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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.

##### MSC:
 55U10 Simplicial sets and complexes in algebraic topology 16E45 Differential graded algebras and applications (associative algebraic aspects)
Full Text:
##### References:
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