Unital ${A}_\infty$-categories

A$_{\infty}$-范畴范畴中余等化子不存在的一个例子

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。

A closed model structure on the category of weakly unital dg categories, II

In this paper, which is subsequent to our previous paper [PS] (but can be read independently from it), we continue our study of the closed model structure on the category $\mathrm{Cat}_{\mathrm{dgwu}}(\Bbbk)$ of small weakly unital dg categories (in the sense of Kontsevich-Soibelman [KS]) over a field $\Bbbk$. In [PS], we constructed a closed model structure on the category of weakly unital dg categories, imposing a technical condition on the weakly unital dg categories, saying that $\mathrm{id}_x\cdot \mathrm{id}_x=\mathrm{id}_x$ for any object $x$. Although this condition led us to a great s…

单位 k-限制无穷操作范畴

Unital k-Restricted Infinity-Operads

本文通过元数限制研究单位无穷操作范畴，建立单位 k-限制无穷操作范畴的完全 Segal 预层模型，并证明限制函子有完全忠实的左右伴随，从而给出过滤与余过滤结构。

比较有向同伦论的非幺半与幺半情形

Comparing the non-unital and unital settings for directed homotopy

本文比较流的 q-模型结构与拓扑富集小范畴的 Ilias 模型结构，证明二者间的提升性质及恒等函子非 Quillen 伴随，并给出极小模型结构的同伦范畴。

On the characterization of Jónsson-Tarski and of subtractive varieties

We investigate unital, subtractive and strongly unital regular categories with enough projectives and give characterizations of their projective covers. The categorical equation "strongly unital = unital + subtractive" is explored: this leads to proofs of their varietal characterizations in terms of the categorical properties of the corresponding algebraic theories.

Multitensor lifting and strictly unital higher category theory

In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result -- the lifting theorem for multitensors -- enables us to see the Gray tensor product of 2-categories and the Crans tensor product of Gray categories as part of this framework. We define weak n-categories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category …

Functors (between oo-categories) that aren't strictly unital

Let C and D be quasi-categories (a.k.a. infinity-categories). Suppose also that one has an assignment sending commutative diagrams of C to commutative diagrams of D which respects face maps, but not necessarily degeneracy maps. (This is akin to having an assignment which respects all compositions, but may not send identity morphisms to identity morphisms.) When does this assignment give rise to an actual functor? We show that if the original assignment can be shown to respect identity morphisms up to homotopy, then there exists an honest functor of infinity-categories which respects the origin…

Centrality and the commutativity of finite products with coequalisers

We study centrality of morphisms in a setting derived from that of a pointed category in which binary products commute with coequalisers. The main results of this paper show that much of the behaviour of central morphisms for unital categories is retained in our setting, including categories which are (weakly) unital, but also categories outside of the unital setting.

Localizations of the category of $A_\infty$ categories and internal Homs

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…

关于 ($\mbox{A$_{\infty}$Cat}$,$W^{\tiny\mbox{A}_{\infty}}_{\tiny\mbox{qe}}$) 的纤维性的一点注记

A remark on fibrancy of ($\mbox{A$_{\infty}$Cat}$,$W^{\tiny\mbox{A}_{\infty}}_{\tiny\mbox{qe}}$)

本文证明严格单位A∞范畴中满足性质F1的函子沿任意函子的拉回存在，从而肯定回答了Pascaleff关于(A∞Cat,W^{A∞}_{qe})在RelCat中是否为纤维对象的问题。

余笛卡尔范畴的幺半群范畴刻画

Magmal characterisations of cocartesian categories

本文综述了用幺半群范畴（含幺半群范畴）刻画余笛卡尔范畴的方法，证明了幺半群范畴$(\mathcal M, \otimes)$是余笛卡尔幺半群、对象有幺半群结构且态射为同态、张量积函子有右伴随三者等价，细化了经典刻画。

环上 $A_\infty$ 范畴与内部 Hom 范畴的局部化

Localizations of the categories of $A_\infty$ categories and internal Homs over a ring

本文证明任意交换环上 dg 范畴及各类 $A_\infty$ 范畴关于拟等价的局部化均等价，并借此为 dg 范畴同伦范畴内部 Hom 的存在性提供新证明，完整证实了 Kontsevich 和 Keller 的断言。

作为结合代数范畴化的模幺半范畴

Module Monoidal Categories as Categorification of Associative Algebras

本文引入非单位模幺半范畴的概念以范畴化结合代数的刻画，并证明在有单位情况下该定义与先前模张量范畴定义的2-范畴等价性。

关于单位代数作用的可表示性

On the representability of actions of unital algebras

本文研究了在理想精确范畴中单位非结合代数作用的可表示性，证明此类范畴一般不可表示，并通过外部弱演员构造证明单位代数的可表示性，最后通过显式构造证明单位泊松代数的可表示性。

Category of A_infinity-categories

We define natural A_infinity-transformations and construct A_infinity-category of A_infinity-functors. The notion of non-strict units in an A_infinity-category is introduced. The 2-category of (unital) A_infinity-categories, (unital) functors and transformations is described.

