nLab Moore path category

Moore path categories

Moore path categories

Idea

The classical Moore path category of a topological space XX is a variant on the usual space of paths IXI \to X, but one which yields a strict category (even an internal \dagger -category in Top).

Definition

Definition

Let XX be a topological space. Its Moore path category MXM X has

  • as space of objects M 0X(MX) 0XM_0 X \coloneqq (M X)_0 \coloneqq X;

  • as space of morphisms M 1X(MX) 1X [0,]×[0,)M_1 X \coloneqq (M X)_1 \subseteq X^{[0,\infty]} \times [0,\infty) the subspace of pairs (γ,len(γ))(\gamma, len(\gamma)) where γ:[0,)X\gamma \colon [0, \infty) \to X is continuous and constant on [len(γ),][len(\gamma), \infty]; the source s(γ)s(\gamma) of γ\gamma is γ(0)\gamma(0) and the target t(γ)t(\gamma) of γ\gamma is γ(len(γ))=γ()\gamma(len(\gamma))=\gamma(\infty);

  • as composition :M 1X× M 0XM 1XM 1X{\circ} \colon M_1 X \times_{M_0 X} M_1 X \to M_1 X, given by len(γγ)=len(γ)+len(γ)len(\gamma' \circ \gamma) = len (\gamma) + len (\gamma'), (γγ)(t)=γ(t)(\gamma' \circ \gamma)(t)=\gamma(t) for t[0,len(γ)]t\in[0, len (\gamma)] and (γγ)(t)=γ(tlen(γ))(\gamma' \circ \gamma)(t)=\gamma'(t - len (\gamma)) otherwise;

  • as identity-assigning map XM 1XX \to M_1 X the map sending xXx\in X to the pair (tx,0)(t \mapsto x, 0);

  • as path-reversal map (γ,len(γ)) (tγ(len(γ)t),len(γ))(\gamma, len (\gamma))^\dagger \coloneqq (t \mapsto \gamma(len (\gamma) - t), len (\gamma)).

The reference below defines M *(X)M_*(X) as a strict cubical ω\omega-category. It also has connections, which satisfy all the laws except cancellation of Γ i \Gamma^-_i and Γ i +\Gamma^+_i under composition. This structure seems a sensible home for nn-paths in XX for all n0n \geq 0, and has the advantage over simplicial or globular versions of “\infty-groupoids” of easily encompassing multiple compositions.

Definition

Suppose f:XYf \colon X \to Y is a continuous map. The Moore mapping path space Γf\Gamma f is the pullback

(Barthel–Riehl 2013, definition 3.2)

Properties

Functorial factorization

Proposition

The Moore mapping path space construction Γ:Top Top\Gamma\colon Top^\to \to Top yields a functorial factorization of maps f:XYf\colon X\to Y as a composition

XLfΓfRfY X \stackrel{L f}{\to} \Gamma f \stackrel{R f}{\to} Y

of a (closed) trivial Hurewicz cofibration LfL f and a Hurewicz fibration RfR f.

(Barthel–Riehl 2013, section 3.1 and corollary 3.12)

Proposition

A morphism of TopTop is

(Barthel–Riehl 2013, section 3.3)

Reference

Last revised on May 30, 2022 at 12:17:16. See the history of this page for a list of all contributions to it.