Erratum, 11 July 2022: This is an updated version of the original paper \cite{Moore1} in which the notion of reparametrization category was incorrectly axiomatized. Details on the changes to the original paper are provided in the Appendix.A reparametrization category is a small topologically enriched semimonoidal category such that the semimonoidal structure induces a structure of a semigroup on objects, such that all spaces of maps are contractible and such that each map can be decomposed (not necessarily in a un…