  We elucidate the relation between smooth Calabi-Yau structures and
pre-Calabi-Yau structures. We show that, from a smooth Calabi-Yau structure on
an $A_\infty$-category $A$, one can produce a pre-Calabi-Yau structure on $A$;
as defined in our previous work, this is a shifted noncommutative version of an
polyvector field satisfying a Maurer-Cartan equation. We explain how this
relation is an analogue of the Legendre transform, and how it admits a lift to
a weak equivalence of two natural simplicial sets, one whose points are
non-degenerate negative cyclic chains, and another whose points are
nondegenerate pre-CY structures.
