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 of showing that definable types are uniformly definable, a property which implies pro-definability using an argument due to E. Hrushovski and F. Loeser. Uniform definability of definable types is finally achieved by studying classes of stably embedded pairs.
1. Introduction
In Reference 13, building on the model theory of algebraically closed valued fields ($\mathrm{ACVF}$), E. Hrushovski and F. Loeser developed a theory which provides a model-theoretic account of the Berkovich analytification of algebraic varieties. Most notably, they obtained results concerning the homotopy type of quasi-projective varieties which were only known under strong algebro-geometric hypothesis on $X$ by results in Reference 1.
One of the difficulties to study Berkovich spaces from a model-theoretic point of view is that such spaces do not seem to generally have (in $\mathrm{ACVF}$) the structure of a definable set –where usual model-theoretic techniques can be applied– but rather canonically the structure of a space of types. Part of the novelty of Hrushovski-Loeser’s work lies on the fact that their spaces can be equipped with the structure of a (strict) pro-definable set, which granted them back the use of different classical model-theoretic tools. It is thus tempting to ask if such a structural result holds for other distinguished subsets of definable types and even for other first-order theories. It turned out this question is closely related to classical topics in model theory such as the model theory of pairs and uniform definability of types. In this article we give a positive answer in various contexts. Formally, we obtain the following result
The fact that pro-definability follows from uniform definability of types goes back to an argument of E. Hrushovski and F. Loeser in Reference 13 which we present in Proposition 4.1. In return, uniform definability of types is obtained from the following criterion which relates it to stably embedded pairs of models of $T$.
Ensuring point $(i)$ for the above-listed theories makes use of characterizations of definable types à la Marker-Steinhorn. In the case of $\mathrm{RCVF}$ and ${p}\mathrm{CF}_{e,f}$, we prove such a characterization (Theorem 5.9) in the spirit of Reference 6.
As a corollary we also obtain pro-definability of some distinguished subspaces of the space of definable types which have a geometric interpretation. In particular, we aim to show that there are spaces of definable types in $\mathrm{ACVF}$ that can mimic Huber’s analytification of an algebraic variety in a similar way the space of generically stable types mimics its Berkovich analytification. We hope this can serve as a basis towards a model theory of adic spaces. In the same spirit, working in $\mathrm{RCVF}$, there are spaces of definable types which can be seen as the model-theoretic counterpart of the analytification of semi-algebraic sets as recently defined by P. Jell, C. Scheiderer and J. Yu in Reference 14. This article aims to lay down a foundation for a model-theoretic study of such spaces. In a sequel, we will further explore structural properties of some of these spaces.
A natural question to ask is if such spaces are also strict pro-definable (i.e., pro-definable with surjective transition maps). Obtaining strictness is much more subtle and is often related to completions of theories of stably embedded pairs. Results towards a positive answer to this question will also be addressed in subsequent work.
2. Preliminaries and notation
2.1. Model theoretic background
Let $\mathcal{L}$ be a first order language (possibly multi-sorted) and $T$ be a complete $\mathcal{L}$-theory. The sorts of $\mathcal{L}$ are denoted by bold letters $\mathbf{D}$. Given a variable $x$, we let $\mathbf{D}_x$ denote the sort where $x$ ranges. If $M$ is a model of $T$ and $\mathbf{D}$ be a sort of $\mathcal{L}$, we let $\mathbf{D}(M)$ denote the set of elements of $M$ which are of sort $\mathbf{D}$. Given an ordered tuple of variables $x=(x_i)_{i\in I}$ (possibly infinite), we extend this notation and set
Let $C$ be a subset of $M$ (i.e. the union of all $\mathbf{D}(M)$,$\mathbf{D}$ a $\mathcal{L}$-sort). The language $\mathcal{L}(C)$ is the language $\mathcal{L}$ together with constant symbols for every element in $C$. Given an $\mathcal{L}(C)$-definable subset $X\subseteq \mathbf{D}_x(M)$, we say that a type $p\in S_x(C)$concentrates on $X$ if $p$ contains a formula defining $X$. We denote by $S_X(C)$ the subset of $S_x(C)$ consisting of those types concentrating on $X$. For a $C$-definable function $f\colon X\to Y$ the pushforward of $f$ is the function $f_*\colon S_X(C)\to S_Y(C)$ sending $tp(a/C)$ to $tp(f(a)/C)$.
