Unbased calculus for functors to chain complexes

Maria Basterra, Kristine Bauer, Agnes Beaudry, Rosona Eldred, Brenda Johnson, Mona Merling, Sarah Yeakel

Published 2014-09-04Version 1

Recently, the Johnson-McCarthy discrete calculus for homotopy functors was extended to include functors from a simplicial model category to spectra. This paper completes the constructions needed to ensure that there exists a discrete calculus tower for functors to chain complexes. Much of the construction of the Taylor tower for functors to spectra carries over in this context. However, one of the essential and most delicate steps in the construction consists in 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 from this model we demonstrate the needed properties directly. We use the explicit models to provide concrete infinite deloopings of the first terms in the resulting Taylor towers when evaluated at the initial object in the source category.