  Derived D-Geometry is considered as a convenient language for a
coordinate-free investigation of nonlinear partial differential equations up to
symmetries. One of the first issues one meets in the functor of points approach
to derived D-Geometry, is the question of a model structure on the category C
of differential non-negatively graded quasi-coherent commutative algebras over
the sheaf D of differential operators of an appropriate underlying variety. In
[BPP15a], we described a cofibrantly generated model structure on C via the
definition of its weak equivalences and its fibrations. In the present article
- the second of a series of works on the BV-formalism - we characterize the
class of cofibrations, give explicit functorial cofibration-fibration
factorizations, as well as explicit functorial fibrant and cofibrant
replacement functors. We then use the latter to build a model categorical
Koszul-Tate resolution for D-algebraic on-shell function algebras.
