# Documentation

Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits

# PresheafedSpace C has colimits. #

If C has limits, then the category PresheafedSpace C has colimits, and the forgetful functor to TopCat 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 AlgebraicGeometry.PresheafedSpace.map_id_c_app {J : Type u'} [] {C : Type u} (j : J) (U : TopologicalSpace.Opens (F.obj j)) :
(F.map ).c.app () = CategoryTheory.CategoryStruct.comp ((TopCat.Presheaf.Pushforward.id (F.obj j).presheaf).inv.app ()) ((TopCat.Presheaf.pushforwardEq (_ : CategoryTheory.CategoryStruct.id (F.obj j) = (F.map ).base) (F.obj j).presheaf).hom.app ())
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theorem AlgebraicGeometry.PresheafedSpace.map_comp_c_app {J : Type u'} [] {C : Type u} {j₁ : J} {j₂ : J} {j₃ : J} (f : j₁ j₂) (g : j₂ j₃) (U : TopologicalSpace.Opens (F.obj j₃)) :
(F.map ).c.app () = CategoryTheory.CategoryStruct.comp ((F.map g).c.app ()) (CategoryTheory.CategoryStruct.comp ((TopCat.Presheaf.pushforwardMap (F.map g).base (F.map f).c).app ()) (CategoryTheory.CategoryStruct.comp ((TopCat.Presheaf.Pushforward.comp (F.obj j₁).presheaf (F.map f).base (F.map g).base).inv.app ()) ((TopCat.Presheaf.pushforwardEq (_ : CategoryTheory.CategoryStruct.comp (F.map f).base (F.map g).base = (F.map ).base) (F.obj j₁).presheaf).hom.app ())))
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theorem AlgebraicGeometry.PresheafedSpace.componentwiseDiagram_obj {J : Type u'} [] {C : Type u} (j : Jᵒᵖ) :
= (F.obj j.unop).presheaf.obj (Opposite.op ((TopologicalSpace.Opens.map ().base).obj U))
@[simp]
theorem AlgebraicGeometry.PresheafedSpace.componentwiseDiagram_map {J : Type u'} [] {C : Type u} {j : Jᵒᵖ} {k : Jᵒᵖ} (f : j k) :
= CategoryTheory.CategoryStruct.comp ((F.map f.unop).c.app (Opposite.op ((TopologicalSpace.Opens.map ().base).obj U))) ((F.obj k.unop).presheaf.map (CategoryTheory.eqToHom (_ : (TopologicalSpace.Opens.map (F.map f.unop).base).op.obj (Opposite.op ((TopologicalSpace.Opens.map ().base).obj U)) = Opposite.op ((TopologicalSpace.Opens.map ().base).obj U))))

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.

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@[simp]
theorem AlgebraicGeometry.PresheafedSpace.pushforwardDiagramToColimit_obj {J : Type u'} [] {C : Type u} (j : J) :
= Opposite.op ( _* (F.obj j).presheaf)

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.

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def AlgebraicGeometry.PresheafedSpace.colimit {J : Type u'} [] {C : Type u} [∀ (X : TopCat), ] :

Auxiliary definition for AlgebraicGeometry.PresheafedSpace.instHasColimits.

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theorem AlgebraicGeometry.PresheafedSpace.colimit_presheaf {J : Type u'} [] {C : Type u} [∀ (X : TopCat), ] :
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theorem AlgebraicGeometry.PresheafedSpace.colimitCocone_ι_app_c {J : Type u'} [] {C : Type u} [∀ (X : TopCat), ] (j : J) :
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theorem AlgebraicGeometry.PresheafedSpace.colimitCocone_ι_app_base {J : Type u'} [] {C : Type u} [∀ (X : TopCat), ] (j : J) :
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def AlgebraicGeometry.PresheafedSpace.colimitCocone {J : Type u'} [] {C : Type u} [∀ (X : TopCat), ] :

Auxiliary definition for AlgebraicGeometry.PresheafedSpace.instHasColimits.

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def AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.descCApp {J : Type u'} [] {C : Type u} [∀ (X : TopCat), ] (s : ) (U : (TopologicalSpace.Opens s.pt)ᵒᵖ) :
s.pt.presheaf.obj U ().obj U

Auxiliary definition for AlgebraicGeometry.PresheafedSpace.colimitCoconeIsColimit.

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theorem AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.desc_c_naturality {J : Type u'} [] {C : Type u} [∀ (X : TopCat), ] (s : ) {U : (TopologicalSpace.Opens s.pt)ᵒᵖ} {V : (TopologicalSpace.Opens s.pt)ᵒᵖ} (i : U V) :

Auxiliary definition for AlgebraicGeometry.PresheafedSpace.colimitCoconeIsColimit.

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theorem AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.desc_fac {J : Type u'} [] {C : Type u} [∀ (X : TopCat), ] (s : ) (j : J) :

Auxiliary definition for AlgebraicGeometry.PresheafedSpace.instHasColimits.

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• AlgebraicGeometry.PresheafedSpace.instPreservesColimitsOfShapePresheafedSpaceCategoryOfPresheafedSpacesTopCatInstTopCatLargeCategoryForget = CategoryTheory.Limits.PreservesColimitsOfShape.mk

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

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

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• AlgebraicGeometry.PresheafedSpace.forgetPreservesColimits = CategoryTheory.Limits.PreservesColimitsOfSize.mk

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

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@[simp]
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