Craig Pastro, Ross Street
We develop the theory of weak bimonoids in braided monoidal categories and show them to be quantum categories in a certain sense. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid R leads to a weak Hopf monoid R \otimes R.
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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 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…
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…
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.
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。
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.
Skew-monoidal categories arise when the associator and the left and right units of a monoidal category are, in a specific way, not invertible. We prove that the closed skew-monoidal structures on the category of right R-modules are precisely the right bialgebroids over the ring R. These skew-monoidal structures induce quotient skew-monoidal structures on the category of R-R-bimodules and this leads to the following generalization: Opmonoidal monads on a monoidal category correspond to skew-monoidal structures with the same unit object which are compatible with the ordinary monoidal structure b…
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…
We prove Heyneman-Radford Theorem in the framework of Monoidal Categories.
We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. We apply it to several new monoidal categories that appeared recently in the theory of Koszul duality for operads and props. This gives a conceptual explanation of the form of the free operad, free dioperad, free properad and free prop.
We prove a coherence theorem for actions of groups on monoidal categories. As an application we prove coherence for arbitrary braided $G$-crossed categories.
Weak braiding for algebras in braided monoidal categories
本文提出弱辫子张量范畴的定义，证明辫子G交叉范畴、辫子张量范畴中交换代数的模范畴均属此类，且合理假设下经扭曲函子得到的弱辫子范畴为群交叉辫子化范畴。
This paper is about skew monoidal tensored V-categories (= skew monoidal hommed V-actegories) and their categories of modules. A module over <M,*,R> is an algebra for the monad T = R * _ on M. We study in detail the skew monoidal structure of M^T and construct a skew monoidal forgetful functor from M^T to the category of E-objects in M where E=M(R,R) is the endomorphism monoid of the unit object R. Then we give conditions for the forgetful functor to be strong monoidal and for the category M^T of modules to be monoidal. In formulating these conditions a notion of `self-cocomplete' subcategorie…
The analogy between Yetter's deformation theory form (lax) monoidal functors and Gerstenahaber's deformation theory for associative algebras is solidified by shown that under reasonable conditions the category of functors with an action of a lax monoidal functor is abelian, that an analogue of the Hochschild cohomology of an algebra with coefficients in a bimodule exists for monoidal functors, and is given by right derived functors. The deformation cohmology of a monoidal natural transformation is shown to be a special case.
Skew monoidal categories are monoidal categories with non-invertible `coherence' morphisms. As shown in a previous paper bialgebroids over a ring R can be characterized as the closed skew monoidal structures on the category Mod R in which the unit object is R. This offers a new approach to bialgebroids and Hopf algebroids. Little is known about skew monoidal structures on general categories. In the present paper we study the one-object case: skew monoidal monoids (SMM). We show that they possess a dual pair of bialgebroids describing the symmetries of the (co)module categories of the SMM. Thes…
This article represents a preliminary attempt to link Kan extensions, and some of their further developments, to Fourier theory and quantum algebra through *-autonomous monoidal categories and related structures.
Division algebras in monoidal categories
针对幺半范畴中的除代数定义问题，将域上除代数的等价定义适配到幺半范畴场景，构建多类除代数定义并给出相关示例与推论。
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。
The braided monoidal structure of tube algebra representations
针对球面半单多张量范畴的管代数，构造其表示的带扭辫子幺半群结构，并证明该范畴与对应ind-范畴的Drinfeld中心辫子张量等价，拓展了线性等价结果。
We study the basic monoidal properties of the category of Hopf modules for a coquasi Hopf algebra. In particular we discuss the so called fundamental theorem that establishes a monoidal equivalence between the category of comodules and the category of Hopf modules. We present a categorical proof of Radford's $S^4$ formula for the case of a finite dimensional coquasi Hopf algebra, by establishing a monoidal isomorphism between certain double dual functors.
We introduce the notions of a half-balanced braided monoidal category and of its contraction. These notions give rise to an explicit description of the action of the Galois group of QQ on Teichmueller groupoids in genus 0, equivalent to that of L. Schneps. We also show that a prounipotent version of this action is equivalent to a graded action.
