# mathlib3documentation

algebraic_geometry.presheafed_space.has_colimits

# PresheafedSpace C has colimits. #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

If C has limits, then the category PresheafedSpace C has colimits, and the forgetful functor to Top preserves these colimits.

When restricted to a diagram where the underlying continuous maps are open embeddings, this says that we can glue presheaved spaces.

Given a diagram F : J ⥤ PresheafedSpace C, we first build the colimit of the underlying topological spaces, as colimit (F ⋙ PresheafedSpace.forget C). Call that colimit space X.

Our strategy is to push each of the presheaves F.obj j forward along the continuous map colimit.ι (F ⋙ PresheafedSpace.forget C) j to X. Since pushforward is functorial, we obtain a diagram J ⥤ (presheaf C X)ᵒᵖ of presheaves on a single space X. (Note that the arrows now point the other direction, because this is the way PresheafedSpace C is set up.)

The limit of this diagram then constitutes the colimit presheaf.

@[simp]
theorem algebraic_geometry.PresheafedSpace.map_id_c_app {J : Type u'} {C : Type u} (F : J ) (j : J) (U : topological_space.opens ((F.obj j).carrier)) :
@[simp]
theorem algebraic_geometry.PresheafedSpace.map_comp_c_app {J : Type u'} {C : Type u} (F : J ) {j₁ j₂ j₃ : J} (f : j₁ j₂) (g : j₂ j₃) (U : topological_space.opens ((F.obj j₃).carrier)) :
(F.map (f g)).c.app (opposite.op U) = (F.map g).c.app (opposite.op U) (F.map f).c).app (opposite.op U) (F.map f).base (F.map g).base).inv.app (opposite.op U) (F.obj j₁).presheaf).hom.app (opposite.op U)
@[simp]
@[simp]
theorem algebraic_geometry.PresheafedSpace.componentwise_diagram_map {J : Type u'} {C : Type u} (F : J ) (j k : Jᵒᵖ) (f : j k) :
noncomputable def algebraic_geometry.PresheafedSpace.componentwise_diagram {J : Type u'} {C : Type u} (F : J )  :

Given a diagram of PresheafedSpace Cs, its colimit is computed by pushing the sheaves onto the colimit of the underlying spaces, and taking componentwise limit. This is the componentwise diagram for an open set U of the colimit of the underlying spaces.

Equations
@[simp]
@[simp]
theorem algebraic_geometry.PresheafedSpace.pushforward_diagram_to_colimit_map {J : Type u'} {C : Type u} (F : J ) (j j' : J) (f : j j') :
noncomputable def algebraic_geometry.PresheafedSpace.pushforward_diagram_to_colimit {J : Type u'} {C : Type u} (F : J ) :

Given a diagram of presheafed spaces, we can push all the presheaves forward to the colimit X of the underlying topological spaces, obtaining a diagram in (presheaf C X)ᵒᵖ.

Equations
noncomputable def algebraic_geometry.PresheafedSpace.colimit {J : Type u'} {C : Type u} [ (X : Top), ] (F : J ) :

Auxiliary definition for PresheafedSpace.has_colimits.

Equations
@[simp]
@[simp]
@[simp]
noncomputable def algebraic_geometry.PresheafedSpace.colimit_cocone {J : Type u'} {C : Type u} [ (X : Top), ] (F : J ) :

Auxiliary definition for PresheafedSpace.has_colimits.

Equations
@[simp]
@[simp]
theorem algebraic_geometry.PresheafedSpace.colimit_cocone_ι_app_base {J : Type u'} {C : Type u} [ (X : Top), ] (F : J ) (j : J) :

Auxiliary definition for PresheafedSpace.colimit_cocone_is_colimit.

Equations
noncomputable def algebraic_geometry.PresheafedSpace.colimit_cocone_is_colimit.desc {J : Type u'} {C : Type u} [ (X : Top), ] (F : J )  :

Auxiliary definition for PresheafedSpace.colimit_cocone_is_colimit.

Equations

Auxiliary definition for PresheafedSpace.has_colimits.

Equations
@[protected, instance]
@[protected, instance]
Equations
@[protected, instance]

When C has limits, the category of presheaved spaces with values in C itself has colimits.

@[protected, instance]

The underlying topological space of a colimit of presheaved spaces is the colimit of the underlying topological spaces.

Equations

The components of the colimit of a diagram of PresheafedSpace C is obtained via taking componentwise limits.

Equations
@[simp]
@[simp]