zbMATH — the first resource for mathematics

Differential Lie modules over curved colored coalgebras and \(\infty\)-simplicial modules. (English. Russian original) Zbl 1332.55010
Math. Notes 96, No. 6, 698-715 (2014); translation from Mat. Zametki 96, No. 5, 709-731 (2014).
This paper begins with a review of the notion of colored differential modules and algebras, as well as colored graded coalgebras. The author then introduces definitions of quadratic-scalar and quadratic colored algebras and curved colored coalgebras, together with the notion of Koszul duality between the two structures and a cobar construction. These definitions lead up to the concept of a differential Lie module over a curved colored coalgebra. With an appropriate notion of homotopy of morphisms of such objects, this structure is shown to be homotopy invariant. The author then goes on to incorporate simplicial structure and to define \(\infty\)-simplicial modules and to prove an analogous homotopy invariance result for these structures.

55U10 Simplicial sets and complexes in algebraic topology
16E45 Differential graded algebras and applications (associative algebraic aspects)
Full Text: DOI
[1] Smirnov, V A, \(A\)_{∞}-simplicial objects and \(A\)_{∞}-topological groups, Mat. Zametki, 66, 913-919, (1999)
[2] Gugenheim, V K A M; Lambe, L A; Stasheff, J D, Perturbation theory in differential homological algebra. II, Illinois J. Math., 35, 357-373, (1991) · Zbl 0727.55012
[3] Smirnov, V A, Homology of \(B\)-constructions and of co-\(B\)-constructions, Izv. Ross. Akad. Nauk Ser. Mat., 58, 80-96, (1994) · Zbl 0924.55018
[4] Positsel’skii, L E, Nonhomogeneous quadratic duality and curvature, Funktsional. Anal. i Prilozhen., 27, 57-66, (1993) · Zbl 0826.16041
[5] J.-L. Loday and B. Vallette, Algebraic Operads, in Grundlehren Math. Wiss. (Springer-Verlag, Berlin, 2012), Vol. 346. · Zbl 1260.18001
[6] Lapin, S V, Homotopy properties of differential Lie modules over curved coalgebras and Koszul duality, Mat. Zametki, 94, 354-372, (2013) · Zbl 1295.55018
[7] Smirnov, V A, Lie algebras over operads and their application to homotopy theory, Izv. Ross. Akad. Nauk Ser. Mat., 62, 121-154, (1998)
[8] V. A. Smirnov, Simplicial and Operad Methods in Algebraic Topology (Faktorial, Moscow, 2002; Translations of Mathematical Monographs, 198, AMS, Providence, RI, 2001). · Zbl 0964.55001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.