Sub-Domain Equivalences

• Sub-domain equivalences are mappings between substructures that preserve invariants, unifying insights from category theory, algebra, logic, and geometry.
• Categorical approaches employ Galois connections, reflective-coreflective pairs, and adjunctions to rigorously relate subdomains with their global counterparts.
• Applications span operator algebras, metric geometry, and program semantics, demonstrating how local properties can be effectively transferred across diverse systems.

Sub-domain equivalences encompass a range of categorical, algebraic, logical, and geometric constructions where equivalence relations, functors, morphisms, or categorical structures—restricted or localized to "subdomains" (subcategories, subalgebras, subsets, etc.)—encode meaningful invariances or transfer properties across mathematical and computational frameworks. This theme recurs in domain theory, categorical algebra, operator algebras, metric geometry, logic, module theory, and probabilistic systems, revealing deep interconnections between global and local behavior.

1. Categorical and Galois-Theoretic Equivalences for Substructures

In domain theory and measurable dynamics, subdomain equivalence is formalized via Galois connections between posets of substructures. For a measurable space $(X, \mathcal{M})$, there exists a Galois connection between the poset of sub-$\sigma$-algebras and the poset of equivalence relations on $X$:

• For each equivalence relation $\sim$ on $X$, the sub-$\sigma$-algebra $\mathcal{M}_{\sim}$ is given by the $\sim$-invariant sets: $$\mathcal{M}_{\sim} = \{A \in \mathcal{M} \mid \forall x \sim y \Rightarrow (x \in A \Leftrightarrow y \in A)\}$$.
• Conversely, each sub-$\sigma$-algebra $\mathcal{A} \subseteq \mathcal{M}$ determines an equivalence relation $$x \sim_{\mathcal{A}} y \iff \forall A \in \mathcal{A}\ (x \in A \Leftrightarrow y \in A)$$.

The duality is formal: $\mathcal{M}_{\sim} \supseteq \mathcal{A}$ iff $\sim \leq \sim_{\mathcal{A}}$.

This structure generalizes further, where equivalence relations on the state space of a stochastic system correspond to congruences (i.e., behavioral equivalences persisting under the system's structure), making subdomain equivalence a bridge between algebraic invariants and measurable or probabilistic semantics (Battenfeld, 2010).

In enriched category theory, Lawvere's structure-semantics adjunction and its modern generalizations equate the category of theories (axiomatizations restricting to a subcategory of arities via a dense functor $j : J \hookrightarrow \mathcal{C}$) with the category of "ary" monads on $\mathcal{C}$. This equivalence hinges on specifying which subcategory of arities is used, such as the corestricted Yoneda embedding for $\Phi$-theories, resulting in an equivalence between small $T$-theories and $\Phi$-accessible monads for locally $\Phi$-presentable categories (Lucyshyn-Wright et al., 2023).

2. Reflective-Coreflective and Subcategory Equivalences

Categorical instances of subdomain equivalence arise in the equivalence of reflective and coreflective subcategories. Reflective subcategories $\mathcal{N} \subseteq \mathcal{C}$ admit left adjoints (reflectors), coreflective subcategories $\mathcal{M} \subseteq \mathcal{C}$ admit right adjoints (coreflectors), and under suitable splitting of the unit and counit, these restricted adjunctions yield adjoint equivalences or isomorphisms of subdomains.

Key examples include:

• The equivalence between maximal and normal coactions in the theory of $C^*$-dynamical systems;
• The equivalence of reduced and universal compact quantum groups, where the duality is realized via the Yoneda structure and the adjoint functors encode universal properties (Bédos et al., 2010).

In these settings, subdomain equivalence means that properties, invariants, and object classes can be transported without loss across apparently distinct models once restricted/reflected to the relevant subcategory.

