Ieke Moerdijk, Ittay Weiss
The category of dendroidal sets is an extension of that of simplicial sets, suitable for defining nerves of operads rather than just of categories. In this paper, we prove some basic properties of inner Kan complexes in the category of dendroidal sets. In particular, we extend fundamental results of Boardman and Vogt, of Cordier and Porter, and of Joyal to dendroidal sets.
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Dendroidal sets have been introduced as a combinatorial model for homotopy coherent operads. We introduce the notion of fully Kan dendroidal sets and show that there is a model structure on the category of dendroidal sets with fibrant objects given by fully Kan dendroidal sets. Moreover we show that the resulting homotopy theory is equivalent to the homotopy theory of connective spectra.
We prove a Dold-Kan type correspondence between the category of dendroidal abelian groups and a suitably constructed category of dendroidal complexes. Our result naturally extends the classical Dold-Kan correspondence between the category of simplicial abelian groups and the category of non-negatively graded chain complexes.
We introduce the concept of a dendroidal set. This is a generalization of the notion of a simplicial set, specially suited to the study of operads in the context of homotopy theory. We define a category of trees, which extends the category $Δ$ used in simplicial sets, whose presheaf category is the category of dendroidal sets. We show that there is a closed monoidal structure on dendroidal sets which is closely related to the Boardman-Vogt tensor product of operads. Furthermore we show that each operad in a suitable model category has a coherent homotopy nerve which is a dendroidal set, extend…
The homotopy theory of infinity-operads is defined by extending Joyal's homotopy theory of infinity-categories to the category of dendroidal sets. We prove that the category of dendroidal sets is endowed with a model category structure whose fibrant objects are the infinity-operads (i.e. dendroidal inner Kan complexes). This extends the theory of infinity-categories in the sense that the Joyal model category structure on simplicial sets whose fibrant objects are the infinity-categories is recovered from the model category structure on dendroidal sets by simply slicing over the point.
A new model structure on dendroidal spaces for the theory of $\infty$-operads
本文提出了一种新的 dendroidal 空间模型结构，用于 $\infty$-操作符理论，并证明其与 Cisinski-Moerdijk 模型结构的 Quillen 等价性。
We study simplicial action of groups on one vertex Kan complexes. We show that every semi-direct product of the fundamental group of an one vertex Kan complex with a finite group can be simplicially realized. We also calculate the cohomology of the fixed point set of a finite $p-$group action on an one vertex aspherical Kan complex.
Homotopy Limits and Homotopy Colimits of Chain Complexes
本文通过cobar和bar构造给出链复形图的同调极限与余极限公式，并证明Bousfield-Kan公式在任何带框架模型范畴中计算同调余极限。
This article shows several new methods for proofs on Kan complexes while using them to give a compact introduction to the homotopy groups of these complexes. Then more advanced objects are studied starting with homology and the Hurewicz homomorphism. Eilenberg-Mac Lane-spaces are constructed explicitly and can then be used to define spectral cohomology. In the end the equivalence to the usual simplicial cohomology is shown.
In this paper three results are established: firstly, that the homotopy function complexes of Dwyer and Kan can be defined as certain total right derived functors; secondly, that they functorially compute the homotopy type of the hom-spaces in the simplicial localisation; and thirdly, that they can be computed by fibrant replacements in a suitable left Bousfield localisation of the projective model structure on simplicial presheaves.
Examples and cofibrant generation of effective Kan fibrations
该研究针对单纯同伦理论中的有效Kan纤维化，证明单纯Malcev代数是有效Kan复形且其为代数弱因子分解系统右类，还提出两种强化形式并验证结果适用。
An analogous construction of simplicial homotopy group for Kan complex can be applied to saturated complicial sets to give monoids. In this paper, we investigate how the construction of loop spaces of Kan complexes lifts to the complicial setting and relates to such monoids.
We exhibit a Quillen equivalence between two model categories encoding the homotopy theory of stratified spaces : the model category of filtered simplicial sets, and that of filtered spaces. Additionally, we introduce a new class of filtered spaces, that of vertical filtered CW-complexes, providing a nice model for the homotopy category of stratified spaces.
A monoidal Dold-Kan correspondence for comodules
本文构造了余模范畴中基于 Postnikov 塔的归纳纤维性替换，证明单纯余模与连通微分分次余模 Quillen 等价，并推出单连通空间的有理 A-理论等价于相应余模结构完美链复形的 K-理论。
On Simplicial Principal Bundles in Descent Categories
该研究将光滑主G-丛的典范上闭链构造推广到下降范畴的单纯对象主丛语境，特例化得到约化Kan复形k-不变量的简化构造，并用于构造极小Kan复形以阐明选择公理的作用。
A horn-like characterization of the fibrant objects in the minimal model structure on simplicial sets
本文证明了单纯集极小模型结构中纤维对象可由类似角包含的映射的提升条件刻画。
Beck-Chevalley Conditions in Simplicial Sets
本研究介绍Dmitry Skvortsov与Valentin Shehtman提出的一类单纯集并命名为Skvortsov-Shehtman复形，还给出了非Kan复形的该类复形实例，丰富了单纯集相关研究案例。
Viewing Kan complexes as $\infty$-groupoids implies that pointed and connected Kan complexes are to be viewed as $\infty$-groups. A fundamental question is then: to what extent can one "do group theory" with these objects? In this paper we develop a notion of a finite $\infty$-group: an $\infty$-group with finitely many non-trivial homotopy groups which are all finite. We prove a homotopical analog of the Sylow theorems for finite $\infty$-groups. We derive two corollaries: the first is a homotopical analog of the Burnside's fixed point lemma for $p$-groups and the second is a "group-theoretic…
We give an elementary construction of a certain class of model structures. In particular, we rederive the Kan model structure on simplicial sets without the use of topological spaces, minimal complexes, or any concrete model of fibrant replacement such as Kan's Ex^infinity functor. Our argument makes crucial use of the glueing construction developed by Cohen et al. in the specific setting of certain cubical sets.
