Homotopy theory of Moore flows (I)

TL;DR

This paper develops a homotopy-theoretic framework for Moore flows, a type of enriched semicategory, establishing a model category structure and a Quillen equivalence with flows, advancing the understanding of their homotopical properties.

Contribution

It introduces the concept of Moore flows and constructs a model category structure, proving a Quillen equivalence with the category of flows, linking these structures in homotopy theory.

Findings

Constructed the q-model category of Moore flows

Proved Quillen equivalence with the q-model category of flows

Paved the way for relating multipointed d-spaces to flows

Abstract

Erratum, 11 July 2022: This is an updated version of the original paper in which the notion of reparametrization category was incorrectly axiomatized. Details on the changes to the original paper are provided in the Appendix. A reparametrization category is a small topologically enriched semimonoidal category such that the semimonoidal structure induces a structure of a semigroup on objects, such that all spaces of maps are contractible and such that each map can be decomposed (not necessarily in a unique way) as a tensor product of two maps. A Moore flow is a small semicategory enriched over the biclosed semimonoidal category of enriched presheaves over a reparametrization category. We construct the q-model category of Moore flows. It is proved that it is Quillen equivalent to the q-model category of flows. This result is the first step to establish a zig-zag of Quillen equivalences between the q-model structure of multipointed $d$-spaces and the q-model structure of flows.