Partitioned Groupoid Assemblies

• Partitioned groupoid assemblies are a generalization of partitioned assemblies where chosen realizers are attached to both objects and identifications via groupoidal or internal-groupoid structures.
• They implement a homotopical BHK interpretation by realizing identifications as paths or isomorphisms, thereby replacing extensional equality with nontrivial homotopical data.
• This framework supports weak cartesian closure and 2-categorical finiteness properties, offering a refined model of intensional type theory and a homotopical bridge to classical realizability toposes.

Partitioned groupoid assemblies are realizability-theoretic structures in which chosen realizers are attached not only to objects but also, in a groupoidal sense, to identifications. The available literature uses two different but closely related definitions. In one setting, a partitioned groupoidal assembly is a triple $(X,A,\|-\|_X)$ consisting of a groupoid $X$, a realizer type $A$, and a functor $\|-\|_X:X\to \Pi A$ into the fundamental groupoid of $A$. In another, a partitioned groupoid assembly is an internal groupoid in assemblies whose assembly of objects is partitioned. Both settings generalize ordinary partitioned assemblies by replacing set-based realization with groupoidal or internal-groupoid data and by treating identifications as realized by paths or isomorphisms rather than by extensional equality alone (Speight, 2024, Agwu, 21 Jul 2025).

1. Ordinary partitioned assemblies as the antecedent

The immediate background is the classical category of partitioned assemblies. For a partial combinatory algebra $A$, the category $\mathbf{PAsm}(A)$ has objects $(I,\varphi)$ with $\varphi:I\to A$, and a morphism

$f:(I,\varphi)\to (J,\psi)$

is a function $X$0 such that there exists $X$1 with

$X$2

In the special case induced by a pca, this is the usual tracking condition: each element $X$3 carries a single chosen realizer $X$4, and a map is tracked by an element of $X$5. This is exactly why they are called partitioned assemblies: unlike a general assembly, where an element can have a set of realizers, each element comes equipped with one distinguished realizer (Frey, 2014).

The same idea appears concretely over Kleene’s first pca. There an assembly $X$6 is partitioned precisely when the total relation $X$7 is single-valued, hence a function $X$8. The full subcategory $X$9 has finite limits, finite coproducts, weak exponentials, and a parameterized natural numbers object $A$0. It also plays the role of the projective base from which richer realizability categories are recovered by completion: $A$1, and the exact completion $A$2 is the effective topos $A$3 (Maietti et al., 2018).

This classical material supplies the conceptual template for groupoidal variants. In both later developments, the “partitioned” condition continues to mean that realization data are chosen rather than merely inhabited, but the target of realization is no longer just a set of realizers.

2. Partitioned groupoidal assemblies over a realizer category

A first formulation appears in realizability models of intensional type theory based on a typed realizer category $A$4. Here $A$5 is a cartesian closed category equipped with an interval $A$6 in the sense of an internal cogroupoid. The interval provides endpoint maps, cocomposition, and coinverse, and hence supports internal notions of path, homotopy, and fundamental groupoid. The resulting functor is

$A$7

In this setting, a partitioned groupoidal assembly is a triple $A$8, where $A$9, $\|-\|_X:X\to \Pi A$0, and

$\|-\|_X:X\to \Pi A$1

is a functor. Objects of the base groupoid are realized by points of $\|-\|_X:X\to \Pi A$2, and isomorphisms in the base groupoid are realized by paths in $\|-\|_X:X\to \Pi A$3. The stated motivation is a homotopical BHK interpretation in which “evidence for an identification is a path.” This is the basic departure from ordinary realizability: identifications are not semantically trivial, but carry non-discrete homotopical information (Speight, 2024).

Morphisms are likewise groupoidal. A morphism

$\|-\|_X:X\to \Pi A$4

is a functor $\|-\|_X:X\to \Pi A$5 such that there exists $\|-\|_X:X\to \Pi A$6 and a natural isomorphism

$\|-\|_X:X\to \Pi A$7

Thus implementation by realizers is required only up to natural isomorphism. This is one of the central structural differences from ordinary partitioned assemblies, where tracking is strict. A partitioned groupoidal assembly is called modest when $\|-\|_X:X\to \Pi A$8 is fully faithful; in that case, realizers of arrows determine arrows uniquely, and every path between realizers comes from a unique isomorphism in the base groupoid (Speight, 2024).

