# Left properness of flows

## TL;DR

This paper provides a correct proof of the left properness of the q-model structure of flows using Reedy techniques, correcting previous inaccuracies and exploring interactions with various cofibration notions across different topological categories.

## Contribution

It offers a rigorous proof of left properness for flows' q-model structure and extends the analysis to interactions with cofibrations in multiple topological categories.

## Key findings

- Corrected proof of left properness of flows
- Analysis of path space functor interactions with cofibrations
- Applicability to various topological space categories

## Abstract

Using Reedy techniques, this paper gives a correct proof of the left properness of the q-model structure of flows. It fixes the preceding proof which relies on an incorrect argument. The last section is devoted to fix some arguments published in past papers coming from this incorrect argument. These Reedy techniques also enable us to study the interactions between the path space functor of flows with various notions of cofibrations. The proofs of this paper are written to work with many convenient categories of topological spaces like the ones of $k$-spaces and of weakly Hausdorff $k$-spaces and their locally presentable analogues, the $\Delta$-generated spaces and the $\Delta$-Hausdorff $\Delta$-generated spaces.