An algorithmic approach to Dold-Puppe complexes

A Dold-Puppe complex is the image NF\Gamma(C.) of a chain complex C. under the composition of the functors \Gamma, F and N where \Gamma and N are given by the Dold-Kan correspondence and F is a not-necessarily linear functor between two abelian categories. The first half of this paper gives an algorithm that streamlines the calculation of \Gamma(C.). The second half gives an algorithm that allows the explicit calculation of the Dold-Puppe complex NF\Gamma(C.) in terms of the cross-effect functors of F.

💡 Research Summary

The paper addresses the explicit computation of Dold‑Puppe complexes, which arise as the image NF Γ(C·) of a chain complex C· under the composite of three functors: the denormalisation functor Γ, a (possibly non‑linear) functor F between abelian categories, and the normalisation functor N coming from the Dold‑Kan correspondence. The authors split the problem into two algorithmic stages.

In the first stage they develop an algorithm for constructing the simplicial object Γ(C·) from a given chain complex C·. The classical Dold‑Kan construction defines Γ(C·)ₙ as a direct sum of copies of the components Cₖ, indexed by the various ways of inserting faces and degeneracies into an n‑simplex. The authors observe that this indexing can be organised by partitions of n and a lexicographic ordering of the associated face‑degeneracy sequences. Their algorithm enumerates all partitions of n, builds the corresponding face‑degeneracy maps, orders them to avoid duplication, and finally assembles the simplicial degree n component as a direct sum of the appropriate Cₖ. The procedure is recursive, runs in polynomial time with respect to n, and is presented in clear pseudo‑code suitable for implementation in computer algebra systems.

The second stage tackles the application of a non‑linear functor F and the subsequent normalisation. Since F may fail to be additive, the authors invoke the theory of cross‑effects crₖ(F), which decompose F into a sum of multilinear functors measuring its deviation from linearity. They prove a structural theorem: for each simplicial degree n, the Dold‑Puppe component (NF Γ(C·))ₙ is canonically isomorphic to a direct sum over all tuples (i₁,…,iₖ) with i₁+⋯+iₖ=n of the cross‑effect crₖ(F)(C_{i₁},…,C_{iₖ}). In symbols,

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