  We introduce a notion generalizing Calabi-Yau structures on A-infinity
algebras and categories, which we call pre-Calabi-Yau structures. This notion
does not need either one of the finiteness conditions (smoothness or
compactness) which are required for Calabi-Yau structures to exist. In terms of
noncommutative geometry, a pre-CY structure is as a polyvector field satisfying
an integrability condition with respect to a noncommutative analogue of the
Schouten-Nijenhuis bracket. We show that a pre-CY structure defines an action
of a certain PROP of chains on decorated Riemann surfaces. In the language of
the cobordism perspective on TQFTs, this gives a partially defined extended
2-dimensional TQFT, whose 2-dimensional cobordisms are generated only by
handles of index one. We present some examples of pre-CY structures appearing
naturally in geometric and topological contexts.
