Univalent Object Classifier

• Univalent object classifier is a universe in a path category that classifies a restricted class of fibrations with an inherent univalence property.
• It features explicit classifiers for discrete propositional and set fibrations within realizability frameworks, supporting impredicative quantification and resizing.
• The work highlights limitations by showing that full discrete fibrations do not admit a univalent classifier, and it suggests extensions to higher path categories.

Searching arXiv for the specified paper and closely related context. arxiv_search.query({"^^^^^^^^2search_query2 OR ti:\2^^^^^^^^"Univalent polymorphism\"","max_results":5,"sort_by":"relevance","sort_order":"descending"}) A univalent object classifier, in the sense of "Univalent polymorphism," is a univalent universe PRESERVED_PLACEHOLDER_^^^^2search_query2^^^^ that classifies a specified class PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^ of fibrations, or families. The central result is not the existence of a single universe for all discrete fibrations, but a pair of restricted realizability-style classifiers: one for an impredicative class of discrete propositional fibrations in the path category $\mathbb{EFF}$, and one for an impredicative class of discrete set fibrations in the richer path category $\mathbb{EFF}_1$. The same work also shows that a univalent classifier for the full class of discrete fibrations cannot exist in these settings (&&&^^^^2search_query2^^^^&&&).

^^^^2id:(Berg, 2018) OR ti:\2^^^^. Path categories and the effective topos

A path category is a category $C$ equipped with fibrations and equivalences, satisfying Brown’s axioms for a category of fibrant objects together with two additional properties. In the formulation used here, the structure includes: closure of fibrations under composition; a terminal object $1$ with every map $X\to 1$ a fibration; pullbacks of fibrations along arbitrary maps; the $6$-for-$2$ property for equivalences; path objects factoring the diagonal $X\to X\times X$ as PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2search_query2^^^^ with the first map an equivalence and the second a fibration; pullback stability of trivial fibrations; and sections for trivial fibrations (&&&^^^^2search_query2^^^^&&&).

Homotopy is defined by path objects: maps PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2id:(Berg, 2018) OR ti:\2^^^^ are homotopic if there exists PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\22^^^^ with PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^3. The homotopy relation is a congruence, and a map becomes an isomorphism in the homotopy category PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^4 if and only if it is an equivalence in PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^5. Consequently, PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^6 is obtained by quotienting maps by homotopy.

The path category PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^7 is built from “locally codiscrete bigroupoids” on a set PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^8 equipped with a realizer map PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^9, sets of $\mathbb{EFF}$^^^^2search_query2^^^^-cells $\mathbb{EFF}$^^^^2id:(Berg, 2018) OR ti:\2^^^^, and effective operations for identities, inverses, and composition of $\mathbb{EFF}$2-cells. Morphisms carry effective action on $\mathbb{EFF}$3- and $\mathbb{EFF}$4-cells, fibrations are given by an effective lifting property for $\mathbb{EFF}$5-cells together with a second lifting clause for parallel $\mathbb{EFF}$6-cells, and equivalences are homotopy equivalences. The resulting homotopy category recovers realizability semantics: there is a functor $\mathbb{EFF}$7 that is full, essentially surjective, and inverts homotopy, so that $\mathbb{EFF}$8 (&&&^^^^2search_query2^^^^&&&).

This setting is essential for the notion of univalent object classifier used here. The classifier is not introduced in an arbitrary higher topos, but inside a path category whose objects and morphisms carry explicit realizability data.

2. Discrete fibrations, impredicativity, and resizing

In $\mathbb{EFF}$9, a fibration $\mathbb{EFF}_1$^^^^2search_query2^^^^ is standard discrete if, for all $\mathbb{EFF}_1$^^^^2id:(Berg, 2018) OR ti:\2^^^^, the conditions $\mathbb{EFF}_1$2 and $\mathbb{EFF}_1$3 imply $\mathbb{EFF}_1$4. A fibration is discrete if it is equivalent, by pre- or post-composition with an equivalence, to a standard discrete fibration (&&&^^^^2search_query2^^^^&&&).

