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Rectification of enriched $\infty$–categories

Rune Haugseng

Algebraic & Geometric Topology 15 (2015) 1931–1982

We prove a rectification theorem for enriched –categories: if V is a nice monoidal model category, we show that the homotopy theory of –categories enriched in V is equivalent to the familiar homotopy theory of categories strictly enriched in V. It follows, for example, that –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 (,n)–categories defined by iterated –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.

enriched higher categories, enriched infinity-categories
Mathematical Subject Classification 2010
Primary: 18D2, 55U35
Secondary: 18D50, 55P48
Received: 19 February 2014
Revised: 31 October 2014
Accepted: 2 November 2014
Published: 10 September 2015
Rune Haugseng
Max Planck Institut für Mathematik
Vivatsgasse 7
53111 Bonn