Six model categories for directed homotopy

Philippe Gaucher

doi:10.52547/cgasa.15.1.145

Abstract

We construct a q-model structure, a h-model structure and a m-model structure on multipointed $d$-spaces and on flows. The two q-model structures are combinatorial and coincide with the combinatorial model structures already known on these categories. The four other model structures (the two m-model structures and the two h-model structures) are accessible. We give an example of multipointed $d$-space and of flow which are not cofibrant in any of the model structures. We explain why the m-model structures, Quillen equivalent to the q-model structure of the same category, are better behaved than the q-model structures.

Highlights

1 Introduction Presentation This paper belongs to our series of papers which aims at comparing the model category Flow of flows introduced in [11] and the model category GdTop of multipointed d-spaces introduced in [15]
Using the notion of topological graph and the Garner Hess Kędziorek Riehl Shipley theorem [20] [10] about accessible rightinduced model structures, we introduce a categorical construction which takes as input an accessible model structure on the category Top of ∆generated spaces satisfying some mild conditions and which gives as output an accessible model structure on multipointed dspaces and on flows
We have proved that the model category Gph(V) is an accessible model category

Summary

Introduction

Presentation This paper belongs to our series of papers which aims at comparing the model category Flow of flows introduced in [11] (with some updated proofs in [18] using Isaev’s work [23]) and the model category GdTop of multipointed d-spaces introduced in [15]. It is expounded the theorem we are going to use to right-induce accessible model structures (Theorem 2.1).

Accessible model category

Accessible Grothendieck bifibration

Delta-generated space

Topological graph

Multipointed d-space

Path space functor and m-cofibrancy