C T ] 2 2 M ar 2 02 0 A ∞-categories , their ∞-category , and their localizations

We collect some foundational results regarding the homotopy theory of A∞-categories. Our two main results are (1) An equivalence between the ∞-category of dg-categories and the ∞category of A∞-categories, and (2) A proof that two models for quotients of A∞-categories (as constructed by Lyubashenko-Manzyuk and Lyubashenko-Ovisienko) satisfy the universal property of quotients in the ∞-categorical sense. Our aims are to give succinct accounts of ∞categorical language for users of A∞-categories, and to exhibit concrete models of localizations of A∞-categories. Indeed, we apply the results here in [OT19] to prove a Liouville version of a conjecture of Teleman from the 2014 ICM.

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