Abstract
In the algebra case, a twisting morphism is defined as a particular map from a dg coassociative coalgebra to a dg associative algebra. Starting with dg associative algebras, why should one consider the category of dg coassociative coalgebras? The conceptual explanation is given by the Koszul duality theory for operads: the operad As is Koszul and its Koszul dual operad is itself. In order to generalize the notion of twisting morphism to dg 
-algebras, one needs to work with dg 
-coalgebras. Such a phenomenon has already been noticed in the literature. For instance, in rational homotopy theory the case 
and 
was treated by Quillen in 1969 and the case 
and 
was treated by Sullivan in 1977.
In this chapter, we extend to dg 
-algebras the notions of twisting morphism, bar and cobar constructions. When the operad 
is Koszul, this allows us to define functorial quasi-free resolutions for 
-algebras and 
-algebras. They will be used in the next chapter to compute homology groups.
La mathématique est une science dangereuse: elle dévoile les supercheries et les erreurs de calcul.
Galileo Galilei
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Loday, JL., Vallette, B. (2012). Bar and Cobar Construction of an Algebra over an Operad. In: Algebraic Operads. Grundlehren der mathematischen Wissenschaften, vol 346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30362-3_11
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DOI: https://doi.org/10.1007/978-3-642-30362-3_11
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