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The Leibniz adjunction in homotopy type theory, with an application to simplicial type theory

This paper demonstrates that simplicial type theory can be formulated as homotopy type theory with a postulated interval type by proving that unique fillers for $(2,1)$ -horns imply unique fillers for all inner horns via the Leibniz adjunction in the wild category of types, a result that has been formalized in Cubical Agda.

Original authors: Tom de Jong, Nicolai Kraus, Axel Ljungström

Published 2026-06-18

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CC BY 4.0

Original authors: Tom de Jong, Nicolai Kraus, Axel Ljungström

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 build a complex, multi-layered city where the roads aren't just flat lines, but have direction, traffic rules, and even "traffic jams" that can be resolved in specific ways. This paper is about building a better set of blueprints for that city, specifically for a mathematical world called Homotopy Type Theory (HoTT).

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

1. The Problem: Building a City with One-Way Streets

In standard mathematics (and standard HoTT), roads are like two-way streets. If you can go from point A to point B, you can always go back. It's like a group of friends where everyone is equally connected.

But the authors want to build a city with one-way streets (directed morphisms). In this city, you can go from A to B, but maybe not back. This is the world of Simplicial Type Theory.

However, there's a catch. In a normal city, if you have a road from A to B and another from B to C, you can easily combine them to make a road from A to C. But in this high-tech mathematical city, simply saying "we can combine them" isn't enough. You have to prove that the combination works perfectly, and that if you combine three roads in different orders, you end up at the same place.

In the "old" way of doing this (the Riehl-Shulman framework), these rules were written in a separate "meta-language" (like a rulebook written outside the city). The authors wanted to write the rules inside the city itself, using a special tool they call an Interval Type (think of it as a ruler that measures direction).

2. The Big Discovery: The "Leibniz Adjunction"

The paper's main technical achievement is proving a powerful rule they call the Leibniz Adjunction.

The Analogy: The "Push-Pull" Machine
Imagine you have two machines:

The Pushout-Product Machine (The Push): This machine takes two one-way roads and combines them to create a new, more complex road structure. It's like taking two Lego bricks and snapping them together side-by-side to make a wider base.
The Pullback-Hom Machine (The Pull): This machine does the reverse. It looks at a complex road structure and asks, "How many ways can I fit a specific smaller road inside this?" It's like asking, "How many different ways can I slide a specific puzzle piece into this larger puzzle?"

The authors proved that these two machines are perfectly linked.

If you know how the "Push" machine works, you automatically know how the "Pull" machine works.
They are two sides of the same coin.

Why is this hard?
Usually, in simple math, this link is obvious. But in this "wild" mathematical world (where roads can twist and turn in infinite ways), proving this link is like trying to tie a knot in a rope that keeps changing shape. The authors had to be incredibly careful to make sure the "knots" (the mathematical proofs) held together without falling apart.

3. The Shortcut: Switching from Maps to Families

One of the clever tricks the authors used was changing their perspective.

The Hard Way: Trying to prove the rule by looking at individual "maps" (specific roads from A to B). This is like trying to fix a traffic jam by looking at every single car individually. It gets messy and confusing very quickly.
The Easy Way: They realized that looking at "families" (groups of roads organized by a starting point) was much cleaner. It's like looking at the traffic flow of an entire neighborhood instead of individual cars.

They proved that the "Map" world and the "Family" world are actually the same thing (thanks to a rule called Univalence). By switching to the "Family" view, the messy knot-tying became much easier to solve.

4. The Result: Solving the "Composition" Puzzle

Once they had their "Push-Pull" machine working, they applied it to a specific problem: Segal Types.

The Problem:
A "Segal Type" is a city where you can combine roads (compose them). But for the city to be stable, you need to ensure that:

Combining roads works.
Combining them in different orders gives the same result (associativity).
All the higher-level "glue" holding these rules together is perfect.

In the past, mathematicians had to check these rules one by one, like checking every single brick in a wall.

The Old Result: They knew the first few layers of bricks were solid (for small shapes like triangles and squares).
The New Result: The authors used their "Push-Pull" machine to prove that if the first layer of bricks is solid, then all the layers above it are solid automatically.

They showed that if a city has a simple rule for combining two roads (a "horn" shape), it automatically has the perfect rules for combining any number of roads, no matter how complex the shape gets.

5. The "Formalization" (The Computer Proof)

Finally, the authors didn't just write this on paper. They built a digital model of their entire theory using a computer program called Cubical Agda.

Think of this as building a virtual simulation of their city.
They ran the code, and the computer checked every single step of their logic to ensure there were no bugs or loose ends.
This proves that their "Push-Pull" machine and their "all-layers-are-solid" result are mathematically 100% correct.

Summary

In short, the authors built a new, internal way to handle "one-way streets" in mathematics. They discovered a powerful "Push-Pull" relationship between combining roads and analyzing them. Using this relationship, they proved that if a mathematical structure works for simple shapes, it automatically works for all complex shapes, saving mathematicians from having to check every single possibility by hand. They verified all of this using a computer to ensure absolute precision.

