1 Introduction
A standard technique in countable model theory associates, to each (possibly infinitary) first-order theory
$\mathcal {T}$
in some language
$\mathcal {L}$
, a topological space X parametrizing all of its countable models up to isomorphism. This enables the application to model theory of powerful tools from topology, dynamics, descriptive set theory, and computability theory, such as Baire category techniques and Borel complexity theory. See [Reference Becker and Kechris7], [Reference Gao25, Sections 3.6 and 11.4], [Reference Kechris33, Section 16.C], [Reference Montalbán49].
In fact, there are several variants of such a “topological space of models” X. Perhaps the best-known consists of all models on a fixed countably infinite set such as
$\mathbb {N}$
, regarded as a subspace of a Cantor space
$X \subseteq \prod _i 2^{\mathbb {N}^{n_i}}$
specifying the interpretations of each relation and function symbol in the language
$\mathcal {L}$
; see [Reference Gao25, Section 3.6]. By taking instead models on (certain) subsets of
$\mathbb {N}$
, one can also encompass finite models up to isomorphism; see [Reference Chen14]. One can also broaden (e.g., to all finitary first-order) or narrow (e.g., to only atomic) the class of formulas which define an open subset of X; see [Reference Bazhenov, Fokina, Rossegger, Soskova and Vatev5], [Reference Gao25, Section 11.4]. Alternatively, one can consider marked models over a fixed generating set, e.g., groups generated by
$\mathbb {N}$
, hence quotients of the free group 
by a normal subgroup, represented as the space of all normal subgroups of 
; see, e.g., [Reference Thomas54].
This landscape is clarified by the observation that such X, as a space of codes of countable structures, may be meaningfully distinguished from the structures themselves. That is, rather than a “space of models” X, one really has a topological space X together with a “continuous map”
$$ \begin{align*} X &\longrightarrow \{\text{all }\mathcal{L}\text{-structures}\} \\ x &\longmapsto \mathcal{M}_x \end{align*} $$
where the right-hand side is not a topological space, but a higher-order analog thereof, crucially differing in that two “points” (i.e., structures) may be “equal” (i.e., isomorphic) in more than one way. Such a generalized “space” is made precise by the concept of the classifying topos of
$\mathcal {L}$
-structures. Classifying toposes have been widely influential in such areas as algebraic geometry, topology, and category theory; see [Reference Artin, Grothendieck and Verdier2], [Reference Johnstone30], [Reference Lane and Moerdijk40]. However, to our knowledge, topos-theoretic ideas are not very well-known or used in the countable model theory literature, perhaps due to the substantial amount of category theory needed to define and work with them.Footnote
1
1.1 Étale structures
The goal of this article is to give a self-contained development of the above perspective from a classical model-theoretic and descriptive set-theoretic angle, minimizing the category theory needed. The central concept of a “continuous family of structures”
$(\mathcal {M}_x)_{x \in X}$
parametrized over a topological space X, as above, may be represented more concretely as an étale bundle of structures
$\mathcal {M} \to X$
, where
$\mathcal {M}$
is the disjoint union of the structures
$\mathcal {M}_x$
equipped with a global topology that captures the “continuity” of the fibers
$\mathcal {M}_x$
over
$x \in X$
. This concept is well-known in topos theory, perhaps more commonly in the equivalent form of a sheaf of structures. However, by taking an étale structure
$\mathcal {M} \to X$
as the formal meaning of a “continuous map”
$X \to \{\mathcal {L}\text {-structures}\}$
as above, one may develop much of classifying topos theory (for countable structures and theories) in a point-set topological manner, without reference to toposes, sheaves, or other categorical notions.
A recurring theme of our account is that many standard concepts and constructions in point-set topology and descriptive set theory have precise analogs for étale structures, thought of as “continuous families of structures” as above. Often, these analogies yield more precise formulations of well-known folklore connections. For instance, it is well-known that the classical Baire category theorem is closely related to the omitting types theorem in (infinitary) first-order logic, and in fact these two results may be used to prove each other; see, e.g., [Reference Eagle and Tall23]. In Section 7, we explain how the omitting types theorem is the literal first-order generalization of the Baire category theorem; and we show how the former may be reduced to the latter via an étale structure, as a higher-order instance of the fact that a continuous open surjection (is category-preserving, hence) may be used to transfer Baire category from its domain to its codomain. Table 1 depicts various other topological concepts with an étale model-theoretic analog that we will discuss.
Table 1 Correspondence between topological and étale model-theoretic notions, when Y is replaced with the “space of all structures”
In the course of developing this dictionary, we also prove generalizations to arbitrary étale parametrizations of classical descriptive set-theoretic results known for specific parametrizations. For instance, the Lopez-Escobar theorem [Reference Lopez-Escobar38] and its strengthening by Vaught [Reference Vaught55] show that every Borel isomorphism-invariant set of models, in the classical space of countably infinite models on
$\mathbb {N}$
, is axiomatizable by an infinitary formula, of the same quantifier complexity as the Borel complexity of the given set. Recently in [Reference Bazhenov, Fokina, Rossegger, Soskova and Vatev5], the authors prove a similar result, for a different parametrizing space, that takes positive (i.e., negation-free) formulas into account.Footnote
2
In [Reference Chen14], we proved such a result that also takes finite models into account. One could also ask whether a Lopez-Escobar theorem holds for the space of marked structures. We prove a result encompassing all of these as special cases.
Theorem 1.1 (Lopez-Escobar for étale structures; see Theorem 10.2).
For any étale parametrization of countable structures
$\mathcal {M} \to X$
with
$\Sigma _1$
saturations, every isomorphism-invariant
$\boldsymbol {\Sigma }^0_\alpha $
set of tuples
$A \subseteq M^n_X$
is defined by a
$\Sigma _\alpha $
formula.
Here
$\Sigma _\alpha $
refers to the
$\alpha $
th level of the complexity hierarchy of the countably infinitary logic
$\mathcal {L}_{\omega _1\omega }$
. To say that the étale structure
$\mathcal {M} \to X$
has
$\Sigma _1$
saturations means that the isomorphism saturation of each open set in M (or a finite power) is
$\Sigma _1$
-definable; see Definition 5.1. This is the key property on an étale structure that implies it is a “nice” parametrization of countable models obeying the usual results, and holds for all of the aforementioned “spaces of countable models” commonly considered in practice (subject to minor conditions on the theory); see Section 6.
As shown in Table 1, this condition is analogous to a continuous map between topological spaces
$f : X \to Y$
being open, or more precisely, being open onto its image
$f(X) \subseteq Y$
, since that is equivalent to each open
$U \subseteq X$
having open saturation by the equivalence relation
$\ker (f) \subseteq X^2$
induced by f. Such maps play an important role in descriptive set theory: for nice (e.g., Polish) spaces
$X, Y$
, the image
$f(X) \subseteq Y$
must be
$\boldsymbol {\Pi }^0_2$
, by a combination of classical results of Sierpiński [Reference Kechris33, Theorem 8.19] (that an open
$T_3$
quotient of a Polish space is Polish) and Alexandrov [Reference Kechris33, Theorem 3.11] (that Polish subspaces are
$\boldsymbol {\Pi }^0_2$
). We prove the analogous result for étale structures.
Theorem 1.2 (see Theorem 5.4).
For any étale parametrization of countable structures
$\mathcal {M} \to X$
with
$\Sigma _1$
saturations, over a (quasi-)Polish base X, the class of parametrized models (i.e., those isomorphic to a fiber of
$\mathcal {M}$
) is axiomatizable among countable structures by a
$\Pi _2$
sentence.
Here quasi-Polish refers to a generalization [Reference de Brecht19] of Polish spaces that we find more convenient; one definition is that they are precisely the open
$T_0$
quotients of Polish spaces. See Definition 2.4.
In fact, Theorem 1.2 generalizes not only the aforementioned classical results of Sierpiński and Alexandrov in topology but also (by taking
$X = 1$
) the classical model-theoretic fact that an atomic model has a
$\Pi _2$
Scott sentence (relative to a countable fragment of
$\mathcal {L}_{\omega _1\omega }$
; see [Reference Gao25, Theorem 11.5.7]). That these three classical results are connected at all may come as a surprise, and is an example of the conceptual clarification we believe is provided by the perspective of étale structures.
1.2 The Joyal–Tierney and Moerdijk representation theorems
Classically, given a continuous open surjection
$f : X \twoheadrightarrow Y$
between topological spaces, the topology on Y is necessarily the quotient topology induced by the topology on X. This fact means that if one is interested in studying a class of spaces Y, which can all be parametrized as open quotients of a restricted class of spaces X, then in some sense it “suffices” to study the spaces X together with the equivalence relations
$\ker (f) \subseteq X^2$
. For example, this provides one formal explanation of the idea that to do descriptive set theory, one really only needs to consider the Baire space
$\mathbb {N}^{\mathbb {N}}$
, whose open quotients yield all Polish spaces.
While this topological fact is rather easy, the analogous fact for étale structures with
$\Sigma _1$
saturations, i.e., “continuous open maps
$X \to \{\text {all structures}\}$
”, is much deeper, since the “space” we are parametrizing is a much more complicated object than the genuine space of parameters X. The analog of the equivalence relation
$\ker (f) \subseteq X^2$
, which consists of pairs of points
$x, y \in X$
such that
$f(x) = f(y)$
, is the isomorphism groupoid 
of an étale structure
$\mathcal {M} \to X$
, consisting of
$x, y \in X$
together with an isomorphism
$g : \mathcal {M}_x \cong \mathcal {M}_y$
. There is a natural “pointwise convergence” topology on 
(generalizing the pointwise convergence topology on
$\operatorname {\mathrm {Aut}}(\mathcal {M})$
for a single countable structure
$\mathcal {M}$
) that turns it into a topological groupoid with space of objects X; see Definition 4.15. We now have one of the central results of topos theory, which we state and prove for the restricted context of countable structures and
$\mathcal {L}_{\omega _1\omega }$
in Section 8:
Theorem 1.3 (Joyal–Tierney [Reference Joyal and Tierney31]; see Theorem 8.4 and Remark 8.6).
Let
$\mathcal {M} \to X$
be an étale parametrization of countable structures with
$\Sigma _1$
saturations, and let
$\mathcal {T}$
be the
$\Pi _2$
theory of its fibers from Theorem 1.2. Then the
$\Sigma _1$
imaginaries over
$\mathcal {T}$
are canonically equivalent (as a category) to the continuous étale actions of the groupoid 
.
Here by
$\Sigma _1$
imaginaries, we mean the usual model-theoretic concept [Reference Hodges28, Chapter 5] adapted to the infinitary setting: namely, a
$\Sigma _1$
-definable quotient of a countable disjoint union of
$\Sigma _1$
-definable sets (represented syntactically as families of
$\Sigma _1$
formulas); see Definition 8.2. Analogously to the topological setting, this result really says that étale structures in some sense “fully parametrize” countable models, since the theory
$\mathcal {T}$
of the parametrized models may be “recovered” from the topological groupoid 
. More precisely, we recover
$\mathcal {T}$
up to
$\Sigma _1$
bi-interpretability, since a theory is determined up to bi-interpretability by its imaginaries.
While Theorem 1.3 concerns the topological context, meaning on the model-theoretic side that we restrict to
$\Sigma _1$
formulas, there is an analogous result in the Borel context, which shows that a theory may be recovered up to
$\mathcal {L}_{\omega _1\omega }$
bi-interpretability from its Borel groupoid of isomorphisms. The proof combines Theorem 1.3 with a (Becker–Kechris-type) topological realization theorem for groupoid actions from [Reference Chen18] in order to translate between the topological and Borel settings. (As explained in Section 1.3 below, this generalizes results from [Reference Chen14] and [Reference Harrison-Trainor, Miller and Montalbán27].)
Theorem 1.4 (see Theorem 10.7 and Remark 10.8).
Let
$\mathcal {M} \to X$
be an étale parametrization of countable structures with
$\Sigma _1$
saturations, and let
$\mathcal {T}$
be the theory of its fibers. Then the
$\mathcal {L}_{\omega _1\omega }$
imaginaries over
$\mathcal {T}$
are canonically equivalent (as a category) to the fiberwise countable Borel actions of the groupoid 
.
As we explain in Section 9, it is in fact possible to make this recovery of a theory from its groupoid of models fully precise, provided that one is willing to make full use of category-theoretic language (which is why we postpone this discussion until Section 9). We are saying that
$$ \begin{align*} \{\text{theories}\} &\longrightarrow \{\text{groupoids}\} \\ \mathcal{T} &\longmapsto \text{isomorphism groupoid of some parametrization of }\mathcal{T} \end{align*} $$
is an “embedding”: we can recover a theory
$\mathcal {T}$
from its isomorphism groupoid. To make this precise, we should specify the kind of structure on the domain and codomain of this map. Namely, they are 2-categories: between theories
$\mathcal {T}_1, \mathcal {T}_2$
, we have interpretations
$\mathcal {F} : \mathcal {T}_1 \to \mathcal {T}_2$
as morphisms; and between parallel interpretations, we have definable isomorphisms as 2-cells (all either
$\Sigma _1$
or
$\mathcal {L}_{\omega _1\omega }$
):
Similarly, between groupoids, we have functors, between which we have natural isomorphisms (continuous or Borel, respectively). See Definition 9.2 and Remark 9.7 for details.
Theorem 1.5 (see Corollaries 11.13 and 11.14).
We have contravariant equivalences of 2-categories
taking a theory to the isomorphism groupoid of some parametrization of it.
As alluded to above, the “injectivity” of these maps (more precisely, that they are fully faithful 2-functors) boils down to the Joyal–Tierney Theorem 1.3 and its Borel analog, Theorem 1.4.
The “surjectivity,” that every topological groupoid satisfying certain conditions listed on the right-hand side above (see Definition 11.1 for details) can be represented as an isomorphism groupoid, boils down to a topos-theoretic argument of Moerdijk [Reference Moerdijk47], which constructs from an abstract topological groupoid G (satisfying certain conditions) an étale structure
$\mathcal {M}$
over the objects of G. We present a self-contained proof in Theorem 11.5, and show that in the countable case, via a Baire category argument, we get a full representation of the groupoid G as 
; see Theorem 11.11. This result simultaneously generalizes the Yoneda lemma from category theory (which shows that every discrete groupoid is canonically an isomorphism groupoid, via the left action on itself), and the classical result that non-Archimedean Polish groups are precisely the automorphism groups of countable structures (see, e.g., [Reference Gao25, Theorem 2.4.4]).
1.3 Connections to other work
This article was largely motivated by several recent works in the countable model theory literature showing connections between the syntax and semantics of infinitary logic. The earliest inspiration for this line of work seems to be [Reference Ahlbrandt and Ziegler1], showing that the automorphism group of an
$\aleph _0$
-categorical (in the usual finitary first-order sense) structure determines it up to (finitary) bi-interpretability. Analogously, [Reference Harrison-Trainor, Miller and Montalbán27] showed that the automorphism group of an arbitrary countable structure determines it up to
$\mathcal {L}_{\omega _1\omega }$
bi-interpretability. In that paper, respectively [Reference Harrison-Trainor, Melnikov, Miller and Montalbán26], the authors also showed that Borel, respectively continuous, functors between the groupoids of all isomorphic copies of two structures on underlying set
$\mathbb {N}$
correspond to
$\mathcal {L}_{\omega _1\omega }$
, respectively
$\Sigma _1$
, interpretations between those structures.
From the étale perspective, these results can be seen as special cases of the Joyal–Tierney Theorem 1.3 and its Borel version, Theorem 1.4, for specific examples of étale parametrizations
$\mathcal {M} \to X$
(whose fibers are all isomorphic).2 The special case of bi-interpretations follows from general categorical considerations, once Joyal–Tierney is recast into its 2-categorical “injectivity” form of Theorem 1.5. In [Reference Chen14], whose main result was yet another special case of Theorem 1.4 (again for a specific parametrization), we made this categorical perspective explicit, including the connections to the Joyal–Tierney and Becker–Kechris theorems (as noted above Theorem 1.4).
We should also note that, in the categorical literature, there have been previous point-set topological presentations of the Joyal–Tierney theorem, such as [Reference Butz and Moerdijk11] and [Reference Awodey and Forssell4]. The latter paper also used the Joyal–Tierney machinery to give a 2-categorical “duality” between theories and groupoids, as in Theorem 1.5, but for finitary first-order logic, and again considering only a specific étale parametrization (or rather, its sheaf-theoretic formulation).
We view this article as a sequel to [Reference Chen14], that is informed by the realization that working with an abstract étale structure, rather than a specific “space of countable models” as was done in that paper, yields a cleaner and more general presentation. In the preprint [Reference Chen15], we proved the Borel version of Theorem 1.5 using [Reference Chen14] and the aforementioned Moerdijk groupoid representation, in the process developing the needed parts of countable étale model theory in a rather dense fashion; the present article is a more comprehensive and (hopefully) readable rewrite of the first half of [Reference Chen15]. The second half of [Reference Chen15], dealing with continuous logic for metric structures [Reference Ben Yaacov, Berenstein, Henson and Usvyatsov8], will be subsumed in a future work that also includes a metric version of the Joyal–Tierney theorem and [Reference Chen14].
Finally, as mentioned in Footnote 2, several of the aforementioned works that inspired this article were rooted in computable structure theory, and proved effective strengthenings of the Lopez-Escobar theorem, Joyal–Tierney theorem, and related results, albeit for specific parametrizations; see [Reference Montalbán49] for a detailed survey. While we work entirely in the boldface setting in this article, it would be interesting to develop an effective version of the theory of étale structures presented here.
2 Preliminaries on topology
Recall that a Polish space is a second-countable, completely metrizable topological space. We will be working in a more general non-Hausdorff setting, that obeys better closure properties.
Definition 2.1 (Selivanov [Reference Selivanov52]).
