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The $\infty$ -category of $\infty$ -categories in simplicial type theory

This paper constructs the $\infty$ -category of $\infty$ -categories within simplicial type theory by adapting cubical type theory techniques, thereby enabling a purely type-theoretic proof of the straightening–unstraightening theorem and demonstrating new applications of the structure homomorphism principle.

Original authors: Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz

Published 2026-02-03

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CC BY 4.0

Original authors: Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Building a "Library of Libraries"

Imagine you are a librarian. You have a massive building (the Universe) filled with books. Each book represents a different kind of mathematical structure.

For a long time, mathematicians using a specific system called Simplicial Type Theory (STT) could write down rules for how to organize these books into "libraries" (which they call categories). They could prove that a specific book was a library, or that two libraries were similar.

However, there was one missing piece of furniture: The Catalog.

They could talk about individual libraries, but they couldn't build a single, giant "Library of Libraries" that contained all the libraries as its own books. In their system, if you tried to put all the libraries into one big box, the box would break or behave strangely. It was like trying to build a map that includes itself; the map gets too big to fit on the paper.

This paper solves that problem. The authors, Daniel Gratzer, Jonathan Weinberger, and Ulrik Buchholtz, have successfully built this "Library of Libraries" (which they call Cat) inside their mathematical system. They didn't just build the shelf; they proved that the shelf itself is a perfect, well-organized library.

The Tools: A New Kind of Ruler

To build this, they had to invent a new way of measuring things.

In standard math, if you have two points, A and B, the path between them is usually just a line. But in this "directed" math, paths have a direction (like a one-way street). You can go from A to B, but not necessarily back.

The authors used a special tool called a "modal operator" (think of it as a magical filter or a lens).

The Problem: When they tried to define the "Library of Libraries," the rules got messy because the "direction" of the paths got confused with the "shape" of the libraries.
The Solution: They used a special lens (called $\flat$ ) that lets them look at the "global" shape of a library without getting distracted by the tiny, wiggly paths inside it. This allowed them to define the rules for the "Library of Libraries" without the system collapsing.

The Main Achievement: The "Directed Univalence"

In standard math, there is a famous rule called Univalence. It says: "If two things are equivalent (basically the same), you can treat them as identical."

The authors discovered a "Directed Univalence" rule for their new Library of Libraries.

The Analogy: Imagine you have two different blueprints for a house. In normal math, if the blueprints result in the same house, they are the same blueprint.
The Twist: In this directed world, the "Library of Libraries" has a special rule: The space of all possible "maps" (functors) between two libraries is exactly the same as the space of all possible "directional paths" between them.

This is a huge deal because it proves that their "Library of Libraries" isn't just a random collection of items; it is a perfectly structured, self-consistent mathematical object.

The "Straightening" Trick

One of the most famous results in this field is called Straightening and Unstraightening.

The Metaphor: Imagine you have a tangled ball of yarn (a complex structure) and you want to lay it out flat on a table (a simple list of rules).
- Unstraightening: Taking a flat list of rules and wrapping it into a 3D shape.
- Straightening: Taking a 3D shape and flattening it out into a list of rules.

The authors proved that in their new "Library of Libraries," you can always do this. You can take any complex, tangled structure and prove it is exactly the same as a simple, flat list of rules, and vice versa. They did this purely using the logic of their type theory, without needing to rely on external, messy geometric models.

Why This Matters (According to the Paper)

Completing the Puzzle: This is the final missing piece for the foundations of this specific type of math. Now, they have a complete system where they can talk about categories, and even talk about the category of all categories.
New Examples: Because they have this "Library of Libraries," they can now easily build other complex structures. For example, they showed how to build "Marked Categories" (libraries where some books are highlighted) and "Monoidal Categories" (libraries that have a special way of combining books).
The Structure Identity Principle: They showed that if you define a structure using the rules of this "Library of Libraries," the system automatically knows how to handle the relationships between those structures. It's like having a blueprint that automatically knows how to build the doors and windows once you draw the walls.

Summary

Think of the authors as architects who finally built the central hub for a massive city of mathematical structures. Before, they could build houses (categories) and neighborhoods, but they couldn't build the city center that held all the neighborhoods together.

They used a special "directional lens" to solve the problem of the city center being too big to fit. Once built, they proved the city center is stable, follows all the rules of a perfect city, and allows them to easily translate between 3D shapes and 2D maps. This opens the door for them to build even more complex mathematical cities in the future.

