nLab nice simplicial topological space

Idea

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.

Definition

Definition

Let $X : \Delta^{op} \to Top$ be a simplicial topological space.

Such $X$ is called

• good if all the degeneracy maps $X_{n-1} \hookrightarrow X_n$ are all closed cofibrations;

• proper if the inclusion $s X_n \hookrightarrow X_n$ of the degenerate simplices is a closed cofibration, where $s X_n = \bigcup_i s_i(X_{n-1})$.

In other words this says: $X_\bullet$ is proper if it is cofibrant in the Reedy model structure $[\Delta^{op}, Top_{Strom}]_{Reedy}$ on simplicial objects with respect to the Strøm model structure on Top.

The notion of good simplicial topological space goes back to (Segal 1973), that of proper simplicial topological space to (May).

Properties

General

Proposition

A good simplicial topological space is proper.

A proof appears as Lewis, corollary 2.4 (b). A generalization of this result is in RobertsStevenson.

Proposition

For $X_\bullet$ any simplicial topological space, then ${|Sing X_\bullet|}$ is good, hence proper, and the natural morphism

$${|Sing X_\bullet|} \to X_\bullet$$

is degreewise a weak homotopy equivalence.

This follows by results in (Lewis).

Proof

Since for $X \in Top$ the map $|Sing X| \to X$ is a cofibrant resolution in the standard Quillen model structure on topological spaces, we have that

$$|Sing X_\bullet| \to X_\bullet$$

is a degreewise weak homotopy equivalence. In particular each space $|Sing X_n|$ 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 $|Sing X_\bullet|$ are closed cofibrations.

Models for the homotopy colimit

Proposition

Let $X_\bullet$ be a simplicial topological space. Then there is a natural weak homotopy equivalence

$$\Vert X_\bullet\Vert \simeq hocolim_{n \in \Delta} X_n$$

from its fat geometric realization of simplicial topological spaces to the homotopy colimit over the simplicial diagram $X : \Delta^{op} \to Top$.

If moreover $X_\bullet$ is proper, then the natural morphism ${\Vert X\Vert} \to {|X|}$ is a weak homotopy equivalence, and hence also the ordinary geometric realization is a model for the homotopy colimit.

Proof

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.

Remark

In the case $X_\bullet$ that is a good simplicial topological space, a direct (i.e., not using the fact that goodness implies properness) proof that $\Vert X\Vert \to |X|$ is a weak homotopy equivalence has been sketched by Graeme Segal and then refined by Tammo tom Dieck.

Related concepts

• well-pointed simplicial topological group

• simplicial topological space

• simplicial topological group

• geometric realization of simplicial spaces.

References

The definition of proper simplicial space goes back to

• Peter May, The Geometry of Iterated Loop Spaces , Lecture Notes in Mathematics, 1972, Volume 271(1972), 100-112 (pdf)

May originally said strictly proper for what now is just called proper .

The definition of good simplicial space goes back to

• Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. (doi:10.1007/BF01390197, pdf, MR 0331377)

The implication $good \Rightarrow proper$ seems to be a folk theorem. Its origin is maybe in

• L. Gaunce Lewis, Jr., When is the natural map $X \to \Omega \Sigma X$ a cofibration? , Trans. Amer. Math. Soc. 273 (1982), 147–155.

A generalization of the statement that good implies proper to other topological concrete categories and a discussion of the geometric realization of $W G \to \bar W G$ for $G$ a simplicial topological group is in

• David Roberts, Danny Stevenson, Simplicial principal bundle in parameterized spaces (arXiv:1203.2460)

• Danny Stevenson, Classifying theory for simplicial parametrized groups (arXiv:1203.2461)

Comments on the relation between properness and cofibrancy in the Reedy model structure on $[\Delta^{op}, Set]$ are made in

• Paul Goerss, Kristen Schemmerhorn, Model Categories and Simplicial Methods (arXiv:math.AT/0609537).