I am wondering: Are there any general theorems or principles relating the theory of Z-graded dg objects and the theory of Z/2Z-graded dg objects? I am mainly interested in dg algebras, dg Lie algebras, and dg categories over fields of characteristic zero.

-
Z/2-dgas are equivalent to algebras over the commutative Z-dga $\mathbb{Z}[u^{\pm 1}]$, where $|u| = 2$. Based on this you have a number of standard constructions associated to a map of commutative dgas $R \to S$, such as tensoring up (equivalent to what Steve Huntsman proposed), forgetting (which gives 2-periodic Z-dgas), and various derived adjunctions which simplify because the range is flat over the domain. Is this something like what you are interested in? –  Tyler Lawson Jul 28 '10 at 2:39
Putting a bounty on this question, hoping that Tyler (or someone else) can elaborate... –  Kevin H. Lin Aug 13 '10 at 9:49
This is discussed, at least for dg-categories, in section 4 of Dyckerhoff's thesis: front.math.ucdavis.edu/0904.4713 where it is needed to understand LG models associated to isolated hypersurface singularities. It elaborates on Tyler's comment. –  Matthew Ballard Aug 13 '10 at 14:53
Propsition 3.1 in the paper arxiv.org/abs/1004.0687 by Dyckerhoff and Murfet gives explicit examples of some of the constructions mentioned by Tyler. –  Jesse Burke Aug 13 '10 at 15:50
A more general question concerning the behavior of (C)DG-categories and (C)DG-modules with respect to a change of the grading group is discussed in Section 2.5 of our new preprint arxiv.org/abs/1010.0982 –  Leonid Positselski Oct 6 '10 at 8:51