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Found 40 declarations mentioning CategoryTheory.MorphismProperty.Over.map.

CategoryTheory.MorphismProperty.Over.map 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] {X Y : T} [P.IsStableUnderComposition] {f : X ⟶ Y} (hPf : P f) : CategoryTheory.Functor (P.Over Q X) (P.Over Q Y)
CategoryTheory.MorphismProperty.Over.mapPullbackAdj 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] (P Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] {X Y : T} [P.IsStableUnderComposition] [Q.IsStableUnderBaseChange] (f : X ⟶ Y) [P.HasPullbacksAlong f] [P.IsStableUnderBaseChangeAlong f] [Q.HasOfPostcompProperty Q] (hPf : P f) (hQf : Q f) : CategoryTheory.MorphismProperty.Over.map Q hPf ⊣ CategoryTheory.MorphismProperty.Over.pullback P Q f
CategoryTheory.MorphismProperty.Over.map.congr_simp 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] {X Y : T} [P.IsStableUnderComposition] {f f✝ : X ⟶ Y} (e_f : f = f✝) (hPf : P f) : CategoryTheory.MorphismProperty.Over.map Q hPf = CategoryTheory.MorphismProperty.Over.map Q ⋯
CategoryTheory.MorphismProperty.Over.map.eq_1 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] {X Y : T} [P.IsStableUnderComposition] {f : X ⟶ Y} (hPf : P f) : CategoryTheory.MorphismProperty.Over.map Q hPf = CategoryTheory.MorphismProperty.Comma.mapRight (CategoryTheory.Functor.id T) (CategoryTheory.Discrete.natTrans fun x => f) ⋯
CategoryTheory.MorphismProperty.instIsLeftAdjointOverTopMapOfHasPullbacksAlongOfIsStableUnderBaseChangeAlong 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] (P : CategoryTheory.MorphismProperty T) {X Y : T} [P.IsStableUnderComposition] (f : X ⟶ Y) [P.HasPullbacksAlong f] [P.IsStableUnderBaseChangeAlong f] (hPf : P f) : (CategoryTheory.MorphismProperty.Over.map ⊤ hPf).IsLeftAdjoint
CategoryTheory.MorphismProperty.Over.mapCongr 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] [P.IsStableUnderComposition] [Q.RespectsIso] {X Y : T} {f g : X ⟶ Y} (hfg : f = g) (hf : P f) : CategoryTheory.MorphismProperty.Over.map Q hf ≅ CategoryTheory.MorphismProperty.Over.map Q ⋯
CategoryTheory.MorphismProperty.Over.map_obj_left 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] {X Y : T} [P.IsStableUnderComposition] {f : X ⟶ Y} (hPf : P f) (X✝ : CategoryTheory.MorphismProperty.Comma (CategoryTheory.Functor.id T) (CategoryTheory.Functor.fromPUnit X) P Q ⊤) : ((CategoryTheory.MorphismProperty.Over.map Q hPf).obj X✝).left = X✝.left
CategoryTheory.MorphismProperty.Over.mapId 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] [P.IsStableUnderComposition] [P.IsMultiplicative] [Q.RespectsIso] (X : T) (f : X ⟶ X := CategoryTheory.CategoryStruct.id X) (hf : f = CategoryTheory.CategoryStruct.id X := by cat_disch) : CategoryTheory.MorphismProperty.Over.map Q ⋯ ≅ CategoryTheory.Functor.id (P.Over Q X)
CategoryTheory.MorphismProperty.Over.map_comp 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] {X Y Z : T} [P.IsStableUnderComposition] {f : X ⟶ Y} (hf : P f) {g : Y ⟶ Z} (hg : P g) : CategoryTheory.MorphismProperty.Over.map Q ⋯ = (CategoryTheory.MorphismProperty.Over.map Q hf).comp (CategoryTheory.MorphismProperty.Over.map Q hg)
CategoryTheory.MorphismProperty.Over.mapComp 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] {X Y Z : T} [P.IsStableUnderComposition] {f : X ⟶ Y} (hf : P f) {g : Y ⟶ Z} (hg : P g) [Q.RespectsIso] (fg : X ⟶ Z := CategoryTheory.CategoryStruct.comp f g) (hfg : fg = CategoryTheory.CategoryStruct.comp f g := by cat_disch) : CategoryTheory.MorphismProperty.Over.map Q ⋯ ≅ (CategoryTheory.MorphismProperty.Over.map Q hf).comp (CategoryTheory.MorphismProperty.Over.map Q hg)
CategoryTheory.MorphismProperty.Over.map_obj_hom 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] {X Y : T} [P.IsStableUnderComposition] {f : X ⟶ Y} (hPf : P f) (X✝ : CategoryTheory.MorphismProperty.Comma (CategoryTheory.Functor.id T) (CategoryTheory.Functor.fromPUnit X) P Q ⊤) : ((CategoryTheory.MorphismProperty.Over.map Q hPf).obj X✝).hom = CategoryTheory.CategoryStruct.comp X✝.hom f
