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Found 38 declarations mentioning CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.Hom.fst.
- CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.Hom.fst ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {X : Type uโ} [CategoryTheory.Category.{vโ, uโ} X] {x y : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} (self : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.Hom X x y) : x.fst โถ y.fst - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.id_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] (x : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X) (Xโ : X) : (CategoryTheory.CategoryStruct.id x).fst.app Xโ = CategoryTheory.CategoryStruct.id (x.fst.obj Xโ) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.fstFunctor_map ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] {Xโ Yโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} (f : Xโ โถ Yโ) : (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.fstFunctor F G X).map f = f.fst - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.Hom.ext ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} {instโ : CategoryTheory.Category.{vโ, uโ} A} {instโยน : CategoryTheory.Category.{vโ, uโ} B} {instโยฒ : CategoryTheory.Category.{vโ, uโ} C} {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {X : Type uโ} {instโยณ : CategoryTheory.Category.{vโ, uโ} X} {x y : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} {xโ yโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.Hom X x y} (fst : xโ.fst = yโ.fst) (snd : xโ.snd = yโ.snd) : xโ = yโ - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.Hom.ext_iff ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} {instโ : CategoryTheory.Category.{vโ, uโ} A} {instโยน : CategoryTheory.Category.{vโ, uโ} B} {instโยฒ : CategoryTheory.Category.{vโ, uโ} C} {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {X : Type uโ} {instโยณ : CategoryTheory.Category.{vโ, uโ} X} {x y : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} {xโ yโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.Hom X x y} : xโ = yโ โ xโ.fst = yโ.fst โง xโ.snd = yโ.snd - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.hom_ext ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] {S S' : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} {f g : S โถ S'} (hโ : f.fst = g.fst) (hโ : f.snd = g.snd) : f = g - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.hom_ext_iff ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {X : Type uโ} [CategoryTheory.Category.{vโ, uโ} X] {S S' : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} {f g : S โถ S'} : f = g โ f.fst = g.fst โง f.snd = g.snd - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.comp_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] {Xโ Yโ Zโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} (f : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.Hom X Xโ Yโ) (g : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.Hom X Yโ Zโ) (Xโยน : X) : (CategoryTheory.CategoryStruct.comp f g).fst.app Xโยน = CategoryTheory.CategoryStruct.comp (f.fst.app Xโยน) (g.fst.app Xโยน) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.Hom.w ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {X : Type uโ} [CategoryTheory.Category.{vโ, uโ} X] {x y : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} (self : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.Hom X x y) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight self.fst F) y.iso.hom = CategoryTheory.CategoryStruct.comp x.iso.hom (CategoryTheory.Functor.whiskerRight self.snd G) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.Hom.w_assoc ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {X : Type uโ} [CategoryTheory.Category.{vโ, uโ} X] {x y : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} (self : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.Hom X x y) {Z : CategoryTheory.Functor X B} (h : y.snd.comp G โถ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight self.fst F) (CategoryTheory.CategoryStruct.comp y.iso.hom h) = CategoryTheory.CategoryStruct.comp x.iso.hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight self.snd G) h) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.w_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] {S S' : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} (ฯ : S โถ S') (x : X) : CategoryTheory.CategoryStruct.comp (F.map (ฯ.fst.app x)) (S'.iso.hom.app x) = CategoryTheory.CategoryStruct.comp (S.iso.hom.app x) (G.map (ฯ.snd.app x)) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.mkIso_hom_fst ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {X : Type uโ} [CategoryTheory.Category.{vโ, uโ} X] {S S' : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} (eโ : S.fst โ S'.fst) (eแตฃ : S.snd โ S'.snd) (w : CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight eโ.hom F) S'.iso.hom = CategoryTheory.CategoryStruct.comp S.iso.hom (CategoryTheory.Functor.whiskerRight eแตฃ.hom G) := by cat_disch) : (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.mkIso eโ eแตฃ w).hom.fst = eโ.hom - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.mkIso_inv_fst ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {X : Type uโ} [CategoryTheory.Category.{vโ, uโ} X] {S S' : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} (eโ : S.fst โ S'.fst) (eแตฃ : S.snd โ S'.snd) (w : CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight eโ.hom F) S'.iso.hom = CategoryTheory.CategoryStruct.comp S.iso.hom (CategoryTheory.Functor.whiskerRight eแตฃ.hom G) := by cat_disch) : (CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.mkIso eโ eแตฃ w).inv.fst = eโ.inv - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.w_app_assoc ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] {S S' : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} (ฯ : S โถ S') (x : X) {Z : B} (h : G.obj (S'.snd.obj x) โถ Z) : CategoryTheory.CategoryStruct.comp (F.map (ฯ.fst.app x)) (CategoryTheory.CategoryStruct.comp (S'.iso.hom.app x) h) = CategoryTheory.CategoryStruct.comp (S.iso.hom.app x) (CategoryTheory.CategoryStruct.comp (G.map (ฯ.snd.app x)) h) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjId_hom_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] (Xโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X) (Xโยน : X) : ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjId F G X).hom.app Xโ).fst.app Xโยน = CategoryTheory.CategoryStruct.id (Xโ.fst.obj Xโยน) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjId_inv_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] (Xโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X) (Xโยน : X) : ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjId F G X).inv.app Xโ).fst.app Xโยน = CategoryTheory.CategoryStruct.id (Xโ.fst.obj Xโยน) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjId_hom_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) (Xโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X) (Xโยน : X) : ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjId X F G).hom.app Xโ).fst.app Xโยน = CategoryTheory.CategoryStruct.id (Xโ.fst.obj Xโยน) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjId_inv_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) (Xโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X) (Xโยน : X) : ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjId X F G).inv.app Xโ).fst.app Xโยน = CategoryTheory.CategoryStruct.id (Xโ.fst.obj Xโยน) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose_obj_map_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) {X : Type uโ} {Y : Type uโ } [CategoryTheory.Category.{vโ, uโ} X] [CategoryTheory.Category.{vโ , uโ } Y] (U : CategoryTheory.Functor X Y) {S S' : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G Y} (ฯ : S โถ S') (Xโ : X) : (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U).map ฯ).fst.app Xโ = ฯ.fst.app (U.obj Xโ) - CategoryTheory.Limits.CategoricalPullback.functorEquiv_counitIso_hom_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] (Xโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X) (Xโยน : X) : ((CategoryTheory.Limits.CategoricalPullback.functorEquiv F G X).counitIso.hom.app Xโ).fst.app Xโยน = CategoryTheory.CategoryStruct.id (Xโ.fst.obj Xโยน) - CategoryTheory.Limits.CategoricalPullback.functorEquiv_counitIso_inv_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] (Xโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X) (Xโยน : X) : ((CategoryTheory.Limits.CategoricalPullback.functorEquiv F G X).counitIso.inv.app Xโ).fst.app Xโยน = CategoryTheory.CategoryStruct.id (Xโ.fst.obj