Left properness of Moore flows

Abstract

We introduce the notion of a reparametrization category with cuts. For every such reparametrization category $\mathcal{P}$, we prove the tensor lemma, namely that the tensor product of two objectwise weak homotopy equivalences of $\mathcal{P}$-spaces is a weak equivalence, and then the left properness of the q-model structure of $\mathcal{P}$-flows. Finally, we prove that the interval reparametrization categories $\mathcal{G}$, $\mathcal{M}$, as well as the final category $1$, have cuts. The last example recovers the left properness of the q-model structure of ordinary flows.

23 pages; v2: for completeness, example of a reparametrization category without cuts added