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.
Let $X$ be a topological space. Its Moore path category $M(X)$ has
$Obj(M(X)) = X$.
Its set of all morphisms consists of pairs $a = (f,r)$ where $f\colon [0, \infty) \to X$ is continuous, $r \geq 0$ and $f$ is constant on $[r, \infty)$. 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 $t \leq r$ and $h(t) = g(t-r)$ for $t \geq r$. 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 $n \geq 0$, and has the advantage over simplicial or globular versions of “$\infty$-groupoids” of easily encompassing multiple compositions.