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.
References
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{#quinn} F Quinn. Homotopically stratified sets. J. Amer. Math. Soc. 1 (1988), 441–499.
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{#aft} David Ayala, John Francis, and Hiro Lee Tanaka. Local structures on stratified spaces. Advances in Mathematics 307 (2017): 903-1028.
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{#lurieHA} Jacob Lurie. Higher Algebra. 2012