Nick Gurski
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.
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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.
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.
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…
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 is an account of some work of Markus Rost and his students Dominik Boos and Susanne Maurer. We adapt it to the braided monoidal setting.
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…
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 show that the word problem for braided monoidal categories is at least as hard as the unknotting problem. As a corollary, so is the word problem for Gray categories. We conjecture that the word problem for Gray categories is decidable.
In this paper, we state and prove precise theorems on the classification of the category of (braided) categorical groups and their (braided) monoidal functors, and some applications obtained from the basic studies on monoidal functors between categorical groups.
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…
The monoidal centre as a limit
幺半范畴的中心是辫状幺半范畴。幺半范畴是范畴的monoidal双范畴中的monoidal对象（或伪对象）。本文提供了一个通用的构造在一个编织monoidal双范畴，产生一个编织monoidal对象从任何monoidal对象。研究了该构造的一些性质和存在的充分条件。
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 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…
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…
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。
Poly-bicategories generalise planar polycategories in the same way as bicategories generalise monoidal categories. In a poly-bicategory, the existence of enough 2-cells satisfying certain universal properties (representability) induces coherent algebraic structure on the 2-graph of single-input, single-output 2-cells. A special case of this theory was used by Hermida to produce a proof of strictification for bicategories. No full strictification is possible for higher-dimensional categories, seemingly due to problems with 2-cells that have degenerate boundaries; it was conjectured by C. Simpso…
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。
We introduce the notion of a braiding on a skew monoidal category, whose curious feature is that the defining isomorphisms involve three objects rather than two. These braidings are shown to arise from, and classify, cobraidings (also known as coquasitriangular structures) on bialgebras. Using a multicategorical approach we also describe examples of braidings on skew monoidal categories arising from 2-category theory.
Tannaka Duality describes the relationship between algebraic objects in a given category and their representations; an important case is that of Hopf algebras and their categories of representations; these have strong monoidal forgetful "fibre functors" to the category of vector spaces. We simultaneously generalize the theory of Tannaka duality in two ways: first, we replace Hopf algebras with weak Hopf algebras and strong monoidal functors with separable Frobenius monoidal functors; second, we replace the category of vector spaces with an arbitrary braided monoidal category. To accomplish thi…
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.
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…
Higher Witt Groups for 2-Categories I: Centralizers
本文研究 multifusion 2-范畴中三类2-函子的中心化子，证明其封闭性，刻画非退化条件，并建立局部模与自由模的关系及分解性质。
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 start from any small strict monoidal braided Ab-category and extend it to a monoidal nonstrict braided Ab-category which contains braided bialgebras. The objects of the original category turn out to be modules for these bialgebras
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。
Multiplier bimonoids (or bialgebras) in arbitrary braided monoidal categories are defined. They are shown to possess monoidal categories of comodules and modules. These facts are explained by the structures carried by their induced functors.
We show that the homotopy colimit construction for diagrams of categories with an operad action, recently introduced by Fiedorowicz, Stelzer and Vogt, has the desired homotopy type for diagrams of weak braided monoidal categories. This provides a more flexible way to realize E-2 spaces categorically.
We introduce the notion of a monoidal category enriched in a braided monoidal category $\mathcal V$. We set up the basic theory, and prove a classification result in terms of braided oplax monoidal functors to the Drinfeld center of some monoidal category $\mathcal T$. Even the basic theory is interesting; it shares many characteristics with the theory of monoidal categories enriched in a symmetric monoidal category, but lacks some features. Of particular note, there is no cartesian product of braided-enriched categories, and the natural transformations do not form a 2-category, but rather sat…
We construct an equivalence between the 2-categories VMonCat of rigid V-monoidal categories for a braided monoidal category V and VModTens of oplax braided functors from V into the Drinfeld centers of ordinary rigid monoidal categories. The 1-cells in each are the respective lax monoidal functors, and the 2-cells are the respective monoidal natural transformations. Our proof also gives an equivalence in the case that we consider only strong monoidal 1-cells on both sides. The 2-categories VMonCat and VModTens have G-graded analogues. We also get an equivalence of 2-categories between G-extensi…
We develop the Witt group for certain braided monoidal categories with duality. In case of a braided fusion category over an algebraically closed field of characteristic zero, we explicitly describe this structure. We then use this description to prove that this tool provides an invariant for finite isocategorical groups. As an application, we show that all groups of order less than 64 are categorically rigid.
