From homotopy links to stratified homotopy theories

TL;DR

This paper establishes an equivalence between homotopy theories of stratified spaces and stratified simplicial sets, connecting different frameworks and extending classical results to stratified contexts.

Contribution

It proves that the unmodified adjunction induces an equivalence between global homotopy theories of stratified spaces and simplicial sets, building on previous stratified homotopy link results.

Findings

The unmodified adjunction induces an equivalence between stratified space and simplicial set homotopy theories.

Classical homotopy theory of conically stratified spaces embeds fully-faithfully into the broader stratified spaces.

Homotopy links are key to connecting different stratified homotopy frameworks.

Abstract

In previous work, the first author defined homotopy theories for stratified spaces from a simplicial and a topological perspective. In both frameworks stratified weak-equivalences are detected by suitable generalizations of homotopy links. These two frameworks are connected through a stratified version of the classical adjunction between the realization and the functor of singular simplices. Using a modified version of this adjunction, the first author showed that over a fixed poset of strata the two homotopy theories were equivalent. Building on this result we now show that the unmodified adjunction induces an equivalence between the global homotopy theories of stratified spaces and of stratified simplicial sets. We do so through an in depth study of the homotopy links. As a consequence, we prove that the classical homotopy theory of conically stratified spaces embeds fully-faithfully into the homotopy theory of all stratified spaces.