Rectification of enriched ∞–categories

Haugseng, Rune

doi:10.2140/agt.2015.15.1931

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Rectification of enriched $\infty$–categories

Rune Haugseng

Algebraic & Geometric Topology 15 (2015) 1931–1982

DOI: 10.2140/agt.2015.15.1931

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 $(\infty, n)$ –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.

Keywords

enriched higher categories, enriched infinity-categories

Mathematical Subject Classification 2010

Primary: 18D2, 55U35

Secondary: 18D50, 55P48

References

Publication

Received: 19 February 2014

Revised: 31 October 2014

Accepted: 2 November 2014

Published: 10 September 2015

Authors

Rune Haugseng
Max Planck Institut für Mathematik Vivatsgasse 7 53111 Bonn Germany
http://people.mpim-bonn.mpg.de/haugseng/

Volume 15, issue 4 (2015)