[Paper Review] A Type-Theoretical Definition of Weak {\omega}-Categories

Eric Finster, Samuel Mimram|arXiv (Cornell University)|Jun 9, 2017

Homotopy and Cohomology in Algebraic Topology9 references3 citations

TL;DR

This paper introduces a dependent type theory that provides a canonical, syntax-based characterization of weak ∞-categories by formalizing pasting schemes as well-formed contexts. It extends Brunerie's type-theoretic definition of ∞-groupoids to ∞-categories using a coherator-based approach inspired by Maltsiniotis and Grothendieck, enabling mechanizable proof checking and meta-theoretic reasoning via a type system grounded in globular sets and coherence rules.

ABSTRACT

We introduce a dependent type theory whose models are weak {\\omega}-categories, generalizing Brunerie's definition of {\\omega}-groupoids. Our type theory is based on the definition of {\\omega}-categories given by Maltsiniotis, himself inspired by Grothendieck's approach to the definition of {\\omega}-groupoids. In this setup, {\\omega}-categories are defined as presheaves preserving globular colimits over a certain category, called a coherator. The coherator encodes all operations required to be present in an {\\omega}-category: both the compositions of pasting schemes as well as their coherences. Our main contribution is to provide a canonical type-theoretical characterization of pasting schemes as contexts which can be derived from inference rules. Finally, we present an implementation of a corresponding proof system.

Motivation & Objective

• To provide a type-theoretic syntax for weak ∞-categories that captures Maltsiniotis' coherator-based definition.
• To formalize pasting schemes as well-formed contexts in dependent type theory, a key extension beyond ∞-groupoid definitions.
• To enable mechanizable proof checking for coherences in ∞-categories through a custom type system.
• To support meta-theoretic reasoning via induction on term structure, simplifying proofs in higher category theory.
• To bridge categorical definitions of ∞-categories with practical, implementable type systems.

Proposed method

• The theory is built on a type system for globular sets, with context and type formation rules defining the syntax of cells and their compositions.
• Pasting schemes are characterized as contexts satisfying specific structural constraints, ensuring they represent valid higher-dimensional compositions.
• Coherences are encoded as typed constants (coh) with dependent product types, where the context defines the source and target of the coherence.
• The system uses bi-directional type checking with normalization by evaluation, ensuring derivability of judgments like $\Gamma \vdash \mathsf{coh} : A$.
• The implementation supports explicit rules for identities, composition, unitors, associativity, and the exchange law, with strict checks on context validity.
• The system enforces side conditions on contexts (e.g., algebraic source/target) to ensure only valid pasting schemes are admitted.

Experimental results

Research questions

• RQ1 How can pasting schemes in ∞-categories be canonically represented as contexts in a dependent type theory?
• RQ2 Can a type-theoretic framework capture the full structure of weak ∞-categories using only inference rules and context formation?
• RQ3 How can coherence cells be encoded as typed constants with dependent types, ensuring correct source and target structure?
• RQ4 What is the relationship between the syntactic pasting schemes in the type theory and their semantic counterparts (e.g., Batanin trees)?
• RQ5 Can this system be implemented in a way that supports automated type checking and meta-theoretic reasoning?

Key findings

• The paper successfully defines a type theory where pasting schemes are represented as well-formed contexts, enabling a canonical syntactic characterization of ∞-categories.
• The system supports a full set of coherence rules, including identities, composition, unitors, associativity, and the exchange law, all checked via type checking.
• An implementation in Haskell demonstrates that the system supports normalization by evaluation and bi-directional type checking for coherence derivability.
• The system detects invalid constructions—such as attempting to define an inverse for a 1-cell—by enforcing side conditions on source and target contexts.
• The theory is conjectured to model Maltsiniotis’ definition of ∞-categories, with a clear path toward formal comparison via the combinatorics of the $\triangleleft$-relation.
• The framework provides a foundation for extending type theory with higher-dimensional coherences, potentially enabling future integration with $\Sigma$ and $\Pi$ types.