# A graphical category for higher modular operads

@article{Hackney2019AGC,
title={A graphical category for higher modular operads},
author={Philip Hackney and Marcy Robertson and Donald Yau},
journal={arXiv: Algebraic Topology},
year={2019}
}
• Published 2019
• Mathematics
• arXiv: Algebraic Topology
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 $\mathbf{U… Expand #### Figures from this paper Modular operads and the nerve theorem • Mathematics • 2020 Abstract We describe a category of undirected graphs which comes equipped with a faithful functor into the category of (colored) modular operads. 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Using this result we construct higher composition operations on (hairy) graph homology which characterize its rational homotopyExpand Twisted arrow categories, operads and Segal conditions We introduce twisted arrow categories of operads and of algebras over operads. Up to equivalence of categories, the simplex category ∆, Segal’s category Γ, Connes cyclic category Λ, Moerdijk–WeissExpand On rectification and enrichment of infinity properads • Mathematics • 2020 We develop a theory of infinity properads enriched in a general symmetric monoidal infinity category. These are defined as presheaves, satisfying a Segal condition and a Rezk completeness condition,Expand #### References SHOWING 1-10 OF 58 REFERENCES Modular Operads We develop a \higher genus" analogue of operads, which we call modular operads, in which graphs replace trees in the deenition. We study a functor F on the category of modular operads, the FeynmanExpand Higher cyclic operads • Mathematics • Algebraic & Geometric Topology • 2019 We introduce a convenient definition for weak cyclic operads, which is based on unrooted trees and Segal conditions. More specifically, we introduce a category$\Xi$of trees, which carries a tightExpand Modular operads • Mathematics, Physics • 1998 We develop a ‘higher genus’ analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the FeynmanExpand Modular operads and the nerve theorem • Mathematics • 2020 Abstract We describe a category of undirected graphs which comes equipped with a faithful functor into the category of (colored) modular operads. The associated singular functor from modular operadsExpand Dwyer–Kan homotopy theory for cyclic operads • Mathematics • Proceedings of the Edinburgh Mathematical Society • 2021 Abstract We introduce a general definition for coloured cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from coloured cyclicExpand Homotopy theory for algebras over polynomial monads • Mathematics • 2013 We study the existence and left properness of transferred model structures for "monoid-like" objects in monoidal model categories. These include genuine monoids, but also all kinds of operads as forExpand Operads of genus zero curves and the Grothendieck–Teichmüller group • Mathematics • Geometry & Topology • 2019 We show that the group of homotopy automorphisms of the profinite completion of the genus zero surface operad is isomorphic to the (profinite) Grothendieck-Teichm\"{u}ller group. Using a result ofExpand Cyclic operads and algebra of chord diagrams • Mathematics • 2000 Abstract. We prove that the algebra$ \cal A \$ of chord diagrams, the dual to the associated graded algebra of Vassiliev knot invariants, is isomorphic to the universal enveloping algebra of aExpand
∞-Categories for the Working Mathematician
homotopy theory C.1. Lifting properties, weak factorization systems, and Leibniz closure C.1.1. Lemma. Any class of maps characterized by a right lifting property is closed under composition,Expand