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Definition of <presheaf>
Topological space X
Arbitrary opened set U, V U < V
Commutative group F ( U )
Homomorphism τUV : F ( V ) → F ( U )
Given conditions
(1)
F ( 0 ) = { 0 }
(2)
τUV = idU (Identity map)
(3)
U ⊂ V ⊂ W τUW =τUV oτVV
Presheaf of commutative ring on X { F ( U ), τUV }
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Definition of <sheaf>
Presheaf F, G on X
Homomorphism ψ : F → G
Given conditions
(1)
Arbitrary opened set U
Homomorphism φ ( U ) : F ( U ) → G ( U )
(2)
Opened sets U < V
Below makes commutative diagram.
F ( V ) , G ( V ), F ( U ) , G ( U )
φ ( V ), φ.( U )
τUV,
(3)
F ( U ) ∋ s
Open covering U
{ U i }i ∈ I , r Ui, U = 0 i ∈ I ⇒ s = 0
(4)
Open covering U
{ U i }i ∈ I
F ( U ) ∋ si i ∈ I
rUi ∩Uj Ui (si ) = r Ui ∩Uj (sj) (i, j ∈ I ) ⇒ rUi, U (s) = si ( i ∈ I )
Presheaf is sheaf.
Tokyo September 17, 2007
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