Shigeru Yamagami
Free products of semisimple tesnor categories are constructed with the help of polygonal presentation. The semisimplicity criterion is obtained for the Bisch-Jones' planar algebras as a byproduct.
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We give a 3-universal property for the Karoubi envelope of a 2-category. Using this, we show that the 3-categories of finite semisimple 2-categories (as introduced in arXiv:1812.11933) and of multifusion categories are equivalent.
Compact Semisimple 2-Categories
本文定义了任意域上的紧半单2-范畴，证明其等价于可分模1-范畴的2-范畴，验证了在特定域上的有限性，并将特征零代数闭域上的相关理论推广至该范畴。
Tannaka duals of Hopf algebras inside semisimple tensor categories are used to construct orbifold tensor categories, which are shown to include the Tannaka dual of the dual Hopf algebras. The second orbifolds are then canonically isomorphic to the initial tensor categories.
We classify semisimple module categories over the tensor category of representations of quantum SL(2) extending previous results to the roots of unity and positive characteristic cases.
Finite Semisimple Module 2-Categories
该研究针对多融合2-范畴$\boldsymbol{\frak{C}}$上的有限半单模2-范畴，证明其具有典范$\boldsymbol{\frak{C}}$-富集结构，并进一步证明这类模2-范畴等价于$\boldsymbol{\frak{C}}$中刚性代数的模2-范畴。
Notes on semisimple tensor categories of rank two
该研究证明代数闭域上存在无穷多个秩为二的半单张量范畴，丰富了低秩幺半范畴的分类认识。
Skein Categories in Non-semisimple Settings
本文提出了一种非半单设定下的 skein 分类，通过张量理想扩展了现有理论，并揭示了因子化同调与 skein 分类之间的关系。
We classify semisimple rigid monoidal categories with two isomorphism classes of simple objects over the field of complex numbers. In the appendix written by P.Etingof it is proved that the number of semisimple Hopf algebras with a given finite number of irreducible representations is finite.
Compact Semisimple Tensor 2-Categories are Morita Connected
本文证明紧致半单张量2-范畴在任意特征零域上具有Morita连通性，推广了融合2-范畴的相应结果，并由此构造了由第四Galois上同调群索引的辫融合1-范畴。
Derived Reconstruction of Semisimple Tensor and Module Categories
本文研究半单张量范畴及其模范畴的有界导出范畴，通过整值特征和稳定化子子群分类幺半与模$t$-结构，证明导出等价可重构原范畴。
We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our joint work with D.Nikshych. In particular, we generalize to the categorical setting the Hopf and quasi-Hopf algebra freeness theorems due to Nichols-Zoeller and Schauenburg, respectively. We also give categorical versions of the theory of distinguished group-like elements in a finite dimensional Hopf algebra, of Lorenz's result on degeneracy of the Cartan matrix, and of the absence of primitive eleme…
Category theoretic aspects of non-rational conformal field theories are discussed. We consider the case that the category C of chiral sectors is a finite tensor category, i.e. a rigid monoidal category whose class of objects has certain finiteness properties. Besides the simple objects, the indecomposable projective objects of C are of particular interest. The fusion rules of C can be block-diagonalized. A conjectural connection between the block-diagonalization and modular transformations of characters of modules over vertex algebras is exemplified with the case of the (1,p) minimal models.
On braided simple extensions and braided non-semisimple near-group categories
本文研究了编织简单扩展和非半单近群范畴的性质，证明了非退化编织非半单近群范畴与sRep(W⊕W*)的扩展关系，并揭示了其与皮卡群的联系。
On the Dualizability of Fusion 2-Categories
本文在任意域上构造了可分模2-范畴的相对2-Deligne张量积，证明了可分紧半单张量2-范畴在Morita 4-范畴中完全可对偶，并推广到特征零域，给出场论解释。
We classify ribbon semisimple monoidal categories with three isomorphism classes of simple objects over the field of complex numbers.
