nLab
simplicial topological group

Definition

Definition

A simplicial topological group is a simplicial object in the category of topological groups.

For various applications the ambient category Top of topological spaces is taken specifically to be

• the category of compactly generated weakly Hausdorff spaces, or

• or the category of k-spaces.

We take Top to be the category of k-spaces in the following.

Definition

A simplicial topological group $G$ is called well-pointed if for $*$ the trivial simplicial topological group and $i:*\to G$ the unique homomorphism, all components $i_n:*\to G_n$ are closed cofibrations.

For $B\in \mathrm{Top}$ a fixed base object, it is often desireable to work in ”$B$-parameterized families”, hence in the over-category $\mathrm{Top}/B$ (see MaySigurdson). There is the relative Strøm model structure on $\mathrm{Top}/B$.

Definition

A simplicial group in $G$ in $\mathrm{Top}/B$ is called well-sectioned if for $B$ the trivial simplicial topological group over $B$ and $i:B\to G$ the unique homomorphism, all components $i_n:B\to G_n$ are $\overline{f}$-cofibration.

Properties

Recall for a discrete simplicial group $G$ the notation $\overline{W}G\to WG$ for the Kan complex presentation of the universal principal infinity-bundle $EG\to BG$ from simplicial group. These constructions for discrete simplicial groups have immediate analogs for simplicial topological groups.

Definition

Let $G$ be a simplicial topological group. Write $\overline{W}G\in {\mathrm{Top}}^{{\Delta }^{\mathrm{op}}}$ for the simplicial topological space whose topological space of $n$-simplices is the product

$$\overline{W}{G}_n:=G_{n-1}\times G_{n-2}\cdots \times G_0$$\bar W G_n := G_{n-1} \times G_{n-2} \cdots \times G_{0}

in Top, equipped wwith the evident (…) face and degeneracy maps.

Definition

We say a morphism $f:X\to Y$ of simplicial topological spaces is a global Kan fibration if for all $n\in \mathbb{N}$ and $0\le k\le n$ the canonical morphism

$$X_n\to Y_n{\times }_{\mathrm{sTop}({\Lambda }_k^n,Y)}\mathrm{sTop}({\Lambda }_k^n,X)$$X_n \to Y_n \times_{sTop(\Lambda^n_k, Y)} sTop(\Lambda^n_k, X)

in Top has a section, where

• ${\Lambda }_k^n\in$ sSet $\hookrightarrow {\mathrm{Top}}^{{\Delta }^{\mathrm{op}}}$ is the $k$th $n$-horn regarded as a discrete simplicial topological space:

• $\mathrm{sTop}(-,-):{\mathrm{sTop}}^{\mathrm{op}}\times \mathrm{sTop}\to \mathrm{Top}$ is the Top-hom object.

We say a simplicial topological space $X_{•}\in {\mathrm{Top}}^{{\Delta }^{\mathrm{op}}}$ is (global) Kan simplicial space if the unique morphism $X_{•}\to *$ is a global Kan fibration, hence if for all $n\in \mathbb{N}$ and all $0\le i\le n$ the canonical continuous function

$$X_n\to \mathrm{sTop}({\Lambda }_k^n,X)$$X_n \to sTop(\Lambda^n_k, X)

into the topological space of $k$th $n$-horns admits a section.

This global notion of Kan simplicial spaces is considered for instance in (BrownSzczarba) and (May).

Proposition

Let $G$ be a simplicial topological group. Then

1. $G$ is a globally Kan simplicial topological space;

2. $\overline{W}G$ is a globally Kan simplicial topological space;

3. $WG\to \overline{W}G$ is a global Kan fibration.

Proof

The first statement appears as (BrownSzczarba, theorem 3.8), the second is noted in (RobertsStevenson), the third as (BrownSzczarba, lemma 6.7).

Proposition

If $G$ is a well-pointed simplicial topological group, then

1. $G$ is a good simplicial topological space;

2. the geometric realization $\mid G\mid$ is well-pointed;

3. $\overline{W}G$ is a proper simplicial topological space.

Proof

The statement about $\overline{W}G$ is proven in (RobertsStevenson). The other statements are referenced there.

Related concepts

• simplicial topological space, nice simplicial topological space

• simplicial topological group

• geometric realization of simplicial topological spaces

References

Basics theory of simplicial topological groups is in

• E. H. Brown and R. H. Szczarba, Continuous cohomology and real homotopy type , Trans. Amer. Math. Soc. 311 (1989), no. 1, 57 (pdf)

and

• Peter May, Geometry of iterated loop spaces , SLNM 271, Springer-Verlag, 1972 (pdf)

Their principal ∞-bundles and geometric realization is discussed in

• David Roberts, Danny Stevenson, Simplicial principal bundles in parametrized spaces (web)

Discussion of homotopy theory over a base $B$ is in

• Peter May, J. Sigurdsson, Parametrized homotopy theory (web)