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.
Keywords: enriched higher categories, enriched infinity-categories
Haugseng, Rune 1
@article{10_2140_agt_2015_15_1931,
author = {Haugseng, Rune},
title = {Rectification of enriched \ensuremath{\infty}{\textendash}categories},
journal = {Algebraic and Geometric Topology},
pages = {1931--1982},
year = {2015},
volume = {15},
number = {4},
doi = {10.2140/agt.2015.15.1931},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2015.15.1931/}
}
Haugseng, Rune. Rectification of enriched ∞–categories. Algebraic and Geometric Topology, Tome 15 (2015) no. 4, pp. 1931-1982. doi: 10.2140/agt.2015.15.1931
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