Combinatorial categorical equivalences of Dold–Kan type

• Stephen Lack ^[1] ; Ross Street ^[1]
  1. [1] Macquarie University

     Macquarie University

     Australia

     
• Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 219, Nº 10 (October 2015), 2015, págs. 4343-4367
• Idioma: inglés
• DOI: 10.1016/j.jpaa.2015.02.020
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• Resumen

  • We prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let X denote an additive category with finite direct sums and splitting idempotents. The class includes (a) the Dold–Puppe–Kan theorem that simplicial objects in X are equivalent to chain complexes in X; (b) the observation of Church, Ellenberg and Farb [9] that X-valued species are equivalent to X-valued functors from the category of finite sets and injective partial functions; (c) a result T. Pirashvili calls of “Dold–Kan type”; and so on. When X is semi-abelian, we prove the adjunction that was an equivalence is now at least monadic, in the spirit of a theorem of D. Bourne.