A. We define extension quasi-categories for exact quasi-categories in an analogous way to the special case of ordinary exact categories. We show that these form an Omegaspectrum, generalizing a theorem of Retakh. In so doing, we give an explicit Kan fibrant resolution of these extension quasi-categories, which relies on a factorization property. Finally, we show that the homotopy groups of extension quasi-categories are naturally isomorphic to the higher extension groups of the extriangulated category given by the homotopy category.

We introduce Reeb complexes in order to capture how generators of homology flow along sections of a real valued continuous function. This intuition suggests a close relation of Reeb complexes to established methods in topological data analysis such as levelset zigzags and persistent homology. We make this relation precise and in particular explain how Reeb complexes and levelset zigzags can be extracted from the first pages of respective spectral sequences with the same termination.

A theory of sections of simplicial height functions is developed. At the core of this theory lies the section complex, which is assembled from higher section spaces. The latter encode flow lines along the height, as well as their homotopies, in a combinatorial way. The section complex has an associated spectral sequence, which computes the homology of the height functions domain. We extract Reeb complexes from the spectral sequence. These provide a first order approximation of how homology generators flow along height levels. Our theory in particular models topological section spaces of piecewise linear functions in a completely combinatorial way.