (Opposite.op x)).app x)) ⋯
• CategoryTheory.Limits.end_.map_π 📋 Mathlib.CategoryTheory.Limits.Shapes.End
  {J : Type u} [CategoryTheory.Category.{v, u} J] {C : Type u'} [CategoryTheory.Category.{v', u'} C] {F : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} [CategoryTheory.Limits.HasEnd F] {F' : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} [CategoryTheory.Limits.HasEnd F'] (f : F ⟶ F') (j : J) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.end_.map f) (CategoryTheory.Limits.end_.π F' j) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.end_.π F j) ((f.app (Opposite.op j)).app j)
• CategoryTheory.Limits.end_.map_π_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.End
  {J : Type u} [CategoryTheory.Category.{v, u} J] {C : Type u'} [CategoryTheory.Category.{v', u'} C] {F : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} [CategoryTheory.Limits.HasEnd F] {F' : CategoryTheory.Functor Jᵒᵖ (CategoryTheory.Functor J C)} [CategoryTheory.Limits.HasEnd F'] (f : F ⟶ F') (j : J) {Z : C} (h : (F'.obj (Opposite.op j)).obj j ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.end_.map f) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.end_.π F' j) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.end_.π F j) (CategoryTheory.CategoryStruct.comp ((f.app (Opposite.op j)).app j) h)

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 ff04530 serving mathlib revision 8623f65