Strong Embeddings & Isomorphism Classes

• Isomorphism classes of strong embeddings are defined by rigid morphisms that equate countable structures under Borel reducibility.
• The topic employs combinatorial and normal trees to encode analytic quasi-orders, establishing universality in classification.
• Results highlight that embedding relations offer a framework to reinterpret intricate analytic classification problems in model and descriptive set theory.

Isomorphism classes of strong embeddings are central to classification problems in model theory, descriptive set theory, and related areas. This notion captures not only when two countable structures are "the same" up to isomorphism, but also when they are mutually embeddable by "strong" morphisms such as embeddings, homomorphisms, or other rigid maps. Analytic equivalence relations—those definable by a Σ^1_1 subset of a Polish space—serve as the backdrop; here, the relative complexity of isomorphism versus strong embeddability is measured by Borel reducibility. The two cited papers (Friedman et al., 2011, Ros, 2011) formulate and answer, in remarkable generality, the problem of characterizing the landscape of isomorphism classes attached to strong embeddings, with deep connections to orbit equivalence relations, coding techniques, and effective classification theory.

1. Key Notions and Definitions

Bi-embeddability, denoted $A = B$ when $A$ embeds into $B$ and $B$ embeds into $A$, is an equivalence relation that is generally coarser than isomorphism, which requires a bijective structure-preserving map. In the context of analytic classes (e.g., models of a sentence in $L_{\omega_1 \omega}$) studied as standard Borel spaces, these relations admit rigorous complexity measurement via Borel reducibility. For analytic equivalence relations $E$ on $X$ and $F$ on $Y$, $E \leq_B F$ signifies the existence of a Borel map $f:X\to Y$ such that $x_1 E x_2$ iff $f(x_1) F f(x_2)$.

Crucial for technical developments is "classwise Borel isomorphism": $E \cong_{cB} F$ if there are Borel reductions whose induced maps on quotient spaces are bijections and inverses of each other.

2. Universality and Completeness Results

The primary result of (Friedman et al., 2011) is that for many natural classes—countable graphs, combinatorial trees, metric spaces, separable Banach spaces, topological spaces—the equivalence relation induced by bi-embeddability is universal for analytic equivalence relations under Borel reducibility. That is, for any analytic equivalence relation $E$, there exists a class (often definable by an $L_{\omega_1 \omega}$-sentence) in which $E$ is Borel equivalent to the bi-embeddability relation.

Formally, for any analytic quasi-order $R$, there is an $L_{\omega_1 \omega}$-sentence $\varphi$ such that $R$ is Borel equivalent (and explicitly Borel isomorphic) to the embeddability relation on the class $\mathrm{Mod}_\varphi$ of countable models of $\varphi$. Since bi-embeddability for $R$ is $R \circ R^{-1}$, every analytic equivalence relation arises as bi-embeddability in this sense.

In contrast, isomorphism for many elementary classes is much less complex: it may have countably many equivalence classes (smoothness), or be far from complete in the Borel hierarchy.

3. Encoding and Reduction Techniques

The proofs rely on "coding" analytic relations into embeddability of structures built from normal trees—combinatorial devices on $2 \times \omega$. Structures $G_T$ (combinatorial trees or their variants) are assigned to trees $T$, with embeddability mirroring the relation $R$ among trees. Advanced Borel reduction machinery ensures these codings are injective, open on range, or even recursive at low complexity (e.g., $4\Sigma_1$-recursive).

A notable technical point: on such coded classes and trees, homomorphisms, weak-homomorphisms, and embeddings collapse—every such morphism is, in fact, a proper embedding. This property ("rigidity") allows the translation of analytic quasi-orders or equivalence relations into structural embeddings in these coded classes.

