Contents

# Contents

## 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

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 notation $\bar W G \to W G$ for the 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 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 $0 \leq k \leq n$ 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 (BrownSzczarba) 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 (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 $|G|$ is well-pointed;

3. $\bar W G$ is a proper simplicial topological space.

###### Proof

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

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

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

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