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The Lascar groups and the first homology groups in model theory

2017
*
Annals of Pure and Applied Logic
*

*THE*

*LASCAR*

*GROUPS*

*AND*

*THE*

*FIRST*

*HOMOLOGY*

*GROUPS*

*IN*

*MODEL*

*THEORY*3

*the*characterization theorem (Theorem 4.4) of

*the*

*first*

*homology*

*group*

*and*give a criterion for Lstp≡stp. ...

*In*Section 5, we state that any connected compact abelian

*group*can appear as

*the*

*first*

*homology*

*group*of

*the*type of a

*model*(Theorem 5.2), which follows from a result by Bouscaren,

*Lascar*, Pillay,

*and*... For

*the*general

*theory*of

*model*

*theory*, of

*the*

*Lascar*

*groups*,

*and*of rosy

*theories*, we refer to [6] , [11] ,

*and*[2] , respectively. For

*the*

*homology*

*theory*

*in*

*model*

*theory*, see [4, 3] . ...

##
###
The Lascar groups and the 1st homology groups in model theory
[article]

2017
*
arXiv
*
pre-print

We also notice that

arXiv:1511.03410v3
fatcat:jozdwht6szcipgx27ugvxrmrs4
*the*map factors naturally via a surjection from*the*'relativised'*Lascar**group*of*the*type (which we define*in*analogy with*the**Lascar**group*of*the**theory*) onto*the**homology**group*, ... We also argue how any abelian connected compact*group*can appear as*the**first**homology**group*of*the*type of a*model*. ... For*the*general*theory*of*model**theory*, of*the**Lascar**groups*,*and*of rosy*theories*, we refer to [6] , [11] ,*and*[2] , respectively. For*the**homology**theory**in**model**theory*, see [4, 3] . ...##
###
Lascar groups and the first homology groups of strong types in rosy theories
[article]

2015
*
arXiv
*
pre-print

As a consequence, we show that

arXiv:1504.07721v4
fatcat:yxf5tyy3yjhgbjb4e6wjavb4gm
*the**first**homology**groups*of strong types*in*rosy*theories*have*the*cardinalities of one or at least 2^_0. ... For a rosy*theory*, we give a canonical surjective homomorphism from a*Lascar**group*over A=^eq(A) to a*first**homology**group*of a strong type over A,*and*we describe its kernel by an invariant equivalence ...*Lascar**group**and**the**first**homology**groups*. ...##
###
The relativized Lascar groups, type-amalgamation, and algebraicity
[article]

2020
*
arXiv
*
pre-print

We apply compact

arXiv:2004.11309v1
fatcat:y4v63g6sobddnhundoijha6ija
*group**theory*to obtain some*model*-theoretic results about*the*relativized*Lascar*Galois*group*of a strong type. ... Let us recall now*the*key definitions*in**the**homology**theory**in**model**theory*. ... Let us recall*the*definitions of*the**Lascar**groups**and*types. These are well-known notions*in**model**theory*. ...##
###
Page 3989 of Mathematical Reviews Vol. , Issue 2000f
[page]

2000
*
Mathematical Reviews
*

To aid

*in**the*exposition we develop a variant of Baum’s (M, E, f)*model*for K-*homology*. Our*model*removes*the*need for Spin® structures*in**the*description of geo- metric K -*homology*.” ...*Model**theory*of*groups**and*automorphism*groups*( Blaubeuren, 1995), 115-125, London Math. Soc. Lecture Note Ser., 244, Cambridge Univ. Press, Cambridge, 1997. ...##
###
Classifying spaces and the Lascar group
[article]

2021
*
arXiv
*
pre-print

We show that

arXiv:1808.04915v2
fatcat:nt75jzc725ejnberbhvdcjn5im
*the**Lascar**group*Gal_L(T) of a*first*order*theory*T is naturally isomorphic to*the*fundamental*group*π_1(|Mod(T)|) of*the*classifying space of*the*category of*models*of T*and*elementary embeddings ...*The**Lascar**group*.*In*[Las82] ,*Lascar*introduced a notion of a Galois*group*of a complete*first*-order*theory*T , now known as*the**Lascar**group*Gal L (T ). ...*In*[GKK13] , a notion of*homology*of types was defined*and**in*[DKL17] , it was shown that*the**first**homology**group*of a given type p is given by*the*abelianization of*the*relativized*Lascar**groups*. ...##
###
Model-Theoretic Algebra With particular emphasis on fields, rings, modules

1993
*
Bulletin of the London Mathematical Society
*

On

doi:10.1112/blms/25.2.195
fatcat:pnriywskevbfxmguff6z3dewky
*the*other hand, I feel that*the*author missed an opportunity by not including any account of*the*development*in**the*last ten years of connections between weak arithmetic*and*computational complexity ... As a whole, this book is highly recommended for*the*serious student of*models*of arithmetic. ... Two are quite closely related,*and*they display*the*breadth*and*depth of techniques currently*in**and*around*the**model**theory*of*groups*. ...##
###
Page 4473 of Mathematical Reviews Vol. , Issue 2004f
[page]

