Delooping Generated Groups in Homotopy Type Theory

Homotopy type theory is a logical setting based on Martin-Löf type theory in which one can perform geometric constructions and proofs in a synthetic way. Namely, types can be interpreted as spaces (up to continuous deformation) and proofs as homotopy invariant constructions. In this context, loop spaces of pointed connected groupoids provide a natural representation of groups, and any group can be obtained as the loop space of such a type, which is then called a delooping of the group. There are two main methods to construct the delooping of an arbitrary group G. The first one consists in describing it as a pointed higher inductive type, whereas the second one consists in taking the connected component of the principal G-torsor in the type of sets equipped with an action of G. We show here that, when a presentation is known for the group, simpler variants of those constructions can be used to build deloopings. The resulting types are more amenable to computations and lead to simpler meta-theoretic reasoning. We also investigate, in this context, an abstract construction for the Cayley graph of a generated group and show that it encodes the relations of the group. Most of the developments performed in the article have been formalized using the cubical version of the Agda proof assistant.