Simplicial Homotopy Type Theory (sHoTT)

• Simplicial Homotopy Type Theory is a framework integrating directed simplicial structure into HoTT to axiomatize internal ∞-categories.
• It introduces strict directed intervals and internal Segal and completeness conditions to formalize higher-dimensional compositional data.
• The theory supports diverse semantic models—including simplicial sets, topos theory, and constructive approaches—validating univalence and cocartesian structures.

Searching arXiv for papers on simplicial homotopy type theory, directed univalence, and internal higher category theory. Search query: "simplicial homotopy type theory directed univalence Rezk type" Simplicial Homotopy Type Theory, usually abbreviated sHoTT, is a type-theoretic foundation for doing $\infty$-category theory internally. It extends homotopy type theory by adding directed simplicial structure, so that one can speak not only about homotopy types and univalence, but also about directed higher-dimensional compositional data, such as composition, Segal conditions, completeness, cocartesian structure, and internal complete Segal objects. In the original Riehl–Shulman program, this internal notion of $\infty$-category is captured by Rezk types, while the motivating semantics comes from simplicial objects such as $M^{\Delta^{op}}$ and ultimately from the simplicial-set interpretation of HoTT and univalence (Rasekh, 11 Aug 2025, Pelayo et al., 2012).

1. Historical formation and foundational aim

The immediate prehistory of sHoTT lies in the simplicial-set interpretation of intensional type theory. In Voevodsky’s simplicial model, types are interpreted as Kan complexes and dependent types as Kan fibrations, and the decisive structural feature is that identity in the universe corresponds to weak equivalence of types, yielding the Univalence Axiom (Pelayo et al., 2012). This simplicial reading established the now-standard homotopical interpretation of type theory and supplied the foundational background against which later simplicial and directed extensions were formulated.

sHoTT was introduced by Riehl and Shulman as an extension of HoTT designed specifically to support an internal theory of $\infty$-categories. The motivating problem is that, unlike ordinary categories, every established definition of $\infty$-category involves an infinite hierarchy of data, and classical presentations such as quasicategories, complete Segal spaces, and simplicial categories are usually tied to a particular ambient foundation. sHoTT addresses this by making $\infty$-categorical structure available axiomatically and internally, rather than only through an external simplicial-set metatheory (Rasekh, 11 Aug 2025).

This foundational aim also has a formalization aspect. The desire is not only to describe $\infty$-categories abstractly, but to express them directly in proof assistants. The paper on sHoTT explicitly points to the proof assistant Rzk as a realization of this program, while earlier HoTT work had already emphasized the practical role of systems such as Coq for univalent mathematics (Rasekh, 11 Aug 2025, Pelayo et al., 2012).

2. Core formal apparatus

In the form emphasized by Riehl–Shulman, the key move is to add a strict directed interval $2$ with two unequal points $0_2,1_2$. From this interval one internally generates simplicial shapes—intervals, cubes, simplices, tetrahedra, and related subshapes—and then formulates internal analogues of the Segal condition and completeness condition. An internal $\infty$-category is then a Rezk type, namely the type-theoretic analogue of a complete Segal space (Rasekh, 11 Aug 2025).

A closely related internal language appears in work on generalized Chevalley criteria. There, simplicial shapes are built from a directed bi-pointed interval, and one works with extension types for inclusions $\infty$0, written

$\infty$1

which encode strict extensions of partial sections. This permits synthetic definitions of directed hom-types, cocartesian structure, and shape-indexed lifting problems entirely inside sHoTT (Weinberger, 2024).

A later line of work shows that much of this structure can be developed from a very weak interval postulate. In that setting, simplicial type theory is taken to be HoTT with a postulated interval type $\infty$2 equipped only with bounded distributive lattice structure. One then defines

$\infty$3

and the horn inclusions $\infty$4. A Segal type is defined by unique fillers for the basic horn $\infty$5, and the paper proves that unique fillers for $\infty$6-horns imply unique fillers for all inner horns $\infty$7 with $\infty$8 (Jong et al., 29 Jan 2026). This identifies a core coherence principle: directed composition determines all inner higher coherences.

