Combinatorial categorical equivalences of Dold-Kan type
Stephen Lack, Ross Street
备注：
We found a gap in the proof of Proposition 6.5 in version 4, which we have fixed with a new Assumption 2.6 in version 5. This holds in all our examples as shown in Proposition 2.9 of version 5. This new assumption is used only in the proof of what is now Theorem 6.5. Some other tidying up and minor corrections have also been made
发表信息：
Journal of Pure and Applied Algebra 219 (2015), pp. 4343-4367
We prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let $\mathscr{X}$ denote an additive category with finite direct sums and split idempotents. The class includes (a) the Dold-Puppe-Kan theorem that simplicial objects in $\mathscr{X}$ are equivalent to chain complexes in $\mathscr{X}$; (b) the observation of Church, Ellenberg and Farb that $\mathscr{X}$-valued species are equivalent to $\mathscr{X}$-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 $\mathscr{X}$ is semi-abelian, we prove the adjunction that was an equivalence is now at least monadic, in the spirit of a theorem of D. Bourn.
By using the Dold-Kan correspondence we construct a Quillen adjunction between the model categories of non-cocommutative coassociative simplicial and differential graded coalgebras over a field. We restrict to categories of connected coalgebras and prove a Quillen equivalence between them.
We establish a large class of homotopy coherent Morita-equivalences of Dold-Kan type relating diagrams with values in any weakly idempotent complete additive $\infty$-category; the guiding example is an $\infty$-categorical Dold-Kan correspondence between the $\infty$-categories of simplicial objects and connective coherent chain complexes. Our results generalize many known 1-categorical equivalences such as the classical Dold-Kan correspondence, Pirashvili's Dold-Kan type theorem for abelian $Γ$-groups and, more generally, the combinatorial categorical equivalences of Lack and Street.
In this work, we establish a categorification of the classical Dold-Kan correspondence in the form of an equivalence between suitably defined $\infty$-categories of simplicial stable $\infty$-categories and connective chain complexes of stable $\infty$-categories. The result may be regarded as a contribution to the foundations of an emerging subject that could be termed categorified homological algebra.
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Self-Revising Discovery Systems for Science: A Categorical Framework for Agentic Artificial Intelligence
Laboratory for Atomistic and Molecular Mechanics（原子分子力学实验室）；Department of Biological Engineering（生物工程系）；Massachusetts Institute of Technology（麻省理工学院）；Department of Civil and Environmental Engineering（土木与环境工程系）；Department of Mechanical Engineering（机械工程系）；Center for Computational Science and Engineering（计算科学与工程中心）；Schwarzman College of Computing（施瓦茨曼计算学院）
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