Rune Haugseng
In this short note we prove that two definitions of (co)ends in $\infty$-categories, via twisted arrow $\infty$-categories and via $\infty$-categories of simplices, are equivalent. We also show that weighted (co)limits, which can be defined as certain (co)ends, can alternatively be described as (co)limits over left and right fibrations, respectively.
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We prove and explain several classical formulae for homotopy (co)limits in general (combinatorial) model categories which are not necessarily simplicially enriched. Importantly, we prove versions of the Bousfield-Kan formula and the fat totalization formula in this complete generality. We finish with a proof that homotopy-final functors preserve homotopy limits, again in complete generality.
We generalize the notion of ends and coends in category theory to the realm of module categories over finite tensor categories. We call this new concept "module (co)end". This tool allows us to give different proofs to several known results in the theory of representations of finite tensor categories. As a new application, we present a description of the relative Serre functor for module categories in terms of a module coend, in a analogous way as a Morita invariant description of the Nakayama functor of abelian categories presented in [J. Fuchs, G. Schaumann and C. Schweigert, Eilenberg-Watts…
Grothendieck $\infty$-groupoids, and still another definition of $\infty$-categories
本文的目的是提出一个简化版本的概念$\infty$ -广群的Grothendieck在“追求堆栈”，并介绍了一个定义的$\infty$ -范畴的启发Grothendieck的方法。
A simple model for twisted arrow $\infty$-categories
本文提出一个简单模型用于 $(\infty,2)$-范畴的扭曲箭 $\infty$-范畴，扩展了 Lurie 的原始工作。
Hammocks and fractions in relative $\infty$-categories
本文研究了单纯空间的$\infty$-范畴$sS$中的$\infty$ -范畴的同伦理论.也就是说，我们认为$sS$ -丰富的$\infty$ -范畴是普通$\infty$ -范畴通过“局部”几何实现函子$Cat_{sS} \to Cat_\infty$的表示，并且我们证明了它们的同伦理论表示$\infty$ -范畴的$\infty$ -范畴，即这个函子从$sS$ -丰富的$\infty$-范畴的$\infty$-范畴的局部化导出等价$Cat_{sS} [[ W_{DK}^{-1} ]] \xrightarrow{\sim} Cat_\infty$.在Dwyer-Kan之后，我们定义了一个从相对$\infty$ -范畴到$sS$ -丰富的$\infty$ -范畴的 *hammock localization* 函子，从而提供了丰富的$sS$ -丰富的$\infty$ -范畴的例子.同时对他们的一个关键结果进行了解包和推广，证明了给定一个包含同伦三箭头演算的相对$\infty$ -范畴，可以用一种更明确和更容易理解的方式来描述$\infty$ -范畴中由其吊床局部化所表示的hom-space.作为这个框架的应用，我们给出了相对$\infty$ -范畴的Rezk神经是（完全）Segal空间的充分条件，推广了与Low的联合工作.
In this paper we introduce a notion of $\mathbf{O}$-monoidal $\infty$-categories for a finite sequence $\mathbf{O}^{\otimes}$ of $\infty$-operads, which is a generalization of the notion of higher monoidal categories in the setting of $\infty$-categories. We show that the $\infty$-category of coCartesian $\mathbf{O}$-monoidal $\infty$-categories and right adjoint lax $\mathbf{O}$-monoidal functors is equivalent to the opposite of the $\infty$-category of Cartesian $\mathbf{O}_{\rm rev}$-monoidal $\infty$-categories and left adjoint oplax $\mathbf{O}_{\rm rev}$-monoidal functors, where $\mathbf…
The goal of this paper is to prove an equivalence between the $(\infty,2)$-category of cartesian factorization systems of $\infty$-categories and that of pointed cartesian fibrations of $\infty$-categories. This generalizes a similar result known for ordinary categories and sheds some light on the interplay between these two seemingly distant concepts.
