Stratified Homotopy Theory and Generalized Simple Homotopy Theory: Foundations, Applications and Intersections

Stratified Homotopy Theory and Generalized Simple Homotopy Theory I will present some of the main results of my PhD thesis, which is concerned with stratified homotopy theory—a version of homotopy theory particularly adapted to the study of singular spaces—as well as an axiomatic, model-categorical approach to generalized simple homotopy theory. In particular, I want to present two new results in stratified homotopy theory, which parallel foundational results of classical homotopy theory: a stratified analogue of the Quillen model structure, as well as a stratified version of the classical Kan–Quillen equivalence between the homotopy theory of spaces and simplicial sets. While the former structure establishes a strong connection between stratified homotopy theory and classical geometric examples of stratified spaces, the latter entails that stratified homotopy theory can equivalently be performed in a purely combinatorial setting. My thesis explores several applications of these results, ranging from stratified topological data analysis, over a stratified version of the homotopy hypothesis relating stratified homotopy types to certain infininity-categories, all the way to stratified versions of simple homotopy theory. Time permitting, I particularly want to present one application that is closely related to the generalized simple homotopy theory part of the thesis: a stratified analogue of the classical Whitehead group and a decomposition theorem which expresses the latter in terms of ordinary Whitehead groups of strata and links.