also helpful. I could not have carried out this project without the generous support of the Simons Center for Geometry and Physics. Further support was received through NSF grant DMS-1005288.

Families of objects\[sec:theory\]
=================================

Suppose that we are given an $A_\infty$-category $A$, and a class in its degree $1$ Hochschild cohomology $\mathit{HH}^1(A,A)$. This class determines one in $H^1(\mathit{hom}_A(X,X))$ for every object $X$, hence an infinitesimal first order deformation of $X$. Deformations coming from Hochschild cohomology have additional properties, for instance (in characteristic $0$) they can be extended to arbitrarily high orders in a formal parameter. However, instead of looking at the infinitesimal theory, we want to consider global deformations. For the sake of illustration, take the parameter space to be the affine line. One then looks for families ${\mathscr{X}}= \{{\mathscr{X}}_s\}$ depending on one algebraic variable $s$, whose fibre at the origin is fixed, ${\mathscr{X}}_0 {\cong}X$, and whose first order deformation behaviour at any value of $s$ is the element of $H^1(\mathit{hom}_A({\mathscr{X}}_s,{\mathscr{X}}_s))$ induced by our Hochschild class. We are mainly concerned with the uniqueness of such families. This, while not totally straightforward, turns out to be much easier than existence issues. An elementary parallel would be the question of integrating a 