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Found 43 declarations mentioning CategoryTheory.Limits.colimit.desc.

• CategoryTheory.Limits.colimit.desc 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.HasColimit F] (c : CategoryTheory.Limits.Cocone F) : CategoryTheory.Limits.colimit F ⟶ c.pt
• CategoryTheory.Limits.colimit.desc_cocone 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] : CategoryTheory.Limits.colimit.desc F (CategoryTheory.Limits.colimit.cocone F) = CategoryTheory.CategoryStruct.id (CategoryTheory.Limits.colimit F)
• CategoryTheory.Limits.colimit.isColimit_desc 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] (c : CategoryTheory.Limits.Cocone F) : (CategoryTheory.Limits.colimit.isColimit F).desc c = CategoryTheory.Limits.colimit.desc F c
• CategoryTheory.Limits.colimit.coconeMorphism_hom 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] (c : CategoryTheory.Limits.Cocone F) : (CategoryTheory.Limits.colimit.coconeMorphism c).hom = CategoryTheory.Limits.colimit.desc F c
• CategoryTheory.Limits.colimit.desc_extend 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] (c : CategoryTheory.Limits.Cocone F) {X : C} (f : c.pt ⟶ X) : CategoryTheory.Limits.colimit.desc F (c.extend f) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.desc F c) f
• CategoryTheory.Limits.colimit.ι_desc 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] (c : CategoryTheory.Limits.Cocone F) (j : J) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι F j) (CategoryTheory.Limits.colimit.desc F c) = c.ι.app j
• CategoryTheory.Limits.colimit.pre_desc 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] {C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.HasColimit F] (E : CategoryTheory.Functor K J) [CategoryTheory.Limits.HasColimit (E.comp F)] (c : CategoryTheory.Limits.Cocone F) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.pre F E) (CategoryTheory.Limits.colimit.desc F c) = CategoryTheory.Limits.colimit.desc (E.comp F) (CategoryTheory.Limits.Cocone.whisker E c)
• CategoryTheory.Limits.colimit.map_desc 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [CategoryTheory.Category.{v, u} C] {F G : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] [CategoryTheory.Limits.HasColimit G] (c : CategoryTheory.Limits.Cocone F) (α : G ⟶ F) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimMap α) (CategoryTheory.Limits.colimit.desc F c) = CategoryTheory.Limits.colimit.desc G ((CategoryTheory.Limits.Cocone.precompose α).obj c)
• CategoryTheory.Limits.colimit.ι_desc_assoc 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] (c : CategoryTheory.Limits.Cocone F) (j : J) {Z : C} (h : c.pt ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι F j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.desc F c) h) = CategoryTheory.CategoryStruct.comp (c.ι.app j) h
• CategoryTheory.Limits.HasColimit.isoOfNatIso_hom_desc 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [CategoryTheory.Category.{v, u} C] {F G : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] [CategoryTheory.Limits.HasColimit G] (t : CategoryTheory.Limits.Cocone G) (w : F ≅ G) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.HasColimit.isoOfNatIso w).hom (CategoryTheory.Limits.colimit.desc G t) = CategoryTheory.Limits.colimit.desc F ((CategoryTheory.Limits.Cocone.precompose w.hom).obj t)
• CategoryTheory.Limits.HasColimit.isoOfNatIso_inv_desc 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [CategoryTheory.Category.{v, u} C] {F G : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] [CategoryTheory.Limits.HasColimit G] (t : CategoryTheory.Limits.Cocone F) (w : F ≅ G) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.HasColimit.isoOfNatIso w).inv (CategoryTheory.Limits.colimit.desc F t) = CategoryTheory.Limits.colimit.desc G ((CategoryTheory.Limits.Cocone.precompose w.inv).obj t)
• CategoryTheory.Limits.colimit.post_desc 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor J C) {D : Type u'} [CategoryTheory.Category.{v', u'} D] [CategoryTheory.Limits.HasColimit F] (G : CategoryTheory.Functor C D) [CategoryTheory.Limits.HasColimit (F.comp G)] (c : CategoryTheory.Limits.Cocone F) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.post F G) (G.map (CategoryTheory.Limits.colimit.desc F c)) = CategoryTheory.Limits.colimit.desc (F.comp G) (G.mapCocone c)