4. Orthogonality and Realization of Pairs

Paper (Ros, 2011) generalizes this to pairs of equivalence and quasi-order relations $(E, F)$: under mild conditions, for any such analytic pair, one may find an $L_{\omega_1 \omega}$-elementary class $\mathcal{C}$ so that, up to Borel bireducibility, $(E, F)$ arise as $(\cong_{\mathcal{C}}, \equiv_{\mathcal{C}})$—isomorphism and bi-embeddability in $\mathcal{C}$. The construction uses trees and graphs with extra relations, leveraging classwise Borel isomorphism for a strong match at the quotient level.

This result confirms an "orthogonality" in complexity: for most analytic $E$ and $F$, their complexities can be tuned independently as realized on some class of countable structures.

5. Implications for Classification and Applications

These results significantly advance the analysis of invariants in model theory and descriptive set theory. For instance:

• They allow any analytic classification problem to be recast as bi-embeddability (or embeddability) in a relevant class of structures.
• They apply in analysis, e.g., showing the complete analytic nature of isometric embeddability on Polish metric spaces, ultrametric spaces, or Banach spaces.
• Via Polish monoid (or group) actions, analytic quasi-orders induced by actions correspond to embeddability relations coded above, tying dynamical systems to descriptive set-theoretic complexity.

In practice, the "universality" of strong embeddings means that for applications requiring reduction of complex equivalence relations, it is always possible to code the problem into bi-embeddability (or some related morphism) for suitable countable models.

6. Open Problems and Future Directions

Several open problems remain:

• Whether for every analytic quasi-order there is an $L_{\omega_1 \omega}$-elementary class in which (weak) epimorphism is complete analytic.
• Whether completeness extends to elementary embeddability, or more generally to relational morphisms beyond embeddings and homomorphisms.
• In analysis, whether the resulting "Borel" classes from such codings can be saturated or closed under the full isomorphism group (isometries, homeomorphisms, etc.).
• Whether all analytic quasi-orders may be realized as induced by continuous actions of Polish monoids—inviting deeper classification via canonical group-theoretic or dynamical invariants.

7. Mathematical Formulations and Key Theorems

Some central results and formulas:

Theorem/Formulation	Statement	Context
Borel Reducibility	$\forall x_1,x_2,\; x_1 R x_2 \iff f(x_1) S f(x_2)$ for Borel $f$	Definition for analytic relations
Theorem 3.3 (Friedman et al., 2011)	For every analytic quasi-order $R$ on standard Borel space $X$, there is an $L_{\omega_1 \omega}$-sentence $\varphi$	Universality of embeddability coding
Theorem 3.9 ("Graph version")	Same as above, but for graph language; embeddability on graphs is complete for analytic quasi-orders	Applicability to ordinary graphs
Proposition 3.10	For combinatorial trees, every homomorphism/weak-homomorphism is an embedding	Rigidity result enabling collapse of morphisms
Classwise Borel Isomorphism	$E \cong_{cB} F$ iff $E \leq_{cB} F \leq_{cB} E$	Finer matching of quotient structures

8. Significance and Contextualization

These results place bi-embeddability and related strong morphism relations at the center of complexity theory for classification problems on countable structures. While classical isomorphism enjoys certain tameness (often countable or smooth), strong embeddings universally realize the full spectrum of analytic equivalence relations, allowing arbitrary analytic problems to be coded as embeddability. This facilitates, for example, the transfer of difficulties in orbit equivalence relations to questions about strong embeddings of structures, and vice versa. The landscape is sharply demarcated: strong embedding relations capture universal complexity, while isomorphism relations remain strictly subordinate in the hierarchy.

9. Future Research Trajectories

Ongoing work will likely clarify:

• The exact reach of completeness results for morphisms beyond embedding.
• The interplay between model-theoretic conjectures (e.g. Vaught’s Conjecture) and the complexity of classification for strong embeddings.
• The role of saturation/closure and symmetry considerations in analytic classes arising in geometric analysis and topological dynamics.
• The maximal generality possible when representing analytic quasi-orders via functional actions of Polish monoids and whether canonical forms can be achieved.

These investigations will deepen the understanding of the structure and stratification of isomorphism classes of strong embeddings, with consequences spanning logic, analysis, and dynamical systems.