Every Q-shaped derived category, under mild assumptions, is equivalent to the derived category of differential graded bimodules over a dg category built from the shape Q.
Silting mutation in triangulated categories
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$Q$-shaped derived categories as derived categories of differential graded bimodules
Every Q-shaped derived category, under mild assumptions, is equivalent to the derived category of differential graded bimodules over a dg category built from the shape Q.