  Centers of categories capture the natural operations on their objects.
Homotopy coherent centers are introduced here as 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. Among many other things, 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. Nontrivial classes of
examples are given as illustration throughout.
