Canonaco, Alberto; Ornaghi, Mattia; Stellari, Paolo

Localizations of the Category of \(A_\infty\) Categories and Internal Homs

Doc. Math. 24, 2463-2492 (2019)
DOI: 10.25537/dm.2019v24.2463-2492
Communicated by Henning Krause


We prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital \(A_\infty\) categories with respect to the corresponding classes of quasi-equivalences are all equivalent. Moreover we show that the last two localizations are equivalent to the corresponding quotients by the relation of being isomorphic in the cohomology of the \(A_\infty\) category of \(A_\infty\) functors. As an application we give a complete proof of a claim by Kontsevich stating that the category of internal Homs for two dg categories can be described as the category of strictly unital \(A_\infty\) functors between them.

Mathematics Subject Classification

18D20, 18E35, 18G55, 57T30


dg categories, \(A_\infty\) categories


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Canonaco, Alberto
Universita degli Studi di Pavia, Dipartimento di Matematica ``F. Casorati'', Via Ferrata 5, Pavia 27100, Italy
Ornaghi, Mattia
Ben Gurion University, Department of Mathematics, Be'er Sheva 84105, Israel
Stellari, Paolo
Dipartimento di Matematica ``F. Enriques'', Universita degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy