nLab
James construction

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

Idea

The free topological monoid FXF X on a topological space XX is canonically filtered by the length of words. Given instead a pointed topological space (X,x)(X,x), there is also a reduced version by taking FXF X and identifying xx with the identity of FXF X. This latter filtered topological space is known as the James construction J(X,x)J(X,x) (James 55).

Details

The James construction J(X,x)J(X,x) may be constructed homotopy-theoretically (Brunerie 13, Brunerie 17). Recall that, for KK a finite simplicial complex, for (X,A)(X,A) a pair of spaces, its polyhedral product (X,A) K(X,A)^K is defined as the union σS(K)(X,A) σ\bigcup_{\sigma\in S(K)}(X,A)^\sigma as a subspace of the Cartesian product X V(K)X^{V(K)}. Here, for σS(K)\sigma\in S(K) a simplex of KK, the subspace (X,A) σ(X,A)^\sigma consist of those xX V(K)x\in X^{V(K)} such that, for each vertex vv in the complement of σ\sigma, the coordinate projection proj vx\proj_v x lies in AA. Equivalently, the polyhedral product (X,A) K(X,A)^K can be considered as a homotopy colimit of these (X,A) σ(X,A)^\sigma over the poset S(K)S(K) of simplexes σ\sigma of KK, where the maps are the respective inclusions.

Definition

For XX a space equipped with a basepoint xx, define a filtered space fil fil_\bullet as follows. Set fil 0\fil_0 as {x}\{x\}. For k1k\ge 1, require that the following square is homotopy pushout:

(X,x) Δ[k1] inc X k fil k1 p k j k fil k \array{ &&&& (X,x)^{\partial \Delta[k-1]} &&&& \\ & && inc \swarrow & & \searrow && \\ && X^k &&&& fil_{k-1} \\ & && {}_{p_k}\searrow & & \swarrow_{j_k} && \\ &&&& fil_k &&&& }

where the unlabeled arrow is the homotopy colimit of a morphism of diagrams over S(Δ[k1])S(\partial \Delta[k-1]) given by the maps

(X,x) σX dim(σ)+1p dim(σ)+1fil dim(σ)+1(X,x)^\sigma \xrightarrow{\sim} X^{\dim(\sigma)+1} \xrightarrow{p_{\dim(\sigma)+1}} fil_{\dim(\sigma)+1}

for each simplex σ\sigma of the boundary simplicial complex Δ[k1]\partial \Delta[k-1] of the standard (k1)(k-1)-simplex. The homotopy colimit fil fil_\infty of the sequence of maps fil 0j 1fil 1j 2fil_0 \stackrel{j_1}{\to} fil_1 \stackrel{j_2}{\to} \ldots is called the James construction on (X,x)(X, x).

Proposition

For (X,x)(X,x) a pointed space, if (X,x)(X,x) is path-connected, then fil ΩΣXfil_\infty \simeq \Omega\Sigma X.

Properties

Relation to configuration spaces

The James construction of XX is homotopy equivalent to the configuration space C( 1,X)C(\mathbb{R}^1, X) of points in the real line with “charges” taking values in XX.

(e.g. Bödigheimer 87, Example 9)

References

The construction is due to

  • Ioan James, Reduced product spaces, Annals of Mathematics, Second Series, 62: 170–197 (1955)

Review includes

Discussion via configuration spaces includes

Discussion via homotopy type theory includes the following

Last revised on October 28, 2018 at 13:00:18. See the history of this page for a list of all contributions to it.