2004
*
Mathematical Reviews
*

K.)
2004f:20002 20-06 03C45 03C60 51E24
Tits buildings

*and**the**model**theory*of*groups*. Papers from*the*workshop held*in*Wiirzburg, September 2000. Edited by Katrin Tent. ...*The*main technical point is that*the**first*differential*in**the*spectral sequence is*the*Steenrod square Sq? of Voevodsky*and*Brosnan. ...##
###
Page 3077 of Mathematical Reviews Vol. , Issue 91F
[page]

1991
*
Mathematical Reviews
*

3077
higher K-

*theory**groups**in*a future paper.*The*third part studies*the*case when*the*algebras are no longer unital but H-unital (i.e. unital*in*a*homological*sense) as*in*a paper by M. ... It is proved that if G = N x H is a sharply 2-transitive*group*which is a semidirect product as written where H is*the*stabilizer of a point*and*if G is superstable (*in**the**model*-theoretic sense) of finite ...##
###
On NIP and invariant measures
[article]

2009
*
arXiv
*
pre-print

We study forking,

arXiv:0710.2330v2
fatcat:xfsoqnaaazfajiw3ocn36dnll4
*Lascar*strong types, Keisler measures*and*definable*groups*, under an assumption of NIP (not*the*independence property), continuing aspects of math.LO/0607442. ... compact commutative*groups**in*o-minimal expansions of real closed fields. ... We conclude*the*paper by proving that*the*topologies on J*and*G/G 00 coincide. Our proof will make use of a little more "*theory*" some of which is of interest*in*its own right. ...##
###
On NIP and invariant measures

2011
*
Journal of the European Mathematical Society (Print)
*

We study forking,

doi:10.4171/jems/274
fatcat:peg6azkqwzh2nmisi5ireedruu
*Lascar*strong types, Keisler measures*and*definable*groups*, under an assumption of NIP (not*the*independence property), continuing aspects of*the*paper [16] . ... for (definably compact) commutative*groups**in*o-minimal expansions of real closed fields. ... A*first*version of this paper was written*in*October 2007. ...##
###
A classification of 2-chains having 1-shell boundaries in rosy theories
[article]

2015
*
arXiv
*
pre-print

*In*particular, we prove that

*in*a rosy

*theory*, every 1-shell of a

*Lascar*strong type is

*the*boundary of some 2-chain, hence making

*the*1st

*homology*

*group*trivial. ... We also show that, unlike

*in*simple

*theories*,

*in*rosy

*theories*there is no upper bound on

*the*minimal lengths of 2-chains whose boundary is a 1-shell. ... Introduction

*In*[5] , [6] , J. Goodrick, A. Kolesnikov

*and*

*the*

*first*author developed a

*homology*

*theory*for any amenable collection of functors

*in*a very general context. ...

##
###
Amalgamation functors and homology groups in model theory
[article]

2011
*
arXiv
*
pre-print

We compute

arXiv:1105.2921v1
fatcat:s3mhtpk7qnacppv5lwwhknwxzm
*the**group*H_2 for strong types*in*stable*theories**and*show that*in*this context,*the*class of possible*groups*H_2 is precisely*the*profinite abelian*groups*. ... We present definitions of*homology**groups*associated to a family of amalgamation functors. We show that if*the*generalized amalgamation properties hold, then*the**homology**groups*are trivial. ...*the*composition of morphisms, are definable*in**models*of a*first*-order*theory*. ...##
###
Representation of Subaltern Identity: A Study of Othering in Sea of Poppies

2020
*
Global Language Review
*

*The*present study is designed to explore

*the*identity construction/reconstruction

*in*Sea of Poppies. It is investigated

*in*

*the*backdrop of postcolonial

*theory*. ... Norman Faircloughs Three-dimensional approach of critical discourse analysis is used to examine

*the*construction

*and*reconstruction of different types of (us/them) identities

*in*

*the*colonial era reflected ... It is investigated

*in*

*the*backdrop of postcolonial

*theory*. ...

##
###
Stable domination and independence in algebraically closed valued fields
[article]

2006
*
arXiv
*
pre-print

*In*Part B, we show that

*the*general

*theory*applies to ACVF. Over a sufficiently rich base, we show that every type is stably dominated over its image

*in*

*the*value

*group*. ...

*In*Part A, we develop a general

*theory*of stably dominated types, showing they enjoy an excellent independence

*theory*, as well as a

*theory*of definable types

*and*germs of definable functions. ...

*The*

*first*takes place entirely

*in*

*the*value

*group*(so

*in*

*the*

*theory*of divisible ordered abelian

*groups*), but could be encoded into ACVF. ...

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