Dold–Kan Correspondence, Revisited

We describe Dold–Kan correspondence for an idempotent complete additive category A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathscr {A}}}$$\end{document}. Our approach is based on a family of idempotents in ZΔ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}\Delta $$\end{document}. We represent the obtained normalised complex equivalence of the category of simplicial objects in A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathscr {A}}}$$\end{document} and the category of non-negatively graded chain complexes in A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathscr {A}}}$$\end{document}, N:sA→Ch⩾0(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N:s{{\mathscr {A}}}\rightarrow \text {Ch}_{\geqslant 0}({{\mathscr {A}}})$$\end{document}, as a coend. Explicit formulae for the right adjoint equivalence K:Ch⩾0(A)→sA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K:\text {Ch}_{\geqslant 0}({{\mathscr {A}}})\rightarrow s{{\mathscr {A}}}$$\end{document} are obtained. It is shown that the functors N, K preserve the homotopy relation. Similar results are obtained for cosimplicial objects.