CategoryTheory.MorphismProperty.Over.pullbackMapHomPullback 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P Q : CategoryTheory.MorphismProperty T} [Q.IsMultiplicative] [P.IsStableUnderComposition] {X Y Z : T} (f : X ⟶ Y) (hPf : P f) (hQf : Q f) (g : Y ⟶ Z) [P.IsStableUnderBaseChangeAlong f] [P.IsStableUnderBaseChangeAlong g] [Q.IsStableUnderBaseChange] [CategoryTheory.Limits.HasPullbacks T] (fg : X ⟶ Z := CategoryTheory.CategoryStruct.comp f g) (hfg : CategoryTheory.CategoryStruct.comp f g = fg := by cat_disch) : (CategoryTheory.MorphismProperty.Over.pullback P Q fg).comp (CategoryTheory.MorphismProperty.Over.map Q hPf) ⟶ CategoryTheory.MorphismProperty.Over.pullback P Q g
CategoryTheory.MorphismProperty.Over.mapCongr_hom_app_left 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] [P.IsStableUnderComposition] [Q.RespectsIso] {X Y : T} {f g : X ⟶ Y} (hfg : f = g) (hf : P f) (X✝ : P.Over Q X) : ((CategoryTheory.MorphismProperty.Over.mapCongr Q hfg hf).hom.app X✝).left = CategoryTheory.CategoryStruct.id X✝.left
CategoryTheory.MorphismProperty.Over.mapCongr_inv_app_left 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] [P.IsStableUnderComposition] [Q.RespectsIso] {X Y : T} {f g : X ⟶ Y} (hfg : f = g) (hf : P f) (X✝ : P.Over Q X) : ((CategoryTheory.MorphismProperty.Over.mapCongr Q hfg hf).inv.app X✝).left = CategoryTheory.CategoryStruct.id X✝.left
CategoryTheory.MorphismProperty.Over.mapId_hom_app_left 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] [P.IsStableUnderComposition] [P.IsMultiplicative] [Q.RespectsIso] (X : T) (f : X ⟶ X := CategoryTheory.CategoryStruct.id X) (hf : f = CategoryTheory.CategoryStruct.id X := by cat_disch) (X✝ : P.Over Q X) : ((CategoryTheory.MorphismProperty.Over.mapId Q X f hf).hom.app X✝).left = CategoryTheory.CategoryStruct.id X✝.left
CategoryTheory.MorphismProperty.Over.mapId_inv_app_left 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] [P.IsStableUnderComposition] [P.IsMultiplicative] [Q.RespectsIso] (X : T) (f : X ⟶ X := CategoryTheory.CategoryStruct.id X) (hf : f = CategoryTheory.CategoryStruct.id X := by cat_disch) (X✝ : P.Over Q X) : ((CategoryTheory.MorphismProperty.Over.mapId Q X f hf).inv.app X✝).left = CategoryTheory.CategoryStruct.id X✝.left
CategoryTheory.MorphismProperty.Over.pullbackMapHomPullback_app 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P Q : CategoryTheory.MorphismProperty T} [Q.IsMultiplicative] [P.IsStableUnderComposition] {X Y Z : T} (f : X ⟶ Y) (hPf : P f) (hQf : Q f) (g : Y ⟶ Z) [P.IsStableUnderBaseChangeAlong f] [P.IsStableUnderBaseChangeAlong g] [Q.IsStableUnderBaseChange] [CategoryTheory.Limits.HasPullbacks T] (fg : X ⟶ Z := CategoryTheory.CategoryStruct.comp f g) (hfg : CategoryTheory.CategoryStruct.comp f g = fg := by cat_disch) (A : P.Over Q Z) : (CategoryTheory.MorphismProperty.Over.pullbackMapHomPullback f hPf hQf g fg hfg).app A = CategoryTheory.MorphismProperty.Over.homMk (CategoryTheory.Limits.pullback.map A.hom fg A.hom g (CategoryTheory.CategoryStruct.id A.left) f (CategoryTheory.CategoryStruct.id Z) ⋯ ⋯) ⋯ ⋯
CategoryTheory.MorphismProperty.Over.mapComp_hom_app_left 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] {X Y Z : T} [P.IsStableUnderComposition] {f : X ⟶ Y} (hf : P f) {g : Y ⟶ Z} (hg : P g) [Q.RespectsIso] (fg : X ⟶ Z := CategoryTheory.CategoryStruct.comp f g) (hfg : fg = CategoryTheory.CategoryStruct.comp f g := by cat_disch) (X✝ : P.Over Q X) : ((CategoryTheory.MorphismProperty.Over.mapComp Q hf hg fg hfg).hom.app X✝).left = CategoryTheory.CategoryStruct.id X✝.left
CategoryTheory.MorphismProperty.Over.mapComp_inv_app_left 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] {X Y Z : T} [P.IsStableUnderComposition] {f : X ⟶ Y} (hf : P f) {g : Y ⟶ Z} (hg : P g) [Q.RespectsIso] (fg : X ⟶ Z := CategoryTheory.CategoryStruct.comp f g) (hfg : fg = CategoryTheory.CategoryStruct.comp f g := by cat_disch) (X✝ : P.Over Q X) : ((CategoryTheory.MorphismProperty.Over.mapComp Q hf hg fg hfg).inv.app X✝).left = CategoryTheory.CategoryStruct.id X✝.left
