A filtered space is called a connected filtered space if it satisfies the following condition:
: The function induced by inclusion is surjective for all ; and, for all , for all and .
There are two other forms of this condition which are useful under different circumstances.
The free topological monoid on a space filtered by the length of words. Given a based space , there is also a reduced version by taking and identifying with the identity of . This latter filtered space is known as the James construction , after Ioan James.
The James construction can be constructed homotopy-theoretically, following ideas of Brunerie. Recall that, for a finite simplicial complex, for a pair of spaces, its polyhedral product is defined as the union as a subspace of the Cartesian product . Here, for a simplex of , the subspace consist of those such that, for each vertex in the complement of , the coordinate projection lies in . Equivalently, the polyhedral product can be considered as a homotopy colimit of these over the poset of simplexes of , where the maps are the respective inclusions.
For a space equipped with a basepoint , define a filtered space as follows. Set as . For , require that the following square is homotopy pushout:
where the unlabeled arrow is the homotopy colimit of a morphism of diagrams over given by the maps
for each simplex of the boundary simplicial complex of the standard -simplex.
For a pointed space, if is path-connected, then .
A similar example to the last using free groups instead of free monoids.
A similar example to the last using free groupoids on topological graphs.
A similar example to the last using the universal topological groupoid induced from a topological groupoid by a continuous function to a space .
Examples of connected filtered spaces are:
The skeletal filtration of a CW-complex.
The word length filtration of the James construction for a space with base point such that is a closed cofibration.
The filtration of the classifying space of a crossed complex, filtered using skeleta of .
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 by a sequence of inclusions.