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Stream: theory: algebraic topology

Topic: fully faithful Exit path functor

Lukas Heidemann (Jul 29 2022 at 15:52):

Suppose that $X$ and $Y$ are piecewise linear conically stratified spaces with path connected strata. Since $X$ and $Y$ are conical the simplicial sets $\textup{Exit}(X)$ and $\textup{Exit}(Y)$ (consisting of stratified maps from the stratified standard simplices) are quasicategories and a stratified map $f : X \to Y$ then induces a functor $\textup{Exit}(f) : \textup{Exit}(X) \to \textup{Exit}(Y)$. Does the assignment $f \mapsto \textup{Exit}(f)$ induce an equivalence between the space of stratified maps $X \to Y$ and the space of functors $\textup{Exit}(X) \to \textup{Exit}(Y)$?

This might be provable using Lemma 5.3. in From Homotopy Links to Stratified Homotopy Theories by expressing the stratified spaces as the realisation of the poset of simplices for some triangulation. But perhaps there is a more direct proof?