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Hypercubical manifolds in homotopy type theory

This paper introduces a synthetic construction of the hypercubical manifold in homotopy type theory, validates it as the homotopy quotient of the 3-sphere under the quaternion group action using combinatorial techniques, and extends the framework to higher-dimensional cellular approximations converging toward a delooping of the quaternion group.

Original authors: Samuel Mimram, Émile Oleon

Published 2026-06-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Samuel Mimram, Émile Oleon

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

Imagine you are trying to describe a very strange, multi-dimensional shape to a friend who has never seen it. You have two different ways to explain it:

The "Glue" Method: You take a solid block (like a cube), cut it up, and glue the opposite faces together after twisting them.
The "Shadow" Method: You imagine a giant, perfect sphere (like a 3D ball) and spin it around in a very specific, complex pattern. If you squint and look at the "shadow" or the result of all those spins, you get the same strange shape.

This paper is about proving that these two very different ways of describing a shape called the Hypercubical Manifold are actually the same thing, but doing it inside a special kind of mathematics called Homotopy Type Theory (HoTT).

Here is a breakdown of what the authors did, using simple analogies:

1. The Two Ways to Build the Shape

The shape in question is a 3D object that mathematicians have known about since 1895.

Way A (The Cube): Imagine a standard cardboard cube. Now, imagine taking the front face and gluing it to the back face, but first, you twist it 90 degrees. You do this for all pairs of opposite faces. When you glue them all together, you get this "Hypercubical Manifold."
Way B (The Sphere): Imagine a perfect 3D sphere. There is a group of 8 special numbers (called the Quaternion group, $Q$ ) that can spin this sphere around. If you spin the sphere using all 8 moves and then "squash" the sphere down so that every point that lands on top of another point becomes a single point, you get the same Hypercubical Manifold.

2. The Problem with the New Math Language

The authors are working in Homotopy Type Theory. Think of this as a new programming language for mathematics where shapes are built out of code.

Way A is easy to code. You just tell the computer: "Make a cube, glue these sides, twist them." The computer builds it immediately.
Way B is hard to code. To tell the computer to "spin the sphere with these 8 moves," you need to define exactly how those moves work on the sphere. In this new language, defining that "spin" action directly is like trying to describe a dance move without having a body to dance with. It's very difficult to define the rules of the spin without already having the shape.

3. The "Magic Trick" (The Solution)

The authors' main achievement is showing how to bridge this gap. They didn't try to define the spin first. Instead, they did it in reverse:

Step 1: They built the shape using the easy "Glue" method (Way A) in their code.
Step 2: They asked the computer: "If we look at this shape, what is the 'shadow' it casts onto the group of 8 spins?"
Step 3: They used a clever mathematical tool (called the Flattening Lemma) to peel back the layers of their glued shape. They calculated what the "inside" of the shape looks like.
The Result: When they peeled it back, they found that the "inside" was exactly the perfect 3D sphere ( $S^3$ ).

This proved that their "Glue" shape is exactly the same as the "Sphere Spin" shape. They showed that the shape they built is indeed the result of spinning a sphere with those 8 moves.

4. Why This Matters (The "Lego" Analogy)

The authors didn't stop at just this one shape. They realized they could build bigger, more complex versions of this shape.

Imagine you have a small Lego model of a house.
The authors showed you can build a "bigger" version of this house that is a better approximation of a perfect sphere.
Then an even bigger one, and an even bigger one.

Each new version is a better "cellular approximation" of the group of 8 spins. As you keep building bigger and bigger versions, they get closer and closer to a perfect mathematical object that represents the group itself.

Summary

The paper is a success story of synthetic geometry.

The Goal: Prove that a shape built by gluing a cube is the same as a shape built by spinning a sphere.
The Challenge: The math language they used makes "spinning" very hard to define directly.
The Solution: They built the shape by gluing, then mathematically "unfolded" it to prove it contains a sphere inside.
The Bonus: They showed this trick works for building infinite families of shapes that get closer and closer to perfect mathematical ideals.

They successfully translated a complex geometric idea into a computer-verifiable proof, showing that the "glue" definition and the "spin" definition are two sides of the same coin.

