Nerve Theorems for Cyclic Operads

The classical nerve theorem establishes that categories can be characterized as simplicial sets satisfying the Segal condition, providing a bridge between categorical and simplicial perspectives. Given a simplicial set $X$, it is the nerve of a category if and only if the following Segal map

$$X_n\to X_1{\times }_{X_0}\mathrm{\cdots }{\times }_{X_0}{X}_1$$

is an isomorphism for all $n\ge 2$. This characterization has proven fundamental in higher category theory and has been extended to various algebraic structures. We establish analogous nerve theorems for (augmented) cyclic multicategories and cyclic operads [2, 4], allowing one to profitably view these structures as presheaves on a category of (planar) trees. Here “cyclic” refers to a rotational symmetry on operations that blurs the distinction between inputs and outputs. Cyclic multicategories and operads play important roles in multivariable adjunctions [2], cyclic homology, graph complexes, and topological field theories, among other applications.

We employ the general frameworks for abstract nerve theorems developed by Leinster and Weber. Specifically, we apply the machinery of monads with arities of Berger, Melliès, and Weber [1], it provides a systematic method for computing minimal dense generators of categories of algebras over strongly cartesian monads. We instantiate this for monads on the category of cyclic multigraphs to establish the theorems; in the case of the free cyclic operad monad, this was anticipated in Elliott’s thesis [3].