Directed univalence for simplicial objects in an $\infty$-topos

Abstract: A fundamental component of homotopy type theory, a synthetic theory of $\infty$-groupoids, is Voevodsky's univalence axiom. Univalence characterizes the identity types in the universal fibration, a classifier for small type families: identity types in the universe are equivalent to types of equivalences. The directed univalence axiom plays a similar foundational role in simplicial type theory, a synthetic theory of $\infty$-categories. In its original form, which does not include universes or directed univalence, the simplicial type theory has semantics in categories of simplicial objects in an $\infty$-topos, with synthetic $\infty$-categories corresponding to internal $\infty$-categories. We verify that directed univalence holds in this semantic setting, constructing an equivalence between hom types in the universal left fibration and function types. In fact, we verify a higher version of this result, constructing an equivalence between homotopy coherent composites in the universal left fibration and composable sequences of functions between types. Using the technique of weighted limits, we reduce this theorem for simplicial objects in an arbitrary $\infty$-topos to calculations "on the left" with simplicial sets.

• The paper introduces directed univalence by establishing an internal equivalence between hom-types and composable function sequences in an ∞-topos.
• It employs model categorical and weighted limit-colimit techniques to prove functorial equivalences for simplicial objects.
• The results bridge homotopy type theory with ∞-category theory, providing synthetic foundations for internal higher categorical semantics.

Directed Univalence for Simplicial Objects in an $\infty$-Topos

Introduction

This paper establishes a strongly functorial and highly structured form of “directed univalence” in the context of simplicial objects in an $\infty$-topos. The authors generalize the univalence axiom pioneered in homotopy type theory (HoTT) to a directed context, targeting the setting of simplicial type theory—a synthetic language for $\infty$-categories. While Voevodsky’s univalence axiom identifies identities in the universe with equivalences, directed univalence characterizes morphisms (homs) in a directed universe with function types, thereby bridging the gap between higher groupoid and higher category semantics.

Background and Foundational Context

Univalence for $\infty$-groupoids (types as spaces) has been foundational in HoTT, with semantics established in model categories of simplicial sets and, more generally, in any $\infty$-topos [Kapulkin-Lumsdaine, Shulman]. The shift from traditional (undirected) univalence to a directed variant is motivated by the desire to give type-theoretic accounts of $\infty$-categories, wherein types admit additional structure via “hom-types,” and composition/coherence must be accounted for homotopically.

Simplicial type theory (RSTT), developed by Riehl and Shulman, extends HoTT to $\infty$-categories, capturing both homotopy-theoretic (groupoid) and categorical (directed) aspects. The appropriate semantic domain for this theory is the $\infty$-topos of simplicial objects, specifically those satisfying Segal and Rezk completeness conditions, supporting internal $\infty$-categories as first-class citizens.

Main Technical Contributions

Universes and Directed Univalence

The paper’s first key step is a detailed construction, in any $\infty$-topos $\infty$0, of the universal left fibration

$\infty$1

classifying “small” left fibrations (covariant universes), leveraging machinery due to Shulman for building universal classifiers in this setting. The left fibrations, being fibrations whose fibers are $\infty$2-groupoids and which are functorial with respect to the base, are precisely the covariant analog of “type families” in HoTT, but with categorical directionality.

Alongside $\infty$3, the authors construct a family of internal categories $\infty$4, each classifying composable sequences of $\infty$5 functions between objects classified by $\infty$6, using techniques of weighted limits colimits and model-theoretic presentations. The core object is a pair of pointwise equivalences—natural up to higher coherent homotopy—between the hom-types of the universal left fibration and composable sequences of functions, expressed as

$\infty$7

where $\infty$8 denotes the exponentiations of $\infty$9 by the standard cosimplicial simplex.

Model-Categorical and $\infty$0-Topos-Level Realization

At the heart of the paper are explicit, functorially well-behaved equivalences at the level of model topoi, proven first in the context of simplicial sets and then generalized via right Quillen functors to categories of simplicial objects in any fibration-extensive type-theoretic model topos (and hence any $\infty$1-topos).

The authors prove that for any $\infty$2 and context $\infty$3 (simplicial object), the functors

$\infty$4

and its inverse

$\infty$5

restrict to equivalences on left fibrations, where $\infty$6 denotes the ordinal category and the equivalence is realized at the level of right Quillen functors between covariant model structures. The homotopical properties are controlled by classical weighted limit/colimit theory and combinatorics of left anodyne extensions, leveraging the stability and extensivity of the $\infty$7-topos setting.

