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arXiv · 1811.07830

Localizations of the category of $A_\infty$ categories and internal Homs

Alberto Canonaco·Mattia Ornaghi·Paolo Stellari
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Abstract

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.

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Alberto Canonaco, Mattia Ornaghi, Paolo Stellari. 2020-03-13. Localizations of the category of $A_\infty$ categories and internal Homs. https://arxiv.org/abs/1811.07830

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