TITLE: Smooth dg algebras (and perfect dg modules and compact dg modules)
QUESTION [16 upvotes]: Katzarkov-Kontsevich-Pantev define a smooth dg ($\mathbb{C}$-)algebra $A$ to be a dg algebra which is a perfect $A \otimes A^{op}$-module. They say that an $A$-module $M$ is perfect if the functor $Hom(M,-)$ from ($A$-modules) to ($\mathbb{C}$-modules) preserves small homotopy colimits. (Definition 2.23 of KKP) *
Kontsevich-Soibelman define smooth dg (or $A_\infty$) algebras in the same way; but they define perfect $A$-modules to be ones which are quasi-isomorphic to a direct summand of an extension of a sequence of modules each of which is quasi-isomorphic to a shift of $A$. (Definition 8.1.1 of KS) **
First question: I have trouble wrapping my head around what it means for a functor to preserve small homotopy colimits. I don't understand the definition from Kontsevich-Soibelman, either. What are the "moral" meanings of these conditions?
Second question: Presumably these two conditions are equivalent. How do you prove this?
Third question: If $A$ is a commutative $(\mathbb{C}$-)algebra, then presumably smoothness of $A$ as a dg algebra is equivalent to smoothness of $\operatorname{Spec} A$ as a ($\mathbb{C}$-)scheme. (Maybe you need $A$ to be finite type?) How do you prove this?  Furthermore, Example 8.1.4 of KS gives some more examples of dg algebras that are allegedly smooth (for instance free algebras $k\langle x_1, \dots, x_n \rangle$). Again, I don't know how to prove that these are in fact smooth, and Kontsevich-Soibelman don't seem to provide proofs; or maybe I overlooked something.

*The answers below indicate that this is the standard definition of compact (also known as small) module.
**The answers below indicate that this is the standard definition of perfect module.

REPLY [4 votes]: As David explained above we have two notions: perfect objects and compact(small) objects.
They coincide for derived categories of DG modules over a DG algebra and, more generaly, 
over a DG category. Proof can be found, for example, in Bernhard Keller paper
link text Section 5.3.
For answer on the third question you can look paper of Rouquier
link text and Lemma 7.2 there which says that if $A$ is a finite dimensional $k$-algebra or a commutative $k$-algebra essentially of finite type such  that for given $V$ a simple A-module, $Z(End_A(V ))$ is a separable extension of $k$. Then, $pdim_{A^{en}} A = gldim A$.