Homotopy coherent centers versus centers of homotopy categories

Centers of categories capture the natural operations defined on their objects. Homotopy coherent centers are an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as Hochschild cohomology type invariants in contexts that are not necessarily linear or stable, and we argue that they are more appropriate to higher categorical contexts than the centers of their homotopy or derived categories. We present an obstruction theory for realizing elements in the centers of homotopy categories, and a Bousfield-Kan type spectral sequence that computes the homotopy groups. Several non-trivial classes of examples are given as illustrations of the general theory throughout.