Comparing operadic theories of $n$-category

We give a framework for comparing on the one hand theories of n-categories that are weakly enriched operadically, and on the other hand n-categories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (Cheng-Gurski) and examples of the latter are the definition by Batanin and variants (Leinster). We will show how to take a theory of n-categories of the former kind and produce a globular operad whose algebras are the n-categories we started with. We first provide a generalisation of Trimble’s original theory that allows for the use of other parametrising operads in a very general way, via the notion of categories weakly enriched in V where the weakness is parametrised by an operad P in the category V. We define weak n-categories by iterating the weak enrichment construction using a series of parametrising operads P_i. We then show how to construct from such a theory an n-dimensional globular operad for each $n \geq 0$ whose algebras are precisely the n-categories we constructed by iterated weak enrichment, and we show that the resulting globular operad is contractible precisely when the operads P_i are contractible. We then show how the globular operad associated with Trimble’s topological definition is related to the globular operad used by Batanin to define fundamental n-groupoids of spaces.

💡 Research Summary

The paper establishes a systematic bridge between two dominant approaches to defining higher categories: (1) operadic weak enrichment, exemplified by Trimble’s definition and its variants (e.g., Cheng‑Gurski), and (2) algebras for contractible globular operads, as used by Batanin and Leinster. The authors first generalize Trimble’s original construction by allowing an arbitrary parametrising operad P in a symmetric monoidal category V. They introduce the notion of a “P‑weakly enriched V‑category,” where the weakness of the enrichment is governed by the operad P. This abstraction subsumes Trimble’s topological enrichment (where P is the little‑interval operad) and also accommodates other choices of P, thereby covering a broad spectrum of existing models.

Iterating this weak enrichment yields a hierarchy of categories: at each level i a new parametrising operad P_i is employed, producing an n‑fold weakly enriched V‑category. The resulting objects are what the authors call “iterated weakly enriched n‑categories.” The key technical achievement is a construction that, from any such iterated weak enrichment, produces an n‑dimensional globular operad O_n whose algebras are precisely the n‑categories obtained by the enrichment process.

The construction of O_n proceeds by interpreting the operations of each P_i as globular pasting diagrams. For each i, the authors define a collection of globular cells indexed by the operations of P_i and then assemble these collections across all levels to form a single globular operad. The operadic composition in O_n mirrors the composition in the original P_i’s, ensuring that an algebra over O_n encodes exactly the same coherence data as the iterated weak enrichment.

A central theorem states that O_n is contractible if and only if every parametrising operad P_i is contractible. Contractibility here means that the operad has a unique operation up to coherent homotopy in each arity, which guarantees that the resulting higher category has “as many equivalences as possible.” The proof proceeds in two directions: (a) assuming each P_i is contractible, the authors exhibit explicit homotopies that contract the globular operad; (b) conversely, if O_n is contractible, they extract from its contraction homotopies that witness the contractibility of each P_i. This bi‑directional result shows that the homotopical flexibility of the enrichment side is exactly captured by the globular operadic side.

The paper culminates with a detailed comparison of Trimble’s topological definition and Batanin’s operadic definition of fundamental n‑groupoids. By taking P_i to be the operad of little i‑cubes (or, equivalently, the topological interval operad) at each stage, the associated globular operad O_n coincides, up to equivalence, with the contractible globular operad used by Batanin to model fundamental n‑groupoids of spaces. Consequently, the fundamental n‑groupoid obtained via Trimble’s iterated weak enrichment is exactly the same object as the one obtained from Batanin’s globular operad.

Overall, the work provides a robust, functorial translation between operadic weak enrichment and globular operad algebras. It clarifies how different “flavors” of n‑category theory are related, offers a method to generate contractible globular operads from any suitable family of parametrising operads, and demonstrates that many existing models are instances of a single unified framework. This unification not only deepens our conceptual understanding of higher categories but also opens the door to constructing new models by selecting appropriate parametrising operads tailored to specific applications in topology, algebra, or mathematical physics.