3. Semantic models and model-building

The original semantic intuition behind the word “simplicial” is straightforward. Riehl–Shulman’s models are built from categories of simplicial objects of the form

$\infty$9

where the directed interval $M^{\Delta^{op}}$0 is interpreted by the walking arrow $M^{\Delta^{op}}$1. In these models, the internal Segal and completeness conditions become the usual categorical ones, and Rezk types correspond to complete Segal objects internal to $M^{\Delta^{op}}$2 (Rasekh, 11 Aug 2025).

The simplicial-set model remains a central semantic benchmark. It gives a univalent model in which types are Kan complexes and dependent types are Kan fibrations, and it also supports richer inductive structure than the minimal core. In particular, if

$M^{\Delta^{op}}$3

is a Kan fibration between Kan complexes, then the associated W-type satisfies

$M^{\Delta^{op}}$4

again as a Kan fibration. This shows that the simplicial model validates W-types as genuine fibrant types, not merely as raw presheaf objects (Berg et al., 2013).

The semantic scope of sHoTT also extends to topos-theoretic and sheaf-theoretic settings. For simplicial objects in a Grothendieck topos, internal weak homotopy equivalence coincides with Kan weak equivalence, and in simplicial presheaves the local notions agree with the standard Jardine and Dugger–Isaksen formulations: local Kan fibrations are precisely the expected local lifting maps, and local weak homotopy equivalences are the usual local weak equivalences (Low, 2014). This is significant because many type-theoretic semantics for homotopical mathematics are inherently local rather than global.

Constructive semantics has also become part of the picture. One constructive program develops internal simplicial homotopy inside a $M^{\Delta^{op}}$5-pretopos with natural numbers object, replacing unrestricted cofibrations by a decidable subclass and equipping fibrations with filler operators, with the result that the category of Kan complexes becomes a model category and universal modest fibrations are univalent candidates for HoTT semantics (Stekelenburg, 2016). Another constructive line builds higher sheaf models of univalent type theory with higher inductive types and names simplicial homotopy type theory among its intended applications, providing constructive model-building technology for higher presheaf and sheaf semantics (Coquand et al., 14 May 2026).

4. Internal $M^{\Delta^{op}}$6-category theory and universal constructions

One of the major developments inside sHoTT is the internal treatment of cocartesian and related structures. For an arbitrary shape inclusion

$M^{\Delta^{op}}$7

Weinberger defines $M^{\Delta^{op}}$8-LARI cells, $M^{\Delta^{op}}$9-LARI families, and $\infty$0-LARI functors. The generalized Chevalley criterion states that a family has enough $\infty$1-LARI lifts precisely when the Leibniz cotensor map

$\infty$2

has a left adjoint right inverse. This generalizes classical endpoint-based cocartesian criteria to arbitrary simplicial shape inclusions (Weinberger, 2024).

A second foundational advance is the construction of internal universes with nontrivial homomorphisms. In triangulated type theory, a directed-univalent universe

$\infty$3

is constructed whose points are simplicial groupoid-like types and whose directed paths are ordinary functions. The central equivalence is

$\infty$4

This gives the first complete examples of the directed structure identity principle and makes it possible to define internally categories such as finite sets, pointed spaces, monoids, partial orders, simplex-category-like objects, and presheaf categories, with internal morphisms coinciding with the expected mathematical ones (Gratzer et al., 2024).

The program then extends from a universe of spaces to a universe of categories. In a variant of STT called triangulated type theory, the internal $\infty$5-category of $\infty$6-categories,

$\infty$7

is constructed, and it classifies cocartesian fibrations via the universal property

$\infty$8

From this, the paper derives directed univalence for $\infty$9 and an internal straightening–unstraightening theorem (Gratzer et al., 2 Feb 2026). This marks a shift from sHoTT as a language for formal reasoning about arbitrary $\infty$0-categories to sHoTT as a setting where nontrivial internal examples such as the $\infty$1-category of $\infty$2-categories are themselves constructible.

5. Beyond the simplicial-object paradigm

A central recent development is that sHoTT is not exhausted by simplicial-object semantics. The original expectation was that every model of sHoTT should arise as a category of simplicial objects in some suitable ambient $\infty$3-category, because the original semantics has exactly that form. The main theorem of “Simplicial Homotopy Type Theory is not just Simplicial” refutes this expectation. For a non-principal filter $\infty$4 on $\infty$5,

$\infty$6

is a model of sHoTT, but for every $\infty$7-category $\infty$8 there is no equivalence

$\infty$9

Thus there are genuine models of sHoTT whose underlying $\infty$0-categories are not categories of simplicial objects in any $\infty$1-category (Rasekh, 11 Aug 2025).

The construction uses filter quotients and filter products. In the broader HoTT setting, filter quotients preserve the model-categorical properties needed for type formers, universes, and higher inductive structure, while often destroying external properties such as cocompleteness, local presentability, and cofibrant generation. In particular, filter products of the Kan model structure on simplicial sets remain HoTT models even though they may lack infinite colimits and cease to be locally presentable (Rasekh, 11 Aug 2025). The sHoTT paper adapts this method to model categories with $\infty$2-shapes and strict intervals, proving that filter products preserve the semantic structure needed for simplicial homotopy type theory (Rasekh, 11 Aug 2025).

The conceptual reason for nonsimpliciality is tied to internal natural numbers. In the filter-product model, an internal natural number is represented by a sequence $\infty$3 modulo eventual equality, and the set of such equivalence classes is externally uncountable. This yields “too many” internal simplices for the model to be controlled by the ordinary countable simplex category $\infty$4 (Rasekh, 11 Aug 2025). A common misconception is that this invalidates simplicial semantics; it does not. The result shows only that simplicial-object models are important but not exhaustive.

6. Adjacent frameworks, constructive variants, and unresolved directions

The internal-language side of the subject has also branched into adjacent frameworks aimed at coherence problems that ordinary HoTT and basic STT do not solve directly. Displayed Type Theory (dTT) introduces discrete and simplicial modes together with a guarded display primitive, and uses this to define a type $\infty$5 of semi-simplicial types coinductively. Its central destructors are

$\infty$6

so that a semi-simplicial type consists of a type of vertices together with, for each vertex, a displayed semi-simplicial type over it. The discrete part of $\infty$7 yields the usual infinite indexed definition of semi-simplicial types, both syntactically and semantically (Kolomatskaia et al., 2023).

Constructive work also exposes a persistent methodological issue: many classical simplicial arguments rely on excluded middle, choice, unrestricted cofibrations, or fibrant replacement, while constructive model-building often requires structured fillers rather than mere existence of fillers. The internal simplicial homotopy program in a $\infty$8-pretopos therefore restricts cofibrations, equips fibrations with filler operators, and builds homotopy theory from fibrant objects directly (Stekelenburg, 2016). This does not replace sHoTT, but it clarifies which parts of simplicial homotopy are robust under constructive reinterpretation.

The present landscape therefore contains at least three interacting strands. One strand treats sHoTT as an internal language for Rezk types and synthetic $\infty$9-category theory. A second strand develops model theory, ranging from simplicial sets and simplicial sheaves to constructive higher sheaf models (Coquand et al., 14 May 2026). A third strand studies the exact scope of the word “simplicial,” including semi-simplicial internal languages and models that are provably not simplicial-object categories (Kolomatskaia et al., 2023, Rasekh, 11 Aug 2025).

Taken together, these developments show that sHoTT is simplicial in origin, deeply tied to simplicial-set semantics and complete Segal intuition, but no longer reducible to a single simplicial-object paradigm. Its current form is a research program spanning internal syntax, categorical semantics, constructive model theory, and the comparative study of foundations for $\infty$0-categories.