Linguistic Premise Premise of Algebraic Linguistics 1-3 / September 17, 2007

Linguistic Premise

 Premise of Algebraic Linguistics 1-3

　　  TANAKA Akio

15

Definition of <presheaf>

Topological space     X

Arbitrary opened set     U, V    U < V

Commutative group     F  ( U )

Homomorphism     τ_UV : F  ( V ) → F  ( U )

Given conditions

(1)

F  ( 0 ) = { 0 }

(2)

τ_UV  = id_U     (Identity map)

(3)

U ⊂ V ⊂ W     τ_UW  =τ_UV oτ_VV

Presheaf of commutative ring on X     { F ( U ), τ_UV }

16

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 )

τ_UV,

(3)

F  ( U ) ∋ s

Open covering U

{ U _i }_i ∈ I , r U_i, U = 0   i ∈ I   ⇒ s = 0

(4)

Open covering U

{ U _i }_i ∈ I

F ( U ) ∋ s_i     i ∈ I

rU_i ∩U_j U_i (s_i ) = r U_i ∩U_j (s_j) (i, j ∈ I )   ⇒ rU_i, U (s) = s_i ( i ∈ I )

Presheaf is sheaf.

Tokyo September 17, 2007

Sekinan Research Field of Language