Frobenius-Schur Indicators of Dual Fusion Categories and Semisimple Partially Dualized Quasi-Hopf Algebras
本文研究球面融合范畴的对偶范畴中对象的Frobenius-Schur指标，建立特定对象指标间的关系，并推导出半单Hopf代数及其部分对偶化quasi-Hopf代数的指标公式与指数等式。
Finite-dimensional quantum groups of type Super A and non-semisimple modular categories
本文研究了超A类型有限维量子群的构造及其非半单模范畴的分类，通过辫状德林费尔德双和非半单球形结构，提出了新的链接不变量并区分了某些结。
A semisimple algebraic tensor category over an algebraically closed field k of characteristic zero is the representation category of all finite dimensional twisted super representations of an affine reductive supergroup G over k. Such a supergroup is reductive if and only if its connected component is reductive. The connected component is reductive if and only if the Lie superalgebra divided by its center is a product of simple Lie algebras of classical type and Lie superalgebras spo(1,2r) of the orthosymplectic types BC_r.
Starting from an abelian category A such that every object has only finitely many subobjects we construct a semisimple tensor category T. We show that T interpolates the categories Rep(Aut(p),K) where p runs through certain projective (pro-)objects of A. The main example is A=finite dimensional F_q-vector spaces. Then T can be considered as the category of representations of GL(n,F_q) where n is not a natural number. This work extends a construction of Deligne for symmetric groups.
Twisted Deligne products of semisimple tensor categories
探讨半单张量范畴$\mathcal C$、$\mathcal D$的扭曲德利涅积分类，给出特定条件下唯一扭曲德利涅积是普通积的结论，借助相关工作实现群论分类及融合范畴精确分解，附录引入范畴$n$-上链概念并回答相关问题。
Locally finitely presented Grothendieck categories and the pure semisimplicity conjecture
本文研究局部有限呈现的纯半单格罗滕迪克范畴与左纯半单环之间的等价关系，并扩展了Auslander和Ringel-Tachikawa对应到左纯半单环类。
We show that every unitarizable fusion category, and more generally every semisimple C*-tensor category, admits a unique unitary structure. Our proof is based on a categorified polar decomposition theorem for monoidal equivalences between such categories. We prove analogous results for unitarizable braided fusion categories and module categories.
A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with respect to a module category. Whenever the dual of a pointed semisimple category with respect to a module category is pointed, we give explicit formulas for the Grothendieck ring and for the associator of the dual. This leads to the definition of categorical Morita equivalence on the set of all finite groups and on the set of all pairs (G, w), where G is a finite group and w is a 3-cocycle on G. A gr…
Unit inclusion in a non-semisimple braided tensor category and non-compact relative TQFTs
研究辫子张量范畴中单位包含诱导的 1-态射的可对偶性判据，证明其在半单与非半单模范畴下分别诱导出相对 3 维 TQFT，并猜想可恢复 WRT 等理论。
We give a formula for the relative Deligne tensor product of two indecomposable finite semisimple module categories over a pointed braided fusion category over an algebraically closed field.
For each finite semisimple tensor category, we associate a quantum group (face algebra) whose comodule category is equivalent to the original one, in a simple natural manner. To do this, we also give a generalization of the Tannaka-Krein duality, which assigns a face algebra for each tensor category equipped with an embedding into a certain kind of bimodule category.
Universal Semisimple Reconstruction and Obstruction Theory
针对有限长度阿贝尔范畴建立半单障碍理论，构造典范障碍子对象与半单化反射子，其迭代可还原经典根基滤过，揭示经典根基理论是范畴框架的模论实例。
A semisimple subcategory of Khovanov's Heisenberg category
本文研究了 Khovanov 的 Heisenberg 分类中的半单子类，展示了其在非交换张量三角几何中的应用，揭示了 Balmer 谱的张量积性质与单侧厚张量理想分类的矛盾。
In conformal field theory the understanding of correlation functions can be divided into two distinct conceptual levels: The analytic properties of the correlators endow the representation categories of the underlying chiral symmetry algebras with additional structure, which in suitable cases is the one of a finite tensor category. The problem of specifying the correlators can then be encoded in algebraic structure internal to those categories. After reviewing results for conformal field theories for which these representation categories are semisimple, we explain what is known about represent…
We study exact module categories over the representation categories of finite-dimensional quasi-Hopf algebras. As a consequence we classify exact module categories over some families of pointed tensor categories with cyclic group of invertible objets of order p, where p is a prime number.