3. Equivalences via Logical, Algebraic, and Frame-theoretic Representations

Domain theory leverages logical, algebraic, and lattice-theoretic representations to formalize subdomain equivalences:

• Information systems and logical frames: Generalizations of Scott's information systems (with consistency relativized) capture all continuous domains. The category of information frames—objects with localized consistency predicates—is shown equivalent to the category of continuous domains. Similarly, the category of CF-approximation spaces (from Wu and Xu) is equivalent to information frames and (hence) to domains (Spreen, 28 Feb 2025).
• Synthetic and locale-theoretic approaches: In synthetic domain theory, locales (especially $\sigma$-frames) and their associated topoi classify domain-like structures. For the free $\sigma$-frame with countably many generators, the space of points (in the sense of locales) is isomorphic to a powerset of the Sierpiński space, establishing a duality between combinatorial data and synthetic domain models (Sterling et al., 19 May 2025).
• Sequent calculi and syntactic representations: Algebraic $L$-domains are characterized by their underlying logical states in a disjunctive sequent calculus, and consequence relations between calculi correspond precisely to Scott-continuous functions, enabling a categorical equivalence between logic-based and order-theoretic perspectives (Wang et al., 2019).
• Covariant logic: In information-system terms, subdomains correspond to subcollections of theories (deductively closed, consistent sets of tokens) with intensified entailment relations, while morphisms are represented by approximable mappings (input-output inference engines), yielding highly structured subdomain inclusions and equivalences (Bukatin, 2015).

4. Metric, Geometric, and Dynamical Subdomain Equivalences

In metric and dynamical contexts, subdomain equivalence investigates correspondence and monotonicity properties between metrics, dynamics, or combinatorial invariants defined on domains and their subdomains:

• Hyperbolic-type metrics: The quasihyperbolic and distance ratio metrics for a domain $G \subset \mathbb{R}^n$ exhibit domain monotonicity: metrics on a subdomain $D \subset G$ always dominate those on $G$. Quantitative refinements yield inequalities with sharp constants—dependent on geometric parameters—enabling precise transference of geometric and analytic estimates under domain inclusion (Klén et al., 2012).

Metric $m$	Domain monotonicity	Reverse inequality (under hypotheses)
$k_D(x,y)$, $j_D(x,y)$	$m_D(x, y) \geq m_G(x, y)$	$k_G(x, y) \leq c \cdot j_G(x, y)$

• Dynamical systems: Subspace restrictions yield "local" notions of shadowing and expansivity. The paper provides conditions under which notions like h-shadowing, s-limit shadowing, and classical shadowing are equivalent or implied by local ball expensivity and openness. For interval maps, strong local shadowing properties are established on subdomains away from critical points, demonstrating robust local-to-global transfer within the dynamics (Barwell et al., 2011).

5. Algebraic and Module-Theoretic Subdomain Equivalences

In algebra and homological algebra, subdomain equivalence formalizes the stability of invariants and categorical structures under equivalence relations weaker than Morita equivalence:

• Separable equivalences: For module categories (e.g., $\operatorname{Mod}\text{-}\Lambda$ and $\operatorname{Mod}\text{-}\Gamma$), separable equivalence via exact functors/tensoring, where the identity functor splits off as a direct summand, ensures that Gorenstein categories, Wakamatsu tilting modules, Gorenstein projective and injective modules, and Auslander/Bass classes are preserved under equivalence. This includes the preservation of orthogonal classes and (co)resolution subcategories (Zhao et al., 29 Jun 2025).
• Module-theoretic subcategories:

Category	Preservation under separable equivalence
Gorenstein category $\mathcal{G}(C)$	$\mathcal{G}(C)$ equivalent to $\mathcal{G}(T_N(C))$
Wakamatsu tilting modules	Image under $N\otimes_\Lambda C$ is again Wakamatsu tilting
$\operatorname{G}_C$-projectives/injectives	Equivalence as above
Auslander/Bass classes	Invariant under functors $T_N, T_M$ with the appropriate conditions

This invariance applies to representation theoretic invariants, Gorenstein homological dimensions, and the "internal" structure of module categories.

