Segal conditions for generalized operads

TL;DR

This paper explores Segal conditions for presheaves on categories of graphs, generalizing dendroidal sets to identify various types of generalized operads like wheeled properads and modular operads.

Contribution

It introduces a framework for Segal conditions on graph-based presheaves, extending dendroidal sets to broader classes of operads and analyzing their homotopy-coherent versions.

Findings

Segal conditions characterize generalized operads on graph categories.

Adjunctions between different operad types are realized via Kan extensions.

Implications for homotopy-coherent generalized operads are discussed.

Abstract

This note is an introduction to several generalizations of the dendroidal sets of Moerdijk--Weiss. Dendroidal sets are presheaves on a category of rooted trees, and here we consider indexing categories whose objects are other kinds of graphs with loose ends. We examine the Segal condition for presheaves on these graph categories, which is one way to identify those presheaves that are a certain kind of generalized operad (for instance wheeled properad or modular operad). Several free / forgetful adjunctions between different kinds of generalized operads can be realized at the presheaf level using only the left Kan extension / restriction adjunction along a functor of graph categories. These considerations also have bearing on homotopy-coherent versions of generalized operads, and we include some questions along these lines.