  We prove a rectification theorem for enriched infinity-categories: If V is a
nice monoidal model category, we show that the homotopy theory of
infinity-categories enriched in V is equivalent to the familiar homotopy theory
of categories strictly enriched in V. It follows, for example, that
infinity-categories enriched in spectra or chain complexes are equivalent to
spectral categories and dg-categories. A similar method gives a comparison
result for enriched Segal categories, which implies that the homotopy theories
of n-categories and (infinity,n)-categories defined by iterated
infinity-categorical enrichment are equivalent to those of more familiar
versions of these objects. In the latter case we also include a direct
comparison with complete n-fold Segal spaces. Along the way we prove a
comparison result for fibrewise simplicial localizations potentially of
independent use.