In an arbitrary topological space X, possibly non-metrizable (hence closed sets may not be
$G_\delta $
), we define the Borel hierarchy as follows:
-
• A
$\boldsymbol {\Sigma }^0_1$
set is an open set. -
• For
$\alpha \ge 2$
, a
$\boldsymbol {\Sigma }^0_\alpha $
set is a countable union where each
$A_i, B_i$
is
$\boldsymbol {\Sigma }^0_\beta $
for some
$\beta < \alpha $
. -
• A
$\boldsymbol {\Pi }^0_\alpha $
set is the complement of a
$\boldsymbol {\Sigma }^0_\alpha $
set, thus for
$\alpha \ge 2$
is of the form
$\bigcap _i (A_i \Rightarrow B_i)$
, where
$A_i, B_i \in \boldsymbol {\Sigma }^0_\beta $
for some
$\beta < \alpha $
, and
$(A_i \Rightarrow B_i) := \neg A_i \cup B_i$
. (Here
$\neg $
denotes set complement.) -
• As usual, a
$\boldsymbol {\Delta }^0_\alpha $
set is a set that is both
$\boldsymbol {\Sigma }^0_\alpha $
and
$\boldsymbol {\Pi }^0_\alpha $
.
where each 
Example 2.2. In a second-countable
$T_0$
space X, the equality relation is
$\boldsymbol {\Pi }^0_2$
in
$X^2$
:
$$ \begin{align*} x = y \iff \forall \text{ basic open } U\, (x \in U \iff y \in U). \end{align*} $$
Definition 2.3. Sierpiński space is the topological space
$\mathbb {S} := 2$
with
$\{1\}$
open but not closed.
Definition 2.4 (de Brecht [Reference de Brecht19]Footnote 3 ).
A quasi-Polish space is a topological space homeomorphic to a
$\boldsymbol {\Pi }^0_2$
subspace of
$\mathbb {S}^{\mathbb {N}}$
.
In [Reference de Brecht19], de Brecht introduced quasi-Polish spaces and proved that they satisfy almost all of the usual descriptive set-theoretic properties of Polish spaces, including the following which we will freely use. For proofs, see [Reference de Brecht19] or [Reference Chen13].
-
(2.5) All quasi-Polish spaces are
$T_0$
, second-countable, and obey the Baire category theorem (which automatically then holds also for intersections of dense
$\boldsymbol {\Pi }^0_2$
sets, not just dense
$G_\delta $
). -
(2.6) A topological space is Polish iff it is quasi-Polish and regular (
$T_3$
). -
(2.7) If X is a quasi-Polish space, and
$A_i \subseteq X$
are countably many
$\boldsymbol {\Sigma }^0_\alpha $
sets, then there is a finer quasi-Polish topology containing each
$A_i$
and contained in
$\boldsymbol {\Sigma }^0_\alpha (X)$
. In more detail,-
(a) adjoining a single
$\boldsymbol {\Delta }^0_2$
set to the topology of X preserves quasi-Polishness; -
(b) if the intersection of countably many quasi-Polish topologies contains a quasi-Polish topology, then their union generates a quasi-Polish topology.
-
-
(2.8) A quasi-Polish space can be made zero-dimensional Polish by adjoining countably many closed sets to the topology.
-
(2.9) Countable products of quasi-Polish spaces are quasi-Polish.
-
(2.10) A space with a countable cover by open quasi-Polish subspaces is quasi-Polish. In particular, a countable disjoint union of quasi-Polish spaces is quasi-Polish.
-
(2.11) A subspace of a quasi-Polish space is quasi-Polish iff it is
$\boldsymbol {\Pi }^0_2$
. -
(2.12) A continuous open
$T_0$
quotient of a quasi-Polish space is quasi-Polish. Conversely, every quasi-Polish space is an open quotient of a zero-dimensional Polish space.
Definition 2.13. For a topological space X, its lower powerspace
$\mathcal {F}(X)$
is the space of all closed subsets
$F \subseteq X$
equipped with the topology generated by the open subbasis consisting of
$$ \begin{align*} \Diamond U := \{F \in \mathcal{F}(X) \mid F \cap U \ne \emptyset\} \end{align*} $$
for each open
$U \subseteq X$
. Equivalently, fix any open basis
$\mathcal {U}$
for X; then the topology on
$\mathcal {F}(X)$
is induced via the embedding
$$ \begin{align} \mathcal{F}(X) &\longrightarrow \mathbb{S}^{\mathcal{U}} \nonumber\\ F &\longmapsto \{U \in \mathcal{U} \mid F \cap U \ne \emptyset\}. \end{align} $$
If X is quasi-Polish, then so is
$\mathcal {F}(X)$
[Reference de Brecht and Kawai20] (see also (5.5) below).
We will also need the fiberwise lower powerspace
$\mathcal {F}_X(Y)$
of a continuous map
$p : Y \to X$
(thought of as a continuous bundle of spaces
$Y_x := p^{-1}(x)$
indexed over X), which is the disjoint union of the lower powerspaces of the fibers
$\mathcal {F}(Y_x)$
, i.e.,
equipped with the topology generated by the first coordinate projection
$\mathcal {F}_X(Y) \to X$
as well as
$$ \begin{align*} \Diamond_X U := \{(x, F) \in \mathcal{F}_X(Y) \mid F \cap U \ne \emptyset\} \end{align*} $$
for each open
$U \subseteq Y$
. When
$X, Y$
are quasi-Polish, so is
$\mathcal {F}_X(Y)$
[Reference Chen18, Proposition 2.5.9].
3 Preliminaries on infinitary logic
We now review some conventions on infinitary first-order logic. Throughout,
$\mathcal {L}$
will denote a countable first-order language. We note that we allow both nullary function symbols (i.e., constants) and nullary relation symbols (i.e., atomic propositions). In particular,
$\mathcal {L}$
could consist entirely of nullary relation symbols, in which case we are working essentially with propositional logic. Everything we do will also generalize straightforwardly to multi-sorted languages, although we will rarely bother to spell out the details of such a generalization.
The countably infinitary first-order logic
$\boldsymbol{\mathcal {L}}_{\omega _1\omega }$
is constructed like the usual finitary first-order logic, but allowing countable conjunctions
$\bigwedge _i \phi _i(\vec {x})$
and disjunctions
$\bigvee _i \phi _i(\vec {x})$
of formulas. We will use the symbols
$\top , \bot $
for nullary conjunction and disjunction. We adopt the convention that each
$\mathcal {L}_{\omega _1\omega }$
formula
$\phi (\vec {x})$
may only have finitely many free variables
$\vec {x}$
. For general background on
$\mathcal {L}_{\omega _1\omega }$
, see [Reference Ash and Knight3, Chapter 6], [Reference Gao25, Chapter 11], [Reference Marker42].
For a first-order
$\mathcal {L}$
-structure
$\mathcal {M}$
and formula
$\phi (x_0,\dotsc ,x_{n-1})$
, we write
$\phi ^{\mathcal {M}} \subseteq M^n$
for the interpretation of
$\phi $
in
$\mathcal {M}$
. We always allow structures to be empty.
There are completeness theorems for various deductive systems for
$\mathcal {L}_{\omega _1\omega }$
[Reference Karp32], [Reference Lopez-Escobar38], [Reference Fourman and Grayson24], which say that an
$\mathcal {L}_{\omega _1\omega }$
sentence has a proof iff it is true in every countable
$\mathcal {L}$
-structure.
Definition 3.1 (complexity classes of
$\mathcal {L}_{\omega _1\omega }$
formulas).
-
• A basic formula is a finite conjunction of atomic formulas (including equality, but no negations, not even
$\ne $
). -
• A positive-primitive formula is one built from atomic formulas using
$\wedge , \top , \exists $
. Up to logical equivalence, these may all be written in the prenex normal form
$\exists \vec {y}\, \phi (\vec {x},\vec {y})$
where
$\phi $
is basic; we will also call this a
$\exists {\wedge }$
formula. -
• A
$\Sigma _1$
formula is built from atomic formulas using
$\wedge , \top , \exists , \bigvee $
. These have a
${\bigvee }\exists {\wedge }$
normal form. -
• For
$\alpha \ge 2$
, a
$\Sigma _\alpha $
formula is built from
$\Sigma _\beta $
formulas and their negations, for
$\beta < \alpha $
, using
$\wedge , \top , \exists , \bigvee $
. These have a normal form where each
$\phi _i, \psi _i$
is
$\Sigma _\beta $
for some
$\beta < \alpha $
.
-
• For
$\alpha \ge 2$
, a
$\Pi _\alpha $
formula is the dual of a
$\Sigma _\alpha $
formula, or equivalently, for
$\phi _i,\psi _i$
as above,
Convention 3.2. We will assume whenever convenient that every
$\mathcal {L}_{\omega _1\omega }$
formula is built using
$\wedge , \top , \exists , \bigvee , \neg $
only, with
$\bigwedge , \forall $
treated as abbreviations for
$\neg \bigvee \neg , \neg \exists \neg $
, respectively. Thus, every formula is
$\Sigma _\alpha $
in the above sense for some
$\alpha $
.
The above definitions of
$\Sigma _\alpha , \Pi _\alpha $
are not the standard ones found in e.g., [Reference Ash and Knight3], which count all finitary quantifier-free formulas as
$\Sigma _1$
. (They are almost the “positive
$\Sigma ^p_\alpha $
,
$\Pi ^p_\alpha $
formulas” of [Reference Bazhenov, Fokina, Rossegger, Soskova and Vatev5], except that we do not even admit
$\ne $
as
$\Sigma _1$
.) While the standard definitions interact well with the Borel hierarchy of the usual zero-dimensional Polish space parametrizing
$\mathcal {L}$
-structures on
$\mathbb {N}$
(see [Reference Gao25, Chapter 11], [Reference Kechris33, Section 16.C], and Example 6.1 below), the above definitions are the natural counterpart to Selivanov’s non-Hausdorff Borel hierarchy (Definition 2.1) for quasi-Polish spaces.
Just as the non-Hausdorff Borel hierarchy generalizes the usual Borel hierarchy for Polish spaces, so too can the above “non-Hausdorff” hierarchy of formulas be regarded as generalizing the usual hierarchy. For given a theory
$\mathcal {T}$
, we may replace it with another
$\mathcal {T}'$
whose
$\Sigma _\alpha , \Pi _\alpha $
formulas, in the above sense, correspond to the usual
$\Sigma _\alpha , \Pi _\alpha $
formulas of
$\mathcal {T}'$
, via the following standard device.
Definition 3.3. A fragment of
$\mathcal {L}_{\omega _1\omega }$
is a set
$\mathcal {F}$
of formulas which contains all atomic formulas and is closed under variable substitutions and subformulas. (Note that this generalizes Sami’s notion of fragment from [Reference Sami51] which requires closure under
$\neg $
, which in turn generalizes the more common notion requiring closure under all finitary operations [Reference Gao25, Definition 11.2.3].)
The Morleyization of a fragment
$\mathcal {F}$
of
$\mathcal {L}_{\omega _1\omega }$
consists of an expanded language
$\mathcal {L}' \supseteq \mathcal {L}$
with a new n-ary relation symbol
$R_\phi $
for each
$\phi (x_0,\dotsc ,x_{n-1}) \in \mathcal {F}$
, together with the following
$\Pi _2 \mathcal {L}'$
-theory
$\mathcal {T}'$
which says that this is an expansion by definitions:
whenever the formulas in the subscripts are in
$\mathcal {F}$
, and adopting here Convention 3.2. Then each
$\mathcal {L}$
-structure
$\mathcal {M}$
has a unique expansion to a model
$\mathcal {M}'$
of
$\mathcal {T}'$
, where
$R_\phi ^{\mathcal {M}'} := \phi ^{\mathcal {M}}$
; and
$\Sigma _\alpha\ \mathcal {L}'$
-formulas
$\phi '$
up to equivalence mod
$\mathcal {T}'$
are in bijection with
$\mathcal {L}$
-formulas
$\phi $
which are “
$\boldsymbol{\Sigma} _\alpha $
over
$\boldsymbol{\mathcal {F}}$
,” meaning constructed as in
$\Sigma _\alpha $
-formulas but replacing atomic formulas with formulas in
$\mathcal {F}$
. We also call such an expansion
$\mathcal {M}'$
a Morleyization of the structure
$\boldsymbol{\mathcal {M}}$
(over the fragment
$\mathcal {F}$
).
If we start with an
$\mathcal {L}$
-theory
$\mathcal {T}$
, then we may instead add the above axioms to
$\mathcal {T}$
to obtain a new theory
$\mathcal {T}' \supseteq \mathcal {T}$
, whose models correspond bijectively to models of
$\mathcal {T}$
. We may furthermore replace each original axiom
$\phi \in \mathcal {T}$
which is “
$\Pi _\alpha $
over
$\mathcal {F}$
” with the equivalent
$\Pi _\alpha\ \mathcal {L}'$
-axiom
$\phi '$
.
Example 3.4. If we Morleyize the fragment
$\mathcal {F}$
consisting of all atomic and negations of atomic formulas, we obtain a theory
$\mathcal {T}'$
modulo which the
$\Sigma _\alpha , \Pi _\alpha\ \mathcal {L}'$
-formulas, in our “non-Hausdorff” sense, are equivalently the
$\Sigma _\alpha , \Pi _\alpha\ \mathcal {L}$
-formulas in the traditional sense.
Example 3.5. Suppose
$\mathcal {L}$
is a countable propositional language, consisting only of nullary relation symbols. It is best to regard such a language as being
$0$
-sorted, so that an “
$\mathcal {L}$
-structure” has no underlying set, and is simply a truth assignment
$\mathcal {M} : \mathcal {L} \to 2$
. The
$\Sigma _1$
formulas define the open sets of a topology, namely, the Sierpiński power topology on
$\mathbb {S}^{\mathcal {L}}$
(also known as the Scott topology). The
$\Sigma _\alpha , \Pi _\alpha $
formulas define precisely the
$\boldsymbol {\Sigma }^0_\alpha , \boldsymbol {\Pi }^0_\alpha $
subsets of
$\mathbb {S}^{\mathcal {L}}$
. Thus, quasi-Polish spaces are up to homeomorphism precisely the spaces of models of countable
$\Pi _2$
propositional theories.
The Morleyization of the preceding remark corresponds in this case to changing topology on
$\mathbb {S}^{\mathbb {N}}$
to turn it into
$2^{\mathbb {N}}$
, as in (2.8). More generally, Morleyization of an arbitrary countable fragment
$\mathcal {F}$
corresponds to changing topology to make countably many Borel sets open as in (2.7).
(In Definition 4.12, we relate Morleyization of non-propositional languages to change of topology.)
Remark 3.6. More generally, given an arbitrary countable language
$\mathcal {L}$
and countable
$\Pi _2 \mathcal {L}$
-theory
$\mathcal {T}$
, and
$n \in \mathbb {N}$
, the
$\Sigma _1$
formulas
$\phi (x_0,\dotsc ,x_{n-1})$
up to
$\mathcal {T}$
-provable equivalence yield a quasi-Polish topology on the space of
$\Sigma _1\ n$
-types of
$\boldsymbol{\mathcal {T}}$
, meaning countably prime (i.e., complement closed under countable disjunctions) filters of such formulas, or equivalently (by the completeness theorem) complete
$\Sigma _1$
theories of n-tuples in countable models of
$\mathcal {T}$
. When
$n = 0$
, this space has equivalent names in other contexts: the localic reflection of a topos (see [Reference Johnstone30, Proposition A4.6.12]), and the topological ergodic decomposition of a Polish group(oid) action (see [Reference Kechris35, Corollary 10.3], [Reference Chen16, Section 2]).
4 Étale spaces and structures
Definition 4.1. Let X be a topological space. An étale space over X is another topological space Y equipped with a continuous map
$p : Y \to X$
which is a local homeomorphism, i.e., Y has an open cover consisting of open sets
$S \subseteq Y$
such that
$p|S : S \to X$
is an open embedding. We will refer to such an
$S \subseteq Y$
as an open section, and to p as the projection map.
We may think of an étale space as a bundle of discrete sets indexed over X, namely, the fibers
$Y_x := p^{-1}(x)$
, with the discreteness witnessed uniformly across all fibers by the open sections. For this reason, when discussing étale spaces, we usually emphasize the space Y rather than the map p. For general background on étale spaces (and their close relatives, sheaves), see [Reference Tennison53].
Remark 4.2. By (2.10), for a quasi-Polish base space X, an étale space Y over it is quasi-Polish iff it is second-countable, in which case it has a countable basis of open sections.
Definition 4.3. If
$p : Y \to X$
is an étale space, and
$f : Z \to X$
is a continuous map, we write
$f^*(Y) = Z \times _X Y$
for the pullback or fiber product
$$ \begin{align*} f^*(Y) = Z \times_X Y := \{(z,y) \in Z \times Y \mid f(z) = p(y)\} \end{align*} $$
fitting into a commutative square:
Here
$f^*(p)$
is the first coordinate projection, and turns
$f^*(Y)$
into an étale space, with a cover by open sections
$f^*(S)$
for open sections
$S \subseteq Y$
. The second projection yields canonical bijections
$$ \begin{align*} f^*(Y)_z \cong Y_{f(z)}. \end{align*} $$
If Z is also étale over X, then so is
$f^*(Y) = Z \times _X Y$
, with open sections consisting of
$T \times _X S$
for open sections
$T \subseteq Z$
and
$S \subseteq Y$
. We also write
$$ \begin{align*} Y^n_X := Y \times_X \dotsb \times_X Y \end{align*} $$
for the n-fold fiber power. When
$n = 0$
, by convention
$Y^0_X := X$
. Sometimes, it is convenient to use the following alternative (canonically isomorphic to the above) definition of
$Y^n_X$
that explicitly records the basepoint in X, hence extends “uniformly” to the case
$n = 0$
:
Remark 4.5. It is easily seen that an equivalent characterization of local homeomorphism
$p : Y \to X$
is that both p and the diagonal embedding
$Y \hookrightarrow Y \times _X Y$
are open maps. The latter can be thought of as saying that the equality relation is open in each fiber, uniformly across all fibers.
Remark 4.6. If
$p : Y \to X$
and
$q : Z \to X$
are two étale spaces over the same base, and
$f : Y \to Z$
is a continuous map over X, i.e., making the triangle
commute, then f is automatically open, since for an open p-section
$S \subseteq Y$
, we have
where T varies over open q-sections in Z. It follows that each such S is an open f-section, whence f is in fact étale.
Remark 4.7. If
$p : Y \to X$
is a second-countable étale space, then p (is not only open but) maps
$\boldsymbol {\Sigma }^0_\alpha $
sets onto
$\boldsymbol {\Sigma }^0_\alpha $
sets, since this is clearly true for subsets of each open section
$S \subseteq Y$
.
It follows that in the preceding remark, assuming Y is second-countable, f also maps
$\boldsymbol {\Sigma }^0_\alpha $
sets onto
$\boldsymbol {\Sigma }^0_\alpha $
sets.
