Rectification of enriched ∞–categories

Abstract

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.