nLab Moore path category

Idea

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

Definition

Definition

Let $X$ be a topological space. Its Moore path category $M X$ has

• as space of objects $M_0 X \coloneqq (M X)_0 \coloneqq X$;

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

• as composition ${\circ} \colon M_1 X \times_{M_0 X} M_1 X \to M_1 X$, given by $len(\gamma' \circ \gamma) = len (\gamma) + len (\gamma')$, $(\gamma' \circ \gamma)(t)=\gamma(t)$ for $t\in[0, len (\gamma)]$ and $(\gamma' \circ \gamma)(t)=\gamma'(t - len (\gamma))$ otherwise;

• as identity-assigning map $X \to M_1 X$ the map sending $x\in X$ to the pair $(t \mapsto x, 0)$;

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

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

Definition

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

(Barthel–Riehl 2013, definition 3.2)

Properties

Functorial factorization

Proposition

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

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

of a (closed) trivial Hurewicz cofibration $L f$ and a Hurewicz fibration $R f$.

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

Proposition

A morphism of $Top$ is

• a Hurewicz fibration iff it admits an algebra structure for the pointed endofunctor $R$;

• a (closed) trivial Hurewicz cofibration iff it admits a coalgebra structure for the copointed endofunctor $L$ (a Moore strong deformation retraction).

(Barthel–Riehl 2013, section 3.3)

Related concepts

• schedule

Reference

• Tobias Barthel, Emily Riehl, On the construction of functorial factorizations for model categories, Algebr. Geom. Topol. 13 (2013) 1089-1124 (arXiv:1204.5427, doi:10.2140/agt.2013.13.1089, euclid:agt/1513715550)

• R. Brown, Moore hyperrectangles on a space form a strict cubical omega-category, arXiv 0909.2212v2