CategoryTheory.MorphismProperty.Over.mapPullbackAdj_unit_app 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] (P Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] {X Y : T} [P.IsStableUnderComposition] [Q.IsStableUnderBaseChange] (f : X ⟶ Y) [P.HasPullbacksAlong f] [P.IsStableUnderBaseChangeAlong f] [Q.HasOfPostcompProperty Q] (hPf : P f) (hQf : Q f) (X✝ : P.Over Q X) : (CategoryTheory.MorphismProperty.Over.mapPullbackAdj P Q f hPf hQf).unit.app X✝ = CategoryTheory.MorphismProperty.Over.homMk (CategoryTheory.Limits.pullback.lift (CategoryTheory.CategoryStruct.id X✝.left) X✝.hom ⋯) ⋯ ⋯
CategoryTheory.MorphismProperty.Over.map_map_left 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] {X Y : T} [P.IsStableUnderComposition] {f : X ⟶ Y} (hPf : P f) {X✝ Y✝ : CategoryTheory.MorphismProperty.Comma (CategoryTheory.Functor.id T) (CategoryTheory.Functor.fromPUnit X) P Q ⊤} (f✝ : X✝ ⟶ Y✝) : ((CategoryTheory.MorphismProperty.Over.map Q hPf).map f✝).left = (CategoryTheory.MorphismProperty.Comma.Hom.hom f✝).left
CategoryTheory.MorphismProperty.Over.mapPullbackAdj_counit_app 📋 Mathlib.CategoryTheory.MorphismProperty.OverAdjunction
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] (P Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] {X Y : T} [P.IsStableUnderComposition] [Q.IsStableUnderBaseChange] (f : X ⟶ Y) [P.HasPullbacksAlong f] [P.IsStableUnderBaseChangeAlong f] [Q.HasOfPostcompProperty Q] (hPf : P f) (hQf : Q f) (Y✝ : P.Over Q Y) : (CategoryTheory.MorphismProperty.Over.mapPullbackAdj P Q f hPf hQf).counit.app Y✝ = CategoryTheory.MorphismProperty.Over.homMk (CategoryTheory.Limits.pullback.fst Y✝.hom f) ⋯ ⋯
CategoryTheory.MorphismProperty.Over.mapPullbackAdj.congr_simp 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] (P Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] {X Y : T} [P.IsStableUnderComposition] [Q.IsStableUnderBaseChange] (f : X ⟶ Y) [P.HasPullbacksAlong f] [P.IsStableUnderBaseChangeAlong f] [Q.HasOfPostcompProperty Q] (hPf : P f) (hQf : Q f) : CategoryTheory.MorphismProperty.Over.mapPullbackAdj P Q f hPf hQf = CategoryTheory.MorphismProperty.Over.mapPullbackAdj P Q f hPf hQf
CategoryTheory.MorphismProperty.Over.mapId.congr_simp 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] [P.IsStableUnderComposition] [P.IsMultiplicative] [Q.RespectsIso] (X : T) (f : X ⟶ X) (hf : f = CategoryTheory.CategoryStruct.id X) : CategoryTheory.MorphismProperty.Over.mapId Q X f hf = CategoryTheory.MorphismProperty.Over.mapId Q X f hf
CategoryTheory.MorphismProperty.Over.mapComp.congr_simp 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P : CategoryTheory.MorphismProperty T} (Q : CategoryTheory.MorphismProperty T) [Q.IsMultiplicative] {X Y Z : T} [P.IsStableUnderComposition] {f : X ⟶ Y} (hf : P f) {g : Y ⟶ Z} (hg : P g) [Q.RespectsIso] (fg : X ⟶ Z) (hfg : fg = CategoryTheory.CategoryStruct.comp f g) : CategoryTheory.MorphismProperty.Over.mapComp Q hf hg fg hfg = CategoryTheory.MorphismProperty.Over.mapComp Q hf hg fg hfg
CategoryTheory.MorphismProperty.Over.pullbackMapHomPullback.congr_simp 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{T : Type u_1} [CategoryTheory.Category.{v_1, u_1} T] {P Q : CategoryTheory.MorphismProperty T} [Q.IsMultiplicative] [P.IsStableUnderComposition] {X Y Z : T} (f : X ⟶ Y) (hPf : P f) (hQf : Q f) (g : Y ⟶ Z) [P.IsStableUnderBaseChangeAlong f] [P.IsStableUnderBaseChangeAlong g] [Q.IsStableUnderBaseChange] [CategoryTheory.Limits.HasPullbacks T] (fg : X ⟶ Z) (hfg : CategoryTheory.CategoryStruct.comp f g = fg) : CategoryTheory.MorphismProperty.Over.pullbackMapHomPullback f hPf hQf g fg hfg = CategoryTheory.MorphismProperty.Over.pullbackMapHomPullback f hPf hQf g fg hfg
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.transitionCocone 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.IsStableUnderBaseChange] [P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1} [CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover} [CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] (d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) {i j : 𝒰.I₀} (hij : i ⟶ j) : CategoryTheory.Limits.Cocone (D.comp ((CategoryTheory.MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).comp (CategoryTheory.MorphismProperty.Over.map ⊤ ⋯)))
