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 Ωn Σn 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.

References

Classically, in spaces, models are known for $\Omegan \Sigman X$. It remains to be understood if such models can also be described in homotopy type theory, or something like the elementary setting of this paper.

— Path spaces of pushouts  (2402.12339 - Wärn, 2024) in Section 8 (Outlook)

Although we do not yet have a convincing construction, we can point to two partial examples in the case $F_2 = W$.

— An abelian model for the Goodwillie tower of the circle  (2609.31269 - Nervo, 25 Sep 2026) in Section 5, subsection “Simpler approximations”