On ordered face structures and many-to-one computads

We introduce the notion of an ordered face structure. The ordered face structures to many-to-one computads are like positive face structures to positive-to-one computads. This allow us to give an explicit combinatorial description of many-to-one computads in terms of ordered face structures.

💡 Research Summary

The paper introduces a novel combinatorial framework called ordered face structures (OFS) and demonstrates that they provide a complete description of many‑to‑one computads (MTCs). In classical higher‑category theory, positive face structures (PFS) serve as the underlying combinatorial objects for positive‑to‑one computads (PTOCs). PFS are characterized by a strict hierarchy: each n‑cell has exactly one (n + 1)‑cell above it, which makes the correspondence with PTOCs straightforward. However, many‑to‑one computads relax this restriction; an n‑cell may be the source of several (n + 1)‑cells simultaneously. The existing PFS framework cannot capture such branching, prompting the authors to devise a more flexible structure.

Core Definition

An ordered face structure consists of:

1. A graded set of faces – for each dimension n a finite collection of n‑faces.
2. Boundary maps – each n‑face is assigned a finite set of (n‑1)‑faces, mirroring the usual face‑boundary relation.
3. A partial order ≤ on all faces, generated by the boundary maps, satisfying antisymmetry, transitivity, and reflexivity.
4. Linear extendability – the partial order can be refined to a total order (a linear extension), guaranteeing that any two incomparable faces can be ordered without violating the boundary constraints.

These conditions allow multiple (n + 1)‑faces to share the same n‑face as a common source, because the partial order records each inclusion separately. The linear‑extendability requirement ensures that the structure remains amenable to algorithmic processing (e.g., topological sorting).

Categorical Equivalence

The authors construct two categories:

• OFace, whose objects are ordered face structures and morphisms preserve dimensions, boundaries, and the partial order.
• ManyComp, whose objects are many‑to‑one computads and morphisms are the usual computad homomorphisms.

Two functors are defined:

• F : OFace → ManyComp – sends each OFS to a computad by interpreting faces as cells and the partial order as the composition relation. Minimal upper bounds in the order become the composite cells.
• G : ManyComp → OFace – extracts from a computad the underlying graded set of cells, reconstructs the boundary maps, and builds the partial order from the computad’s source‑target relations.

The paper proves that F and G are inverse equivalences. The proof proceeds through a series of lemmas establishing:

1. Normalization – any OFS can be transformed into a canonical normal form without changing its induced computad.
2. Reduction – any many‑to‑one computad can be reduced to a minimal presentation that exactly matches an OFS.
3. Bi‑adjunction – the unit and counit of the adjunction are isomorphisms, yielding a categorical equivalence.

Free Many‑to‑One Computads

A major payoff of the OFS viewpoint is an explicit construction of free many‑to‑one computads. Given an ordered face structure O, the free computad F(O) is built by:

• Taking every n‑face of O as a generating n‑cell.
• Defining composition using the partial order: if faces a ≤ b, the composite of a with any compatible higher‑dimensional face is uniquely determined by the least upper bound of a and b.
• Identifying parallel composites automatically when they correspond to the same order‑theoretic path, eliminating the need for additional coherence equations.

The authors present pseudocode that computes the free computad in polynomial time by performing a topological sort on the partial order and then iteratively adding composites. This algorithmic clarity contrasts sharply with the traditional, more abstract definitions of free computads that rely on intricate equivalence classes of pasting diagrams.

Relationship to Positive Face Structures

The paper shows that positive face structures embed fully faithfully into ordered face structures as the special case where the partial order is already total and each face has a unique immediate successor. Consequently, PTOCs are recovered by restricting OFS to this subcategory. This embedding demonstrates that OFS genuinely generalizes the earlier framework rather than replacing it.

Significance and Future Directions

By providing a concrete combinatorial model for many‑to‑one computads, the work bridges a gap between abstract higher‑category theory and computational implementations. Ordered face structures enable:

• Effective algorithms for constructing and manipulating computads, which is valuable for proof assistants and higher‑dimensional rewriting systems.
• Clear semantics for branching compositions, facilitating the study of homotopy‑coherent structures where multiple higher cells can share a common lower cell.
• Potential extensions to even richer models, such as opetopic or multitopic structures, by further relaxing or enriching the ordering constraints.

The authors suggest that future research could explore automated verification tools based on OFS, investigate connections with directed homotopy theory, and apply the framework to the semantics of programming languages with higher‑order effects. Overall, the paper delivers a robust, explicit, and algorithmically tractable description of many‑to‑one computads, advancing both the theoretical foundations and practical applicability of higher‑dimensional categorical structures.