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Homotopy limits in the category of dg-categories in terms of \( \mathrm {A}_{\infty } \)-comodules

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We apply an explicit construction of a simplicial powering in dg-categories, due to Holstein (Properness and simplicial resolutions for the model category dgCat, 2014. arXiv:1403.4381) and Arkhipov and Poliakova (Homol Homotopy Appl 22(2):151–162, 2019), as well as our own results on homotopy ends (Arkhipov and Ørsted in Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories, 2018. arXiv:1807.03266), to obtain an explicit model for the homotopy limit of a cosimplicial system of dg-categories. We apply this to obtain a model for homotopy descent in terms of \(\mathrm {A}_{\infty }\)-comodules, proving a conjecture by Block et al. (Homol Homotopy Appl 19(2):343–371, 2017) in the process.

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We would like to thank Edouard Balzin for his interest in the project, his guidance, suggestions, and for reading an early draft of the paper. We also thank Daria Poliakova for stimulating discussions.

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Correspondence to Sebastian Ørsted.

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Arkhipov, S., Ørsted, S. Homotopy limits in the category of dg-categories in terms of \( \mathrm {A}_{\infty } \)-comodules. European Journal of Mathematics 7, 671–705 (2021).

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