余模的幺半 Dold-Kan 对应

对于平行循环模的Dold-Kan定理

The Dold-Kan theorem for paracyclic modules

本文通过直接关联Karoubi算子与归一化子复形，证明了平行循环模的Dold-Kan定理。

通过表示论的$\boldsymbol{\text{∞}}$-Dold-Kan对应

$\infty$-Dold-Kan correspondence via representation theory

本研究用纯导出子论重新表述并证明Happel和Ladkani的经典结果，得到有界链复形的$\text{∞}$-Dold-Kan对应导出子论版本，搭建同伦论与表示论桥梁，且该等价可在通用倾斜理论中实现为谱双模作用。

A bialgebra axiom and the Dold-Kan correspondence

We introduce a bialgebra axiom for a pair $(c,\ell)$ of a colax-monoidal and a lax-monoidal structures on a functor $F\colon \mathscr{M}_1\to \mathscr{M}_2$ between two (strict) symmetric monoidal categories. This axiom can be regarded as a weakening of the property of $F$ to be a strict symmetric monoidal functor. We show that this axiom transforms well when passing to the adjoint functor or to the categories of monoids. Rather unexpectedly, this axiom holds for the Alexander-Whitney colax-monoidal and the Eilenberg-MacLane lax-monoidal structures on the normalized chain complex functor in th…

Coalgebras in the Dwyer-Kan localization of a model category

We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $\infty$-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories. The result will induce a Dold-Kan correspondence of coalgebras in $\infty$-categories. Moreover, it shows that Shipley's zig-zag of Quillen equivalences lifts to an explicit symmetric monoidal equivalence of $\infty$-categories for the stable Dold-Kan correspondence. We study homotopy coherent coalgebras associated to a monoidal monoidal category. We show examples when these coalgebras cannot be rigi…

同伦 Dold–Kan 对应

Homotopical Dold-Kan Correspondences

论文围绕 test category 推广 Dold–Kan 对应，定义 Whitehead 范畴并构造模型结构，证明同调函子在局部化后给出范畴等价。

高阶范畴的同调

Homology of higher categories

本文基于Eilenberg-Steenrod公理定义高阶范畴的同调，通过范畴论类比公理建立Dold-Kan对应，利用Street神经网络作为奇异复形，推导出具有乘法结构的高阶同调计算。

Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence

The (dual) Dold-Kan correspondence says that there is an equivalence of categories $K:\cha\to \Ab^Δ$ between nonnegatively graded cochain complexes and cosimplicial abelian groups, which is inverse to the normalization functor. We show that the restriction of $K$ to $DG$-rings can be equipped with an associative product and that the resulting functor $DGR^*\to\ass^Δ$, although not itself an equivalence, does induce one at the level of homotopy categories. The dual of this result for chain $DG$ and simplicial rings was obtained independently by S. Schwede and B. Shipley through different method…

A categorified Dwyer-Kan correspondence

The classical Dold-Kan correspondence is known to admit a categorification in the form of an equivalence between the $\infty$-categories of $2$-simplicial stable $\infty$-categories and connective chain complexes of stable $\infty$-categories. In this work, we extend these concepts to give a categorification of the classical Dwyer-Kan correspondence by showing that the $\infty$-category of $2$-duplicial stable $\infty$-categories is equivalent to the $\infty$-category of conective chain complexes with right adjoints.

Simplicial Homotopy Theory, Link Homology and Khovanov Homology

The purpose of this note is to point out that simplicial methods and the well-known Dold-Kan construction in simplicial homotopy theory can be fruitfully applied to convert link homology theories into homotopy theories. Dold and Kan prove that there is a functor from the category of chain complexes over a commutative ring with unit to the category of simplicial objects over that ring such that chain homotopic maps go to homotopic maps in the simplicial category. Furthermore, this is an equivalence of categories. In this way, given a link homology theory, we construct a mapping taking link diag…

Differential graded versus Simplicial categories

We construct a zig-zag of Quillen adjunctions between the homotopy theories of differential graded and simplicial categories. In an intermediate step we generalize Shipley-Schwede's work on connective DG algebras by extending the Dold-Kan correspondence to a Quillen equivalence between categories enriched over positive graded chain complexes and simplicial k-modules. As an application we obtain a conceptual explanation of Simpson's homotopy fiber construction.

关于骨架的乘积

On products of skeleta

该研究在相容有限余极限的对称幺半∞-范畴中，证明骨架滤过函子是典范松弛对称幺半的，同时建立了∞-前幺半范畴的局部化等新结论，为同伦论中相关构造提供了范畴论基础。

Equivalences of monoidal model categories

We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000), 491-511]. As an application we extend the Dold-Kan equivalence to show that the model categories of simplicial rings, modules and algebras are Quillen equivalent to the associated model categories of connected differential g…

A Riemann--Hilbert correspondence for infinity local systems

We describe an $A_\infty$-quasi-equivalence of dg-categories between the first authors' $\mathcal{P}_{\mathcal{A}}$ ---the category of category of prefect $A^0$-modules with flat $\Z$-connection, corresponding to the de Rham dga $\mathcal{A}$ of a compact manifold $M$--- and the dg-category of \emph{infinity-local systems} on $M$ ---homotopy coherent representations of the smooth singular simplicial set of $M$, $\Pinf$. We understand this as a generalization of the Riemann--Hilbert correspondence to $\Z$-connections ($\Z$-graded superconnections in some circles). In one formulation an infinity…