Homotopy coherent theorems of Dold-Kan type (Q2078857)

Given an abelian category \(A\), the classical Dold-Kan correspondence [\textit{D. M. Kan}, Trans. Am. Math. Soc. 87, 330--346 (1958; Zbl 0090.39001); \textit{A. Dold}, Ann. Math. (2) 68, 54--80 (1958; Zbl 0082.37701)] gives an equivalence of categories \[ \mathrm{Fun}(\Delta^{\mathrm{op}},A)\simeq\mathrm{Ch}_{\geq 0}(A) \] between simplicial objects in \(A\) and connective chain complexes in \(A\). By replacing the simplex category \(\Delta\) by other categories of similar combinatorial nature, many related equivalences have been constructed [\textit{T. Church} et al., Duke Math. J. 164, No. 9, 1833--1910 (2015; Zbl 1339.55004); \textit{R. Helmstutler}, J. Pure Appl. Algebra 218, No. 7, 1302--1323 (2014; Zbl 1286.55008); \textit{S. Lack} and \textit{R. Street}, J. Pure Appl. Algebra 219, No. 10, 4343--4367 (2015; Zbl 1317.18016); \textit{S. Lack} and \textit{R. Street}, J. Pure Appl. Algebra 224, No. 3, 1364--1366 (2020; Zbl 1423.18032); \textit{T. Pirashvili}, Math. Ann. 318, No. 2, 277--298 (2000; Zbl 0963.18006); \textit{J. Słomińska}, J. Algebra 274, No. 1, 118--137 (2004; Zbl 1042.18010); \textit{J. Słomińska}, Bull. Pol. Acad. Sci., Math. 59, No. 1, 33--40 (2011; Zbl 1225.18001)]. The principal objective in this paper is to simultaneously generalize these equivalences in the homotopy coherent context of \(\infty\)-categories. To this end, categories \(B\) endowed with the structure \[ \mathbb{B}=(B,E,E^{\vee}) \] of a so-called \textit{DK-triple} (Definition 3.1.1) are studied, while a pointed category \(N_{0}=N_{0}(\mathbb{B})\) is associated to each DK-triple \(\mathbb{B}\). The princpal result is the following theorem. Theorem. For each weakly idempotent complete additive \(\infty\)-category \(\mathcal{A}\), the DK-triple \(\mathbb{B}\) induces a natural equivalence \[ \mathrm{Fun}(B,\mathcal{A})\simeq\mathrm{Fun}^{0}(N_{0},\mathcal{A}) \] between the \(\infty\)-categories of diagrams \(B\rightarrow\mathcal{A}\) and of pointed diagrams \(N_{0}\rightarrow\mathcal{A}\). A synopsis of the paper goes as follows. \begin{itemize} \item[\S 2] gives preliminaries concerning \(\infty\)-categorical notation and tools (\S 2.1), pointed \(\infty\)-categories (\S 2.2), quotient categories and coherent chain complexes (\S 2.3), additive and semiadditive \(\infty\)-categories (\S 2.4) and weakly idempotent complete \(\infty\)-categories (\S 2.5). \item[\S 3] states the main theorem (\S 3.3) after DK-triples (\S 3.1) and key constructions (\S 3.2). \item[\S 4] is concerned with examples of the main theorem. \S 4.1 explains how to equip the simplex category \(B=\Delta^{\mathrm{op}}\) with the structure of a DK-triple, while a similar discussion can be seen in [\textit{S. Lack} and \textit{R. Street}, J. Pure Appl. Algebra 219, No. 10, 4343--4367 (2015; Zbl 1317.18016), Example 3.2]. \S 4.2 addresses an important class of examples of diagonalizable DK-triples out of consideration for partial maps with respect to certain factorization systems after [loc. cit., Example 3.1]. \item[\S 5] gives the proof of the main theorem (\S 5.3) after cofinality lemmas (\S 5.1) and inductive construction in the reduced case (\S 5.2). \item[\S 6] deals with comparison with other works. \S 6.1 argues that when \(\mathcal{A}\) is an idempotent complete additive ordinary category, the main theorem recovers the general Dold-Kan type equivalence of Lack and Street [loc. cit., Theorem 6.8]. \S 6.2 compares the author's approach with Lurie's stable Dold-Kan correspondence [\url{https://people.math.harvard.edu/~lurie/papers/HA.pdf}, Theorem 1.2.4.1] claiming that \[ \mathrm{Fun}(\Delta^{\mathrm{op}},\mathcal{D})\simeq \mathrm{Fun}(\mathbb{N},\mathcal{D}) \] where \(\mathcal{D}\) is a stable \(\infty\)-category, and we have \[ \mathrm{Fun}(\mathbb{N},\mathcal{D})\simeq\mathrm{Ch}_{\geq 0}(D) \] as was shown by \textit{S. Ariotta} [``Coherent cochain complexes and Beilinson t-structures, with an appendix by Achim Krause'', Preprint, \url{arXiv:2109.01017}]. \item[\S 7] addresses further tools and formulas. \end{itemize}