Models for iterated loop–suspension Ω^n Σ^n X in homotopy type theory

Develop explicit homotopy type-theoretic (or elementary ∞-topos) models for the iterated loop–suspension Ω^n Σ^n X that parallel the classical models known in spaces, and prove their correctness and naturality within these constructive frameworks.

Background

Classically, topologists have concrete models for Ω^n Σ^n X (e.g., the Milgram and Gils constructions) that describe iterated loop spaces of suspensions in the category of spaces. The paper suggests extending such modeling to homotopy type theory or an elementary ∞-categorical setting, where expressing and manipulating higher structures and colimits can be subtle.

Achieving this would bridge classical homotopy-theoretic constructions with the type-theoretic framework, potentially enabling new formalized results and computational tools for higher homotopy operations in HoTT.