Definition 4.8. Let
$\mathcal {L}$
be a countable language, X be a topological space. An étale
$\boldsymbol{\mathcal {L}}$
-structure
$\mathcal {M}$
over X is defined by “internalizing the usual definition of first-order structure into the category of étale spaces over X.” Concretely,
$\mathcal {M}$
consists of the following data:
-
• an underlying étale space
$p : M \to X$
(in place of an underlying set); -
• for each n-ary function symbol
$f \in \mathcal {L}$
, a continuous map
$f^{\mathcal {M}} : M^n_X \to M$
over X; -
• for each n-ary relation symbol
$R \in \mathcal {L}$
, an open set
$R^{\mathcal {M}} \subseteq M^n_X$
.
By an abuse of notation, we will often refer to these data as “the étale structure
$p : \mathcal {M} \to X$
.”
Given such
$\mathcal {M}$
, we may inductively interpret each term
$\tau (x_0,\dotsc ,x_{n-1})$
in the obvious manner to get a continuous map
$\tau ^{\mathcal {M}} : M^n_X \to M$
over X. We may then interpret each
$\Sigma _\alpha\ \mathcal {L}_{\omega _1\omega }$
formula
$\phi (x_0,\dotsc ,x_{n-1})$
to get a subset
$\phi ^{\mathcal {M}} \subseteq M^n_X$
, which is
$\boldsymbol {\Sigma }^0_\alpha $
assuming that M is second-countable, by an easy induction using the three preceding remarks in the
$=, \exists $
cases.
If
$\phi $
is a sentence, then
$\phi ^{\mathcal {M}} \subseteq M^0_X = X$
(recall Definition 4.3); if
$\phi ^{\mathcal {M}} = X$
, then we call
$\mathcal {M}$
an étale model of
$\boldsymbol{\phi} $
, denoted
$\mathcal {M} \models \phi $
.
Remark 4.9. For an étale structure
$p : \mathcal {M} \to X$
, for each
$x \in X$
, we may restrict everything to the fiber over x to get an ordinary structure
$\mathcal {M}_x$
. Thus, we may think of
$\mathcal {M}$
as a “continuous bundle of discrete structures, indexed over X,” or as a “continuous map” (as explained in the introduction):
$$ \begin{align*} \mathcal{M} : X &\longrightarrow \{\text{all }\mathcal{L}\text{-structures}\} \\ x &\longmapsto \mathcal{M}_x. \end{align*} $$
Here the right-hand side is meant in a vague, informal sense; we do not literally mean a continuous map to a topological space whose elements are
$\mathcal {L}$
-structures. The whole point is that
$\mathcal {M}$
provides a way to regard X itself as a “space of structures.” (One may make the above map precise using the classifying topos of
$\mathcal {L}$
-structures; see [Reference Johnstone30, Chapter D3].)
For each n-ary formula
$\phi $
, the fibers
$\phi ^{\mathcal {M}}_x$
of
$\phi ^{\mathcal {M}} \subseteq M^n_X$
are the ordinary interpretations
$\phi ^{\mathcal {M}_x}$
. In particular, for a sentence
$\phi $
, to say that
$\mathcal {M} \models \phi $
just means
$\mathcal {M}_x \models \phi $
for each x.
Example 4.10. For a countable propositional language
$\mathcal {L}$
, regarded as a 0-sorted first-order language as in Example 3.5, an étale
$\mathcal {L}$
-structure
$p : \mathcal {M} \to X$
consists simply of an open set
$P^{\mathcal {M}} \subseteq X$
for each 0-ary relation symbol
$P \in \mathcal {L}$
, hence literally corresponds to a continuous map
$X \to \mathbb {S}^{\mathcal {L}}$
(whose Pth coordinate is the indicator function of
$P^{\mathcal {M}}$
).
Remark 4.11. If
$p : \mathcal {M} \to X$
is an étale structure, and
$f : Z \to X$
is a continuous map, then we have a pullback structure
$f^*(\mathcal {M}) \to Z$
, given by pulling back the underlying étale space M and the interpretations of all the symbols (Definition 4.3). Clearly, for
$z \in Z$
and a formula
$\phi $
,
$$ \begin{align*} f^*(\mathcal{M})_z &\cong \mathcal{M}_{f(z)}, \\ \phi^{f^*(\mathcal{M})} &= f^*(\phi^{\mathcal{M}}). \end{align*} $$
In particular, if
$\mathcal {M}$
satisfies some theory, then so does
$f^*(\mathcal {M})$
.
Definition 4.12. Let
$p : \mathcal {M} \to X$
be a second-countable étale structure, and
$\mathcal {F}$
be a countable fragment of
$\mathcal {L}_{\omega _1\omega }$
as in Definition 3.3, with Morleyization
$\mathcal {T}'$
. We may then Morleyize each fiber
$\mathcal {M}_x$
over
$\mathcal {F}$
as in Definition 3.3; however, the resulting bundle may no longer be an étale structure in the original topologies on
$M, X$
. Instead, we refine the topologies as follows.
Let
$X'$
be X with the finer topology generated by all subbasic open sets of the form
where
$U \subseteq M^n_X$
is a (basic) open set and
$\phi \in \mathcal {F}$
is an n-ary formula. This topology indeed refines that of X, since we may take
$n = 0$
and
$\phi = \top $
. Since for m-ary
$\phi $
and n-ary
$\psi $
,
a basis for this topology consists of all sets
$p(U \cap \phi ^{\mathcal {M}})$
as above but now with
$\phi $
“basic over
$\mathcal {F}$
,” i.e., a finite conjunction of formulas in
$\mathcal {F}$
.
Now let
$p' : \mathcal {M}' \to X'$
be the pullback of
$\mathcal {M}$
along the identity
$X' \to X$
. Thus, for each
$n \in \mathbb {N}$
, an open basis for
$M^{\prime n}_{X'}$
consists of
$$ \begin{align*} U \cap p^{-1}(p(V \cap \phi^{\mathcal{M}})) \end{align*} $$
for
$U \subseteq M^n_X$
(basic) open,
$V \subseteq M^m_X$
(basic) open for some other
$m \in \mathbb {N}$
, and
$\phi $
a finite conjunction of m-ary formulas in
$\mathcal {F}$
. For
$U = V \subseteq M^n_X$
an open section, the above set becomes simply
$U \cap \phi ^{\mathcal {M}}$
; taking a union over all U yields that
$\phi ^{\mathcal {M}} \subseteq M^{\prime n}_{X'}$
is open for each n-ary
$\phi \in \mathcal {F}$
, so that we may canonically expand
$\mathcal {M}'$
to an étale model of
$\mathcal {T}'$
over
$X'$
, which we call the Morleyization of the étale structure
$\boldsymbol{\mathcal {M}}$
(over
$\mathcal {F}$
).
Lemma 4.13. If X above is quasi-Polish, then so is
$X'$
.
Proof Adjoining each
$p(U \cap \phi ^{\mathcal {M}})$
(in the notation above) to the topology of X yields a quasi-Polish topology, by an induction on
$\phi $
, using that fragments are closed under subformulas (Definition 3.3), and using (2.7)(a) and (b) to handle
$\neg , \bigvee , \exists $
.
Remark 4.14. From the perspective of Remark 4.9, the “space of all models of
$\mathcal {T}'$
” can be seen as the “space of all
$\mathcal {L}$
-structures, with a finer topology (generated by
$\mathcal {F}$
).” The Morleyized base space
$X'$
above is then the result of “pulling back this finer topology along
$\mathcal {M} : X \to \{\text {all }\mathcal {L}\text {-structures}\}$
”:
Definition 4.15. Let
$p : \mathcal {M} \to X$
be an étale structure. Its isomorphism groupoid 
is the set of all triples
$(x,y,g)$
where
$x, y \in X$
and
$g : \mathcal {M}_x \cong \mathcal {M}_y$
is an isomorphism. We equip it with the topology generated by the first and second projections to X as well as the subbasic open sets
for open
$U, V \subseteq M$
. Equivalently, 
has an open basis consisting of the sets 
, defined in the same way, for all open
$U, V \subseteq M^n_X$
where
$n \in \mathbb {N}$
.
Since the expression 
is easily seen to preserve unions in each of the variables
$U, V$
, it is enough to consider
$U, V$
in any basis, e.g., a basis of open sections. This reveals the topology on 
to be a generalization of the usual pointwise convergence topology on the automorphism group of a single structure, to which 
reduces when
$X = 1$
.
As its name suggests, the isomorphism groupoid 
is (the space of morphisms of) a topological groupoid with space of objects X. The domain and codomain of a morphism are
These maps 
and the evident composition, identity, and inverse operations are easily seen to be continuous. (For the abstract notion of topological groupoid, see Definition 11.1.)
We have a natural action of 
on the underlying étale space
${M \to X}$
:
In other words, each morphism 
acts as the bijection
$g : M_x \to M_y$
. This action is easily seen to be jointly continuous. Similarly, 
acts continuously on the fiber powers
$M^n_X$
. (For the abstract notion of groupoid action, again see Definition 11.1.)
We will often abuse notation and refer to 
simply as
$g : \mathcal {M}_x \cong \mathcal {M}_y$
.
Lemma 4.16. If
$\mathcal {L}$
is a countable language and
$p : \mathcal {M} \to X$
is a second-countable étale
$\mathcal {L}$
-structure over a quasi-Polish space X, then 
is quasi-Polish.
Proof First, consider the case where
$\mathcal {L} = \emptyset $
, so that
$\mathcal {M}$
is just a second-countable étale space and 
. The definition of the topology on 
amounts to embedding it in the fiberwise lower powerspace (Definition 2.13) of
$p \times p : M \times M \to X \times X$
via
So we need only check that the image of this embedding is
$\boldsymbol {\Pi }^0_2$
. Fix a countable basis
$\mathcal {S}$
of open sections in M. For
$(x,y,F) \in \mathcal {F}_{X \times X}(M \times M)$
,
${F \subseteq M_x \times M_y}$
is the graph of a function iff
(which say that for each
$a \in M_x$
, there is at least, resp., at most, one
$b \in M_y$
such that
$(a,b) \in F$
); these conditions are
$\boldsymbol {\Pi }^0_2$
in the topology on
$\mathcal {F}_{X \times X}(M \times M)$
. Similarly, to say that
$F^{-1}$
is the graph of a function is a
$\boldsymbol {\Pi }^0_2$
condition.
Now if
$R \in \mathcal {L}$
is an n-ary relation symbol, then to say that
$g : M_x \cong M_y$
preserves R means
Similarly for
$g^{-1}$
to preserve R, and for preservation of functions.
Remark 4.17. If X and M are both zero-dimensional Polish, then so is 
. Indeed, we may find countable bases for each
$M^n_X$
consisting of clopen sections
$S \subseteq M^n_X$
such that
$p(S) \subseteq X$
is also clopen. If
$S, T \subseteq M^n_X$
are two such sections, then the basic open 
has open complement 
.
Remark 4.18. We have the following evident generalization of 
: if
$p : \mathcal {M} \to X$
and
$q : \mathcal {N} \to Y$
are two étale structures over two base spaces, then we have a space 
of isomorphisms between a fiber of
$\mathcal {M}$
and a fiber of
$\mathcal {N}$
. If
$\mathcal {M}, \mathcal {N}$
are both second-countable over quasi-Polish
$X, Y$
, then 
is quasi-Polish (zero-dimensional if
$X, Y, M, N$
are).
5 Étale structures with
$\Sigma _1$
saturations
We now come to the crucial condition on an étale
$\mathcal {L}$
-structure, that implies that it is a “nice” parametrization of countable
$\mathcal {L}$
-structures obeying the usual model-theoretic results.
Definition 5.1. Let
$\mathcal {M}$
be a second-countable étale
$\mathcal {L}$
-structure over a space X. We say that
$\mathcal {M}$
has
$\boldsymbol{\Sigma} _1$
-definable saturations of open sets, or more succinctly has
$\boldsymbol{\Sigma} _1$
saturations, if for every
$n \in \mathbb {N}$
and (basic) open set
$U \subseteq M^n_X$
, the saturation 
under the natural action of 
(recall Definition 4.15) is equal to
$\phi ^{\mathcal {M}}$
for some
$\Sigma _1$
formula
$\phi (x_0,\dotsc ,x_{n-1})$
.
Remark 5.2. From the perspective of Remark 4.9, this means that
$$ \begin{align*} \mathcal{M} : X \longrightarrow \{\text{all }\mathcal{L}\text{-structures}\} \end{align*} $$
is a “continuous open map onto its image.” (This is literally true for a propositional language
$\mathcal {L}$
via Example 4.10, and is made precise for general first-order
$\mathcal {L}$
by the notion of an open geometric surjection between toposes; see [Reference Johnstone30, C3.1], [Reference Lane and Moerdijk40, IX.6].)
Remark 5.3. Definition 5.1 may also be regarded as saying that
$\mathcal {M}$
is “uniformly
$\Sigma _1$
-atomic”: when
$X = 1$
, it means that every
$\mathcal {L}_{\omega _1\omega }$
-type realized in
$\mathcal {M}$
is axiomatized by a
$\Sigma _1$
formula.
Theorem 5.4. If
$\mathcal {L}$
is a countable language and
$\mathcal {M}$
is a second-countable étale
$\mathcal {L}$
-structure with
$\Sigma _1$
saturations over a quasi-Polish space X, then the class of countable
$\mathcal {L}$
-structures isomorphic to a fiber of
$\mathcal {M}$
is axiomatizable by a countable
$\Pi _2$
theory.
This result can be regarded as a common generalization of the well-known special cases alluded to by the two preceding remarks: that a continuous map between (quasi-)Polish spaces which is open onto its image has
$\boldsymbol {\Pi }^0_2$
image (the combination of (2.12) and (2.11)), and that an atomic model has a
$\Pi _2$
Scott sentence (relative to a fragment; see [Reference Gao25, Theorem 11.5.7]). The following proof can be seen as combining these two special cases, namely, the proofs of (2.12), (2.11) in [Reference Chen13, Theorems 10.1 and 4.1] and the usual back-and-forth construction of Scott sentences in, e.g., [Reference Montalbán48, Section 3.3].
Proof For each
$n \in \mathbb {N}$
, let
$\mathcal {U}_n$
be a countable open basis for the fiber power
$M^n_X$
. We use the fact that every quasi-Polish space has a countable posite representation; see [Reference Chen13, Section 8] for details. Briefly, each
$\vec {a} \in M^n_X$
is determined by its neighborhood filter basis
$\mathcal {N}_{\vec {a}} \subseteq \mathcal {U}_n$
; and this map
$\vec {a} \mapsto \mathcal {N}_{\vec {a}}$
yields an embedding
$M^n_X \hookrightarrow \mathbb {S}^{\mathcal {U}_n}$
, which thus has
$\boldsymbol {\Pi }^0_2$
image (2.11). Using that
$\mathcal {U}_n$
is a basis, one can manipulate a
$\boldsymbol {\Pi }^0_2$
definition of this image into the following form: for a countable binary relation
$\rhd _n$
consisting of certain pairs
$(\mathcal {V}, U)$
of open covers
$\mathcal {V} \subseteq \mathcal {U}_n$
and
$U \in \mathcal {U}_n$
with
$\bigcup \mathcal {V} = U$
, such that
$$ \begin{align*} \mathcal{V} \rhd_n U \supseteq U' \in \mathcal{U}_n \implies \exists \mathcal{V}' \rhd_N U'\, \forall V' \in \mathcal{V}'\, \exists V \in \mathcal{V}\, (V \supseteq V') \end{align*} $$
(“every cover of U refines to one of
$U' \subseteq U$
”), we have that
$\mathcal {A} \subseteq \mathcal {U}_n$
is a neighborhood filter of some
$\vec {a} \in M^n_X$
iff it is a
$\rhd _n$
-prime filter, meaning a filter satisfying the further
$\boldsymbol {\Pi }^0_2$
condition
It then follows that the
$\mathcal {C} \subseteq \mathcal {U}_n$
which are merely upward-closed and satisfy this condition correspond via (2.14) to arbitrary closed sets
$F \in \mathcal {F}(M^n_X)$
; see [Reference Chen13, proof of Theorem 9.2].
For each
$U \in \mathcal {U}_n$
, let
$\phi _U(x_0,\dotsc ,x_{n-1})$
be a
$\Sigma _1$
formula defining 
. We claim that the following
$\Pi _2$
sentences axiomatize the countable
$\mathcal {L}$
-structures isomorphic to a fiber of
$\mathcal {M}$
. For each n, let
$\pi _n : M^{n+1}_X \to M^n_X$
denote the projection onto the first n coordinates.
$$ \begin{align} \top &\to \bigvee_{U \in \mathcal{U}_0} \phi_U. \end{align} $$
It is straightforward to check that each
$\mathcal {M}_x$
indeed satisfies these axioms.
Now let
$\mathcal {N}$
be another countable
$\mathcal {L}$
-structure satisfying these axioms. For each
$\vec {b} \in N^n$
,
is upward-closed and satisfies (5.5) by (5.6) and (5.7), thus corresponds to a closed
$F_{\vec {b}} \subseteq M^n_X$
such that
$$ \begin{align*} F_{\vec{b}} \cap U \ne \emptyset \iff U \in \mathcal{C}_{\vec{b}} \iff \phi_U^{\mathcal{N}}(\vec{b}) \qquad \forall U \in \mathcal{U}_n. \end{align*} $$
Using this equivalence, for
$\mathcal {N}$
to satisfy the other five axioms above means the following.
-
• (5.8) means
$F_{\vec {b}} \cap \psi ^{\mathcal {M}} \ne \emptyset \implies \psi ^{\mathcal {N}}(\vec {b})$
for each basic formula
$\psi $
, which means that each
$\vec {a} \in F_{\vec {b}}$
yields a well-defined partial homomorphism
$\mathcal {M}_x \rightharpoonup \mathcal {N}$
given by
$a_i \mapsto b_i$
. (Well-definedness follows by taking
$\psi (\vec {x})$
to be an equality
$x_i = x_j$
, so that
$a_i = a_j \implies b_i = b_j$
.) -
• (5.9) means
, i.e., the set of
$\vec {a} \in F_{\vec {b}}$
for which the above partial homomorphism
$a_i \mapsto b_i$
is a partial isomorphism (onto its image) is dense
$G_\delta $
. -
• (5.10) means
$(\bigcup _{d \in N} F_{\vec {b},d}) \cap U \ne \emptyset \iff F_{\vec {b}} \cap \pi _n(U) \ne \emptyset $
, i.e.,
$\overline {\bigcup _{d \in N} F_{\vec {b},d}} = \pi _n^{-1}(F_{\vec {b}})$
. -
• In particular,
$\pi _n : M^{n+1}_X \to M^n_X$
restricts to a map
$F_{\vec {b},d} \to F_{\vec {b}}$
; (5.11) says it has dense image. -
• (5.12) means
$F_\emptyset \ne \emptyset $
.