3. Weak exponentials, path structure, and type-theoretic semantics

The category $\|-\|_X:X\to \Pi A$9 of partitioned groupoidal assemblies is developed as a realizability-enriched groupoid semantics. Its terminal object is

$A$0

products are formed by

$A$1

and a central result is that $A$2 is weakly cartesian closed. The weak exponential has the form

$A$3

where $A$4 is a groupoid of realized functors and realized natural isomorphisms. Moreover, the weak exponential object $A$5 is modest whenever $A$6 is modest (Speight, 2024).

The same paper equips $A$7 with a 2-categorical structure. It inherits an interval object of coarrows, becomes a $A$8-category, and, under the hypothesis that the boundary functor

$A$9

is an isomorphism, is finitely complete as a $A$0-category. In that regime, pseudopullbacks are constructed explicitly, and 2-dimensional realizers are used to witness the comparison between the two composites in a pseudopullback square (Speight, 2024).

These categorical properties are then organized into a path-category semantics. Fibrations are taken to be internal isofibrations, and a morphism in $A$1 is a fibration if and only if it is an isofibration in $A$2. Path objects are given by weak exponentials,

$A$3

and the category has weak $A$4-types. The semantic conclusion stated in the paper is that, under mild conditions on the realizer category, the ensuing category of partitioned groupoidal assemblies models intensional $A$5-truncated type theory without function extensionality. The failure of function extensionality is tied to the weakness of the $A$6-types: the $A$7-principle is not generally available, and the construction requires choosing realizers only up to existence rather than strict uniqueness (Speight, 2024).

In the untyped case, where the realizer category has a universal object $A$8 such that every $A$9 is a pseudoretract of $\mathbf{PAsm}(A)$0, the same framework yields an impredicative universe of $\mathbf{PAsm}(A)$1-types. The relevant class is that of modest fibrations, and the paper proves that $\mathbf{PAsm}(A)$2 has a representation for modest fibrations and that these are closed under dependent products. This is described as a groupoidal analogue of the traditional realizability universe (Speight, 2024).

4. Partitioned groupoid assemblies internal to assemblies

A second formulation works directly over a pca $\mathbf{PAsm}(A)$3 via internal groupoids in the category of assemblies. The ambient category is

$\mathbf{PAsm}(A)$4

the category of internal groupoids and functors in $\mathbf{PAsm}(A)$5. Here an assembly is given concretely by a pair $\mathbf{PAsm}(A)$6, and a morphism $\mathbf{PAsm}(A)$7 is tracked by some $\mathbf{PAsm}(A)$8 such that for all $\mathbf{PAsm}(A)$9 and $(I,\varphi)$0, $(I,\varphi)$1 is defined and

$(I,\varphi)$2

A partitioned assembly is an assembly $(I,\varphi)$3 isomorphic to one $(I,\varphi)$4 where each $(I,\varphi)$5 is a singleton set. The full subcategory of such assemblies is denoted $(I,\varphi)$6 (Agwu, 21 Jul 2025).

The paper then defines $(I,\varphi)$7 as the full subcategory of $(I,\varphi)$8 whose underlying assembly of objects is a partitioned assembly. Concretely, an object of $(I,\varphi)$9 is an internal groupoid

$\varphi:I\to A$0

in $\varphi:I\to A$1 such that $\varphi:I\to A$2 is partitioned. The supplied text does not introduce a stronger condition requiring the arrow assembly $\varphi:I\to A$3 to be partitioned as well; the repeated description is specifically that the underlying assembly of objects is partitioned. Because $\varphi:I\to A$4 is a full subcategory, its morphisms are simply the same internal functors as in $\varphi:I\to A$5 (Agwu, 21 Jul 2025).