Xโยน) - CategoryTheory.Limits.CategoricalPullback.toCatCommSqOver_map_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] {J J' : CategoryTheory.Functor X (CategoryTheory.Limits.CategoricalPullback F G)} (Fโ : J โถ J') (Xโ : X) : ((CategoryTheory.Limits.CategoricalPullback.toCatCommSqOver F G X).map Fโ).fst.app Xโ = (Fโ.app Xโ).fst - CategoryTheory.Limits.CategoricalPullback.functorEquiv_functor_map_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] {J J' : CategoryTheory.Functor X (CategoryTheory.Limits.CategoricalPullback F G)} (Fโ : J โถ J') (Xโ : X) : ((CategoryTheory.Limits.CategoricalPullback.functorEquiv F G X).functor.map Fโ).fst.app Xโ = (Fโ.app Xโ).fst - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.toFunctorToCategoricalPullback_map_app_fst ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] {S S' : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} (ฯ : S โถ S') (x : X) : (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.toFunctorToCategoricalPullback F G X).map ฯ).app x).fst = ฯ.fst.app x - CategoryTheory.Limits.CategoricalPullback.functorEquiv_inverse_map_app_fst ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] {S S' : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} (ฯ : S โถ S') (x : X) : (((CategoryTheory.Limits.CategoricalPullback.functorEquiv F G X).inverse.map ฯ).app x).fst = ฯ.fst.app x - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjComp_hom_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) {X : Type uโ} {Y : Type uโ } {Z : Type uโ} [CategoryTheory.Category.{vโ, uโ} X] [CategoryTheory.Category.{vโ , uโ } Y] [CategoryTheory.Category.{vโ, uโ} Z] (U : CategoryTheory.Functor X Y) (V : CategoryTheory.Functor Y Z) (Xโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G Z) (xโ : X) : ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjComp F G U V).hom.app Xโ).fst.app xโ = CategoryTheory.CategoryStruct.id (Xโ.fst.obj (V.obj (U.obj xโ))) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjComp_inv_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) {X : Type uโ} {Y : Type uโ } {Z : Type uโ} [CategoryTheory.Category.{vโ, uโ} X] [CategoryTheory.Category.{vโ , uโ } Y] [CategoryTheory.Category.{vโ, uโ} Z] (U : CategoryTheory.Functor X Y) (V : CategoryTheory.Functor Y Z) (Xโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G Z) (xโ : X) : ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjComp F G U V).inv.app Xโ).fst.app xโ = CategoryTheory.CategoryStruct.id (Xโ.fst.obj (V.obj (U.obj xโ))) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjComp_hom_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {Aโ : Type uโ} {Bโ : Type uโ } {Cโ : Type uโ} [CategoryTheory.Category.{vโ, uโ} Aโ] [CategoryTheory.Category.{vโ , uโ } Bโ] [CategoryTheory.Category.{vโ, uโ} Cโ] {Fโ : CategoryTheory.Functor Aโ Bโ} {Gโ : CategoryTheory.Functor Cโ Bโ} {Aโ : Type uโ} {Bโ : Type uโ} {Cโ : Type uโ} [CategoryTheory.Category.{vโ, uโ} Aโ] [CategoryTheory.Category.{vโ, uโ} Bโ] [CategoryTheory.Category.{vโ, uโ} Cโ] {Fโ : CategoryTheory.Functor Aโ Bโ} {Gโ : CategoryTheory.Functor Cโ Bโ} (X : Type uโโ) [CategoryTheory.Category.{vโโ, uโโ} X] (ฯ : CategoryTheory.Limits.CatCospanTransform F G Fโ Gโ) (ฯ' : CategoryTheory.Limits.CatCospanTransform Fโ Gโ Fโ Gโ) (Xโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X) (xโ : X) : ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjComp X ฯ ฯ').hom.app Xโ).fst.app xโ = CategoryTheory.CategoryStruct.id (ฯ'.left.obj (ฯ.left.obj (Xโ.fst.obj xโ))) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjComp_inv_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {Aโ : Type uโ} {Bโ : Type uโ } {Cโ : Type uโ} [CategoryTheory.Category.{vโ, uโ} Aโ] [CategoryTheory.Category.{vโ , uโ } Bโ] [CategoryTheory.Category.{vโ, uโ} Cโ] {Fโ : CategoryTheory.Functor Aโ Bโ} {Gโ : CategoryTheory.Functor Cโ Bโ} {Aโ : Type uโ} {Bโ : Type uโ} {Cโ : Type uโ} [CategoryTheory.Category.{vโ, uโ} Aโ] [CategoryTheory.Category.{vโ, uโ} Bโ] [CategoryTheory.Category.{vโ, uโ} Cโ] {Fโ : CategoryTheory.Functor Aโ Bโ} {Gโ : CategoryTheory.Functor Cโ Bโ} (X : Type uโโ) [CategoryTheory.Category.