Braided-enriched monoidal categories were introduced in work of Morrison-Penneys, where they were characterized using braided central functors. Recent work of Kong-Yuan-Zhang-Zheng and Dell extended this characterization to an equivalence of 2-categories. Since their introduction, braided-enriched fusion categories have been used to describe certain phenomena in topologically ordered systems in theoretical condensed matter physics. While these systems are unitary, there was previously no general notion of unitarity for enriched categories in the literature. We supply the notion of unitarity fo…
Weak Hopf monoids in braided monoidal categories
我们发展了辫状幺半范畴范畴中的弱双序理论，并证明了它们在一定意义上是量子范畴。弱Hopf幺半群被证明是量子群胚。每个可分的Frobenius幺半群R导致一个弱Hopf幺半群R \otimes R。
Monoidal categories enriched in a braided monoidal category $\mathcal{V}$ are classified by braided oplax monoidal functors from $\mathcal{V}$ to the Drinfeld centers of ordinary monoidal categories. In this article, we prove that this classifying functor is strongly monoidal if and only if the original $\mathcal{V}$-monoidal category is tensored over $\mathcal{V}$. We then define a completion operation which produces a tensored $\mathcal{V}$-monoidal category $\overline{\mathcal{C}}$ from an arbitrary $\mathcal{V}$-monoidal category $\mathcal{C}$, and we determine many equivalent conditions w…
In this paper we introduce the notion of weak operator and the theory of Yetter-Drinfeld modules over a weak braided Hopf algebra with invertible antipode in a strict monoidal category. We prove that the class of such objects constitutes a non strict monoidal category. It is also shown that this category is not trivial, that is to say that it admits objects generated by the adjoint action (coaction) associated to the weak braided Hopf algebra.
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.
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…
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。
Universal pseudomorphisms, with applications to diagrammatic coherence for braided and symmetric monoidal functors
本文提出通用伪态射理论，证明在一定假设下伪态射融贯性等价于严格代数融贯性，并应用于普通、对称与辫子幺半函子的图解融贯性。
Fixed point theory and trace for bicategories
莱夫谢茨不动点定理很容易从莱夫谢茨数与不动点指数的同一性中得出。这种识别是对称幺半范畴中迹的函性的一个结果。有改进的莱夫谢茨数和不动点指数，给一个逆的Lefschetz不动点定理。这个定理的一个重要部分是识别这些不同的不变量。本文将对称幺半范畴中的迹推广到具有阴影的双范畴中。我们显示的不变量中使用的Lefschetz不动点定理的逆是这种跟踪的例子，并提供了一些必要的标识的functoriality的跟踪。这里使用的方法不使用单纯技术，因此很容易推广到其他环境。
We describe a class of examples of braided monoidal categories which are built from Hopf algebras in symmetric categories. The construction is motivated by a calculation in two-dimensional conformal field theory and is tailored to contain the braided monoidal categories occurring in the study of the Ising model, their generalisation to Tamabara-Yamagami categories, and categories occurring for symplectic fermions.
Presentations for unbraided, braided and symmetric pseudomonoids are defined. Biequivalences characterising the semistrict bicategories generated by these presentations are proven. It is shown that these biequivalences categorify results in the theory of monoids and commutative monoids, and generalise standard coherence theorems for braided and symmetric monoidal categories.
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。
Generalized Yetter-Drinfeld modules, the center of bi-actegories and groupoid-crossed braided bicategories
本文研究双作用范畴的 $E$-中心，证明其等价于广义 Yetter-Drinfeld 模范畴，并引入双群胚交叉辫状双范畴概念，推广 Turaev 的群交叉辫状幺半范畴。
Weak vertical composition II: totalities
本文研究仅纵向复合具有弱性的半严格三范畴，将双退化情形组织为带弱函子和变换的2-范畴，并证明其与辫子弱幺半范畴的2-范畴双伴随双等价。
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.
We prove that the classical result asserting that the relative Picard group of a faithfully flat extension of commutative rings is isomorphic to the first Amitsur cohomology group stills valid in the realm of symmetric monoidal categories. To this end, we built some group exact sequences from an adjunction in a bicategory, which are of independent interest. As a particular byproduct of the evolving theory, we prove a version of Hilbert's theorem 90 for cocommutatvie coalgebra coextensions (=surjective homomorphisms).
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.
Small Bénabou's bicategories and, in particular, Mac Lane's monoidal categories, have well-understood classifying spaces, which give geometric meaning to their cells. This paper contains some contributions to the study of the relationship between bicategories and the homotopy types of their classifying spaces. Mainly, generalizations are given of Quillen's Theorems A and B to lax functors between bicategories.
We define the Drinfeld center of a monoidal category enriched over a braided monoidal category, and show that every modular tensor category can be realized in a canonical way as the Drinfeld center of a self-enriched monoidal category. We also give a generalization of this result for important applications in physics.
Frobenius functors and one-sided Hopf algebras in braided monoidal categories
本文在辫子幺半范畴中证明：双幺半群为单侧Hopf幺半群且对映体为反同态，当且仅当自由Hopf模函子是Frobenius的，并给出自由单侧Hopf幺半群的构造及例子。
Local Modules in Braided Monoidal 2-Categories
本文在辫子幺半 2-范畴中细化了局部模概念，证明了局部模 2-范畴具有辫子幺半结构与对偶，并在辫子融合 2-范畴中确立其为辫子多融合 2-范畴，最后探讨了 Lagrangian 代数及其与拓扑物相边界分类的联系。
Homological Algebra in Abelian Framed Bicategories: Exact Sequences and Embedding Theorems
本文提出阿贝尔框架双范畴，用于开发有向结构的同调与上同调理论，并证明了与经典定理类似的嵌入定理，拓展了有向同调的应用范围。
Braided Bialgebras of Type One
交换辫幺半范畴中的1型辫双代数被刻画为辫分次双代数，它们作为代数和余代数都是强$\mathbb{N}$ -分次的.
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