Michael A. Shulman
We present a method of constructing symmetric monoidal bicategories from symmetric monoidal double categories that satisfy a lifting condition. Such symmetric monoidal double categories frequently occur in nature, so the method is widely applicable, though not universally so.
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We prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories. We show how coherence for these structures can be interpretted topologically using up-to-homotopy operad actions and the algebraic classification of surface braids.
We present a method of constructing monoidal, braided monoidal, and symmetric monoidal bicategories from corresponding types of monoidal double categories that satisfy a lifting condition. Many important monoidal bicategories arise naturally in this way, and applying our general method is much easier than explicitly verifying the coherence laws of a monoidal bicategory for each example. Abstracting from earlier work in this direction, we express the construction as a functor between locally cubical bicategories that preserves monoid objects; this ensures that it also preserves monoidal functor…
We define bicategories internal to 2-categories. When the ambient 2-category is symmetric monoidal categories, this provides a convenient framework for encoding the structures of a symmetric monoidal 3-category. This framework is well suited to examples arising in geometry and algebra, such as the 3-category of bordisms or the 3-category of conformal nets.
Framed bicategories and monoidal fibrations
在某些双范畴中，1-胞元是0-胞元之间的“态射”，例如范畴之间的函子，但在另一些双范畴中，1-胞元是0-胞元上的“对象”，例如双模、跨度、分配器或参数化谱。许多双范畴概念在这些情况下不能很好地工作，因为“0-胞元之间的态射”，如环同态，缺失了。我们可以通过使用伪双范畴来包含它们，但通常这些态射也会诱导作用于1-胞体的碱基变化函子。我们避免复杂的连贯性问题，通过描述基地的变化“非代数”，使用分类纤维化。由此产生的“框架双范畴”组装成2-范畴，伴随着等价、附加等概念，这些概念比通常的双范畴概念更适合我们的例子。然后，我们描述了两种方法来构建框架bicategories。一个是环和双模的类似物，它从一个框架双范畴开始并构建另一个。另一个从一个“monoidal纤维”，这意味着一个参数化的家庭的幺半范畴，并产生一个类似的框架双范畴的跨度。结合这两者，我们得到一个结构，其中包括丰富和内部范畴作为特殊情况。
We introduce the notion of a relative pseudomonad, which generalises the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient method to define pseudomonas on the Kleisli bicategory of a relative pseudomonad. The results are applied to define several pseudomonads on the bicategory of profunctors in an homogeneous way, thus providing a uniform approach to the definition of bicategories that are of interest in operad theory, mathematical logic, and theoretical computer science.
Coherence theorems are fundamental to how we think about monoidal categories and their generalizations. In this paper we revisit Mac Lane's original proof of coherence for monoidal categories using the Grothendieck construction. This perspective makes the approach of Mac Lane's proof very amenable to generalization. We use the technique to give efficient proofs of many standard coherence theorems and new coherence results for bicategories with shadow and for their functors.
Symmetry shifting for monoidal bicategories
本文研究了单子双范畴中辫子结构对单子结构的影响，扩展了Joyal和Street的经典定理，并利用无穷操作的加法定理简化了证明过程。
Monoidal Kleisli Bicategories and the Arithmetic Product of Coloured Symmetric Sequences
将算术积推广至着色对称序列，证明其构成正规oplax幺半结构，利用幺半双范畴高效验证相干条件。
The higher algebra and geometry of monoidal bicategories
该研究通过分析小立方操作符的几何，揭示了辫状、硅勒克和对称单调双范畴作为$\mathsf{E}_k$-单子的结构特性。
This paper proves three different coherence theorems for symmetric monoidal bicategories. First, we show that in a free symmetric monoidal bicategory every diagram of 2-cells commutes. Second, we show that this implies that the free symmetric monoidal bicategory on one object is equivalent, as a symmetric monoidal bicategory, to the discrete symmetric monoidal bicategory given by the disjoint union of the symmetric groups. Third, we show that every symmetric monoidal bicategory is equivalent to a strict one. We give two topological applications of these coherence results. First, we show that t…
This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well understood simple geometric realizations, and we here deal with homotopy types represented by lax diagrams of bicategories, that is, lax functors to the tricategory of bicategories. In this paper, it is proven that, when a certain bicategorical Grothendieck construction is performed on a lax diagram of bicategories, then the classifying space of the resulting bicategory can be though…
Eilenberg-Moore Bicategories for Opmonoidal Pseudomonads
本文在二维背景下分析了单子与张量结构的兼容性，定义了伪单子的 Gray 张量积，并描述了张量结构提升至 Eilenberg-Moore 伪代数的充分条件，进而将结果扩展至辫、syllapses 和对称性。
