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 K(R; M[X]) as a functor of X. Resolving general R-bimodules by bimodules of the form M[X], this also determines the Goodwillie Taylor tower of K(R; M) as a functor of M. The towers converge when X or M is connected. This also gives the Goodwillie Taylor tower of K(R ∝ M) approx- K(R; B.M) as a functor of M. For a functor with smash product F and an F-bimodule P, we construct an invariant W(F; P) which is an analog of TR(F) with coefficients. We study the structure of this invariant and its finite-stage approximations W_n(F; P) and conclude that the functor sending X → W_n(R; M[X]) is the n-th stage of the Goodwillie calculus Taylor tower of the functor which sends X → K(R; M[X]). Thus the functor X → W(R; M[X]) is the full Taylor tower, which converges to K(R; M[X]) for connected X.
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机译：对于R是离散环，M是简单R-双模，而X是简单集，我们构造了参数化内同构K（R; M [X]）的简化K理论的Goodwillie Taylor塔，作为X的函子。一般R-bimodules由M [X]形式的双模数决定，这也将K（R; M）的Goodwillie Taylor塔确定为M的函子。当X或M连接时，这些塔会收敛。这也给出了K（R ∝ M）的Goodwillie Taylor塔，作为M的函子。对于具有粉碎乘积F和F-双模数P的函子，我们构造了不变的W（F; P），它是TR（F）具有系数的类似物。我们研究了该不变量的结构及其有限阶近似W_n（F; P），并得出结论，发送X→W_n（R; M [X]）的函子是该Goodwellie微积分泰勒塔的n阶。发送X→K（R; M [X]）的函子。因此，函子X→W（R; M [X]）是完整的泰勒塔，对于连接的X，它收敛到K（R; M [X]）。
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