\(D\)-structures and derived Koszul duality for unital operad algebras. (English) Zbl 1331.18011
Koszul duality for quadratic operads and their algebras, introduced by V. Ginzburg and M. Kapranov [Duke Math. J. 76, No. 1, 203–272 (1994; Zbl 0855.18006)], has two versions, one derived and one non derived. The problem is that it does not interact in a correct way with units. The aim of this paper is the presentation of a derived Koszul duality for unital algebras over unital operads, and also to provide a framework for unifying the different algebraic contexts for Koszul duality.
The (non-unital) homology is considered, via the Ginzburg-Kapranov construction. Even if it may have a 0 chain homology, it has an additional construction, named the D-structure, a concept due to Lipshitz, Ozsváth and Thurston [R. Lipshitz et al., Geom. Topol. 19, No. 2, 525–724 (2015; Zbl 1315.57036)], which gives rise to an equivalence of derived categories. This is thought as a unital version of the Koszul duality using non-unital Quillen homology. The multi-sorted case is also treated in an appendix.
The (non-unital) homology is considered, via the Ginzburg-Kapranov construction. Even if it may have a 0 chain homology, it has an additional construction, named the D-structure, a concept due to Lipshitz, Ozsváth and Thurston [R. Lipshitz et al., Geom. Topol. 19, No. 2, 525–724 (2015; Zbl 1315.57036)], which gives rise to an equivalence of derived categories. This is thought as a unital version of the Koszul duality using non-unital Quillen homology. The multi-sorted case is also treated in an appendix.
Reviewer: Loïc Foissy (Calais)
MSC:
| 18D50 | Operads (MSC2010) |
| 55N35 | Other homology theories in algebraic topology |
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