We let $\mathcal{U}$ be a monster model of $T$. As usual, a set is said to be small if it is of cardinality smaller than $|\mathcal{U}|$. A type $p(x)$ is a global type if $p\in S_x(\mathcal{U})$. Following Shelah’s terminology, a subset $X\subseteq \mathbf{D}_x(M)$ is $*$-$C$-definable if there is a small collection $\Theta$ of $\mathcal{L}(C)$-formulas$\varphi (x)$ (where only finitely many $x_i$ occur in each formula) such that $X=\{a\in \mathbf{D}_x(M)\mid M\models \varphi (a), \varphi \in \Theta \}$.
We let $\mathrm{dcl}$ and $\mathrm{acl}$ denote the usual definable and algebraic closure model-theoretic operators. Given a tuple of variables $x$ of length $n$, a $C$-definable set $X\subseteq \mathbf{D}_x(\mathcal{U})$ and a subset $A$ of $\mathcal{U}$, we let $X(A)\coloneq X\cap \mathrm{dcl}(A)^n$.
2.2. Definable types
We will mainly study definable types over models of $T$. Recall that given a subset $A\subseteq M$, a type $p\in S_x(M)$ is $A$-definable (or definable over $A$) if for every $\mathcal{L}$-formula$\varphi (x,y)$ there is an $\mathcal{L}(A)$-formula$d_p(\varphi )(y)$ such that for every $c\in \mathbf{D}_y(M)$
The map $\varphi (x,y)\mapsto d_p(\varphi )(y)$ is called a scheme of definition for $p$, and the formula $d_p(\varphi )(y)$ is called a $\varphi$-definition for $p$. We say $p\in S_x(M)$ is definable if it is $M$-definable. Given any set $B$ containing $M$, we use $p|B$ to denote the type $\{\varphi (x,b)\mid N\models d_p(\varphi )(b)\}$, where $N$ is any model of $T$ containing $B$. We refer the reader to Reference 18, Section 1 for proofs and details of these facts. The following is folklore.
Let $\varphi (x;y)$ be a partitioned formula. A formula $\psi (y,z_\varphi )$ is a uniform definition for $\varphi$ (in $T$) if for every model $M$ of $T$ and every definable type $p\in S_x(M)$ there is $c=c(p,\varphi )\in \mathbf{D}_{z_\varphi }(M)$ such that $\psi (y,c)$ is a $\varphi$-definition for $p$. We say $T$ has uniform definability of types if every partitioned $\mathcal{L}$-formula$\varphi (x;y)$ has a uniform definition, which we write $d(\varphi )(y,z_\varphi )$. The following is a routine coding exercise.
2.3. Background in valued fields
For a valued field $(K,v)$ we let $\Gamma _K$ denote the value group, $\mathcal{O}_K$ its valuation ring, $k_K$ the residue field and $\mathrm{res}\colon \mathcal{O}_K\to k_K$ the residue map. Given a valued field extension $(K\subseteq L, v)$ and a subset $A\coloneq \{a_1,\ldots ,a_n\}\subseteq L$, we say that $A$ is $K$-valuation independent if for every $K$-linear combination $\sum _{i=1}^n c_ia_i$ with $c_i\in K$,$v(\sum _{i=1}^n c_ia_i) = \min _{i}(v(c_ia_i))$. The extension $L|K$ is called $vs$-defectlessFootnote1 if every finitely generated $K$-vector subspace $V$ of $L$ admits a $K$-valuation basis, that is, a $K$-valuation independent set which spans $V$ over $K$.
1
This is the same as “separated” in W. Baur’s and F. Delon’s terminology.
