When we are dealing with ordinary things or dg things (where thing = algebra or category), I think I understand how HH^2 corresponds to 1st order deformations and HH^3 corresponds to obstructions.


One often hears (or at least I often hear) that HH^\* corresponds to A-infinity deformations. I am wondering whether there is any reference which works this out precisely. **EDIT: This seems to be incorrect (depending on what we mean by "deformation"). See Damien's answer. And see David Ben-Zvi's comment.**



Hochschild cohomology and A-infinity deformations
Well. Even in the case of a DG (or $A\_\infty$) algebra $A$, infinitesimal (i.e. 1st order) deformations are classified by $HH^2(A,A)$. Namely, the structure maps (a-k-a Taylor components) of an $A\_\infty$-algebra, viewed as elements of the Hochschild cochain complex, do have total degree $2$. 


I think that one recovers the full Hochschild cohomolgy $HH^\*(A,A)$ by considering "derived" infinitesimal deformations (namely, deformations for which the deformation parameter is allowed to have non zero degree). 


In other words, and making use of funny words, $HH^\*(A,A)$ is the tangent to the derived stack of associative (better, $A\_\infty$) algebras at the point $A$. While $HH^2(A,A)$ can be viewed as the tangent to the coarse moduli space. As an indermediate statement between those two, in his PhD thesis Mathieu Anel computed the tangent complex to the 2-stack of associative algebras (not in the derived context): he found that it is precisely a 2 step complex, obtain as a truncation of the Hochschild complex. See <http://arxiv.org/abs/math/0607385> (in french, sorry). 