{vโโ, uโโ} X] (ฯ : CategoryTheory.Limits.CatCospanTransform F G Fโ Gโ) (ฯ' : CategoryTheory.Limits.CatCospanTransform Fโ Gโ Fโ Gโ) (Xโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X) (xโ : X) : ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjComp X ฯ ฯ').inv.app Xโ).fst.app xโ = CategoryTheory.CategoryStruct.id (ฯ'.left.obj (ฯ.left.obj (Xโ.fst.obj xโ))) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjTransformObjSquare_iso_hom_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {Aโ : Type uโ} {Bโ : Type uโ } {Cโ : Type uโ} [CategoryTheory.Category.{vโ, uโ} Aโ] [CategoryTheory.Category.{vโ , uโ } Bโ] [CategoryTheory.Category.{vโ, uโ} Cโ] {Fโ : CategoryTheory.Functor Aโ Bโ} {Gโ : CategoryTheory.Functor Cโ Bโ} {X : Type uโ} {Y : Type uโ} [CategoryTheory.Category.{vโ, uโ} X] [CategoryTheory.Category.{vโ, uโ} Y] (ฯ : CategoryTheory.Limits.CatCospanTransform F G Fโ Gโ) (U : CategoryTheory.Functor X Y) (Xโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G Y) (xโ : X) : ((CategoryTheory.CatCommSq.iso ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ฯ) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ฯ) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose Fโ Gโ).obj U)).hom.app Xโ).fst.app xโ = CategoryTheory.CategoryStruct.id (ฯ.left.obj (Xโ.fst.obj (U.obj xโ))) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precomposeObjTransformObjSquare_iso_inv_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {Aโ : Type uโ} {Bโ : Type uโ } {Cโ : Type uโ} [CategoryTheory.Category.{vโ, uโ} Aโ] [CategoryTheory.Category.{vโ , uโ } Bโ] [CategoryTheory.Category.{vโ, uโ} Cโ] {Fโ : CategoryTheory.Functor Aโ Bโ} {Gโ : CategoryTheory.Functor Cโ Bโ} {X : Type uโ} {Y : Type uโ} [CategoryTheory.Category.{vโ, uโ} X] [CategoryTheory.Category.{vโ, uโ} Y] (ฯ : CategoryTheory.Limits.CatCospanTransform F G Fโ Gโ) (U : CategoryTheory.Functor X Y) (Xโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G Y) (xโ : X) : ((CategoryTheory.CatCommSq.iso ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ฯ) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ฯ) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose Fโ Gโ).obj U)).inv.app Xโ).fst.app xโ = CategoryTheory.CategoryStruct.id (ฯ.left.obj (Xโ.fst.obj (U.obj xโ))) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjPrecomposeObjSquare_iso_hom_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {Aโ : Type uโ} {Bโ : Type uโ } {Cโ : Type uโ} [CategoryTheory.Category.{vโ, uโ} Aโ] [CategoryTheory.Category.{vโ , uโ } Bโ] [CategoryTheory.Category.{vโ, uโ} Cโ] {Fโ : CategoryTheory.Functor Aโ Bโ} {Gโ : CategoryTheory.Functor Cโ Bโ} {X : Type uโ} {Y : Type uโ} [CategoryTheory.Category.{vโ, uโ} X] [CategoryTheory.Category.{vโ, uโ} Y] (U : CategoryTheory.Functor X Y) (ฯ : CategoryTheory.Limits.CatCospanTransform F G Fโ Gโ) (Xโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G Y) (xโ : X) : ((CategoryTheory.CatCommSq.iso ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ฯ) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose Fโ Gโ).obj U) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ฯ)).hom.app Xโ).fst.app xโ = CategoryTheory.CategoryStruct.id (ฯ.left.obj (Xโ.fst.obj (U.obj xโ))) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transformObjPrecomposeObjSquare_iso_inv_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {Aโ : Type uโ} {Bโ : Type uโ } {Cโ : Type uโ} [CategoryTheory.Category.{vโ, uโ} Aโ] [CategoryTheory.Category.{vโ , uโ } Bโ] [CategoryTheory.Category.{vโ, uโ} Cโ] {Fโ : CategoryTheory.Functor Aโ Bโ} {Gโ : CategoryTheory.Functor Cโ Bโ} {X : Type uโ} {Y : Type uโ} [CategoryTheory.Category.{vโ, uโ} X] [CategoryTheory.Category.{vโ, uโ} Y] (U : CategoryTheory.Functor X Y) (ฯ : CategoryTheory.Limits.CatCospanTransform F G Fโ Gโ) (Xโ : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G Y) (xโ : X) : ((CategoryTheory.CatCommSq.iso ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform Y).obj ฯ) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).obj U) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose Fโ Gโ).obj U) ((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ฯ)).inv.app Xโ).fst.app xโ = CategoryTheory.CategoryStruct.id (ฯ.left.obj (Xโ.fst.obj (U.obj xโ))) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform_obj_map_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {Aโ : Type uโ} {Bโ : Type uโ } {Cโ : Type uโ} [CategoryTheory.Category.{vโ, uโ} Aโ] [CategoryTheory.Category.{vโ , uโ } Bโ] [CategoryTheory.Category.