Localizations of the Category of AA_\infty Categories and Internal Homs

  • Alberto Canonaco

    Universita degli Studi di Pavia, Dipartimento di Matematica ``F. Casorati'', Via Ferrata 5, Pavia 27100, Italy
  • Mattia Ornaghi

    Ben Gurion University, Department of Mathematics, Be'er Sheva 84105, Israel
  • Paolo Stellari

    Dipartimento di Matematica ``F. Enriques'', Universita degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy
Localizations of the Category of $A_\infty$ Categories and Internal Homs cover
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We prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital AA_\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 AA_\infty category of AA_\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 AA_\infty functors between them.

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Alberto Canonaco, Mattia Ornaghi, Paolo Stellari, Localizations of the Category of AA_\infty Categories and Internal Homs. Doc. Math. 24 (2019), pp. 2463–2492

DOI 10.4171/DM/731