Technical Summary: The Leibniz Adjunction in Homotopy Type Theory

Problem Statement
Simplicial type theory (STT) extends homotopy type theory (HoTT) to provide a synthetic framework for higher categories by equipping types with directed morphisms via an interval. A central challenge in this field is defining "Segal types"—types where directed morphisms can be composed. While composition is straightforward, ensuring that this composition is associative and unital requires an infinite hierarchy of higher coherences. In the standard formulation of STT by Riehl and Shulman, these coherences are often handled via meta-theoretic constructions, making full formalization difficult. Specifically, proving that a type with unique fillers for $(2,1)$ -horns (the basic condition for Segal types) implies unique fillers for all inner $(n,k)$ -horns (generalizing composition associativity and unitality) is non-trivial. Previous results, such as those by Riehl and Shulman, covered specific low-dimensional cases ( $n=3$ ), but a general proof for all $n$ and $k$ within a unified HoTT framework remained an open technical hurdle.

Methodology
The authors adopt a "plain" homotopy type theory approach, following Gratzer, Weinberger, and Buchholtz, rather than the meta-theoretic interval approach of Riehl and Shulman. They postulate an interval type $I$ that forms a bounded distributive lattice, avoiding the need for total orders or specific modalities.

The core technical innovation is the internalization of the Leibniz adjunction within the "wild category of types."

Wild Categories: The authors work with "wild categories," which are naive translations of category theory into HoTT. These lack higher coherence data (associativity and identity laws hold definitionally or are not tracked), making them distinct from $(\infty, 1)$ -categories but sufficient for the authors' purposes.
The Adjunction: They establish that the pushout-product (denoted $\hat{\times}$ $\hat{\times}$ ) is left adjoint to the pullback-hom (denoted $\pitchfork$ $⋔$ ) in the wild category of maps.
- For maps $f: A \to B$ and $g: X \to Y$ , the pushout-product $f \hat{\times} g$ is the induced map from the pushout of $f \times Y$ and $A \times g$ .
- The pullback-hom $f \pitchfork g$ is the induced map between function types.
- The adjunction states: $\text{Map}(f \hat{\times} g, h) \simeq \text{Map}(f, g \pitchfork h)$ .
Maps vs. Families: A direct proof of this adjunction in the wild category of maps is intractable due to the complexity of tracking coherence data (7-tuples of maps and identifications). To overcome this, the authors utilize the univalence axiom to establish an equivalence between the wild category of maps and the wild category of families (dependent types). They prove that the pushout-product and pullback-hom correspond to the fiberwise join and a specific dependent product construction in the category of families, where the adjunction is easier to verify.

Key Contributions and Results

The Leibniz Adjunction (Theorem 3.16): The paper proves that the pushout-product and pullback-hom form an adjoint pair in the wild category of types. This is a significant technical result because, while well-known for 1-categories, proving it for higher structures (even in the "wild" setting) requires careful handling of the equivalence between maps and families to avoid intractable coherence bookkeeping.
Generalization of Segal Type Properties (Theorem 4.9): Using the Leibniz adjunction, the authors prove that if a type has unique fillers for $(2,1)$ $(2, 1)$ -horns (i.e., it is a Segal type), it automatically has unique fillers for all inner $(n,k)$ $(n, k)$ -horns ( $0 < k < n$ $0 < k < n$ ).
- This generalizes Riehl and Shulman's result (which covered $n=3$ ) and provides a version of Lurie's result for simplicial type theory.
- The proof relies on showing that inner horn inclusions $\lambda^n_k$ are "inner anodyne" (left orthogonal to all Segal types). This is achieved by demonstrating that $\lambda^n_k$ is a retract of the pushout-product $\lambda^n_k \hat{\times} \lambda^2_1$ , and then applying the closure properties of orthogonality derived from the Leibniz adjunction.
Formalization: All results are formalized in Cubical Agda. The code is available and serves as a verification of the complex combinatorial arguments involving indices and pushouts.

Significance and Claims
The paper claims that its primary contribution is the successful internalization of the Leibniz adjunction within HoTT, which serves as a powerful tool for deriving higher coherences in simplicial type theory.

Synthetic Framework: By working with an internal interval type, the authors demonstrate that all higher coherences for Segal types can be stated and derived synthetically within HoTT, without resorting to meta-theoretic quantification.
Formalization Feasibility: The authors highlight that their approach allows for a complete formalization in Agda. They note that while the Riehl–Shulman meta-theoretic approach might make the statement of the theorem easier (due to set-based reasoning at the meta-level), the proof of the general case in their internal framework is non-trivial and required the development of the Leibniz adjunction to manage the "overwhelming majority of the challenge" regarding coherence tracking.
Technical Utility: The equivalence between the wild category of maps and families is presented as a crucial methodological tool that simplifies proofs involving the pushout-product and pullback-hom, making the derivation of orthogonality properties feasible.

In summary, the paper provides a rigorous, formalized foundation for simplicial type theory, proving that the basic Segal condition implies full higher categorical coherence through the application of the Leibniz adjunction in a wild categorical setting.