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 begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their applications to 4d topological quantum field theories and 2-tangles (surfaces embedded in 4-dimensional space). Then we give concise definitions of semistrict monoidal 2-categories and braided monoidal 2-categories, and show how these may be unpacked to give long explicit definitions similar to, but not quite the same as, those given by Kapranov and Voevodsky. Finally, we describe how to construct a semistrict braided monoidal 2-category Z(C) as the `center' of a semistrict monoidal…
We introduce monoidal categories whose monoidal products of any positive number of factors are lax coherent and whose nullary products are oplax coherent. We call them $\mathsf{Lax}^+\mathsf{Oplax}^0$-monoidal. Dually, we consider $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal categories which are oplax coherent for positive numbers of factors and lax coherent for nullary monoidal products. We define $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$-duoidal categories with compatible $\mathsf{Lax}^+\mathsf{Oplax}^0$- and $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal structures. We introduce comonoids in $\mathsf{La…
In our recent paper [Sh1] a version of the "generalized Deligne conjecture" for abelian $n$-fold monoidal categories is proven. For $n=1$ this result says that, given an abelian monoidal $k$-linear category $\mathscr{A}$ with unit $e$, $k$ a field of characteristic 0, the dg vector space $\mathrm{RHom}_{\mathscr{A}}(e,e)$ is the first component of a Leinster 1-monoid in $\mathscr{A}lg(k)$ (provided a rather mild condition on the monoidal and the abelian structures in $\mathscr{A}$, called homotopy compatibility, is fulfilled). In the present paper, we introduce a new concept of a ${\it graded}…
Given a monoidal category C, an ordinary category M, and a monad T in M, the lifts in a strict sense of a fixed action of C on M to an action of C on the Eilenberg-Moore category of T-modules in M are in a bijective correspondence with certain families of natural transformations, indexed by the objects in the monoidal category C. These families are analogues of distributive laws between monads.
A linear Gr-category is a category of finite-dimensional vector spaces graded by a finite group together with natural tensor product. We classify the braided monoidal structures of a class of linear Gr-categories via explicit computations of the 3-cocycles and quasi-bicharacters of finite abelian groups which are direct product of two cyclic groups.
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…
In this paper, we define and study weak monoidal Hom-Hopf algebras, which generalize both weak Hopf algebras and monoidal Hom-Hopf algebras. If $H$ is a weak monoidal Hom-Hopf algebra with bijective antipode and let $Aut_{wmHH}(H)$ be the set of all automorphisms of $H$. Then we introduce a category ${_{H}\mathcal{WMHYD}^{H}}(α,β)$ with $α,β\in Aut_{wmHH}(H)$ and construct a braided $T$-category $\mathcal{WMHYD}(H)$ that having all the categories ${_{H}\mathcal{WMHYD}^{H}}(α,β)$ as components.
Hopf (bi-)modules and crossed modules over a bialgebra B in a braided monoidal category C are considered. The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). Bialgebra projections and Hopf bimodule bialgebras over a Hopf algebra in C are found to be isomorphic categories. As a consequence a generalization of the Radford-Majid criterion for a braided Hopf algebra to be a cross product is obtained. The results of this paper turn out to be fundamental for the construction of (bicovariant) differential calculi on braided Hopf alg…
Braided deformations of (symmetric) monoidal categories are related to Vassiliev theory by a direct generalization of well-known results relating "quantum" knot invariants to Vassiliev invariants. The deformation theory of braidings is subsumed by the deformation theory of monoidal functors, which proves surprisingly rich: the deformation complex of a monoidal functor has the same structure as the deformation complex of an algebra, including a pre-Lie structure, from which it is see that the problem of deforming monoidal functors (including braidings) is purely cohomological in nature.
We extend the formalism of Hopf cyclic cohomology to the context of braided categories. For a Hopf algebra in a braided monoidal abelian category we introduce the notion of stable anti-Yetter-Drinfeld module. We associate a para-cocyclic and a cocyclic object to a braided Hopf algebra endowed with a braided modular pair in involution in the sense of Connes and Moscovici. When the braiding is symmetric the full formalism of Hopf cyclic cohomology with coefficients can be extended to our categorical setting.
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。
Let $k$ be a field, $k^*=k\setminus\{0\}$ and $C_2$ the cyclic group of order 2. In this note we compute all the braided monoidal structures on the category of $k$-vector spaces graded by the Klein group $C_2\times C_2$. Actually, for the monoidal structures we will compute the explicit form of the 3-cocycles on $C_2\times C_2$ with coefficients in $k^*$, while for the braided monoidal structures we will compute the explicit form of the abelian 3-cocycles on $C_2\times C_2$ with coefficients in $k^*$. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided…
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.
The chiral conformal field theory of free super-bosons is generated by weight one currents whose mode algebra is the affinisation of an abelian Lie super-algebra h with non-degenerate super-symmetric pairing. The mode algebras of a single free boson and of a single pair of symplectic fermions arise for even|odd dimension 1|0 and 0|2 of h, respectively. In this paper, the representations of the untwisted mode algebra of free super-bosons are equipped with a tensor product, a braiding, and an associator. In the symplectic fermion case, i.e. if h is purely odd, the braided monoidal structure is e…
We show that the equivalence between several possible characterizations of Frobenius algebras, and of symmetric Frobenius algebras, carries over from the category of vector spaces to more general monoidal categories. For Frobenius algebras, the appropriate setting is the one of rigid monoidal categories, and for symmetric Frobenius algebras it is the one of sovereign monoidal categories. We also discuss some properties of Nakayama automorphisms.
