Tuesday, 24 May 2016

Derived Category Language 3

Derived Category language 3

Language stability and triangulated category


[Epitome]
T. Bridgeland defined stability conditions on triangulated categories in 2007.
Language's stability presented by Sergej Karcevskij in 1928.
Bridgeland's stability seems to hint for  Karcevskij's stability.

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1.
[Overview of Bridgeland 2002]

Stability conditions on triangulated categories 

A is Abelian category.
K group K(A) is defined as the next.
Here is generated at complete exact sequence as the next.  
And 2  ~  E 1 + E 3.
Group homomorphism .
D is triangulated category.
Stability condition on D is defined as by the next.
Data ( Z, A ) satisfies the next condition.
  is bounded kernel of t-structure. gives stability condition on A .

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2.
[Paper on Karcevskij conjecture by TANAKA Akio 2011]




Condition of Meaning
TANAKA Akio
September 11, 2011
[Preparation]
Graded differential algebra 
Minimal model of graded differential algebra 
Degree of homogenious element x of graded differential algabra |x|
Basis of linear space is given by homogenious and elements x1, ....., xn
N  (V) =  L (V) k  = N  ( x 1 , .....,  x n  )
Operation of minimal model 
<Example>
Spherical surface Sn, n≥2
de Rham complex *(Sn)
When n iseaven number,
Volume element of S
Mn = Λ (x), |x| = ndx = 0,
M2n-1 gives minimal model Sn to de Rham complex  .
When n is odd number,
Mn gives minimal model Sn to de Rham complex  .
[Interpretation]
Word is given by spherical surface.
Meaning of word is given by elements x1, ....., xn.
Word has minimal model.
Word becomes formal.
Fundamental group of word contains free group of rank b1(M).
Here KARCEVSKIJ's "stable part" is identified to fundamental group and " mobile part" is identified to free group.

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3.
[Paper on description of language by TANAKA Akio 2011]

Description of Language
TANAKA Akio
September 9, 2011
<Preparation>
Manifold M
Cup product map of 
Dual map of  
Free Lie algebra that  generates £ (  )
 is identified as the partial space of £(  ) that quadrastic Lie bracket of  generates.
Ideal of £(  ) that Im η generates a 
Holonomy Lie algebra of M 
Completion of holonomy Lie algebra 
<Theorem>
If M has quadrastic homology connection, Malcev completion  becomes isomorphic with holonoly Lie algebra completion  .
<Therorem>
Primary minimum model M(1)of differential manifold M is isomorphic with Malcev completion of  of M's fundamental group.
<Inrterpretation>
For the description of a language model there is a need primary minimum model M(1) of differential manifold M.

This paper has been published by Sekinan Research Field of Language.
All rights reserved.
© 2011 by The Sekinan Research Field of Language

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4.
[Note]
If word satisfies Bridgeland 2002's Data{Z, A] , word has a stability in language.
For the problem of additional meaning refer to Karcevskij conjecture 1928 and Kawamata conjecture 2002.

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#This paper is unfinished

Tokyo
25 May 2016
SRFL Theory

Monday, 23 May 2016

Language stability and triangulated category

T. Bridgeland defined stability conditions on triangulated categories in 2007.
Language's stability presented by Sergej Karcevskij in 1928.
Bridgeland's stability seems to hint for  Karcevskij's stability.

This paper is unfinished.

Tokyo
23 May 2016
SRFL Theory

Sekinan Archive - Invitation to language universals

    Sekinan Archive - Invitation to language universals






 Invitation to Essays and Papers of Sekinan Library


Sekinan Archive shows the invitational data on language universals to the researchers.

Half farewell to Sergej Karcevskij and the Linguistic Circle of Prague
Road to Language Universals

​Tokyo
27 December 2015


Present Work​
Picture
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About

About

Author:
 TANAKA Akio

Born:
 1947 in Tokyo

Foundation:
1986 at Tachikawa, Tokyo as Sekinan Library
vide: 
Tachikawa, Youth Days               
Road to Language Universals
2003 at Hakuba, Nagano as SRFL Sekinan Research Field of Language
vide: 
Road to Language Universals                
     
Source of Sekinan:
Sekinan means shrub, rhododendron in Japanese. 

