In general, we have functors . If is a -algebra, then is an equivalence of -categories, is fully faithful, and the essential image of consists of the connective objects of (that is, those algebras having for ).
What is the explicit functor ? I suppose that the natural thing would be to take a simplicial -algebra and assign it to
and take a map and assign it to
but as far as I could find this isn't stated explicitly in DAG. Is this the case, and if so, do you have a source or proof? And how does one show that is a differential graded algebra?