Nonsmooth Calabi-Yau structures for algebras and coalgebras

We show that generalised Calabi-Yau dg (co)algebras are Koszul dual to generalised symmetric dg (co)algebras, without needing to assume any smoothness or properness hypotheses. Similarly, we show that Gorenstein and Frobenius are Koszul dual properties. As an application, we give a new characterisation of Poincaré duality spaces, which extends a theorem of Félix- Halperin-Thomas to the non-simply connected setting.