Ind- and Pro- definable sets

Topological properties of definable sets in ordered Abelian groups of burden 2

We obtain some new results on the topology of unary definable sets in densely ordered Abelian groups of burden groups of burden 2. In the special case in which the structure has dp-rank 2, we show that the existence of an infinite definable discrete set precludes the definability of a set which is dense and codense in an interval, or of a set which is topologically like the Cantor middle-third set. If the structure has burden 2 and both an infinite discrete set D and a dense-codense set X are definable, then translates of X must witness the Independence Property. In the last section, an explic…

Definable categories and T-motives

Making use of Freyd's free abelian category on a preadditive category we show that if $T:D\rightarrow \mathcal{A}$ is a representation of a quiver $D$ in an abelian category $\mathcal{A}$ then there is an abelian category $\mathcal{A} (T)$, a faithful exact functor $F_T: \mathcal{A} (T) \to \mathcal{A}$ and an induced representation $\tilde T: D \to \mathcal{A} (T)$ such that $F_T\tilde T= T$ universally. We then can show that $\mathbb{T}$-motives as well as Nori's motives are given by a certain category of functors on definable categories.

Trees of definable sets over the p-adics

To a definable subset of Z_p^n (or to a scheme of finite type over Z_p) one can associate a tree in a natural way. It is known that the corresponding Poincare series P(X) = \sum_i N_i X^i is rational, where N_i is the number of nodes of the tree at depth i. This suggests that the trees themselves are far from arbitrary. We state a conjectural, purely combinatorial description of the class of possible trees and provide some evidence for it. We verify that any tree in our class indeed arises from a definable set, and we prove that the tree of a definable set (or of a scheme) lies in our class in…

Definable subgroups in $SL\_2$ over a p-adically closed field

The aim of this paper is to describe all definable subgroups of SL2(K), for K a p-adically closed field. We begin by giving some "frame subgroups" which contain all nilpotent or solvable subgroups of SL2(K). A complete description is givien for K a p-adically closed field, some results are generalizable to K a field elementarily equivalent to a finite extension of Qp and an almost complete description is given for Q an p . We give also some indications about genericity and generosity in SL2(K).

Amenability, connected components, and definable actions

We study amenability of definable and topological groups. Among our main technical tools is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and some results around measures. As an application we show that if $G$ is an amenable topological group, then the Bohr compactification of $G$ coincides with a certain "weak Bohr compactification" introduced in [24]. Formally, $G^{00}_{topo} = G^{000}_{topo}$. We also prove wide generalizations of this result, implying in particular its extension to a "definable-topological" context, confirming the main conjec…

Definable and Non-definable Notions of Structure

Definability is a key notion in the theory of Grothendieck fibrations that characterises when an external property of objects can be accessed from within the internal logic of the base of a fibration. In this paper we consider a generalisation of definability from properties of objects to structures on objects, introduced by Shulman under the name local representability. We first develop some general theory and show how to recover existing notions due to Bénabou and Johnstone as special cases. We give several examples of definable and non definable notions o structure, focusing on algebraic we…

在NSOP₁中寻找稳定子群

Looking for stabilizers in NSOP$\_1$

该研究针对NSOP₁理论中的可定义与型可定义群，定义Kim-稳定子群，结合Hrushovsky稳定子群证明满足假设的NSOP₁域上可定义群的有界指型可定义子群可有限对一嵌入代数群，且ω-自由PAC域的可定义群满足该条件。

Vapnik-Chervonenkis density in some theories without the independence property, II

We study the Vapnik-Chervonenkis (VC) density of definable families in certain stable first-order theories. In particular we obtain uniform bounds on VC density of definable families in finite U-rank theories without the finite cover property, and we characterize those abelian groups for which there exist uniform bounds on the VC density of definable families.

Groups definable in o-minimal structures: structure theorem, G^000, definable amenability and bounded orbits

We settle some open problems in the special case of groups in o-minimal structures, such as the equality of G^00 and G^000 and the equivalence of definable amenability and existence of a type with bounded orbit. We prove almost exactness of the G to G^00 functor. We ask further questions about types with bounded orbits in NIP theories.