{vโ, uโ} Cโ] {Fโ : CategoryTheory.Functor Aโ Bโ} {Gโ : CategoryTheory.Functor Cโ Bโ} (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] (ฯ : CategoryTheory.Limits.CatCospanTransform F G Fโ Gโ) {x y : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X} (f : x โถ y) (Xโ : X) : (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).obj ฯ).map f).fst.app Xโ = ฯ.left.map (f.fst.app Xโ) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose_map_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) {X : Type uโ} {Y : Type uโ } [CategoryTheory.Category.{vโ, uโ} X] [CategoryTheory.Category.{vโ , uโ } Y] {U V : CategoryTheory.Functor X Y} (ฮฑ : U โถ V) (x : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G Y) (Xโ : X) : (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).map ฮฑ).app x).fst.app Xโ = x.fst.map (ฮฑ.app Xโ) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose_map_app_snd_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C B) {X : Type uโ} {Y : Type uโ } [CategoryTheory.Category.{vโ, uโ} X] [CategoryTheory.Category.{vโ , uโ } Y] {U V : CategoryTheory.Functor X Y} (ฮฑ : U โถ V) (x : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G Y) (Xโ : X) : (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.precompose F G).map ฮฑ).app x).snd.app Xโ = x.snd.map (ฮฑ.app Xโ) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform_map_app_fst_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {Aโ : Type uโ} {Bโ : Type uโ } {Cโ : Type uโ} [CategoryTheory.Category.{vโ, uโ} Aโ] [CategoryTheory.Category.{vโ , uโ } Bโ] [CategoryTheory.Category.{vโ, uโ} Cโ] {Fโ : CategoryTheory.Functor Aโ Bโ} {Gโ : CategoryTheory.Functor Cโ Bโ} (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] {ฯ ฯ' : CategoryTheory.Limits.CatCospanTransform F G Fโ Gโ} (ฮท : ฯ โถ ฯ') (S : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X) (y : X) : (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).map ฮท).app S).fst.app y = ฮท.left.app (S.fst.obj y) - CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform_map_app_snd_app ๐ Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic
{A : Type uโ} {B : Type uโ} {C : Type uโ} [CategoryTheory.Category.{vโ, uโ} A] [CategoryTheory.Category.{vโ, uโ} B] [CategoryTheory.Category.{vโ, uโ} C] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor C B} {Aโ : Type uโ} {Bโ : Type uโ } {Cโ : Type uโ} [CategoryTheory.Category.{vโ, uโ} Aโ] [CategoryTheory.Category.{vโ , uโ } Bโ] [CategoryTheory.Category.{vโ, uโ} Cโ] {Fโ : CategoryTheory.Functor Aโ Bโ} {Gโ : CategoryTheory.Functor Cโ Bโ} (X : Type uโ) [CategoryTheory.Category.{vโ, uโ} X] {ฯ ฯ' : CategoryTheory.Limits.CatCospanTransform F G Fโ Gโ} (ฮท : ฯ โถ ฯ') (S : CategoryTheory.Limits.CategoricalPullback.CatCommSqOver F G X) (y : X) : (((CategoryTheory.Limits.CategoricalPullback.CatCommSqOver.transform X).map ฮท).app S).snd.app y = ฮท.right.app (S.snd.obj y)
About
Loogle searches Lean and Mathlib definitions and theorems.
You can use Loogle from within the Lean4 VSCode language extension
using the Loogle command from the command palette. 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:
By constant:
๐Real.sin
finds all lemmas whose statement somehow mentions the sine function.By lemma name substring:
๐"differ"
finds all lemmas that have"differ"somewhere in their lemma name.By subexpression:
๐_ * (_ ^ _)
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
๐Real.sqrt ?a * Real.sqrt ?aIf the pattern has parameters, they are matched in any order. Both of these will find
List.map:
๐(?a -> ?b) -> List ?a -> List ?b
๐List ?a -> (?a -> ?b) -> List ?bBy main conclusion:
๐|- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of allโandโ) 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,
๐|- _ < _ โ tsum _ < tsum _
will findtsum_lt_tsumeven though the hypothesisf i < g iis not the last.You can filter for definitions vs theorems: Using
โข (_ : Type _)finds all definitions which provide data whileโข (_ : Prop)finds all theorems (and definitions of proofs).
If you pass more than one such search filter, separated by commas
Loogle will return lemmas which match all of them. The
search
๐ Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ โ _
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. Please review the Lean FRO Terms of Use and Privacy Policy.
This is Loogle revision 9f11169 serving mathlib revision ce5dd8c