The cotriple resolution of differential graded algebras (Q2821731)

In the context of algebras over operads in a homotopical setting, it is often necessary to take cofibrant resolutions. While these resolutions exist for formal reasons, it can be difficult to describe them explicitly. The purpose of this paper is to describe a cofibrant resolution of differential graded algebras over the \(E_\infty\) operad (more specifically, the Barratt-Eccles operad) and the commutative operad. The main result is that the geometric realization of the cotriple resolution of a dg-algebra \(A\) is in fact a cofibrant resolution of, and in particular weakly equivalent to, the original dg-algebra \(A\). This kind of construction has been established previously in the context of algebras in spectra, but the approaches used there cannot be applied in the dg-algebra setting.NEWLINENEWLINEAfter an initial section developing cosimplicial framings in the non-unitary dg Barratt-Eccles operad, the main result of the paper is proved in this context and then generalized to the unitary version; the proof is then adapted to the unitary commutative operad. The paper concludes with an outline of how this argument might be generalized to more general operads.