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Segal conditions for generalized operads

Philip Hackney

TL;DR

This work develops a unified framework for extending the Segal condition from categories and operads to a family of graph-based indexing categories that model generalized operads (e.g., dioperads, properads, wheeled/properads, modular operads, cyclic operads). It introduces multiple graph categories, formulates Segal presheaves on them, and shows how generalized operads arise as these Segal objects, with nerve theorems tying operadic theories to Segal presheaves. The paper also provides explicit mechanisms (left Kan extensions/restrictions) for transporting Segal structures across different graph-categorical settings and discusses how these ideas support homotopy-coherent versions and future extensions to disconnected graphs and model structures. Overall, it lays a comprehensive groundwork for homotopical and categorical treatments of a broad spectrum of operadic-like structures through Segal presheaf theory.

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.

Segal conditions for generalized operads

TL;DR

This work develops a unified framework for extending the Segal condition from categories and operads to a family of graph-based indexing categories that model generalized operads (e.g., dioperads, properads, wheeled/properads, modular operads, cyclic operads). It introduces multiple graph categories, formulates Segal presheaves on them, and shows how generalized operads arise as these Segal objects, with nerve theorems tying operadic theories to Segal presheaves. The paper also provides explicit mechanisms (left Kan extensions/restrictions) for transporting Segal structures across different graph-categorical settings and discusses how these ideas support homotopy-coherent versions and future extensions to disconnected graphs and model structures. Overall, it lays a comprehensive groundwork for homotopical and categorical treatments of a broad spectrum of operadic-like structures through Segal presheaf theory.

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.
Paper Structure (16 sections, 17 theorems, 33 equations, 11 figures)

This paper contains 16 sections, 17 theorems, 33 equations, 11 figures.

Table of Contents

  1. Introduction
  2. What is a generalized operad?
  3. Graphs
  4. Étale maps and embeddings
  5. Boundary-compatibility
  6. Categories of simply-connected graphs
  7. On several categories of graphs
  8. Properadic graphical category
  9. The Segal condition and generalized operads
  10. The general situation
  11. Nerves
  12. Interactions
  13. Proofs
  14. Full tree maps preserve intersections
  15. Tree maps and full tree maps coincide
  16. ...and 1 more sections

Key Result

Proposition 3.2

The category $\mathbf{U}_{0}$ is equivalent to the full subcategory of augmented cyclic operads $\mathbf{Cyc}^+$ on the objects $\mathsf{C}(G)$ (where $G$ ranges over all trees). This equivalence restricts to an equivalence between $\mathbf{U}_{\mathrm{cyc}}$ and the full subcategory of cyclic opera

Figures (11)

  • Figure 1: Operations $f\colon w,r,y \to p,q$ and $g\colon u,v,c,x \to a,x,r$ along with composition $f\circ_r g$ and contraction $\xi_x(g)$.
  • Figure 2: An undirected graph from \ref{['example undirected graph']}
  • Figure 3: Subgraph determined by one vertex and four edges
  • Figure 4: The stars $\medstar_5$ and $\medstar_{4,2}$
  • Figure 5: The first five linear graphs $L_0, L_1, L_2, L_3, L_4$
  • ...and 6 more figures

Theorems & Definitions (61)

  • Definition 2.1: Undirected graphs
  • Definition 2.2: Directed graphs
  • Definition 2.3
  • Example 2.4
  • Remark 2.5
  • Definition 2.6: Subgraph
  • Example 2.7: Edge
  • Example 2.8: Stars or corollas
  • Example 2.9: Linear graphs
  • Definition 2.10: paths, cycles, trees
  • ...and 51 more