File: Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean

full_name: AlgebraicGeometry.SheafedSpace.isColimit_exists_rep
code: theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
    ∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x 
goal: ∃ i y, (c.ι.app i).base y = x

full_name: AlgebraicGeometry.SheafedSpace.colimit_exists_rep
code: theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
    ∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x 
goal: ∃ i y, (CategoryTheory.Limits.colimit.ι F i).base y = x

full_name: AlgebraicGeometry.SheafedSpace.instEpiTopCatBaseπ
code: instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base 
goal: CategoryTheory.Epi (CategoryTheory.Limits.coequalizer.π f g).base

full_name: AlgebraicGeometry.LocallyRingedSpace.coproduct
code: /-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace 
goal: AlgebraicGeometry.LocallyRingedSpace

full_name: AlgebraicGeometry.LocallyRingedSpace.coproductCofan
code: /-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F 
goal: CategoryTheory.Limits.Cocone F

full_name: AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
code: /-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) 
goal: CategoryTheory.Limits.IsColimit (AlgebraicGeometry.LocallyRingedSpace.coproductCofan F)

full_name: AlgebraicGeometry.LocallyRingedSpace.instHasCoproducts
code: instance : HasCoproducts.{u} LocallyRingedSpace.{u} 
goal: CategoryTheory.Limits.HasCoproducts AlgebraicGeometry.LocallyRingedSpace

full_name: AlgebraicGeometry.LocallyRingedSpace.instPreservesColimitsOfShapeSheafedSpaceCommRingCatDiscreteForgetToSheafedSpace
code: noncomputable instance (J : Type _) :
    PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} 
goal: CategoryTheory.Limits.PreservesColimitsOfShape (CategoryTheory.Discrete J)
  AlgebraicGeometry.LocallyRingedSpace.forgetToSheafedSpace

full_name: AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
code: instance coequalizer_π_app_isLocalRingHom
    (U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
    IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) 
goal: IsLocalRingHom ((CategoryTheory.Limits.coequalizer.π f.val g.val).c.app (Opposite.op U))

full_name: AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
code: /-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y 
goal: TopologicalSpace.Opens ↑Y.toTopCat

full_name: AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
code: theorem imageBasicOpen_image_preimage :
    (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
      (imageBasicOpen f g U s).1 
goal: ⇑(CategoryTheory.Limits.coequalizer.π f.val g.val).base ⁻¹'
    (⇑(CategoryTheory.Limits.coequalizer.π f.val g.val).base ''
      (AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen f g U s).carrier) =
  (AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen f g U s).carrier

full_name: AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
code: theorem imageBasicOpen_image_open :
    IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) 
goal: IsOpen
  (⇑(CategoryTheory.Limits.coequalizer.π f.val g.val).base ''
    (AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen f g U s).carrier)

full_name: AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
code: instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
    IsLocalRingHom ((coequalizer.π f.val g.val : _).stalkMap x) 
goal: IsLocalRingHom (AlgebraicGeometry.PresheafedSpace.Hom.stalkMap (CategoryTheory.Limits.coequalizer.π f.val g.val) x)

full_name: AlgebraicGeometry.LocallyRingedSpace.coequalizer
code: /-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace 
goal: AlgebraicGeometry.LocallyRingedSpace

full_name: AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
code: /-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g 
goal: CategoryTheory.Limits.Cofork f g

full_name: AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
code: theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
    (h : IsLocalRingHom (f.stalkMap x)) :
    IsLocalRingHom (g.stalkMap x) 
goal: IsLocalRingHom (AlgebraicGeometry.PresheafedSpace.Hom.stalkMap g x)

full_name: AlgebraicGeometry.LocallyRingedSpace.coequalizerCoforkIsColimit
code: /-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) 
goal: CategoryTheory.Limits.IsColimit (AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork f g)

full_name: AlgebraicGeometry.LocallyRingedSpace.instHasCoequalizer
code: instance : HasCoequalizer f g 
goal: CategoryTheory.Limits.HasCoequalizer f g

full_name: AlgebraicGeometry.LocallyRingedSpace.instHasCoequalizers
code: instance : HasCoequalizers LocallyRingedSpace 
goal: CategoryTheory.Limits.HasCoequalizers AlgebraicGeometry.LocallyRingedSpace

full_name: AlgebraicGeometry.LocallyRingedSpace.preservesCoequalizer
code: noncomputable instance preservesCoequalizer :
    PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} 
goal: CategoryTheory.Limits.PreservesColimitsOfShape CategoryTheory.Limits.WalkingParallelPair
  AlgebraicGeometry.LocallyRingedSpace.forgetToSheafedSpace

full_name: AlgebraicGeometry.LocallyRingedSpace.instHasColimits
code: instance : HasColimits LocallyRingedSpace 
goal: CategoryTheory.Limits.HasColimits AlgebraicGeometry.LocallyRingedSpace

full_name: AlgebraicGeometry.LocallyRingedSpace.preservesColimits_forgetToSheafedSpace
code: noncomputable instance preservesColimits_forgetToSheafedSpace :
    PreservesColimits LocallyRingedSpace.forgetToSheafedSpace.{u} 
goal: CategoryTheory.Limits.PreservesColimits AlgebraicGeometry.LocallyRingedSpace.forgetToSheafedSpace