Homotopy theory of Moore flows (II)

TL;DR

This paper establishes a Quillen equivalence between Moore flows and multipointed d-spaces, providing new insights into their homotopy theories and the categorization functor's structure.

Contribution

It proves the Quillen equivalence of q-model structures for Moore flows and multipointed d-spaces, and offers a novel proof of the categorization functor's derived equivalence.

Findings

Quillen equivalence between Moore flows and multipointed d-spaces

New proof of the categorization functor's derived equivalence

Inverse up to homotopy of the categorization functor using Moore flows

Abstract

This paper proves that the q-model structures of Moore flows and of multipointed $d$-spaces are Quillen equivalent. The main step is the proof that the counit and unit maps of the Quillen adjunction are isomorphisms on the q-cofibrant objects (all objects are q-fibrant). As an application, we provide a new proof of the fact that the categorization functor from multipointed $d$-spaces to flows has a total left derived functor which induces a category equivalence between the homotopy categories. The new proof sheds light on the internal structure of the categorization functor which is neither a left adjoint nor a right adjoint. It is even possible to write an inverse up to homotopy of this functor using Moore flows.