Such as an algebra $A$ itself, $A$-(bi)modules, differential forms $\Omega^\bullet A$, the Hochschild (co)homology of $A$; anything based on $A$ and especially a non-commutative counterpart of some structure in the usual algebraic geometry.

Any module category is fine; I just wanted an interpretation from the derived perspective. Now that you've mentioned the isomorphism between Harrison cohomology and the André-Quillen one, I'm fairly contented. Thank you.

@ManuelRivera Thank you for the comment. I'll take a look at your preprint.