Naturality, Higher Coherence, and Strong Univalence

The equivalences above are proven not only at the level of objects but are shown to be part of a natural transformation of simplicial objects, up to homotopy, and satisfying higher coherence relations required for internal categorical semantics. This is essential for applications to synthetic $\infty$8-category theory, such as verifying the Segal and Rezk completeness conditions internally for the universal objects constructed.

Directed univalence is thus promoted from a mere existence statement about an equivalence of hom-spaces to a strongly natural, structurally robust equivalence, compatible with all categorical and type-theoretic operations required for further internalization and derivation of type-theoretic principles (e.g., structure identity or homomorphism principles).

Consequences, Implications, and Examples

The results immediately imply that $\infty$9 is an internal $\infty$0-category (i.e., a Segal and Rezk complete object), and the univalence equivalence provides a direct, internal characterization of arrows and higher cells in spaces of interest (e.g., magmas, internal $\infty$1-groupoids). Various standard results in the theory (e.g., the structure identity and structure homomorphism principles) are obtained as corollaries.

The approach shows that the conceptual leap from undirected to directed univalence is accompanied by an increased level of technical sophistication—particularly with regard to functoriality and handling coherence—but also leads to a more robust synthetic account of $\infty$2-category theory in the type-theoretic tradition.

Extensions and Future Directions

The authors note that their techniques dualize to the universal right fibration and, by further development, apply to cocartesian and cartesian fibrations, describing families of $\infty$3-categories varying over a base. There is promise, both by direct categorical arguments and in the style of Sattler–Wärn, of deriving analogous univalence results for these fibrations, which would have broad implications for the semantics of more general dependent type theories and synthetic higher category theory, especially in settings involving cubical/categorical objects and constructive metatheories.

There is also a close relationship highlighted with Lurie’s straightening–unstraightening equivalence and recent work on universal classifier objects for various types of (co)cartesian fibrations in higher category theory.

Numerical and Structural Highlights

• Pointwise equivalence of functors: The functors $\infty$4 and $\infty$5 are proven to be homotopical inverses for all $\infty$6, not just on 1-cells.
• Higher coherence: The constructions assemble into natural transformations of simplicial objects, with the higher homotopies required for full internal categorical semantics.
• Strong universality properties: The universal left fibration built is univalent, fibrant, and classifying, with the univalence condition upgraded in the directed case to an equivalence between homs and mapping spaces.

Contrasts and Claims

• Directed univalence fails for the “absolute” universe: It is explicitly claimed that the universe classifying all types (i.e., the universal Reedy fibration) is not directed univalent; its homs correspond to spans rather than functions and are not characterizable internally.
• Weighted limit–colimit machinery: It is boldly asserted (and realized in formal proofs) that “it's all in the weights”—the core technical work is done by the combinatorics and properties of weights for (co)ends and their interaction with the (left) anodyne/covariant model structures.
• Applicability to any $\infty$7-topos: The semantic constructions and equivalences are verified not just in bisimplicial sets, but for simplicial objects in arbitrary $\infty$8-topoi, leveraging extensive, type-theoretic model topos machinery [Shulman].

Implications and Future Prospects

This work provides a comprehensive semantic foundation for extending the univalence axiom to the directed setting, thus facilitating coherent type-theoretic treatments of $\infty$9-categories. Practically, this opens the door for foundational developments in synthetic higher category theory, internal to arbitrary $\infty$0-topoi, with direct implications for computational and logical frameworks (e.g., proof assistants, constructive mathematics) that aim to integrate higher categorical (directed homotopy) reasoning alongside standard type-theoretic tools.

Theoretically, the results lay groundwork for further generalizations—to cartesian/cocartesian fibrations and beyond—paving the way for a universal algebraic account of dependent type theory in the fully higher-category-dynamic sense, connecting semantic, model-theoretic, and proof-theoretic advances in a unified architecture.

Conclusion

This paper establishes a fully structured, highly coherent form of directed univalence for simplicial objects in any $\infty$1-topos, providing both explicit constructions and deep structural results. The approach is firmly grounded in contemporary techniques from categorical homotopy theory and model categories, and it enables a new tier of synthetic $\infty$2-category theory in the language of type theory. The results have significant implications for both the foundations of mathematics and the applied development of categorical type theories in formal and computational systems.

Reference: "Directed univalence for simplicial objects in an $\infty$3-topos" (2607.02420)