A graphical category for higher modular operads

Abstract

We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial presheaves for a certain category of undirected graphs. This new category of undirected graphs, denoted $\mathbf{U}$, plays a similar role for modular operads that the dendroidal category $\Omega$ plays for operads. We carefully study properties of $\mathbf{U}$, including the existence of certain factorization systems. Related structures, such as cyclic operads and stable modular operads, can be similarly treated using categories derived from $\mathbf{U}$.

arXiv commentMinor expository changes throughout, including a new section on "Further directions", Remark 4.3, Example 5.4, Figures 2 and 4

References in corpus (16)

• Higher Topos Theory
• Dendroidal Sets
• Homotopy theory for algebras over polynomial monads
• On an extension of the notion of Reedy category
• Infinity Properads and Infinity Wheeled Properads
• Modular operads and Batalin-Vilkovisky geometry
• Coloured Koszul duality and strongly homotopy operads
• Operads of genus zero curves and the Grothendieck-Teichmüller group
• Dwyer--Kan homotopy theory for cyclic operads
• Modular operads and the nerve theorem
• Higher cyclic operads
• The 2-Chu-Dialectica construction and the polycategory of multivariable adjunctions
• Group actions on Segal operads
• On factorizations of graphical maps
• Brane actions, Categorification of Gromov-Witten theory and Quantum K-theory
• Functors between Reedy model categories of diagrams

Cited by in corpus (8)

• Dwyer--Kan homotopy theory for cyclic operads
• Modular operads and the nerve theorem
• Graphical combinatorics and a distributive law for modular operads
• On rectification and enrichment of infinity properads
• Categories of graphs for operadic structures
• Envelopes for Algebraic Patterns
• Segal conditions for generalized operads
• Brauer diagrams, modular operads, and a graphical nerve theorem for circuit algebras