Natural homotopy of multipointed d-spaces

TL;DR

This paper extends the concept of natural homotopy to multipointed d-spaces, establishing a connection with directed spaces, and explores invariance properties and bisimilarity in this context.

Contribution

It introduces a reflection from multipointed d-spaces to directed spaces, extending natural systems and analyzing homotopy invariance and bisimilarity.

Findings

Reflection coincides with standard realization for precubical sets

Natural systems are invariant under globular subdivision

Bisimilarity up to homotopy is established for cellular multipointed d-spaces

Abstract

We identify Grandis' directed spaces as a full reflective subcategory of the category of multipointed $d$-spaces. When the multipointed $d$-space realizes a precubical set, its reflection coincides with the standard realization of the precubical set as a directed space. The reflection enables us to extend the construction of the natural system of topological spaces in Baues-Wirsching's sense from directed spaces to multipointed $d$-spaces. In the case of a cellular multipointed $d$-space, there is a discrete version of this natural system which is proved to be bisimilar up to homotopy. We also prove that these constructions are invariant up to homotopy under globular subdivision. These results are the globular analogue of Dubut's results. Finally, we point the apparent incompatibility between the notion of bisimilar natural systems and the q-model structure of multipointed $d$-spaces and we give some suggestions for future works.