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/-
Presheaf of rings.
https://stacks.math.columbia.edu/tag/006N
Author: Ramon Fernandez Mir
-/
import sheaves.presheaf
universes u v
-- Definition of a presheaf of rings.
structure presheaf_of_rings (α : Type u) [topological_space α]
extends presheaf α :=
(Fring : ∀ (U), comm_ring (F U))
(res_is_ring_hom : ∀ (U V) (HVU : V ⊆ U), is_ring_hom (res U V HVU))
instance {α : Type u} [topological_space α]
: has_coe (presheaf_of_rings α) (presheaf α)
:= ⟨λ F, F.to_presheaf⟩
attribute [instance] presheaf_of_rings.Fring
attribute [instance] presheaf_of_rings.res_is_ring_hom
instance presheaf_of_rings.comm_ring {α : Type u} [topological_space α]
(F : presheaf_of_rings α) (U : topological_space.opens α) : comm_ring (F U) :=
F.Fring U
namespace presheaf_of_rings
variables {α : Type u} {β : Type v} [topological_space α] [topological_space β]
-- Morphism of presheaf of rings.
structure morphism (F G : presheaf_of_rings α)
extends presheaf.morphism F.to_presheaf G.to_presheaf :=
(ring_homs : ∀ (U), is_ring_hom (map U))
local infix `⟶`:80 := morphism
def identity (F : presheaf_of_rings α) : F ⟶ F :=
{ ring_homs := λ U, is_ring_hom.id,
..presheaf.id F.to_presheaf }
-- Isomorphic presheaves of rings.
local infix `⊚`:80 := presheaf.comp
structure iso (F G : presheaf_of_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_rings
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