On the $\mathbb{Z} D_\infty$-category
Michel Van den Bergh
本文给出了Idun Reiten和作者在具有Serre对偶的Noether遗传范畴的分类中引入的$\ZZ D_\infty$范畴的性质的一个直接证明。
In this paper we give a direct proof of the properties of the $\ZZ D_\infty$ category which was introduced in the classification of noetherian, hereditary categories with Serre duality by Idun Reiten and the author.
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Cosheaves are a dual notion of sheaves. In this paper, we prove existence of a dual of sheafifications, called \textit{cosheafifications}, in the $\infty$-category theory. We also prove that the $\infty$-category of $\infty$-cosheaves is presentable and equivalent to an $\infty$-category of left adjoint functors.
An $\infty$-cosmos is a setting in which to develop the formal category theory of $(\infty,1)$-categories. In this paper, we explore a few atypical examples of $\infty$-cosmoi whose objects are 2-categories or bicategories rather than $(\infty,1)$-categories and compare the formal category theory that is so-encoded with classical 2-category and bicategory theory. We hope this work will inform future explorations into formal $(\infty,2)$-category theory.
We show that the strict 1-category $\square$ of cubes -- defined to be the full subcategory of strict $ω$-categories whose objects are the Gray tensor powers of the arrow category -- are dense in the $(\infty,1)$-category $\mathsf{Cat}_ω$ of weak $(\infty,\infty)$-categories, in both Rezk-complete and incomplete variants. More precisely, we show that Joyal's category $Θ$ is contained in the idempotent completion of $\square$, and in fact that the idempotent completion of $\square$ is closed under suspensions and wedge sums. This result extends a theorem of Campbell and Maehara in dimension 2. …
We extend the notion of a factorization system in a category to the realm of $\infty$-categories. To this end, we provide a description of the category of $\infty$-categories with factorization systems as the category of presheaves of spaces on a certain category that satisfy a version of the Segal condition. Additionally, we study the notion of distributive laws between monads and, more generally, lax functors in the context of categories with factorization systems. In particular, we also characterize categories with factorization systems as distributive laws in the bicategory of spans.
Univalence was first defined in the setting of homotopy type theory by Voevodsky, who also (along with Kapulkin and Lumsdaine) adapted it to a model categorical setting, which was subsequently generalized to locally Cartesian closed presentable $\infty$-categories by Gepner and Kock. These definitions were used to characterize various $\infty$-categories as models of type theories. We give a definition for univalent morphisms in finitely complete $\infty$-categories that generalizes the aforementioned definitions and completely focuses on the $\infty$-categorical aspects, characterizing it via…
Given an $\infty$-bicategory $\mathbb{D}$ with underlying $\infty$-category $\mathcal{D}$, we construct a Cartesian fibration $\operatorname{Tw}(\mathbb{D})\to \mathcal{D} \times \mathcal{D}^{\operatorname{op}}$, which we call the enhanced twisted arrow $\infty$-category, classifying the restricted mapping category functor $\operatorname{Map}_{\mathbb{D}}:\mathcal{D}^{\operatorname{op}}\times \mathcal{D} \to \mathbb{D}^{\operatorname{op}} \times \mathbb{D} \to \operatorname{Cat}_{\infty}$. With the aid of this new construction, we provide a description of the $\infty$-category of natural trans…
On the fibrewise effective Burnside $\infty$-category
有效Burnside $\infty$ -范畴是等变稳定同伦理论的$\infty$ -范畴方法的核心。在这篇练习曲中，我们回忆了扭曲箭头$\infty$ -范畴的构造，并利用Joyal-Tierney的一个非常有用的论证修正，给出了它是一个$\infty$ -范畴的新证明。扭曲的箭头$\infty$ -范畴又被用来构造有效的Burnside $\infty$ -范畴。我们采用这个主题的一个变化，构建一个纤维有效的伯恩赛德$\infty$范畴。为了表明这一constuctionworks fibrewise，我们介绍了一个片段的理论，我们称之为大理石纹单纯集，我们使用了一个更进一步修改形式的Joyal-蒂尔尼参数。
In this paper, we apply an explicit construction of a simplicial powering in dg-categories, due to Holstein (2016) and Arkhipov and Poliakova (2018), as well as our own results on homotopy ends (Arkhipov and Ørsted 2018), to obtain an explicit model for the homotopy limit of a cosimplicial system of dg-categories. We apply this to obtain a model for homotopy descent in terms of $\mathrm{A}_{\infty}$-comodules, proving a conjecture by Block, Holstein, and Wei (2017) in the process.