, i.e., the set of 
Consider now the forest of all
$\vec {a} \in F_{\vec {b}}$
, each of which determines a partial homomorphism as above, as well as all infinite branches through this forest, which we arrange into a space as follows:
For
$m \le n \le \omega $
, let
$\pi _{nm} : Y_n \to Y_m$
denote the projection. Put a topology on Y with an open basis consisting of, for each
$m \in \mathbb {N}$
and (basic) open
$U \subseteq Y_m$
(where
$Y_m$
has the disjoint union topology),
This topology is quasi-Polish; see [Reference Chen16, Lemma A.2], where it was called the lax colimit of the levels
$Y_n$
. Since each
$\pi _n : F_{\vec {b},d} \to F_{\vec {b}}$
has dense image by (5.11), the closure in Y of each
$F_{\vec {b}} \subseteq Y_n \subseteq Y$
is all
$F_{\vec {b}|m}$
for initial segments
$\vec {b}|m$
of
$\vec {b}$
. Thus each 
is dense, and so 
is dense
$G_\delta $
. And by (5.12),
$Y_0 = F_\emptyset \ne \emptyset $
, i.e., the forest has at least one root.
We now verify that the set of all
$(\vec {b},\vec {a}) \in Y$
for which the partial homomorphism
$a_i \mapsto b_i$
is an isomorphism
$\mathcal {M}_x \cong \mathcal {N}$
is comeager, which will complete the proof.
-
• The set of
$(\vec {b},\vec {a}) \in Y$
for which the partial homomorphism
$a_i \mapsto b_i$
is a partial isomorphism is dense
$\boldsymbol {\Pi }^0_2$
, since it is the intersection, over all
$n \in \mathbb {N}$
, of the union of the lower levels
$Y_0, \dotsc , Y_{n-1}$
(which is closed) and of the disjoint union over all
$\vec {b} \in N^n$
of those
$\vec {a} \in \smash {F_{\vec {b}}}$
for which
$a_i \mapsto b_i$
is a partial isomorphism; and this latter is dense
$G_\delta $
in by (5.9). -
• The set of
$(\vec {b},\vec {a}) \in Y$
for which
$\vec {b}$
enumerates N is clearly
$G_\delta $
, and is dense because for a nonempty basic open where
$U \in \mathcal {U}_n$
, also
$\smash {F_{\vec {b},d}} \cap \pi _n^{-1}(U) \ne \emptyset $
by (5.11). -
• Finally, the set of
$(\vec {b},\vec {a}) \in Y$
for which
$\vec {a} \in M^n_X$
enumerates the fiber
$M_x$
in which it lies is dense
$\boldsymbol {\Pi }^0_2$
: we will show that for each open section
$S \in \mathcal {U}_1$
, the set of
$(\vec {b},\vec {a})$
such that if
$\vec {a}$
lies in a fiber over
$p(S)$
, then
$\vec {a}$
includes a point in S (which is a
$\boldsymbol {\Pi }^0_2$
set), is dense, which suffices because each point in M is in some open section. For a nonempty basic open where
$U \in \mathcal {U}_n$
, pick any
$\vec {a} \in F_{\vec {b}} \cap U$
, say lying in the fiber
$M_x$
. If
$x \in p(S)$
, then
$U \times _X S \subseteq M^{n+1}_X$
is an open set such that
$F_{\vec {b}} \cap \pi _n(U \times _X S) \ne \emptyset $
, since it contains
$\vec {a}$
. Thus by the
$\Longleftarrow $
direction of (5.10), for some
$d \in N$
, we have
$F_{\vec {b},d} \cap (U \times _X S) \ne \emptyset $
; an element
$(\vec {a}',c)$
of this intersection then belongs to (since
$\vec {a}' \in U$
) and includes
$c \in S$
.
of the disjoint union over all 
is dense 
by (
where 
where 
(since 
Definition 5.13. If
$\mathcal {M} \to X$
is a second-countable étale structure, and a theory
$\mathcal {T}$
axiomatizes the countable structures isomorphic to some
$\mathcal {M}_x$
, we say that X parametrizes models of
$\boldsymbol{\mathcal {T}}$
via
$\boldsymbol{\mathcal {M}}$
.
Thus, Theorem 5.4 says that if X is quasi-Polish and
$\mathcal {M}$
has
$\Sigma _1$
saturations, then X parametrizes models of a
$\Pi _2$
theory. Conversely, every
$\Pi _2$
theory has a quasi-Polish parametrization with
$\Sigma _1$
saturations; see Example 6.9.
Lemma 5.14. If
$p : \mathcal {M} \to X$
is a second-countable étale structure with
$\Sigma _1$
saturations, and
$f : Z \to X$
is a continuous open map, then the pullback structure
$f^*(\mathcal {M}) \to Z$
(Remark 4.11) also has
$\Sigma _1$
saturations.
Via Remark 5.2, this says that “the composite of open maps
$\mathcal {M} \circ f$
is still open”.
Proof Let
$\pi _2 : f^*(M) \to M$
be the pullback projection (see the diagram in Definition 4.3), which is open because it is a pullback of f. It is easily seen that for
$U \subseteq f^*(M)^n_Z$
, we have
Thus if
$\phi $
defines 
, then
$\phi $
also defines 
.
Remark 5.15. If
$p : \mathcal {M} \to X$
is a second-countable étale structure with
$\Sigma _1$
saturations, and
$Z = \{x \in X \mid \mathcal {M}_x \models \mathcal {T}\}$
for a theory
$\mathcal {T}$
, then the restriction
$\mathcal {M}|Z$
, i.e., pullback of
$\mathcal {M}$
along the inclusion
$Z \hookrightarrow X$
, clearly also has
$\Sigma _1$
saturations, as witnessed by the same
$\Sigma _1$
formulas.
(This is not an instance of the preceding lemma, since
$Z \subseteq X$
need not be open.)
Results such as 5.14 and 5.15 clearly work just as well if “
$\Sigma _1$
saturations” is replaced with “
$\Sigma _\alpha $
saturations” throughout, for some
$\alpha < \omega _1$
. The following calculation provides a rigorous way of systematically making such generalizations.
Lemma 5.16. Let
$p : \mathcal {M} \to X$
be a second-countable étale structure, and
$\mathcal {F}$
be a countable fragment of
$\mathcal {L}_{\omega _1\omega }$
. Suppose that saturations of open sets in
$\mathcal {M}$
are definable by formulas which are
$\Sigma _1$
over
$\mathcal {F}$
. Then the Morleyized étale structure
$p' : \mathcal {M}' \to X'$
(Definition 4.12) has
$\Sigma _1$
saturations.
Proof From Definition 4.12,
$M^{\prime n}_{X'}$
has an open basis of sets of the form
$$ \begin{align*} U \cap p^{-1}(p(V \cap \phi^{\mathcal{M}})) &= \pi_1(U \times_X (V \cap \phi^{\mathcal{M}})) \\ &= \pi_1((U \times_X V) \cap \phi(x_n, \dotsc, x_{n+m-1})^{\mathcal{M}}) \end{align*} $$
where
$U \subseteq M^n$
is open,
$V \subseteq M^m$
is open for some other
$m \in \mathbb {N}$
,
$\phi $
is a finite conjunction of m-ary formulas in
$\mathcal {F}$
,
$\pi _1 : M^n \times _X M^m \to M^n$
is the first coordinate projection, and
$\phi (x_n, \dotsc , x_{n+m-1})^{\mathcal {M}} = M^n_X \times _X \phi ^{\mathcal {M}}$
; since the last is isomorphism-invariant, we get
where
$\psi $
is a formula defining 
which is
$\Sigma _1$
over
$\mathcal {F}$
, hence may be replaced in
$\mathcal {M}'$
by an equivalent
$\Sigma _1$
formula in the Morleyized language
$\mathcal {L}'$
.
Corollary 5.17. If
$p : \mathcal {M} \to X$
is a second-countable étale structure over quasi-Polish X with
$\Sigma _\alpha $
saturations, then the class of structures isomorphic to one of its fibers is
$\Pi _{\alpha +1}$
-axiomatizable.
This is the
$\Sigma _\alpha $
version of Theorem 5.4, and generalizes (U1)
$\implies $
(U2) of Montalbán’s characterization of Scott rank [Reference Montalbán48] (see Theorem 7.4 below for other parts of the characterization).
Proof Let
$\mathcal {F}$
be the countable fragment generated by
$\Sigma _\alpha $
formulas defining the saturations of basic open sets in
$\mathcal {M}$
. By Lemma 5.16, the Morleyized
$p' : \mathcal {M}' \to X'$
has
$\Sigma _1$
saturations. Thus by Theorem 5.4, its fibers are
$\Pi _2$
-axiomatizable in the Morleyized language
$\mathcal {L}'$
, hence
$\Pi _{\alpha +1}$
-axiomatizable in the original language
$\mathcal {L}$
.
Lemma 5.18. If
$p : \mathcal {M} \to X$
is a second-countable étale structure with
$\Sigma _1$
saturations, then 
is an open topological groupoid, i.e., 
are open.
Proof For open
$U, V \subseteq M^n_X$
, we have 
; similarly for 
.
Remark 5.19. More generally, if
$p : \mathcal {M} \to X$
and
$q : \mathcal {N} \to Y$
are two étale structures, and
$\mathcal {M}$
has
$\Sigma _1$
saturations, then 
(recall Remark 4.18) is open.
6 Examples of parametrizations
We begin by recasting the standard zero-dimensional Polish parametrization of countably infinite models in terms of an étale structure; see [Reference Kechris33, Definition 16.5], [Reference Gao25, Section 3.6].
Example 6.1 (space of structures on
$\mathbb {N}$
).
Let
$\mathcal {L}$
be a countable language,
$$ \begin{align*} X := \prod_{n\text{-ary fn } f \in \mathcal{L}} \mathbb{N}^{\mathbb{N}^n} \times \prod_{n\text{-ary rel } R \in \mathcal{L}} 2^{\mathbb{N}^n} \end{align*} $$
be the zero-dimensional Polish space of
$\mathcal {L}$
-structures on
$\mathbb {N}$
, with the product topology, and
$$ \begin{align*} M := X \times \mathbb{N} \end{align*} $$
equipped with the first projection
$p : M \to X$
; this is clearly étale. On each fiber
$M_x = \{x\} \times \mathbb {N} \cong \mathbb {N}$
of M, we may put the structure
$\mathcal {M}_x := (M_x,x)$
; that is,
$$ \begin{align*} P^{\mathcal{M}}(x, a_0, \dotsc, a_{n-1}) &:= x(P)(a_0, \dotsc, a_{n-1}) \end{align*} $$
for each symbol (function or relation)
$P \in \mathcal {L}$
, using here the “based” representation (4.4) of
$M^n_X$
. This defines an étale
$\mathcal {L}$
-structure
$\mathcal {M}$
over X. The isomorphism groupoid is
where
$S_\infty $
is the infinite symmetric group of all bijections
$\mathbb {N} \cong \mathbb {N}$
, acting on X via pushforward of structure (the logic action). Up to topological groupoid isomorphism, we may forget about the second coordinate y above, yielding 
(the action groupoid of the logic action).
This
$\mathcal {M}$
does not have
$\Sigma _1$
saturations, however, but only
$\Sigma _2$
saturations. The complement of the diagonal in
$M^2_X$
is open and 
-invariant, but not
$\Sigma _1$
-definable, since
$\ne $
is not
$\Sigma _1$
; similarly for negations of any relation symbols in
$\mathcal {L}$
. In fact, this is the only issue: by definition of the product topology on X, a basic open set in
$M^n_X \cong X \times \mathbb {N}^n$
is of the form
for some fixed
$\vec {a} \in \mathbb {N}^{n+m}$
and finite conjunction
$\phi (y_n, \dotsc , y_{n+m-1})$
of atomic
$\mathcal {L}$
-formulas and their negations; the 
-saturation of this set is easily seen to be defined by the
$\Sigma _2$
formula
(This is the base case of the proof of the classical Lopez-Escobar theorem; see [Reference Kechris33, Proposition 16.9].)
We may thus Morleyize all negated atomic formulas, as in Example 3.4, to obtain an expanded language
$\mathcal {L}' \supseteq \mathcal {L}$
and a
$\Pi _2\ \mathcal {L}'$
-theory
$\mathcal {T}'$
. Let
$\mathcal {M}'$
be the unique expansion of
$\mathcal {M}$
to an étale model of
$\mathcal {T}'$
, by interpreting the newly added relation symbols as the complements of the atomic formulas interpreted in
$\mathcal {M}$
(which are clopen). Then
$\mathcal {M}'$
has
$\Sigma _1$
saturations, namely, given by the Morleyized formulas
$\psi '$
corresponding to the above
$\psi $
(as in Definition 3.3). A
$\Pi _2$
axiomatization of the fibers of
$\mathcal {M}'$
is given by
$\mathcal {T}'$
together with the “infinite models” axioms for each
$n \in \mathbb {N}$
$$ \begin{align} \exists x_0, \dotsc, x_{n-1} \bigwedge_{i \ne j} (x_i \ne x_j). \end{align} $$
If we started with a
$\Pi _2\ \mathcal {L}$
-theory
$\mathcal {T}$
which already has only infinite models, then we may restrict this
$\mathcal {M}'$
to the models of
$\mathcal {T}$
(as in Remark 5.15) to obtain a parametrization of the models of the Morleyized
$\mathcal {T}$
; if furthermore
$\mathcal {T}$
already implied that every negated atomic formula is equivalent to a
$\Sigma _1$
formula, then there is no need to Morleyize.
The “infinite models” restriction in the above example may be lifted, by enlarging the parametrizing space X to include finite models. There are several ways to do so, however, that affect the saturation complexity of the resulting
$\mathcal {M}$
, i.e., the amount of Morleyization needed to obtain
$\Sigma _1$
saturations. The simplest approach is the following:
Example 6.3 (space of structures on
$\le \mathbb {N}$
with
$\Delta _1$
sizes).
For each
$N \le \mathbb {N}$
(i.e.,
$N \in \mathbb {N} \cup \{\mathbb {N}\}$
), define an étale structure
$\mathcal {M}_N \to X_N$
over the space
$X_N$
of
$\mathcal {L}$
-structures on N, exactly as above. We may then simply take the disjoint unions
$X := \bigsqcup _{N \le \mathbb {N}} X_N$
and
$\mathcal {M} := \bigsqcup _{N \le \mathbb {N}} \mathcal {M}_N$
. The groupoid 
consists of the disjoint union of the logic actions of each symmetric group
$S_N$
on
$X_N$
.
However, each
$X_N \subseteq X = M^0_X$
is clopen invariant, while “the model has size N” is
$\Sigma _3$
-definable for finite N (by a conjunction of a sentence (6.2) and a negation of such a sentence) and
$\Pi _3$
-definable for infinite N (by the infinite conjunction of all (6.2)); thus this
$\mathcal {M}$
only has
$\Sigma _4$
saturations. Even if we first Morleyize
$\ne $
, i.e., use the traditional
$\Sigma _1$
that includes
$\ne $
, we still only get
$\Sigma _3$
saturations. (It is easy to see that Morleyizing the above sentences (6.2), their subformulas, their negations, and their conjunction, thereby rendering each
$X_N \subseteq X\ \Sigma _1$
-definable, is sufficient to yield
$\Sigma _1$
saturations, since each
$\mathcal {M}_N$
individually has
$\Sigma _1$
saturations by the same argument as in the preceding example.)
The worst feature of this approach is that infinite models are clopen in X but only
$\Pi _3$
-definable. In order to reduce this gap, we need a coarser topology on the disjoint union
$X = \bigsqcup _N X_N$
that allows finite models to converge to infinite ones. In the next few examples, we assume for simplicity that
$\mathcal {L}$
is relational; there is little loss of generality in doing so, since a
$\Pi _2$
theory may be used to impose that a relation is the graph of a function.
Example 6.4 (space of structures on
$\le \mathbb {N}$
with
$\Delta _1$
finite sizes).
Let
$\overline {\mathbb {N}} := \{0, 1, 2, \dotsc , \mathbb {N}\}$
, with the topology of the one-point compactification of
$\mathbb {N}$
, i.e.,
$0, 1, 2, \dotsc $
are isolated points converging to
$\mathbb {N}$
. We may realize the disjoint union X in the preceding example as
regarded now as a closed subspace of the compact zero-dimensional space
$\overline {\mathbb {N}} \times \prod _{R \in \mathcal {L}} 2^{\mathbb {N}^n}$
, and
$$ \begin{gather*} M := \{(N, x, a) \in X \times \mathbb{N} \mid a \in N\} \subseteq \overline{\mathbb{N}} \times \prod_{R \in \mathcal{L}} 2^{\mathbb{N}^n} \times \mathbb{N}, \\ R^{\mathcal{M}}(N, x, a_0, \dotsc, a_{n-1}) := x(R)(a_0, \dotsc, a_{n-1}) \quad \text{for }R \in \mathcal{L}. \end{gather*} $$
A basic open set in
$\overline {\mathbb {N}}$
is some finite Boolean combination of the sets
$\{n, n+1, \dotsc , \mathbb {N}\}$
which are axiomatized by the sentences (6.2). Thus, a basic open set in
$M^n_X$
is
for some
$\vec {a} \in \mathbb {N}^{n+m}$
, finite conjunction
$\phi (y_n, \dotsc , y_{n+m-1})$
of atomic
$\mathcal {L}$
-formulas and their negations, and finite conjunction
$\psi $
of the sentences (6.2) and their negations. Note that in addition to the condition “
$a_0, \dotsc , a_{n-1} \in N$
”, for each non-negated atomic formula in
$\phi $
, each of its arguments among
$a_n, \dotsc , a_{n+m-1}$
is also implicitly required to be in N. (Once we remember these implicit conditions, we may assume
$\phi $
contains no equalities, which have constant truth value given
$\vec {a}$
.) All of these conditions of the form “
$a_i \in N$
” together mean
$N \ge \max _i (a_i+1)$
, which can itself be expressed by a single sentence
$\theta $
of the form (6.2). Thus the saturation of the above set is defined by
which is
$\Sigma _3$
; it suffices to Morleyize negations of atomic formulas and of (6.2). As before, if we are only interested in parametrizing models of a
$\Pi _2$
theory modulo which negations of atomic formulas and “the model has size
$\le n$
” are already
$\Sigma _1$
-definable, then there is no need to Morleyize anything.
