Spaces of smooth embeddings and configuration categories

In the homotopical study of spaces of smooth embeddings, the functor calculus method (Goodwillie–Klein–Weiss manifold calculus) has opened up important connections to operad theory. Using this and a few simplifying observations, we arrive at an operadic description of the obstructions to deforming smooth immersions into smooth embeddings. We give an application which in some respects improves on recent results of Arone–Turchin and Dwyer–Hess concerning high-dimensional variants of spaces of long knots.