Enriched $\infty$categories via nonsymmetric $\infty$operads
Abstract
We set up a general theory of weak or homotopycoherent enrichment in an arbitrary monoidal $\infty$category $\mathcal{V}$. Our theory of enriched $\infty$categories has many desirable properties; for instance, if the enriching $\infty$category $\mathcal{V}$ is presentably symmetric monoidal then $\mathrm{Cat}^\mathcal{V}_\infty$ is as well. These features render the theory useful even when an $\infty$category of enriched $\infty$categories comes from a model category (as is often the case in examples of interest, e.g. dgcategories, spectral categories, and $(\infty,n)$categories). This is analogous to the advantages of $\infty$categories over more rigid models such as simplicial categories  for example, the resulting $\infty$categories of functors between enriched $\infty$categories automatically have the correct homotopy type. We construct the homotopy theory of $\mathcal{V}$enriched $\infty$categories as a certain full subcategory of the $\infty$category of "manyobject associative algebras" in $\mathcal{V}$. The latter are defined using a nonsymmetric version of Lurie's $\infty$operads, and we develop the basics of this theory, closely following Lurie's treatment of symmetric $\infty$operads. While we may regard these "manyobject" algebras as enriched $\infty$categories, we show that it is precisely the full subcategory of "complete" objects (in the sense of Rezk, i.e. those whose space of objects is equivalent to its space of equivalences) which are local with respect to the class of fully faithful and essentially surjective functors. Lastly, we present some applications of our theory, most notably the identification of associative algebras in $\mathcal{V}$ as a coreflective subcategory of pointed $\mathcal{V}$enriched $\infty$categories as well as a proof of a strong version of the BaezDolan stabilization hypothesis.
 Publication:

arXiv eprints
 Pub Date:
 December 2013
 arXiv:
 arXiv:1312.3178
 Bibcode:
 2013arXiv1312.3178G
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Category Theory;
 18D20;
 18D50;
 55P48;
 55U35
 EPrint:
 Added erratum