Develop the higher homotopy theory of ruptured Kan complexes

Establish a higher-level homotopy theory for ruptured Kan complexes by constructing an appropriate model structure, defining higher obstruction invariants and characteristic classes for ruptured fibrations, formulating a ruptured ∞-topos with internal logic and type theory, and analyzing how truncation interacts with gaps to yield a ruptured n-type theory.

Background

OHTT introduces gaps at all homotopical levels, but the paper leaves the corresponding higher-categorical and homotopical foundations undeveloped.

The authors explicitly list open directions—model structures, obstruction invariants, a ruptured ∞-topos, and truncation interactions—and then state that the higher theory remains open.