6. Equivalences in Descriptive Set Theory, Computability, and Operator Algebras

Subdomain equivalences also manifest in descriptive set theory, categorical logic, and operator algebra:

• Categories of equivalence relations: In the category of equivalence relations $\mathbb{E}q$ (and subcategories defined by computability, such as ceers, coceers, and dark ceers), categorical properties and constructions (products, coproducts, coequalizers) are linked to computability-theoretic subdomains. Epimorphism and monomorphism characterizations, as well as closure properties, can fail or succeed depending on the subdomain, revealing subtleties in categorical structure with computational flavor (Rose et al., 2021).
• Operator-theoretic context: The domain equivalence relation for self-adjoint operators, $$A \mathrel{E_{\operatorname{dom}}^{\mathrm{SA}(H)}} B \iff \operatorname{dom}(A) = \operatorname{dom}(B)$$, is found to be $F_\sigma$ and continuously bireducible with the orbit equivalence $E_{\ell^\infty}^{\mathbb{R}^\mathbb{N}}$; it is universal for $K_\sigma$ equivalence relations in the sense of descriptive set theory. This universality establishes the complexity of subdomain classification in the space of self-adjoint operators, and demonstrates strong limitations on classifiability by countable invariants (Ando et al., 2014).

7. Subdomain Equivalences in Program Semantics and System Verification

In computer science and formal verification:

• Component-based equivalences: Open pNets, generalizations of automata with holes and hierarchical composition, facilitate strong and weak bisimulations on subarchitectures of a system (i.e., subdomains of the process structure). These bisimulations are defined at the level of open automata and shown to be congruences, making bisimilarity a robust notion for component-based verification; substitution of equivalent components in a pNet preserves overall behavioral equivalence (Ameur-Boulifa et al., 2020).
• Denotational/program semantics: Logical states in sequent calculi correspond to points in algebraic domains; consequence relations between calculi encode Scott-continuous functions, thus tying the semantics of domain-based computation to syntactically defined subdomains and categorical equivalence (Wang et al., 2019).

Summary Table: Subdomain Equivalence Concepts Across Areas

Setting	Subdomain Equivalence Realization	Reference(s)
Domain/measure theory	Galois connection between sub-$\sigma$-algebras and equivalence relations	(Battenfeld, 2010)
Categorical algebra	Reflective-coreflective subcategory equivalence, enriched structure-semantics	(Bédos et al., 2010, Lucyshyn-Wright et al., 2023)
Information/logic frames	Categorical equivalence: information frames, continuous domains, CF-approximation spaces	(Spreen, 28 Feb 2025)
Synthetic locale theory	Points of free $\sigma$-frames correspond to powers of the Sierpiński space	(Sterling et al., 19 May 2025)
Geometric analysis	Quantitative monotonicity of hyperbolic-type metrics in subdomains	(Klén et al., 2012)
Operator algebras	Domain equivalence of SAOs continuously bireducible with $\ell^\infty$-orbit equivalence	(Ando et al., 2014)
Computability/categorical logic	Categories of equivalence relations and subcategory closure properties	(Rose et al., 2021)
Program verification	Bisimulation congruence for open pNets/components	(Ameur-Boulifa et al., 2020)
Homological algebra	Gorenstein/tilting categories and class invariants under separable equivalence	(Zhao et al., 29 Jun 2025)
Metric geometry	Subdomain monotonicity/equivalences for metrics (quasihyperbolic, j-distance)	(Klén et al., 2012)
Shadowing theory/dynamics	Local/global (subdomain) equivalences for expansivity and shadowing properties	(Barwell et al., 2011)

Conclusion

Sub-domain equivalences unify a broad collection of mathematical and computational phenomena where structure-preserving correspondences or adjunctions—often realized via Galois connections, reflective/coreflective pairs, information systems, logical functors, or geometric monotonicity—ensure that properties, invariants, or behaviors of substructures carry over or determine their ambient domains. The continued development of these equivalences, as seen in modern categorical, algebraic, analytic, and geometric settings, uncovers new opportunities for rigorously transferring complex properties between global systems and their syntactic, algebraic, or spatial subdomains, with significant implications for both theory and applied domains.