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.

Definition

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

• $\mathrm{Obj}(M(X))=X$.

• Its set of all morphisms consists of pairs $a=(f,r)$ where $f:[0,\mathrm{\infty })\to X$ is continuous, $r\ge 0$ and $f$ is constant on $[r,\mathrm{\infty })$. The source of $a$ is $f(0)$ and the target of $a$ is $f(r)$. The number $r$ may be called the shape of $a$. We may compose $a=(f,r)$ and $b=(g,s)$ to obtain $a\circ b=(h,r+s)$ where $h(t)=f(t)$ for $t\le r$ and $h(t)=g(t-r)$ for $t\ge r$. Identities are paths of shape $0$.

The advantage of this definition as pairs is partly in giving a topology on $M(X)$, but also in iteration.

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 $n\ge 0$, and has the advantage over simplicial or globular versions of ”$\mathrm{\infty }$-groupoids” of easily encompassing multiple compositions.

Related concepts

• schedule

Reference

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