In this expository paper, we discuss and compare the notions of braided and coboundary monoidal categories. Coboundary monoidal categories are analogues of braided monoidal categories in which the role of the braid group is replaced by the cactus group. We focus on the categories of representations of quantum groups and crystals and explain how while the former is a braided monoidal category, this structure does not pass to the crystal limit. However, the categories of representations of quantum groups of finite type also possess the structure of a coboundary category which does behave well in…
The Relative Monoidal Center and Tensor Products of Monoidal Categories
本文建立增广幺半范畴理论，定义平衡双模，构造相对张量积与相对幺半中心并证明其Morita对偶性，还在量子代数等实例中验证了相对中心的范畴等价性。
We study the bialgebra structures on quiver coalgebras and the monoidal structures on the categories of locally nilpotent and locally finite quiver representations. It is shown that the path coalgebra of an arbitrary quiver admits natural bialgebra structures. This endows the category of locally nilpotent and locally finite representations of an arbitrary quiver with natural monoidal structures from bialgebras. We also obtain theorems of Gabriel type for pointed bialgebras and hereditary finite pointed monoidal categories.
We show that, with some technical conditions, an abelian category can be embedded into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail.
In this paper we give an expository account of quasistrict symmetric monoidal 2-categories, as introduced by Schommer-Pries. We reformulate the definition using a graphical calculus called wire diagrams, which facilitates computations and emphasizes the central role played by the interchangor coherence isomorphisms.
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。
Braided cohomology of quasi-triangular bialgebras and braided Morita invariance
引入线性幺半范畴中编织余代数的编织上链复形和编织上同调，证明在有限维条件下拟三角双代数的编织上链复形是编织Morita不变量。
Braided commutative algebras over quantized enveloping algebras
本研究推广Davydov全中心构造，从B-中心幺半范畴的代数出发在相对幺半中心中构建辫子交换代数，将其应用于量子包络代数表示范畴得到辫子交换模代数，还构造了B-中心幺半范畴中代数的Morita不变量。
Concepts of first order differential calculus (FODC) on a monoidal Hom-algebra and left-covariant FODC over a left Hom-quantum space with respect to a monoidal Hom-Hopf algebra are presented. Then, extension of the universal FODC over a monoidal Hom-algebra to a universal Hom-differential calculus is described. Next, concepts of left(right)-covariant and bicovariant FODC over a monoidal Hom-Hopf algebra are studied in detail. Subsequently, notion of quantum Hom-tangent space associated to a bicovariant Hom-FODC is introduced and equipped with an analogue of Lie bracket (commutator) through Wor…
We apply the notion of relative adjoint functor to generalise closed monoidal categories. We define representations in such categories and give their relation with left actions of monoids. The translation of these representations under lax monoidal functors is investigated. We introduce tensor product of representations of bimonoids as a functorial binary operation and show how symmetric lax monoidal functors act on this product. Finally we apply the general theory to classical and quantum representations.
Two Hopf algebras are called monoidally Morita equivalent if module categories over them are equivalent as linear monoidal categories. We introduce monoidal Morita invariants for finite-dimensional Hopf algebras based on certain braid group representations arising from the Drinfeld double construction. As an application, we show, for any integer $n$, the number of elements of order $n$ is a monoidal Morita invariant for finite group algebras. We also describe relations between our construction and invariants of closed 3-manifolds due to Reshetikhin and Turaev.
We give a definition of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded $R$-modules to become a monoidal categorification of a quantum cluster algebra, where $R$ is a symmetric Khovanov-Lauda-Rouquier algebra. Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions once the first-step mutations are possible. In the course of the study, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.
We introduce and investigate new invariants on the pair of modules $M$ and $N$ over quantum affine algebras $U_q'(\mathfrak{g})$ by analyzing their associated R-matrices. From new invariants, we provide a criterion for a monoidal category of finite-dimensional integrable $U_q'(\mathfrak{g})$-modules to become a monoidal categorification of a cluster algebra.
Covariant Hom-bimodules are introduced and the structure theory of them in the Hom-setting is studied in a detailed way. The category of bicovariant Hom-bimodules is proved to be a (pre)braided monoidal category and its structure theory is also provided in coordinate form. The notion of Hom-Yetter-Drinfel'd modules is presented and it is shown that the category of Hom-Yetter-Drinfel'd modules is a (pre)braided tensor category as well. As one of the main results, a (pre)braided monoidal equivalence between these tensor categories is verified, which extends the fundamental theorem of Hom-Hopf mo…
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