We may further reduce the need for Morleyization, provided we are willing to coarsen the topology to be quasi-Polish.
Example 6.7 (space of structures on
$\le \mathbb {N}$
with
$\Sigma _1$
lower bounds on size).
Modify the preceding example by putting the Scott topology on
$\overline {\mathbb {N}}$
, with open sets
$\{n, n+1, \dotsc , \mathbb {N}\}$
. Then the
$\psi $
in (6.6) above will only be a conjunction of the sentences (6.2), no longer their negations, and so the final formula defining the saturation will be
$\Sigma _2$
; it suffices to Morleyize negations of atomic formulas.
If we also replace
$2$
with
$\mathbb {S}$
in the definition of X, then the
$\phi $
in (6.6) will become basic (and equality-free); and so we only need to Morleyize
$\ne $
. In other words, we obtain a parametrization of any
$\Pi _2$
theory modulo which
$\ne $
is
$\Sigma _1$
-definable. (This parametrization was used in [Reference Chen14]; its restriction to the countably infinite models was used in [Reference Bazhenov, Fokina, Rossegger, Soskova and Vatev5].)
Remark 6.8. In fact, the quasi-Polishness is not essential here: by (2.12), we may find a continuous open surjection
$Z \twoheadrightarrow X$
from a zero-dimensional Polish Z (e.g.,
$\sum : 2^{\mathbb {N}} \twoheadrightarrow \overline {\mathbb {N}}$
), and then pull back
$\mathcal {M} \to X$
from above to an étale structure over Z, with underlying étale space
$\subseteq Z \times \mathbb {N}$
which is clearly also zero-dimensional. Thus every
$\Pi _2$
theory modulo which
$\ne $
is
$\Sigma _1$
-definable has a zero-dimensional Polish parametrization via a zero-dimensional étale structure with
$\Sigma _1$
saturations.
The final step of removing the need to Morleyize
$\ne $
requires a more drastic change. We need to render the diagonal in
$M^2_X$
open but not closed, which clearly cannot be achieved by merely coarsening the topology on the base X while keeping the fibers subsets of some fixed set such as
$\mathbb {N}$
.
Example 6.9 (space of partially enumerated structures).
Let
$\mathcal {L}$
be a countable relational language, and let
$\mathcal {L}' := \{{\sim }\} \sqcup \mathcal {L}$
where
$\sim $
is a binary relation symbol. Construct as in the second part of Example 6.7 a parametrization
$\mathcal {M}' \to X'$
of all
$\mathcal {L}'$
-structures which has
$\Sigma _1$
saturations modulo
$\ne $
. Let
$X \subseteq X'$
be the structures in which
$\sim $
is an equivalence relation and all relations in
$\mathcal {L}$
are
$\sim $
-invariant. Then
$\sim ^{\mathcal {M}'}$
, which is open in
$(M')^2_{X'}$
, restricted to the fibers over X is an open fiberwise equivalence relation on the étale space
$M'|X \to X$
, i.e., only equating pairs of elements in the same fiber. It follows that the quotient
$M := (M'|X)/({\sim ^{\mathcal {M}'}}|X) \to X$
is also an étale space, and that the quotient map
$M'|X \twoheadrightarrow M$
is open. Let
$\mathcal {M}$
be the étale structure on M descended from
$\mathcal {M}'|X$
.
Explicitly, the space X consists of tuples
$(N, {\sim }, x)$
, consisting of
$N \in \overline {\mathbb {N}}$
(with the Scott topology), an equivalence relation
$\sim $
on N, and a
$\sim $
-invariant
$\mathcal {L}$
-structure x on N (both with the Sierpiński topology), so that x descends to the quotient
$N/{\sim }$
; this descended structure is then the fiber
$\mathcal {M}_{(N, {\sim }, x)}$
of
$\mathcal {M} \to X$
. It is clear that
$\mathcal {M}$
parametrizes all countable
$\mathcal {L}$
-structures up to isomorphism, since for any countable
$\mathcal {L}$
-structure, we may enumerate it with an initial segment
$N \le \mathbb {N}$
and then lift the structure to N.
We now check that
$\mathcal {M}$
has
$\Sigma _1$
saturations. A basic open set in
$M^n_X$
is the image under the quotient map of one in
$(M')^n_{X'}$
as in (6.5) restricted to X, where (in the notation of that example)
$\vec {a}$
is fixed,
$\phi $
is a basic equality-free formula and
$\psi $
only asserts that “the model has size
$\ge \dotsb $
” as in Example 6.7, since we used the Scott topology for
$X'$
. We claim that the saturation of such an image in
$M^n_X$
is defined by (6.6) with all occurrences of
$\sim $
in
$\phi $
replaced with
$=$
and all
$\ne $
replaced with
$\top $
. Suppose
$([b_0],\dotsc ,[b_{n-1}]) \in (N'/{\sim '})^n = M^n_{(N',{\sim '},x')}$
satisfies this modified version of (6.6), as witnessed by
$([b_n],\dotsc ,[b_{n+m-1}]) \in (N'/{\sim '})^m = M^m_{(N',{\sim '},x')}$
; we must find
$N, {\sim }, x$
such that
$(N, {\sim }, x, a_0, \dotsc , a_{n-1})$
is in (6.5) and there is an isomorphism of the quotient structures
$(N,x)/{\sim } \cong (N',x')/{\sim '}$
mapping
$[a_i] \mapsto [b_i]$
for
$i < n$
. Let
$N \subseteq \mathbb {N}$
be an initial segment such that
-
(i) if
$N' = \emptyset $
, then
$N = \emptyset $
; -
(ii) each
$a_0, \dotsc , a_{n+m-1} \in N$
(this is consistent with (i) since
$[b_0], \dotsc , [b_{n+m-1}] \in N'/{\sim '}$
); -
(iii) N is at least as big as required by
$\psi $
(consistent with (i) since
$N'/{\sim '} \models \psi $
); -
(iv)
is at least as big as .
is at least as big as 
.By virtue of the
$\bigwedge _{a_i = a_j} (y_i = y_j)$
in (6.6),
$a_i \mapsto [b_i]$
is a well-defined function
$\{a_0,\dotsc ,a_{n+m-1}\} \to \{[b_0],\dotsc ,[b_{n+m-1}]\} \subseteq N'/{\sim '}$
; extend it to a surjection
$h : N \twoheadrightarrow N'/{\sim '}$
, which is easily seen to be possible using the above conditions. Now define 
, and lift the rest of the quotient structure on
$(N',x')/{\sim '}$
to the structure x on N. Then h descends to an isomorphism
$N/{\sim } \cong N'/{\sim '}$
mapping
$[a_i] \mapsto [b_i]$
, whence
$(N, {\sim }, x, a_0, \dotsc , a_{n-1})$
is in (6.5).
As usual, by further restricting this construction to the models of a
$\Pi _2$
theory
$\mathcal {T}$
, we obtain that every
$\Pi _2$
theory has a quasi-Polish parametrization with
$\Sigma _1$
saturations. As in Remark 6.8, we may further pull back to a zero-dimensional Polish
$Z \twoheadrightarrow X$
; however, unlike there, we can no longer assume that the underlying étale space M is also zero-dimensional.
Remark 6.10. Minor variations of Examples 6.3, 6.4, 6.7 and 6.9 are to allow arbitrary subsets
$N \subseteq \mathbb {N}$
, not just initial segments. (Such a variation of Example 6.7 was essentially used in [Reference Awodey and Forssell4].)
Remark 6.11 (space of totally enumerated structures).
A further variation of Example 6.9 is to again require
$N = \mathbb {N}$
, as in Example 6.1 (while still using the Sierpiński space for relations including the equivalence relation
$\sim $
). In the above verification that
$\mathcal {M}$
has
$\Sigma _1$
saturations, the possibility of
$N \ne \mathbb {N}$
only comes into play in (i), and is clearly not needed if also
$N' = \mathbb {N}$
. The minor downside is that now
$\mathcal {M}$
only parametrizes nonempty models. (This was the original parametrization used by Joyal–Tierney [Reference Joyal and Tierney31, VII Section 3 Theorem 1].)
Finally, we consider a parametrization of a somewhat different flavor, which only differs from that in Remark 6.11 for languages with function symbols. This parametrization or a variant thereof is widely used for studying classification of algebraic structures such as groups; see, e.g., [Reference Thomas54].
Example 6.12 (space of marked structures).
Let
$\mathcal {L}$
be an arbitrary countable language, possibly with function symbols. Let T be the set of
$\mathcal {L}$
-terms over countably many variables
$a_0, a_1, \dotsc $
. Let
$X'$
be the space of enumerated
$\mathcal {L}$
-structures as in Remark 6.11, where function symbols in
$\mathcal {L}$
are encoded via their graphs, and where the role of the index set
$\mathbb {N}$
is replaced by T; let
$\mathcal {M}' \to X'$
be the corresponding étale structure. Thus a point in
$X'$
consists essentially of an equivalence relation
${\sim }$
on T and an
$\mathcal {L}$
-structure on
$T/{\sim }$
. Let
$X \subseteq X'$
be the
$\boldsymbol {\Pi }^0_2$
subspace where
$\sim $
is a congruence with respect to the usual syntactic action of function symbols on terms, and where the quotient structure on
$T/{\sim }$
interprets each function symbol via this syntactic action. In other words,
Let
$\mathcal {M} := \mathcal {M}'|X$
, so that
$\mathcal {M}_{(\sim ,x)} = (T/{\sim }, \widetilde {x})$
is said generated structure.
Note that
$X \subseteq X'$
is not 
-invariant; thus we cannot immediately conclude that
$\mathcal {M}$
has
$\Sigma _1$
saturations by Remark 5.15. Nonetheless, this is the case, as we now check. Chasing through Remark 6.11, Example 6.9, and ultimately (6.5), a basic open set in
$M^n_X$
is
for some fixed terms
$t_0, \dotsc , t_{n+m-1} \in T$
and basic
$(\mathcal {L} \cup \{\sim \})$
-formula
$\phi $
not mentioning
$=$
; by absorbing function applications into
$t_n, \dotsc , t_{n+m-1}$
, we may assume
$\phi $
only mentions relation symbols in
$\mathcal {L}$
and
$\sim $
. Let
$k \in \mathbb {N}$
be sufficiently large so that
$t_0, \dotsc , t_{n+m-1}$
only mention the variables
$a_0, \dotsc , a_{k-1}$
. Then the 
-saturation of (6.13) is defined by
where
$\phi '$
is
$\phi $
with all occurrences of
$\sim $
replaced with
$=$
. Indeed, if
$\mathcal {N}$
is a countable nonempty (which the fibers of
$\mathcal {M}$
all are)
$\mathcal {L}$
-structure which satisfies this formula under an assignment
$\vec {x} \mapsto \vec {b} \in N^n$
, as witnessed by some
$h : \{a_0, \dotsc , a_{k-1}\} \to N$
, we may extend h to the remaining variables
$a_k, a_{k+1}, \dotsc $
to yield an enumeration of N, and then to all terms to obtain a surjective homomorphism
$h : T \twoheadrightarrow N$
with respect to the function symbols in
$\mathcal {L}$
, and finally pull back the relations as well as
$=$
in
$\mathcal {N}$
along h to obtain
$({\sim },x) \in X$
such that
$h : T \twoheadrightarrow N$
descends to an isomorphism
$\mathcal {M}_{({\sim },x)} \cong \mathcal {N}$
and
$({\sim },x,[t_0],\dotsc ,[t_{n-1}])$
is in (6.13) where
$t_0, \dotsc , t_{n-1}$
are any h-preimages of
$\vec {b}$
.
Remark 6.15. For finite
$N \in \mathbb {N}$
, one could analogously construct a space
$X_N$
as above starting from only N variables
$a_0, \dotsc , a_{N-1}$
, parametrizing N-generated structures. However, saturations would only be
$\Sigma _3$
, since in the defining formulas (6.14) (where without loss we can take
$k := N$
), we need to add a clause
$\forall y \bigvee _{t \in T} (y = t)$
saying that
$a_0, \dotsc , a_{N-1}$
are generators.
7 Omitting types and Scott rank
The classical Baire category theorem states that in sufficiently nice topological spaces, a countable intersection of dense open (or even
$\boldsymbol {\Pi }^0_2$
) sets is still dense. The omitting types theorem in model theory, and its infinitary variants, state that a countable conjunction of “dense
$\Sigma _1$
(or even
$\Pi _2$
) formulas” is still “dense.” It is well-known folklore that these two results are closely related, and in fact omitting types can also be reduced to Baire category via some standard coding tricks; see, e.g., [Reference Eagle and Tall23] and the references therein. As an application of the “continuous open map” perspective of Remark 5.2, we give here an easy and transparent version of such a reduction.
Theorem 7.1 (omitting types).
Let
$\mathcal {T}$
be a satisfiable countable
$\Pi _2$
theory.
-
(a) Let
$\phi _i(\vec {x}_i)$
be countably many
$\Pi _2$
formulas of various arities
$n_i \in \mathbb {N}$
, such that for each
$\Sigma _1\ \theta (\vec {x}_i)$
satisfiable in some model of
$\mathcal {T}$
,
$\phi _i \wedge \theta $
is also satisfiable in some model of
$\mathcal {T}$
. Then
$\mathcal {T} \cup \{\forall \vec {x}_i\, \phi _i(\vec {x}_i)\}_i$
is satisfiable. -
(b) Let
$\psi _i(\vec {x}_i)$
be countably many
$\Sigma _2$
formulas of various arities
$n_i \in \mathbb {N}$
, such that for each
$\Sigma _1\ \theta (\vec {x}_i)$
, if
$\mathcal {T} \models \forall \vec {x}_i\, (\theta (\vec {x}_i) \to \psi _i(\vec {x}_i))$
, then
${\mathcal {T} \models \forall \vec {x}_i\, \neg \theta (\vec {x}_i)}$
. Then
$\mathcal {T} \cup \{\forall \vec {x}_i\, \neg \psi _i(\vec {x}_i)\}_i$
is satisfiable.
Proof The statements are dual; we prove (a). By Example 6.9, there is a quasi-Polish parametrization
$p : \mathcal {M} \to X$
of
$\mathcal {T}$
with
$\Sigma _1$
saturations, i.e., a “continuous open map
$\mathcal {M} : X \twoheadrightarrow \{\text {models of }\mathcal {T}\}$
.” Then “the
$\mathcal {M}$
-preimage of each dense
$\Pi _2\ \phi _i$
is dense
$\boldsymbol {\Pi }^0_2$
”, i.e., each
$\phi _i^{\mathcal {M}} \subseteq M^{n_i}_X$
is dense
$\boldsymbol {\Pi }^0_2$
, since for each open
$\emptyset \ne U \subseteq M^{n_i}_X$
, letting
$\theta $
be
$\Sigma _1$
with 
,
$\theta $
is satisfiable in a model of
$\mathcal {T}$
, whence so is
$\phi \wedge \theta $
, whence 
since
$\phi ^{\mathcal {M}}$
is 
-invariant. Hence, each
$\phi _i^{\mathcal {M}} \subseteq M^{n_i}_X$
is comeager. It follows that each
$(\forall \vec {x}_i\, \phi _i(\vec {x}_i))^{\mathcal {M}} \subseteq X$
is comeager, since dually, the p-image of each meager set in
$M^{n_i}_X$
is easily seen to be meager in X, by considering a countable basis of open sections in
$M^{n_i}_X$
. Thus
$\bigcap _i (\forall \vec {x}_i\, \phi _i(\vec {x}_i))^{\mathcal {M}}$
is also comeager, hence dense by Baire category; picking any x in this set, we have
$\mathcal {M}_x \models \mathcal {T} \cup \{\forall \vec {x}_i\, \phi _i(\vec {x}_i)\}_i$
.
Example 7.2. Taking
$\mathcal {T}$
to be a propositional theory, as in Example 3.5 (and the arities
$n_i$
above to all be
$0$
), we recover the Baire category theorem for quasi-Polish spaces as a special case.
Remark 7.3. Via Morleyization, Theorem 7.1 generalizes to an omitting types theorem for an arbitrary countable fragment
$\mathcal {F}$
, which generalizes the classical omitting types for
$\mathcal {L}_{\omega _1\omega }$
(see, e.g., [Reference Marker42, Theorem 4.9]) to our more general fragments (Definition 3.3) not necessarily closed under
$\neg , \wedge , \vee , \forall , \exists $
.
In particular, Theorem 7.1 is sufficiently general to imply Montalbán’s omitting types theorem for
$\Pi _\alpha $
[Reference Montalbán48, Lemma 3.2], using which we may generalize most parts of Montalbán’s characterization of Scott rank [Reference Montalbán48, Theorem 1.1] to our weaker notions of
$\Sigma _\alpha , \Pi _\alpha $
.
Theorem 7.4 (characterization of Scott rank).
Let
$p : \mathcal {M} \to X$
be a second-countable étale structure with
$\Sigma _1$
saturations over a quasi-Polish space X, and let
$x \in X$
. The following are equivalent:
-
(i) Every automorphism orbit of
$\mathcal {M}_x$
is
$\Sigma _\alpha $
-definable without parameters. -
(ii)
$\mathcal {M}_x$
has a
$\Pi _{\alpha +1}$
Scott sentence
$\phi $
. -
(iii)
is
$\boldsymbol {\Pi }^0_{\alpha +1}$
. -
(iv) For every
$\Pi _\alpha $
formula
$\psi (\vec {z})$
with
$\psi ^{\mathcal {M}_x} \ne \emptyset $
, there is a
$\Sigma _\alpha $
formula
$\theta (\vec {z})$
with
$\emptyset \ne \theta ^{\mathcal {M}_x} \subseteq \psi ^{\mathcal {M}_x}$
.
is 
To recover [Reference Montalbán48, Theorem 1.1], Morleyize negated atomic formulas as in Example 3.4 and take the standard parametrization from Example 6.1. The proof is the same as in [Reference Montalbán48] but with results used therein replaced by their generalizations from this article, namely, Corollary 5.17 for (i)
$\implies $
(ii), the Lopez-Escobar Theorem 10.2 for (ii)
$\iff $
(iii), and Theorem 7.1 for (ii)
$\implies $
(iv).
