On the Taylor tower of relative $K$-theory

Abstract

For $R$ a discrete ring, $M$ a simplicial $R$–bimodule, and $X$ a simplicial set, we construct the Goodwillie Taylor tower of the reduced $K$–theory of parametrized endomorphisms $\widetilde{K}\bigg(R; \widetilde{M}\big[X\big]\bigg)$ as a functor of $X$. Resolving general $R$–bimodules by bimodules of the form $\widetilde{M}\big[X\big]$, this also determines the Goodwillie Taylor tower of $\widetilde{K}\bigg(R; M\bigg)$ as a functor of $M$. The towers converge when $X$ or $M$ is connected. This also gives the Goodwillie Taylor tower of $\widetilde{K}\big(R⋉M\big)\simeq\widetilde{K}\big(R;B.M\big)$ as a functor of $M$.

For a functor with smash product $F$ and an $F$–bimodule $P$, we construct an invariant $W\big(F;P\big)$ which is an analog of $TR\big(F\big)$ with coefficients. We study the structure of this invariant and its finite-stage approximations $W_{n}\big(F;P\big)$ and conclude that the functor sending $X\mapsto W_{n}\big(R;\widetilde{M}\big[X\big]\big)$ is the n–th stage of the Goodwillie calculus Taylor tower of the functor which sends $X\mapsto\widetilde{K}\big(R;\widetilde{M}\big[X\big]\big)$. Thus the functor $X\mapsto W\big(R;\widetilde{M}\big[X\big]\big)$ is the full Taylor tower, which converges to $\widetilde{K}\big(R;\widetilde{M}\big[X\big]\big)$ for connected $\text{X}$.