Unbased calculus for functors to chain complexes

TL;DR

This paper develops a discrete calculus tower for homotopy functors from unbased simplicial model categories to chain complexes, extending previous spectral functor calculus to a new algebraic context.

Contribution

It completes the construction of the Taylor tower for functors to chain complexes, adapting methods from spectral calculus to the chain complex setting.

Findings

Constructed an explicit model for iterated fibers.

Proved a functor is a cotriple using identities involving homotopy limits.

Provided concrete infinite deloopings of initial terms in Taylor towers.

Abstract

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.