Remark 7.5. The omitting types theorem has an analog in topos theory, called the
$\neg \neg $
-subtopos, which is the “theory of models of
$\mathcal {T}$
omitting all non-isolated
$\Pi _1$
types”; see [Reference Johnstone30, Example A4.5.9].
8 Imaginary sorts and the Joyal–Tierney theorem
Definition 8.1. For an étale structure
$p : \mathcal {M} \to X$
, an étale
$\boldsymbol{\mathrm{Iso}}_{\boldsymbol{X}}(\boldsymbol{\mathcal {M}})$
-space over X is an étale space
$q : A \to X$
equipped with a (jointly) continuous action of the topological groupoid
$\mathrm{Iso}_X(\mathcal {M})$
, where each 
acts via a bijection
$A_x \cong A_y$
. (See Definition 11.1 for the formal definition of an abstract groupoid action.)
The prototypical étale 
-space is M (recall Definition 4.15). We may build other étale 
-spaces via the following basic operations:
-
• Each fiber power
$M^n_X$
is also an étale -space. More generally, a fiber product of finitely many étale -spaces is again such a space, under the diagonal action. -
• If
$A \to X$
is an étale -space, and
$U \subseteq A$
is an open set invariant under the action of , then U is itself an étale -space. If
$A = M^n_X$
, and
$\mathcal {M}$
has
$\Sigma _1$
saturations, then such U are precisely the
$\Sigma _1$
-definable n-ary relations
$\phi ^{\mathcal {M}} \subseteq M^n_X$
. -
• A (possibly infinite) disjoint union of étale
-spaces is again such. -
• If
$p : A \to X$
is an étale -space, and
${\sim } \subseteq A \times _X A$
is an -invariant open (in
$A \times _X A$
) equivalence relation, then p and the action descend to the quotient space
$A/{\sim }$
which again becomes an étale -space.
-space. More generally, a fiber product of finitely many étale 
-spaces is again such a space, under the diagonal action.
-space, and 
, then U is itself an étale 
-space. If 
-spaces is again such.
-space, and 
-invariant open (in 
-space.We now introduce syntactic names for these last two operations, generalizing formulas.
Definition 8.2. For a countable
$\mathcal {L}_{\omega _1\omega }$
theory
$\mathcal {T}$
, a
$\Sigma _1$
imaginary sort over
$\boldsymbol{\mathcal {T}}$
is an expression
where
$i, j$
run over the same countable index set, each
$\phi _i$
is a
$\Sigma _1$
formula in some number of free variables
$n_i \in \mathbb {N}$
, each
$\epsilon _{ij}$
is a
$\Sigma _1$
formula in
$n_i+n_j$
variables, and
$\mathcal {T}$
proves the sentences
Equivalently by the completeness theorem, in each
$\mathcal {M} \models \mathcal {T}$
,
We then define the interpretation of
$\boldsymbol{\Phi} $
in
$\boldsymbol{\mathcal {M}}$
to be the quotient set
Similarly, if
$p : \mathcal {M} \to X$
is an étale model of
$\mathcal {T}$
, then we have an étale 
-space
$\Phi ^{\mathcal {M}} \to X$
defined in the same way, as the quotient of
$\bigsqcup _i \phi _i^{\mathcal {M}} \to X$
by the open equivalence relation
$\bigsqcup _{i,j} \epsilon _{ij}^{\mathcal {M}}$
.
Remark 8.3. Thus, a
$\Sigma _1$
imaginary is a syntactic name for a quotient of a countable disjoint union of
$\Sigma _1$
-definable subsets of finite products
$M^n$
of copies of the underlying set or étale space M of an (étale) model
$\mathcal {M}$
. We may always distribute these four types of operations over each other to put them into this order; thus, imaginaries are themselves closed under these operations.
-
• If
$\Phi = (\bigsqcup _i \phi _i)/(\bigsqcup _{i,j} \epsilon _{ij})$
and
$\Psi = (\bigsqcup _k \psi _k)/(\bigsqcup _{k,l} \eta _{kl})$
are imaginaries, we may define their product sort
$\Phi \times \Psi := (\bigsqcup _{i,k} \phi _i \wedge \psi _k)/(\bigsqcup _{i,j,k,l} \epsilon _{ij} \wedge \eta _{kl})$
. There is also the singleton sort (nullary product) where here
$\top $
is regarded as having no free variables. -
• If
$\Phi = (\bigsqcup _i \phi _i)/(\bigsqcup _{i,j} \epsilon _{ij})$
is a
$\Sigma _1$
imaginary sort over
$\mathcal {T}$
, a
$\Sigma _1$
-definable subsort
$\Psi \subseteq \Phi $
is given by countably many
$\Sigma _1$
formulas
$\psi _i$
such that “
$\bigsqcup _i \psi _i \subseteq \bigsqcup _i \phi _i$
is
$\mathcal {T}$
-provably
$(\bigsqcup _{i,j} \epsilon _{ij})$
-invariant,” which can be expressed by certain
$\Pi _2$
sentences much as in Definition 8.2. Such
$\Psi $
can also be thought of as a
$\Sigma _1$
imaginary in its own right, namely,
$\Psi = (\bigsqcup _i \psi _i)/(\bigsqcup _{i,j} \epsilon _{ij} \wedge \psi _i \wedge \psi _j)$
, such that
$\Psi ^{\mathcal {M}} \subseteq \Phi ^{\mathcal {M}}$
for every
$\mathcal {M} \models \mathcal {T}$
. -
• A countable disjoint union of imaginaries
$\bigsqcup _i \Phi _i$
is given by simply merging the formulas. -
• Finally, if
$\Phi = (\bigsqcup _i \phi _i)/(\bigsqcup _{i,j} \epsilon _{ij})$
is an imaginary, and
$H \subseteq \Phi \times \Phi $
is a
$\Sigma _1$
-definable subsort which is
$\mathcal {T}$
-provably an equivalence relation, then we may define the quotient sort
$\Phi /H$
by replacing the
$\epsilon _{ij}$
’s with the formulas
$\eta _{ij}$
defining H.
where here 
Theorem 8.4 (Joyal–Tierney [Reference Joyal and Tierney31]).
Let
$\mathcal {T}$
be a countable
$\mathcal {L}_{\omega _1\omega }$
theory,
${p : \mathcal {M} \to X}$
a second-countable étale structure with
$\Sigma _1$
saturations parametrizing models of
$\mathcal {T}$
. Then every second-countable étale 
-space
$q : A \to X$
is isomorphic to
$\Phi ^{\mathcal {M}}$
for some
$\Sigma _1$
imaginary
$\Phi $
over
$\mathcal {T}$
.
The following proof is a streamlined version of the proofs in [Reference Awodey and Forssell4, Section 1.4] and [Reference Chen14, Theorem 8.1] for two specific parametrizations (namely, those in Remark 6.10 and Example 6.7).
Proof Let
$a \in A_x$
, and let
$a \in S \subseteq A$
be an open section. Since the identity 
fixes a, by continuity of the action, there are open neighborhoods
$a \in S' \subseteq S$
and 
, where we may assume
$T, T' \subseteq M^n_X$
are open sections, such that 
. Since 
, we have
$x \in p(T \cap T')$
. By replacing S with
$S' \cap q^{-1}(p(T \cap T'))$
and
$T, T'$
with
$T \cap T' \cap p^{-1}(q(S'))$
, we get
$p(T) = q(S)$
and 
.
So we have shown that A has a basis of open sections
$S \subseteq A$
for which there exists an open section
$T \subseteq M^n_X$
for some n such that
$p(T) = q(S)$
and 
. For such
$S, T$
, consider
(Here we are abusing notation by writing
$T_y$
for the unique element of
$T_y$
, etc.) We claim that this is a well-defined, continuous 
-equivariant map, whose image contains S. Well-definedness is because if
$T_y \ne \emptyset $
then
$S_y \ne \emptyset $
as
$p(T) \subseteq q(S)$
, and if
$g \cdot T_y = g' \cdot T_{y'}$
, where
$g : \mathcal {M}_y \cong \mathcal {M}_z$
and
$g' : \mathcal {M}_{y'} \cong \mathcal {M}_z$
and both
$y, y' \in p(T)$
, then 
and so
$g \cdot S_y = g' \cdot S_{y'}$
. Equivariance is clear; and
$S \subseteq \operatorname {\mathrm {im}}(h_{S,T})$
by taking
$g = 1$
, using
$p(T) = q(S)$
. For continuity: suppose
$$ \begin{align*} h_{S,T}(g \cdot T_y) = g \cdot S_y \in S', \end{align*} $$
where
$S' \subseteq A$
is another open section for which there is an open section
$T' \subseteq M^{n'}_X$
such that
$p(T') = q(S') \ni z$
and 
. Let
$g^{-1} \cdot T^{\prime }_z \in T" \subseteq M^{n'}_X$
be another open section, and let
Then 
whence
$y \in U$
whence 
; and
Now since A is second-countable, hence Lindelöf, it has a countable cover by open sections
$S_i \subseteq A$
for which there exist corresponding
$T_i \subseteq M^{n_i}_X$
as above. For each i, let
$\phi _i$
be a
$\Sigma _1$
formula defining 
. We then have a continuous equivariant surjection
Its congruence kernel
${\sim } = \{(\vec {a},\vec {b}) \in (\bigsqcup _i \phi _i^{\mathcal {M}})^2_X \mid h(\vec {a}) = h(\vec {b})\} \subseteq (\bigsqcup _i M^{n_i}_X)^2_X \cong \bigsqcup _{i,j} M^{n_i+n_j}_X$
is an invariant open set, thus there are
$\Sigma _1$
formulas
$\epsilon _{ij}$
defining
${\sim }$
restricted to each
$M^{n_i+n_j}_X$
. Then
$\Phi := (\bigsqcup _i \phi _i)/(\bigsqcup _{i,j} \epsilon _{ij})$
is a
$\Sigma _1$
imaginary over
$\mathcal {T}$
, because the interpretation of
$\bigsqcup _{i,j} \epsilon _{ij}$
in each countable model of
$\mathcal {T}$
, which is isomorphic to some fiber
$\mathcal {M}_x$
, is the equivalence relation
$\sim _x$
on
$\bigsqcup _i \phi _i^{\mathcal {M}_x}$
. And h descends to an isomorphism of étale 
-spaces
$\Phi ^{\mathcal {M}} = (\bigsqcup _i \phi _i^{\mathcal {M}})/{\sim } \cong A$
.
Example 8.5. If
$X = 1$
and
$\mathcal {M}$
is a countable structure with
$\Sigma _1$
-definable automorphism orbits, as in Remark 5.3, then Theorem 8.4 says that every continuous action of
$\operatorname {\mathrm {Aut}}(\mathcal {M})$
on a countable discrete space A is named by some
$\Sigma _1$
imaginary of
$\mathcal {M}$
. To see this directly: for every
$a \in A$
, the stabilizer
$\operatorname {\mathrm {Aut}}(\mathcal {M})_a \subseteq \operatorname {\mathrm {Aut}}(\mathcal {M})$
is a clopen subgroup, hence contains some basic clopen subgroup
$\operatorname {\mathrm {Aut}}(\mathcal {M},\vec {b})$
, the automorphism group of
$\mathcal {M}$
expanded with finitely many fixed constants
$\vec {b} \in M^n$
; then the orbit
$\operatorname {\mathrm {Aut}}(\mathcal {M}) \cdot a$
is a definable quotient of the
$\Sigma _1$
-definable orbit
$\operatorname {\mathrm {Aut}}(\mathcal {M}) \cdot \vec {b}$
. The above proof can be seen as the natural generalization of this to étale structures.
Remark 8.6. Theorem 8.4 says that the map
is surjective up to isomorphism. The full Joyal–Tierney theorem says that it is in fact an equivalence of categories; Theorem 8.4 above is merely the key ingredient. Namely, the category on the left has:
-
•
$\Sigma _1$
imaginary sorts
$\Phi $
as objects; -
• subobjects of
$\Phi $
are
$\Sigma _1$
-definable subsorts
$\Psi \subseteq \Phi $
(Remark 8.3), which (modulo
$\mathcal {T}$
-provable equivalence) correspond bijectively to open invariant subsets of
$\Phi ^{\mathcal {M}}$
by
$\Sigma _1$
saturations; -
• morphisms
$\Theta : \Phi \to \Psi $
are
$\Sigma _1$
-definable functions, i.e.,
$\Sigma _1$
-definable subsorts
$\Theta \subseteq \Phi \times \Psi $
which are
$\mathcal {T}$
-provably the graph of a function. We usually identify definable functions modulo
$\mathcal {T}$
-provable equivalence.
For a countable
$\Pi _2$
theory
$\mathcal {T}$
, this category of
$\Sigma _1$
imaginaries is called the syntactic
$\boldsymbol{\sigma} $
-pretopos (also known as classifying
$\boldsymbol{\sigma} $
-pretopos) of the theory
$\mathcal {T}$
, and can be thought of as a canonical algebraic representation of the syntax of
$\mathcal {T}$
; see [Reference Johnstone30, Section D1.4], [Reference Chen14, Section 10]. Now the above equivalence boils down to the standard category-theoretic fact [Reference Johnstone30, Lemma D3.5.6] that a functor preserving finite categorical limits is an equivalence iff it is surjective on objects up to isomorphism and bijective on subobjects (in the usual categorical parlance, essentially surjective, conservative, and full on subobjects).
9 Interpretations and functors
Taking a more global view, the equivalence of categories in Remark 8.6 says that the map
is “injective”: we may recover a theory from the groupoid of its parametrization. To make this statement precise, we need to discuss the appropriate structure on the class of all theories.
Definition 9.2. Let
$\mathcal {T}_1, \mathcal {T}_2$
be countable
$\Pi _2$
theories in two respective countable languages
$\mathcal {L}_1, \mathcal {L}_2$
. A
$\Sigma _1$
interpretation
$\mathcal {F} : \mathcal {T}_1 \to \mathcal {T}_2$
is a “model of
$\mathcal {T}_1$
within the category of
$\Sigma _1$
imaginaries over
$\mathcal {T}_2$
.”
-
• First, one defines an interpretation
$\mathcal {F} : \mathcal {L}_1 \to \mathcal {T}_2$
to be an “
$\mathcal {L}_1$
-structure in
$\mathcal {T}_2$
-imaginaries,” consisting of an underlying
$\Sigma _1$
imaginary F over
$\mathcal {T}_2$
, an interpretation of each n-ary
$R \in \mathcal {L}_1$
as a
$\Sigma _1$
-definable subsort
$R^{\mathcal {F}} \subseteq F^n$
, and an interpretation of each n-ary
$f \in \mathcal {L}_1$
as a
$\Sigma _1$
-definable function
$f^{\mathcal {F}} : F^n \to F$
. -
• Next, one inductively defines, in the usual way, interpretations of
$\mathcal {L}_1$
-terms and formulas as definable functions and subsorts over
$\mathcal {T}_2$
. -
• One then says that
$\mathcal {F}$
satisfies an
$\mathcal {L}_1$
-sentence
$\phi \in \mathcal {T}_1$
if its interpretation
$\phi ^{\mathcal {F}}$
, which will be a definable subsort of the singleton sort over
$\mathcal {T}_2$
(recall Remark 8.3), is the entirety of . -
• Finally, one can extend the interpretation of
$\mathcal {L}_1$
-formulas to imaginaries
$\Phi $
over
$\mathcal {T}_1$
, which become interpreted as imaginaries
$\Phi ^{\mathcal {F}}$
over
$\mathcal {T}_2$
.
over 
.
The categorical viewpoint of Remark 8.6 is particularly useful when dealing with interpretations: a
$\Sigma _1$
interpretation
$\mathcal {F} : \mathcal {T}_1 \to \mathcal {T}_2$
is simply a functor
$\Phi \mapsto \Phi ^{\mathcal {F}}$
from the syntactic
$\sigma $
-pretopos of
$\mathcal {T}_1$
to the syntactic
$\sigma $
-pretopos of
$\mathcal {T}_2$
, preserving finite limits and countable colimits. For instance, this makes it easy to define the composition of interpretations
$\mathcal {T}_1 \to \mathcal {T}_2 \to \mathcal {T}_3$
. See [Reference Chen14, Section 10].
A
$\Sigma _1$
-definable isomorphism
$h : \mathcal {F} \cong \mathcal {G} : \mathcal {T}_1 \to \mathcal {T}_2$
between two interpretations is defined in the same way as a usual isomorphism, but replacing the underlying function with a
$\Sigma _1$
-definable function over
$\mathcal {T}_2$
. Equivalently, it is a natural isomorphism between functors.
An interpretation
$$ \begin{align*} \mathcal{F} : \mathcal{T}_1 &\longrightarrow \mathcal{T}_2 \end{align*} $$
can be thought of as a syntactic specification for a map
Namely, given a (countable/étale)
$\mathcal {M} \models \mathcal {T}_2$
, the model
$\mathcal {F}^*(\mathcal {M}) = \mathcal {F}^{\mathcal {M}} \models \mathcal {T}_1$
has underlying set/étale space
$F^{\mathcal {M}}$
and each symbol
$P \in \mathcal {L}_1$
interpreted as
$(P^{\mathcal {F}})^{\mathcal {M}}$
. Similarly, a
$\Sigma _1$
-definable isomorphism
$h : \mathcal {F} \cong \mathcal {G}$
between interpretations yields, for each
$\mathcal {M} \models \mathcal {T}_2$
, an
$\mathcal {L}_1$
-isomorphism
$h^{\mathcal {M}} : \mathcal {F}^{\mathcal {M}} \cong \mathcal {G}^{\mathcal {M}}$
.
For detailed background on interpretations and imaginaries (albeit in the context of finitary first-order logic), see [Reference Hodges28, Chapter 5]. We first give a familiar finitary example:
Example 9.4. The construction from each integral domain
$\mathcal {R}$
of its field of fractions is specified by an interpretation
$\mathcal {F} : (\text {theory of fields}) \to (\text {theory of integral domains})$
.