We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal $\infty$-category $\mathcal{V}$. Our theory of enriched $\infty$-categories has many desirable properties; for instance, if the enriching $\infty$-category $\mathcal{V}$ is presentably symmetric monoidal then $\mathrm{Cat}^\mathcal{V}_\infty$ is as well. These features render the theory useful even when an $\infty$-category of enriched $\infty$-categories comes from a model category (as is often the case in examples of interest, e.g. dg-categories, spectral categories, and $(\infty,n)$-categories). This…
Factorization Systems on $\infty$-Categories: Un/Straightening and Monadicity
该研究证明了∞-范畴上分解系统的两个结构性结果：一是对一类∞-范畴纤维化全空间的分解系统分类，二是相关遗忘函子为单子右伴随，丰富了∞-范畴的结构理论。
Cores and localizations of $(\infty,\infty)$-categories
本文研究了$(\infty,\infty)$-范畴在极限情况下的核与局部化，探讨了其作为反射局部化的关系及中间局部化的性质。
We use Lurie's symmetric monoidal envelope functor to give two new descriptions of $\infty$-operads: as certain symmetric monoidal $\infty$-categories whose underlying symmetric monoidal $\infty$-groupoids are free, and as certain symmetric monoidal $\infty$-categories equipped with a symmetric monoidal functor to finite sets (with disjoint union as tensor product). The latter leads to a third description of $\infty$-operads, as a localization of a presheaf $\infty$-category, and we use this to give a simple proof of the equivalence between Lurie's and Barwick's models for $\infty$-operads.
The Grothendieck Construction for $\infty$-Categories Fibered over Categorical Patterns
本文通过相对化Grothendieck构造，为纤维化于范畴模式中的∞-范畴族建立纤维化处理框架，并构造万有共笛卡尔纤维化的类比，给出极限与余极限的计算方法。
Comparison of models for $(\infty, n)$-categories, I
虽然目前正在使用$(\infty,1)$类别的许多不同模型，但已知它们彼此是Quillen等效的。它们的几个高阶类似物正在被开发作为$(\infty, n)$ -范畴的模型。在本文中，我们建立模型结构的一些自然产生的范畴的对象，应被认为是$(\infty,n)$ -范畴。此外，我们建立了它们之间的Quillen等价。
On duoidal $\infty$-categories
本文在$\infty$-范畴框架下引入双幺半$\infty$-范畴，定义三类函子及相应的三种范畴表述，并据此定义双幺半群等结构。
Models for rational $(\infty, 1)$-categories
本研究引入有理$(\infty, 1)$-范畴的概念，提出有理完全Segal空间与有理Segal范畴两种模型，并证明二者等价，为有理高阶同伦理论提供新的范畴化模型框架。
We introduce a notion of bimodule in the setting of enriched $\infty$-categories, and use this to construct a double $\infty$-category of enriched $\infty$-categories where the two kinds of 1-morphisms are functors and bimodules. We then consider a natural definition of natural transformations in this context, and show that in the underlying $(\infty,2)$-category of enriched $\infty$-categories with functors as 1-morphisms the 2-morphisms are given by natural transformations.
