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

Definition

A simplicial group in $G$ in $Top/B$ 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 $\bar f$-cofibrations.

Properties

Recall for a discrete simplicial group $G$ the simplicial classifying space coprojection $W G \to \overline{W} G$, being a Kan complex presentation of the universal principal infinity-bundle $\mathbf{E}G \to \mathbf{B}G$ 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 $\bar W G \in Top^{\Delta^{op}}$ for the simplicial topological space whose topological space of $n$-simplices is the product

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

in Top, equipped with the evident face and degeneracy maps (see at simplicial classifying space).

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 $0 \leq k \leq n$ the canonical morphism

$$X_n \to Y_n \times_{sTop(\Lambda^n_k, Y)} sTop(\Lambda^n_k, X)$$

in Top has a section, where

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

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

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

$$X_n \to sTop(\Lambda^n_k, X)$$

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

This global notion of Kan simplicial spaces is considered for instance in (Brown & Szczarba 1989) and (May).

Proposition

Let $G$ be a simplicial topological group. Then

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

2. $\bar W G$ is a globally Kan simplicial topological space;

3. $W G \to \bar W G$ is a global Kan fibration.

Proof

The first statement appears as (Brown & Szczarba 1989, theorem 3.8), the second is noted in (Roberts & Stevenson 2012), the third as (Brown & Szczarba 1989, lemma 6.7).

Proposition

If $G$ is a well-pointed simplicial topological group (Def. ), then

1. its underlying simplicial topological space is good;

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

3. the geometric realization $|G|$ is well-pointed.

(Roberts & Stevenson 2012, Prop. 3, for the last item see also Baez & Stevenson 2008, Lem. 1)

Related concepts

• simplicial topological space, nice simplicial topological space

• simplicial group, simplicial abelian group

• geometric realization of simplicial topological spaces

References

Basic discussion of simplicial topological groups:

• Edgar H. Brown, R. H. Szczarba?, Continuous cohomology and real homotopy type , Trans. Amer. Math. Soc. 311 (1989), no. 1, 57 (doi:10.1090/S0002-9947-1989-0929667-6, pdf)

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

Discussion of their geometric realization and principal $\infty$-bundles:

• John Baez, Danny Stevenson, The Classifying Space of a Topological 2-Group, In: Baas N., Friedlander E., Jahren B., Østvær P. (eds.) Algebraic Topology_. Abel Symposia, vol 4. Springer 2009 (arXiv:0801.3843, doi:10.1007/978-3-642-01200-6_1)

• David Roberts, Danny Stevenson, Simplicial principal bundles in parametrized spaces, New York Journal of Mathematics Volume 22 (2016) 405-440 (arXiv:1203.2460, nyjm:22-19)

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

• Peter May, Johann Sigurdsson, Parametrized Homotopy Theory (web)