Let $\mathcal{L}_\mathrm{ring}$ be the language of rings and $\mathcal{L}_\mathrm{div}$ be its extension by a binary predicate $\mathrm{div}$. We let $\mathrm{ACVF}$ be the $\mathcal{L}_\mathrm{div}$-theory of algebraically closed (non-trivially) valued fields, where $\mathrm{div}$ is interpreted in a valued field $(K,v)$ by $\mathrm{div}(x,y)\Leftrightarrow v(x)\leqslant v(y)$. Recall this theory has quantifier elimination (see also Reference 12).
We let ${p}\mathrm{CF}_{e,f}$ be the $\mathcal{L}$-theory of a finite extension $K$ of $\mathbb{Q}_p$ with $p$-ramification index $e$ and residue degree $f$. Here $\mathcal{L}$ is $\mathcal{L}_\mathrm{div}$ together with $d=ef$ new constants symbols interpreted in any model by elements which modulo $p\mathcal{O}$ form an $\mathbb{F}_p$-basis of $\mathcal{O}_K/p\mathcal{O}_K$ (see Reference 20). Recall this theory admits quantifier elimination by adding predicates for $n^\mathrm{th}$-powers (see Reference 20, Theorem 5.6).
Finally, the theory $\mathrm{RCVF}$ is the theory of real closed (non-trivially) valued fields in which the valuation ring is convex with respect to the ordering. It has quantifier elimination by results in Reference 3.
3. Completions by definable types
3.1. The definable completion
Let $X\subseteq \mathbf{D}_x(\mathcal{U})$ be a $C$-definable set and $A$ be a small set containing $C$. The definable completion of $X$ over $A$, denoted ${S^\mathrm{def}_{X, \mathcal{U}}}(A)$, is the space of $A$-definable global types which concentrate on $X$. For a tuple of variables $x$ we write ${S^\mathrm{def}_{x, \mathcal{U}}}(A)$ for ${S^\mathrm{def}_{\mathbf{D}_x(\mathcal{U}), \mathcal{U}}}(A)$.
Given a $C$-definable function $f\colon X\to Y$, it is easy to see that the image of a definable type under $f_*$ is again definable. We will therefore use the more functorial notation $f^{\mathrm{def}}\colon {S^\mathrm{def}_{X}}(A)\to {S^\mathrm{def}_{Y}}(A)$ for every small set $A$ containing $C$.
3.2. Other completions by definable types
3.2.1. The bounded completion
Let $T$ be an o-minimal theory and $M$ be a model of $T$. Given an elementary extension $M\preceq N$, we say that $N$ is bounded by $M$ if for every $b\in N$, there are $c_1, c_2\in M$ such that $c_1\leqslant b\leqslant c_2$. Let $A$ be a small subset of $\mathcal{U}$ and $X$ be a definable set. A type $p\in {S^\mathrm{def}_{X}}(A)$ is bounded if for any small model $M$ containing $A$ and every realization $a\models p|M$, there is an elementary extension $M\preceq N$ with $a\in X(N)$ and such that $N$ is bounded by $M$.
Let $T$ be either $\mathrm{RCVF}$ or a completion of $\mathrm{ACVF}$. Let $A$ be a small subset of $\mathcal{U}$. A type $p\in {S^\mathrm{def}_{X}}(A)$ is bounded if for any small model $M$ containing $A$ and every realization $a\models p|M$,$\Gamma (\mathrm{acl}(Ma))$ is bounded by $\Gamma (M)$, where $\Gamma$ denotes the value group sort.
Finally, let $T$ be either an o-minimal theory, a completion of $\mathrm{ACVF}$ or $\mathrm{RCVF}$. Let $A$ be a small subset of $\mathcal{U}$ and $X$ be a definable set. The bounded completion of $X$ over $A$, denoted $\widetilde{X}(A)$, is the set of bounded global $A$-definable types.
3.2.2. The orthogonal completion
Let $T$ be either a completion of $\mathrm{ACVF}$ or $\mathrm{RCVF}$. Let $A$ be a small subset of $\mathcal{U}$. A type $p\in {S^\mathrm{def}_{x}}(A)$ is said to be orthogonal to $\Gamma$ if for every model $M$ containing $A$ and every realization $a\models p|M$,$\Gamma (M)=\Gamma (\mathrm{acl}(Ma))$. Given a definable set $X$, the orthogonal completion of $X$, denoted by $\widehat{X}(A)$, is the set of global $A$-definable types concentrating on $X$ which are orthogonal to $\Gamma$.
