CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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.
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 $Top/B$.
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.
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.
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
in Top, equipped wwith the evident (…) face and degeneracy maps.
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
$\Lambda^n_k \in$ 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
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).
Let $G$ be a simplicial topological group. Then
$G$ is a globally Kan simplicial topological space;
$\bar W G$ is a globally Kan simplicial topological space;
$W G \to \bar W G$ is a global Kan fibration.
The first statement appears as (BrownSzczarba, theorem 3.8), the second is noted in (RobertsStevenson), the third as (BrownSzczarba, lemma 6.7).
If $G$ is a well-pointed simplicial topological group, then
$G$ is a good simplicial topological space;
the geometric realization $|G|$ is well-pointed;
$\bar W G$ is a proper simplicial topological space.
The statement about $\bar W G$ is proven in (RobertsStevenson). The other statements are referenced there.
simplicial topological space, nice simplicial topological space
simplicial topological group
Basics theory of simplicial topological groups is in
and
Their principal ∞-bundles and geometric realization is discussed in
Discussion of homotopy theory over a base $B$ is in