Rectification of enriched $\infty$–categories

Abstract

We prove a rectification theorem for enriched $\infty$–categories: if $V$ is a nice monoidal model category, we show that the homotopy theory of $\infty$–categories enriched in $V$ is equivalent to the familiar homotopy theory of categories strictly enriched in $V$. It follows, for example, that $\infty$–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 $\left(\infty ,n\right)$–categories defined by iterated $\infty$–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 fiberwise simplicial localizations potentially of independent use.