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Abstract

This thesis is concerned with the homotopy theory of stratified spaces as well as generalized simple homotopy theory. On the side of stratified homotopy theory, we establish a series of foundational results concerning several different versions of stratified homotopy theory that have been suggested in the literature. A central question in stratified homotopy theory is how the latter interacts with classical, geometrical examples of stratified spaces. With the aim of answering this question, we prove the existence of semi-model structures on the category of stratified spaces that present, respectively, the homotopy theories of stratified spaces suggested by Doteau and Henriques, Haine and Nand-Lal. Importantly, these structures are such that classical geometrical examples of stratified spaces are bifibrant, and they are furthermore strongly related to Quinn's approach to stratified homotopy theory. To prove the existence of these structures, we perform a detailed investigation of combinatorial approaches to stratified homotopy theory, develop a theory of generalized regular neighborhoods in stratified spaces, and use the latter to obtain cellular models of generalized stratified homotopy links of stratified cell complexes. Furthermore, we prove stratified analogues of the classical Kan-Quillen equivalence between simplicial sets and topological spaces. As a consequence of our investigations, we obtain a presentation of a stratified version of the homotopy hypothesis, as conjectured by Ayala, Francis and Rozenblyum: We prove that Lurie's construction of the infinity-category of Exit-paths defines a Quillen equivalence, between a semi-model category of stratified spaces and the left Bousfield localization of the Joyal model structure that presents the homotopy theory of such small infinity-categories in which every endomorphism is invertible. We apply our theoretical results in the topological data analysis of stratified spaces, proving a sampling theorem that guarantees the recovery of persistent stratified homotopy-theoretic information from large classes of two-strata Whitney stratified spaces.

On the side of generalized simple homotopy theory, we develop an axiomatic framework that allows for the investigation of the latter in the context of (semi-)model categories that are equipped with appropriate notions of generating boundary inclusions and elementary expansions. We show that the resulting theory behaves much like the classical simple homotopy theory of spaces or chain complexes, and encompasses these frameworks. We furthermore perform a detailed investigation of the simple homotopy theory of diagram categories, proving a decomposition theorem for their Whitehead groups and establishing results on the compatibility of simple equivalences with certain colimits. We then apply our general framework to some of the (semi-)model categories for stratified homotopy theory which we investigated earlier in this thesis. In particular, we prove a decomposition theorem for the resulting stratified Whitehead groups associated to Douteau's theory in terms of classical Whitehead groups of strata and generalized homotopy links.