  Suppose one has found a non-empty sub-category $\mathcal{A}$ of the Fukaya
category of a compact Calabi-Yau manifold $X$ which is homologically smooth in
the sense of non-commutative geometry, a condition intrinsic to $\mathcal{A}$.
Then, we show $\mathcal{A}$ split-generates the Fukaya category and moreoever,
that our hypothesis implies (and is therefore equivalent to the assertion that)
$\mathcal{A}$ satisfies Abouzaid's geometric generation criterion [Abo]. An
immediate consequence of earlier work [G1, GPS1, GPS2] is that the open-closed
and closed-open maps, relating quantum cohomology to the Hochschild invariants
of the Fukaya category, are also isomorphisms. Our result continues to hold
when $c_1(X) \neq 0$ (for instance, when $X$ is monotone Fano), under a further
hypothesis: the 0th Hochschild cohomology of $\mathcal{A}$
$\mathrm{HH}^0(\mathcal{A})$ should have sufficiently large rank: $\mathrm{rk}\
\mathrm{HH}^0(\mathcal{A}) \geq \mathrm{rk}\ \mathrm{QH}^0(X)$. Our proof
depends only on formal properties of Fukaya categories and open-closed maps,
the most recent and crucial of which, compatibility of the open-closed map with
pairings, was observed independently in ongoing joint work of the author with
Perutz and Sheridan [GPS2] and by Abouzaid-Fukaya-Oh-Ohta-Ono [AFO+]; a proof
in the simplest settings appears here in an Appendix. Because categories Morita
equivalent to categories of coherent sheaves or matrix factorizations are
homologically smooth, our result applies to resolve the split-generation
question in homological mirror symmetry for compact symplectic manifolds
(generalizing a result of Perutz-Sheridan [PS2] proven in the case $c_1(X) =
0$): any embedding of coherent sheaves or matrix factorizations into the
split-closed derived Fukaya category is automatically a Morita equivalence when
it has large enough $\mathrm{HH}^0$ (which it always does if $c_1(X)=0$).