具有正规投影的范畴中的阿贝尔对象

Abelian objects in categories with normal projections

本文证明在具有正规投影的任意范畴中，内部阿贝尔群仍保持简单行为，即与减法代数等价、结构唯一且态射保持结构，将幺半与减法范畴的结论推广。

Extensivity of categories of relational structures

We prove that the category of models of any relational Horn theory satisfying a mild syntactic condition is infinitely extensive. Central examples of such categories include the categories of preordered sets and partially ordered sets, and the categories of small $\mathscr{V}$-categories, (symmetric) pseudo-$\mathscr{V}$-metric spaces, and (symmetric) $\mathscr{V}$-metric spaces for a commutative unital quantale $\mathscr{V}$. We also explicitly characterize initial sources and final sinks in such categories, and in particular embeddings and quotients.

Moment categories and operads

A moment category is endowed with a distinguished set of split idempotents, called moments, which can be transported along morphisms. Equivalently, a moment category is a category with an active/inert factorisation system fulfilling two simple axioms. These axioms imply that the moments of a fixed object form a monoid, actually a left regular band. Each locally finite unital moment category defines a specific type of operad which records the combinatorics of partitioning moments into elementary ones. In this way the notions of symmetric, non-symmetric and $n$-operad correspond to unital moment…

幺半DG范畴中的$A_{\infty}$-结构与强同伦幺性

$A_{\infty}$-structures in monoidal DG categories and strong homotopy unitality

本文在任意幺半DG范畴中借助无界扭曲复形定义A∞代数、余代数、模与余模，引入强同伦幺性，构造自由-遗忘同伦伴随及导出范畴，并证明A∞代数与余代数的同伦伴随给出导出模-余模等价，应用于A∞单子与增强正合单子。

Derived $(\infty,1)$-categories of two kinds

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.

On $\infty$-Categories

A new definition for the notion of a (general) $\infty$-category is given.

严格结合且含单位的 $\infty$-范畴的一种语法

A Syntax for Strictly Associative and Unital $\infty$-Categories

本文提出严格结合且含单位的 $\infty$-范畴的首个类型论语法，以 insertion 计算规则保证严格性并获得相等性判定程序，且通过 OCaml 实现与高阶同伦示例验证。

Cubes are dense in $(\infty,\infty)$-categories

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. …

$(\infty, \infty)$-范畴的同伦理论作为关于富化的泛不动点

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。

Euler characteristic for weak n-categories and $\left(\infty,1\right)$-categories

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.

严格$\infty$-范畴的一个归纳模型结构

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。

Factorization systems in $\infty$-categories

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.

$(\infty,ω)$-范畴的复形模型

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。

Constructing Coproducts in locally Cartesian closed $\infty$-Categories

We prove that every locally Cartesian closed $\infty$-category with subobject classifier has a strict initial object and disjoint and universal binary coproducts.

$A_\infty$范畴的同伦理论

The Homotopy Theory of $A_\infty$Categories

本文研究了$A_\infty$范畴的同伦理论，引入半自由范畴概念并证明非单位范畴的分解性质。

$(\ninfty,n)$-范畴的形变理论

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。

Central extensions of associative algebras and weakly action representable categories

A central extension is a regular epimorphism in a Barr exact category $\mathscr{C}$ satisfying suitable conditions involving a given Birkhoff subcategory of $\mathscr{C}$ (joint work with G. M. Kelly, 1994). In this paper we take $\mathscr{C}$ to be the category of (not-necessarily-unital) algebras over a (unital) commutative ring and consider central extensions with respect to the category of commutative algebras. We propose a new approach that avoids the intermediate notion of central extension due to A. Fröhlich in showing that $α:A\to B$ is a central extension if and only if $aa'=a'a$ for …

Parsummable categories as a strictification of symmetric monoidal categories

We prove that the homotopy theory of parsummable categories (as defined by Schwede) with respect to the underlying equivalences of categories is equivalent to the usual homotopy theory of symmetric monoidal categories. In particular, this yields a model of symmetric monoidal categories in terms of categories equipped with a strictly commutative, associative, and unital (but only partially defined) operation.

Exact triangles as modules over an $A_\infty$-category

This note describes a strictly-unital $A_\infty$-category whose representations are exact triangles such that the three-fold symmetry on exact triangles is manifest on the $A_\infty$-category.

Grothendieck $\infty$-groupoids, and still another definition of $\infty$-categories

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.

一个用于扭曲箭 $\infty$-范畴的简单模型

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

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…

On higher monoidal $\infty$-categories

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…

Cartesian factorization systems and pointed cartesian fibrations of $\infty$-categories

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.

Enriched $\infty$-categories via non-symmetric $\infty$-operads

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…

Vopěnka's principle in $\infty$-categories

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.

On (co)ends in $\infty$-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.

∞-范畴上的分解系统：非/拉直与单子性

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。

Strict $\infty $-categories. Concrete Duality

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.

$(\infty,\infty)$-范畴的核与局部化

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。

$\infty$-operads as symmetric monoidal $\infty$-categories

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.

针对纤维化于范畴模式之上的∞-范畴的Grothendieck构造

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

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.

论双幺半$\infty$-范畴

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。

有理$(\infty, 1)$-范畴的模型

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。

Bimodules and natural transformations for enriched $\infty$-categories

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.

关于双富集$\infty$-范畴

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。

正合∞范畴的Retakh定理与高阶扩张函子

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。

($\infty,n)$-范畴的Gray张量积

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。