Identities among relations for higher-dimensional rewriting systems

We generalize the notion of identities among relations, well known for presentations of groups, to presentations of n-categories by polygraphs. To each polygraph, we associate a track n-category, generalizing the notion of crossed module for groups, in order to define the natural system of identities among relations. We relate the facts that this natural system is finitely generated and that the polygraph has finite derivation type.

💡 Research Summary

The paper extends the classical notion of “identities among relations”—originally formulated for group presentations—to the setting of higher‑dimensional rewriting systems, namely presentations of n‑categories by polygraphs. The authors begin by recalling that, for a group presentation ⟨X | R⟩, the relations R may satisfy higher‑order equations (identities among relations) which are captured algebraically by a crossed module. Their goal is to replace the crossed module by a genuine n‑dimensional analogue that works for any polygraph Σ presenting an n‑category.

To achieve this, they construct from a given polygraph Σ a track n‑category T(Σ). The free n‑category F(Σ) generated by Σ contains cells of dimensions 0 through n. In T(Σ) each (k − 1)‑cell is equipped with its automorphism group as an object, and each k‑cell is regarded as a morphism (an isomorphism) between such objects. Consequently T(Σ) is a 2‑category whose objects are (k − 1)‑cells, 1‑morphisms are k‑cells, and 2‑morphisms are the isomorphisms (or “tracks”) between parallel k‑cells. This structure generalises the crossed module: in dimension 2 the track 2‑category coincides with the usual crossed module, while in higher dimensions it provides a coherent system of vertical and horizontal compositions satisfying the interchange law.

Within T(Σ) the authors define a natural system of identities among relations. Roughly, for each (k − 1)‑cell a, the set of tracks from a to itself forms an abelian group (or more generally a module) that records all higher‑dimensional equations satisfied by the generating k‑cells. Collecting these groups over all (k − 1)‑cells yields a functorial object—called the identity module—that encodes all “identities among relations” present in the presentation Σ.

The central technical contribution is the proof of an equivalence between two finiteness properties:

1. Finite generation of the identity module: there exists a finite set of tracks whose vertical and horizontal composites generate all tracks in T(Σ).
2. Finite derivation type (FDT) of the polygraph: the rewriting system induced by Σ admits a finite homotopy basis, i.e., a finite set of critical branchings whose confluence resolves all ambiguities.

The authors show that if the identity module is finitely generated, then one can extract from its generators a finite homotopy basis for the rewriting system, establishing FDT. Conversely, assuming Σ has FDT, the finite homotopy basis provides a finite set of generating tracks, proving the identity module is finitely generated. The proof relies heavily on the explicit description of vertical (∘ᵥ) and horizontal (∘ₕ) compositions of tracks and on the interchange law (a∘ᵥb)∘ₕ(c∘ᵥd) = (a∘ₕc)∘ᵥ(b∘ₕd). This law guarantees that any composite of tracks can be rearranged into a normal form built from the chosen generators.

Beyond the core equivalence, the paper situates the construction within broader contexts. In dimension 2 the track n‑category reduces to the classical crossed module, confirming that the new framework truly generalises known algebraic models. In dimensions 3 and higher the structure aligns with coherent groupoids and higher‑dimensional categorical groups, offering a concrete algebraic model for homotopical data. From a computational perspective, the finiteness results imply that, for polygraphs of finite derivation type, the higher‑dimensional confluence problem is decidable and can be handled by algorithms that manipulate a finite set of generating identities.

Potential applications include:

• Higher‑dimensional rewriting: automated confluence checking, critical pair analysis, and construction of homotopy bases for rewriting systems used in proof assistants.
• Homotopy type theory: the identity module provides a syntactic counterpart to higher homotopies, suggesting a route to encode coherence conditions algebraically.
• Algebraic topology: the track n‑category can be viewed as a combinatorial model of a space’s Postnikov tower, with the identity module representing k‑invariants.

In conclusion, the paper delivers a robust algebraic apparatus that lifts the classical theory of identities among relations to arbitrary dimensions, establishes a precise correspondence between algebraic finiteness (finite generation) and rewriting finiteness (finite derivation type), and opens avenues for both theoretical exploration and practical algorithmic implementation in higher‑dimensional category theory and rewriting systems.