On bi-enriched $\infty$-categories
本文将Lurie的富集$\infty$-范畴推广为左富集、右富集和双富集$\infty$-范畴，并利用双富集结构研究富集函子范畴、富集Kan扩张及相关end/coend公式，发展了独立的富集$\infty$-范畴理论。
A theorem of Retakh for exact $\infty$-categories and higher extension functors
该研究针对正合∞范畴，通过双纤维化定义扩张∞范畴，证明其稳定包络不变且构成Ω谱以推广Retakh定理，还验证了其同伦群与对应外三角范畴的高阶扩张群自然同构。
The Gray tensor product of $(\infty,n)$-categories
本文基于粘贴定理构造了$(\infty,n)$-范畴的新模型，并借助Day卷积与反射实现了其Gray张量积，进而用不依赖模型的泛性质加以刻画。
Enriched $\infty$-categories as marked module categories
本研究证明富化$\u221e$-范畴可由带可表预层标记的富化预层范畴完全刻画，建立了其与带$\u221e$-群胚函子的可展示$\u2113$-模范畴满子范畴的等价，借此构造新的富化$\u221e$-范畴张量积并重新表述单值性概念。
Homotopy Posets, Postnikov Towers, and Hypercompletions of $\infty$-Categories
研究同伦格在∞-范畴中的扩展，揭示其作为纤维化长正合序列的层次，并探讨Postnikov塔的收敛性及超完备∞-范畴的构造。
We show that the homotopy theory of strict 2-categories embeds in that of $(\infty,2)$-categories in the form of 2-precomplicial sets. More precisely, we construct a nerve-categorification adjunction that is a Quillen pair between Lack's model structure for 2-categories and Riehl-Verity's model structure for 2-complicial sets. Furthermore, we show that Lack's model structure is transferred along this nerve and that the nerve is homotopically fully faithful.
A survey of (\infty, 1)-categories
本文给出了所谓的（\infty，1）-范畴的不同定义之间的比较。（\infty，1）-范畴被认为是n-态射都可逆的\infty -范畴的模型。从同伦理论的观点来看，它们也是同伦理论中的同伦理论的模型。这四种不同的结构都是等价的，它们是单纯范畴、西格尔范畴、完备西格尔空间和拟范畴。
Model $\infty$-categories II: Quillen adjunctions
我们证明了模型$\infty$ -范畴上的各种结构在它们的局部化上下降到相应的结构：（i）Quillen算子;（ii）两变量Quillen算子;（iii）monoidal和对称monoidal模型结构;和（iv）丰富的模型结构。
This paper is the second in a series of two papers about generalizing Quillen's Theorem A to strict $\infty$-categories. In the first one, we presented a proof of this Theorem A of a simplicial nature, direct but somewhat ad hoc. In the current paper, we give a conceptual proof of an $\infty$-categorical nature of the same theorem. This proof is based on the theory of join and slices for strict $\infty$-categories developed by the authors in a previous paper, and on a comma construction for strict $\infty$-categories generalizing classical comma categories and Gray's comma 2-categories. This $…
An example of the non-existence of coequalizers in the category of the A$_{\infty}$-categories
本文证明严格单位与非单位的A∞-代数及A∞-范畴范畴在相应态射下均无余等化子，给出反例。
Goerss--Hopkins obstruction theory is a powerful tool for constructing structured ring spectra from purely algebraic data. Using the formalism of model $\infty$-categories, we provide a generalization that applies in an arbitrary presentably symmetric monoidal stable $\infty$-category (such as that of equivariant spectra or of motivic spectra).
An equivalence between enriched $\infty$-categories and $\infty$-categories with weak action
该研究证明Lurie与Gepner-Haugseng两种意义下的$\u2118$-富化∞-范畴等价，建立了对应松弛幺半函子限制关系，并给出稳定∞-范畴嵌入与富化Yoneda嵌入两项应用。
We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of $(\infty,1)$-categories and $\infty$-operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structu…
We prove the compatibility between the suspension construction and the complicial nerve of $ω$-categories. As a motivating application, we produce a Quillen pair between the models of $(\infty,n)$-categories given by Rezk's complete Segal $Θ_n$-spaces and Verity's $n$-complicial sets.
We define for each $n \geq 1$ a symmetric monoidal $(\infty, n+1)$-category $n\mathrm{Pr}^L$ whose objects we call presentable $(\infty,n)$-categories, generalizing the usual theory of presentable $(\infty,1)$-categories. We show that each object $\mathcal{C}$ in $n\mathrm{Pr}^L$ has an underlying $(\infty,n)$-category $ψ_n(\mathcal{C})$ which admits all conical colimits, and that conical colimits of right adjointable diagrams in $ψ_n(\mathcal{C})$ can be computed in terms of conical limits after passage to right adjoints.