A Morse Mayer-Vietoris Sequence for Cosheaf Homology on Simplicial Complexes
该研究针对有限单纯复形上余层同调，构造迈耶-菲托里斯短正合序列并结合离散莫尔斯理论得到莫尔斯链复形的对应序列，建立其与标准序列的拟同构，还涉及范畴论视角的相关探讨。
For every simplicial complex X, we construct a locally CAT(0) cubical complex T_X, a cellular isometric involution i on T_X and a map t_X from T_X to X with the following properties: t_Xi = t_X; t_X is a homology isomorphism; the induced map from the quotient of T_X by the involution i to X is a homotopy equivalence; the induced map from the fixed point subspace for i in T_X to X is a homology isomorphism. The construction is functorial in X. One corollary is an equivariant Kan-Thurston theorem: every connected proper G-CW-complex has the same equivariant homology as the classifying space for …
For a category $\mathcal E$ with finite limits and well-behaved countable coproducts, we construct a model structure, called the effective model structure, on the category of simplicial objects in $\mathcal E$, generalising the Kan--Quillen model structure on simplicial sets. We then prove that the effective model structure is left and right proper and satisfies descent in the sense of Rezk. As a consequence, we obtain that the associated $\infty$-category has finite limits, colimits satisfying descent, and is locally Cartesian closed when $\mathcal E$ is, but is not a higher topos in general.…
The Hurewicz model structure on simplicial $R$-modules
本文基于Christensen-Hovey定理和Dold-Kan对应，描述了单纯R-模上的Hurewicz模型结构，并证明其与链复形情形均为幺半模型范畴。
Thomason-Type Model Structures on Simplicial Complexes and Graphs
本文研究了单纯复形和图上的Thomason型模型结构，证明其与Kan-Quillen模型结构相关，并展示这些结构的共纤生成性和恰当性。
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。
We construct a symmetric monoidal category $LIE^{MC}$ whose objects are shifted L-infinity algebras equipped with a complete descending filtration. Morphisms of this category are "enhanced" infinity morphisms between shifted L-infinity algebras. We prove that any category enriched over $LIE^{MC}$ can be integrated to a simplicial category whose mapping spaces are Kan complexes. The advantage gained by using enhanced morphisms is that we can see much more of the simplicial world from the L-infinity algebra point of view. We use this construction in a subsequent paper to produce a simplicial mod…
From Nilspace Theory to Simplicial Homotopy Theory
本文构造了从立方空间范畴到凝聚单纯集范畴的忠实函子，将零空间的弱结构定理转化为单纯集的Postnikov塔，并揭示了零空间结构群与Kan复形同伦群的对应关系。
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…
This paper develops the foundations of a simplicial theory of weak omega-categories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets provides a common generalisation of the theories of (strict) omega-categories, Kan complexes and Joyal's quasi-categories. We generalise a number of results due to the current author with regard to complicial sets and strict omega-categories to provide an armoury of well behaved technical devices, such as joins and Gray tensor products, which will be used to s…
Given any modular category $\mathcal{C}$ over an algebraically closed field $k$, we extract a sequence $(M_g)_{g\geq 0}$ of $\mathcal{C}$-bimodules. We show that the Hochschild chain complex $CH(\mathcal{C};M_g)$ of $\mathcal{C}$ with coefficients in $M_g$ carries a canonical homotopy coherent projective action of the mapping class group of the surface of genus $g+1$. The ordinary Hochschild complex of $\mathcal{C}$ corresponds to $CH(\mathcal{C};M_0)$. This result is obtained as part of the following more comprehensive topological structure: We construct a symmetric monoidal functor $\mathfra…
In this paper we study the question of how to transfer homotopic structure from the category sD of simplicial objects in a fixed category D to D. To this end we use a sort of homotopy colimit s : sD --> D, which we call simple functor. For instance, the Bousfield-Kan homotopy colimit in a Quillen simplicial model category is an example of simple functor. As a remarkable example outside the setting of Quillen models we include Deligne simple of mixed Hodge complexes. We prove here that the simple functor induces an equivalence on the corresponding localized categories. We also describe a natura…
Strictification of $\infty$-Groupoids is Comonadic
本文研究了弱$\infty$-广群到严格$\infty$-广群的泛严格化伴随，证明任何单纯集合均能通过该伴随诱导的典范余单纯分解总化在同伦弱等价意义下恢复，并由此得出该严格化伴随在拟范畴间构成余单子伴随。
Simplicial lists in operad theory I
本文定义了单纯列表范畴 $\mathsf{sList}$，通过构造完全忠实神经函子证明其支持非对称算畴组合学，为非对称 $\infty$-算畴理论提供合理模型，并研究了其预层结构、连贯神经函子及同调群。
In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterize its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove that any adjunction of quasi-categories extends to a homotopy coherent adjunction and furthermore that these extensions are homotopically unique in the sense that the relevant spaces of extensions are contr…
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…
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