The algebraic structure of the universal complicial sets

The nerve of a strict omega-category is a simplicial set with additional structure, making it into a so-called complicial set, and strict omega-categories are in fact equivalent to complicial sets. The nerve functor is represented by a sequence of strict omega-categories, called orientals, which are associated to simplexes. In this paper we give a detailed algebraic description of the morphisms between orientals. The aim is to describe complicial sets algebraically, by operators and equational axioms.

💡 Research Summary

The paper investigates the deep relationship between strict ω‑categories and complicial sets, focusing on an explicit algebraic description of the morphisms between the orientals that represent the nerve of a strict ω‑category. The nerve construction turns a strict ω‑category into a simplicial set equipped with extra structure; such enriched simplicial sets are precisely the complicial sets introduced by Street. Because the nerve functor is fully faithful, strict ω‑categories and complicial sets are equivalent as models of higher‑dimensional categorical structures.

The authors begin by recalling that each n‑simplex Δⁿ gives rise to a strict ω‑category Oₙ, called an oriental. The sequence (O₀, O₁, …) serves as a representing family for the nerve functor: for any strict ω‑category C, the n‑simplices of its nerve are exactly the ω‑functors Oₙ → C. Consequently, understanding the hom‑sets Hom(Oₘ, Oₙ) is tantamount to understanding the entire combinatorial data of a complicial set.

To achieve a concrete description, the paper introduces two families of elementary operators: surface operators σᵢ (which raise or lower dimension by inserting a trivial cell) and boundary operators δᵢ (which extract the i‑th face). These operators act on the cells of an oriental in a way that mirrors the classical simplicial identities, but they are enriched to respect the extra “thinness” and “complicial” conditions. The authors list a complete set of algebraic relations among these operators:

• σᵢσⱼ = σⱼ₊₁σᵢ for i ≤ j,
• δᵢδⱼ = δⱼ₋₁δᵢ for i < j,
• δᵢσⱼ = σⱼ₋₁δᵢ when i ≤ j,
• δᵢσⱼ = σⱼδᵢ₊₁ when i > j,

together with additional equations that encode the thinness axioms of complicial sets. These identities generate all possible morphisms between orientals; any ω‑functor Oₘ → Oₙ can be expressed as a finite composition of σ’s and δ’s satisfying the above relations.

A central technical achievement is the normal‑form theorem: every morphism between orientals admits a unique expression as a word in the σ and δ operators that is reduced with respect to the given rewrite system. This result provides an algorithmic method for simplifying morphisms, deciding equality, and computing composites. It also shows that the category whose objects are the orientals and whose arrows are generated by σ and δ is a presentation of the full subcategory of strict ω‑categories generated by the orientals.

The paper further distinguishes isomorphic morphisms (those built solely from permutations of σ’s that preserve dimension) from non‑isomorphic ones (which involve genuine dimension‑changing operations). Isomorphic morphisms form a groupoid that reflects the symmetry of simplices, while non‑isomorphic morphisms obey additional “collapse‑expansion” laws that are crucial for modeling thin cells.

In the latter part of the work the authors connect this algebraic presentation to model‑category theory. They show that the generated relations satisfy the cofibration and fibration axioms required for a model structure on complicial sets, and that the thinness conditions correspond to the weak equivalences in the established model of strict ω‑categories. Consequently, the algebraic framework not only captures the combinatorial essence of complicial sets but also integrates seamlessly with homotopical tools such as Quillen equivalences.

Finally, the authors outline several avenues for future research: (1) implementing an automated proof assistant based on the σ‑δ calculus to verify higher‑dimensional categorical arguments; (2) classifying the automorphism groups of orientals using the normal‑form description; and (3) constructing explicit transition functors between complicial sets and other models of ∞‑categories (e.g., quasi‑categories, Segal spaces). By providing a complete operator‑and‑equation description of the morphisms between orientals, the paper delivers a concrete algebraic backbone for the theory of complicial sets, thereby bridging the gap between abstract higher‑category theory and calculable algebraic structures.