# Decomposition vs filtration vs stratification

Are there accepted/standard definitions of "decomposition", "filtration", and "stratification" of a topological space (or of a manifold, or of an algebraic variety) $X$?

I tend to understand "decomposition" simply as the partition of $X$ into disjoint subspaces, perhaps with some qualification (e.g. "locally closed decomposition"). I understand "filtration" as a realization of $X$ as an increasing/decreasing (perhaps required to be finite) union of closed subspaces. It's not very clear to me what a "stratification" (whithout any further specification) is supposed to be; I think in many case it could be useful to define it as a decomposition such that the collection of the unions of the "pieces" of index $\leq i$ form a filtration (by closed subspaces).

Are there contexts in which the definitions differ significantly from what I've suggested?

• I'm sure that at least casually, I have talked about the Mayer-Vietoris sequence as something obtained from a decomposition of a space. So I do not think of the pieces of a decomposition as being disjoint. – Dan Ramras Sep 7 '14 at 20:19
• One area of literature where overlapping pieces in "decompositions" is standard is metric geometric, e.g. the notion of "finite decomposition complexity". – Dan Ramras Sep 7 '14 at 20:26
• For stratifications there is the notion of a Whitney stratification (en.wikipedia.org/wiki/Whitney_conditions). – Qiaochu Yuan Sep 7 '14 at 21:53
• You should be interested in the comments of Grothendieck in section 5 of his "Esquisse d-un Programme" (1984) (matematicas.unex.es/~navarro/res/esquisseeng.pdf) which criticises the notion of topological space as inadequate for geometry, and discusses stratifications. See my preprint page pages.bangor.ac.uk/~mas010/brownpr.html for slides of a talk in June in Paris discussing the use of filtrations to define a strict cubical higher homotopy groupoid. – Ronnie Brown Sep 8 '14 at 10:17

At the risk of sounding (oxy?)moronic, I'd say that the term "stratification" is locally standard. Meaning, there exist (at least) three communities which agree internally on what the term means, but these definitions are not consistent across these communities. When you see the term "stratification of $$X$$" without further qualifications (have you ever?) it is most likely to be one of these three.

## Filtrations

In each case, one first needs to define a "filtration" of $$X$$, which the category theory types will want to define as a functor. Perhaps this is not a bad idea: the indexing category $$I$$ should be a well-ordered set (usually $$\mathbb{N}$$ or $$\mathbb{R}$$), and the target $$T(X)$$ is a suitable subcategory of the category of subsets of $$X$$ (always ordered by inclusion). Here there are many choices: if $$X$$ is a smooth manifold, maybe you'd like the target category to be smooth sub-manifolds. But maybe $$X$$ is any old topological space and you only care about the open sets, or only the closed sets, etc. So already, there are tons of choices, and even when $$X$$ is a manifold it is not clear whether a filtration of $$X$$ must only involve closed sub-manifolds.

So, a filtration of $$X$$ is a functor $$F: I \to T(X)$$ whose colimit equals $$X$$. Whenever $$i \leq j$$ we get at least an inclusion $$F(i) \hookrightarrow F(j)$$ with whatever additional structure you've decided to tack on into $$T(X)$$ (for instance, it is typical but not universal to assume that these inclusions are cofibrations). And the choice of $$T(X)$$ is not even standard when $$X$$ is a compact manifold (closed subsets, open subsets or both?). If $$X$$ is a simplicial complex, then of course things are more straightforward because sub-complexes must be closed. In general people are careful to say things like "let blah be a filtration of $$X$$ by closed smooth sub-manifolds" or some such phrase.

## Stratifications

A stratification is a filtration $$F:I \to T(X)$$ where one imposes some additional regularity on the complements $$F(j) \setminus F(i)$$ whenever $$j \geq i$$. You might ask each one to be a smooth or PL sub-manifold of $$X$$ for instance. Here are the three types of stratifications that you're most likely to encounter, in increasing order of generality and (therefore?) decreasing order of popularity.

Whitney Stratifications: If $$X$$ is a smooth variety, then typically a stratification of $$X$$ is a Whitney stratification. Here, the indexing category if $$\mathbb{N}$$ and each $$F(i+1) \setminus F(i)$$ is a smooth manifold subject to Whitney's Condition B which you can find in Qiaochu's Wikipedia link (see comment under the question).

Thom-Mather Stratifications: If $$X$$ is a space which shows up in singularity theory or intersection homology, then its stratifications are likely to be Thom-Mather. The precise definition, as you can see here, is wicked since it inducts on dimension.

Quinn Stratifications: I've only ever seen these in the surgery theory literature (eg work of Weinberger, Cappell-Shaneson etc). Weinberger's book in particular contains a nice exposition. Here's the basic idea: for each pair of topological spaces $$Z \subset Y$$, we let $$[Y,Z]$$ be the class of stratified maps from $$([0,1],\{0\})$$ to $$(Y,Z)$$. There is an obvious map $$[Y,Z] \to Z$$ given by evaluating at zero. The Quinn conditions ask that each such map $$[F(i) \cup F(j), F(i)] \to F(i)$$ is a fibration whenever $$i \leq j$$.