Dwyer--Kan homotopy theory for cyclic operads

TL;DR

This paper develops a comprehensive framework for colored cyclic operads within a symmetric monoidal category, establishing model structures and adjunctions that extend existing operad theories.

Contribution

It introduces a new definition for colored cyclic operads, provides explicit formulas for adjoints, and lifts model structures from colored operads to cyclic operads.

Findings

Defined colored cyclic operads over symmetric monoidal categories.

Derived explicit formulas for adjoint functors between cyclic and ordinary operads.

Lifted the model structure from colored operads to cyclic operads in simplicial sets.

Abstract

We introduce a general definition for colored cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from colored cyclic operads to colored operads has both adjoints, each of which is relatively simple. Explicit formulae for these adjoints allow us to lift the Cisinski--Moerdijk model structure on the category of colored operads enriched in simplicial sets to the category of colored cyclic operads enriched in simplicial sets.