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A type theory for invertibility in weak $ω$ -categories

This paper introduces ICaTT, a conservative extension of the CaTT type theory that incorporates coinductive invertibility to facilitate the concise formalization of equivalences and $\omega$ -equifibrations, supported by an implementation and a semantic interpretation in marked weak $\omega$ -categories.

Original authors: Thibaut Benjamin, Camil Champin, Ioannis Markakis

Published 2026-02-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Thibaut Benjamin, Camil Champin, Ioannis Markakis

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 building a massive, multi-dimensional Lego structure. In the world of mathematics, this structure is called a weak $\omega$ -category. It's a way of organizing shapes, arrows, and connections that can go on forever in every direction.

For a long time, mathematicians had a rulebook (a "type theory" called CaTT) for building these structures. This rulebook was great at describing how pieces fit together, but it had a major blind spot: it didn't know how to handle invertibility.

In everyday life, "invertible" means you can do something and then undo it perfectly. If you turn left, you can turn right to get back. In these complex mathematical worlds, proving something is "invertible" is incredibly hard because it requires an infinite amount of proof. You have to show not just that you can undo the move, but that you can undo the undoing, and the undoing of that, forever.

This paper introduces a new, upgraded rulebook called ICaTT. Think of it as adding a special "Undo Button" feature to your Lego instruction manual.

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

1. The Problem: The Infinite "Undo" Chain

In a normal world, if you have a key (a morphism), you can check if it has a matching lock (an inverse).

Normal Category: Key fits lock. Done.
Weak $\omega$ -Category: The key fits the lock, but the fit is slightly wobbly. So, you need a shim (a 2D cell) to make it fit. But that shim is also wobbly, so you need a gasket (a 3D cell) to fix the shim. And that gasket needs a seal... and so on, forever.

Previously, the old rulebook (CaTT) couldn't easily write down "this key is invertible" because it would require writing an infinite list of instructions.

2. The Solution: The "Magic Tag" (ICaTT)

The authors created ICaTT. They added a new type of instruction called Inv.

Think of Inv as a Magic Tag you can stick on any Lego piece.
If you stick this tag on a piece, the rulebook automatically assumes: "Okay, this piece has an infinite chain of undo buttons attached to it."
The rulebook gives you tools to check the tag (destructors) and create new tags (constructors).
Crucially, they added a "Recursion" tool (rec). This is like a "Copy-Paste" button that says, "To prove the next level of undoing, just copy the proof from the level below." This allows the system to handle the infinite chain without writing it all out.

3. The "Walking Equivalence" (The Ultimate Test)

To prove their new system works, they built a specific, famous structure called the "Walking Equivalence."

Analogy: Imagine a "Walking" robot. It's a theoretical robot that just walks back and forth between two points, proving it can go there and come back.
In the old rulebook, describing this robot was messy and required a huge, complex context.
In the new ICaTT rulebook, they can describe this robot in a single, elegant line of code. It's like going from drawing a 50-page blueprint to just writing "Robot: Walk."

4. Why Does This Matter? (The "Fibrant" World)

The authors didn't just write a new rulebook; they showed it connects to the real world of mathematics.

They proved that ICaTT is a "conservative extension." This is a fancy way of saying: "We added new features, but we didn't break any of the old rules. If you could prove something before, you can still prove it now."
They showed that if you take a model built with ICaTT, you can turn it into a "Marked $\omega$ -category."
- Analogy: Imagine a map of a city. The "Marked" cells are like highlighting all the "One-Way Streets" that actually allow you to turn around.
- The new system ensures that every "highlighted" street is truly a two-way street (invertible). This helps mathematicians build a "Model Structure," which is like a universal framework for comparing different mathematical worlds.

5. The Implementation (The Proof Assistant)

The authors didn't just talk about this; they built a computer program (a proof assistant) to test it.

They used this program to re-prove several difficult mathematical theorems about invertibility.
The Result: The program did it with much less effort and fewer lines of code than before. It's like upgrading from a manual typewriter to a word processor with "Auto-Correct" and "Templates."

Summary

Think of CaTT as a basic instruction manual for building complex 3D shapes. It was great, but it couldn't easily describe shapes that could be perfectly reversed.

ICaTT is the Pro Version of that manual. It adds a "Reversibility" module that handles the infinite complexity of "undoing" automatically.

It makes describing "equivalences" (things that are the same but look different) much easier.
It allows mathematicians to build a solid foundation for the "homotopy theory" of these shapes (the study of how they can stretch and twist).
It proves that the new system is safe, consistent, and ready for the next generation of mathematical discoveries.

In short: They gave mathematicians a better language to talk about "undoing" things in infinite-dimensional spaces, making a previously impossible task manageable and elegant.

1. Problem Statement

The paper addresses the challenge of formally reasoning about invertibility in weak $\omega$ -categories.