Teacher:
CHINO Eiichi
vide:
Coffee shop named California         
Under the Dim Light          
CHINO Eiichi and Golden Prague

                 
Inherited thinking:
The Linguistic Circle of Prague in 1920s
vide:
Linguistic Circle of Prague
Prague in 1920s, The Linguistic Circle of Prague and Sergej Karcevskij's paper "Du dualisme asymetrique du signe linguistique"


Tokyo
7 April 2015
Sekinan Study       



Read more: http://srflnote.webnode.com/about/

Additional meaning and embedding

Additional meaning and embedding


1.Derived category



Category theoretic mirror symmetry conjecture

1.
Fukaya category is
Fuk(X
ω).
2.
General symplectic manifold is
  .

3.
Derived category of A   is   .
4.
Conjecture
When
   and  
have physical mirror relation, 
there exists the next triangle category's equivalence



2.Embedding


Derived Category Language 2
Kawamata Conjecture

Conjecture

is birational map between smooth objective algebraic manifolds.
And
.
At This condition,
there exists next fully faithful embedding.



3.Stable and mobile



Condition of Meaning
TANAKA Akio
September 11, 2011
[Preparation]
Graded differential algebra 
Minimal model of graded differential algebra 
Degree of homogenious element x of graded differential algabra |x|
Basis of linear space is given by homogenious and elements x1, ....., xn
Λ (V) = Λ(V)k =Λ (x1, ....., xn )
Operation of minimal model 
<Example>
Spherical surface Sn, n≥2
de Rham complex *(Sn)
When n iseaven number,
Volume element of S
Mn = Λ (x), |x| = ndx = 0,
M2n-1 gives minimal model Sn to de Rham complex  .
When n is odd number,
Mn gives minimal model Sn to de Rham complex  .
[Interpretation]
Word is given by spherical surface.
Meaning of word is given by elements x1, ....., xn.
Word has minimal model.
Word becomes formal.
Fundamental group of word contains free group of rank b1(M).
Here KARCEVSKIJ's "stable part" is identified to fundamental group and " mobile part" is identified to free group.


This paper has been published by Sekinan Research Field of Language.
All rights reserved.
© 2011 by 
The Sekinan Research Field of Language


