nLab
simplicial topological group

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

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 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 simplicial classifying space coprojection WGW¯GW G \to \overline{W} G, being a 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 with the evident face and degeneracy maps (see at simplicial classifying space).

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 (Brown & Szczarba 1989) 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 (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 GG is a well-pointed simplicial topological group (Def. ), then

  1. its underlying simplicial topological space is good;

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

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

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

References

Basic discussion of simplicial topological groups:

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

Discussion of homotopy theory over a base BB is in

Last revised on September 23, 2021 at 04:42:18. See the history of this page for a list of all contributions to it.