Lecture Notes on Infinity-Properads
These are notes for three lectures on higher properads given at a program at the mathematical institute MATRIX in Australia in June 2016. The first lecture covers the case of operads, and provides a brief introduction to the Moerdijk-Weiss theory of dendroidal sets. The second lecture extends the discussion to properads and our work with Donald Yau on graphical sets. These two lectures conclude with models for higher (pr)operads given by an inner horn filling condition. Finally, in the last lecture, we explore some properties of the graphical category and use them to propose a Segal-type model for higher properads.
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These lectures were given in the inaugural workshop at the mathematical research institute MATRIX in Australia called “Higher Structures in Geometry and Physics” in June 2016; needless to say, these notes would not exist had MATRIX not supported us and allowed us to host the program in the first place. We would like to thank all the participants of the workshop for asking interesting questions and forcing us to refine these ideas, and also to Jon Beardsley, Julie Bergner, and Joachim Kock for offering feedback on earlier drafts of these notes. A special thank you goes to Gabriel C. Drummond-Cole who generously shared his liveTE Xed notes which formed the backbone of this document. We are also grateful to the Hausdorff Research Institute for Mathematics and the Max Planck Institute for Mathematics for their hospitality while we were finishing the writing and editing of these notes.
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