We use the basic expected properties of the Gray tensor product of $(\infty,2)$-categories to study (co)lax natural transformations. Using results of Riehl-Verity and Zaganidis we identify lax transformations between adjunctions and monads with commutative squares of (monadic) right adjoints. We also identify the colax transformations whose components are equivalences (generalizing the "icons" of Lack) with the 2-morphisms that arise from viewing $(\infty,2)$-categories as simplicial $\infty$-categories. Using this characterization we identify the $\infty$-category of monads on a fixed object …
We construct a flagged $\infty$-category ${\sf Corr}$ of $\infty$-categories and bimodules among them. We prove that ${\sf Corr}$ classifies exponentiable fibrations. This representability of exponentiable fibrations extends that established by Lurie of both coCartesian fibrations and Cartesian fibrations, as they are classified by the $\infty$-category of $\infty$-categories and its opposite, respectively. We introduce the flagged $\infty$-subcategories ${\sf LCorr}$ and ${\sf RCorr}$ of ${\sf Corr}$, whose morphisms are those bimodules which are \emph{left final} and \emph{right initial}, re…
From Hopf Algebras in Model Categories to Hopf Algebras in $\infty$-categories
本文证明了对代数无余纤维性或纤维性假设时，模型范畴中的代数对象可转移到 $\infty$-范畴中，并在温和假设下将该对应推广至交换双代数和交换 Hopf 代数。
Categorical spectra as pointed $(\infty,\mathbb{Z})$-categories
该研究给出弱$\u2124$-范畴（即$(\u221e,\u2124)$-范畴）的$\u221e$-范畴定义，证明分类谱等价于带基点的$(\u221e,\u2124)$-范畴，验证二者胞元的一致性并还原Lessard对谱的相关描述。
Fully faithful functors and pushouts of $\infty$-categories
本文研究了完全忠实函子的稳定性，计算了沿其的$\infty$-范畴推出中的映射anima，并应用于Dwyer函子和Reedy范畴的推出。
On the tensor product of enriched $\infty$-categories
本文证明充实∞-范畴的张量积保持余极限并修正了前人错误，同时证明本质满射与完全忠实函子构成分解系统且与张量积及内部Hom兼容。
For most models of $(\infty,2)$-categories an embedding of the $\infty$-category of 2-categories into that of $(\infty,2)$-categories has been constructed in the form of a nerve construction of some flavor. We prove that all those nerve embeddings induce equivalent functors, modulo change of model. We also show that all the nerve embeddings realize the $\infty$-category of 2-categories as the sub-$\infty$-category of $(\infty,2)$-categories that are local with respect to a certain class of maps.
We define and discuss lax and weighted colimits of diagrams in $\infty$-categories and show that the coCartesian fibration associated to a functor is given by its lax colimit. A key ingredient, of independent interest, is a simple characterization of the free Cartesian fibration associated to a a functor of $\infty$-categories. As an application of these results, we prove that lax representable functors are preserved under exponentiation, and also that the total space of a presentable Cartesian fibration between $\infty$-categories is presentable, generalizing a theorem of Makkai and Paré to t…
Double categorical model of $(\infty,1)$-categories
基于已有工作，本文在双范畴上构造了四种模型结构，分别建模单纯空间、Segal空间、$(\infty,1)$-范畴和$\infty$-群胚，并利用Grothendieck构造给出了计算同伦余极限的显式公式。
Monoidal Relative Categories Model Monoidal $\infty$-Categories
本文证明单调相对范畴的同调理论等价于单调∞-范畴的同调理论，并在对称单调情况下同样成立。通过应用，证明了每个可呈现的单调或可呈现的对称单调∞-范畴都由单调或对称单调模型范畴呈现。
The homotopy bicategory of $(\infty, 1)$-categories
给出了$(\infty, 1)$ -范畴的$(\infty, 2)$-范畴的同伦双范畴的Joyal-Riehl-Verity构造的正确性的证据;特别地，证明了用完备Segal空间代替拟范畴的类似构造产生了双范畴等价的2-范畴.
