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arXiv · 1906.01143

A graphical category for higher modular operads

Philip Hackney·Marcy Robertson·Donald Yau
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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 $\mathbf{U}$.

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Philip Hackney, Marcy Robertson, Donald Yau. 2019-06-04. A graphical category for higher modular operads. https://doi.org/10.1016/j.aim.2020.107044

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