Exact triangles as modules over an $A_\infty$-category
本文描述了一个严格酉的$A_\infty$ -范畴，它的表示是正合三角形，使得正合三角形上的三重对称性在$A_\infty$ -范畴上表现出来.
On autoequivalences of the (\infty, 1)-category of \infty-operads
研究了\infty -运算的自等价的（\infty，1）-范畴.利用Toën，Lurie，Barwick和Schommer-Pries引入的技巧，我们证明了这个（\infty，1）-范畴是一个可收缩的\infty -广群.我们的计算是基于模型的完全树状Segal空间介绍的Cisinski和Moerdijk。类似地，我们证明了非对称\infty -运算的自等价的（\infty，1）-范畴是与Z/2 Z相关联的离散幺半范畴.我们还在Rezk的Θ_n-空间上计算了（\infty，n）-范畴的自等价的（\infty，1）-范畴.
The $\infty$-category of $\infty$-categories in simplicial type theory
本文通过仿射类型论构造了∞-范畴的∞-范畴，恢复了∞-范畴理论的核心结果，并展示了其在结构同构原理中的应用。
$\infty$-Open-multicommutativity in the category $\comp$
本文考虑紧T2空间范畴中函子的$\infty$ -开多交换性的概念。这个性质是开多交换性在无限图情形下的推广。证明了每个开多交换函子都是$\infty$ -开多交换函子。
Deformation Theory for $(\infty,n)$-categories
本文为$(\ninfty,n)$-范畴建立形变理论，通过定义TwAr范畴并证明稳定化同构，应用于lax-idempotent单子的$\infty$-范畴刻画。
Formal category theory in $\infty$-equipments I
本文将proarrow equipments推广至$\infty$-范畴，定义$\infty$-equipments，并研究其例子及内部范畴论基本概念如余极限和Kan扩张。
Colimits in the $\infty$-category of $\infty$-topoi and étale morphisms
该研究提出Lurie关于∞-拓扑斯∞-范畴中平展态射宽子范畴对小余极限封闭结论的新证明，基于平展态射与泛性质族的新刻画且不依赖更大宇宙，还给出泛性质完备化的初等构造。
Free ${A}_\infty$-categories
对于微分分次k-Q，我们定义了由Q生成的自由A-无穷范畴FQ。主要结果是：对任意A-无穷范畴A，限制A-无穷函子A_ \infty（FQ，A）-> A_1（Q，A）是等价的，其中最后一个A-无穷范畴的对象是微分分次k-箭图Q -> A的态射.
