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Friday, March 18, 2016

**Abstract:** An atomic discrete module is a functor from finite sets to chain complexes of $R$-modules that is completely determined by its value at a particular set. For a general discrete module, that is, a functor from finite sets to chain complexes of $R$-modules, one can use left Kan extensions to construct a filtration by atomic discrete modules. Robinson gave an explicit description of a bicomplex for computing the stable homology of a general discrete module, in which the rows are given by the stable homology of the associated atomic discrete modules. Goodwillie’s calculus of functors gives us a way to approximate functors that is analogous to the Taylor series for real functions. The stable homology of a functor is the homology of the Goodwillie derivative of the functor. This fact inspires us to generalize Robinson's bicomplex to one for computing the higher order polynomial approximations produced by Goodwillie calculus. To this end, we give an explicit bicomplex for atomic functors such that truncation by rows allows us to compute all of the Goodwillie polynomial approximations.