TITLE: Definition of Hochschild (co)homology of a (dg or A-infinity) category
QUESTION [12 upvotes]: How do you define the Hochschild (co)homology of a dg category or an A-infinity category? I've only seen it defined when the category is equivalent to a category of modules over a dg algebra; then the Hochschild (co)homology is just that of the algebra. But more generally?

REPLY [3 votes]: Maybe this answer comes five years too late, but I have recently put an elementary construction of the Hochschild (co)homology of $A_\infty$-algebras on the arXiv:
http://arxiv.org/abs/1601.03963