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

From homotopy links to stratified homotopy theories