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.transitionCocone_pt 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.IsStableUnderBaseChange] [P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1} [CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover} [CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] (d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) {i j : 𝒰.I₀} (hij : i ⟶ j) : (d.transitionCocone hij).pt = (d.cocone j).pt
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.transitionMap 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.IsStableUnderBaseChange] [P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1} [CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover} [CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] (d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) {i j : 𝒰.I₀} (hij : i ⟶ j) : (CategoryTheory.MorphismProperty.Over.map ⊤ ⋯).obj (d.cocone i).pt ⟶ (d.cocone j).pt
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.trans 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.IsStableUnderBaseChange] [P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1} [CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover} [CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] (d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) {i j : 𝒰.I₀} (hij : i ⟶ j) : D.comp ((CategoryTheory.MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).comp (CategoryTheory.MorphismProperty.Over.map ⊤ ⋯)) ⟶ D.comp (CategoryTheory.MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.transitionMap_id 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.IsStableUnderBaseChange] [P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1} [CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover} [CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] (d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) (i : 𝒰.I₀) : d.transitionMap (CategoryTheory.CategoryStruct.id i) = (CategoryTheory.MorphismProperty.Over.mapId ⊤ (𝒰.X i) (AlgebraicGeometry.Scheme.Cover.trans 𝒰 (CategoryTheory.CategoryStruct.id i)) ⋯).hom.app (d.cocone i).pt
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.transitionMap.eq_1 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.IsStableUnderBaseChange] [P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1} [CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover} [CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] (d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) {i j : 𝒰.I₀} (hij : i ⟶ j) : d.transitionMap hij = (CategoryTheory.Limits.isColimitOfPreserves (CategoryTheory.MorphismProperty.Over.map ⊤ ⋯) (d.isColimit i)).desc (d.transitionCocone hij)
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.functor_map 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.IsStableUnderBaseChange] [P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1} [CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover} [CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] (d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) {i j : 𝒰.I₀} (hij : i ⟶ j) : d.functor.map hij = (d.transitionMap hij).left
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.transitionCocone.eq_1 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.IsStableUnderBaseChange] [P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1} [CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover} [CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] (d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) {i j : 𝒰.I₀} (hij : i ⟶ j) : d.transitionCocone hij = (CategoryTheory.Limits.Cocones.precompose (d.trans hij)).obj (d.cocone j)
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.trans_app_left 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.IsStableUnderBaseChange] [P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1} [CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover} [CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] (d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) {i j : 𝒰.I₀} (hij : i ⟶ j) (X : J) : ((d.trans hij).app X).left = CategoryTheory.Limits.pullback.map (D.obj X).hom (𝒰.f i) (D.obj X).hom (𝒰.f j) (CategoryTheory.CategoryStruct.id (D.obj X).left) (AlgebraicGeometry.Scheme.Cover.trans 𝒰 hij) (CategoryTheory.CategoryStruct.id S) ⋯ ⋯
