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?
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1$\begingroup$ Have you looked into arxiv.org/abs/1308.0135? $\endgroup$– SashaCommented May 14, 2015 at 7:40
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$\begingroup$ No, I haven't, will try to make sense of it. Is there a place there that I should be specifically focusing on? $\endgroup$– Lev BorisovCommented May 14, 2015 at 10:49
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$\begingroup$ I thought Preygel's thesis was another reference for this, but I guess it's more $\infty$ than dg. $\endgroup$– bananastackCommented May 14, 2015 at 14:36
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1$\begingroup$ Are you referring to arxiv.org/pdf/1101.5834.pdf ? $\endgroup$– Lev BorisovCommented May 14, 2015 at 14:47
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1$\begingroup$ @Meow Yes, this looks like what I was asking about. Now, if I could only remember why ... :) $\endgroup$– Lev BorisovCommented Sep 2, 2021 at 20:52
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1 Answer
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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 $f$.