Quasi Modular Operads
Michelle Strumila
TL;DR
Quasi Modular Operads develops a graphical framework for infinity modular operads by equipping graphical sets with an inner Kan condition and proving its equivalence to the Segal condition. It constructs the graphical nerve and its left adjoint (modular realisation), then proves a Nerve Theorem linking graphical sets satisfying Segal or inner Kan conditions to modular operads, thereby providing a homotopy-coherent model for higher-genus operadic structures. The approach generalizes dendroidal and asteroidal formalisms to the modular setting and, with cyclic operads treated in Appendix, establishes a foundational bridge between graphical presheaves and classical operadic notions. The results enable robust, model-categorical treatments of infinity modular operads and their cyclic variants, with potential applications to surface-based operad theories and related algebraic topology constructs.
Abstract
Modular operads are an extension of operads. In the same way that operads, as dendroidal sets, can be considered as presheaves over the category of trees, so can modular operads be considered as presheaves over a category of graphs. This paper contains a definition of the Kan condition for infinity modular operads, as well as a proof of the Nerve Theorem for modular operads, and the equivalence of the modular Kan and Segal conditions. Appendix A contains the same material for cyclic operads.