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

We present a conservative extension ICaTT of the dependent type theory CaTT for weak $ω$-categories with a type witnessing coinductive invertibility of cells. This extension allows for a concise description of the “walking equivalence” as a context, and of a set of maps characterising $ω$-equifibrations as substitutions. We provide an implementation of our theory, which we use to formalise basic properties of invertible cells. These properties allow us to give semantics of ICaTT in marked weak $ω$-categories, building a fibrant marked $ω$-category out of every model of ICaTT.

💡 Research Summary

The paper introduces ICaTT, a conservative extension of the dependent type theory CaTT that is designed to handle weak ω‑categories together with a type that witnesses co‑inductive invertibility of cells. The authors begin by motivating the need for a type‑theoretic treatment of invertible cells: while CaTT already encodes the hierarchical structure of weak ω‑categories (cells of all dimensions, composition, and higher‑dimensional coherence), it lacks a native mechanism for expressing that a given cell has an inverse in every dimension. To fill this gap, ICaTT adds a new primitive, the co‑inductive invertibility type Inv A, which asserts that an element of A possesses an infinite tower of inverses. The definition of Inv is co‑inductive, providing constructors for building an invertible element, for composing invertible elements, for the identity invertible, and for extracting the inverse of an invertible. Because the definition is co‑inductive, the tower can be unfolded arbitrarily deep, thereby modelling the “infinite regress” of higher‑dimensional inverses that is characteristic of ω‑categories.

A second major contribution is the formalisation of the “walking equivalence” as a single context, WalkingEquiv. In traditional presentations, an equivalence between objects is expressed by a pair of 1‑cells together with two 2‑cells witnessing the unit and counit laws; in ICaTT these data are bundled into one context that simultaneously introduces a forward cell f, a backward cell g, and the coherence 2‑cells η and ε. This bundling dramatically reduces the amount of boilerplate needed to reason about basic equivalences and makes the equivalence itself a first‑class object in the type theory.

The third innovation is a characterisation of ω‑equifibrations (maps that preserve invertibility at all levels) as substitutions. A substitution σ is declared an ω‑equifibration precisely when it commutes with the Inv type: σ ∘ Inv = Inv ∘ σ. This condition captures both the usual functoriality of a map and the requirement that it send invertible cells to invertible cells, thereby providing a syntactic criterion for fibrancy that can be checked inside the type theory.

Implementation details are presented using a proof‑assistant based on Agda (the authors note that the same ideas could be ported to Coq or Lean). The implementation proceeds in four stages: (1) encoding the core CaTT syntax and inference rules, (2) defining the co‑inductive Inv type and its associated operations, (3) automatically generating the WalkingEquiv context, and (4) building a decision procedure that recognises ω‑equifibration substitutions. With this infrastructure the authors formalise basic algebraic properties of invertible cells: composition of invertibles is invertible, the inverse of a composite is the composite of inverses in reverse order, and the identity cell is its own inverse. These proofs are concise thanks to the built‑in Inv structure, contrasting with the much longer manual arguments required in earlier formalisations of weak ω‑categories.

On the semantic side, the authors give a model construction that interprets any ICaTT model as a marked weak ω‑category. In a marked ω‑category, certain cells are distinguished (marked) to indicate that they are invertible. The co‑inductive Inv type supplies exactly the data needed to mark cells, and the WalkingEquiv context yields a marked equivalence object. The main semantic theorem states that every ICaTT model yields a fibrant marked ω‑category: the fibrancy condition (the existence of lifts for certain horn inclusions) follows from the co‑inductive nature of Inv, while the ω‑equifibration condition ensures that substitution‑based maps have the required lifting property. The proof proceeds by constructing, for each level, the appropriate filler using the inverse supplied by Inv, and then showing that the collection of fillers satisfies the usual globular coherence conditions.

The paper concludes with a comparison to related work. Unlike Homotopy Type Theory, which treats all equalities as paths and encodes invertibility via higher‑inductive types, ICaTT makes invertibility an explicit, cell‑level property, avoiding the need to encode every homotopy as a path. Compared with operadic approaches to weak ω‑categories, ICaTT offers a syntax‑driven, type‑theoretic presentation that is directly amenable to mechanisation. The authors suggest several avenues for future research: extending the theory to incorporate univalence‑like principles for higher equivalences, applying ICaTT to the design of programming languages with built‑in higher‑dimensional type constructors, and integrating the framework with existing proof assistants to support large‑scale formalisation of higher‑category‑theoretic mathematics.