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simplicial topological group

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

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

Definition

A simplicial group in GG in Top/BTop/B is called well-sectioned if for BB the trivial simplicial topological group over BB and i:BGi : B \to G the unique homomorphism, all components i n:BG ni_n : B \to G_n are f¯\bar f-cofibrations.

Properties

Recall for a discrete simplicial group GG the notation W¯GWG\bar W G \to W G for the Kan complex presentation of the universal principal infinity-bundle EGBG\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 GG be a simplicial topological group. Write W¯GTop Δ op\bar W G \in Top^{\Delta^{op}} for the simplicial topological space whose topological space of nn-simplices is the product

W¯G n:=G n1×G n2×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:XYf : X \to Y of simplicial topological spaces is a global Kan fibration if for all nn \in \mathbb{N} and 0kn0 \leq k \leq n the canonical morphism

X nY n× sTop(Λ k n,Y)sTop(Λ 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

We say a simplicial topological space X Top Δ opX_\bullet \in Top^{\Delta^{op}} is (global) Kan simplicial space if the unique morphism X *X_\bullet \to * is a global Kan fibration, hence if for all nn \in \mathbb{N} and all 0in0 \leq i \leq n the canonical continuous function

X nsTop(Λ k n,X) X_n \to sTop(\Lambda^n_k, X)

into the topological space of kkth nn-horns admits a section.

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

Proposition

Let GG be a simplicial topological group. Then

  1. GG is a globally Kan simplicial topological space;

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

  3. WGW¯GW 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 GG is a well-pointed simplicial topological group, then

  1. GG is a good simplicial topological space;

  2. the geometric realization |G||G| is well-pointed;

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

Proof

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

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

Discussion of homotopy theory over a base BB is in

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

Revised on January 20, 2014 03:26:26 by Urs Schreiber (89.204.135.222)