In Reference 13, the set $\widehat{X}(A)$ is called the stable completion of $X$ over $A$. The name arises since, in this context, $\widehat{X}(A)$ also corresponds to the set of definable types over $A$ which are stably dominated, and equivalently, which are generically stable (see Reference 13, Proposition 2.9.1). However, such an equivalence does not hold in $\mathrm{RCVF}$: every generically stable type (resp. stably dominated) must be a realized type.
3.2.3. Geometric interpretation
For $T$ a completion of $\mathrm{ACVF}$, let $V$ be a variety over a complete rank 1 valued field $F$. In Reference 13, $\widehat{V}$ is introduced as a model-theoretic analogue of the Berkovich analytification $V^{\mathrm{an}}$ of $V$. Similarly, our aim is to view $\widetilde{V}$ as a model-theoretic analogue of the Huber analytification of $V$. When $T$ is $\mathrm{RCVF}$,$\widehat{V}$ is a good candidate to be the model-theoretic counterpart of the analytification of semi-algebraic sets defined by Jell, Scheiderer and Yu in Reference 14. The set $\widehat{V}$ is also tightly related to the set of residue field dominated types as defined by Ealy, Haskell and Maříková in Reference 9. The space $\widetilde{V}$ (in $\mathrm{RCVF}$) seems to suggest there is an analogue of Huber’s analytification of semi-algebraic sets. Finally, ${S^\mathrm{def}_{V}}$ can be viewed as a model-theoretic analogue of the “space of valuations on $V$”. As mentioned in the introduction, we will present more structural results concerning these spaces in a sequel of this article.
4. Spaces of definable types as pro-definable sets
4.1. Pro-definable sets and morphisms
Let $(I,\leq )$ be a small upwards directed partially ordered set and $C$ be a small subset of $\mathcal{U}$. A $C$-definable projective system is a collection $(X_i, f_{ij})$ such that:
(1)
for every $i\in I$,$X_i$ is a $C$-definable set;
(2)
for every $i, j\in I$ such that $i\geqslant j$;$f_{ij}\colon X_i\to X_j$ is $C$-definable;
(3)
$f_{ii}$ is the identity on $X_i$ and $f_{ik} = f_{jk}\circ f_{ij}$ for all $i \geqslant j \geqslant k$.
A pro-$C$-definable set$X$ is the projective limit $X\coloneq \varprojlim _{i\in I} X_i$ of a $C$-definable projective system $(X_i, f_{ij})$. We say that $X$ is pro-definable if it is pro-$C$-definable for some small set of parameters $C$. Pro-definable sets can also be seen as $\ast$-definable sets. By a result of Kamensky Reference 15, we may identify $X$ and $X(\mathcal{U})$.
Let $X=\varprojlim _{i\in I} X_i$ and $Y=\varprojlim _{j\in J} Y_j$ be two pro-$C$-definable sets with associated $C$-definable projective systems $(X_i,f_{ii'})$ and $(Y_j, g_{jj'})$. A pro-$C$-definable morphism is the data of a monotone function $d\colon J \to I$ and a family of $C$-definable functions $\{\varphi _{ij}\colon X_i\to Y_j \mid i\geqslant d(j)\}$ such that, for all $j\geqslant j'$ in $J$ and all $i \geqslant i'$ in $I$ with $i\geqslant d(j)$ and $i'\geqslant d(j')$, it holds that $\varphi _{i'j'}\circ f_{ii'} =g_{jj'}\circ \varphi _{ij}$.
4.2. Completions by definable types as pro-definable sets
What does it mean that a completion by definable types is pro-definable? Let us give the precise meaning of this in the case of the definable completion. The other completions are handled analogously.