• CategoryTheory.Limits.colimit.pre_desc_assoc 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] {C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.HasColimit F] (E : CategoryTheory.Functor K J) [CategoryTheory.Limits.HasColimit (E.comp F)] (c : CategoryTheory.Limits.Cocone F) {Z : C} (h : c.pt ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.pre F E) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.desc F c) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.desc (E.comp F) (CategoryTheory.Limits.Cocone.whisker E c)) h
• CategoryTheory.Limits.colimit.map_desc_assoc 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [CategoryTheory.Category.{v, u} C] {F G : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] [CategoryTheory.Limits.HasColimit G] (c : CategoryTheory.Limits.Cocone F) (α : G ⟶ F) {Z : C} (h : c.pt ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimMap α) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.desc F c) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.desc G ((CategoryTheory.Limits.Cocone.precompose α).obj c)) h
• CategoryTheory.Limits.HasColimit.isoOfNatIso_hom_desc_assoc 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [CategoryTheory.Category.{v, u} C] {F G : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] [CategoryTheory.Limits.HasColimit G] (t : CategoryTheory.Limits.Cocone G) (w : F ≅ G) {Z : C} (h : t.pt ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.HasColimit.isoOfNatIso w).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.desc G t) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.desc F ((CategoryTheory.Limits.Cocone.precompose w.hom).obj t)) h
• CategoryTheory.Limits.HasColimit.isoOfNatIso_inv_desc_assoc 📋 Mathlib.CategoryTheory.Limits.HasLimits
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [CategoryTheory.Category.{v, u} C] {F G : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] [CategoryTheory.Limits.HasColimit G] (t : CategoryTheory.Limits.Cocone F) (w : F ≅ G) {Z : C} (h : t.pt ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.HasColimit.isoOfNatIso w).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.desc F t) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.desc G ((CategoryTheory.Limits.Cocone.precompose w.inv).obj t)) h
• CategoryTheory.Limits.cokernelBiproductιIso_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {J : Type w} {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (f : J → C) (i : J) [CategoryTheory.Limits.HasBiproduct f] [CategoryTheory.Limits.HasBiproduct (Subtype.restrict (fun j => j ≠ i) f)] : (CategoryTheory.Limits.cokernelBiproductιIso f i).hom = CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.biproduct.ι f i) 0) (CategoryTheory.Limits.CokernelCofork.ofπ (CategoryTheory.Limits.biproduct.toSubtype f fun j => ¬j = i) ⋯)
• CategoryTheory.Limits.cokernelBiproductFromSubtypeIso_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.Biproducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K : Type} [Finite K] [CategoryTheory.Limits.HasFiniteBiproducts C] (f : K → C) (p : K → Prop) : (CategoryTheory.Limits.cokernelBiproductFromSubtypeIso f p).hom = CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.parallelPair (CategoryTheory.Limits.biproduct.fromSubtype f p) 0) (CategoryTheory.Limits.CokernelCofork.ofπ (CategoryTheory.Limits.biproduct.toSubtype f pᶜ) ⋯)
• CategoryTheory.Limits.cokernelBiprodInlIso_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : CategoryTheory.Limits.cokernelBiprodInlIso.hom = CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.parallelPair CategoryTheory.Limits.biprod.inl 0) (CategoryTheory.Limits.biprod.inlCokernelCofork X Y)
• CategoryTheory.Limits.cokernelBiprodInrIso_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : CategoryTheory.Limits.cokernelBiprodInrIso.hom = CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.parallelPair CategoryTheory.Limits.biprod.inr 0) (CategoryTheory.Limits.biprod.inrCokernelCofork X Y)
• CategoryTheory.Limits.colimit.ι_desc_apply 📋 Mathlib.CategoryTheory.ConcreteCategory.Elementwise
  {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] (c : CategoryTheory.Limits.Cocone F) (j : J) {F✝ : C → C → Type uF} {carrier : C → Type w} {instFunLike : (X Y : C) → FunLike (F✝ X Y) (carrier X) (carrier Y)} [inst : CategoryTheory.ConcreteCategory C F✝] (x : carrier (F.obj j)) : (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.colimit.desc F c)) ((CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.colimit.ι F j)) x) = (CategoryTheory.ConcreteCategory.hom (c.ι.app j)) x
• CategoryTheory.Limits.Types.Colimit.ι_desc_apply 📋 Mathlib.CategoryTheory.Limits.Types.Colimits
