A differential graded algebra is semifree (or semi-free) if the underlying graded algebra is free: if after forgetting the differential, it is isomorphic as a graded algebra to a (polynomial) tensor algebra of some (super)graded vector space.
A differential graded-commutative algebra is semifree (or semi-free) if the underlying graded-commutative algebra is free: if after forgetting the differential, it is isomorphic as a graded-commutative algebra to a Grassmann algebra of some graded vector space .
to an -coring with a grouplike element associate its Amitsur complex with underlying graded module where and differential linearly extending the formulas for and
conversely, to a semi-free dga one associates the -coring where isa new group-like indeterminate; this is by definition a direct sum of left -modules with a right -module structure given by
Moreover flat connections for a semi-free dga are in - correspondence with the comodules over the corresponding coring with a group-like element.
At least when the algebra in degree is of the form for some space , which then is the space of objects of the Lie infinity-algebroid. But if it is a more general algebra in degree one can think of a suitably generalized -algebroid, for instance with a noncommutative space of objects. This generalizes the step from Lie algebroids to Lie–Rinehart pairs.
Sometimes semi-free DGAs are called quasi-free, but this is in collision with the terminology about formal smoothness of noncommutative algebras, i.e. quasi-free algebras in the sense of Cuntz and Quillen (and with extensions to homological smootheness of dg-algebras by Kontsevich).