Technical Summary: The $\infty$ -category of $\infty$ -categories in Simplicial Type Theory

Problem Statement
Simplicial Type Theory (STT), introduced by Riehl and Shulman, provides a synthetic framework for $(\infty, 1)$ -category theory by extending Homotopy Type Theory (HoTT) with a directed interval $I$ . While STT successfully defines $\infty$ -categories as types satisfying specific properties (Segal and Rezk conditions) and supports constructions like limits, colimits, and fibered categories, a foundational gap remained: the universe of types $U$ itself is not an $\infty$ -category within STT. Specifically, the subtype of types satisfying the $\infty$ -category predicate ( $\text{isCat}$ ) fails to form a category because synthetic morphisms (functions $I \to \sum_{A:U} \text{isCat}(A)$ ) do not compose or faithfully represent functors. Consequently, the "category of categories" ( $\text{Cat}$ ) was conjectured to exist as a constructible subtype of the universe but had not been realized. This absence prevented the internalization of fundamental results like the straightening–unstraightening theorem and hindered the application of directed type theory to higher algebraic structures.

Methodology
The authors construct $\text{Cat}$ by adapting techniques from cubical type theory and modal dependent type theory (MTT) to the context of Simplicial Type Theory. The methodology proceeds through several key steps:

Triangulated Type Theory ( $TT^\triangle$ ): The authors work within $TT^\triangle$ , a relaxation of STT where the directed interval $I$ is a bounded distributive lattice rather than a totally ordered set. This model (Dedekind cubical spaces) admits an "amazing right adjoint" to the exponential functor $I \to (-)$ , a feature missing in standard simplicial spaces but crucial for defining universal fibrations.
Modalities: The theory is extended with modalities, specifically the $\flat$ modality (discrete functor) and the $\sharp$ modality (codiscrete functor). These allow the internalization of global elements and the distinction between discrete groupoids and general categories.
Universal Cocartesian Family: Following Licata et al. and Weaver and Licata, the authors define $\text{Cat}$ not by a direct predicate on types, but as the base of a universal cocartesian family. They utilize Lemma 2.9 to transpose predicates defined over $I^n$ (such as being an inner fibration or having locally cocartesian lifts) into predicates over the universe $U$ .
Definition of $\text{Cat}$ : $\text{Cat}$ is defined as the subtype of $U$ consisting of types that are Rezk, inner (satisfying the relative Segal condition), and possess locally cocartesian lifts that compose.
Directed Univalence: The authors prove that $\text{Cat}$ satisfies a "directed univalence" principle, establishing an equivalence between synthetic morphisms in $\text{Cat}$ and functions between the underlying types, mediated by the $\flat$ modality.

Key Contributions and Results

Construction of $\text{Cat}$ : The primary contribution is the first construction of the $\infty$ -category of $\infty$ -categories within STT. The authors define $\text{Cat}$ as a subtype of the universe and prove it is itself a category (i.e., it is simplicial, Segal, and Rezk).
Universal Property (Theorem 4.2): The paper establishes that $\text{Cat}$ is the base of the universal cocartesian family. For any category $C$ (specifically $\flat$ -annotated), there is an equivalence:
$\langle \flat \mid C \to \text{Cat} \rangle \simeq \langle \flat \mid \sum_{E: C \to U} \text{isCocart}(E) \rangle$
This characterizes $\text{Cat}$ via the Grothendieck correspondence (straightening–unstraightening) purely type-theoretically.
Directed Univalence (Corollary 5.5): The authors prove that for $\flat$ -elements $A, B$ of $\text{Cat}$ , the type of synthetic morphisms is equivalent to the type of functions between them:
$\text{hom}_{\text{Cat}}(A, B) \simeq \langle \flat \mid A \to B \rangle$
This result distinguishes $\text{Cat}$ from the category of spaces (discrete categories), where the equivalence holds without the $\flat$ modality.
Synthetic Straightening–Unstraightening: By analyzing the fibers of the unstraightening map $U: (C \to \text{Cat}) \to \text{Cat}/C$ , the authors provide a fully synthetic proof of Lurie's straightening–unstraightening theorem. They show that the image of this map consists precisely of cocartesian families and cocartesian functors.
Structure Homomorphism Principle (SHP): The construction enables new examples of the directed version of the structure identity principle. The authors demonstrate how to construct complex categories, such as the category of monoidal categories and the category of marked categories, by combining $\text{Cat}$ with type-theoretic connectives and modalities.

Significance and Claims
The paper claims to close a major gap in the foundations of Simplicial Type Theory. By constructing $\text{Cat}$ , the authors enable:

Internalization of Higher Category Theory: Major foundational results, previously requiring external model-categorical arguments, can now be proved synthetically within the type theory.
Semantics for Directed Type Theory: The existence of $\text{Cat}$ allows for a syntactic translation of "fully directed" type theories into STT, providing a model for these theories that adequately captures $\infty$ -categories.
New Constructions: The framework facilitates the definition of complex algebraic structures (e.g., monoidal categories, $E_1$ -monoids) and the application of algebraic K-theory concepts (via Quillen's definition of K-groups) within a synthetic setting.

The authors emphasize that their approach is model-independent in its derivation (relying on axioms of $TT^\triangle$ ) while being validated by a model in cubical spaces. They view this work as a step toward realizing higher algebra and operad theory synthetically, noting that while the construction of $\text{Cat}$ is complete, further work is needed to fully explore its applications in algebraic K-theory and to develop a computational account of the theory.