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.isPullback 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.IsStableUnderBaseChange] [P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1} [CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover} [CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] (d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) [∀ {i j : 𝒰.I₀} (hij : i ⟶ j), CategoryTheory.Limits.PreservesColimitsOfShape J (CategoryTheory.MorphismProperty.Over.pullback P ⊤ (AlgebraicGeometry.Scheme.Cover.trans 𝒰 hij))] {i j : 𝒰.I₀} (hij : i ⟶ j) : CategoryTheory.IsPullback (d.transitionMap hij).left (d.cocone i).pt.hom (d.cocone j).pt.hom (AlgebraicGeometry.Scheme.Cover.trans 𝒰 hij)
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.transitionCocone_ι_app 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.IsStableUnderBaseChange] [P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1} [CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover} [CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] (d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) {i j : 𝒰.I₀} (hij : i ⟶ j) (X : J) : (d.transitionCocone hij).ι.app X = CategoryTheory.CategoryStruct.comp ((d.trans hij).app X) ((d.cocone j).ι.app X)
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.transitionMap_comp 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.IsStableUnderBaseChange] [P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1} [CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover} [CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] (d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) {i j k : 𝒰.I₀} (hij : i ⟶ j) (hjk : j ⟶ k) : d.transitionMap (CategoryTheory.CategoryStruct.comp hij hjk) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.MorphismProperty.Over.mapComp ⊤ ⋯ ⋯ (AlgebraicGeometry.Scheme.Cover.trans 𝒰 (CategoryTheory.CategoryStruct.comp hij hjk)) ⋯).hom.app (d.cocone i).pt) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.MorphismProperty.Over.map ⊤ ⋯).map (d.transitionMap hij)) (d.transitionMap hjk))
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.cocone_ι_transitionMap 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.IsStableUnderBaseChange] [P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1} [CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover} [CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] (d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) {i j : 𝒰.I₀} (hij : i ⟶ j) (a : J) : CategoryTheory.CategoryStruct.comp ((CategoryTheory.MorphismProperty.Over.map ⊤ ⋯).map ((d.cocone i).ι.app a)) (d.transitionMap hij) = CategoryTheory.CategoryStruct.comp ((d.trans hij).app a) ((d.cocone j).ι.app a)
AlgebraicGeometry.Scheme.Cover.ColimitGluingData.cocone_ι_transitionMap_assoc 📋 Mathlib.AlgebraicGeometry.ColimitsOver
{P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.IsStableUnderBaseChange] [P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1} [CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover} [CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰] (d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) {i j : 𝒰.I₀} (hij : i ⟶ j) (a : J) {Z : P.Over ⊤ (𝒰.X j)} (h : (d.cocone j).pt ⟶ Z) : CategoryTheory.CategoryStruct.comp ((CategoryTheory.MorphismProperty.Over.map ⊤ ⋯).map ((d.cocone i).ι.app a)) (CategoryTheory.CategoryStruct.comp (d.transitionMap hij) h) = CategoryTheory.CategoryStruct.comp ((d.trans hij).app a) (CategoryTheory.CategoryStruct.comp ((d.cocone j).ι.app a) 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:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
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
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 36960b0 serving mathlib revision 9a4cf1d