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/-
Presheaf of toplogical rings.
-/
import topology.algebra.ring
import sheaves.presheaf_of_rings
universes u v
open topological_space
-- Definition of a presheaf of topological rings.
structure presheaf_of_topological_rings (α : Type u) [topological_space α]
extends presheaf_of_rings α :=
(Ftop : ∀ (U), topological_space (F U))
(Ftop_ring : ∀ (U), topological_ring (F U))
(res_continuous : ∀ (U V) (HVU : V ⊆ U), continuous (res U V HVU))
instance presheaf_of_topological_rings.has_coe {α : Type u} [topological_space α] :
has_coe (presheaf_of_topological_rings α) (presheaf α) :=
⟨λ F, F.to_presheaf⟩
instance presheaf_of_topological_rings.topological_space_sections {α : Type u} [topological_space α]
(F : presheaf_of_topological_rings α) (U : opens α) : topological_space (F U) :=
F.Ftop U
attribute [instance] presheaf_of_topological_rings.Ftop
attribute [instance] presheaf_of_topological_rings.Ftop_ring
instance presheaf_of_topological_rings.comm_ring {X : Type u} [topological_space X]
(F : presheaf_of_topological_rings X) (U : opens X) : comm_ring (F U) :=
F.Fring U
namespace presheaf_of_topological_rings
variables {α : Type u} {β : Type v} [topological_space α] [topological_space β]
-- Morphism of presheaf of topological rings.
structure morphism (F G : presheaf_of_topological_rings α)
extends presheaf.morphism F.to_presheaf G.to_presheaf :=
(ring_homs : ∀ (U), is_ring_hom (map U))
(continuous_homs : ∀ (U), continuous (map U))
local infix `⟶`:80 := morphism
def identity (F : presheaf_of_topological_rings α) : F ⟶ F :=
{ ring_homs := λ U, is_ring_hom.id,
continuous_homs := λ U, continuous_id,
..presheaf.id F.to_presheaf }
-- Isomorphic presheaves of rings.
local infix `⊚`:80 := presheaf.comp
structure iso (F G : presheaf_of_topological_rings α) :=
(mor : F ⟶ G)
(inv : G ⟶ F)
(mor_inv_id : mor.to_morphism ⊚ inv.to_morphism = presheaf.id F.to_presheaf)
(inv_mor_id : inv.to_morphism ⊚ mor.to_morphism = presheaf.id G.to_presheaf)
end presheaf_of_topological_rings
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