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Categories of graphs for operadic structures

Philip Hackney

TL;DR

This work develops a uniform, category-theoretic framework for graph-based shape categories used to model homotopy-coherent generalized operads (e.g., modular, cyclic, dioperadic, and wheeled structures). By introducing a new, pair-based description of graphical maps and proving an equivalence with the classical graphical maps, it streamlines composition and active–inert factorizations. It then connects undirected and directed graph categories via orientation data, and establishes nerve theorems that relate modular operads and wheeled properads to Segal presheaves on oriented graph categories, with a precise left Kan extension formula showing Segality is preserved under forgetful functors. The results provide a coherent foundation for comparing a range of operadic theories and set the stage for future infinity-operad interpretations and homotopical refinements in a unified presheaf setting.

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.

Categories of graphs for operadic structures

TL;DR

This work develops a uniform, category-theoretic framework for graph-based shape categories used to model homotopy-coherent generalized operads (e.g., modular, cyclic, dioperadic, and wheeled structures). By introducing a new, pair-based description of graphical maps and proving an equivalence with the classical graphical maps, it streamlines composition and active–inert factorizations. It then connects undirected and directed graph categories via orientation data, and establishes nerve theorems that relate modular operads and wheeled properads to Segal presheaves on oriented graph categories, with a precise left Kan extension formula showing Segality is preserved under forgetful functors. The results provide a coherent foundation for comparing a range of operadic theories and set the stage for future infinity-operad interpretations and homotopical refinements in a unified presheaf setting.

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.

Paper Structure

This paper contains 16 sections, 56 theorems, 76 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lemmas concerning embeddings
  4. Unions of embeddings
  5. A new description of graphical maps
  6. From new graph maps to graphical maps
  7. From graphical maps to new graph maps
  8. Directed graphs and the wheeled properadic graphical category
  9. Directed graphs
  10. Wheeled properads and the oriented graphical category
  11. Hypermoment categories, algebraic patterns, Segal presheaves
  12. Left Kan extension and Segality
  13. Categories of trees for cyclic operads
  14. Acyclic graphs and the properadic graphical category
  15. Nodeless loops and the extended graphical category
  16. ...and 1 more sections

Key Result

Theorem 1

Let $G$ and $G'$ be undirected connected graphs. A graphical map $\varphi \colon G \to G'$ is the same thing as a pair consisting of an involutive function $\varphi_0\colon A_G \to A_{G'}$ and a function $\hat{\varphi} \colon \operatorname{Emb}(G) \to \operatorname{Emb}(G')$, so that this pair is co

Figures (8)

  • Figure 1: A graph with loose ends
  • Figure 2: Functors between graph categories
  • Figure 3: An embedding $\medstar_5 \rightarrowtail G$
  • Figure 4: Each of $\ell_1$, $\ell_2$, and $\mathop{\mathrm{id}}\nolimits_G$ is a union for $h$ and $k$.
  • Figure 5: The embedding $h=k \colon \medstar_2 \rightarrowtail G$ from \ref{['ex loop vertex']}
  • ...and 3 more figures

Theorems & Definitions (171)

  • Theorem 1: \ref{['old new equivalence']} and \ref{['thm extended graph cat']}
  • Theorem 1': \ref{['thm oriented wheeled equiv']} and \ref{['thm oriented wheeled equiv extended']}
  • Theorem 2: \ref{['prop restriction segal']} and \ref{['thm lke segal']}
  • Definition 2: Graphs
  • Example 2
  • Definition 2
  • Remark 2
  • Example 2: Examples of embeddings
  • Lemma 2
  • proof
  • ...and 161 more