-
• The underlying imaginary is
$F = \phi /\epsilon $
where
$\phi , \epsilon $
are the formulas in the language of rings:
$$ \begin{align*} \phi(x,y) &:= (y \ne 0), & \epsilon(x_1,y_1,x_2,y_2) := (x_1y_2 = x_2y_1). \end{align*} $$
Thus given an integral domain
$\mathcal {R}$
, the field of fractions
$\mathcal {F}^{\mathcal {R}}$
has underlying set -
• The operation
$+$
of fields is interpreted as the definable function
$+^{\mathcal {F}} = \psi /\epsilon ^3$
where
$$ \begin{align*} \psi(x_1,y_1,x_2,y_2,x_3,y_3) &:= \epsilon(x_1y_2+x_2y_1, y_1y_2, x_3, y_3). \end{align*} $$
In other words, for an integral domain
$\mathcal {R}$
, this defines the graph of
$+$
on
$\mathcal {F}^{\mathcal {R}}$
:
$$ \begin{align*} [(a_1,b_1)] + [(a_2,b_2)] = [(a_3,b_3)] \iff (a_1b_2+a_2b_1, b_1b_2) \sim (a_3,b_3). \end{align*} $$
Similarly for the other operations. (This is not a
$\Sigma _1$
interpretation, unless we include a unary relation symbol for “nonzero” in the language of integral domains.)
Next, we give an example that uses the availability of countable disjoint unions.
Example 9.5. We have an interpretation from the theory of groups to the theory of sets, that specifies the construction from a set X of the free group over X. Its underlying imaginary is a countable disjoint union
$\bigsqcup _{\vec {s} \in \{\pm 1\}^{<\omega }} \phi _s$
where for each
$\vec {s} = (s_0,\dotsc ,s_{n-1})$
, the formula
$\phi _{\vec {s}}(x_0,\dotsc ,x_{n-1}) := \bigwedge _{s_i \ne s_{i+1}} (x_i \ne x_{i+1})$
says “
$x_0^{s_0} \dotsm x_{n-1}^{s_{n-1}}$
is a reduced word in the free group.”
Example 9.6. The notion of
$\Sigma _1$
interpretation subsumes that of étale structure, which is essentially the same thing as an interpretation into a propositional theory. Indeed, recall from Example 3.5 that a quasi-Polish space X is the space of models of a countable
$\Pi _2$
propositional theory
$\mathcal {T}_0$
. A second-countable étale space
$p : A \to X$
then corresponds to a
$\Sigma _1$
imaginary over
$\mathcal {T}_0$
(this is the trivial case of the Joyal–Tierney Theorem 8.4, for the trivial étale structure
$X \to X$
). Thus, a
$\Sigma _1$
interpretation
$\mathcal {M}$
from another countable
$\Pi _2$
theory
$\mathcal {T}$
into
$\mathcal {T}_0$
is a model of
$\mathcal {T}$
in
$\mathcal {T}_0$
-imaginaries, i.e., étale spaces over X, i.e., an étale model of
$\mathcal {T}$
over X. The induced map
$\mathcal {M}^*$
as in (9.3), from models
$x \in X$
of
$\mathcal {T}_0$
to models of
$\mathcal {T}$
, is just the “continuous map
$x \mapsto \mathcal {M}_x$
” from Remark 4.9.
Remark 9.7. Thus, a natural structure to put on the class of all countable
$\Pi _2$
theories, on the left-hand side of (9.1), is that of a 2-category, whose objects are countable
$\Pi _2$
theories
$\mathcal {T}$
, morphisms are
$\Sigma _1$
interpretations
$\mathcal {F} : \mathcal {T}_1 \to \mathcal {T}_2$
, and 2-cells are
$\Sigma _1$
-definable isomorphismsFootnote
4
between interpretations
$h : \mathcal {F} \cong \mathcal {G} : \mathcal {T}_1 \to \mathcal {T}_2$
. We may visualize this 2-category as follows:
For general background on 2-categories, see [Reference Johnstone30, Section B1.1], [Reference Borceux9, I Chapter 7].
However, there remain annoying technicalities in defining a map (technically, a 2-functor) from this 2-category of theories to groupoids, as in (9.1). This is due to non-canonical coding choices. To define such a 2-functor on objects, we have to pick, for each countable
$\Pi _2$
theory
$\mathcal {T}$
, some particular quasi-Polish parametrization
$\mathcal {M} \to X$
of it, and then take its isomorphism groupoid 
. Worse yet, to define the 2-functor on morphisms, we have to pick, for an interpretation between theories
${\mathcal {F} : \mathcal {T}_1 \to \mathcal {T}_2}$
, some map
$f : X_2 \to X_1$
between the respective chosen parametrizing spaces which realizes the operation on models specified by
$\mathcal {F}$
, i.e., so that
$(\mathcal {M}_1)_{f(x)} \cong \mathcal {F}^{(\mathcal {M}_2)_x}$
.
It is possible to make all of these coding choices in some explicit ad hoc manner, and this was done in [Reference Harrison-Trainor, Melnikov, Miller and Montalbán26] and [Reference Chen14]. Here, we take the opportunity to illustrate a more abstract approach, via a standard trick in higher-dimensional category theory: instead of defining the 2-functor (9.1) directly, we first “cover” its domain with a bigger 2-category that is equivalent, but contains isomorphic copies that include all possible coding choices beforehand.
Definition 9.8. A quasi-Polish parametrized
$\boldsymbol{\Pi} _\mathbf{2}$
theory will mean a tuple
$(\mathcal {L},\mathcal {T},X,\mathcal {M},p)$
, consisting of a countable
$\Pi _2$
theory
$\mathcal {T}$
in some countable language
$\mathcal {L}$
, together with some quasi-Polish parametrization X of models of
$\mathcal {T}$
via an étale model
$p : \mathcal {M} \to X$
with
$\Sigma _1$
saturations. We will often abbreviate the tuple
$(\mathcal {L},\mathcal {T},X,\mathcal {M},p)$
to
$(\mathcal {T},X,\mathcal {M})$
.
A parametrized
$\boldsymbol{\Sigma} _\mathbf{1}$
interpretation
$(\mathcal {F},f) : (\mathcal {L}_1,\mathcal {T}_1,X_1,\mathcal {M}_1,p_1) \to (\mathcal {L}_2,\mathcal {T}_2,X_2,\mathcal {M}_2,p_2)$
consists of a
$\Sigma _1$
interpretation
$\mathcal {F} : \mathcal {T}_1 \to \mathcal {T}_2$
, a continuous map
$f : X_2 \to X_1$
, and a (specified, but left nameless) isomorphism of étale structures (over
$X_2$
)
$f^*(\mathcal {M}_1) \cong \mathcal {F}^{\mathcal {M}_2}$
:
Regarding
$\mathcal {M}_1, \mathcal {M}_2$
as “continuous maps to countable models” as in Remark 4.9, the picture becomes
Two consecutive parametrized interpretations may be composed in the obvious manner. A
$\Sigma _1$
-definable isomorphism between parametrized interpretations
$h : (\mathcal {F},f) \cong (\mathcal {G},g)$
is simply one between the underlying intepretations
$h : \mathcal {F} \cong \mathcal {G}$
.
We thus get a 2-category of quasi-Polish parametrized
$\Pi _2$
theories, parametrized
$\Sigma _1$
interpretations, and
$\Sigma _1$
-definable isomorphisms, which admits canonical maps to both sides of (9.1):
The left leg simply forgets about the parametrizations. The right leg, a contravariant 2-functor, takes a parametrized theory
$(\mathcal {T},X,\mathcal {M})$
to the isomorphism groupoid 
; takes a parametrized interpretation
$(\mathcal {F},f) : (\mathcal {T}_1,X_1,\mathcal {M}_1) \to (\mathcal {T}_2,X_2,\mathcal {M}_2)$
to the continuous functor
and takes a
$\Sigma _1$
-definable isomorphism
$h : (\mathcal {F},f) \cong (\mathcal {G},g)$
to the continuous natural isomorphism
between the continuous functors
$(\mathcal {F},f)^*$
and
$(\mathcal {G},g)^*$
as defined above.
Theorem 9.12. In the above diagram (9.9):
-
(a) The right leg is a full and faithful contravariant 2-functor, i.e., restricts to an equivalence between each hom-groupoid of parametrized interpretations
$(\mathcal {T}_1,X_1,\mathcal {M}_1) \to (\mathcal {T}_2,X_2,\mathcal {M}_2)$
and the corresponding hom-groupoid of continuous functors . -
(b) The left leg restricts to an equivalence of 2-categories between the full sub-2-category of zero-dimensional Polish parametrized
$\Pi _2$
theories (meaning the base space X of the parametrization is zero-dimensional Polish, not necessarily the étale space M), and all countable
$\Pi _2$
theories.
.
Thus, we have a composite full and faithful 2-functor, i.e., “embedding of 2-categories,”
$$ \begin{align} \{\text{ctbl }\Pi_2\text{ theories}\} \simeq \{\text{0-d Pol parametrized }\Pi_2\text{ theories}\} \longrightarrow \{\text{q-Pol groupoids}\} \end{align} $$
taking each countable
$\Pi _2$
theory to the isomorphism groupoid of any of its zero-dimensional Polish parametrizations with
$\Sigma _1$
saturations.
Proof (a) follows from the Joyal–Tierney theorem. Indeed, for a parametrized interpretation
$(\mathcal {F},f) : (\mathcal {T}_1,X_1,\mathcal {M}_1) \to (\mathcal {T}_2,X_2,\mathcal {M}_2)$
,
$\mathcal {F}$
is a model of
$\mathcal {T}_1$
in the category of
$\Sigma _1$
imaginaries over
$\mathcal {T}_2$
, which by Joyal–Tierney in the form of Remark 8.6 is equivalently a model
$\mathcal {F}^{\mathcal {M}_2}$
in the category of second-countable étale 
-spaces, i.e., an étale model of
$\mathcal {T}_1$
over
$X_2$
equipped with a continuous action of 
via isomorphisms; this action corresponds, via the isomorphism
$f^*(\mathcal {M}_1) \cong \mathcal {F}^{\mathcal {M}_2}$
, to the extension of
$f : X_2 \to X_1$
to a functor 
via (9.10). Conversely, given such a continuous functor 
, the unique-up-to-definable-isomorphism interpretation
$\mathcal {F} : \mathcal {T}_1 \to \mathcal {T}_2$
inducing f in the above manner takes a
$\Sigma _1$
imaginary over
$\mathcal {T}_1$
, which is equivalently by Remark 8.6 an étale 
-space
$q : A \to X_1$
, to the
$\Sigma _1$
imaginary over
$\mathcal {T}_2$
corresponding to the pullback 
-space
$f^*(A)$
; then
$\mathcal {F}^{\mathcal {M}_2} \cong f^*(\mathcal {M}_1)$
by definition. It is straightforward to check that this correspondence is functorial via (9.11).
For (b), the left leg is essentially surjective, i.e., every countable
$\Pi _2$
theory has a zero-dimensional Polish parametrization, by Example 6.9; and it is locally full and faithful, i.e., bijective on 2-cells between each fixed pair of morphisms, by definition of said 2-cells as
$\Sigma _1$
-definable isomorphisms. It remains only to check that it is locally (essentially) surjective, i.e., for two zero-dimensional Polish parametrized theories
$(\mathcal {T}_1,X_1,\mathcal {M}_1), (\mathcal {T}_2,X_2,\mathcal {M}_2)$
, every
$\Sigma _1$
interpretation
$\mathcal {F} : \mathcal {T}_1 \to \mathcal {T}_2$
can be parametrized via some continuous map
$f : X_2 \to X_1$
and isomorphism
$f^*(\mathcal {M}_1) \cong \mathcal {F}^{\mathcal {M}_2}$
. That is, for each fiber
$(\mathcal {M}_2)_x$
of
$\mathcal {M}_2$
, we know the model
$\mathcal {F}^{(\mathcal {M}_2)_x} \models \mathcal {T}_1$
is isomorphic to some fiber
$(\mathcal {M}_1)_{f(x)}$
of
$\mathcal {M}_1$
; we need to find the fiber
$f(x)$
and the isomorphism in a continuous manner. Form the space
as in Remark 4.18. By Remark 5.19, the second projection 
is open, and it is also surjective, since as noted before, each
$\mathcal {F}^{(\mathcal {M}_2)_x}$
is isomorphic to some
$(\mathcal {M}_1)_y$
. Thus by Michael’s selection theorem [Reference Michael44, Corollary 1.4] (see also [Reference Michael45], [Reference de Brecht, Pauly and Schröder21]; to deal with the fact that 
is quasi-Polish instead of Polish, use the latter paper or (2.12)), 
has a continuous section 
, whose first coordinate yields f and third coordinate yields the isomorphism
$f^*(\mathcal {M}_1) \cong \mathcal {F}^{\mathcal {M}_2}$
.
Remark 9.14. If one is only interested in certain restricted kinds of theories, then it suffices in Theorem 9.12 to restrict to a full sub-2-category of zero-dimensional Polish parametrizations which are known to parametrize all such theories.
For example, if one is only interested in theories with no finite models, and modulo which negated atomic formulas are equivalent to
$\Sigma _1$
formulas (e.g., from Morleyizing to recover the traditional definition of
$\Sigma _1$
as in Example 3.4), then it suffices to consider the usual Polish space of models on
$\mathbb {N}$
as in Example 6.1, thereby recovering the boldface version of [Reference Harrison-Trainor, Melnikov, Miller and Montalbán26, Theorem 1.5].
10 The Lopez-Escobar theorem and
$\mathcal {L}_{\omega _1\omega }$
imaginaries
An important tool in descriptive set theory is the Baire category quantifier
$\exists ^*$
(and its dual
$\forall ^*$
); see [Reference Kechris33, Section 8.J, Exercise 22.22], [Reference Melleray and Tsankov43, Section A], [Reference Chen18, Section 2.3–4]. Given a continuous open map
$f : X \to Y$
between quasi-Polish spaces, each
$A \in X$
has a “Baire-categorical image” under f:
The usefulness of
$\exists ^*_f$
is largely because it, unlike ordinary image, preserves Borel sets; in fact, it preserves
$\boldsymbol {\Sigma }^0_\alpha $
sets for all
$\alpha $
.
We have an analogous result for étale structures
$\mathcal {M} \to X$
with
$\Sigma _1$
saturations, thought of as “maps
$x \mapsto \mathcal {M}_x : X \to \{\text {all structures}\}$
” as in Remark 5.2. The “fibers” of such a map should be thought of as the isomorphism orbits 
of each
$\vec {a} \in M^n_X$
(and not the fiber structures
$\mathcal {M}_x$
, which are the “values” of the map
$x \mapsto \mathcal {M}_x$
). Thus, “image” becomes “saturation.”
Definition 10.1. For an étale structure
$p : \mathcal {M} \to X$
, open 
, and any
$A \subseteq M^n_X$
, the Vaught transform is the “Baire-categorical saturation”
(Here 
is the codomain map; recall Definition 4.15.)
The more common notation for
$U * A$
is
$A^{\triangle U^{-1}}$
, including in [Reference Lupini39] where it was first studied for groupoids; the above notation suggestive of the ordinary saturation
$U \cdot A$
is from [Reference Chen18, Definition 4.2.1].
Theorem 10.2 (Lopez-Escobar for étale structures).
Let
$\mathcal {L}$
be a countable language,
$p : \mathcal {M} \to X$
be a second-countable étale
$\mathcal {L}$
-structure with
$\Sigma _1$
saturations over a quasi-Polish X. For any
$\boldsymbol {\Sigma }^0_\alpha $
set
$A \subseteq M^n_X$
, there is a
$\Sigma _\alpha $
formula defining 
.
Remark 10.3. It follows from this statement that more generally, for any basic open 
, where
$U, V \subseteq M^m_X$
are open,
is the fiberwise inverse image of V under a binary relation
$\phi ^{\mathcal {M}} \subseteq M^n_X \times _X M^m_X$
, for some
$\Sigma _\alpha $
formula
$\phi $
with
$n+m$
variables depending only on U.
Proof of Theorem 10.2
By induction on
$\alpha $
. For
$\alpha = 1$
, this is just the fact that
$\mathcal {M}$
has
$\Sigma _1$
saturations. For a countable union of sets for which the result holds, we may take the disjunction of the formulas. By Definition 2.1 of the Borel hierarchy, it thus remains to show, assuming the result holds for all
$\boldsymbol {\Sigma }^0_\alpha $
sets, that for any two
$A, B \in \boldsymbol {\Sigma }^0_\alpha (M^n_X)$
, the result holds for 
. Fix countable bases
$\mathcal {U}_m$
for each
$M^m_X$
, so that 
for
$U, V \in \mathcal {U}_m$
form a basis for 
. For
$\vec {a} \in M^n_x$
,
by the property of Baire (see [Reference Kechris33, Proposition 8.26]), this holds iff 
is comeager in some nonempty basic open set 
, and so this is
by the induction hypothesis and preceding remark, there are
$\Sigma _\alpha $
formulas
$\phi _U, \psi _U$
such that this is
where
$\Longleftarrow $
is because
$\vec {b}$
belongs to some open section
$V \in \mathcal {U}_m$
. So this is defined by the
$\Sigma _{\alpha +1}$
formula
Example 10.4. Applying Theorem 10.2 to the standard Polish space of models on
$\mathbb {N}$
(Example 6.1) recovers the classical Lopez-Escobar theorem [Reference Lopez-Escobar38], or rather its strengthening adapted levelwise to the Borel hierarchy by Vaught [Reference Vaught55] (see also [Reference Kechris33, Theorem 16.8], [Reference Gao25, Theorem 11.3.6]).
Applying it instead to the parametrization of Example 6.7 yields the version of the Lopez-Escobar theorem used in [Reference Chen14], which is the boldface version of the effective “positive” (but still admitting
$\ne $
as
$\Sigma _1$
) Lopez-Escobar theorem in [Reference Bazhenov, Fokina, Rossegger, Soskova and Vatev5].
We now have the Borel analogs of the material from the two preceding sections.
Definition 10.5. For a second-countable étale structure
$p : \mathcal {M} \to X$
over quasi-Polish X, a (fiberwise) countable Borel
$\mathbf{\mathrm{Iso}} _\mathbf{X}(\boldsymbol{\mathcal {M}})$
-space over X is a standard Borel space A equipped with a countable-to-1 Borel map
${q : A \to X}$
and a Borel action of 
.
