The rank filtration and Robinson’s complex

For a functor from the category of finite sets to abelian groups, Robinson constructed a bicomplex in [A. Robinson, Gamma homology, Lie representations and E∞ multiplications, Invent. Math. 152 (2) (2003) 331–348] which computes the stable derived invariants of the functor as defined by Dold–Puppe in [A. Dold, D. Puppe, Homologie nicht-additiver Funktoren. Anwendungen., Ann. Inst. Fourier (Grenoble) 11 (1961) 201–312]. We identify a subcomplex of Robinson’s bicomplex which is analogous to a normalization and also computes these invariants. We show that this new bicomplex arises from a natural filtration of the functor obtained by taking left Kan approximations on subcategories of bounded cardinality.