We introduce simple models for associative algebras and bimodules in the context of non-symmetric $\infty$-operads, and use these to construct an $(\infty,2)$-category of associative algebras, bimodules, and bimodule homomorphisms in a monoidal $\infty$-category. By working with $\infty$-operads over $Δ^{n,\text{op}}$ we iterate these definitions and generalize our construction to get an $(\infty,n+1)$-category of $E_{n}$-algebras and iterated bimodules in an $E_{n}$-monoidal $\infty$-category. Moreover, we show that if $\mathcal{C}$ is an $E_{n+k}$-monoidal $\infty$-category then the $(\infty…
We construct a left adjoint $\mathcal{H}^\text{st}\colon \mathbf{Ex}_{\infty} \rightarrow \mathbf{St}_{\infty}$ to the inclusion $\mathbf{St}_{\infty} \hookrightarrow \mathbf{Ex}_{\infty}$ of the $\infty$-category of stable $\infty$-categories into the $\infty$-category of exact $\infty$-categories, which we call the stable hull. For every exact $\infty$-category $\mathcal{E}$, the unit functor $\mathcal{E} \rightarrow \mathcal{H}^\text{st}(\mathcal{E})$ is fully faithful and preserves and reflects exact sequences. This provides an $\infty$-categorical variant of the Gabriel-Quillen embedding …
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…
We prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital $A_\infty$ categories with respect to the corresponding classes of quasi-equivalences are all equivalent. Moreover we show that the last two localizations are equivalent to the corresponding quotients by the relation of being isomorphic in the cohomology of the $A_\infty$ category of $A_\infty$ functors. As an application we give a complete proof of a claim by Kontsevich stating that the category of internal Homs for two dg categories can be described as the category of strictly un…
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。
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。
A Whirlwind Tour of the World of $(\infty,1)$-categories
这篇对更高范畴理论的介绍旨在给读者一种直觉，即什么是$(\infty,1)$ -范畴，它们什么时候是一个合适的工具，它们如何适应高阶范畴的景观，普通范畴论的概念如何推广到这个新的设置，以及人们把这个理论用在什么地方。它是一个广阔领域的粗略指南，专注于想法和动机，省略了几乎所有的证明和技术细节，并提供了许多参考。
Fibrations in $\infty$-category theory
在这篇简短的笔记中，我们用大量的例子讨论了拟范畴理论中的纤维化动物。我们强调简单和清晰的结构，这些纤维化提供给最终用户的高阶范畴论。
The homotopy type of the $\infty$-category associated to a simplicial complex
本文是关于严格$n$ -范畴同伦理论的系列论文之一。在本系列的第一篇文章中，我们给出了保证严格$n$ -范畴上存在一个严格的模型范畴结构的条件。本文的主要目的是展示这些条件之一。为此，我们将任何单纯复形与一个由计算机生成的严格$\infty$范畴相关联。我们猜想这个$\infty$ -范畴与相应的单纯复形具有相同的同伦类型，并且当单纯复形来自偏序集时，我们证明了这个猜想.我们引入了一个准始对象的$\infty$范畴的概念，我们表明，街的东方人承认这样一个对象。一个主要的工具，在这方面的文本是施泰纳的理论增广定向复合体。
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.
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。
We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of $\infty$-categories and we use this to prove many classical results about monads in the $\infty$-categorical framework. Amongst other things, we show that the category of algebras for an accessible monads on a locally presentable $\infty$-category $\mathcal{E}$ is again locally presentable, and that a diagram of accessible monads on a locally presentable $\infty$-category admits a colimit. Our results also provide a new and simpler way to construct and describe monads in terms of theories.
