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Non-commutative rings and algebras, non-associative algebras. Can be used in combination with ra.rings-and-algebras

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What is the motivation behind the definition for a smooth differential graded category?

Let $\mathcal{A}$ be an $\mathbb{F}$-linear differential graded category. It is said to be smooth if it is a perfect complex over the differential graded category $\mathcal{A}^\circ\otimes_\mathbb{F}\ …
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