Enriched ∞\infty-operads

In this paper we initiate the study of enriched $\infty$-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 $\infty$-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 $\infty$-operads enriched in the $\infty$-category arising from a nice symmetric monoidal model category is equivalent to the homotopy theory of strictly enriched operads.Comment: Accepted version, 59 page

Smooth one-dimensional topological field theories are vector bundles with connection

We prove that smooth 1-dimensional topological field theories over a manifold are the same as vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism category, which encodes cutting laws rather than gluing laws. We make this idea precise through a smooth generalization of Rezk's complete Segal spaces. With such a definition in hand, we analyze the category of field theories using a combination of descent, a smooth version of the 1-dimensional cobordism hypothesis, and standard differential geometric arguments.Comment: 20 pages. Comments and questions are very welcom

Adding inverses to diagrams encoding algebraic structures

We modify a previous result, which showed that certain diagrams of spaces are essentially simplicial monoids, to construct diagrams of spaces which model simplicial groups. Furthermore, we show that these diagrams can be generalized to models for Segal groupoids. We then modify Segal's model for simplicial abelian monoids in such a way that it becomes a model for simplicial abelian groups.Comment: 24 pages, final version; erratum included at the end. arXiv admin note: text overlap with arXiv:math/050841