A nice simplicial topological space is a simplicial topological space that satisfies certain extra properties that make it well behaved in homotopy theory, notably so that its geometric realization of simplicial spaces is its homotopy colimit.
Let be a simplicial topological space.
Such is called
good if all the degeneracy maps are all closed cofibrations;
proper if the inclusion of the degenerate simplices is a closed cofibration, where .
A good simplicial topological space is proper.
For any simplicial topological space, then is good, hence proper, and the natural morphism
is degreewise a weak homotopy equivalence.
This follows by results in (Lewis).
is a degreewise weak homotopy equivalence. In particular each space is a CW-complex, hence in particular a locally equi-connected space. By (Lewis, p. 153) inclusions of retracts of locally equi-connected spaces are closed cofibrations, and since degeneracy maps are retracts, this means that the degeneracy maps in are closed cofibrations.
from its fat geometric realization of simplicial topological spaces to the homotopy colimit over the simplicial diagram .
That the geometric realization of simplicial topological spaces of a proper simplicial space is is homotopy colimit follows from the above fact that proper spaces are Reedy cofibrant, and using the general statement discussed at homotopy colimit about description of homotopy colimits by coends.
In the case that is a good simplicial topological space, a direct (i.e., not using the fact that goodness implies properness) proof that is a weak homotopy equivalence has been sketched by Graeme Segal and then refined by Tammo tom Dieck.
simplicial topological space, nice simplicial topological space
The definition of proper simplicial space goes back to
May originally said strictly proper for what now is just called proper .
The definition of good simplicial space goes back to
The implication seems to be handled like a folk theorem. Its origin is maybe in
Comments on the relation between properness and cofibrancy in the Reedy model structure on are made in