Homotopy theory of Moore flows (III)

Abstract

The previous paper of this series shows that the q-model categories of G-
multipointed d-spaces and of G-flows are Quillen equivalent. In this paper,
the same result is established by replacing the reparametrization category G by
the reparametrization category M. Unlike the case of G, the execution paths
of a cellular M-multipointed d-space can have stop intervals. The technical
tool to overcome this obstacle is the notion of globular naturalization. It is the
globular analogue of Raussen’s naturalization of a directed path in the geometric
realization of a precubical set. The notion of globular naturalization working
both for G and M, the proof of the Quillen equivalence we obtain is valid for the
two reparametrization categories. Together with the results of the first paper of
this series, we then deduce that G-multipointed d-spaces and M-multipointed
d-spaces have Quillen equivalent q-model structures. Finally, we prove that
the saturation hypothesis can be added without any modification in the main
theorems of the paper.