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

Abstract

In this paper, we apply an explicit construction of a simplicial powering in dg-categories, due to Holstein (2016) and Arkhipov and Poliakova (2018), as well as our own results on homotopy ends (Arkhipov and Ørsted 2018), 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 A ∞-comodules, proving a conjecture by Block, Holstein, and Wei (2017) in the process.

Homotopy characters as a homotopy limit

“…Here we are considering reduced Cobar construction for the sake of comparing with the result in [AØ2] and with the computation in DGAlg(k). Reduced and non-reduced Cobar constructions are coMorita equivalent by Proposition 5.7 (though not Morita equivalent).…”

“…Recall the notion of an A ∞ -comodule over a DG-coalgebra (A ∞ -comodules can be considered over any A ∞ -coalgebra, but this generality will not be needed). For detailed exposition see [AØ2] or, on the dual side, [Kel].…”

“…In the papers [BHW] and [AØ2] the authors realized homotopy limits in DGCat(k) ∆ op as derived totalizations. Below we cite Prop.…”

“…Below we cite Prop. 4.0.2 from [AØ2], with formulas written in their most explicit form. To simplify the notation, we denote by ∂ (i 1 ...i k ) an inclusion with image i 1 , .…”

“…The note is devoted to an explicit calculation of a homotopy limit for a certain cosimplicial diagram in the model category of DG-categories. Recall that a general construction for representatives of such derived limits was given in the papers [BHW] and [AØ2]. Below we consider a baby example where the answer appears to be both explicit and meaningful.…”