Solvable groups definable in o-minimal structures

Let N be an o-minimal structure. In this paper we develop group extension and group cohomology theory over N and use it to describe the N-definable solvable groups. We prove an o-minimal analogue of the Lie-Kolchin-Mal'cev theorem and we describe the N-definable G-modules and the N-definable rings.

o-minimal结构中可定义集合族的基数

Cardinality of definable families of sets in o-minimal structures

本文证明o-minimal结构中可定义无限集的可定义子集族基数不超过该集本身，并导出可定义类型计数与可定义拓扑空间存在性的推论。

Locally definable homotopy

In "On o-minimal homotopy groups", o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to the category of locally definable spaces, for which we introduce homology and homotopy functors. We also study the concept of connectedness in V-definable groups -- which are examples of locally definable spaces. We show that the various concepts of connectedness associated to these groups, which have appeared in the literature, are non-equivalent.

Tame topology over definable uniform structures

A visceral structure on M is given by a definable base for a uniform topology on its universe in which all basic open sets are infinite and any infinite definable subset X of M has non-empty interior. This context includes o-minimal ordered groups, p-adic fields, and other examples. Assuming only viscerality, we show that the definable sets in M satisfy some desirable topological tameness conditions. For example, any definable unary function on M has a finite set of discontinuities; any definable function on a Cartesian power of M is continuous on a nonempty open set; and assuming definable fi…

一些无法合并的可定义类型

Some definable types that cannot be amalgamated

本文构造了一个理论，证明其中的可定义类型不具备合并性质，从而回答了模型论中关于类型合并的一个问题。

On countable cofinality and decomposition of definable thin orderings

We prove that in some cases definable thin sets (including chains) of Borel partial orderings are necessarily countably cofinal. This includes the following cases: analytic thin sets, ROD thin sets in the Solovay model, and $Σ^1_2$ thin sets in the assumption that $ω_1^{L[x]}<ω_1$ for all reals $x$. We also prove that definable thin wellorderings admit partitions into definable chains in the Solovay model.

关于全超越环境中的可定义伽罗瓦理论与可定义伽罗瓦上同调

On definable Galois theory and definable Galois cohomology in the totally transcendental setting

本文在全超越一阶理论中建立可定义伽罗瓦理论与上同调的联系，给出微分伽罗瓦上同调的余极限公式，并引入微分域和参数集的有界性新概念。

Definable $C^r$ vector bundles and bilinear spaces in an o-minimal structure and their homotopy theorems

Consider an o-minimal structure on the real field. Let $M$ be a definable $C^r$ manifold, where $r$ is a nonnegative integer. We first demonstrate an equivalence of the category of definable $C^r$ vector bundles over $M$ with the category of finitely generated projective modules over the ring $C_{\text{df}}^r(M)$. Here, the notation $C_{\text{df}}^r(M)$ denotes the ring of definable $C^r$ functions on $M$. We also show an equivalence of the category of definable $C^r$ bilinear spaces over $M$ with the category of bilinear spaces over the ring $C_{\text{df}}^r(M)$. The main theorems of this pap…

$Σ_1(κ)$-definable subsets of $\mathrm{H}(κ^+)$

We study $Σ_1(ω_1)$-definable sets (i.e. sets that are equal to the collection of all sets satisfying a certain $Σ_1$-formula with parameter $ω_1$) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is $Σ_1(ω_1)$-definable, the set of all stationary subsets of $ω_1$ is not $Σ_1(ω_1)$-definable and the complement of every $Σ_1(ω_1)$-definable Bernstein subset of ${}^{ω_1}ω_1$ is not $Σ_1(ω_1)$-definable. In contrast, we show that the existence of a Woodin cardinal is compatible …

Definable maximal cofinitary groups

Using countable support iteration of $S$-proper posets, for some appropriate stationary set $S$, we obtain a generic extension of the constructible universe, in which $\mathfrak{b}=\mathfrak{c}=\aleph_2$ and there is a maximal cofinitary group with a $Π^1_2$-definable set of generators.