Equivalence of models for equivariant $(\infty, 1)$-categories
在本文中，我们证明了$(\infty, 1)$ -范畴的已知模型都可以扩展到任何离散群$G$的等变版本。我们表明，在两个模型中，我们也可以考虑任何单纯组$G$的行动。
In this paper we study duoidal structures on $\infty$-categories of operadic modules. Let $\mathcal{O}^{\otimes}$ be a small coherent $\infty$-operad and let $\mathcal{P}^{\otimes}$ be an $\infty$-operad. If a $\mathcal{P}\otimes\mathcal{O}$-monoidal $\infty$-category $\mathcal{C}^{\otimes}$ has a sufficient supply of colimits, then we show that the $\infty$-category ${\rm Mod}_A^{\mathcal{O}}(\mathcal{C})$ of $\mathcal{O}$-$A$-modules in $\mathcal{C}^{\otimes}$ has a structure of $(\mathcal{P},\mathcal{O})$-duoidal $\infty$-category for any $\mathcal{P}\otimes\mathcal{O}$-algebra object $A$.
Free fibrations, lax colimits and Kan extensions for $(\infty,2)$-categories
本文研究了$(\infty,2)$-范畴的纤维化、余限和Kan扩展，提出了自由纤维化的构造，并给出了部分懒惰余限和Kan扩展的纤维化描述。
A homotopy coherent nerve for $(\infty,n)$-categories
本文为 $(\infty,n)$-范畴构造同伦相干神经，证明其给出严格丰富范畴与 Segal 范畴对象模型间的右 Quillen 等价，从而统一两种同伦相干图定义。
A Quillen's Theorem A for strict $\infty$-categories I: the simplicial proof
本文的目的是证明Quillen著名定理A在严格$\infty$ -范畴上的一个推广。这个结果是核心的同伦理论的严格$\infty$范畴的作者。这里提出的证明是一个单纯的性质，并使用施泰纳的理论增广定向复形。在随后的论文中，我们将证明相同的结果，纯粹的$\infty$ -范畴的方法。
Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper we prove general adjoint functor theorems for functors between $\infty$-categories. One of our main results is an $\infty$-categorical generalization of Freyd's classical General Adjoint Functor Theorem. As an application of this result, we recover Lurie's adjoint functor theorems for presentable $\infty$-categories. We also discuss the comparison between adjunctions of $\infty$-categories and homotopy adjunctions, and give a treatment of Brown representability for $\infty$-catego…
Derived $\infty$-categories as exact completions
本文发展了正则 $\infty$-范畴的正合完备化理论，证明其可恢复阿贝尔范畴导出 $\infty$-范畴的连通部分，并扩展了阿贝尔范畴的经典刻画及建立了相关定理的 $\infty$-范畴版本。
Limits of $(\infty, 1)$-categories with structure and their lax morphisms
本研究将Riehl与Verity关于准范畴Eilenberg-Moore对象保极限的结果推广到多类带结构的$(\n\infty,1)$-范畴模型，证明仅需图中一个态射保结构时就存在含插入子、等化子的松弛态射极限，给出了这类极限存在的最小条件与清晰解释。
Sketchable infinity categories
本文将草图理论同伦相干地推广到$\infty$-范畴，证明了可展示$\infty$-范畴是极限草图模型，可及$\infty$-范畴是任意草图模型，并为多种$\infty$-范畴给出了具体草图。
Quasi-unital $\infty$-Categories
受Lurie的拟酉代数理论的启发，我们证明了$\infty$ -范畴的一个类似结果.特别是，我们表明，单位结构的$\infty$类可以唯一地恢复从底层的非单位结构，一旦合适的候选单位已被确定。本文的主要结果可以用来证明1维配边假设，在即将发表的论文中描述。
A local-global principle for parametrized $\infty$-categories
本文提出了一种关于参数化∞-范畴的局部-整体原理，通过局部数据和粘合数据描述了∞-范畴的结构，并推导了双∞-范畴CORR的分类。
