# Variants

## Variants

There are many different definitions of stratifications on spaces:

•  stratifications from [[filtered topological space filtrations]],
•  [[poset-stratified space poset-stratifications]],
• Whitney stratifications,
• Thom-Mather stratifications,
•  [[stratifold stratifolds]],
•  [[orbifold orbifolds]],
• homotopical stratifications (as developed by Quinn),
• conically smooth stratifications (as developed by Ayala-Francis-Tanaka,
• …etc.

## Basic features

### Stratification structures

Most of the above variants share the same basic structure, captured by the following definitions.

\begin{terminology} Given a decomposition $f$ of a [[topological space]] $X$ into [[subspace#topological_subspaces|subspaces]] (denoted, in the following, by lower-case letters such as $s$), the exit path preorder $\mathsf{Exit}(f)$ of $f$ is the preorder of subspaces $s$ in $f$ with a generating arrow $s \to r$ whenever the closure of $r$ intersects $s$ non-trivially. \end{terminology}

\begin{defn} A stratification $f$ on $X$ is a decomposition of $X$ such that $\mathsf{Exit}(f)$ is a poset. \end{defn}

The subspaces in a stratification are also called strata. The opposite poset $\mathsf{Entr}(f)\op$ is also called the entrance path poset and denoted by $\mathsf{Entr}(f)$.

\begin{constr} Given a stratification $f$ on $X$, there is a map $X \to \mathrm{Exit}(f)$ mapping points $x \in s$ to $s \in \mathrm{Exit}(f)$, which is sometimes called the characteristic map of $f$, and (abusing notation) denoted by $f : X \to \mathrm{Exit}(f)$. \end{constr}

 The characteristic map need not be continuous in general (unless the stratification is locally finite, see the next remark); it is, however, continuous on the preimage of any finite [[full functor full]] subposet of the exit path poset. It is often convenient to construct stratificiations by constructing their characteristic map.

\begin{eg} (Poset stratifications). Let $X$ be a space, $P$ a poset, and $f : X \to P$ a continuous map (in other words, $f$ is a $P$-[[poset-stratified space|stratification of $X$]]). This determines a stratification $\mathrm{c}(f)$ of $X$ whose strata are the connected components of the preimages $f\inv(x)$, $x \in P$. The map $f$ factors uniquely through the characteristic map $\mathrm{c}(f) : X \to \mathsf{Exit}(\mathrm{c}(f))$ by a [[conservative functor|conservative]] poset map $\mathsf{Exit}(\mathrm{c}(f)) \to P$. (Such (characteristic,conservative)-factorizations are essentially unique.) \end{eg}

\nid The example shows that any poset-stratification determines a unique stratification. (The converse does not hold: many poset-stratifications may determine the same stratification.)

\begin{eg} (Filtered spaces). Any filtered space $X_0 \subset X_1 \subset … \subset X_n$ in which $X_i$ is a closed subspace of $X_{i+1}$ defines a continuous map $X \to [n] = (0 \to 1 \to … \to n)$ mapping points in $X_{i+1} \setminus X_{i}$ to $i$, and thus a stratification by the previous example. As a concrete instance of this example, the filtration by skeleta of any [[cell complex]] defines the “stratification by cells” in this way. \end{eg}

\begin{eg}[Trivial stratification] Every topological space $U$ is trivially a stratification with a single stratum $U$. \end{eg}

\begin{rmk} (Continuity of characteristic map). Let $(X,f)$ be a stratified space. One says that the stratification $f$ is “locally finite” if each stratum $s$ of $f$ has an open neighborhood in $X$ which only contains finitely many strata. (If $(X,f)$ satisfies the frontier condition, see next remark, then, equivalently, $(X,f)$ is locally finite iff each point $x \in X$ has an open neighborhood intersecting only finitely many strata.) If $f$ is locally finite, then the characteristic map $f : X \to \mathsf{Exit}(f)$ is a [[continuous map]]. \end{rmk}

\begin{rmk} (Openness of characteristic map). Let $(X,f)$ be a stratified space. One says the stratification $f$ satisfies the “frontier condition” (or, as an adjective, that it is “frontier-constructible”) if, for any two strata $s, r$, the closure $\overline r$ intersects $s$ non-trivially, then $s \subset \overline r$. The stratification $f$ is frontier-constructible iff the characteristic map $f : X \to \mathsf{Exit}(f)$ is an [[open map]]. \end{rmk}

\nid It is generally very reasonable to assume stratifications to be locally finite and frontier-constructible.

### The category of stratifications

\begin{defn} A stratified map $F : (X,f) \to (Y,g)$ of stratified spaces is a continuous map $F : X \to Y$ which factors through the characteristic maps $f$ and $g$ by a necessarily unique map, denoted by $\mathsf{Exit}(F) : \mathsf{Exit}(f) \to \mathsf{Exit}(g)$. \end{defn}