  {J : Type v} [CategoryTheory.Category.{w, v} J] (F : CategoryTheory.Functor J (Type u)) [CategoryTheory.Limits.HasColimit F] (s : CategoryTheory.Limits.Cocone F) (j : J) (x : F.obj j) : (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.colimit.desc F s)) ((CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.colimit.ι F j)) x) = (CategoryTheory.ConcreteCategory.hom (s.ι.app j)) x
• CategoryTheory.preservesColimitIso_inv_comp_desc 📋 Mathlib.CategoryTheory.Limits.Preserves.Limits
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesColimit F G] [CategoryTheory.Limits.HasColimit F] (t : CategoryTheory.Limits.Cocone F) : CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesColimitIso G F).inv (G.map (CategoryTheory.Limits.colimit.desc F t)) = CategoryTheory.Limits.colimit.desc (F.comp G) (G.mapCocone t)
• CategoryTheory.preservesColimitIso_inv_comp_desc_assoc 📋 Mathlib.CategoryTheory.Limits.Preserves.Limits
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesColimit F G] [CategoryTheory.Limits.HasColimit F] (t : CategoryTheory.Limits.Cocone F) {Z : D} (h : G.obj t.pt ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesColimitIso G F).inv (CategoryTheory.CategoryStruct.comp (G.map (CategoryTheory.Limits.colimit.desc F t)) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.desc (F.comp G) (G.mapCocone t)) h
• CategoryTheory.Limits.colimit.ι_desc_app 📋 Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] (H : CategoryTheory.Functor J (CategoryTheory.Functor K C)) [CategoryTheory.Limits.HasColimit H] (c : CategoryTheory.Limits.Cocone H) (j : J) (k : K) : CategoryTheory.CategoryStruct.comp ((CategoryTheory.Limits.colimit.ι H j).app k) ((CategoryTheory.Limits.colimit.desc H c).app k) = (c.ι.app j).app k
• CategoryTheory.Limits.colimit.ι_desc_app_assoc 📋 Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
  {C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] (H : CategoryTheory.Functor J (CategoryTheory.Functor K C)) [CategoryTheory.Limits.HasColimit H] (c : CategoryTheory.Limits.Cocone H) (j : J) (k : K) {Z : C} (h : c.pt.obj k ⟶ Z) : CategoryTheory.CategoryStruct.comp ((CategoryTheory.Limits.colimit.ι H j).app k) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.Limits.colimit.desc H c).app k) h) = CategoryTheory.CategoryStruct.comp ((c.ι.app j).app k) h
• CategoryTheory.Limits.coconeFiberwiseColimitOfCocone_ι_app 📋 Mathlib.CategoryTheory.Limits.Shapes.Grothendieck
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {F : CategoryTheory.Functor C CategoryTheory.Cat} {H : Type u₂} [CategoryTheory.Category.{v₂, u₂} H] {G : CategoryTheory.Functor (CategoryTheory.Grothendieck F) H} [∀ {X Y : C} (f : X ⟶ Y), CategoryTheory.Limits.HasColimit ((F.map f).toFunctor.comp ((CategoryTheory.Grothendieck.ι F Y).comp G))] (c : CategoryTheory.Limits.Cocone G) (X : C) : (CategoryTheory.Limits.coconeFiberwiseColimitOfCocone c).ι.app X = CategoryTheory.Limits.colimit.desc ((CategoryTheory.Grothendieck.ι F X).comp G) (CategoryTheory.Limits.Cocone.whisker (CategoryTheory.Grothendieck.ι F X) c)
• CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone_isColimit_desc 📋 Mathlib.CategoryTheory.Limits.Constructions.Filtered
  {C : Type u} [CategoryTheory.Category.{v, u} C] {α : Type w} [CategoryTheory.Limits.HasFiniteCoproducts C] [CategoryTheory.Limits.HasColimitsOfShape (Finset (CategoryTheory.Discrete α)) C] (F : CategoryTheory.Functor (CategoryTheory.Discrete α) C) (s : CategoryTheory.Limits.Cocone F) : (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone F).isColimit.desc s = CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.CoproductsFromFiniteFiltered.liftToFinsetObj F) { pt := s.pt, ι := { app := fun x => CategoryTheory.Limits.Sigma.desc fun x_1 => s.ι.app ↑x_1, naturality := ⋯ } }
• CategoryTheory.Functor.pointwiseLeftKanExtension_desc_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasPointwiseLeftKanExtension F] (G : CategoryTheory.Functor D H) (α : F ⟶ L.comp G) (Y : D) : ((L.pointwiseLeftKanExtension F).descOfIsLeftKanExtension (L.pointwiseLeftKanExtensionUnit F) G α).app Y = CategoryTheory.Limits.colimit.desc ((CategoryTheory.CostructuredArrow.proj L Y).comp F) (L.costructuredArrowMapCocone F G α Y)
• CategoryTheory.Functor.pointwiseLeftKanExtension_map 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasPointwiseLeftKanExtension F] {Y₁ Y₂ : D} (f : Y₁ ⟶ Y₂) : (L.pointwiseLeftKanExtension F).map f = CategoryTheory.Limits.colimit.desc ((CategoryTheory.CostructuredArrow.proj L Y₁).comp F) { pt := CategoryTheory.Limits.colimit ((CategoryTheory.CostructuredArrow.proj L Y₂).comp F), ι := { app := fun g => CategoryTheory.Limits.colimit.ι ((CategoryTheory.CostructuredArrow.proj L Y₂).comp F) ((CategoryTheory.CostructuredArrow.map f).obj g), naturality := ⋯ } }
• AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.descCApp 📋 Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
  {J : Type u'} [CategoryTheory.Category.{v', u'} J] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimitsOfShape J TopCat] [∀ (X : TopCat), CategoryTheory.Limits.HasLimitsOfShape Jᵒᵖ (TopCat.Presheaf C X)] [CategoryTheory.Limits.HasLimitsOfShape Jᵒᵖ C] (F : CategoryTheory.Functor J (AlgebraicGeometry.PresheafedSpace C)) (s : CategoryTheory.Limits.Cocone F) (U : (TopologicalSpace.Opens ↑↑s.pt)ᵒᵖ) : s.pt.presheaf.obj U ⟶ ((TopCat.Presheaf.pushforward C (CategoryTheory.Limits.colimit.desc (F.comp (AlgebraicGeometry.PresheafedSpace.forget C)) ((AlgebraicGeometry.PresheafedSpace.forget C).mapCocone s))).obj (CategoryTheory.Limits.limit (AlgebraicGeometry.PresheafedSpace.pushforwardDiagramToColimit F).leftOp)).obj U
• AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.desc_c_naturality 📋 Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
  {J : Type u'} [CategoryTheory.Category.{v', u'} J] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimitsOfShape J TopCat] [∀ (X : TopCat), CategoryTheory.Limits.HasLimitsOfShape Jᵒᵖ (TopCat.Presheaf C X)] [CategoryTheory.Limits.HasLimitsOfShape Jᵒᵖ C] (F : CategoryTheory.Functor J (AlgebraicGeometry.PresheafedSpace C)) (s : CategoryTheory.Limits.Cocone F) {U V : (TopologicalSpace.Opens ↑↑s.pt)ᵒᵖ} (i : U ⟶ V) : CategoryTheory.CategoryStruct.comp (s.pt.presheaf.map i) (AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.descCApp F s V) = CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.descCApp F s U) (((TopCat.Presheaf.pushforward C (CategoryTheory.Limits.colimit.desc (F.comp (AlgebraicGeometry.PresheafedSpace.forget C)) ((AlgebraicGeometry.PresheafedSpace.forget C).mapCocone s))).obj (AlgebraicGeometry.PresheafedSpace.colimitCocone F).pt.presheaf).map i)
• CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.associator_hom_left 📋 Mathlib.CategoryTheory.Monoidal.PushoutProduct
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasPushouts C] [CategoryTheory.MonoidalCategory C] (X₁ X₂ X₃ : CategoryTheory.Arrow C) [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) (CategoryTheory.MonoidalCategory.tensorRight X₃.left)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) (CategoryTheory.MonoidalCategory.tensorRight X₃.right)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom)) (CategoryTheory.MonoidalCategory.tensorLeft X₁.left)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom)) (CategoryTheory.MonoidalCategory.tensorLeft X₁.right)] : (CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.associator X₁ X₂ X₃).hom.left = CategoryTheory.Limits.pushout.desc (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.associator X₁.right X₂.right X₃.left).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.right (CategoryTheory.Limits.pushout.inl (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom))) (CategoryTheory.Limits.pushout.inl (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom (CategoryTheory.Limits.pushout (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom))) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left (CategoryTheory.Limits.pushout.desc (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.right X₃.hom) (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.right) ⋯))))) (CategoryTheory.CategoryStruct.comp ⋯.isoPushout.hom (CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) X₃.right) (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom) X₃.right)) ((CategoryTheory.Limits.Cocone.precompose (CategoryTheory.Limits.spanExt (CategoryTheory.MonoidalCategoryStruct.associator X₁.left X₂.left X₃.right) (CategoryTheory.MonoidalCategoryStruct.associator X₁.right X₂.left X₃.right) (CategoryTheory.MonoidalCategoryStruct.associator X₁.left X₂.right X₃.right) ⋯ ⋯).hom).obj (CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.right (CategoryTheory.Limits.pushout.inr (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom))) (CategoryTheory.Limits.pushout.inl (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom (CategoryTheory.Limits.pushout (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom))) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left (CategoryTheory.Limits.pushout.desc (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.right X₃.hom) (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.right) ⋯)))) (CategoryTheory.Limits.pushout.inr (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom (CategoryTheory.Limits.pushout (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom))) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left (CategoryTheory.Limits.pushout.desc (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.right X₃.hom) (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.right) ⋯))) ⋯)))) ⋯
• CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.associator_inv_left 📋 Mathlib.CategoryTheory.Monoidal.PushoutProduct
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasPushouts C] [CategoryTheory.MonoidalCategory C] (X₁ X₂ X₃ : CategoryTheory.Arrow C) [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) (CategoryTheory.MonoidalCategory.tensorRight X₃.left)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) (CategoryTheory.MonoidalCategory.tensorRight X₃.right)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom)) (CategoryTheory.MonoidalCategory.tensorLeft X₁.left)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom)) (CategoryTheory.MonoidalCategory.tensorLeft X₁.right)] : (CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.associator X₁ X₂ X₃).inv.left = CategoryTheory.Limits.pushout.desc (CategoryTheory.CategoryStruct.comp ⋯.isoPushout.hom (CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.right (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left)) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.right (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom))) ((CategoryTheory.Limits.Cocone.precompose (CategoryTheory.Limits.spanExt (CategoryTheory.MonoidalCategoryStruct.associator X₁.right X₂.left X₃.left).symm (CategoryTheory.MonoidalCategoryStruct.associator X₁.right X₂.right X₃.left).symm (CategoryTheory.MonoidalCategoryStruct.associator X₁.right X₂.left X₃.right).symm ⋯ ⋯).hom).obj (CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.Limits.pushout.inl (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.Limits.pushout.desc (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.right X₂.hom) (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.right) ⋯) X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft (CategoryTheory.Limits.pushout (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) X₃.hom)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.Limits.pushout.inl (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) X₃.right) (CategoryTheory.Limits.pushout.inr (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.Limits.pushout.desc (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.right X₂.hom) (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.right) ⋯) X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft (CategoryTheory.Limits.pushout (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) X₃.hom))) ⋯)))) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.associator X₁.left X₂.right X₃.right).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.Limits.pushout.inr (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) X₃.right) (CategoryTheory.Limits.pushout.inr (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.Limits.pushout.desc (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.right X₂.hom) (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.right) ⋯) X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft (CategoryTheory.Limits.pushout (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) X₃.hom)))) ⋯
• CategoryTheory.Monad.MonadicityInternal.comparisonAdjunction_counit_app 📋 Mathlib.CategoryTheory.Monad.Monadicity
  {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₁, u₂} D] {G : CategoryTheory.Functor D C} {F : CategoryTheory.Functor C D} (adj : F ⊣ G) [∀ (A : adj.toMonad.Algebra), CategoryTheory.Limits.HasCoequalizer (F.map A.a) (adj.counit.app (F.obj A.A))] (B : D) : (CategoryTheory.Monad.MonadicityInternal.comparisonAdjunction adj).counit.app B = CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.parallelPair (F.map (G.map (adj.counit.app B))) (adj.counit.app (F.obj (G.obj B)))) (CategoryTheory.Monad.MonadicityInternal.counitCofork adj B)
• CategoryTheory.MonoidalCategory.Limits.colimit.whiskerLeft_ι_desc 📋 Mathlib.CategoryTheory.Monoidal.Limits.HasLimits
