Lecture notes on infinity-properads

TL;DR

This paper provides lecture notes on higher properads, introducing models and theories including dendroidal and graphical sets, and establishing a Segal-type model for higher properads.

Contribution

It extends operad theory to properads using graphical sets and develops models based on inner horn filling conditions and the graphical category.

Findings

Models for higher properads via inner horn filling

Segal-type model for higher properads

Connections between dendroidal sets and graphical sets

Abstract

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 give a Segal-type model for higher properads.