A key characterization, stated in Freyd-style form, identifies discreteness with an orthogonality condition: for a fibration $\mathbb{EFF}_1$5, discreteness is equivalent to the diagonal square with the object $\mathbb{EFF}_1$6 being a homotopy pullback, and equivalently to the existence of a computable function producing, from $\mathbb{EFF}_1$7 with the same image in $\mathbb{EFF}_1$8 and the same realizer, a $\mathbb{EFF}_1$9-cell $C$^^^^2search_query2^^^^ lifting $C$^^^^2id:(Berg, 2018) OR ti:\2^^^^. This makes the discrete class computationally rigid in a way compatible with the realizability structure.

The class $C$2 of discrete fibrations in $C$3 is impredicative. It is closed under pullback, under composition, and under pushforward along arbitrary fibrations in the sense of right adjoints $C$4 to pullback. More precisely, for every fibration $C$5, the pullback functor on the homotopy category has a homotopy right adjoint $C$6, constructed as homotopy $C$7-types, and $C$8 preserves discreteness. This is the paper’s sense of polymorphism: closure under impredicative quantification (&&&^^^^2search_query2^^^^&&&).

The same framework supports resizing statements. A class $C$9 satisfies propositional resizing if every propositional fibration lies in $1$^^^^2search_query2^^^^. In $1$^^^^2id:(Berg, 2018) OR ti:\2^^^^, assuming the axiom of choice in the metatheory, every propositional fibration is discrete, so discrete propositional fibrations satisfy resizing. The paper also records stability under $1$2 of h-levels: $1$3 preserves $1$4-types, in particular $1$5-types and $1$6-types.

3. Univalent representation as object classification

For a class $1$7 of fibrations in a path category, a univalent representation consists of a fibration $1$8 satisfying two conditions. The first is classification: for any $1$9 in $X\to 1$^^^^2search_query2^^^^, there exists a classifying map $X\to 1$^^^^2id:(Berg, 2018) OR ti:\2^^^^ such that $X\to 1$2 is equivalent, in the slice over $X\to 1$3, to the pullback $X\to 1$4, and $X\to 1$5 is unique up to homotopy. The second is univalence: for $X\to 1$6, if $X\to 1$7 and $X\to 1$8 denote the fibers of $X\to 1$9, then the canonical map

$6$^^^^2search_query2^^^^

is an equivalence (&&&^^^^2search_query2^^^^&&&).

Here $6$^^^^2id:(Berg, 2018) OR ti:\2^^^^ is the homotopy class of paths $6$2 in a path object $6$3, and $6$4 is the space of equivalences between fibers, equivalently isomorphisms in the homotopy category between those fibers. Operationally, univalence asserts that every fiberwise equivalence arises uniquely up to homotopy by transport along a path in $6$5.

A common misunderstanding is to identify a univalent object classifier with a universe for all objects in the ambient category. In this setting, the classifier is always relative to a chosen class of fibrations. The positive results concern discrete propositional fibrations in $6$6 and discrete set fibrations in $6$7, not the full class of discrete fibrations (&&&^^^^2search_query2^^^^&&&).

4. The classifier for discrete propositional fibrations in $6$8

The positive result in $6$9 concerns discrete fibrations whose fibers are propositions in the sense of Homotopy Type Theory. The universe $2$^^^^2search_query2^^^^ has as $2$^^^^2id:(Berg, 2018) OR ti:\2^^^^-cells subsets $2$2, realizer $2$3 for all $2$4, and as $2$5-cells $2$6 pairs of realizers $2$7 and $2$8, given by partial recursive functions total on $2$9 and $X\to X\times X$^^^^2search_query2^^^^ respectively. The total space $X\to X\times X$^^^^2id:(Berg, 2018) OR ti:\2^^^^ has as $X\to X\times X$2-cells pairs $X\to X\times X$3 with $X\to X\times X$4 and $X\to X\times X$5, realizer $X\to X\times X$6, and the same $X\to X\times X$7-cell data as $X\to X\times X$8. The map

$X\to X\times X$9

is projection PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2search_query2search_query2^^^^ (&&&^^^^2search_query2^^^^&&&).

This PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2search_query2id:(Berg, 2018) OR ti:\2^^^^ is a discrete propositional fibration. Classification is explicit. If PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2search_query22^^^^ is a discrete propositional fibration, then using propositional truncation one may assume PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2search_query2^^^^3 with PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2search_query2^^^^4, PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2search_query2^^^^5, and PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2search_query2^^^^6. The classifying map is

PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2search_query2^^^^7

For PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2search_query2^^^^8, transport PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2search_query2^^^^9 gives the action of PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2id:(Berg, 2018) OR ti:\2search_query2^^^^ on PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2id:(Berg, 2018) OR ti:\2id:(Berg, 2018) OR ti:\2^^^^-cells, and by construction PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2id:(Berg, 2018) OR ti:\22^^^^ in the slice over PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2id:(Berg, 2018) OR ti:\2^^^^3.

Univalence is equally concrete. For PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2id:(Berg, 2018) OR ti:\2^^^^4 and PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2id:(Berg, 2018) OR ti:\2^^^^5 in PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2id:(Berg, 2018) OR ti:\2^^^^6, a path PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2id:(Berg, 2018) OR ti:\2^^^^7 is represented by mutually inverse realizers PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2id:(Berg, 2018) OR ti:\2^^^^8 and PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2id:(Berg, 2018) OR ti:\2^^^^9. Transport along such a path induces an equivalence between fibers PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\22search_query2^^^^ and PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\22id:(Berg, 2018) OR ti:\2^^^^. Conversely, any equivalence between fibers yields such a pair PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\222^^^^, so PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\22^^^^3 is an equivalence. Under the axiom of choice in the metatheory, every propositional fibration is discrete, so all propositional fibrations are small with respect to PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\22^^^^4. Modulo the usual coherence problems, this gives a model of the Calculus of Constructions with a univalent type of propositions (&&&^^^^2search_query2^^^^&&&).

5. The classifier for discrete set fibrations in PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\22^^^^5

The path category PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\22^^^^6 extends PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\22^^^^7 by adding a level of PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\22^^^^8-cells. For each pair of parallel PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\22^^^^9-cells there is a set of PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2max_results2search_query2^^^^-cells and effective data for identities, inverses, and compositions; morphisms act on PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2max_results2id:(Berg, 2018) OR ti:\2^^^^-, PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^32-, and PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^33-cells with coherent computational data; and fibrations satisfy an additional lifting clause for PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^34-cells. This yields a path category with homotopy exponentials and PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^35-types in which every fibration is a fibration of groupoids, or PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^36-types (&&&^^^^2search_query2^^^^&&&).

Discrete fibrations in PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^37 are defined as in PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^38 and remain impredicative. The setting also has PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^39-truncation: every fibration factors, up to universal property, through a fibration of sets. The small path category PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2sort_by2search_query2^^^^ consists of discrete objects, essentially the subcategory of PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2sort_by2id:(Berg, 2018) OR ti:\2^^^^-types with computable structure. Its objects are sets PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^42 equipped with hom-sets PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^43 and effective identity, inverse, and composition operations; morphisms are computable homomorphisms with tracking on hom-sets.

The universe PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^44 has as PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^45-cells objects PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^46 of PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^47, realizer PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^48, PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^49-cells given by coded homotopy equivalences PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2relevance2search_query2^^^^ in PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2relevance2id:(Berg, 2018) OR ti:\2^^^^, and PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^52-cells given by homotopies PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^53. The total space PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^54 has as PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^55-cells triples PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^56 with PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^57 and PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^58, realizer PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^59, and PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2sort_order2search_query2^^^^-cells consisting of a coded equivalence PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2sort_order2id:(Berg, 2018) OR ti:\2^^^^ together with a PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^62-cell PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^63. The projection

PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^64

is a fibration of discrete sets (&&&^^^^2search_query2^^^^&&&).