Let $\mathsf{PreOrd}(\mathbb C)$ be the category of internal preorders in an exact category $\mathbb C$. We show that the pair $(\mathsf{Eq}(\mathbb C), \mathsf{ParOrd}(\mathbb C))$ is a pretorsion theory in $\mathsf{PreOrd}(\mathbb C)$, where $\mathsf{Eq}(\mathbb C)$ and $\mathsf{ParOrd}(\mathbb C)$) are the full subcategories of internal equivalence relations and of internal partial orders in $\mathbb C$, respectively. We observe that $\mathsf{ParOrd}(\mathbb C)$ is a reflective subcategory of $\mathsf{PreOrd}(\mathbb C)$ such that each component of the unit of the adjunction is a pullback-s…
A monoidal Grothendieck construction for $\infty$-categories
本文提出了一种在∞-范畴上构造单子Grothendieck构造的方法，证明了coCartesian纤维的∞-范畴与Day卷积单子结构的等价性。
The Morita $(\infty,2)$-category of a monoidal category as a $2$-complicial set
本文为满足极简条件的幺半范畴构造了Morita $(\infty,2)$-范畴，将其实现为$3$-余骨架的$2$-复合单纯集，并明确编码了幺半群、双模及其映射和复合结构。
For a category $\mathbb{C}$, a small category $\mathbb{I}$, and a pre-cover relation $\sqsubset$ on $\mathbb C$ we prove, under certain completeness assumptions on $\mathbb C$, that a morphism $g: B\to C$ in the functor category $\mathbb {C}^{\mathbb I}$ admits an image with respect to the pre-cover relation on $\mathbb C^{\mathbb I}$ induced by $\sqsubset$ as soon as each component of $g$ admits an image with respect to $\sqsubset$. We then apply this to show that if a pointed category $\mathbb{C}$ is: (i) algebraically cartesian closed; (ii) exact protomodular and action accessible; or (iii)…
Towards 2-derivators for formal $\infty$-category theory
本文提出2-导出量的公理，用于形式∞-范畴理论，展示其在多种模型中的适用性及位移操作下的稳定性。
A note on the $(\infty,n)$-category of cobordisms
本文给出了Lurie完全拓广的拓扑域论的精确定义.利用完备$n$ -fold Segal空间作为模型，构造了一个可能具有切结构的$n$ -维配边的$(\infty,n)$ -范畴.我们赋予它一个对称的幺半群结构，并证明我们可以恢复通常的范畴cobordisms $n\operatorname{Cob}$和cobordism双范畴$n\operatorname{Cob}^{ext}$。
Sheaves of $(\infty, \infty)$-categories
本文基于 complete Segal space 对象给出了 $(n, r)$-范畴层的函子式表示，证明了 $(\infty, \infty)$-范畴层与 $\omega$-范畴层在底层 topos 上定义了 distributors，并得出了相应的层化结果。
Formal category theory in $\infty$-equipments II: Lax functors, monoidality and fibrations
本文研究了$\infty$-装备框架，探讨了装备间的lax函子、闭幺半结构和内部纤维化概念，并证明该形式化方法可轻松建立内部$\infty$-范畴论的基础。
Tensor product of A$_{\infty}$-categories
本文定义了A$_\infty$-范畴的张量积及其导出形式，使相关范畴成为闭合对称单子范畴，并提供了内部同态的显式描述。
The $\infty$-Categorical Reflection Theorem and Applications
将Adámek-Rosický反射定理推广到∞-范畴，证明可表现∞-范畴中关于极限和κ-滤过余极限封闭的全子范畴仍是可表现的，并用于分类对称幺半∞-范畴中与幂等代数模范畴等价的子范畴。
Exponentiable Higher Toposes
我们证明了可指数$\infty$ -范畴类：$\mathcal X$可指数当且仅当$\mathcal S\mathrm{h}(\mathcal X)$是连续$\infty$ -范畴.证明的核心是将$\mathcal X$上的$\mathcal C$值层的$\infty$范畴描述为满足有限极限条件和滤余极限条件（而不是纯粹的极限条件）的函子的$\infty$范畴;我们称这样的函子为$ω$ -连续层。作为一个应用，我们证明了当$\mathcal X$是可指数的时，它的稳定层的$\infty$ -范畴$\mathcal S\mathrm{h}(\mathcal X, \mathrm{Sp})$是可表示稳定$\infty$ -范畴的$\infty$ -范畴中的对偶对象.
