# Categorified cyclic operads

## TL;DR

This paper introduces a new concept of categorified cyclic operads, establishing coherence theorems that ensure all canonical isomorphisms commute, and demonstrates applications including a generalized Feynman category.

## Contribution

It defines categorified cyclic operads with coherence results and connects them to existing structures like profunctors and Feynman categories.

## Key findings

- Proves coherence theorems for categorified cyclic operads
- Shows equivalence between entries-only and exchangeable-output styles
- Provides examples using profunctors and Feynman categories

## Abstract

In this paper, we introduce a notion of categorified cyclic operad for set-based cyclic operads with symmetries. Our categorification is obtained by relaxing defining axioms of cyclic operads to isomorphisms and by formulating coherence conditions for these isomorphisms. The coherence theorem that we prove has the form "all diagrams of canonical isomorphisms commute". Our coherence results come in two flavours, corresponding to the "entries-only" and "exchangeable-output" definitions of cyclic operads. Our proof of coherence in the entries-only style is of syntactic nature and relies on the coherence of categorified non-symmetric operads established by Do\v{s}en and Petri\'c. We obtain the coherence in the exchangeable-output style by "lifting" the equivalence between entries-only and exchangeable-output cyclic operads, set up by the second author. Finally, we show that a generalisation of the structure of profunctors of B\' enabou provides an example of categorified cyclic operad, and we exploit the coherence of categorified cyclic operads in proving that the Feynman category for cyclic operads, due to Kaufmann and Ward, admits an odd version.