We say that definable types are pro-definable in $T$ if, for every set of parameters $C$ and for every $\mathcal{L}(C)$-definable set $X$, there is a pro-$C$-definable (possibly in $\mathcal{L}^\mathrm{eq}$) set $P_X$ and a bijection $h_X^M\colon {S^\mathrm{def}_{X}}(M)\to P_X(M)$ for every model $M$ containing $C$. Further, we require some functoriality: if $N$ is an elementary extension of $M$, we have natural maps ${S^\mathrm{def}_{X}}(M)\to {S^\mathrm{def}_{X}}(N)$ and $P_{X}(M)\to P_X(N)$; and if $f\colon X\to Y$ is an $\mathcal{L}(C)$-definable function, then there are pro-$C$-definable morphisms $f'_N\colon P_X(N)\to P_Y(N)$ and $f'_M\colon P_X(N)\to P_Y(N)$ making the following diagram commute
Similarly, we say a subfunctor $\mathcal{C}(\bullet )$ of ${S^\mathrm{def}_{\bullet }}$ is pro-definable if it satisfies the same conditions after replacing ${S^\mathrm{def}_{\bullet }}$ with $\mathcal{C}(\bullet )$ in the above diagram. It is in this sense that we say that the bounded and orthogonal completions are pro-definable (note in both cases we have a subfunctor by Remark 3.3).
The following result, essentially due to E. Hrushovski and F. Loeser Reference 13, Lemma 2.5.1, shows the link between uniform definability of types and pro-definability. We include a proof for the reader’s convenience.
5. Stably embedded pairs and elementarity
Suppose $\mathcal{L}$ is a one-sorted language. Let $\mathcal{L}_P$ be a language extending $\mathcal{L}$ by a new unary predicate $P$. We denote an $\mathcal{L}_P$-structure as a pair $(N, A)$ where $N$ is an $\mathcal{L}$-structure and $A\subseteq N$ corresponds to the interpretation of $P$. Given a complete $\mathcal{L}$-theory$T$, the $\mathcal{L}_P$-theory of elementary pairs of models of $T$, is denoted $T_P$. Given a tuple $x=(x_1,\ldots ,x_m)$, we abuse of notation and write $P(x)$ as an abbreviation for $\bigwedge _{i=1}^n P(x_i)$. When $\mathcal{L}$ is multi-sorted we let $\mathcal{L}_P$ denote the language which extends $\mathcal{L}$ by a new unary predicate $P_\mathbf{D}$ for every $\mathcal{L}$-sort$\mathbf{D}$. Analogously, an $\mathcal{L}_P$-structure$N$ is a model of $T_P$ if the collection of subsets $P_\mathbf{D}(N)$ forms an elementary $\mathcal{L}$-substructure of $N$. We will also denote any such a structure as a pair $(N, M)$ where $M\preceq N\models T$ and for every $\mathcal{L}$-sort$\mathbf{D}$,$P_\mathbf{D}(N)=\mathbf{D}(M)$.
5.1. Stable embedded pairs
Let $M\preceq N$ be an elementary extension of models of $T$. The extension is called stably embedded if for every $\mathcal{L}(N)$-definable subset $X\subseteq \mathbf{D}_x(N)$, the set $X\cap \mathbf{D}_x(M)$ is $\mathcal{L}(M)$-definable in $M$.
The class of stably embedded models of $T$ will be denoted $\mathcal{SE}(T)$. It is a standard exercise to show that $M\preceq N$ is stably embedded if and only if for every tuple $a$ in $N$, the type $tp(a/M)$ is definable.
Our main objective is to show that for various NIP theories $T$,$\mathcal{SE}(T)$ is an $\mathcal{L}_P$-elementary class. Stable theories constitute a trivial example of this phenomenon, since the class of stably embedded pairs coincides with the class of elementary pairs. O-minimal theories constitute a less trivial example. Let us first recall Marker-Steinhorn’s characterizarion of definable types in o-minimal structures.
Since the Dedekind completeness of the small structure of a pair is expressible in $\mathcal{L}_P$, we readily obtain:
The following result of Q. Brouette shows the analogue result for $\mathrm{CODF}$.
The following characterization of definable types in $\mathbb{Z}$-groups (i.e. models of Presburger arithmetic) is due to G. Conant and S. Vojdani (see Reference 5).