Fibrations and Yoneda's lemma in an $\infty$-cosmos
我们使用术语$\infty$ -范畴和$\infty$ -函子来表示$\infty$ -宇宙中的对象和态射：一个满足一些公理的简单丰富范畴，让人想起一个丰富的范畴的对象。拟范畴、Segal范畴、完备Segal空间、标记单纯集、迭代完备Segal空间、$θ_n$ -空间以及它们各自的纤维化版本都是这个意义上的$\infty$ -范畴。本系列以前的工作表明，$\infty$ -范畴和$\infty$ -函子的基本范畴论只能参照$\infty$ -宇宙的公理来发展;实际上，大部分工作是同伦2-范畴、$\infty$ -范畴、$\infty$ -函子和自然变换的严格2-范畴的内部工作。在准范畴的$\infty$ -宇宙中，我们精确地重新获得了由Joyal和Lurie发展的相同的范畴论，尽管我们的定义自然是2-范畴的，没有利用区分每个模型的组合细节。本文介绍了一类$\infty$ -函子--carnivibration及其群胚变体。笛卡尔纤维化形成了一个基石，在抽象处理的“类范畴”结构的一个洛杉矶街，并发挥了重要作用，在卢里的工作准范畴。在建立了它们的基本理论之后，我们陈述并证明了Yoneda引理，它具有可表示纤维化映射的拟范畴与纤维在其表示元上的拟范畴之间的等价形式。一个配套文件将应用这些结果建立一个演算的模块之间的$\infty$类别，这将被用来定义和研究点态Kan扩展沿$\infty$…
Rigid Algebras and Cospans
本文引入刚性代数概念，将其推广到任意对称幺半$(\infty,2)$-范畴，并证明刚性代数的$(\infty,2)$-范畴实际上是$(\infty,1)$-范畴；通过余跨构造与刚性交换代数的$(\infty,1)$-范畴之间的伴随关系，建立了与给定$(\infty,1)$-范畴的典范等同。
We study cocartesian fibrations in the setting of the synthetic $(\infty,1)$-category theory developed in the simplicial type theory introduced by Riehl and Shulman. Our development culminates in a Yoneda Lemma for cocartesian fibrations.
We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal's approach: we identify a meaningful construction for the quasi-category of weighted cones over a diagram in a quasi-category, whose terminal object is the weighted limit of the considered diagram. When the quasi-category arises as the homotopy coherent nerve of a category enriched over Kan complexes, we use techniques by Riehl-Verity to show that the weighted limit …
Stable $\infty$-Operads and the multiplicative Yoneda lemma
我们为每个$\infty$ -运算$\mathcal{O}^\otimes$构造了具有一定有限极限的新的$\infty$ -运算的谱对象和交换群对象在$\mathcal{O}$。我们表明，这些都是普遍稳定的响应。添加剂$\infty$ -从$\mathcal{O}^\otimes$获得的操作物。我们推断，对于一个稳定的（resp。加法）对称monoidal $\infty$ -范畴$\mathcal{C}$ Yoneda嵌入因子通过从$\mathcal{C}$到谱的$\infty$ -范畴的精确、反变函子的$\infty$ -范畴（分别为.连接谱），并承认一定的乘法细化。作为应用，我们证明了具有Smash积的谱的对称幺半群$\infty$ -范畴Sp的正合松弛对称幺半群内函子中的单位函子Sp $\to$ Sp是初始的.
We use the terms "$\infty$-categories" and "$\infty$-functors" to mean the objects and morphisms in an "$\infty$-cosmos." Quasi-categories, Segal categories, complete Segal spaces, naturally marked simplicial sets, iterated complete Segal spaces, $θ_n$-spaces, and fibered versions of each of these are all $\infty$-categories in this sense. We show that the basic category theory of $\infty$-categories and $\infty$-functors can be developed from the axioms of an $\infty$-cosmos; indeed, most of the work is internal to a strict 2-category of $\infty$-categories, $\infty$-functors, and natural tra…
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。
The $(\infty,2)$-category of internal $(\infty,1)$-categories
本文研究了内部$(\infty,1)$-范畴的$(\infty,2)$-范畴的闭包性质及与模型范畴的对应关系。
Model $\infty$-categories I: some pleasant properties of the $\infty$-category of simplicial spaces
单纯集和单纯空间在同伦理论中都被广泛地用作空间的表示，在这两种情况下，我们都通过几何实现来提取“底层空间”。在这方面，我们很好地处理了单纯集的范畴;这是由于在其上存在一个合适的模型结构，这是特别方便的，因为它具有“适当”和“共纤生成”的技术性质。本文致力于表明，如果人们愿意绝对地处理$\infty$，那么人们就可以像处理单纯集一样处理单纯空间。确切地说，这是单纯空间的 * $\infty$ -范畴 * 上的一个适当的、共纤生成的模型结构的形式，我们也在这里介绍它的定义。
In these notes we describe models of globular weak $(\infty,m)$-categories ($m\in\mathbb{N}$) in the Grothendieck style, i.e for each $m\in\mathbb{N}$ we define a globular coherator $Θ^{\infty}_{\mathbb{M}^m}$ whose set-models are globular weak $(\infty,m)$-categories. Then we describe the combinatorics of the small category $Θ_0$ whose objects are cubical pasting diagrams and whose morphisms are morphisms of cubical sets. This provides an accurate description of the monad, on the category of cubical sets (without degeneracies and connections), of cubical strict $\infty$-categories with connec…
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.
