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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 familia
The invertibility hypothesis for a monoidal model category S asks that localizing an S-enriched category with respect to an equivalence results in an weakly equivalent enriched category. This is the most technical among the axioms for S to be an exce
In this paper we present background results in enriched category theory and enriched model category theory necessary for developing model categories of enriched functors suitable for doing functor calculus.
Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as $[0,infty)$-enriched categories. We show that in many cases magnitude can be categorified to a homology theory for enriched categories, which we call
We introduce a notion of bimodule in the setting of enriched $infty$-categories, and use this to construct a double $infty$-category of enriched $infty$-categories where the two kinds of 1-morphisms are functors and bimodules. We then consider a natu
This is the first of a series of papers on enriched infinity categories, seeking to reduce enriched higher category theory to the higher algebra of presentable infinity categories, which is better understood and can be approached via universal proper