For a discrete set fibration PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^65, one may arrange PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^66, take PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^67 to be second projection, and impose PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^68. Then each fiber PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^69 is an object of PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2descending2search_query2^^^^, because between parallel PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2descending2id:(Berg, 2018) OR ti:\2^^^^-cells in PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^72 there is a unique PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^73-cell. The classifying map sends PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^74 to PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^75, sends PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^76 to transport PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^77, and uses the induced homotopy PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^78 on PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^79-cells. By construction, PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2Univalent polymorphism,2search_query2^^^^.

Univalence takes the same conceptual form as in the propositional case but at one higher truncation level. If PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2Univalent polymorphism,2id:(Berg, 2018) OR ti:\2^^^^ and PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^82 is an equivalence over PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^83, then the corresponding fiberwise equivalences assemble, using the path object PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^84 in PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^85, into a homotopy PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^86, and PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^87 is fiberwise homotopic to the transport induced by PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^88. Hence PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^89 is an equivalence. Modulo coherence, this yields a model of the Calculus of Constructions with a univalent type of sets, also containing objects like PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^9^^^^2search_query2^^^^ and PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^9^^^^2id:(Berg, 2018) OR ti:\2^^^^ (&&&^^^^2search_query2^^^^&&&).

6. Obstructions, semantic consequences, and open directions

The paper proves that univalence fails for the full class of discrete fibrations in both ambient categories. In PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^92, every fibration is a fibration of sets, so any putative universe classifying all discrete fibrations would itself be a set. But univalence would then force each small object to have, up to homotopy, a unique self-equivalence, which fails already for the discrete object PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^93, whose identity and swap give distinct self-equivalences. Therefore the class of all discrete fibrations in PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^94 does not admit a univalent representation (&&&^^^^2search_query2^^^^&&&).

In PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^95, every fibration is a fibration of groupoids. If a class had a univalent classifier with base PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^96 a groupoid, then for every small object PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^97 and self-equivalence PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^98, any two homotopies PRESERVED_PLACEHOLDER_^^^^2id:(Berg, 2018) OR ti:\2^^^^99 would have to be equal. The paper gives a discrete object with a nontrivial self-equivalence and distinct such homotopies, so a univalent classifier for all discrete fibrations cannot exist in $\mathbb{EFF}$^^^^2search_query2search_query2^^^^ either.

The distinction between the positive and negative results can be summarized as follows.

Setting	Classified class	Result
$\mathbb{EFF}$^^^^2search_query2id:(Berg, 2018) OR ti:\2^^^^	discrete propositional fibrations	impredicative and univalent
$\mathbb{EFF}$^^^^2search_query22^^^^	all discrete fibrations	no univalent representation
$\mathbb{EFF}$^^^^2search_query2^^^^3	discrete set fibrations	impredicative and univalent
$\mathbb{EFF}$^^^^2search_query2^^^^4	all discrete fibrations	no univalent representation

These results have direct semantic consequences for impredicative type theory. In $\mathbb{EFF}$^^^^2search_query2^^^^5, with small fibrations taken to be discrete propositional fibrations, one obtains, modulo the usual coherence problems, a model of an impredicative Calculus of Constructions with a univalent type of propositions. In $\mathbb{EFF}$^^^^2search_query2^^^^6, with small fibrations taken to be discrete set fibrations, one obtains, again modulo coherence, a model of an impredicative Calculus of Constructions with a univalent type of sets. The qualifier about coherence is essential: the semantics are formulated using path categories and homotopy universal properties, so computational rules are propositional rather than definitional equalities. A fully coherent model would require recasting the semantics in a categories-with-families style framework.

The broader conceptual point is that the work realizes the paradigm of a univalent object classifier inside realizability-oriented path categories, but only for carefully chosen impredicative subclasses. It also leaves open whether the class of discrete fibrations, in the local or internal sense, has a non-univalent representation in $\mathbb{EFF}$^^^^2search_query2^^^^7 or in $\mathbb{EFF}$^^^^2search_query2^^^^8. The authors further sketch a program toward path categories $\mathbb{EFF}$^^^^2search_query2^^^^9 and ultimately $\mathbb{EFF}$^^^^2id:(Berg, 2018) OR ti:\2search_query2^^^^ intended to classify broader classes, such as discrete groupoids, impredicatively with univalent classifiers (&&&^^^^2search_query2^^^^&&&).