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We consider Taylor approximation for functors from the small category of finite pointed
sets to modules and give an explicit description for the homology of the layers
of the Taylor tower. These layers are shown to be fibrant objects in a suitable closed
model category structure. Explicit calculations are presented in characteristic zero
including an application to higher order Hochschild homology. A spectral sequence for the
homology of the homotopy fibres of this approximation is provided.
We study the Taylor towers of the nth symmetric and exterior power functors, Spⁿ and Λⁿ. We obtain a description of the layers of the Taylor towers, and , in terms of the first terms in the Taylor towers of and for t < n. The homology of these first terms is related to the stable derived functors (in the sense of Dold and Puppe) of and . We use stable derived functor calculations of Dold and Puppe to determine the lowest nontrivial homology groups for and .
We show that the dimension of the derived category of an elliptic curve or a tubular weighted projective line is one. We give explicit generators realizing this number, and show that they are in a certain sense minimal.
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