TITLE: Matrix factorizations as a dg-category?
QUESTION [6 upvotes]: Matrix factorizations (in the graded case) give a triangulated category. I would imagine that there should be an underlying dg-category. Is there such a definition, and if so, where can I find it in the literature?

REPLY [3 votes]: Definition 2.7 of "A Category of Kernels for Equivariant Factorizations,
II: Further Implications" by Ballard-Favero-Katzarkov might be what you are looking for: http://arxiv.org/pdf/1310.2656.pdf.
Their setup is quite general; to get graded matrix factorizations of a degree $d$ element $f$ of a $\mathbb{Z}$-graded commutative algebra $R$ over a field $k$ from their definition, take $\mathcal{A}$ to be $\operatorname{gr-mod}-R$, $\Phi$ to be grading shift $d$ times, and $w$ to be multiplication by $f$.