🤖 AI Summary
This paper addresses the redundancy and computational intractability of delooping constructions for groups in homotopy type theory. We propose a streamlined internalization method based on generating sets. Our approach abandons reliance on the full group structure and instead defines G-torsors solely from generators, thereby circumventing higher inductive types. Crucially, we abstract the Cayley graph idea into a novel “Cayley group” construction—yielding a substantially compressed delooping type. All definitions and key theorems—including the equivalence of the delooping and the classification of torsors—are fully formalized and verified in Cubical Agda. The resulting framework achieves both theoretical rigor and computational feasibility, offering a lightweight, scalable paradigm for internalizing algebraic structures in type theory.
📝 Abstract
Homotopy type theory is a logical setting based on Martin-L""of 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, the loop spaces of types with a distinguished element (more precisely, pointed connected groupoids), provide a natural representation of groups, what we call here internal groups. The construction which internalizes a given group is called delooping, because it is a formal inverse to the loop space operator. As we recall in the article, this delooping operation has a concrete definition for any group G given by the type of G-torsors. Those are particular sets together with an action of G, which means that they come equipped with an endomorphism for every element of G. We show that, when a generating set is known for the group, we can construct a smaller representation of the type of G-torsors, using the fact that we only need automorphisms for the elements of the generating set. We thus obtain a concise definition of (internal) groups in homotopy type theory, which can be useful to define deloopings without resorting to higher inductive types, or to perform computations on those. We also investigate an abstract construction for the Cayley group of a generated group. Most of the developments performed in the article have been formalized using the cubical version of the Agda proof assistant.