I've mentioned informally the connection between modular operads and compact closed categories a few times, so it's about time I made it precise.
This is the first of two talks on the subject. This week, I'll review the definition of modular operads and introduce a category of circuit operads. These are modular operads with some extra structure and are closely related to Bar-Natan and Dansco's circuit algebras used in low dimensional topology.
Then I'll describe an monadic adjunction between modular operads and the category of small compact closed categories, and modify it to obtain an idempotent adjunction between circuit operads and small compact closed categories. Next time, I'll use an iterated distributive law to prove a nerve theorem for circuit operads.