Unpacking the combinatorics of modular operads

Abstract: Whilst operads are governed by trees, undirected graphs of arbitrary genus are needed in order to describe modular operads. And this can get complicated. Especially if we're interested in understanding notions of modular operads, such as Joyal and Kock's compact symmetric multicategories, where the combination of the contraction operation and a unital operadic composition presents particular challenges.

I'll describe how to first break the problem into its constituent parts, and then use the classical theory of distributive laws to put the pieces back together. The decomposition allows us to apply Weber's theory to get a fully faithful nerve via completely abstract methods. More interestingly, the proof method makes the combinatorics of modular operads, and especially the fiddly stuff, completely explicit. Hence it provides a roadmap for developing the theory, and the possibility for gaining new conceptual insights into the structures described.

algebraic topologycategory theoryquantum algebra

Audience: learners

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