This chapter, written for "Stable categories and structured ring spectra," edited by Andrew J. Blumberg, Teena Gerhardt, and Michael A. Hill, surveys the history of homotopical categories, from Gabriel and Zisman's categories of fractions to Quillen's model categories, through Dwyer and Kan's simplicial localizations and culminating in $(\infty,1)$-categories, first introduced through concrete models and later re-conceptualized in a model-independent framework. This reader is not presumed to have prior acquaintance with any of these concepts. Suggested exercises are included to fertilize intui…
In this paper we give an example of duoidal $\infty$-categories. We introduce map $\mathcal{O}$-monoidales in an $\mathcal{O}$-monoidal $(\infty,2)$-category for an $\infty$-operad $\mathcal{O}^{\otimes}$. We show that the endomorphism mapping $\infty$-category of a map $\mathcal{O}$-monoidale is a coCartesian $(Δ^{\rm op},\mathcal{O})$-duoidal $\infty$-category. After that, we introduce a convolution product on the mapping $\infty$-category from an $\mathcal{O}$-comonoidale to an $\mathcal{O}$-monoidale. We show that the $\mathcal{O}$-monoidal structure on the duoidal endomorphism mapping $\i…
Comparing lax functors of $(\infty,2)$-categories
本文研究了$(\infty,2)$-范畴的oplax正规化函子，证明了带尺度单纯集与完备Segal对象两种设定下的概念等价，并利用部分lax Kan扩张新技术构建了增强Segal对象模型。
Derived mapping spaces of $\infty$-categories
本文证明了导出映射空间引理，推广了 Cisinski 关于 $\infty$-范畴分数演算的核心定理，为分析范畴局部化中的映射空间提供了统一框架，并给出了立方或单纯范畴是底层范畴同伦等价局部化的充分条件。
Kan extensions and the calculus of modules for $\infty$-categories
$(\infty,1)$ -范畴的各种模型，包括拟范畴、完备Segal空间、Segal范畴和自然标记单纯集，都可以看作是$\infty$ -宇宙的对象。在一般的$\infty$ -宇宙中，我们称其对象为$\infty$ -范畴，我们在$\infty$ -范畴之间引入模（也称为原函子或对应），具体表现为具有群胚纤维的适当定义的纤维化的跨度。顾名思义，从$A$到$B$的模块在适当的意义上是配备有$A$的左动作和$B$的右动作的$\infty$类别。应用米田引理的纤维化形式，我们开发了一个通用演算的模块，证明他们自然组装成一个多范畴的结构称为虚拟设备，这是已知的是一个强大的设置，在其中开发正式的范畴论。使用微积分的模块，它是简单的定义和研究逐点Kan扩展，我们涉及的情况下，carnival封闭的$\infty$ -cosmoi，限制和colimits的图值在一个$\infty$类，如在以前的工作中介绍。
We prove a generalization of Quillen's Theorem B to strict $\infty$-categories. More generally, we show that under similar hypothesis as for Theorem B, the comma construction for strict $\infty$-categories, that we introduced with Maltsiniotis in a previous paper, is the homotopy pullback with respect to Thomason equivalences. We give several applications of these results, including the construction of new models for certain Eilenberg-Mac Lane spaces.
A model-independent Gray tensor product for $(\infty,2)$-categories
本文构造了 $(\infty,2)$-范畴的模型无关 Gray 张量积，并证明其由与经典严格 2-范畴 Gray 张量积在 Gray 立方体上一致的唯一幺半双闭结构所刻画。
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