A graded tensor category over a group $G$ will be called a strongly $G$-graded tensor category if every homogeneous component has at least one multiplicativily invertible object. Our main result is a description of the module categories over a strongly $G$-graded tensor category as induced from module categories over tensor subcategories associated with the subgroups of $G$.
We describe equivalence classes of exact indecomposable module categories over a finite graded tensor category. When applied to a pointed fusion category, our results coincide with the ones obtained in [S. Natale, On the equivalence of module categories over a group-theoretical fusion category, SIGMA, Symmetry Integrability Geom. Methods Appl. 13 Paper 042, 9 p. (2017)].
I review recent developments of "non-semisimple" modular tensor categories in the sense of Lyubashenko and the categorical Verlinde formula for such categories (this is a proceedings article for Meeting for Study of Number theory, Hopf algebras and related topics held at University of Toyama, Japan, 12-15 February 2017).
In this paper we classify all semisimple tensor categories with the same fusion rules as $\operatorname{Rep}(SO(4))$, or one of the associated truncations. We show that such categories are explicitly classified by two non-zero complex numbers. Furthermore we show these tensor categories are always braided, and there exist exactly 8 braidings.
Tensor categories
本文给出幺半范畴与张量范畴的基本定义和构造，并引用原始文献，以群论范畴为例说明。
The purpose of this article is to study the existence of Deligne's tensor product of abelian categories by comparing it with the well-known ten- sor product of finitely cocomplete categories. The main result states that the former exists precisely when the latter is an abelian category, and moreover in this case both tensor products coincide. An example of two abelian categories whose Deligne tensor product does not exist is given.
We introduce the Frobenius-Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius-Schur theorem, including that for semisimple quasi-Hopf algebras, weak Hopf C*-algebras and association schemes. Our framework also clarifies a mechanism how the `twisted' theory arises from the ordinary case. As a demonstration, we give a twisted Frobenius-Schur theorem for semisimple quasi-Hopf algebras. We also give several applications to the quantum SL_2.
Positivstellensatz for C*-tensor categories
本文在刚性半单C*-张量范畴中探讨半预C*-代数，并利用环表示技术将Ozawa的群性质(T)判据推广至该范畴。
We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively 2, 3 and 4-dimensional framed local topological field theories. In particular, we produc…
A graded tensor category over a group $G$ will be called a crossed product tensor category if every homogeneous component has at least one multiplicatively invertible object. Our main result is a description of the crossed product tensor categories, graded monoidal functors, monoidal natural transformations, and braiding in terms of coherent outer $G$-actions over tensor categories.
The Reduced Tensor Product of Braided Tensor Categories Containing a Symmetric Fusion Category
本研究针对含固定对称融合子范畴的辫子张量范畴，借助富集程序构造约化张量积对称幺模结构，并建立其与Drinfeld中心交叉辫子张量范畴2-范畴的等价性，还给出其在极小模扩张配对中的应用。
We define equivariantization of tensor categories under tensor group scheme actions and give necessary and sufficient conditions for an exact sequence of tensor categories to be an equivariantization under a finite group or finite group scheme action. We introduce the notion of central exact sequence of tensor categories and use it in order to present an alternative formulation of some known characterizations of equivariantizations for fusion categories, and to extend these characterizations to equivariantizations of finite tensor categories under finite group scheme actions. In particular, we…
Finite Pre-Tensor Categories that are Morita Equivalent to Finite Tensor Categories
本文研究有限预张量范畴与有限张量范畴的莫里塔等价问题，证明预张量范畴的莫里塔等价当且仅当其Drinfeld中心为有限张量范畴。
We discuss algebraic and representation theoretic structures in braided tensor categories C which obey certain finiteness conditions. Much interesting structure of such a category is encoded in a Hopf algebra H in C. In particular, the Hopf algebra H gives rise to representations of the modular group SL(2,Z) on various morphism spaces. We also explain how every symmetric special Frobenius algebra in a semisimple modular category provides additional structure related to these representations.
