Abstract
We describe Dold–Kan correspondence for an idempotent complete additive category \({{\mathscr {A}}}\). Our approach is based on a family of idempotents in \({\mathbb {Z}}\Delta \). We represent the obtained normalised complex equivalence of the category of simplicial objects in \({{\mathscr {A}}}\) and the category of non-negatively graded chain complexes in \({{\mathscr {A}}}\), \(N:s{{\mathscr {A}}}\rightarrow \text {Ch}_{\geqslant 0}({{\mathscr {A}}})\), as a coend. Explicit formulae for the right adjoint equivalence \(K:\text {Ch}_{\geqslant 0}({{\mathscr {A}}})\rightarrow s{{\mathscr {A}}}\) are obtained. It is shown that the functors N, K preserve the homotopy relation. Similar results are obtained for cosimplicial objects.
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Communicated by D. N. Yetter.
To the memory of Ukrainian mathematician Sergiy Adamovych Ovsienko.
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Lyubashenko, V. Dold–Kan Correspondence, Revisited. Appl Categor Struct 30, 543–567 (2022). https://doi.org/10.1007/s10485-021-09665-7
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DOI: https://doi.org/10.1007/s10485-021-09665-7