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The notion of independence in categories of algebraic structures, part I: Basic properties

1988
*
Annals of Pure and Applied Logic
*

*The*following are some examples. (a)

*In*

*the*

*category*

*of*fields F with field homomorphisms, we have

*the*

*notion*

*of*

*independence*. ... And also that we can de&~ (say,

*in*a given class

*of*

*structures*) an

*independence*relation with respect to a set

*of*'equations'

*in*very much

*the*same manner

*algebraic*

*independence*is

*in*

*the*class

*of*gelds ...

*In*Section 7, we introduce

*the*

*notion*

*of*

*S*-

*minimal*amalgam and relate it to

*S*&ninimal

*extensions*. ...

##
###
Strongly Minimal Steiner Systems II: Coordinatization and Quasigroups
[article]

2021
*
arXiv
*
pre-print

We note that a strongly

arXiv:2106.13704v1
fatcat:z7nu7rrysrgqran6ayxwyjmmie
*minimal*Steiner k-Steiner system (M,R) from (Baldwin-Paolini 2020) can be 'coordinatized'*in**the*sense*of*(Gantner-Werner 1975) by a quasigroup if k is a prime-power. ... Nevertheless, by refining*the*construction, if k is a prime power there is a (2,k)-variety*of*quasigroups which is strongly*minimal*and definably coordinatizes a Steiner k-system. ... 5 and*ii*) to employ a function µ to bound*the*number 0-primitive*extensions**of*each finite*structure*so that closure*in**the*geometry on*the*generic model for*the*resulting class*of*finite*structures*is ...##
###
Deformation theory of representations of prop(erad)s II

2009
*
Journal für die Reine und Angewandte Mathematik
*

To do so, we endow

doi:10.1515/crelle.2009.084
fatcat:urkjk2uenbg6ppzm3zoyqzo5we
*the**category**of*prop(erad)*s*with a model*category**structure*. We provide a complete study*of*models for prop(erad)*s*. ... Their underlying chain complex is endowed with a canonical Lie*algebra*up to homotopy*structure**in*general and a Lie*algebra**structure*only*in**the*Koszul case. ...*The*first author expresses his thanks to*the*Max Planck Institute for Mathematics*in*Bonn and É cole Normale Supérieure*in*Paris for stimulating conditions during*the*work on this project. ...##
###
Derived Algebraic Geometry II: Noncommutative Algebra
[article]

2007
*
arXiv
*
pre-print

We develop a general theory

arXiv:math/0702299v5
fatcat:6tdcjvtqtjctvlqxahu7kvrtcu
*of**algebras**in*a monoidal infinity*category*, which we use to (re)prove some basic results*in**the*theory*of*associative ring spectra. ... We show that*the*infinity*category**of*spectra admits an essentially unique monoidal*structure*(such that*the*tensor product preserves colimits*in*each variable), and thereby recover*the*classical smash-product ... We will also introduce*the**notion**of*an*algebra*object*of*a monoidal ∞-*category*C. ...##
###
Etale and crystalline companions, II
[article]

2022
*
arXiv
*
pre-print

*In*answer to a conjecture

*of*Deligne, we establish that for any prime ℓ≠ p, an ℓ-adic Weil sheaf on X which is

*algebraic*(or irreducible with finite determinant) admits a crystalline companion

*in*

*the*

*category*...

*of*overconvergent F-isocrystals, for which

*the*Frobenius characteristic polynomials agree at all closed points (with respect to some fixed identification

*of*

*the*

*algebraic*closures

*of*ℚ within fixed

*algebraic*... As for

*part*(

*ii*)

*of*Theorem 0.1.1, by restricting to curves one sees that

*the*coefficients

*in*question are all

*algebraic*, but one needs a uniformity argument over these curves to show that

*the*

*extension*...

##
###
Finite approximation properties of C^*-modules II
[article]

2022
*
arXiv
*
pre-print

We introduce and study a

arXiv:2208.05655v1
fatcat:rstdaxmg5rdyvhoumgzr6a5tfm
*notion**of*amenability for vector valued traces. ... We study quasidiagonality and local reflexivity for C^*-*algebras*which are C^*-module over another C^*-*algebra*with compatible actions. ...*In*section 3, we study a*notion**of*quasidiagonality*in**the**category**of*C * -modules and extend Voiculescu theorem (Theorem 3.6). ...##
###
Discrete derived categories II: The silting pairs CW complex and the stability manifold
[article]

2015
*
arXiv
*
pre-print

*In*this article, we define

*the*CW complex

*of*silting pairs for a triangulated

*category*and show that it is contractible

*in*

*the*case

*of*discrete derived

*categories*. ... We obtain that

*the*space

*of*stability conditions

*of*discrete derived

*categories*is contractible. ...

*The*second author acknowledges

*the*financial support ...