An Enriched Approach to the Strictification of $(\infty,1)$-Categories
研究如何对$(\infty,1)$-范畴严格化，核心方法是先将其建模为$\infty$-群胚丰富的范畴，经局部和全局严格化得到$(\omega,1)$-范畴，证明了相关函子保守性，还解释了对同调怀特海德定理的推广。
We show that for any uncountable cardinal $λ$, the category of sets of cardinality at least $λ$ and monomorphisms between them cannot appear as the category of point of a topos, in particular is not the category of models of a $L_{(\infty,ω)}$-theory. More generally we show that for any regular cardinal $κ< λ$ it is neither the category of $κ$-points of a $κ$-topos, in particular, not the category of models of a $L_{(\infty,κ)}$-theory. The proof relies on the construction of a categorified version of the Scott topology, which constitute a left adjoint to the functor sending any topos to its c…
Parametrized higher category theory and higher algebra: Exposé IV -- Stability with respect to an orbital $\infty$-category
本文发展了$G$ -范畴（$G$的轨道范畴上的范畴的预层）的稳定性理论，其中$G$是有限群.我们将Mackey函子刻画为$G$ -交换幺半群，并利用它将$G$ -谱刻画为$G$ -空间的$G$ -稳定化.作为一个应用程序，我们提供了一个定理的另一种证明Guillou和五月。这里的理论是在轨道范畴的更一般的背景下发展起来的。
Parametrized higher category theory and higher algebra: Exposé I -- Elements of parametrized higher category theory
我们介绍了参数化$\infty$理论的基本要素-范畴和它们之间的函子。这些概念被定义为$\infty$ -范畴和它们之间的函子的适当纤维化。在这个阶段，我们尽可能多地举例子。简单的操作，如形成对立和形成函子$\infty$ -范畴，变得稍微更多地参与参数化设置，但我们解释了如何执行这些建设。所有这些结构都可以明确地进行，而不需要采取诸如拉直之类的绝望行为。这个Exposé的主要结果是：（1）$T$ - $\infty$-范畴的$T$ -对象在任何$\infty$ -范畴中的一个普遍特征，（2）$T$ - $\infty$ -范畴的一个内部Hom的存在性，以及（3）一个参数化的Yoneda引理。
The extension of ordinary category theory to $\infty$-categories at the start of the 21st century was a spectacular achievement pioneered by Joyal and Lurie with contributions from many others. Unfortunately, the technical arguments required to solve the infinite homotopy coherence problems inherent in these results make this theory difficult for non-experts to learn. This essay surveys two programs that seek to narrow the gap between $\infty$-category theory and ordinary 1-category theory. The first leverages similarities between the categories in which 1-categories and $\infty$-categories li…
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 $A$.
The purpose of this survey is to present in a uniform way the notion of equivalence between strict $n$-categories or $(\infty,n)$-categories, and inside a strict $(n+1)$-category or $(\infty,n+1)$-category.
A $t$-structure on the $\infty$-category of mixed graded modules
本文以模型无关方式研究混合分次模的 $\infty$-范畴，赋予其左右完备可达 $t$-结构，并将其等同于滤过模范畴上 Beilinson $t$-结构的左完备化，为未来工作提供参考。
The presentable stable envelope of an exact category
本文证明了精确∞-范畴的Gabriel-Quillen嵌入定理的类比，构造了精确∞-范畴的可呈现稳定包络，并讨论了其对称单子结构及乘法性质。
We show that Segal spaces, and more generally category objects in an $\infty$-category $\mathcal{C}$, can be identified with associative algebras in the double $\infty$-category of spans in $\mathcal{C}$. We use this observation to prove that "having identities" is a property of a non-unital $(\infty,n)$-category.
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