Linguistic Premise Premise of Algebraic Linguistics 4-1 / September 29, 2007

Linguistic Premise

 Premise of Algebraic Linguistics 4-1

　　  TANAKA Akio

1 <presheaf>

Topological space     X

Open set of X     U

Abelian group     F ( U )     assumption F ( 0 ) = { 0 }

Element of F ( U )     Section in F over U

Map between sets     U → U ’

Restriction map      Homomorphism  r_UU’ ^F  : F ( U ’ ) → F ( U )     assumption r_UU ^F  = id  _F ( U )  and  r_UU’ ◯ r_{U’U’ ‘}= r_UU’’

Presheaf is contrafunctor from category of open set over X to category of abelian group.

Contrafunctor and category are the terms of category theory. The definitions are omitted now.

2 <sheaf>

Topological space     X

Open set of X     U

Union among finite or infinite open sets     U = ∪_λ∈ΛU_λ

U_λμ := U_λ∩U_μ

F ( U ) = Ker [ П_λ, μ∈Λ : П_λ∈ΛF ( U_λ ) → П_λ, μ∈ΛF ( U_λμ ) ]

Ker is kernel. Refer to

The upper formula expressed function’s globalization and localized functions’ putted globalized situation.

3 <structure sheaf>

Topological space     X

Open set of X     U

Topological manifold     M

Real number valued continuous functions over U     Г( U, O _M )

O _M  is structure sheaf that defines geometric structure. One of the generalized structures is scheme.

4 <stalk and germ>

Topological space     X

Open set of X     U

Abelian group     F ( U )     assumption F ( 0 ) = { 0 }

Point     P ∈ X

Neighborhood of the point     { U ; P ∈ U }

Stalk of presheaf at  P      direct limit F_p = lim _P ∈ U F ( U )

Element of stalk F_p      Germ of presheaf at P     

5 <homomorphism of presheaf>

Presheaf     F ,  G

Open set     U

Homomorphism f ( U ) : F ( U ) → G ( U )    

assumption  r_UU’ ^F  : F ( U ’ ) → F ( U )    F ( U ) ◯ r_UU’ ^F  =  r_UU’ ^G   ◯ F ( U ’ )

F ( U ) is subsheaf of G , when f ( U ) is surjection of subset.

6 <kernel of sheaf>

Sheaf     F ,  G

Open set     U

Homomorphism f ( U ) : F ( U ) → G ( U )

K is subsheaf of F , when K ( U ) = Ker ( f ( U ) ) .

K is kernel ,expressed by Ker ( f ).

7 <quotient sheaf>

Presheaf     G ⊂ F

Quotient sheaf F / G  is  associated from presheaf F ( U ) / G ( U ).

Tokyo September 29, 2007

Sekinan Research Field of Language