5.2. Stable embedded pairs of valued fields
In what follows we gather the corresponding characterization of stably embedded pairs of models for $\mathrm{ACVF}$,$\mathrm{RCVF}$ and ${p}\mathrm{CF}_{e,f}$. We follow the notations introduced in Section 2.3. We need first the following terminology for induced structures. Let $M$ be an $\mathcal{L}$-structure and $\mathbf{D}$ be an imaginary sort in $M^{\mathrm{eq}}$. We let $\mathcal{L}_\mathbf{D}$ be the language having a predicate $P_R$ for every $\mathcal{L}$-definable (without parameters) subset $R\subseteq \mathbf{D}^n(M)$. The structure $(\mathbf{D}(M),\mathcal{L}_\mathbf{D})$ in which every $P_R$ is interpreted as the set $R$ is called the induced structure on $\mathbf{D}(M)$. The sort $\mathbf{D}$ is called stably embedded if every $\mathcal{L}^\mathrm{eq}(M)$-definable subset of a cartesian power of $\mathbf{D}(M)$ is $\mathcal{L}_\mathbf{D}$-definable. The following lemma is left to the reader.
Let us now explain how to show that the class $\mathcal{SE}(T)$ is $\mathcal{L}_P$-elementary for $T$ either $\mathrm{ACVF}$,$\mathrm{RCVF}$ or ${p}\mathrm{CF}_{e,f}$. In all three cases, the result will follow from the following theorem:
For $\mathrm{ACVF}$ the above Theorem is precisely the content of Reference 6, Theorem 1.9. For $\mathrm{RCVF}$ and ${p}\mathrm{CF}_{e,f}$ the result is new and the corresponding proof is presented in Sections 5.2.1 and 5.2.2.
Some of these results were recently extended by P. Touchard to other classes of Henselian fields, see Reference 22.
5.2.1. Stably embedded pairs of real closed valued fields
5.2.2. Stably embedded pairs of models of ${p}\mathrm{CF}_{e,f}$
Let $K$ be a model of ${p}\mathrm{CF}_{e,f}$. We need some preliminary lemmas.
Let $(K\subseteq L,v)$ be a valued field extension. Let $G$ be the convex hull of $\Gamma _K$ in $\Gamma _L$ and $w$ be the valuation on $L$ obtained by composing $v$ with the canonical quotient map $\Gamma _L\to \Gamma _L/G$. Let us denote $k_K^w$ and $k_L^w$ the residue fields of $(K,w)$ and $(L,w)$. As $w$ is trivial on $K$,$K\cong k_K^w$.
An element $a\in L$ is limit over $K$ if the extension $K(a)|K$ is an immediate extension. We let the reader check that if $K$ is a model of ${p}\mathrm{CF}_{e,f}$ and $a$ is limit over $K$, then the type $tp(a/K)$ is not definable.
We summarize the above results in the following theorem.
Note that there are NIP $\mathcal{L}$-theories$T$ for which the class $\mathcal{SE}(T)$ is not $\mathcal{L}_P$-elementary. The following example is due to L. Newelski.
6. Uniform definability via classes of pairs
The following theorem provides an abstract criterion for a theory $T$ to have uniform definability of types.
We expect similar results hold for theories of (tame) valued fields with generic derivations as defined in Reference 7 and for theories of o-minimal fields with a generic derivation as defined in Reference 11.
Acknowledgments
We would like to thank: S. Starchenko and Y. Peterzil for motivating the study spaces of definable types using pairs; M. Hils for interesting discussions; L. Newelski for sharing with us Example 5.15; and E. Kaplan for pointing out a proof of Theorem 6.3 for $\mathrm{CODF}$.
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Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université - Campus Pierre et Marie Curie 4, place Jussieu - Boite Courrier 247 75252 Paris Cedex 05
The first author was partially supported by the ERC project TOSSIBERG (Grant Agreement 637027), ERC project MOTMELSUM (Grant agreement 615722) and individual research grant Archimedische und nicht-archimedische Stratifizierungen höherer Ordnung, funded by the DFG. The second author was partially supported by NSF research grant DMS1500671.
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