Volodymyr Lyubashenko, Oleksandr Manzyuk
For a differential graded k-quiver Q we define the free A-infinity-category FQ generated by Q. The main result is that for an arbitrary A-infinity-category A the restriction A-infinity-functor A_\infty(FQ,A) -> A_1(Q,A) is an equivalence, where objects of the last A-infinity-category are morphisms of differential graded k-quivers Q -> A.
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The aim of this paper is to reformulate the theory of unbounded derived categories, including more recent categories of first and second kind, using the language of $(\infty,1)$-categories.
A new definition for the notion of a (general) $\infty$-category is given.
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. …
Homotopy theories of $(\infty, \infty)$-categories as universal fixed points with respect to enrichment
本文证明了具归纳与余归纳等价的$(\infty, \infty)$-范畴满足弱富化泛性质，分别构成局部可表示不动点子范畴的始对象与富化不动点范畴的终对象。
The Euler characteristic was defined for finite strict n-categories by Leinster using the theory of enriched categories. This was an extension of some of his earlier work, which defined Euler characteristic for finite categories. Building on Leinster's work, we extend the notion of Euler characteristic to certain finite weak 2-categories and present a sketch of a similar extension to weak n-categories. We also discuss the extension of the Euler characteristic to certain finite $\left(\infty,1\right)$-categories.
An inductive model structure for strict $\infty$-categories
本文为标记严格$\infty$-范畴构建了左半模型结构，通过Street神经建立与复集的Quillen伴随，并据此证明了弱$(\infty,\infty)$-范畴存在至少三种不同的非等价概念。
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.
The complicial model of $(\infty,ω)$-categories
研究$(\infty,ω)$-范畴的复形模型，核心方法是证明Verity n-复形集性质，贡献是给出相关模型。
We prove that every locally Cartesian closed $\infty$-category with subobject classifier has a strict initial object and disjoint and universal binary coproducts.
We prove that three definitions of unitality for A-infinity-categories suggested by the first author, by Kontsevich and Soibelman, and by Fukaya are equivalent.
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。
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。
A note on Noetherian $(\infty, \infty)$-categories
本文解决了一般对称幺半 $(\infty, 1)$-范畴上 enrichment 自函子 $(-)\mathbf{Cat}$ 初始代数的存在性猜想，证明 Adámek 构造在受限情形下于 $\lambda$ 步终止，并通过引入 Noetherian 弱有限性刻画了其初始代数。
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.
Limits and colimits of synthetic $\infty$-categories
本文在合成 simplicial 同调类型理论框架下，研究了 ∞-范畴的极限与余极限理论，并展示了空间族的极限可通过依赖积计算。
Categorical Theory of $(\infty,ω)$-Categories
本文系统发展$(\infty,\omega)$-范畴理论，将松弛Grothendieck构造、Yoneda引理、松弛（余）极限及松弛Kan扩张等经典范畴论结果推广到该高阶范畴框架中。
Effectivity of Generalized Double $\infty$-Categories
该研究构造了(n,m)-双∞-范畴与特定滤链间的伴随函子，证明其诱导等价，推广了Rezk和Lurie的结果，还验证了n=0和n=1时的相关结论与猜想。
This introduction to higher category theory is intended to a give the reader an intuition for what $(\infty,1)$-categories are, when they are an appropriate tool, how they fit into the landscape of higher category, how concepts from ordinary category theory generalize to this new setting, and what uses people have put the theory to. It is a rough guide to a vast terrain, focuses on ideas and motivation, omits almost all proofs and technical details, and provides many references.
Theory and models of $(\infty,ω)$-categories
本文建立$(\infty,\omega)$-范畴的两种模型（完备Segal $\Theta$-空间与complicial集合）的Quillen等价，并利用其联系在$(\infty,\omega)$-范畴中实现标准范畴构造，重点研究Grothendieck构造。
In this note, we study the connection between Gray tensor product and suspension. We derive a characterization of weak equivalences as fully faithful and essentially surjective functors. We construct the $co$ duality, a weak involution that reverses the direction of even dimensional cells. We conclude by studying Grothendieck fibrations of complicial sets.
The aim of this paper is to present a simplified version of the notion of $\infty$-groupoid developed by Grothendieck in "Pursuing Stacks" and to introduce a definition of $\infty$-categories inspired by Grothendieck's approach.
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。
We study the *homotopy theory* of $\infty$-categories enriched in the $\infty$-category $sS$ of simplicial spaces. That is, we consider $sS$-enriched $\infty$-categories as presentations of ordinary $\infty$-categories by means of a "local" geometric realization functor $Cat_{sS} \to Cat_\infty$, and we prove that their homotopy theory presents the $\infty$-category of $\infty$-categories, i.e. that this functor induces an equivalence $Cat_{sS} [[ W_{DK}^{-1} ]] \xrightarrow{\sim} Cat_\infty$ from a localization of the $\infty$-category of $sS$-enriched $\infty$-categories. Following Dwyer--Ka…
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.
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 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…
In this article, the interplay between Vopěnka's principle, as well as its weaker counterpart, and presentable $\infty$-categories is studied. Analogous statements, arising after replacing categories with $\infty$-categories in the original ones, are introduced and compared to these. Further, the attention is focused on the question of to what extent the consequences that (weak) Vopěnka's principle have on the detection of reflexive subcategories of a presentable categories can be generalized to $\infty$-categories.
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.
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…
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。
An elementary theory of strict $\infty $-categories with application to concrete duality is given. New examples of first and second order concrete duality are presented.
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。
While many different models for $(\infty,1)$-categories are currently being used, it is known that they are Quillen equivalent to one another. Several higher-order analogues of them are being developed as models for $(\infty, n)$-categories. In this paper, we establish model structures for some naturally arising categories of objects which should be thought of as $(\infty,n)$-categories. Furthermore, we establish Quillen equivalences between them.
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.
In this paper we give a summary of the comparisons between different definitions of so-called (\infty,1)-categories, which are considered to be models for \infty-categories whose n-morphisms are all invertible for n>1. They are also, from the viewpoint of homotopy theory, models for the homotopy theory of homotopy theories. The four different structures, all of which are equivalent, are simplicial categories, Segal categories, complete Segal spaces, and quasi-categories.
We prove that various structures on model $\infty$-categories descend to corresponding structures on their localizations: (i) Quillen adjunctions; (ii) two-variable Quillen adjunctions; (iii) monoidal and symmetric monoidal model structures; and (iv) enriched model structures.
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。
In this paper we define a sequence of monads $\mathbb{T}^(\infty;n)$ $(n\in\mathbb{N})$ on $\infty$-$\mathbb{G}\text{r}$, the category of the $\infty$-graphs. We conjecture that algebras for $\mathbb{T}^(0;n)$ which are defined in a purely algebraic setting, are models of weak $\infty$-groupoids. And for all $n>1$ we conjecture that algebras for $\mathbb{T}^(\infty;n)$ which are defined in a purely algebraic setting, are models of weak $(\infty; n)$-categories.
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。
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