Definition 10.6. For a countable
$\mathcal {L}_{\omega _1\omega }$
theory
$\mathcal {T}$
, a
$\mathcal {L}_{\omega _1\omega }$
imaginary sort
$\boldsymbol{\Phi} = (\boldsymbol{\bigsqcup} _\mathbf{i} \boldsymbol{\phi} _\mathbf{i})/(\boldsymbol{\bigsqcup} _{\mathbf{i},\mathbf{j}} \boldsymbol{\epsilon} _{\mathbf{ij}})$
over
$\boldsymbol{\mathcal {T}}$
is defined exactly as in Definition 8.2, except that the formulas
$\phi _i, \epsilon _{ij}$
may be
$\mathcal {L}_{\omega _1\omega }$
instead of
$\Sigma _1$
. These may be interpreted in a second-countable étale
$p : \mathcal {M} \to X$
over quasi-Polish X to yield a countable Borel 
-space
$\Phi ^{\mathcal {M}} \to X$
.
Theorem 10.7. Let
$\mathcal {T}$
be a countable
$\mathcal {L}_{\omega _1\omega }$
theory,
$p : \mathcal {M} \to X$
be a second-countable étale space with
$\Sigma _1$
saturations parametrizing models of
$\mathcal {T}$
. Then every countable Borel 
-space
$q : A \to X$
is isomorphic to
$\Phi ^{\mathcal {M}}$
for some
$\mathcal {L}_{\omega _1\omega }$
imaginary
$\Phi $
over
$\mathcal {T}$
.
This was proved in [Reference Chen14] for a particular parametrization
$\mathcal {M}$
, namely, that of Example 6.7 (and implicitly in [Reference Harrison-Trainor, Miller and Montalbán27] for the standard parametrization of Example 6.1).
Proof By [Reference Chen18, proofs of Corollaries 4.5.13 and 4.3.9], we may topologically realize A as a second-countable étale 
-space, after refining the topology on the space of objects X of the groupoid 
by adjoining countably many sets of the form 
to the topology, where
$U_i, V_i \subseteq M^{m_i}_X$
are open and
$B_i \subseteq X$
are Borel. By Remark 10.3, each 
for some
$\mathcal {L}_{\omega _1\omega }$
formula
$\phi _i(x_0,\dotsc ,x_{m_i-1})$
. Morleyize the formulas
$\phi _i$
to obtain a new theory
$\mathcal {T}'$
in an expanded language
$\mathcal {L}' \supseteq \mathcal {L}$
, and then Morleyize the étale structure
$\mathcal {M}$
via Definition 4.12, to obtain a new étale structure
$p' : \mathcal {M}' \to X'$
, now with
$\Sigma _1$
saturations by Lemma 5.16, such that
$X'$
is X with a finer topology in which each
$p(U_i \cap \phi _i^{\mathcal {M}})$
becomes open. Pulling back the topologically realized étale space A to
$X'$
, we thus obtain a second-countable étale 
-space, which by the Joyal–Tierney Theorem 8.4 is named by some
$\Sigma _1$
imaginary in
$\mathcal {M}'$
, hence by an
$\mathcal {L}_{\omega _1\omega }$
imaginary in
$\mathcal {M}$
.
Remark 10.8. As in Remark 8.6, it follows that we in fact have an equivalence of categories
where the subobjects on the left, namely,
$\mathcal {L}_{\omega _1\omega }$
-definable subsorts of
$\mathcal {L}_{\omega _1\omega }$
imaginaries, correspond to 
-invariant Borel subspaces by the Lopez-Escobar Theorem 10.2.
This recovers one of the main results of [Reference Chen14] (extending the boldface result of [Reference Harrison-Trainor, Miller and Montalbán27]). To recover the rest, we need to convert Theorem 10.7 into the 2-categorical form of Theorem 9.12.
Definition 10.9. An
$\mathcal {L}_{\omega _1\omega }$
interpretation between two
$\mathcal {L}_{\omega _1\omega }$
theories is defined exactly as in Definition 9.2, except that the formulas and imaginaries used may be
$\mathcal {L}_{\omega _1\omega }$
rather than
$\Sigma _1$
.
Definition 10.10. Recall the notion of quasi-Polish parametrized
$\Pi _2$
theory
$(\mathcal {L}, \mathcal {T}, X, \mathcal {M}, p)$
from Definition 9.8. A quasi-Polish parametrized
$\boldsymbol{\mathcal {L}}_{\boldsymbol{\omega} _\mathbf{1}\boldsymbol{\omega} }$
theory will mean such a tuple where
$\mathcal {T}$
is a countable
$\mathcal {L}_{\omega _1\omega }$
theory in the countable language
$\mathcal {L}$
, and
$p : \mathcal {M} \to X$
is a second-countable étale structure over the quasi-Polish space X with
$\mathcal {L}_{\omega _1\omega }$
saturations of open sets.
A parametrized
$\boldsymbol{\mathcal {L}}_{\boldsymbol{\omega} _\mathbf{1}\boldsymbol{\omega} }$
interpretation
$(\mathcal {F},f) : (\mathcal {L}_1,\mathcal {T}_1,X_1,\mathcal {M}_1,p_1) \to (\mathcal {L}_2,\mathcal {T}_2,X_2,\mathcal {M}_2,p_2)$
between two such parametrized theories consists of an
$\mathcal {L}_{\omega _1\omega }$
interpretation
$\mathcal {F} : \mathcal {T}_1 \to \mathcal {T}_2$
, a Borel map
$f : X_2 \to X_1$
, and a Borel isomorphism of fiberwise countable Borel structures
$f^*(\mathcal {M}_1) \cong \mathcal {F}^{\mathcal {M}_2}$
.
We have the Borel analog of the diagram (9.9):
in which the three 2-categories and both 2-functors are as before (see (9.10) and (9.11)), but with
$\mathcal {L}_{\omega _1\omega }$
interpretations and definable isomorphisms and Borel functors and natural transformations.
Theorem 10.12. In the above diagram:
-
(a) The right leg is a full and faithful 2-functor.
-
(b) The left leg is an equivalence of 2-categories (on its entire domain).
Thus, we have a composite full and faithful 2-functor
$\{\text {ctbl }\mathcal {L}_{\omega _1\omega }\text { theories}\} \to \{\text {quasi-Polish groupoids}\}$
.
Proof As in Theorem 9.12, (a) follows from Theorem 10.7 or rather Remark 10.8. For (b), essential surjectivity follows from the fact that every countable
$\mathcal {L}_{\omega _1\omega }$
theory has a quasi-Polish (or even zero-dimensional Polish) parametrization with
$\mathcal {L}_{\omega _1\omega }$
saturations by Morleyization and any of the examples from Section 6 that admit finite models, and local full faithfulness is trivial as before. For fullness, we again follow the proof of Theorem 9.12; the added ingredient needed is that (in the notation from there) surjectivity of the Borel map 
is already enough to imply the existence of a Borel section, which follows by a straigtforward application of Kechris’s large section uniformization theorem [Reference Kechris34, Theorem 18.6
${}^{\ast }$
] applied to the meager ideal; see [Reference Chen13, Theorem 7.9].
11 Groupoid representations
Theorems 9.12 and 10.12 show that the operation of taking the semantics of a theory is an “embedding” from the 2-category of theories to the 2-category of groupoids (in both the continuous and Borel settings). It is natural to ask what is the image of this 2-functor, i.e., which quasi-Polish groupoids arise as the groupoid of models of a theory. The corresponding question in topos theory was answered by Moerdijk [Reference Moerdijk47], who proved what can be called the topos-theoretic analog of the Yoneda lemma (which says that absent any topological structure, every groupoid is canonically a groupoid of isomorphisms between structures, via the left translation action on itself). In this final section, we adapt Moerdijk’s result to the countable model-theoretic setting.
First, we clarify what we mean by an abstract topological groupoid (e.g., 
) and action.
Definition 11.1. A topological groupoid
$G \rightrightarrows X$
consists of topological spaces
$G, X$
of morphisms and objects, respectively, with continuous domain and codomain maps 
, as well as identity
$1_{(-)} : X \to G$
, inverse
$(-)^{-1} : G \to G$
, and composition
$\circ $
(or simply juxtaposition) of adjacent morphisms, subject to the usual axioms of associativity, identity, and inverse.
As is common when working with topological groupoids, we identify each object
$x \in X$
with the corresponding identity morphism
$1_x \in G$
, so that
$X \subseteq G$
is a subspace.
By an open subgroupoid of G, we mean an open subset
$U \subseteq G$
of morphisms which is closed under composition and inverse. It follows that for each
$g : x \to y \in U$
, the identity morphisms at both its source
$1_x = g^{-1} \circ g$
and target
$1_y = g \circ g^{-1}$
are in U as well, whence we may regard U as the space of morphisms of a groupoid in its own right, with objects 
.
We call G a non-Archimedean topological groupoid if every identity morphism
$1_x \in G$
has a neighborhood basis of open subgroupoids.
A continuous action of the groupoid G on a topological space A equipped with a continuous map
$q : A \to X$
, regarded as a topological bundle over X, is a map
jointly continuous in both variables, taking
$g : x \to y \in G$
and
$a \in A_x$
to
$g \cdot a \in A_y$
, and obeying the usual associativity and identity axioms. We also call
$(A,q)$
equipped with the action a G-space, and an étale G-space if A is an étale space over X.
Definition 11.2. Now suppose
$G \rightrightarrows X$
is an open topological groupoid, i.e., 
are open maps (see Lemma 5.18). For an open subgroupoid
$U \subseteq G$
as above, let (by an abuse of notation)
denote the space of left cosets of U. This is naturally an étale G-space over X via 
and the left multiplication action of G. An open section
$S/U \subseteq G/U$
may be identified with its lift in G, which is an open right-U-invariant subset 
such that
$S^{-1}S \subseteq U$
.
For two open subgroupoids
$U, V \subseteq G$
and an open right-V-invariant 
, we have
$$ \begin{align} &U \subseteq SS^{-1} \\&\quad\iff \text{the right multiplication map } (-)S : G/U \to G/V \text{ is well-defined}.\nonumber \end{align} $$
Note also that such a right multiplication map, when defined, is clearly left-G-equivariant.
Definition 11.4. Let
$G \rightrightarrows X$
be an open non-Archimedean topological groupoid. Fix a family
$\mathcal {U}$
of open subgroupoids forming a neighborhood basis for each identity morphism, and for each
$U \in \mathcal {U}$
, an open cover
$\mathcal {S}_U$
of 
by open right-U-invariant sets S such that
$S^{-1}S \subseteq U$
(corresponding to a cover of
$G/U$
by open sections
$S/U$
). Note that when G is second-countable,
$\mathcal {U}$
and each
$\mathcal {S}_U$
may be chosen to be countable, whence each
$G/U$
is a second-countable étale space over X.
The canonical G-structure
$\mathcal {M} = \mathcal {M}_{G,\mathcal {U},(\mathcal {S}_U)_U}$
(determined by the
$\mathcal {U}, \mathcal {S}_U$
) is the multi-sorted étale structure over X with a sort
$G/U$
for each
$U \in \mathcal {U}$
, and a unary function
$(-)S : G/U \to G/V$
for each
$U, V \in \mathcal {U}$
and
$S \in \mathcal {S}_V$
obeying (11.3). The canonical left translation action of G on each
$G/U$
is an action via isomorphisms between fibers of
$\mathcal {M}_G$
, yielding a canonical functor
which is the identity on objects and sending each
$g : x \to y \in G$
to its action
$\mathcal {M}_x \cong \mathcal {M}_y$
.
Theorem 11.5 (Moerdijk [Reference Moerdijk47]).
For any open non-Archimedean
$T_0$
topological groupoid G with canonical structure
$\mathcal {M} = \mathcal {M}_{G,\mathcal {U},(\mathcal {S}_U)_U}$
as above, the canonical functor 
is a topological embedding with 
-fiberwise dense image.
Proof First, we convert to an alternate description of 
. An isomorphism
$f : \mathcal {M}_x \to \mathcal {M}_y$
in this groupoid is determined by its values
$f(gU)$
for each element
$gU$
of each sort
$(G/U)_x$
of
$\mathcal {M}_x$
. But preservation of the right multiplication maps
$(-)S$
means that f is in fact determined by its values on just the identity cosets
$1_x U$
: indeed, for any other coset
$gU \in (G/U)_x$
, we may find
(This is the part most analogous to the Yoneda lemma.) Moreover, these values
$f(1_x U)$
must obey
$$ \begin{align} 1_x \in U \subseteq V \in \mathcal{U} \implies f(1_x U) V = f(1_x V), \end{align} $$
by finding
$1_x \in S \in \mathcal {S}_U$
and
$1_x \in T \in \mathcal {S}_V$
and then a neighborhood
${1_x \in W \subseteq SS^{-1} \cap TT^{-1}}$
in
$\mathcal {U}$
, so that
$f(1_x U) V = f(1_x W S) V = f(1_x W) SV = f(1_x W) T = f(1_x WT) = f(1_x V)$
. In other words, we have a canonical injection
into the set of all families
$(f(1_x U) \in (G/U)_y)_{1_x \in U \in \mathcal {U}}$
obeying the coherence condition (11.7).
Moreover, the above injections (11.8) together form a topological embedding of 
into the inverse limit topology on the right-hand side, in the following sense. For each
$U \in \mathcal {U}$
and open right-U-invariant 
, the set
which is the preimage under (11.8) of the coherent families containing a fiber of the open section
$S/U \subseteq G/U$
, is open. We claim that these sets together with the maps 
suffice to generate the topology of 
(from Definition 4.15). Indeed, by (11.6), a subbasic open 
, where
$S/U, T/U \subseteq G/U$
are basic open sections, may be written in terms of the sets (11.9) as
(where
$T'/V \subseteq G/V$
is an open section containing
$f(1_x V)$
in (11.6)).
Now the
$\iota $
-preimage of such a set 
consists of all
$g : x \to y \in G$
such that
$1_x \in U$
and
$gU \in S/U$
, which is just the open right-U-invariant
$S \subseteq G$
. Since G is non-Archimedean, its open right-U-invariant sets over all
$U \in \mathcal {U}$
are easily seen to form a basis; thus
$\iota $
is an embedding.
Finally, the image of
$\iota $
is 
-fiberwise dense: from above, 
has an open basis of sets
where
$U_0, \dotsc , U_{n-1} \in \mathcal {U}$
,
$S_i/U_i \subseteq G/U_i$
are open sections, and
$V, W \subseteq X$
are open. If such a set is nonempty in some 
, then it contains some isomorphism
$f : \mathcal {M}_x \to \mathcal {M}_y$
where
$x = 1_x \in V \cap \bigcap _i U_i$
,
$y \in W$
, and each
$f(1_x U_i) \subseteq S_i$
. Let
$U \in \mathcal {U}$
with
$1_x \in U \subseteq V \cap \bigcap _i U_i$
, and pick
${g \in f(1_x U)}$
. Then
$g : x \to y \in G$
with each
$gU_i \supseteq gU \subseteq f(1_x U) \subseteq f(1_x U_i)$
, whence
$gU_i = f(1_x U_i)$
since both are left cosets of
$U_i$
, whence 
.
Remark 11.10. The inverse limits on the right-hand side of (11.8) above may be patched together, over all
$x, y \in X$
, into a topological category, which can be seen as a “left completion” of G analogous to the left completion of a Polish automorphism group [Reference Becker6]. This is studied in depth by Moerdijk [Reference Moerdijk47] in the topos-theoretic context.
Theorem 11.11. For an open non-Archimedean quasi-Polish groupoid G, with canonical second-countable étale structure
$\mathcal {M} = \mathcal {M}_{G,\mathcal {U},(\mathcal {S}_U)_U}$
with respect to some countable
$\mathcal {U}, \mathcal {S}_U$
as above, the canonical functor 
is a topological groupoid isomorphism.
Proof By Theorem 11.5, we may identify G with a 
-fiberwise dense quasi-Polish (hence 
-fiberwise dense
$\boldsymbol {\Pi }^0_2$
) subgroupoid of 
; since
$(-)^{-1}$
is a homeomorphism, 
is also 
-fiberwise dense
$\boldsymbol {\Pi }^0_2$
. By Pettis’s theorem for the left translation action of the open quasi-Polish groupoid G on the bundle 
(proved the same way as Pettis’s theorem for Polish groups; see [Reference Chen18, 4.2.8]), it follows that 
.
Remark 11.12. We note that the last step above of applying Pettis’s theorem also has precedent in topos theory, namely, the “closed subgroupoid theorem” of Johnstone [Reference Johnstone29]; see [Reference Chen17]. In the terminology of [Reference Moerdijk46], [Reference Moerdijk47], we have shown that an open quasi-Polish groupoid is non-Archimedean iff it is étale-complete.
Corollary 11.13. The 2-functor from Theorem 9.12 is an equivalence of 2-categories
$$ \begin{align*} \{\text{countable }\Pi_2\text{ theories}\} \simeq \{\text{open non-Archimedean quasi-Polish groupoids}\} \end{align*} $$
(with continuous functors and natural isomorphisms on the right).
Corollary 11.14. The 2-functor from Theorem 10.12 is an equivalence of 2-categories
$$ \begin{align*} \{\text{countable }\mathcal{L}_{\omega_1\omega}\text{ theories}\} \simeq \{\text{open non-Archimedean quasi-Polish groupoids}\} \end{align*} $$
(with Borel functors and natural isomorphisms on the right).
As before (see Theorem 9.12), these equivalences of 2-categories take a theory to the isomorphism groupoid of any of its parametrizations, hence are not completely canonical (depending on a choice of parametrization). Also, in the latter result, we may replace “quasi-Polish” with “zero-dimensional Polish,” due to the freedom to Morleyize arbitrarily (see Remark 9.14).
Remark 11.15. It may seem a bit strange that Theorem 10.12 and Corollary 11.14 are mostly about the Borel setting, yet still mention quasi-Polish spaces and groupoids. This is an instance of the subtle interactions between topological and Borel structure in the presence of a group(oid) action, as exemplified by results, such as Pettis’s theorem and the Becker–Kechris theorem; see [Reference Chen18] for an extended discussion on this point.
To our knowledge, it is an open problem to give a purely Borel-theoretic characterization of the standard Borel groupoids equivalent to the isomorphism groupoid of some countable
$\mathcal {L}_{\omega _1\omega }$
theory.
Acknowledgments
I would like to thank Anush Tserunyan for providing useful comments on a draft of this article, as well as the anonymous referee for several helpful comments and suggestions.
Funding
Research supported by NSF grant DMS-2224709.
with the homomorphism category, a topological category, which is a more involved notion than a topological groupoid. See Remark 
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