This formulation is tied to the ordinary projective theory of partitioned assemblies. The paper constructs, for any assembly $\varphi:I\to A$6, a partitioned cover

$\varphi:I\to A$7

and states that, assuming Choice, partitioned assemblies are precisely the projective objects in $\varphi:I\to A$8. Under the same assumption, $\varphi:I\to A$9 is the reg/lex completion of $f:(I,\varphi)\to (J,\psi)$0, and $f:(I,\varphi)\to (J,\psi)$1 is the ex/lex completion of $f:(I,\varphi)\to (J,\psi)$2. This gives $f:(I,\varphi)\to (J,\psi)$3 the role of a groupoid-level analogue of working with projective or partitioned presentations before exact completion (Agwu, 21 Jul 2025).

5. Finite bi(co)limits, $f:(I,\varphi)\to (J,\psi)$4-types, and the realizability topos

In the internal-groupoid framework, the main model of type theory is the ambient category $f:(I,\varphi)\to (J,\psi)$5, not $f:(I,\varphi)\to (J,\psi)$6 itself. The paper states that $f:(I,\varphi)\to (J,\psi)$7 carries the structure of a $f:(I,\varphi)\to (J,\psi)$8-tribe, has $f:(I,\varphi)\to (J,\psi)$9-types with reductions, has a univalent object classifier for assemblies and modest assemblies, and has a model structure obtained using the $X$00-types with reductions. Against this background, $X$01 is introduced as a technically well-behaved full subcategory with its own bicategorical finiteness properties (Agwu, 21 Jul 2025).

The specific result about partitioned groupoid assemblies is that $X$02 has finite bilimits and bicolimits. The proof strategy described in the supplied text is indirect: finite limits and colimits are first established in $X$03, and then two biadjunctions are constructed—one between $X$04 and $X$05, and another on $X$06—to transport the finite bi(co)limit structure. The significance of this is bicategorical rather than merely 1-categorical: the subcategory of partitioned presentations is stable enough to support finite bilimit and bicolimit calculations up to equivalence (Agwu, 21 Jul 2025).

The most striking comparison theorem concerns truncation. The paper states that the $X$07-types of $X$08 are the internal equivalence relations of $X$09, and that the homotopy category of the full subcategory of the $X$10-types of $X$11 is $X$12, the realizability topos of $X$13. In the wording quoted in the supplied text, restricting to $X$14 makes the $X$15-types present a category equivalent to the realizability topos by inverting the categorical equivalences. This is a homotopical refinement of the classical completion picture: the ordinary realizability topos reappears as the $X$16-truncated homotopy shadow of a groupoid-assembly construction (Agwu, 21 Jul 2025).

6. Terminological scope and conceptual significance

The available literature does not use a single uniform definition of “partitioned groupoid assemblies.” One line of work studies partitioned groupoidal assemblies over a realizer category with interval, where realization is a functor into a fundamental groupoid and morphisms are realized up to natural isomorphism. Another studies the full subcategory of internal groupoids in assemblies whose object assembly is partitioned. These are distinct constructions, and no theorem in the supplied material identifies them directly (Speight, 2024, Agwu, 21 Jul 2025).

A related terminological caution is that foundational papers on partitioned assemblies do not themselves discuss partitioned groupoid assemblies. The paper on characterizing partitioned assemblies and realizability toposes is explicitly about ordinary $X$17 and $X$18, giving an extensional characterization of both categories in terms of weak local cartesian closure, locality, and a separated discrete generic object. Likewise, the paper on elementary quotient completions, Church’s Thesis, and partitioned assemblies studies ordinary $X$19, its doctrines, and the completions yielding $X$20 and $X$21; it states that it does not discuss “partitioned groupoid assemblies” in any explicit sense [(Frey, 2014); (Maietti et al., 2018)].

What unifies the later groupoidal developments is a common structural shift. Classical partitioned assemblies isolate a presentation layer in which each element carries a chosen realizer. The groupoidal versions retain this “chosen realization” feature, but add nontrivial identity data. In the realizer-category formulation, objects are realized by points and identifications by paths in a fundamental groupoid. In the internal-groupoid formulation, partitioned object assemblies provide rigid presentations inside a larger homotopical category of groupoid assemblies, and the $X$22-truncated homotopy theory recovers the realizability topos. The shared conceptual theme is therefore not a single formal definition, but the combination of realizability, chosen realizers, and groupoidal identity structure (Speight, 2024, Agwu, 21 Jul 2025).