A monad for CSMs
Compact Symmetric Multicategories (also known as coloured modular operads) were introduced by Joyal and Kock in a 2011 note. Unfortunately their construction, though exceedingly elegant, did not account for an important difference in the character of systems governed by graphs that do not admit cycles, and those governed by graphs that may contain cycles. I will show how Joyal and Kock's original construction can be modified to account for this problem. This enables the construction of a monad for CSMs and a (Weber-style) nerve and characterisation of CSMs in terms of a Segal condition. I believe the new construction also gives some insights on the original problem. The issue that Joyal and Kock encountered is one that crops up frequently and in a range of contexts, and I am always curious to hear about new examples and possible applications of this work.
Dr Sophie Raynor , Macquarie University