4.Additional meaning


For the Supposition of KARCEVSKIJ Sergej
Completion of Language
September 23, 2011
[Preparation]
1.
n dimensional complex space Cn
Open set 
Whole holomorphic function over 
Ring sheaf for 
U →Oan(U)
Complex analytic manifold Cann
Algebraic manifold An multinomial of Cann
Ideal of multinomial ring a  [x1x2, ..., xn]
V(a) = {(a1a2, ..., an Cn (a1a2, ..., an) = 0,  a }
Whole closed set of V(a
Fundamental open set D(f) = {(a1a2, ..., an Cn | (a1a2, ..., an) ≠ 0}
Arbitrary family of open set {Ui} 
Easy sheaf F 
Zariski topological space 
Ring sheaf O
Affine space An = (  , O)
Ring R
Set of whole maximum ideal Spm R1
Spm R Spectrum of R
<Proposition>
Spm is Noether- like.
<Proposition>
is integral domain.
Whole of open sets without null set Ux
Quotient field K
Mapping from Uto whole partial set of O
O(V(a)c) =  Rf
c expresses complementary set.
O is easy sheaf of ring over Spm R that is whole set K.
<Definition>
R is finite generative integral domain over k.
Triple (i) (ii) (iii) is called affine algebraic variety.
(i) Set Spm R
(ii) Zariski topology
(iii) Ring's sheaf O
is called structure sheaf of affine algebraic variety.
Ring homomorphism between definite generative integral domains 
Upper is expressed by  .
Ring holomorphism OX(U) → OY((t )-1U)
Morphism from affine algebraic variety Y to ( OX(U) → OY((t )-1U), X)
When  is surjection, t is isomorphism overclosed partial set defined by p= Ker  .
Upper is called to closed immersion.
2.
Ring holomorphism 
Morphism between affine algebraic varieties 
Kernel of  p
Image of  
<Definition>
It is called that when  is injection is dominant.
<Definition>
R is medium ring between S and its quotient field K.
When  that is given by natural injection  is isomorphism over open set,  is called open immersion.
<Definition>
When X is algebraic variety, longitude of maximum chain is equal to transcendental dimension of function field k(X).
It is called dimension of algebraic variety X, expressed by dim X.
<Definition>
Defined generative field over k K
Space ( X, Ox )added ring that is whole sets of K that has open covers {Ui} satisfies next conditions is called algebraic variety.
(i) Each Ui is affine algebraic variety that has quotient K .
(ii) For each i, j  I, intersection  is open partial set of  .
3.
<Definition>
Tensor product between ring and itself becomes ring by each elements products.
Elements  that defines surjective homomorphism is expressed by  .
Image  of closed embedding defined by  is called diagonal.
<Definition>
Field K
Ringed space that have common whole set K (A, OA) (B, OB)
Topological space C
Open embedding 
A and B have common partial set C.
Topological space glued A and B by C 
Easy sheaf over OW
ahere, arbitrary open set Ø ≠ 
Ringed space  is called glue of A and B by C.
<Definition>
Intedgral domains that have common quotient field K R, S
Element R am ≠ 0
Element S bn ≠ 0
Spm T  Spm R, Spm T  Spm S
Glue defined by the upper is called simple.
<Definition>
Affine algebraic varieties U1U2
Common open set of U1UUC
Diagonal embedding 
When the upper is closed set, glue is called separated.
<Proposition>
For simple glue , next is equivalent.
(*) It is separated.
(**) Ring  is generated by R and S.
<Definition>
R and S are integral domains that have common quotient field K.
For partial ring T=RS generated by R and S, when <Definition> simple is satisfied, it is called "Spm R ad Spm S are simple glue."
<Sample>
Projective space Pn is simple glue.
<Definition>
Algebraic Variety's morphism is glue of affine algebraic variety's ring homomorphism image.
Algebraic direct product is direct product of affine algebraic variety.
4.
Affine algebraic variety X
Ring over k R
 is called R value point of X.
Whole  is called set of R value point of X, expressed by X(R).
Ring homomorphism over k 
X(f) := X(R)X(S)
Ring homomorphism 
<Definition>
 is function from ring category over k to category of set.
<Definition>
Functors from ring category to set category F, G
Ring R
Family of  over ring R {}
{} has functional morphism.
Functors F,G have isomorphism ( or natural transformation).
Functor from ring's category to set's category that is isomorphic to algebraic variety, is called representable or represent by X, or fine moduli.
<Definition>
Functor from ring's category to set's category F
When  satisfies the next conditions, X is called coarse moduli.
(i) There is natural transformation  :  .
(ii) Natural transformation  ,
Morphism that satisfies  is existent uniquely.
(iii) For algebraic close field k k, (k') is always bijection.
<Definition>
Algebraic variety G that  is functor to group's category is called algebraic group.
<Definition>
Finite generative ring over k A
When G = Spm A satisfies 3 conditions on the next triad is called affine algebraic group.
Triad
Conditions
(i)  are commutative for .
(ii)There is identity map for A.
(iii) There is coincident with  for A.
5.
Projective space over Pn
(2n+1) dimensional spherical surface {}
Pn has continuous surjection from .
Pn is compact.
<Definition>
Map  is called closed map when  is closed set image  becomes closed set.
<Definition>
Algebraic variety X is called complete when projection  is closed map for arbitrary manifold Y.
<Definition>
Morphism from complete algebraic manifold X to separated algebraic manifold Y is closed map.
<Proposition>
Projective space Pn is complete.
<System>
Algebraic manifold that has closed embedding at Pn is complete.
This algebraic manifold is called projective algebraic manifold.
[Interpretation]
Here language is expressed by Pn.
Word is expressed by projective algebraic manifold.
Meaning of word is expressed by closed embedding.

This paper has been published by Sekinan Research Field of Language.
All rights reserved.
© 2011 by 
The Sekinan Research Field of Language

5.
Note

Provisional philosophic conjecture on additional meaning is the following.

  1. From symplectic geometry to Fukaya category.
  2. From Fukaya category to derived category.
  3. Kawamata conjecture at derived category.
  4. Smooth objective algebraic manifolds satisfies birational map.
  5. Fully faithful embedding exists between two manifolds.

[References]
Extract of Zoho papers at Sekinan Library.
More details refer to Sekinan Zoho.


Tokyo
23 May 2016