A poset-stratified space is a particular way to define a stratified space that is convenient for some purposes.
A poset-stratified space is a continuous map $X\to P$, where $X$ is an arbitrary topological space and $P$ is a $T_0$ Alexandroff space, i.e. a poset with the Alexandroff topology.
The category of stratified spaces is a full subcategory of the arrow category of Top. Note that Alexandroff topologies embed Pos fully-faithfully in Top, so a map of stratified spaces consists of a continuous map $X\to Y$ and a poset map $P\to Q$ making a square commute.
The cone on a stratified space $X\to P$ is the quotient $X\times [0,1) / X\times \{0\}$, stratified by the poset $P^\lhd$ that adjoins a new bottom element to $P$.
The standard stratification of the $n$-simplex $\Delta^n$ is obtained by regarding it as the $(n+1)$-fold cone of $\emptyset$. It is stratified by the poset $[n] = \{0,\dots,n\}$. Note that every stratified map $\Delta^n \to \Delta^m$ has an underlying poset map $[n]\to [m]$; it is a theorem the resulting map $Strat(\Delta^n,\Delta^m) \to Pos([n],[m])$ is a homotopy equivalence. Thus, we have an embedding of the simplex category into $Strat$.
Any manifold $M$ can be equipped with the trivial stratification over the terminal poset. As a special case, the extended simplices $\Delta^n_e = \{ \vec{t} \in \mathbb{R}^{n+1} : \sum t_i = 1\}$ assemble into a cosimplicial object $\Delta^\bullet_e \in cStrat$.
There is a notion of a conically smooth atlas on a stratified space. (The definition is quite involved, inducting on a number of parameters simultaneously; see (Ayala–Francis–Tanaka, section 3).) The category of conically smooth stratified spaces (with conically smooth maps among them) is often again simply called $Strat$.
One can endow this new category $Strat$ with an enrichment in Kan complexes via the extended simplices, by defining $\mathrm{map}_{Strat}(X,Y)_\bullet = \mathrm{hom}_{Strat}(X \times \Delta^\bullet_e,Y)$. This presents the $\infty$-categorical localization $\mathcal{S}trat$ of $Strat$ at the stratified homotopy equivalences (see (Ayala–Francis–Rozenblyum, Theorem 2.4.5)).
For a stratified space $X$, its exit path $\infty$-category is a simplicial space (in the $\infty$-categorical sense) defined by
using the above embedding of $\Delta$ into $Strat$. It is proven in Ayala-Francis-Rozenblyum that this is indeed a complete Segal space. The main result of that paper is the stratified homotopy hypothesis, which is the assertion that this construction defines a fully-faithful embedding $Exit : \mathcal{S}trat \hookrightarrow Cat_\infty$, and that certain sheaves on $\mathcal{S}$trat, known as striation sheaves, are equivalent to $(\infty, 1)$-categories.
Jacob Lurie, Higher Algebra, Appendix A.5
David Ayala, John Francis, Hiro Lee Tanaka, Local structures on stratified spaces, arXiv
David Ayala, John Francis, Nick Rozenblyum, Factorization homology from higher categories, arXiv
David Ayala, John Francis, Nick Rozenblyum, A stratified homotopy hypothesis, arXiv
Peter J. Haine, On the homotopy theory of stratified spaces (arXiv:1811.01119)
whilst an earlier paper on exit paths is
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