##
###
The geometry of Hrushovski constructions, II. The strongly minimal case
[article]

2011
*
arXiv
*
pre-print

We investigate

arXiv:1103.3632v1
fatcat:imt66qkjt5h5bpwwha24izhhfm
*the*isomorphism types*of*combinatorial geometries arising from Hrushovski's flat strongly*minimal**structures*and answer some questions from Hrushovski's original paper. ... But then Y ≤ Y ∪ {b ij } is a simply*algebraic**extension**in*Z 1 . As Y ≤ Z 1 is a*minimally*simply*algebraic**extension*, this implies Y = {b i0 } ∪*s*ij and Z 1 = {b i0 , b ij } ∪*s*ij . ... We let µ be a function from*the*set*of*isomorphism types*of**minimally*simply*algebraic**extensions**in*C 0 (L) to*the*non-negative integers. ...##
###
On p-adic differential equations on semistable varieties II
[article]

2014
*
arXiv
*
pre-print

Given an open variety over a DVR with semistable reduction,

arXiv:1402.0720v1
fatcat:6trebuswjvc7tkuwa5qizgk6yi
*the*author constructed*in*that paper a fully faithful*algebraization*functor from*the**category**of*certain log overconvergent isocrystals on*the*... special fiber to*the**category**of*modules with regular integrable connection on*the*generic fiber. ... (grant-*in*-aid)*of**the*Japanese Society for*the*Promotion*of*Science (JSPS). ...##
###
Derived coisotropic structures II: stacks and quantization

2018
*
Selecta Mathematica, New Series
*

We extend results about n-shifted coisotropic

doi:10.1007/s00029-018-0407-1
fatcat:xh7zcewndrfyxaf7nxspgzygja
*structures*from*part*I*of*this work to*the*setting*of*derived Artin stacks. ... We also compare non-degenerate shifted coisotropic*structures*and shifted Lagrangian*structures*and show that there is a natural equivalence between*the*two spaces*in*agreement with*the*classical result ...*The*work*of*P.S. was supported by*the*EPSRC grant EP/I033343/1. ...##
###
Scaled-free objects II

2015
*
Annals of Functional Analysis
*

This work creates two

doi:10.15352/afa/06-3-18
fatcat:oknu5s2usjg6pdyjeh532wlurm
*categories**of*"array-weighted sets" for*the*purposes*of*constructing universal matrix-normed spaces and*algebras*. ... Moreover,*the*universal matrix-normed*algebra*is used to prove*the*existence*of*a free product for matrix-normed*algebras*using*algebraic*methods. ... Proposition 4 . 8 . 48 If all x ∈ X are array-free*in*X, then N X = {0}. Theorem 4 . 9 ( 49 Array-free and finite sets,*Part**II*). ...##
###
On p-adic differential equations on semistable varieties II

2014
*
Manuscripta mathematica
*

*category*

*of*log convergent isocrystals on X k with exponents

*in*τ (Σ), which is

*the*p-adic semistable version

*of*

*the*canonical logarithmic

*extension*

*of*Deligne, André-Baldassarri. ... Given an open variety over a DVR with semistable reduction,

*the*author constructed

*in*[10] a fully faithful

*algebraization*functor from

*the*

*category*

*of*certain log overconvergent isocrystals on

*the*special ... (grant-

*in*-aid)

*of*

*the*Japanese Society for

*the*Promotion

*of*Science (JSPS). ...

##
###
Sheaves on the alcoves and modular representations II
[article]

2020
*
arXiv
*
pre-print

We relate

arXiv:1801.03958v3
fatcat:ckj5k363oje25fousue3an4yja
*the**category**of*sheaves on alcoves that was constructed*in*"Sheaves on*the*alcoves and modular representations I" to*the*representation theory*of*reductive*algebraic*groups. ...*In*particular, we show that its indecomposable projective objects encode*the*simple rational characters*of*a reductive*algebraic*group*in*all characteristics above*the*Coxeter number. ... As*S*is a full subcategory*of**the*abelian*category**of*sheaves*of*Z-modules on A, it inherits an exact*structure*, i.e. a*notion**of*short exact sequences. ...##
###
Strict algebraic models for rational parametrised spectra II
[article]

2020
*
arXiv
*
pre-print

*In*this article, we extend Sullivan's PL de Rham theory to obtain simple

*algebraic*models for

*the*rational homotopy theory

*of*parametrised spectra. ... While not full,

*the*rational homotopy

*categories*we consider contain a large class

*of*parametrised spectra. ... For any cdga A

*in*this class,

*the*cohomology

*algebra*H • (A) computes

*the*cup product

*structure*

*of*H • (X; Q) and, conversely,

*the*rational homotopy groups

*of*X can be recovered from a

*minimal*model

*of*...

##
###
Weak Hopf Algebras II: Representation theory, dimensions and the Markov trace
[article]

1999
*
arXiv
*
pre-print

*In*

*the*special case

*of*weak Kac

*algebras*we show that I=δ is an integer. ... This

*category*has isomorphic left dual and right dual objects which leads, as usual, to

*the*

*notion*

*of*dimension function. ... for

*Part*

*II*. ...

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