Rigidity of non-negligible objects of moderate growth in braided categories
本文研究了辫状范畴中非可忽略对象的刚性性质，并探讨了其在半单范畴和对数共形场论中的应用。
We aim to study Morita theory for tensor triangulated categories. For two finite tensor categories having no projective simple objects, we prove that their stable equivalence induced by an exact $\Bbbk$-linear monoidal functor can be lifted to a tensor equivalence under some certain conditions.
Given a tensor functor between tensor categories $\mathcal{C}$ and $\mathcal{D}$, we give criteria that, under certain assumptions, the Frobeniusness of $\mathcal{C}$ or $\mathcal{D}$ implies the Frobeniusness of the other one. We also give an affirmative answer to Natale's question asking if the class of Frobenius tensor categories is closed under exact sequences.
Growth problems in diagram categories
研究图示/插值范畴中生成对象张量幂的分解数增长速率的渐进行为，推导出半单情况下的渐近公式。
A C*-tensor category with simple unit object is realized by von Neumann algebra bimodules of finite Jones index if and only if it is rigid.
We focus on the problem of producing new modular tensor categories from Hopf algebras. To do this, we first give a general method to construct factorizable Hopf algebras. Then we apply the method to construct two families of ribbon factorizable Hopf algebras which are not quantum groups or Drinfel'd double. One family are point Hopf algebras which give non-semisimple modular tensor categories, while the other family are C^*-Hopf algebras which give new unitary modular tensor categories. Lastly, we determine the fusion rings of these unitary modular tensor categories.
For C a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show: 1) C always contains a simple projective object; 2) if C is in addition ribbon, the internal characters of projective modules span a submodule for the projective SL(2,Z)-action; 3) the action of the Grothendieck ring of C on the span of internal characters of projective objects can be diagonalised; 4) the linearised Grothendieck ring of C is semisimple iff C is semisimple. Results 1-3 remain true in positive characteristic under an extra assumption. Result 1 implies that t…
We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A=F-Vect, where F is a field. An object X in A with two-sided dual X^ gives rise to a Frobenius alg…
In this expository paper we explain in detail how to construct bicategorical colimits of several kinds of tensor categories, for example essentially small finitely cocomplete K-linear tensor categories. The constructions are direct and elementary.
We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally presentable linear categories, and that the tensor product of locally coherent Grothendieck categories is locally coherent if and only if the Deligne tensor product of their abelian categories of finitely presented objects exists. We describe the tensor product of non-commutative projective schemes in terms of Z-algebras, and show that for project…
We endow the homotopy category of well generated (pretriangulated) dg categories with a tensor product satisfying a universal property. The resulting monoidal structure is symmetric and closed with respect to the cocontinuous RHom of dg categories (in the sense of Toën [26]). We give a construction of the tensor product in terms of localisations of dg derived categories, making use of the enhanced derived Gabriel-Popescu theorem [21]. Given a regular cardinal alpha, we define and construct a tensor product of homotopically alpha-cocomplete dg categories and prove that the well generated tensor…
Following the analogy between algebras (monoids) and monoidal categories the construction of nucleus for non-associative algebras is simulated on the categorical level. Nuclei of categories of modules are considered as an example.
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.
Filtered bicolimit presentations of locally presentable linear categories, Grothendieck categories and their tensor products
本文提出两种将Grothendieck范畴恢复为滤过双余极限的方法，证明第一种与张量积相容而第二种一般不相容，并将Kelly张量积的性质推广到局部可表现线性范畴。
Restriction categories were established to handle maps that are partially defined with respect to composition. Tensor topology realises that monoidal categories have an intrinsic notion of space, and deals with objects and maps that are partially defined with respect to this spatial structure. We introduce a construction that turns a firm monoidal category into a restriction category and axiomatise the monoidal restriction categories that arise this way, called tensor-restriction categories.
We extend the calculus of relations to embed a regular category A into a family of pseudo-abelian tensor categories T(A,d) depending on a degree function d. Under the condition that all objects of A have only finitely many subobjects, our main results are as follows: 1. Let N be the maximal proper tensor ideal of T(A,d). We show that T(A,d)/N is semisimple provided that A is exact and Mal'cev. Thereby, we produce many new semisimple, hence abelian, tensor categories. 2. Using lattice theory, we give a simple numerical criterion for the vanishing of N. 3. We determine all degree functions for w…
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