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 $\mathbf 1$, have cuts. The last example recovers the left properness of the q-model structure of ordinary flows.