Parameterized Realizability Topos

• Parameterized realizability topos is a higher-categorical model that defines realizability via extra parameters like Heyting categories and enriched OPCAs.
• It employs regular functors and assembly constructions to transfer computational and logical structure across different categorical contexts.
• The construction over arbitrary Heyting categories via exact completion ensures modularity and precise classification of geometric morphisms.

A parameterized realizability topos is a higher-categorical model in which the realizability interpretation of logic, computation, and semantics is systematically “parameterized”—that is, relativized to additional data such as a base Heyting category, oracle sets, partial combinatory algebras (pcas) equipped with extra structure, or, more generally, a chosen class of computational resources. This generalizes both classical realizability toposes built on $\mathsf{Set}$ and relative or modified realizability frameworks. The archetypal construction is the relative realizability topos, where the parameters consist of an ordered partial combinatory algebra (OPCA) $(A',A)$ in a Heyting category $\mathcal{E}$. This parameterization enables the robust transfer and comparison of computational and logical structure across categorical contexts.

1. Universal Property and Regular Models

The defining property of a parameterized realizability topos, particularly in the setting of relative realizability, is a strong universal property stated in terms of regular functors and filters. Given a base Heyting category $\mathcal{E}$ and an OPCA pair $(A',A)$ internal to $\mathcal{E}$, a “regular model” is specified by a regular functor $F : \mathcal{E} \to \mathcal{C}$ together with an $F$–filter $C \subseteq F(A)$ satisfying:

• Upward closure: $x \in C$ and $x \leq y$ implies $y \in C$.
• Closure under application: $x, y \in C$ and $xy = z$ implies $z \in C$.
• Intersection: For every subobject $U \subseteq A$ intersecting $A'$, $F(U)$ intersects $C$.

The category of assemblies $\mathrm{Asm}(A',A)$ in $\mathcal{E}$, constructed with its canonical regular functor $V : \mathcal{E} \to \mathrm{Asm}(A',A)$ and distinguished filter, forms the pseudoinitial regular model for these data. Every other regular model $(F, C)$ factors uniquely (up to isomorphism) as a regular functor through this assembly construction, respecting the filter structure.

A typical explicit instance: every representation of a partial arrow $f : U \subseteq A^n \to A$ via

$$[f] = \{ a \in A \mid \forall x \in U,\, \exists y \ (xy\downarrow \wedge y \leq f(x)) \}$$

arises from a unique (up to isomorphism) functor factoring through the canonical assembly construction.

2. Role and Structure of Regular Functors

Regular functors—those preserving finite limits and regular epis—are critical for maintaining both logical and computational properties across parameterized realizability toposes. Key aspects:

• The canonical $V : \mathcal{E} \to \mathrm{Asm}(A',A)$ is a regular, faithful functor, and admits a right adjoint $D : \mathrm{Asm}(A',A) \to \mathcal{E}$ with $D \circ V = \operatorname{id}_\mathcal{E}$, making the assembly construction depend (parameterically) on $\mathcal{E}$.
• Morphisms of regular models $(F,C) \to (G,D)$ are regular functors with natural transformations that also preserve the distinguished filter, enabling structural comparisons between different parameterizations.
• There is a precise correspondence (e.g., Corollary 34) between the poset of $F$–filters and the category of regular extensions of $F$, and thus analysis of regular functors provides control and classification over the “landscape” of parameterized realizability models.

3. Construction over Arbitrary Heyting Categories

Parameterized realizability toposes are not exclusive to the base category $\mathsf{Set}$; the construction uniformly applies to arbitrary (possibly large) base Heyting categories. The process is as follows:

• Assembly objects: For $(A',A)$ in a Heyting category $\mathcal{E}$, an assembly is a pair $(X,Y)$ with $X$ an object of $\mathcal{E}$ and $Y \subseteq A \times X$ such that:
  • For every $x \in X$, there is $a \in A$ with $(a, x) \in Y$.
  • $Y$ is downward closed: $(a, x) \in Y$ and $b \leq a$ imply $(b, x) \in Y$.
• Category properties: $\mathrm{Asm}(A',A)$ is a Heyting category (Lemma 23) and is locally cartesian closed (Theorem 44).
• Exact completion: The parameterized (relative) realizability topos is the exact (ex/reg) completion $$\mathrm{RT}(A',A) = \mathrm{Asm}(A',A)_{\mathrm{ex/reg}}$$.
• The parameterization allows models over syntactic or semantic categories of type theory, extending standard realizability to novel logical environments.