We prove coherence theorems for dualizable objects in monoidal bicategories and for fully dualizable objects in symmetric monoidal bicategories, describing coherent dual pairs and coherent fully dual pairs. These are property-like structures one can attach to an object that are equivalent to the properties of dualizability and full dualizability. We extend diagrammatic calculus of surfaces of Christopher Schommer-Pries to the case of surfaces equipped with a framing. We present two equivalence relations on so obtained framed planar diagrams, one which can be used to model isotopy classes of fr…
We show that in any symmetric monoidal category, if a weight for colimits is absolute, then the resulting colimit of any diagram of dualizable objects is again dualizable. Moreover, in this case, if an endomorphism of the colimit is induced by an endomorphism of the diagram, then its trace can be calculated as a linear combination of traces on the objects in the diagram. The formal nature of this result makes it easy to generalize to traces in homotopical contexts (using derivators) and traces in bicategories. These generalizations include the familiar additivity of the Euler characteristic an…
Monoidal bicategories, differential linear logic, and analytic functors
研究通过引入线性指数余单子和余废弃变换的双范畴对应物发展幺半双范畴理论，将若亚尔解析函子微计算扩展到预层范畴间的解析函子，如同普通微积分的变量扩展。
Symmetric Monoidal Bicategories and Biextensions
本文借助范畴扩张及 Picard 群胚系数的上同调数据研究幺半 2-范畴与双范畴，证明幺半结构生成双扩张，并揭示交换性条件的渐进稳定化与双扩张对称性及 MacLane 同调的立方 Q-构造的对应关系。
Traces in symmetric monoidal categories are well-known and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs "noncommutative" traces, such as the Hattori-Stallings trace for modules over noncommutative rings. In this paper we study a generalization of the symmetric monoidal trace which applies to noncommutative situations; its context is a bicategory equipped with an extra structure called a "shadow." In particular, we prove its functoriality a…
We show how to construct a Gamma-bicategory from a symmetric monoidal bicategory, and use that to show that the classifying space is an infinite loop space upon group completion. We also show a way to relate this construction to the classic Gamma-category construction for a bipermutative category. As an example, we use this machinery to construct a delooping of the K-theory of a bimonoidal category as defined by Baas-Dundas-Rognes.
Quantum categories have been recently studied because of their relation to bialgebroids, small categories, and skew monoidales. This is the first of a series of papers based on the author's PhD thesis in which we examine the theory of quantum categories developed by Day, Lack, and Street. A quantum category is an opmonoidal monad on the monoidale associated to a biduality $R\dashv R^{\circ}$, or enveloping monoidale, in a monoidal bicategory of modules $\mathsf{Mod}(\mathcal{V})$ for a monoidal category $\mathcal{V}$. Lack and Street proved that quantum categories are in equivalence with right…
We treat the problem of lifting bicategories into double categories through categories of vertical morphisms. We make use of a specific instance of the Grothendieck construction to provide, for every bicategory equipped with a possible vertical category, together with a suitable monoidal pre-cosheaf relating these two structures, a double category lifting the decorated bicategory along the category of vertical morphisms provided as set of initial conditions. We prove in particular that every decorated bicategory admits a lift to a double category. We study relations of instances of our constru…
The periodic table of n-categories for low dimensions I: degenerate categories and degenerate bicategories
我们研究了弱n-范畴周期表的低维情况。人们普遍认为，退化范畴产生幺半群，双退化双范畴产生交换幺半群，退化双范畴产生幺半范畴;然而，为了充分理解这种对应，我们考察了这些结构的总体以及它们之间的映射和它们之间的更高映射。范畴自然形成2-范畴{ \bfseries Cat}，因此我们取其0-胞元是退化范畴的完整子2-范畴。幺半群自然地形成一个范畴，但我们把它看作一个离散的2-范畴来进行比较。我们表明，这种结构不会产生一个双等价，得到一个等价，我们忽略了自然变换，只考虑退化范畴的{ \it范畴}。退化双范畴也有类似的情形。这种结构的三范畴并不与幺半范畴等价;我们必须只考虑这种结构的范畴。对于双退化双范畴，其三范畴并不自然地三等价于交换幺半群范畴（被视为三范畴）。然而，在这种情况下，只考虑范畴也不能给出等价性;为了得到等价性，我们考虑双退化双范畴的{ \it双范畴}。最后，我们得出一个假设，如何上述情况下可能概括为n倍退化的n-范畴。
A convenient model category for bicategories
本文引入并研究柔性2-范畴的模型范畴，它与Lack的2-范畴模型范畴Quillen等价，但具有所有对象都是余纤维的、关于笛卡尔闭结构是幺半模型范畴等优良性质，其纤维对象全子范畴等价于双范畴与正规伪函子的范畴。
The purpose of this expository note is to describe duality and trace in a symmetric monoidal category, along with important properties (including naturality and functoriality), and to give as many examples as possible. Among other things, this note is intended as background for the generalizations to the context of bicategories and indexed monoidal categories.
Scalar enrichment and cotraces in bicategories
本文证明封闭幺半双范畴可在标量丰富范畴上丰富化，用余迹替换2-胞腔，统一范畴迹与余迹概念，并揭示其与线性代数迹的联系。
Icons
范畴正统理论认为，普通数学结构的集合，如群、环或空间，形成范畴（如群范畴）;一维范畴结构的集合，如范畴、幺半范畴或有限极限范畴，形成2范畴;二维范畴结构的集合，如2范畴或双范畴，形成3范畴。我们描述了一个有用的方式，把bicategories作为对象的2类。无论是从技术上还是从概念上来说，这都有点令人惊讶。这个2-类别的2-细胞是关键的新成分;它们是标题的图标。这些可以被认为是“其组成部分是恒等式的oplax自然变换”，但我们也将给出一个更基本的描述。我们描述了这些图标的一些性质，并给出应用程序的幺半范畴，2-神经的bicategories，2维Lawvere理论，束的bicategories。
fc-multicategories
FC-多范畴是一种非常普遍的二维结构，包括双范畴、幺半范畴、双范畴和普通多范畴。我们定义它们，并解释它们如何为两个熟悉的分类概念提供一个自然的环境。第一个是双模构造，传统上在适当的共完全双范畴上进行，但也许更自然地在fc-多范畴上进行。二是丰富：有一种关于范畴在FC-多范畴中富集的理论，它扩展了通常的关于范畴在幺半范畴中富集的理论。最后，我们指出这项工作只是一个更大现象中最简单的案例。
We study Quillen model categories equipped with a monoidal skew closed structure that descends to a genuine monoidal closed structure on the homotopy category. Our examples are 2-categorical and include permutative categories and bicategories. Using the skew framework, we adapt Eilenberg and Kelly's theorem relating monoidal and closed structure to the homotopical setting. This is applied to the construction of monoidal bicategories arising from the pseudo-commutative 2-monads of Hyland and Power.
Coherence for indexed symmetric monoidal categories
本文针对索引对称幺半范畴这一双范畴细化结构，拓展现有相干定理，证明由其定义的双范畴上一大类运算具有典范同构，并引入判定同构存在性的严格图形演算。
We study semi-strict tricategories in which the only weakness is in vertical composition. We construct these as categories enriched in the category of bicategories with strict functors, with respect to the cartesian monoidal structure. As these are a form of tricategory it follows that doubly-degenerate ones are braided monoidal categories. We show that this form of semi-strict tricategory is weak enough to produce all braided monoidal categories. That is, given any braided monoidal category $B$ there is a doubly-degenerate ``vertically weak'' semi-strict tricategory whose associated braided m…
Generalized enrichment of categories
我们定义了短语“类别丰富的FC-多类别”，并探讨了一些例子。fc-多范畴是一种非常普遍的二维结构，其特例是双范畴、双范畴、幺半范畴和普通多范畴。在fc-多范畴中的富集扩展了在幺半范畴、双范畴和多范畴中的富集理论（或多或少为人所知）。此外，fc-multicategories提供了一个自然的设置双模建设，传统上进行适当的cocomplete bicategories。虽然本文是基本的和自足的，我们也解释了为什么，从一个角度来看，fc-多范畴是自然的结构，丰富的范畴。
We define a traced pseudomonoid as a pseudomonoid in a monoidal bicategory equipped with extra structure, giving a new characterisation of Cauchy complete traced monoidal categories as algebraic structures in Prof, the monoidal bicategory of profunctors. This enables reasoning about the trace using the graphical calculus for monoidal bicategories, which we illustrate in detail. We apply our techniques to study traced ∗-autonomous categories, proving a new equivalence result between the left ⊗-trace and the right ⅋-trace, and describing a new condition under which traced ͨ…
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…
Thomason's Homotopy Colimit Theorem has been extended to bicategories and this extension can be adapted, through the delooping principle, to a corresponding theorem for diagrams of monoidal categories. In this version, we show that the homotopy type of the diagram can be also represented by a genuine simplicial set nerve associated with it. This suggests the study of a homotopy colimit theorem, for diagrams $\b$ of braided monoidal categories, by means of a simplicial set {\em nerve of the diagram}. We prove that it is weak homotopy equivalent to the homotopy colimit of the diagram, of simplic…
We give an alternative presentation of braided monoidal categories. Instead of the usual associativity and braiding we have just one constraint (the b-structure). In the unital case, the coherence conditions for a b-structure are shown to be equivalent to the usual associativity, unit and braiding axioms. We also discuss the next dimensional version, that is, b-structures on bicategories. As an application, we show how special b-categories result in the Yang-Baxter equation, and how special b-bicategories produce Zamolodchikov's tetrahedron equation. Finally, we define a cohomology theory (the…
We establish an equivalence of homotopy theories between symmetric monoidal bicategories and connective spectra. For this, we develop the theory of $Γ$-objects in 2-categories. In the course of the proof we establish strictfication results of independent interest for symmetric monoidal bicategories and for diagrams of 2-categories.
Constructing linear bicategories
本文研究了线性双范畴的构造与性质，通过量词和量词oids定义了LD-量词及线性量词oids，并展示了多种标准构造在该框架下的应用，从而产生新的线性双范畴例子。
We show how the notion of intercategory encompasses a wide variety of three-dimensional structures from the literature, notably duoidal categories, monoidal double categories, cubical bicategories, double bicategories and Gray categories. Variations on the notion of span provide further examples of interest, an important one being the intercategory of sets. We consider the three kinds of morphism of intercategory as well as the cells binding them with applications to the above structures. In particular hom functors are studied.
Traced monoidal categories as algebraic structures in $\mathbf{Prof}$
本研究提出迹伪幺半群的定义，给出柯西完备迹幺半范畴在$\u003cmath\u003e\mathbf{Prof}\u003c/math\u003e$中的新代数刻画，可借助图形演算推理迹，并将其用于研究迹$*$-自守范畴，得到迹的等价结果与范畴自守的新条件。
We study integrals of Hopf monoids in duoidal endohom categories of naturally Frobenius map monoidales in monoidal bicategories. We prove two Maschke type theorems, relating the separability of the underlying monoid and comonoid, respectively, to the existence of normalized integrals. It covers the examples provided by Hopf monoids in braided monoidal categories, weak Hopf algebras, Hopf algebroids over central base algebras, Hopf monads on autonomous monoidal categories and Hopf categories.
This paper has two objectives. The first is to develop the theory of bicategories enriched in a monoidal bicategory -- categorifying the classical theory of categories enriched in a monoidal category -- up to a description of the free cocompletion of an enriched bicategory under a class of weighted bicolimits. The second objective is to describe a universal property of the process assigning to a monoidal category V the equipment of V-enriched categories, functors, transformations, and modules; we do so by considering, more generally, the assignation sending an equipment C to the equipment of C…
Univalent Monoidal Categories
本文在单值基础中研究幺半范畴，证明单值幺半范畴的双范畴具有单值性，并构造其 Rezk 完备化，且全部结果已在 Coq 的 UniMath 库中形式化。
This book is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, adjunctions and monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double cate…
Kornel Szlachányi recently used the term skew-monoidal category for a particular laxified version of monoidal category. He showed that bialgebroids $H$ with base ring $R$ could be characterized in terms of skew-monoidal structures on the category of one-sided $R$-modules for which the lax unit was $R$ itself. We define skew monoidales (or skew pseudo-monoids) in any monoidal bicategory $\mathscr M$. These are skew-monoidal categories when $\mathscr M$ is $\mathrm{Cat}$. Our main results are presented at the level of monoidal bicategories. However, a consequence is that quantum categories in th…
Interacting Monoidal Structures with Applications in Computing
本文在$n$重幺半范畴中定义$n$重幺半群与余幺半群对象，给出通信协议及布尔半环模代数等实例，服务于并发理论与更高维重写。
Monads and limits in bicategories of circuits
本文研究电路双范畴 KSW(K) 中的单子结构，利用半自由双交叉积给出自由单子的显式描述，并刻画其中的双重余/极限及双范畴特征。
Strictification and non-strictification of monoidal categories
本文综述幺半范畴严格化与非严格化的多种构造方法，证明其给出自由（非）严格幺半范畴并构成自由-遗忘2-伴随，同时展示这些构造与Power融贯定理及双范畴Yoneda嵌入的联系。
We present a weak form of a recognition principle for Quillen model categories due to J.H. Smith. We use it to put a model category structure on the category of small categories enriched over a suitable monoidal simplicial model category. The proof uses a part of the model structure on small simplicial categories due to J. Bergner. We give an application of the weak form of Smith's result to left Bousfield localizations of categories of monoids in a suitable monoidal model category.
Smooth and proper dg-algebras have an Euler class valued in the Hochschild homology of the algebra. This Euler class is worthy of this name since it satisfies many familiar properties including compatibility with the familiar pairing on the Hochschild homology of the algebra and that of its opposite. This compatibility is the Riemann-Roch theorems of Shklyarov and Petit. In this paper we prove a broad generalization of these Riemann-Roch theorems. We generalize from the bicategory of dg-algebras and their bimodules to monoidal bicategories and from Euler class to traces of non identity maps. O…
From Gs-monoidal to Oplax Cartesian Categories: Constructions and Functorial Completeness
本文扩展 gs-monoidal 范畴为带箭头预序的 oplax cartesian 范畴，刻画其 Kleisli 与 span 构造，并证明 Yoneda 嵌入及函子完备性定理。
Equivalence via surjections
本文证明了范畴、幺半范畴、双范畴和双范畴的等价性可由严格满射和忠实映射生成，并指出双范畴的正确等价概念是Campbell的群居双等价。
Premonoidal and Kleisli double categories
本文探讨预单调双范畴的双范畴版本，引入了有趣的积和多范畴结构，并研究了不同结构之间的关系及预单调双范畴的中心性质。
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…
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。
Constructing monoidal structures on fibered categories via factorizations
本研究针对小范畴上纤维化范畴的幺半群结构构造问题，提出利用光滑态射与闭浸入对应的子范畴限制的因子分解方法，证明满足相干条件的限制对可唯一确定整体幺半群结构，并给出适配反常层的逆像替换直像的变体方法。
In this thesis, we present a flexible framework for specifying and constructing operads which are suited to reasoning about network construction. The data used to present these operads is called a \emph{network model}, a monoidal variant of Joyal's combinatorial species. The construction of the operad required that we develop a monoidal lift of the Grothendieck construction. We then demonstrate how concepts like priority and dependency can be represented in this framework. For the former, we generalize Green's graph products of groups to the context of universal algebra. For the latter, we exa…
Double-functorial representation of regular hyperdoctrines
本文通过双范畴视角，证明正则超预层等价于从跨距到五元组的特定松对称幺半伪双函子，刻画了Frobenius性质，并提出了正则双超预层的新概念。
Braided monoidal categories arise naturally as centres of monoidal categories and have been the focus of much recent attention in both mathematics and physics. By suitably restricting the use of the exchange rule, we obtain a sequent calculus whose categorical semantics may be seen as freely constructing the centre of a monoidal category. This calculus is shown to admit a strongly normalising and confluent cut elimination procedure. The resulting logic fits neatly into the landscape of noncommutative logics and is distinguished by possessing a particularly perspicuous semantics.
We study categorical models for the unitless fragment of multiplicative linear logic. We find that the appropriate notion of model is a special kind of promonoidal category. Since the theory of promonoidal categories has not been developed very thoroughly, at least in the published literature, we need to develop it here. The most natural way to do this - and the simplest, once the (substantial) groundwork has been laid - is to consider promonoidal categories as an instance of the general theory of pseudomonoids in a monoidal bicategory. Accordingly, we describe and explain the notions of monoi…
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。
Calculations for Plus Constructions
本文研究了唯一分解范畴(UFC)的不同构造方法，并通过显式计算展示了plus构造如何重现并阐明许多现有构造。
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