Technical Summary: Hypercubical Manifolds in Homotopy Type Theory

Problem Statement
Homotopy type theory (HoTT) offers a synthetic framework for topology where types represent spaces up to homotopy equivalence. While many classical spaces can be defined as Higher Inductive Types (HITs) by specifying their cell structures (e.g., the hypercubical manifold $K$ as a CW-complex), establishing their relationship to group actions is non-trivial. Specifically, the hypercubical manifold is classically defined as the quotient of the 3-sphere $S^3$ by a free action of the quaternion group $Q$ . In HoTT, defining such an action requires a coherent map $\phi: BQ \to \mathcal{U}$ (where $BQ$ is the delooping of $Q$ ) such that the fiber is $S^3$ . However, standard elimination principles for $BQ$ only allow elimination into groupoids, whereas $S^3$ is not $n$ -truncated for any $n$ . Consequently, there is no direct way to define the action of $Q$ on $S^3$ within the theory, making it difficult to prove that the HIT definition of $K$ is equivalent to the quotient definition $S^3/Q$ .

Methodology
The authors employ a two-step strategy leveraging the duality between fibrations and group actions, and the flattening lemma for computing fibers of HITs.

Action-Fibration Duality: Instead of attempting to define the action $\phi: BQ \to \mathcal{U}$ directly, the authors define a map $\phi: K \to BQ$ from the manifold to the delooping. By the action-fibration duality (Proposition 5), a fiber sequence $F \to K \to BQ$ corresponds to a coherent action of $Q$ on $F$ where $K$ is the homotopy quotient. The goal is thus to prove that the fiber of $\phi$ is $S^3$ .
Iterative Fiber Computation: The map $\phi$ $ϕ$ is constructed using the elimination principle of $K$ $K$ (defined as a HIT with 2 points, 4 paths, 3 2-cells, and 1 3-cell). To compute the fiber of $\phi$ $ϕ$ , the authors proceed by induction on the skeleta of $K$ $K$ ( $K_0, K_1, K_2, K_3$ $K_{0}, K_{1}, K_{2}, K_{3}$ ).
- They utilize the Flattening Lemma (Lemma 3) to compute the total space of the fibration over pushouts and coequalizers.
- For each skeleton $K_k$ , they compute the fiber of the restricted map $\phi_k: K_k \to BQ$ .
- The computation reveals that the fiber of the 0-skeleton is a discrete set of 8 points (copies of $Q$ ), the fiber of the 1-skeleton is the 1-skeleton of a tesseract (homotopy equivalent to $S^3$ ), and subsequent steps attach cells to reconstruct the full 3-sphere structure.
Higher-Dimensional Generalization: The authors generalize this construction using the join operation in type theory. They define higher-dimensional manifolds $K_n$ as the iterated join of $n$ copies of $K$ over $BQ$. This construction relies on the property that the join of fiber sequences yields a new fiber sequence, and that joins increase connectivity.

Key Contributions and Results

Synthetic Definition of $K$ : The paper provides a formal definition of the hypercubical manifold $K$ as a Higher Inductive Type in HoTT.
Equivalence of Definitions: The primary result (Theorem 10) establishes a fiber sequence $S^3 \to K \to BQ$ . This proves that the HIT definition of $K$ is indeed the homotopy quotient of $S^3$ by the quaternion group $Q$ , validating the synthetic construction against the classical topological definition.
Fundamental Group: As a consequence of the fiber sequence, the authors prove that the fundamental group of $K$ is $Q$ (Proposition 12).
Universal Cover: The paper demonstrates that $S^3$ is the universal cover of $K$ (Theorem 14), recovering the Galois fibration structure synthetically.
Higher-Dimensional Approximations: The authors introduce a sequence of types $K_n$ (where $K_1 = K$ ) defined via iterated joins. They prove that $K_n$ is the quotient of $S^{4n-1}$ by $Q$ and that the canonical map $K_n \to BQ$ is $(4n-2)$ -connected (Proposition 19).
Convergence to Delooping: In the limit, the sequence converges to a type $K_\infty$ which is equivalent to $BQ$, providing a non-recursive, cellular approximation of the delooping of $Q$ that avoids formal groupoid truncation.

Significance
The paper illustrates the constructive power of homotopy type theory by resolving a subtle combinatorial problem: defining a coherent group action on a non-truncated space without a direct definition of the action. By working backwards from the quotient space to the action via the flattening lemma, the authors show that synthetic methods can recover deep topological facts about group actions and universal covers.

The construction of $K_n$ offers a practical, cellular model for computing the cohomology of $Q$ in low dimensions, as these types are expected to be cellular and amenable to synthetic computation. Furthermore, the methodology is presented as a general approach for defining types corresponding to spaces obtained by gluing faces of polyhedra (such as the Poincaré homology sphere). Finally, the work contributes to the broader program of constructing the group $SU(2)$ and its delooping in HoTT, as the action of $Q$ on $S^3$ is a specific case of the action of $SU(2)$ on itself.

1. The Two Ways to Build the Shape

2. The Problem with the New Math Language

3. The "Magic Trick" (The Solution)

4. Why This Matters (The "Lego" Analogy)

Summary

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