Some constructions on ω-groupoids

Weak ω-groupoids are the higher dimensional generalisation of setoids and are an essential ingredient of the construc- tive semantics of Homotopy Type Theory [10]. Following up on our previous formalisation [3] and Brunerie’s notes [5], we present a new formalisation of the syntax of weak ω- groupoids in Agda using heterogeneous equality. We show how to recover basic constructions on ω-groupoids using sus- pension and replacement. In particular we show that any type forms a groupoid and we outline how to derive higher dimensional composition. We present a possible semantics using globular sets and discuss the issues which arise when using globular types instead