Higher Morita Equivalence and the Structure of Derived Loop Stacks on Stratified Manifolds

The study of loop spaces and their algebraic counterparts, loop stacks, is a cornerstone of modern geometric topology and mathematical physics. However, the classical theory is largely confined to smooth manifolds. This paper develops a comprehensive framework for understanding derived loop stacks on stratified spaces, which are ubiquitous in algebraic geometry, singularity theory, and moduli problems. Our approach leverages the powerful machinery of higher category theory, specifically the theory of (∞, n)-categories and factorization homology. We introduce a notion of stratified factorization algebras and demonstrate their central role in describing the local-to-global structure of these generalized loop stacks. The main result is a structure theorem that describes the derived loop stack LX of a stratified derived stack X as a homotopy limit over the poset of its strata. The terms in this limit decomposition involve the loop stacks of the smooth strata and newly defined ‘exit path’ categories that encode the gluing data at the singular loci. We further establish a higher Morita equivalence between the ∞-category of quasi-coherent sheaves on LX and the ∞-category of modules for a globally defined stratified factorization algebra. This equivalence provides a powerful computational tool, translating a complex geometric problem into a more tractable algebraic one. We conclude with a discussion of potential applications to topological field theories on singular spacetimes and the representation theory of Hecke-type algebras arising from stratified mapping spaces.