\nid Stratified spaces and their maps form the category $\mathbf{Strat}$ of stratification. The construction of exit path posets yields a functor $\mathsf{Exit} : \mathbf{Strat} \to \mathbf{Pos}$ (dually, using $^{\mathrm{op}} : \mathbf{Pos} \to \mathbf{Pos}). The functor has a right inverse, as follows. \begin{constr} Every poset$P$has a classifying stratification$||P||$(also called the stratified realization of$P$), whose underlying space is the classifying space$|P|$of$P$(i.e. the [[geometric realization|realization]] of the [[nerve]] of$P$), and whose characteristic map is the map$|P| \to P$that maps$|P^{\leq x}| \setminus |P^{<x}|$to$x$(here, the full subposets$P^{\leq x} = {y \leq x}$and$P^{<x} = {y < x}$of$P$are the “lower” resp. “strict lower closures” of an element$x$in$P$). Moreover, given a poset map$F : P \to Q$, the realization of its nerve yields a stratified map$||F|| : ||P|| \to ||Q||$. We obtain a functor$||-|| : \mathbf{Pos} \to \mathbf{Strat}$. \end{constr} Every classifying stratification is frontier-constructible. It makes sense to further terminologically distinguish maps of stratifications as follows. \begin{rmk} (Types of stratified maps). Let$F : (X,f) \to (Y,g)$be a stratified map. The stratified map$F$is called: \begin{itemize} \item a substratification if$F : X \to Y$is a subspace and$\mathsf{Exit}(F)$is [[conservative functor|conservative]]; if, moreover,$X = g\inv \circ \mathsf{Exit}(F) \circ f (X)$then one says the substratification is constructible; \item a coarsening if$F : X \to Y$is a homeomorphism (to emphasize the opposite process, one also calls$F$a refinement); \item a stratified homeomorphism (or stratified iso) if$F : X \to Y$is a homeomorphism of spaces and$\mathsf{Exit}(F)$is an isomorphism of posets. \end{itemize} \end{defn} ## Fundamental categories  Just as spaces have [[fundamental infinity-groupoid fundamental]]$\infty$-[[fundamental infinity-groupoid groupoids]], stratified spaces also have “[[fundamental category fundamental categories]]”. However, the role of sets for spaces is now played by posets: the following table illustrates the analogy. (The table is further explained below.) base concept$\infty$-concept presentation [[set sets]]$\equiv(0,0)$-category \infty$-sets $\eqv$ [[topological space spaces]] sets with w.e.
posets $\equiv$ $(0,1)$-categories $\infty$-posets $\eqv$ stratified spaces posets with w.e.
categories $\equiv$ $(1,1)$-categories $\infty$-[[infinity-category categories]] [[category with weak equivalences categories with w.e.]]
 In the table, an “$\infty$-X” is intuitively to be understood as an $(\infty,\infty)$-[[(infinity,infinity)-category category]] which admits a [[conservative functor]] to an X, where X can e.g. stand for “set”, “poset”, or “category”. Yet more generally, X can be an $(n,r)$-[[(n,r)-category category]] for $n,r < \infty$. A “set with weak equivalences” means a poset with weak equivalences in which each arrow is a weak equivalence. The left column is related to the middle column by an “$\infty$-zation functor” (which simply includes $1$-structures into $\infty$-structures), and the middle and right columns are related by an “$\infty$-[[category with weak equivalences#InfCatPres localization functor]]” (which should be a weak equivalence in some sense).

In order to make the above precise, one must work with sufficiiently convenient stratifications. We describe two simple ways of constructing/presenting fundamental $\infty$-posets below.

### Conical stratificiations

\begin{defn} Given stratifications $(X,f)$ and $(Y,g)$ their product is the stratification of $X \times Y$ with characteristic map $f \times g$. \end{defn}

\begin{defn} Given a stratification $(X,f)$, the stratified (open) cone $(\cone(X), \cone(f))$ stratifies the topological open cone $\cone(X) = X \times [0,1) / X \times {0}$ by the product $(X,f) \times (0,1)$ away from the cone point ${0}$ (here, the open interval $(0,1)$ is trivially stratified), and by setting the cone point ${0}$ to be its own stratum. \end{defn}

\begin{defn} A conical stratification $(X,f)$ is a stratification in which each point $x \in X$ has a neighborhood (i.e. an open substratification) that is a stratified product $U \times (\cone(Z),\cone(l))$ with $x \in U \times {0}$. \end{defn}

Every conical stratification is frontier-constructible.

\begin{constr} Given a conical stratification $(X,f)$, then [Lurie]{#LurieHA} constructs entrance path $\infty$-category $\mathcal{E}\mathrm{ntr}(f)$ as a quasicategory: the $k$-simplices of the quasicategory $\mathcal{E}\mathrm{ntr}(f)$ are precisely stratified maps $||[k]|| \to (X,f)$, where $[k] = (0 \to 1 \to … \to k)$. \end{constr}

The construction translates objects in the middle row from the first column to the second column.

### Regular stratifications

\begin{defn}[Regular stratifications] A stratification $(X,f)$ is regular if it admits a refinement $||P|| \to (X,f)$ by the stratified realization of some poset $P$. \end{defn}

\begin{constr} Given a regular stratification $(X,f)$ and a refinement $||P|| \to (X,f)$, one can construct the presented entrance path $\infty$-category $\mathcal{PE}\mathrm{ntr}(f)$ as a poset with weak equivalences with underlying poset $P$ and weak equivalences $\mathsf{Exit}(F)\inv(\id)$. \end{constr}

(Showing that this construction is, in an appropriate sense, independent of the choice of $P$ requires a bit more work…)

The construction translates objects in the middle row from the first column to the third column in the above table.

• {#quinn} F Quinn. Homotopically stratified sets. J. Amer. Math. Soc. 1 (1988), 441–499.

• {#aft} David Ayala, John Francis, and Hiro Lee Tanaka. Local structures on stratified spaces. Advances in Mathematics 307 (2017): 903-1028.

• {#lurieHA} Jacob Lurie. Higher Algebra. 2012