Question

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.

Answer 1

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).

Answer 2

You might want to look at 0705.3719.

Answer 3

I'll have to read it more carefully, but this paper of Penkava and Schwarz seems to do it: http://arxiv.org/abs/hep-th/9408064