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {F : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] (c : CategoryTheory.Limits.Cocone F) (j : J) {Q : C} : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft Q (CategoryTheory.Limits.colimit.ι F j)) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft Q (CategoryTheory.Limits.colimit.desc F c)) = CategoryTheory.MonoidalCategoryStruct.whiskerLeft Q (c.ι.app j)
• CategoryTheory.MonoidalCategory.Limits.colimit.ι_desc_whiskerRight 📋 Mathlib.CategoryTheory.Monoidal.Limits.HasLimits
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {F : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] (c : CategoryTheory.Limits.Cocone F) (j : J) {Q : C} : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.Limits.colimit.ι F j) Q) (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.Limits.colimit.desc F c) Q) = CategoryTheory.MonoidalCategoryStruct.whiskerRight (c.ι.app j) Q
• CategoryTheory.MonoidalCategory.Limits.colimit.whiskerLeft_ι_desc_assoc 📋 Mathlib.CategoryTheory.Monoidal.Limits.HasLimits
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {F : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] (c : CategoryTheory.Limits.Cocone F) (j : J) {Q Z : C} (h : CategoryTheory.MonoidalCategoryStruct.tensorObj Q c.pt ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft Q (CategoryTheory.Limits.colimit.ι F j)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft Q (CategoryTheory.Limits.colimit.desc F c)) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft Q (c.ι.app j)) h
• CategoryTheory.MonoidalCategory.Limits.colimit.ι_desc_whiskerRight_assoc 📋 Mathlib.CategoryTheory.Monoidal.Limits.HasLimits
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {F : CategoryTheory.Functor J C} [CategoryTheory.Limits.HasColimit F] (c : CategoryTheory.Limits.Cocone F) (j : J) {Q Z : C} (h : CategoryTheory.MonoidalCategoryStruct.tensorObj c.pt Q ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.Limits.colimit.ι F j) Q) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.Limits.colimit.desc F c) Q) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight (c.ι.app j) Q) h
• LightCondensed.lanPresheafIso_hom 📋 Mathlib.Condensed.Discrete.Colimit
  {S : LightProfinite} {F : CategoryTheory.Functor LightProfiniteᵒᵖ (Type u)} (hF : CategoryTheory.Limits.IsColimit (F.mapCocone (CategoryTheory.Limits.coconeRightOpOfCone S.asLimitCone))) : (LightCondensed.lanPresheafIso hF).hom = CategoryTheory.Limits.colimit.desc ((CategoryTheory.CostructuredArrow.proj FintypeCat.toLightProfinite.op (Opposite.op S)).comp (FintypeCat.toLightProfinite.op.comp F)) (LightProfinite.Extend.cocone F S)
• LightCondensed.lanPresheafNatIso_hom_app 📋 Mathlib.Condensed.Discrete.Colimit
  {F : CategoryTheory.Functor LightProfiniteᵒᵖ (Type u)} (hF : (S : LightProfinite) → CategoryTheory.Limits.IsColimit (F.mapCocone (CategoryTheory.Limits.coconeRightOpOfCone S.asLimitCone))) (S : LightProfiniteᵒᵖ) : (LightCondensed.lanPresheafNatIso hF).hom.app S = CategoryTheory.Limits.colimit.desc ((CategoryTheory.CostructuredArrow.proj FintypeCat.toLightProfinite.op (Opposite.op (Opposite.unop S))).comp (FintypeCat.toLightProfinite.op.comp F)) (LightProfinite.Extend.cocone F (Opposite.unop S))
• Condensed.lanPresheafIso_hom 📋 Mathlib.Condensed.Discrete.Colimit
  {S : Profinite} {F : CategoryTheory.Functor Profiniteᵒᵖ (Type (u + 1))} (hF : CategoryTheory.Limits.IsColimit (F.mapCocone S.asLimitCone.op)) : (Condensed.lanPresheafIso hF).hom = CategoryTheory.Limits.colimit.desc ((CategoryTheory.CostructuredArrow.proj FintypeCat.toProfinite.op (Opposite.op S)).comp (FintypeCat.toProfinite.op.comp F)) (Profinite.Extend.cocone F S)
• Condensed.lanPresheafNatIso_hom_app 📋 Mathlib.Condensed.Discrete.Colimit
  {F : CategoryTheory.Functor Profiniteᵒᵖ (Type (u + 1))} (hF : (S : Profinite) → CategoryTheory.Limits.IsColimit (F.mapCocone S.asLimitCone.op)) (S : Profiniteᵒᵖ) : (Condensed.lanPresheafNatIso hF).hom.app S = CategoryTheory.Limits.colimit.desc ((CategoryTheory.CostructuredArrow.proj FintypeCat.toProfinite.op (Opposite.op (Opposite.unop S))).comp (FintypeCat.toProfinite.op.comp F)) (Profinite.Extend.cocone F (Opposite.unop S))

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:

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.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : 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
🔍 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. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c