Higher-dimensional categories with finite derivation type

We study convergent (terminating and confluent) presentations of n-categories. Using the notion of polygraph (or computad), we introduce the homotopical property of finite derivation type for n-categories, generalizing the one introduced by Squier for word rewriting systems. We characterize this property by using the notion of critical branching. In particular, we define sufficient conditions for an n-category to have finite derivation type. Through examples, we present several techniques based on derivations of 2-categories to study convergent presentations by 3-polygraphs.

💡 Research Summary

The paper investigates convergent presentations of n‑categories, extending the classical theory of string rewriting systems to the realm of higher‑dimensional category theory. The authors adopt polygraphs (also known as computads) as the syntactic framework for describing generators and relations in each dimension, thereby providing a uniform way to present n‑categories. A presentation is called terminating if every rewriting sequence is finite, and confluent if any two rewriting paths starting from the same (n‑1)‑cell can be joined to a common normal form. When both properties hold, the presentation is convergent, guaranteeing that the associated equivalence relation on cells is decidable and that each cell has a unique normal form.

The central notion of the paper is finite derivation type (FDT), originally introduced by Squier for word‑rewriting systems. The authors generalize FDT to n‑categories by requiring that every equality between parallel n‑cells can be witnessed by a finite set of (n+1)‑cells. In other words, the homotopy 2‑type (or higher homotopy type) of the presented n‑category is generated by finitely many higher‑dimensional rewriting steps. To detect FDT, the paper introduces the concept of critical branchings: a critical branching occurs when two rewriting rules are simultaneously applicable to overlapping parts of an (n‑1)‑cell. The authors prove that if all critical branchings are resolvable—that is, each can be completed to a commuting diagram of (n+1)‑cells—then the whole polygraph is confluent. Moreover, if the set of critical branchings is finite and each is resolvable by a finite family of (n+1)‑cells, the presented n‑category has FDT. This yields a practical, combinatorial criterion for FDT that mirrors Squier’s original theorem but works in any dimension.

A substantial part of the work is devoted to 3‑polygraphs presenting 2‑categories. Here the authors develop a technique based on derivations: they assign a weight (or measure) to each 2‑cell and define a well‑founded preorder on 2‑cells that respects the rewriting rules. This preorder provides a termination argument, while the analysis of critical branchings supplies confluence. By constructing explicit derivations, the authors can certify termination for a wide class of 2‑categories, and then use the critical branching analysis to obtain confluence, thereby establishing convergence of the 3‑polygraph. The derivation method also yields bounds on the length of rewriting sequences, which is useful for algorithmic implementations.

The paper illustrates the theory with several examples:

1. Free 2‑categories generated by a set of 1‑cells and 2‑cells. Even though the underlying syntax is unrestricted, the authors show that a carefully chosen set of rewriting rules yields a finite collection of critical branchings, each resolvable by a finite set of 3‑cells, thus giving the free 2‑category FDT.

2. Monoidal categories with a commutativity constraint. The exchange law introduces non‑trivial interactions between parallel compositions. By encoding the exchange as a rewriting rule in a 3‑polygraph, the authors identify the critical branchings arising from overlapping exchanges and resolve them using a finite family of coherence 3‑cells, demonstrating FDT for this class.

3. Higher‑dimensional cell complexes modeled by 3‑polygraphs. The authors treat a specific cell complex where 2‑cells represent faces and 3‑cells encode attaching maps. Critical branchings correspond to different ways of gluing faces, and the authors construct a finite set of higher‑dimensional coherence cells that resolve all overlaps, thereby establishing FDT.

Through these examples the authors verify that the sufficient conditions they propose—finite critical branchings and finite resolvers—are not merely abstract but can be realized in concrete categorical settings. They also discuss the relationship between these conditions and necessary conditions: any n‑category with FDT must have a finite set of generating (n+1)‑cells that resolves all critical branchings, although the converse need not hold without additional coherence constraints.

In the concluding section the authors outline future directions. They suggest the development of automated tools for detecting critical branchings and constructing resolvers, the exploration of connections between FDT and homological finiteness properties (e.g., finite projective resolutions in the associated chain complexes), and the extension of the derivation technique to even higher dimensions (4‑polygraphs and beyond). They also hint at potential applications in higher‑dimensional rewriting, homotopical algebra, and formal verification, where finite presentations of complex categorical structures are essential.

Overall, the paper provides a robust generalization of Squier’s finite derivation type to arbitrary dimensions, supplies concrete combinatorial criteria based on critical branchings, and demonstrates the practicality of these ideas through detailed 2‑category examples. It bridges rewriting theory, homotopical algebra, and higher‑dimensional category theory, offering both theoretical insight and tools for future computational implementations.