2 APONTE ROM´ LIVERNET, ROBERTSON, WHITEHOUSE, AND ZIEGENHAGEN a commutative ring, however, Kadeishivili’s result no longer holds. If the homol- ogy of the differential graded algebra is not projective over the ground ring there need no longer be a minimal A∞-algebra quasi-isomorphic to the given differential graded algebra. In order to bypass the projectivity assumptions necessary for Kadeishvili’s re- sult, Sagave developed the notion of derived A∞-algebras [Sag10]. While classi- cal A∞-algebras are graded algebras, derived A∞–algebras are bigraded algebras. Sagave establishes a notion of minimal model for differential graded algebras (dgas) whose homology is not necessarily projective by showing that the structure of a derived A∞–algebra arises on some projective resolution of the homology of a dif- ferential graded algebra. In this paper, we continue the work of [LRW13], developing the description of these structures using operads. The operads we use are non-symmetric operads in the category BiCompl v of bicomplexes with zero horizontal differential. We have an operad dAs in this category encoding bidgas, which are simply monoids in bicomplexes. It is shown in [LRW13] that derived A∞-algebras are precisely algebras over the operad dA∞ = (dAs)∞ = Ω((dAs)¡). Here (dAs)¡ is the Koszul dual cooperad of the operad dAs, and Ω denotes the cobar construction. In this manner, we view a derived A∞-algebra as the infinity version of a bidga, just as an A∞-algebra is the infinity version of a dga. We further investigate the operad dAs, in particular studying (dAs)¡-coalgebras. The structure of an As¡-coalgebra is well-known to be equivalent, via a suspension, to that of a usual coassociative coalgebra. Analogously, (dAs)¡-coalgebras are equiv- alent, via a suspension in the vertical direction, to coassociative coalgebras which are equipped with an extra piece of structure. A substantial part of this paper is concerned with representations of derived A∞-algebras. Besides being an important part of the basic operadic theory of these algebras, we will use this theory in subsequent work to develop the Hochschild cohomology of derived A∞-algebras with coefficients. In section 4, we give a general result expressing a representation of a P∞-algebra for any Koszul operad P in terms of a square-zero coderivation. Then we work this out explicitly for the derived A∞ case. We explain how this relates to Sagave’s derived A∞-modules: the operadic notion of representation yields a two-sided version of Sagave’s modules. Finally, we present a new, explicit example of a derived A∞-algebra. The con- struction is based on some examples of A∞-algebras due to Allocca and Lada [AL10]. The paper is organized as follows. In section 2 we begin with a brief review of previous work on derived A∞-algebras and establish our notation and conventions. Sections 3 and 4 cover the material on (dAs)¡-coalgebras, coderivations and repre- sentations. Section 5 presents our new example. A brief appendix establishes the relationship between two standard sign conventions and gives details of cooperadic suspension in our bigraded setting.
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