Last active
May 28, 2018 13:38
-
-
Save kbuzzard/e051858b8e3348e884610ace8cd87c20 to your computer and use it in GitHub Desktop.
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
import algebra.ring | |
structure pre_semi_sheaf_of_rings (α : Type) := | |
(F : Π (U : set α), Type) | |
[Fring : ∀ (U : set α), comm_ring (F U)] | |
attribute [instance] pre_semi_sheaf_of_rings.Fring | |
structure morphism_of_pre_semi_sheaves_of_rings {α : Type} | |
(FPT : pre_semi_sheaf_of_rings α) (GPT : pre_semi_sheaf_of_rings α) := | |
(morphism : ∀ U : set α, (FPT.F U) → GPT.F U) | |
(ring_homs : ∀ U : set α, @is_ring_hom _ _ _ _ (morphism U)) | |
attribute [instance] morphism_of_pre_semi_sheaves_of_rings.ring_homs | |
def is_identity_morphism_of_pre_semi_sheaves_of_rings {α : Type} | |
{FPT : pre_semi_sheaf_of_rings α} (phi: morphism_of_pre_semi_sheaves_of_rings FPT FPT) := | |
∀ (U : set α), phi.morphism U = id | |
def composition_of_morphisms_of_pre_semi_sheaves_of_rings {α : Type} | |
{FPT GPT HPT : pre_semi_sheaf_of_rings α} (fg : morphism_of_pre_semi_sheaves_of_rings FPT GPT) | |
(gh : morphism_of_pre_semi_sheaves_of_rings GPT HPT) : | |
morphism_of_pre_semi_sheaves_of_rings FPT HPT := | |
{ morphism := λ U, gh.morphism U ∘ fg.morphism U, | |
ring_homs := λ U, is_ring_hom.comp _ _ -- composition of two ring homs is a ring hom done by type class inference | |
} | |
def are_isomorphic_pre_semi_sheaves_of_rings {α : Type} | |
(FPR : pre_semi_sheaf_of_rings α) (GPR : pre_semi_sheaf_of_rings α) : Prop := | |
∃ (fg : morphism_of_pre_semi_sheaves_of_rings FPR GPR) (gf : morphism_of_pre_semi_sheaves_of_rings GPR FPR), | |
is_identity_morphism_of_pre_semi_sheaves_of_rings (composition_of_morphisms_of_pre_semi_sheaves_of_rings fg gf) | |
∧ is_identity_morphism_of_pre_semi_sheaves_of_rings (composition_of_morphisms_of_pre_semi_sheaves_of_rings gf fg) | |
definition pre_semi_sheaf_of_rings_pullback | |
{α : Type} | |
{β : Type} | |
(PR : pre_semi_sheaf_of_rings β) | |
(f : α → β) | |
: pre_semi_sheaf_of_rings α := | |
{ F := λ V,PR.F (f '' V) | |
} | |
theorem presheaves_iso (X : Type) (F : pre_semi_sheaf_of_rings X) : | |
are_isomorphic_pre_semi_sheaves_of_rings | |
(pre_semi_sheaf_of_rings_pullback F id) F | |
:= | |
begin | |
constructor,tactic.swap, | |
{ constructor,tactic.swap, | |
{ intros U HU, | |
change (pre_semi_sheaf_of_rings_pullback F id).F U at HU, | |
change (λ V, F.F (id '' V)) U at HU, -- why did I have to do that? | |
sorry | |
},sorry, | |
}, | |
sorry | |
end |
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment