Xiao-Wu Chen
We give an informal introduction to model categories, and treat three important examples in some details: the category of small categories, the category of dg algebras, and the category of small dg categories.
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In this informal introduction to dg categories, the slogan is that dg categories are more rudimentary than triangulated categories. We recall some details on the dg quotient category introduced by Bernhard Keller and Vladimir Drinfeld.
In this informal expository note, we present a universal, formulaic construction of the free product of rigid $C^*$-2-categories. This construction allows for a straightforward, purely categorical formulation of the free composition of subfactors and planar algebras considered by Bisch and Jones. As an application, we explain the results of Tarrago and Wahl on free wreath products of compact quantum groups in this categorical language.
We give a natural-deduction-style type theory for symmetric monoidal categories whose judgmental structure directly represents morphisms with tensor products in their codomain as well as their domain. The syntax is inspired by Sweedler notation for coalgebras, with variables associated to types in the domain and terms associated to types in the codomain, allowing types to be treated informally like "sets with elements" subject to global linearity-like restrictions. We illustrate the usefulness of this type theory with various applications to duality, traces, Frobenius monoids, and (weak) Hopf …
A 2-categories companion
本文是一个相当非正式的指导一些基本理论的2范畴和bicategories，包括概念的限制和colimit，2维泛代数，正式范畴论，神经的bicategories。正如这些事情的方式一样，主题的选择有点个人化。没有试图在严格性或完整性。它也不完全是介绍性的：你找不到双范畴的定义;但你也不需要一个定义来阅读它。为了与范畴论的哲学保持一致，双范畴之间的态射比双范畴本身发挥更大的作用。
Coherence for lenses and open games
计算机科学中的多态透镜范畴，以及合成对策论中的开放对策范畴，都有一种奇怪的结构，让人想起紧凑的封闭范畴，但在某些关键方面有所不同。具体地说，它们有一个态射族，其行为类似于紧致闭范畴的计数器，但没有相应的单位;它们有一个“部分”对偶，其行为类似于紧致闭范畴中的换位，当它被定义时。我们公理化这个结构，我们称之为“目的论范畴”。我们精确地定义了一种适合于这些范畴的图解语言，并证明了它们的一个连贯性定理。这支持了在组合对策论中使用图解推理，这种推理以前只被非正式地使用。
New examples of non-unique enhancements for triangulated categories
本文提出一种构造具有不同增强的三角范畴的方法，展示t-结构的存在不必然导致增强的唯一性。
Categories of layered semirings
我们将[17，19]的结构推广到分层半环，以丰富分层半环的结构，并为算术应用提供有限例（包括有限例）。[19]中的分层范畴论被相应地推广到非可消幺半群。
Are introduced six examples of non-braidable tensor categories which are extensions of the category Comod(H), for H a super-group algebra; and two examples of braided categories where the only possible braiding is the trivial braiding.
Injectivity paucity in AB5 categories of oversize chains
本文构造了无非零注入（或射影）对象的AB5范畴例子，结合Rickard的例子与2-函子构造方法，填补了文献中的空白。
More examples of pseudosymmetric braided categories
我们研究了辫范畴和拟三角霍普夫代数的一些例子，并分别确定它们中的哪一个是伪对称的，哪一个是伪三角的.我们还证明了存在一个泛拟对称辫范畴。
Triangulated categories without models
我们展示的例子，三角范畴既不是一个稳定的范畴的Frobenius范畴，也不是一个完整的三角子范畴的同伦范畴的一个稳定的模型范畴。更重要的是，我们的例子不承认任何非平凡的确切函子，或从这些代数分别拓扑三角范畴。
We propose a construction of a stable category for any pretorsion theory in a lextensive category. We prove the universal property of the stable category, that extends previous results obtained for the stable category of internal preorders in a pretopos. Some examples are provided in the categories of topological spaces and of (small) categories.
Integral categories form a sub-class of pre-abelian categories whose systematic study was initiated by Rump in 2001. In the first part of this article we determine whether several categories of topological and bornological vector spaces are integral. Moreover, we establish that the class of integral categories is not contained in the class of quasi-abelian categories, and that there exist semi-abelian categories that are neither integral nor quasi-abelian. In the last part of the article we show that a category is quasi-abelian if and only if it has admissible intersections, in the sense consi…
Cartesian differential categories come equipped with a differential combinator that formalizes the derivative from multi-variable differential calculus, and also provide the categorical semantics of the differential $λ$-calculus. An important source of examples of Cartesian differential categories are the coKleisli categories of the comonads of differential categories, where the latter concept provides the categorical semantics of differential linear logic. In this paper, we generalize this construction by introducing Cartesian differential comonads, which are precisely the comonads whose coKl…
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。
Centers and homotopy centers in enriched monoidal categories
本文研究了幺半范畴中幺半群的中心和同伦中心的理论，这些幺半群本身在duoidal范畴中是丰富的。Duoidal范畴（由Aguillar和Mahajan引入，名为2-幺半范畴）是具有两个monoidal结构的范畴，它们通过一些不一定可逆的凝聚态射联系在一起。在这个意义上的幺半群的中心包括许多不是“经典的”的例子。特别是，范畴的2-范畴是我们意义上的中心的一个例子。同伦中心（类似于经典的Hochschild复形）的例子包括2-范畴、2-函子和自然变换的Gray范畴Gray和dg-范畴、dg-函子和相干dg-变换的Tamarkin同伦2-范畴。
Association schemes, classical RCFT's, and centres of monoidal functor categories
在这里，我们描述三个简单的例子，什么是所谓的图形傅立叶变换[4]。这些例子中至少有两个可以简单地看作是在合适的monoidal闭函子范畴上的monoidal余单子，但是第三个例子，涉及monoidal闭函子范畴的“中心”，通常不是余单子的。对于前两个示例（即，关联方案和RCFT），在本说明的早期版本中设想了一个更详细的“概率类别”设置，但许多读者错过了要点，因此在下面进行了简化（希望如此）。
We introduce partially lax limits of infinity-categories, which interpolate between ordinary limits and lax limits. Most naturally occurring examples of lax limits are only partially lax; we give examples arising from enriched categories and operads. Our main result is a formula for partially lax limits and colimits in terms of the Grothendieck construction. This generalizes a formula of Lurie for ordinary limits and of Gepner-Haugseng-Nikolaus for fully lax limits.
Factorizable enriched categories and applications
定义了两个丰富范畴的扭张量积，它推广了代数结构的各种“积”，包括群的双交叉积、（余）代数的扭张量积和双代数的双向量积.定义中的关键要素是两个丰富范畴之间的简单扭曲系统的概念。为了给出简单扭曲张量积的例子，我们引入了丰富范畴的匹配对。还讨论了与普通范畴、偏序集和群胚有关的其他几个例子。
Concrete Categories in Homotopy Type Theory
在同伦类型论中引入了一些真正的高阶范畴，定义为类型范畴的良行为子范畴。我们给出了几个例子，一些展示其他事物的技术不是例子。虽然只是所需的一小部分，但它是一个自然的结构，并且对于寻求提供完全通用的结构的人来说可能是有指导意义的。
Infinity categories with duality and hermitian multiplicative infinite loop space machines
本文证明具有对偶的预加性无穷范畴可产生直和埃尔米特K-理论谱，并构造E-无穷环谱，同时给出典范对偶的存在性及对偶的分类与扭曲。
We define the notion of braided Coxeter category, which is informally a tensor category carrying compatible, commuting actions of a generalised braid group B_W and Artin's braid groups B_n on the tensor powers of its objects. The data which defines the action of B_W bears a formal similarity to the associativity constraints in a monoidal category, but is related to the coherence of a family of fiber functors. We show that the quantum Weyl group operators of a quantised Kac-Moody algebra U_h(g), together with the universal R-matrices of its Levi subalgebras, give rise to a braided Coxeter struc…
Some examples of spaces of stability conditions on derived categories
研究了某些射影簇上相干层的导出范畴的稳定性条件。给出了光滑射影曲线的稳定流形的完整描述，并研究了射影空间和del Pezzo曲面的稳定流形的连通开子集.
From Matrices to Morphisms II: Examples of Computational Categories in MATLAB and Octave
本文提出计算范畴概念，以 MATLAB、Octave 数据结构表示多种数学结构，构建统一框架并实现算法化，为范畴论与矩阵类科学计算提供双向视角。
Biset functors for categories
本文将群上的双集函子推广到任意有限范畴，建立 Burnside 环、简单函子、幺半结构与纤维化等理论，并通过 Boolean 格联系对应函子。
Herschend-Liu-Nakaoka introduced the notion of $n$-exangulated categories as higher dimensional analogues of extriangulated categories defined by Nakaoka-Palu. The class of $n$-exangulated categories contains $n$-exact categories and $(n+2)$-angulated categories as examples. In this article, we introduce a notion of Frobenius $n$-exangulated categories which are a generalization of Frobenius $n$-exact categories. We show that the stable category of a Frobenius $n$-exangulated category is an $(n+2)$-angulated category. As an application, this result generalizes the work by Jasso. We provide a c…
Cartesian reverse differential categories (CRDCs) are a recently defined structure which categorically model the reverse differentiation operations used in supervised learning. Here we define a related structure called a monoidal reverse differential category, prove important results about its relationship to CRDCs, and provide examples of both structures, including examples coming from models of quantum computation.
Monoidal 2-categories from foam evaluation
本文提出通过闭合泡沫评估公式构建半严格单向2范畴，应用于链同调理论中的例子，包括gl(N)-泡沫和Bar-Natan装饰边界范畴。
Rota-Baxter Categories
我们介绍Rota-Baxter范畴，并构造了这样的结构的例子。
Factorisation and Cohomology of Higher Categories
我们构造了一个迭代法，用于将小严格n-范畴分解成一个唯一的（直到同构）小1-范畴的集合。在此之后，我们发展的理论，包括一个大类的$\infty$ -范畴。我们使用这种因式分解来定义高阶范畴的上同调理论。最后，我们讨论了低维范畴分解的一些具体例子。
On the non additivity of the trace in derived categories
本文给出了一个完美复形的短正合列的自同态的例子，其中中间映射的迹不等于另外两个映射的迹之和。关键是所涉及的平方只有在同伦下才是可交换的。鉴于这个例子，我发现在1968年，德利涅立即创建了他的“类别spectrales”，不久之后，Illusie介绍了“过滤导出范畴“在一个令人满意的一种加法恢复的痕迹。这篇论文是用法语写的，最后有一个简短的按时间顺序排列的评论。
This is a companion paper of arXiv:1901.09461, where different notions of dimension for triangulated categories are discussed. Here we compute dimensions for some examples of triangulated categories and thus illustrate and motivate material from loc. cit. Our examples include path algebras of finite ordered quivers, orbifold projective lines, some tensor powers of path algebras in Dynkin quivers of type A and categories, generated by an exceptional pair.
This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads, and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of th…
The familial nature of enrichment over virtual double categories
本文研究虚拟双范畴上强化范畴的家族性质，证明相关2-函子是参数化右2-伴随且具有家族性，并通过家族构造形式化其上强化范畴的2-范畴，为更一般结构下的强化范畴提供更好的形式性质与示例覆盖。
Generating linear categories of partitions
本文提出一种逼近划分线性范畴的算法，通过计算机实验首次获得非简单划分线性范畴实例，并证明其新颖性，部分范畴用量子群反交换扭结解释。
Liftable pairs of adjoint functors between braided monoidal categories in the sense of \cite{GV-OnTheDuality} provide auto-adjunctions between the associated categories of bialgebras. Motivated by finding interesting examples of such pairs, we study general pre-rigid monoidal categories. Roughly speaking, these are monoidal categories in which for every object $X$, an object $X^{\ast}$ and a nicely behaving evaluation map from $X^{\ast}\otimes X$ to the unit object exist. A prototypical example is the category of vector spaces over a field, where $X^{\ast}$ is not a categorical dual if $X$ is …
Strict $\infty $-categories. Concrete Duality
给出严格$\infty $范畴的一个初等理论及其在具体对偶中的应用.第一和第二阶具体对偶的新的例子。
Following the pattern from linear logic, the coKleisli category of a differential category is a Cartesian differential category. What then is the coEilenberg-Moore category of a differential category? The answer is a tangent category! A key example arises from the opposite of the category of Abelian groups with the free exponential modality. The coEilenberg-Moore category, in this case, is the opposite of the category of commutative rings. That the latter is a tangent category captures a fundamental aspect of both algebraic geometry and Synthetic Differential Geometry. The general result appli…
Latent fibrations are an adaptation, appropriate for categories of partial maps (as presented by restriction categories), of the usual notion of fibration. The paper initiates the development of the basic theory of latent fibrations and explores some key examples. Latent fibrations cover a wide variety of examples, some of which are partial versions of standard fibrations, and some of which are particular to partial map categories (particularly those that arise in computational settings). Latent fibrations with various special properties are identified: hyperconnected latent fibrations, in par…
The Rosický Tangent Categories of Algebras over an Operad
本文证明任意 operad 的代数范畴及其反范畴均为切范畴，切丛分别由半直积与 Kähler 微分模构造且互为伴随，并由此从 operad 获得丰富的切范畴与 Cartesian 微分范畴例子。
We introduce Manifold tensor categories, which make precise the notion of a tensor category with a manifold of simple objects. A basic example is the category of vector spaces graded by a Lie group. Unlike classic tensor category theory, our setup keeps track of the smooth (and topological) structure of the manifold of simple objects. We set down the necessary definitions for Manifold tensor categories and a generalisation we term Orbifold tensor categories. We also construct a number of examples, most notably two families of examples we call Interpolated Tambara-Yamagami categories and Interp…
A category of FI type is one which is sufficiently similar to finite sets and injections so as to admit nice representation stability results. Several common examples admit a Grothendieck fibration to finite sets and injections. We begin by carefully reviewing the theory of fibrations of categories with motivating examples relevant to algebra and representation theory. We classify which functors between FI type categories are fibrations, and thus obtain sufficient conditions for an FI type category to be the result of a Grothendieck construction.
Non-left-complete derived categories
我们给出了阿贝尔范畴A的导出范畴D（A）不是左完备的例子。也许其中最自然的是其中A是特征p>0的域k上的加法群G_a的表示范畴。
Integral categories were recently developed as a counterpart to differential categories. In particular, integral categories come equipped with an integration operator, known as an integral transformation, whose axioms generalize the basic integration identities from calculus such as integration by parts. However, the literature on integral categories contains no example that captures integration of arbitrary smooth functions: the closest are examples involving integration of polynomial functions. This paper fills in this gap by developing an example of an integral category whose integral trans…
V-graded categories and V-W-bigraded categories: Functor categories and bifunctors over non-symmetric bases
本文研究了在非对称基础上的V-分级范畴和双分级范畴，提出了一种适用于任意单向范畴V的函子范畴和双函子理论，探讨了V-分级模和预射影范畴的例子，并与Garner和López Franco的理论进行了比较。
Cluster categories for topologists
我们考虑三角轨道范畴，与集群范畴的激励的例子，在他们通常的代数三角范畴的背景下，然后从另一个角度提出他们在拓扑三角范畴的框架。
This paper emerged as a result of tackling the following three issues. Firstly, we would like the well known embedding of bicategories into pseudo double categories to be monoidal, which it is not if one uses the usual notion of a monoidal pseudo double category. Secondly, in \cite{Gabi} the question was raised: which would be an alternative notion to intercategories of Grandis and Paré, so that monoids in Böhm's monoidal category $(Dbl,\ot)$ of strict double categories and strict double functors with a Gray type monoidal product be an example of it? We obtain and prove that precisely the mono…
This is a first of a series of two papers. Our motive is to tackle the question raised in Böhm's "The Gray Monoidal Product of Double Categories" from Applied Categorical Structures: which would be an alternative notion to intercategories of Grandis and Paré, so that monoids in Böhm's monoidal category $Dbl$ of strict double categories and double pseudo functors be an example of it? Before addressing this question we observe that although bicategories embed into pseudo double categories, this embedding is not monoidal, with the usual notion of a monoidal pseudo double category. We then prove t…
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.
Monoidal abelian envelopes
本文证明了非阿贝尔幺半群范畴的阿贝尔包络的构造性存在定理，为张量范畴的构造提供新工具，并给出Deligne猜想及正特征下张量范畴的新证明。
In this paper, we investigate the property (P) that finite products commute with arbitrary coequalizers in pointed categories. Examples of such categories include any regular unital or (pointed) majority category with coequalizers, as well as any pointed factor permutable category with coequalizers. We establish a Mal'tsev term condition characterizing pointed varieties of universal algebras satisfying (P). We then consider categories satisfying (P) locally, i.e., those categories for which every fibre $Pt_{\mathbb{C}}(X)$ of the fibration of points $π: Pt_{\mathbb{C}} \rightarrow \mathbb{C}$ …
n-Categories Admissible by n-graph
考虑了n-范畴及相关主题的概念。一个n-范畴被描述为一个具有合成的n-图。提出了一种新的操作数定义。给出了一些说明性的例子。
In this paper we present $2$-category theory from the perspective of Gray-categories using the graphical calculus of separated surface diagrams. As an extended example we consider cones and limits of $2$-functors. Then we use the canonical adjunction between $2$-computads and $2$-categories to interpret the comparison structure of lax functors and extend the surface diagram calculus with compositor sheets in order to represent and reason about them.
Enhanced left triangulated categories
本文研究具有同调核的dg范畴，其同调范畴自然具有左三角结构。通过证明此类dg范畴的稳定化具有自然的dg增强结构。
From operator categories to topological operads
在本文中，我们介绍的概念，一个运营商类别和两个不同的模型同伦理论的$\infty$ -操作在一个运营商类别-其中一个扩展Lurie的理论$\infty$ -操作，其中另一个是全新的，即使在交换设置。我们定义完美的运营商类别，我们描述了一个类别$Λ(Φ)$附加到一个完美的运营商类别$Φ$，提供西格尔地图。我们定义了一个圈积的运营商类别和Boardman-Vogt张量积的一种形式，躺在它。然后，我们给出的运营商类别的例子，提供了泛性质的操作$A_n$和$E_n$（$1\leq n\leq +\infty$），以及收集新的例子。
Maximal exact structures on additive categories
我们证明了每一个具有核和上核的加法范畴都有一个极大正合结构。此外，我们还讨论了由泛函分析产生的后一种类型范畴的两个例子。
We give a self-contained introduction to accessible categories and how they shed light on both model- and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality localized to a given category, as well as stable independence, a generalization of pushouts and model-theoretic forking that may interest mathematicians at large. We give many examples, including recently discovered connections with homotopy theory and homological algebra. We also discuss concrete versions of accessible categories (such as abstract elementary cla…
Quandles with isomorphic representation categories
该研究构造了两类不同构有限拟群的例子，其泛包络群同构，进而二者的加法与张量表示范畴也同构，揭示了拟群与其表示范畴的特殊关系。
Let $n$ be an integer greater or equal than $3$. We give a simultaneous generalization of $(n-2)$-exact categories and $n$-angulated categories, and we call it one-sided $n$-suspended categories. One-sided $n$-angulated categories are also examples of one-sided $n$-suspended categories. We provide a general framework for passing from one-sided $n$-suspended categories to one-sided $n$-angulated categories. Besides, we give a method to construct $n$-angulated quotient categories from Frobenius $n$-prile categories. These results generalize their works by Jasso for $n$-exact categories, Lin for …
Formalizing all indexed mathematics as a benchmark for general reasoning, with the example of implementing dilatations of categories
本文提出将全部已发表数学形式化为机器可验证知识库的基准问题，并以范畴代数中的范畴膨胀为例展示实现过程。
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