Loogle!

Result

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

• CategoryTheory.Limits.pushout.map 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y Z S T : C} (f₁ : S ⟶ W) (f₂ : S ⟶ X) [CategoryTheory.Limits.HasPushout f₁ f₂] (g₁ : T ⟶ Y) (g₂ : T ⟶ Z) [CategoryTheory.Limits.HasPushout g₁ g₂] (i₁ : W ⟶ Y) (i₂ : X ⟶ Z) (i₃ : S ⟶ T) (eq₁ : CategoryTheory.CategoryStruct.comp f₁ i₁ = CategoryTheory.CategoryStruct.comp i₃ g₁) (eq₂ : CategoryTheory.CategoryStruct.comp f₂ i₂ = CategoryTheory.CategoryStruct.comp i₃ g₂) : CategoryTheory.Limits.pushout f₁ f₂ ⟶ CategoryTheory.Limits.pushout g₁ g₂
• CategoryTheory.Limits.pushout.congrHom_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f₁ f₂ : X ⟶ Y} {g₁ g₂ : X ⟶ Z} (h₁ : f₁ = f₂) (h₂ : g₁ = g₂) [CategoryTheory.Limits.HasPushout f₁ g₁] [CategoryTheory.Limits.HasPushout f₂ g₂] : (CategoryTheory.Limits.pushout.congrHom h₁ h₂).hom = CategoryTheory.Limits.pushout.map f₁ g₁ f₂ g₂ (CategoryTheory.CategoryStruct.id Y) (CategoryTheory.CategoryStruct.id Z) (CategoryTheory.CategoryStruct.id X) ⋯ ⋯
• CategoryTheory.Limits.pushout.map_isIso 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y Z S T : C} (f₁ : S ⟶ W) (f₂ : S ⟶ X) [CategoryTheory.Limits.HasPushout f₁ f₂] (g₁ : T ⟶ Y) (g₂ : T ⟶ Z) [CategoryTheory.Limits.HasPushout g₁ g₂] (i₁ : W ⟶ Y) (i₂ : X ⟶ Z) (i₃ : S ⟶ T) (eq₁ : CategoryTheory.CategoryStruct.comp f₁ i₁ = CategoryTheory.CategoryStruct.comp i₃ g₁) (eq₂ : CategoryTheory.CategoryStruct.comp f₂ i₂ = CategoryTheory.CategoryStruct.comp i₃ g₂) [CategoryTheory.IsIso i₁] [CategoryTheory.IsIso i₂] [CategoryTheory.IsIso i₃] : CategoryTheory.IsIso (CategoryTheory.Limits.pushout.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂)
• CategoryTheory.Limits.pushout.map_id 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} [CategoryTheory.Limits.HasPushout f g] : CategoryTheory.Limits.pushout.map f g f g (CategoryTheory.CategoryStruct.id Y) (CategoryTheory.CategoryStruct.id Z) (CategoryTheory.CategoryStruct.id X) ⋯ ⋯ = CategoryTheory.CategoryStruct.id (CategoryTheory.Limits.pushout f g)
• CategoryTheory.Limits.pushout.congrHom_inv 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f₁ f₂ : X ⟶ Y} {g₁ g₂ : X ⟶ Z} (h₁ : f₁ = f₂) (h₂ : g₁ = g₂) [CategoryTheory.Limits.HasPushout f₁ g₁] [CategoryTheory.Limits.HasPushout f₂ g₂] : (CategoryTheory.Limits.pushout.congrHom h₁ h₂).inv = CategoryTheory.Limits.pushout.map f₂ g₂ f₁ g₁ (CategoryTheory.CategoryStruct.id Y) (CategoryTheory.CategoryStruct.id Z) (CategoryTheory.CategoryStruct.id X) ⋯ ⋯
• CategoryTheory.Limits.pushout.map_comp 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z X' Y' Z' X'' Y'' Z'' : C} {f : X ⟶ Y} {g : X ⟶ Z} {f' : X' ⟶ Y'} {g' : X' ⟶ Z'} {f'' : X'' ⟶ Y''} {g'' : X'' ⟶ Z''} (i₁ : X ⟶ X') (j₁ : X' ⟶ X'') (i₂ : Y ⟶ Y') (j₂ : Y' ⟶ Y'') (i₃ : Z ⟶ Z') (j₃ : Z' ⟶ Z'') [CategoryTheory.Limits.HasPushout f g] [CategoryTheory.Limits.HasPushout f' g'] [CategoryTheory.Limits.HasPushout f'' g''] (e₁ : CategoryTheory.CategoryStruct.comp f i₂ = CategoryTheory.CategoryStruct.comp i₁ f') (e₂ : CategoryTheory.CategoryStruct.comp g i₃ = CategoryTheory.CategoryStruct.comp i₁ g') (e₃ : CategoryTheory.CategoryStruct.comp f' j₂ = CategoryTheory.CategoryStruct.comp j₁ f'') (e₄ : CategoryTheory.CategoryStruct.comp g' j₃ = CategoryTheory.CategoryStruct.comp j₁ g'') : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.map f g f' g' i₂ i₃ i₁ e₁ e₂) (CategoryTheory.Limits.pushout.map f' g' f'' g'' j₂ j₃ j₁ e₃ e₄) = CategoryTheory.Limits.pushout.map f g f'' g'' (CategoryTheory.CategoryStruct.comp i₂ j₂) (CategoryTheory.CategoryStruct.comp i₃ j₃) (CategoryTheory.CategoryStruct.comp i₁ j₁) ⋯ ⋯
• CategoryTheory.Limits.pushout.map_comp_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z X' Y' Z' X'' Y'' Z'' : C} {f : X ⟶ Y} {g : X ⟶ Z} {f' : X' ⟶ Y'} {g' : X' ⟶ Z'} {f'' : X'' ⟶ Y''} {g'' : X'' ⟶ Z''} (i₁ : X ⟶ X') (j₁ : X' ⟶ X'') (i₂ : Y ⟶ Y') (j₂ : Y' ⟶ Y'') (i₃ : Z ⟶ Z') (j₃ : Z' ⟶ Z'') [CategoryTheory.Limits.HasPushout f g] [CategoryTheory.Limits.HasPushout f' g'] [CategoryTheory.Limits.HasPushout f'' g''] (e₁ : CategoryTheory.CategoryStruct.comp f i₂ = CategoryTheory.CategoryStruct.comp i₁ f') (e₂ : CategoryTheory.CategoryStruct.comp g i₃ = CategoryTheory.CategoryStruct.comp i₁ g') (e₃ : CategoryTheory.CategoryStruct.comp f' j₂ = CategoryTheory.CategoryStruct.comp j₁ f'') (e₄ : CategoryTheory.CategoryStruct.comp g' j₃ = CategoryTheory.CategoryStruct.comp j₁ g'') {Z✝ : C} (h : CategoryTheory.Limits.pushout f'' g'' ⟶ Z✝) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.map f g f' g' i₂ i₃ i₁ e₁ e₂) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.map f' g' f'' g'' j₂ j₃ j₁ e₃ e₄) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.map f g f'' g'' (CategoryTheory.CategoryStruct.comp i₂ j₂) (CategoryTheory.CategoryStruct.comp i₃ j₃) (CategoryTheory.CategoryStruct.comp i₁ j₁) ⋯ ⋯) h
• CategoryTheory.SmallObject.functorMap.eq_1 📋 Mathlib.CategoryTheory.SmallObject.Construction
  {C : Type u} [CategoryTheory.Category.{v, u} C] {I : Type w} {A B : I → C} (f : (i : I) → A i ⟶ B i) {S T X Y : C} {πX : X ⟶ S} {πY : Y ⟶ T} (τ : CategoryTheory.Arrow.mk πX ⟶ CategoryTheory.Arrow.mk πY) [CategoryTheory.Limits.HasColimitsOfShape (CategoryTheory.Discrete (CategoryTheory.SmallObject.FunctorObjIndex f πX)) C] [CategoryTheory.Limits.HasColimitsOfShape (CategoryTheory.Discrete (CategoryTheory.SmallObject.FunctorObjIndex f πY)) C] [CategoryTheory.Limits.HasPushout (CategoryTheory.SmallObject.functorObjTop f πX) (CategoryTheory.SmallObject.functorObjLeft f πX)] [CategoryTheory.Limits.HasPushout (CategoryTheory.SmallObject.functorObjTop f πY) (CategoryTheory.SmallObject.functorObjLeft f πY)] : CategoryTheory.SmallObject.functorMap f τ = CategoryTheory.Limits.pushout.map (CategoryTheory.SmallObject.functorObjTop f πX) (CategoryTheory.SmallObject.functorObjLeft f πX) (CategoryTheory.SmallObject.functorObjTop f πY) (CategoryTheory.SmallObject.functorObjLeft f πY) τ.left (CategoryTheory.SmallObject.functorMapTgt f τ) (CategoryTheory.SmallObject.functorMapSrc f τ) ⋯ ⋯

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