论可定义 J-集

On definable J-sets

本文研究了可定义群的可定义 J-集与弱生成集的比较，证明了二者重合性质在足够饱和模型上不变且属模型论性质，并在超稳定交换群和 pCF 中给出正面结果及不重合的反例。

Definable topological dynamics

For a group $G$ definable in a first order structure $M$ we develop basic topological dynamics in the category of definable $G$-flows. In particular, we give a description of the universal definable $G$-ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification of $G$, that is to the quotient $G^*/{G^*}^{00}_M$ (where $G^*$ is the interpretation of $G$ in a monster model). More generally, we obtain these results locally, i.e. in the category of $Δ$-definable $G$-flows for any fixed set $Δ$ of formulas of an…

Local metric properties and regular stratifications of p-adic definable sets

We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a $p$-adic field from the metric, differential and measure geometric point of view. We prove that the local density of such sets at each of their points does exist. We then introduce the notion of distinguished tangent cone with respect to some open subgroup with finite index in the multiplicative group of our field and show, as it is the case in the real setting, that, up to some multiplicities, the local density may be computed on this distinguished tangent cone.We also prove that these distinguished tangent…

群的扩张中包含积集的可定义集

Definable sets containing productsets in expansions of groups

本文研究群一阶扩张中可定义集的积集性质，分别在稳定公式、稳定群及 distal 扩张情形下，利用 Banach 密度、coheir 独立性与正则引理给出刻画和可定义版本。

Directed sets and topological spaces definable in o-minimal structures

We study directed sets definable in o-minimal structures, showing that in expansions of ordered fields these admit cofinal definable curves, as well as a suitable analogue in expansions of ordered groups, and furthermore that no analogue holds in full generality. We use the theory of tame pairs to extend the results in the field case to definable families of sets with the finite intersection property. We then apply our results to the study of definable topologies. We prove that all definable topological spaces display properties akin to first countability, and give several characterizations of…

Pro-definability of spaces of definable types

We show pro-definability of spaces of definable types in various classical complete first order theories, including complete o-minimal theories, Presburger arithmetic, $p$-adically closed fields, real closed and algebraically closed valued fields and closed ordered differential fields. Furthermore, we prove pro-definability of other distinguished subspaces, some of which have an interesting geometric interpretation. Our general strategy consists in showing that definable types are uniformly definable, a property which implies pro-definability using an argument due to E. Hrushovski and F. Loese…

On minimal flows and definable amenability in some distal NIP theories

We study the definable topological dynamics $(G(M), S_G(M))$ of a definable group acting on its type space, where $M$ is either an $o$-minimal structure or a $p$-adically closed field, and $G$ a definable amenable group. We focus on the problem raised by Neweslki of whether weakly generic types coincide with almost periodic types, showing that the answer is positive when $G$ has boundedly many global weakly generic types. We also give two "minimal counterexamples" where $G$ has unboundedly many global weakly generic types, extending the main results of "On minimal flows, definably amenable gro…

1-h-minimal 赋值域中的闭有界集

Closed bounded sets in 1-h-minimal valued fields

本文研究 1-h-minimal 赋值域中由值群索引的递减可定义闭有界集族，证明其朴素紧致性，并据此推出局部拓扑可定义群存在由开子群组成的单位元可定义邻域族。

Types, transversals and definable compactness in o-minimal structures

Through careful analysis of types inspired by [AGTW21] we characterize a notion of definable compactness for definable topologies in general o-minimal structures, generalizing results from [PP07] about closed and bounded definable sets in o-minimal expansions of ordered groups. Along the way we prove a parameter version for o-minimal theories of the connection between dividing and definable types known in the more general dp-minimal context [SS14], through an elementary proof that avoids the use of existing forking and VC literature. In particular we show that, if an $A$-definable family of se…

Approximation by $O$-minimal sets in power-bounded $T$-convex valued fields

We show that, for a certain large class of power-bounded $o$-minimal $\mathcal{L}_T$-theories $T$ whose field of exponents is infinite-dimensional as a vector space over the rationals, any definable set in a $T$-convex valued field $(\mathcal{R}, \mathfrak{O})$ is in a precise sense the limit of a family of $\mathcal{L}_T$-definable sets indexed over the residue field. Alternatively, in the mainstream model-theoretic language, this says that if $(\mathcal{R}', \mathfrak{O}')$ is an elementary substructure of $(\mathcal{R}, \mathfrak{O})$ and if the residue field of $\mathfrak{O}$ contains an e…

论$Σ_1$-可定义的闭无界集

On $Σ_1$-Definable Closed Unbounded Sets

该文聚焦集合论中基数$κ$的$Σ_1$-可定义闭无界集，研究其对完全闭无界滤子的逼近程度及对定义复杂度的依赖，发现逼近能力差异大且依赖底层集合论宇宙性质。

Pseudofinite H-structures and groups definable in supersimple H-structures

In this paper we explore some properties of H-structures. We describe a construction of H-structures based on one-dimensional asymptotic classes which preserves pseudo-finiteness. That is, the H-structures we construct are ultraproducts of finite structures. We also prove that under the assumption that the base theory is supersimple of SU-rank one, there are no new definable groups in H-structures.

Cell decomposition and classification of definable sets in p-optimal fields

We prove that for p-optimal fields (a very large subclass of p-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef's paper [Invent. Math, 77 (1984)]. We derive from it the existence of definable Skolem functions and strong p-minimality. Then we turn to strongly p-optimal fields satisfying the Extreme Value Property (a property which in particular holds in fields which are elementarily equivalent to a p-adic one). For such fields K, we prove that every definable subset of KxK^d whose fibers are inverse images by the valuation o…

The structure on the real field generated by the standard part map on an o-minimal expansion of a real closed field

Let R be a sufficiently saturated o-minimal expansion of a real closed field, let O be the convex hull of the rationals in R, and let st: O^n \to \mathbb{R}^n be the standard part map. For X \subseteq R^n define st(X):=st(X \cap O^n). We let \mathbb{R}_{\ind} be the structure with underlying set \mathbb{R} and expanded by all sets of the form st(X), where X \subseteq R^{n} is definable in R and n=1,2,.... We show that the subsets of \mathbb{R}^n that are definable in \mathbb{R}_{\ind} are exactly the finite unions of sets of the form st(X) \setminus st(Y), where X,Y \subseteq R^n are definable…

关于在Hensel最小结构中可定义的闭子集

About closed subsets definable in Hensel minimal structures

本文研究了Hensel最小结构中闭0可定义子集的性质，证明了此类子集可由连续0可定义函数的零点集定义，并给出了非阿基米德版本的蒂泽-乌里松扩展定理。

可微近似连续可定义映射并保持像

Differentiable approximation of continuous definable maps that preserves the image

本文通过 o-最小几何与 PL 几何的结合，证明连续可定义映射可被可微映射近似且保持图像不变。

带有双线性型的向量空间中可定义集合、群与域

Sets, groups, and fields definable in vector spaces with a bilinear form

本文研究带双线性型的向量空间理论中可定义群与域的结构，引入维数概念，证明群具有（代数-由-阿贝尔）-由-代数结构，域同构于标量域，并推广到实闭域情形。

Karp 的 NP 完全问题在一阶可定义结构上的研究

Karp's NP-complete problems over first-order definable structures

本文确定了 Karp 的 NP 完全问题在一阶可定义结构（如轨道有限集）上的可判定性。

在某些赋值域上可定义的简单群

On simple groups definable in some valued fields

本文证明在特定赋值域中，可定义的简单群本质上是线性代数群，并探讨了其在正特征代数闭域中的扩展结果。