nLab
Moore path category

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.

Definition

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

  • Obj(M(X))=XObj(M(X)) = X.

  • Its set of all morphisms consists of pairs a=(f,r)a = (f,r) where f:[0,)Xf\colon [0, \infty) \to X is continuous, r0r \geq 0 and ff is constant on [r,)[r, \infty). The source of aa is f(0)f(0) and the target of aa is f(r)f(r). The number rr may be called the shape of aa. We may compose a=(f,r)a = (f,r) and b=(g,s)b = (g,s) to obtain ab=(h,r+s)a \circ b = (h,r+s) where h(t)=f(t)h(t) = f(t) for tr t \leq r and h(t)=g(tr)h(t) = g(t-r) for trt \geq r. Identities are paths of shape 00.

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

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.

Reference

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

Revised on November 5, 2012 21:23:49 by Stephan Alexander Spahn (79.227.182.186)