Linguistic Premise Premise of Algebraic Linguistics 1-4 / September 17, 2007

Linguistic Premise

 Premise of Algebraic Linguistics 1-4

　　  TANAKA Akio

17

Definition of <stalk and germ>

Topological space      X

Point on X        x

Presheaf of topological space X     F

Set of all the opened neighborhood    U

U, V ⊂ U        U ⊂ V

Order of U         U  ≺ V

r_UV  : F  ( U ) → F  ( V )   

Set of F  ( U )    

direct sum of the set     ⎿⏌F  ( U )

Elements of s, s’

∈ F  ( U )   s’ ∈ F  ( V )    

W ⊂U ∩V

s | _Ｗ = s’ | _Ｗ

s ∼ s’

Equivalence class contained s ∈ F  ( U )   　s_x

Homomorphism F  ( U ) → F  _x

F  _x  is stalk on x of F.

s_x  is germ on x of F.

18

Definition of <sheaf space>

Presheaf of commutative ring     F

F _d =   ⎿⏌_x∈X F _x

Map    p : F _d → X

F _d  ×_XF _d = { a, b | p(a) = p(b) }

F _d ×_X F _d ⊂ F _d × F _d

Conditions of F _d ×_X F _d

(1)

p is local isomorphism.

(2)

a) m : F _d ×_X F _d ∋(a, b) ↦ a + b ∈ F _d

b) l : F _d ∋ - a  ↦ F _d

F _d  is sheaf space of commutative group on X.

Tokyo September 17, 2007

Sekinan Research Field of Language