Types are Internal $\infty$-Groupoids

Abstract

By extending type theory with a universe of definitionally associative and unital polynomial monads, we show how to arrive at a definition of opetopic type which is able to encode a number of fully coherent algebraic structures. In particular, our approach leads to a definition of $\infty$-groupoid internal to type theory and we prove that the type of such $\infty$-groupoids is equivalent to the universe of types. That is, every type admits the structure of an $\infty$-groupoid internally, and this structure is unique.