Left properness of Moore flows

Abstract

We introduce reparametrization categories with cuts. For every such category $\mathcal{P}$, the tensor product of two objectwise weak homotopy equivalences of $\mathcal{P}$-spaces is an objectwise weak homotopy equivalence, and the q-model structure on $\mathcal{P}$-flows is left proper. The reparametrization categories $\mathcal{G}$ and $\mathcal{M}$, as well as the terminal category $1$, have cuts. The terminal category recovers the left properness of the q-model structure on ordinary flows. We also give a symmetric reparametrization category with cuts and a strict unit. This category is not equivalent to $1$.

arXiv comment23 pages; v2: for completeness, example of a reparametrization category without cuts added; v3: presentation improved and proofs expanded