Categories of graphs for operadic structures

TL;DR

This paper reviews various graph categories used in describing homotopy-coherent operads, introduces uniform morphism definitions, and demonstrates how certain adjunctions between operad categories can be realized at the presheaf level.

Contribution

It provides new, uniform definitions for morphisms in graph categories related to operads and shows how to realize key adjunctions at the presheaf level.

Findings

Uniform definitions for morphisms in graph categories

Comparison of categories of generalized operads

Realization of adjunctions at the presheaf level

Abstract

We recall several categories of graphs which are useful for describing homotopy-coherent versions of generalized operads (e.g. cyclic operads, modular operads, properads, and so on), and give new, uniform definitions for their morphisms. This allows for straightforward comparisons, and we use this to show that certain free-forgetful adjunctions between categories of generalized operads can be realized at the level of presheaves. This includes adjunctions between operads and cyclic operads, between dioperads and augmented cyclic operads, and betweeen wheeled properads and modular operads.