Context: Weak $\omega$ -categories are infinite-dimensional structures where composition is associative and unital only up to higher-dimensional invertible cells. The existing type theory CaTT (Categorical Type Theory) by Finster and Mimram provides a syntactic framework for weak $\omega$ -categories but lacks internal mechanisms to assert that a specific cell is invertible.
The Gap: In higher category theory, invertibility is defined coinductively. A cell $f$ is invertible if it possesses a left inverse, a right inverse, and unitors (homotopies) relating them to identities, which themselves must be invertible, ad infinitum.
The Difficulty: Capturing this infinite data syntactically is non-trivial. Standard dependent type theories struggle with coinductive definitions that require infinite unfolding. Furthermore, while the semantics of invertibility are understood in specific models (like marked $\omega$ -categories), there was no internal type-theoretic language to construct and manipulate these structures directly without resorting to complex meta-theoretic arguments.

2. Methodology

The authors introduce ICaTT, a conservative extension of CaTT designed to internalize the notion of invertibility.

A. Syntactic Extension

ICaTT extends the raw syntax of CaTT with:

A New Type Constructor: Inv_A(t), representing the type of invertibility structures over a term $t$ of type $A$ .
Destructors: Six operations to extract the components of an invertibility structure:
- linv, rinv: Left and right inverses.
- lunit, runit: Unitors (homotopies to identity).
- ilunit, irunit: Invertibility structures for the unitors (ensuring the coinductive nature).
Constructors:
- inv: Constructs an invertibility structure from explicit data (inverses and unitors).
- can: A "canonical" constructor that internalizes a theorem stating that coherence cells (generated by coh) are invertible if their constituent variables are invertible.
- rec: A recursive constructor allowing the definition of invertibility structures by induction on the dimension, handling the infinite nature of the structure via an inductive hypothesis context.

B. Semantics and Models

The authors utilize the framework of clans (categories with display maps) to define models.

Conservativity: They prove that ICaTT is a conservative extension of CaTT. The inclusion of the syntactic categories is fully faithful.
Marked $\omega$ -Categories: To provide semantics for the new invertibility types, the authors generalize the concept of marked $\omega$ -categories (where certain cells are distinguished as "marked") to the weak case.
Fibrant Objects: They define fibrant marked $\omega$ -categories, where marked cells are precisely those that are invertible up to marking. This provides a semantic home for the Inv type.

C. Implementation

The authors implemented ICaTT as a proof assistant (built on top of an existing CaTT implementation). This tool includes:

A unification algorithm to handle implicit arguments.
Support for the new inv, rec, and can keywords.
Automated suspension operations to generate higher-dimensional analogues.

3. Key Contributions

The Theory ICaTT: The primary contribution is the definition of ICaTT, which successfully internalizes coinductive invertibility. It allows users to define invertibility structures directly within the type theory rather than as external meta-properties.
Conservative Extension Proof: The authors prove that adding invertibility does not change the provable equalities of the underlying categorical terms (categorical types and terms remain normalizing). This ensures the theory is safe to use for reasoning about standard weak $\omega$ -categories.
Formalization of the "Walking Equivalence":
- The "walking equivalence" is a universal object classifying invertible 1-cells.
- In CaTT, this object cannot be represented as a finite context because it requires infinite generators.
- In ICaTT, the authors construct the walking equivalence as a single context ( $E_1$ ) containing a variable of type Inv. They prove that the model of this context is isomorphic to the colimit of the finite truncations of the walking equivalence (the Ozornova-Rovelli model).
Characterization of Equifibrations: Using the walking equivalence in ICaTT, they recover the characterization of equifibrations (analogous to isofibrations in 1-categories) as maps having the right lifting property against specific inclusions involving the walking equivalence.
Semantics in Fibrant Marked $\omega$ -Categories: The authors construct a functor from the category of ICaTT models to fibrant marked $\omega$ -categories. This establishes that the syntactic Inv type corresponds exactly to the semantic notion of a marked, invertible cell.

4. Results and Formal Proofs

Using their implementation, the authors formalized several key properties of invertibility that were previously difficult to express in CaTT:

Closure under Composition: The composite of two invertible cells is invertible.
Invertibility of Inverses: The left and right inverses of an invertible cell are themselves invertible.
Uniqueness of Inverses: There exists an invertible cell between the left and right inverses of a given invertible cell.
2-of-6 Property: If $f \circ g$ and $g \circ h$ are invertible, then $f, g,$ and $h$ are invertible.
Propagation: If a cell $f: u \to v$ is invertible and $u$ is invertible, then $v$ is invertible.

These proofs were achieved with minimal effort compared to previous meta-theoretic approaches, demonstrating the utility of the internal language.

5. Significance

Bridging Syntax and Semantics: ICaTT provides a rigorous syntactic bridge between the abstract definition of weak $\omega$ -categories and the concrete notion of invertibility required for homotopy theory.
Model Structure Construction: The ability to define and manipulate invertible cells internally is a crucial step toward constructing a model structure on weak $\omega$ -categories. Such a structure is currently an open problem; the paper suggests that ICaTT provides the necessary tools to define weak equivalences and fibrations (equifibrations) syntactically.
Computational Utility: The implementation demonstrates that these complex coinductive structures can be handled computationally, paving the way for automated reasoning in higher category theory.
Generalization: The work generalizes the theory of marked $\omega$ -categories to the weak setting, offering a new perspective on how to handle "directed" invertibility (where only specific cells are invertible, not all).

In summary, this paper presents a foundational step in the formalization of higher category theory by introducing a type theory that natively supports the coinductive definition of invertibility, thereby enabling the internal construction of fundamental homotopical objects like the walking equivalence and equifibrations.