Coherence without unique normal forms

Coherence theorems for covariant structures carried by a category have traditionally relied on the underlying term rewriting system of the structure being terminating and confluent. While this holds in a variety of cases, it is not a feature that is inherent to the coherence problem itself. This is demonstrated by the theory of iterated monoidal categories, which model iterated loop spaces and have a coherence theorem but fail to be confluent. We develop a framework for expressing coherence problems in terms of term rewriting systems equipped with a two dimensional congruence. Within this framework we provide general solutions to two related coherence theorems: Determining whether there is a decision procedure for the commutativity of diagrams in the resulting structure and determining sufficient conditions ensuring that ``all diagrams commute’’. The resulting coherence theorems rely on neither the termination nor the confluence of the underlying rewriting system. We apply the theory to iterated monoidal categories and obtain a new, conceptual proof of their coherence theorem.

💡 Research Summary

The paper revisits the classic coherence problem in category theory from a fresh perspective that does not rely on the usual termination and confluence assumptions of the underlying term‑rewriting system (TRS). Traditionally, coherence theorems are proved by encoding the categorical structure as a TRS, showing that the system is terminating (every rewrite sequence ends) and confluent (any two rewrite sequences from the same term can be joined). These properties guarantee a unique normal form, allowing one to argue that any two parallel morphisms reduce to the same normal form and therefore the corresponding diagram commutes. While this approach works for many familiar structures (e.g., monoidal, braided, symmetric monoidal categories), it fails for more intricate settings where the rewrite rules generate non‑terminating or non‑confluent behaviour.

A prominent counter‑example is the theory of iterated monoidal categories, which model iterated loop spaces. Although a coherence theorem is known for these categories, the associated TRS is not confluent; different rewrite paths can lead to distinct normal forms or even diverge indefinitely. Hence the classic method cannot be applied, exposing a gap in the general theory of coherence.

To bridge this gap, the authors introduce a new framework: a term‑rewriting system equipped with a two‑dimensional congruence. The basic rewrite rules remain unchanged, but a separate equivalence relation “≡” is imposed on rewrite sequences, representing higher‑dimensional cells (2‑cells) that identify different paths. In this setting, the goal is no longer to force all paths to converge to a unique normal form; instead, one seeks to ensure that any two parallel paths are related by the ≡‑congruence. This shift replaces the need for termination/confluence with the requirement that the congruence be sufficiently expressive to connect all relevant paths.

Within this framework the paper addresses two central questions:

1. Decidability of diagram commutativity.
   Given a categorical presentation together with its two‑dimensional congruence, can we algorithmically decide whether a particular diagram commutes? The authors construct a decision procedure that builds a finite “normal‑form exploration graph” where nodes are terms reachable by rewriting, and edges are either rewrite steps or ≡‑identifications. Because ≡‑identifications can collapse infinite rewrite branches, the graph may be finite even when the underlying TRS is non‑terminating. A breadth‑first search on this graph either finds a connecting ≡‑path (proving commutativity) or exhausts the graph (proving non‑commutativity). Thus, termination of the procedure is guaranteed by the finiteness induced by the congruence, not by the TRS itself.

2. Sufficient conditions for global coherence.
   When does a presentation guarantee that all diagrams commute? The authors propose a completeness condition on the two‑dimensional congruence: it must contain, for every basic generator (associator, unitors, braiders, etc.), all possible higher‑dimensional interchange squares that express the usual coherence axioms. If the congruence is “closed under interchange” and “covers every elementary critical pair,” then any pair of parallel morphisms can be linked by a finite chain of ≡‑steps. Consequently, the entire structure is coherent without any reference to normal forms.

The theoretical results are illustrated by applying the framework to iterated monoidal categories. In the classical proof, one must manage a complicated hierarchy of interchange laws and prove confluence by a delicate combinatorial argument. Using the new approach, the authors encode the same generators and relations, then add a single family of 2‑dimensional congruence rules that capture the essential interchange axioms. The decision procedure shows that any diagram built from these generators can be reduced to a common ≡‑class, yielding a concise, conceptual proof of coherence. Moreover, the method scales to other non‑confluent settings, such as double monoidal categories or structures where braiding is only partially defined.

The paper concludes with several avenues for future work. First, the automatic generation and optimization of the two‑dimensional congruence from a given presentation could be explored, potentially leading to tooling for coherence checking. Second, the framework may be integrated into type‑theoretic or programming‑language settings, where coherence conditions correspond to compiler optimizations or proof‑assistant tactics. Third, connections with homotopy‑theoretic models (e.g., ∞‑categories) suggest that the two‑dimensional congruence could be viewed as a truncation of higher homotopies, opening a path toward a unified treatment of coherence across all dimensions.

In summary, the paper reframes coherence from a “unique normal‑form” problem to a “sufficiently expressive two‑dimensional congruence” problem. By doing so, it eliminates the need for termination and confluence, provides concrete decision procedures, and offers a robust sufficient condition for global coherence. This contribution broadens the applicability of coherence theorems to a wide class of categorical structures that were previously out of reach, and it supplies both conceptual clarity and practical tools for researchers in category theory, algebraic topology, and formal methods.