  Recently, the Johnson-McCarthy discrete calculus for homotopy functors was
extended to include functors from an unbased simplicial model category to
spectra. This paper completes the constructions needed to ensure that there
exists a discrete calculus tower for functors from an unbased simplicial model
category to chain complexes over a fixed commutative ring. Much of the
construction of the Taylor tower for functors to spectra carries over to this
context. However, one of the essential steps in the construction requires
proving that a particular functor is part of a cotriple. For this, one needs to
prove that certain identities involving homotopy limits hold up to isomorphism,
rather than just up to weak equivalence. As the target category of chain
complexes is not a simplicial model category, the arguments for functors to
spectra need to be adjusted for chain complexes. In this paper, we take
advantage of the fact that we can construct an explicit model for iterated
fibers, and prove that the functor is a cotriple directly. We use related ideas
to provide concrete infinite deloopings of the first terms in the resulting
Taylor towers when evaluated at the initial object in the source category.
