# Fully dualizability of dg-algebras

I am working in the context of fully extended TQFTs and, at the moment, I am trying to find fully dualizable objects in certain $(\infty, 2)-$categories. In particular I know that for the $(\infty, 2)-$category $dgAlg_2$, (objects: dg-algebras; 1-morphims from $A$ to $B$: $(A, B)-$dg modules: 2-morphisms: interwiners; $\dots$) I shoulg get that fully dualizable objects are $smooth$ and $compact$ dg-algebras (a dg-algebra $A$ is smooth if $\sum_iH^i(A) <\infty$ and it is compact if it is smooth as an $A^e-$module). I have no clue how to prove this, so any hint or suggestion for some references would be really appreciated.

• Suppose $A$ is smooth and proper. Then $A$ is dualizable as a module over $k$ and over $A^e=A\otimes A^{op}$. To prove full dualizability you need to exhibit adjoints to the evaluation map which is $A$ considered as an $(A^e, k)$-bimodule. Both $A^* = Hom(A, k)$ and $A^! = Hom_{A^e}(A, A^e)$ give you candidates for the adjoints. – Pavel Safronov Mar 7 '16 at 15:52
• @Andrea What you call smooth is usually called proper, while smooth means compact (in the categorical sense) as an $A^e$-module. – Marc Hoyois Mar 7 '16 at 15:54