nLab filtered topological space

Definition

A filtered topological space $X_*$ is a filtered object in Top, hence

1. a topological space $X=X_\infty$

2. equipped with a sequence of subspaces

   $$X_*:= \quad X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n \subseteq \cdots \subseteq X_\infty.$$

A filtered space $X_*$ is called a connected filtered space if it satisfies the following condition:

$(\phi)_0$: The function $\pi_0X_0 \to \pi_0 X_r$ induced by inclusion is surjective for all $r \geq 0$; and, for all $i \geq 1$, $(\phi_i): \pi_i(X_r,X_i,v)=0$ for all $r \gt i$ and $v \in X_0$.

There are two other forms of this condition which are useful under different circumstances.

Examples

1. A CW-complex $X$ with its filtration by skeleta $X^n$.

2. A configuration space of points is filtered by number of points.

3. The free topological monoid $F X$ on a topological space $X$ filtered by the length of words. Given a pointed topological space $(X,x)$, there is also a reduced version by taking $F X$ and identifying $x$ with the identity of $F X$. This latter filtered space is known as the James construction $J(X,x)$ (James 55).

   The James construction $J(X,x)$ may be constructed homotopy-theoretically (Brunerie 13, Brunerie 17). Recall that, for $K$ a finite simplicial complex, for $(X,A)$ a pair of spaces, its polyhedral product $(X,A)^K$ is defined as the union $\bigcup_{\sigma\in S(K)}(X,A)^\sigma$ as a subspace of the Cartesian product $X^{V(K)}$. Here, for $\sigma\in S(K)$ a simplex of $K$, the subspace $(X,A)^\sigma$ consist of those $x\in X^{V(K)}$ such that, for each vertex $v$ in the complement of $\sigma$, the coordinate projection $\proj_v x$ lies in $A$. Equivalently, the polyhedral product $(X,A)^K$ can be considered as a homotopy colimit of these $(X,A)^\sigma$ over the poset $S(K)$ of simplexes $\sigma$ of $K$, where the maps are the respective inclusions.

   Definition

   For $X$ a space equipped with a basepoint $x$, define a filtered space $fil_\bullet$ as follows. Set $\fil_0$ as $\{x\}$. For $k\ge 1$, require that the following square is homotopy pushout:

   $$\array{ &&&& (X,x)^{\partial \Delta[k-1]} &&&& \\ & && inc \swarrow & & \searrow && \\ && X^k &&&& fil_{k-1} \\ & && {}_{p_k}\searrow & & \swarrow_{j_k} && \\ &&&& fil_k &&&& }$$

   where the unlabeled arrow is the homotopy colimit of a morphism of diagrams over $S(\partial \Delta[k-1])$ given by the maps

   $$(X,x)^\sigma \xrightarrow{\sim} X^{\dim(\sigma)+1} \xrightarrow{p_{\dim(\sigma)+1}} fil_{\dim(\sigma)+1}$$

   for each simplex $\sigma$ of the boundary simplicial complex $\partial \Delta[k-1]$ of the standard $(k-1)$-simplex.

   Proposition

   For $(X,x)$ a pointed space, if $(X,x)$ is path-connected, then $fil_\infty \simeq \Omega\Sigma X$.

4. A similar example to the last using free groups instead of free monoids.

5. A similar example to the last using free groupoids on topological graphs.

6. A similar example to the last using the universal topological groupoid $U_\sigma(G)$ induced from a topological groupoid $G$ by a continuous function $f: Ob(G) \to Y$ to a space $Y$.

Examples of connected filtered spaces are:

1. The skeletal filtration of a CW-complex.

2. The word length filtration of the James construction for a space with base point such that $\{x\} \to X$ is a closed cofibration.

3. The filtration $(B C)_*$ of the classifying space of a crossed complex, filtered using skeleta of $C$.

This condition occurs in the higher homotopy van Kampen theorem for crossed complexes.

Properties

We thus see that filtered spaces arise from many geometric and algebraic situations, and see also stratified spaces). It is therefore interesting that one can define strict higher homotopy groupoids for filtered spaces more easily than for spaces themselves.

Note also that it is standard to be able to replace, using mapping cylinders, a sequence of maps $Y_n \to Y_{n+1}$ by a sequence of inclusions.

Related concepts

• stratified space

filtered objects

• filtered topological space

  • spectral sequence of a tower of fibrations

• filtered vector space

• filtered chain complex

  • spectral sequence of a filtered complex

  • Hodge filtration

• filtered ring

• filtered object in an (∞,1)-category

• filtered stable homotopy type

  • spectral sequence of a filtered stable homotopy type

• filtered probability space

associated graded objects

• associated graded vector space

• associated graded ring

References

The James construction is due to

• Ioan James, Reduced product spaces, Annals of Mathematics, Second Series, 62: 170–197 (1955)

Review of the James construction includes

• Dylan Wilson, James construction, 2017 (pdf)

• Wikipedia, James reduced product

Discussion of the James construction via homotopy type theory includes the following

• Guillaume Brunerie, The James Construction and $\pi_4(S^3)$, talk at the Institute of Advanced Studies on March 27, 2013 (recording)

• Guillaume Brunerie, The James construction and $\pi_4(S^3)$ in homotopy type theory (arXiv:1710.10307)