# Six model categories for directed homotopy

## TL;DR

This paper develops multiple model structures for multipointed d-spaces and flows, comparing their properties and demonstrating the advantages of m-model structures over q-model structures.

## Contribution

It introduces new accessible m- and h-model structures on these categories and analyzes their relationships and properties.

## Key findings

- q-model structures are combinatorial and match known structures
- m-model structures are Quillen equivalent to q-model structures
- Some objects are not cofibrant in any model structure

## 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.