4. Geometric Morphisms and Classification via Parameters

A central innovation is the precise construction and classification of geometric morphisms (i.e., topos morphisms that are adjunctions where inverse images preserve finite limits) into parameterized realizability toposes:

• Points: Set-valued regular models correspond to points of $\mathrm{RT}(A',A)$; an “ide-filter” condition yields a regular functor $D_c : \mathrm{RT}(A',A) \to \mathcal{E}$ with a right adjoint (Theorem 53).
• Characters: Morphisms from $A$ into the downset object $DP$, preserving combinatory structure, induce morphisms from internal presheaf toposes into $\mathrm{RT}(A',A)$.
• Applicative morphisms: Applicative morphisms between OPCA pairs induce regular functors between assembly categories. If computationally dense, these extend to geometric morphisms between the relative realizability toposes (Theorem 63).

This classification provides fine control over the spectrum of possible topos morphisms and subtoposes, directly in terms of the computational and logical “parameters.”

5. Exact Completion, Modularity, and Decomposition

The exact completion process, applied to the parameterized assembly category, enforces modularity and enables intrinsic decomposition:

• For any regular category, the ex/reg completion yields its “smallest” exact extension; for assemblies over a parameterized OPCA, this produces the full topos structure.
• The process generalizes the effective topos construction to parameterized contexts.
• Geometric morphisms and subtoposes can be understood intrinsically in terms of the structure of the underlying filter and OPCA data, with the assembly category serving as a “centerpiece” in the parameterized theory.

6. Connections, Broader Implications, and Future Directions

Parameterization empowers realizability theory beyond classical $\mathsf{Set}$-based models:

• Universal property: The parameterized realizability topos is the pseudoinitial exact model extending a Heyting category with a “universal” filter subobject, closed under application, intersecting every nontrivial subobject.
• Bridging logic and computation: Regular functors and filters serve as the bridge between logic on the base, computational structure on the fibers, and topos-theoretic properties.
• Interface with sheaf theory and localic toposes: Characters and presheaf constructions reveal connections to localic and sheaf toposes built from computational data.
• New morphisms and expansion: The parameterized perspective suggests abundant new morphisms (geometric, exact, or regular) may be realized, and that new parameters—even in purely logical or type-theoretic base categories—admit systematic extension of realizability.
• Ongoing research avenues: Open questions include developing “completion” constructions in situations lacking split epis or global sections, further analyzing the role of parameterized filters, and exploring connections with higher-categorical and synthetic approaches.

7. Summary Table: Key Elements in Parameterized Realizability Toposes

Component	Structural Role	Parameterization Aspect
Base Heyting Category $\mathcal{E}$	Underlying logic/environment	Varies over $\mathsf{Set}$, type theory
OPCA pair $(A',A)$	Computational structure	Chosen per parameterization
Regular functor $V$	Canonical embedding	Faithful, regular, base-dependent
Assembly category $\mathrm{Asm}(A',A)$	Pre-topos with computational data	Defined in internal logic of $\mathcal{E}$
Exact (ex/reg) completion	Realizability topos construction	Upgrades regularity to exactness
Filter $C$	Realizer-parameter subobject	Controls realizability model
Geometric morphisms	Topos-theoretic structure	Classified via functors, characters, applicative morphisms

References and Technical Foundations

The technical apparatus elaborated above, including the universal property, internal assembly structures, filters, and geometric morphisms, is fully deployed and justified in "Regular Functors and Relative Realizability Categories" (Stekelenburg, 2011). The parameterization arises fundamentally through the explicit dependence on arbitrary base Heyting categories and specified OPCAs, as well as through the systematic use of regular functors and filter data. This framework generalizes, subsumes, and explains a wide class of realizability toposes, enabling rigorous structural comparison and transfer of logical and computational properties between models.