ABSTRACT. We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal ∞-category V. Our theory of enriched ∞-categories has many desirable properties; for instance, if the enriching ∞-category V is presentably symmetric monoidal then Cat V ∞ is as well. These features render the theory useful even when an ∞-category of enriched ∞-categories comes from a model category (as is often the case in examples of interest, e.g. dg-categories, spectral categories, and (∞, n)-categories). This is analogous to the advantages of ∞-categories over more rigid models such as simplicial categories -for example, the resulting ∞-categories of functors between enriched ∞-categories automatically have the correct homotopy type.We construct the homotopy theory of V-enriched ∞-categories as a certain full subcategory of the ∞-category of "many-object associative algebras" in V. The latter are defined using a non-symmetric version of Lurie's ∞-operads, and we develop the basics of this theory, closely following Lurie's treatment of symmetric ∞-operads. While we may regard these "many-object" algebras as enriched ∞-categories, we show that it is precisely the full subcategory of "complete" objects (in the sense of Rezk, i.e. those whose spaces of objects are equivalent to their spaces of equivalences) which are local with respect to the class of fully faithful and essentially surjective functors. We also consider an alternative model of enriched ∞-categories as certain presheaves of spaces satisfying analogues of the "Segal condition" for Rezk's Segal spaces. Lastly, we present some applications of our theory, most notably the identification of associative algebras in V as a coreflective subcategory of pointed V-enriched ∞-categories as well as a proof of a strong version of the Baez-Dolan stabilization hypothesis.

We construct higher categories of iterated spans, possibly equipped with extra structure in the form of higher-categorical local systems, and classify their fully dualizable objects. By the Cobordism Hypothesis, these give rise to framed topological quantum field theories, which are the framed versions of the classical TQFTs considered in the quantization programme of Freed-Hopkins-Lurie-Teleman.Using this machinery, we also construct an (∞, 1)-category of symplectic derived algebraic stacks and Lagrangian correspondences and show that all its objects are dualizable.

We take another look at the construction of double ∞-categories of algebras and bimodules and prove a few supplemental results about these, including a simpler proof of the Segal condition and a comparison between our construction and that of Lurie. We then take a more streamlined, inductive approach to the higher Morita categories of 𝐸 𝑛 -algebras and show that these deloop correctly.

In this paper we initiate the study of enriched ∞-operads. We introduce several models for these objects, including enriched versions of Barwick's Segal operads and the dendroidal Segal spaces of Cisinski and Moerdijk, and show these are equivalent. Our main results are a version of Rezk's completion theorem for enriched ∞-operads: localization at the fully faithful and essentially surjective morphisms is given by the full subcategory of complete objects, and a rectification theorem: the homotopy theory of ∞-operads enriched in the ∞-category arising from a nice symmetric monoidal model category is equivalent to the homotopy theory of strictly enriched operads.Date: July 26, 2017.

Rectification of enriched ∞-categories RUNE HAUGSENGWe 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 fibrewise simplicial localizations potentially of independent use. 18D2, 55U35; 18D50, 55P48 IntroductionIn [13], David Gepner and I set up a general theory of "weakly enriched categories" -more precisely, we introduced a notion of ∞-categories enriched in a monoidal ∞-category, and constructed an ∞-category of these objects where the equivalences are the natural analogue of fully faithful and essentially surjective functors in this context. In this paper we are interested in the situation where the monoidal ∞-category we enrich in can be described by a monoidal model category -this applies to many, if not most, interesting examples of monoidal ∞-categories. If V is a model category, then inverting the weak equivalences W gives an ∞-category V[W −1 ]; if V is a monoidal model category, then V[W −1 ] inherits a monoidal structure, so our theory produces an ∞-category of V[W −1 ]-enriched ∞-categories. On the other hand, there is also often a model structure on ordinary V-enriched categories (cf. [19,6,32,24]) where the weak equivalences are the so-called DK-equivalences, namely the functors that are weakly fully faithful (i.e. given by weak equivalences in V on morphism objects), and essentially surjective (up to homotopy). Our main goal in this paper is to prove a rectification theorem in this setting: In particular, V[W −1 ]-enriched ∞-categories can be rectified to V-categories: every V[W −1 ]-enriched ∞-category is equivalent to one coming from a category enriched in V. We will state and prove a precise version of this result in §5. The precise meaning of "nice" required applies, for example, to the category of chain complexes over a ring with the usual projective model structure, and certain model structures on symmetric spectra. We can therefore conclude that the ∞-category of spectral categories is equivalent to that of spectral ∞-categories, and the ∞-category of dg-categories to that of ∞-categories enriched in the derived ∞-category of abelian groups. We will prove a precise version of this theorem in §6. From this we can conclude that the homotopy theories of n-categories and (∞, n)-categories constructed in [13, §6.1] using iterated enrichment are equivalent to those construct...