“Mathematics is not the rigid and rigidity-producing schema that the layman thinks it is; rather, in it we find ourselves at that meeting point of constraint and freedom that is the very essence of human nature.”
Hermann Weyl
Dold-Kan correspondence is a basic result in homotopy theory that establishes correspondence between simplicial abelian groups and (connective) chain complexes of abelian groups. This is given by functors that form an equivalence of categories between the category of simplicial abelian groups \(\mathbf{SAb}\) and the category of connective chain complexes of abelian groups \(\mathrm{Ch}_{+}(\mathbf{Ab})\). Thus, this correspondence interpolates between homological algebra and (simplicial) homotopy theory.
Theorem. Let \(\mathcal A\) be an abelian category. There is an equivalence of categories
\[N:\mathrm{Fun}(\Delta^{\mathrm{op}}, A)\rightleftharpoons \mathrm{Ch}_+(\mathcal A):\Gamma \]where \(N\) is the normalized chain complex functor, defined as \[(NA)_n=\bigcap_{i=1}^n\mathrm{Ker}(d_i^n)\] with \(\partial_n:=\left.d_0^n\right|_{(N A)_n}:(N A)_n \rightarrow(N A)_{n-1}\). And $\Gamma$ is the simplicialization functor, where \[\Gamma(C)_n=\bigoplus_{n\twoheadrightarrow k}C_k\] where the maps are given in here.
I haven’t have time to finish reading the proof yet. I will write more here when i do finish reading the proof.
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