Ross Street
The centre of a monoidal category is a braided monoidal category. Monoidal categories are monoidal objects (or pseudomonoids) in the monoidal bicategory of categories. This paper provides a universal construction in a braided monoidal bicategory that produces a braided monoidal object from any monoidal object. Some properties and sufficient conditions for existence of the construction are examined.
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We denote the monoidal bicategory of two-sided modules (also called profunctors, bimodules and distributors) between categories by $\mathrm{Mod}$; the tensor product is cartesian product of categories. For a groupoid $\scr{G}$, we study the monoidal centre $\mathrm{ZPs}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$ of the monoidal bicategory $\mathrm{Ps}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$ of pseudofunctors and pseudonatural transformations; the tensor product is pointwise. Alexei Davydov defined the full centre of a monoid in a monoidal category. We define a higher dimensional version: the full monoi…
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.
Here we describe three straightforward examples of what was called a graphic Fourier transformation in [4]. At least two of these examples may be viewed simply as monoidal comonads on suitable monoidal closed functor categories, but the third example, which involves "centres" of monoidal closed functor categories, is generally not comonadic. For the first two examples (i.e., association schemes and RCFT's), a more elaborate "probicategory" set-up was envisaged in an earlier version of this note, but many readers missed the main point so it is simplified (hopefully) below.
We show that the Drinfeld centre of a symmetric fusion category is a bilax 2-fold monoidal category. That is, it carries two monoidal structures, the convolution and symmetric tensor products, that are bilax monoidal functors with respect to each other. We additionally show that the braiding and symmetry for the convolution and symmetric tensor products are compatible with this bilax structure. We establish these properties without referring to Tannaka duality for the symmetric fusion category. This has the advantage that all constructions are done purely in terms of the fusion category struct…
In a recent paper, Daisuke Tambara defined two-sided actions on an endomodule (= endodistributor) of a monoidal V-category A. When A is autonomous (= rigid = compact), he showed that the V-category (that we call Tamb(A)) of so-equipped endomodules (that we call Tambara modules) is equivalent to the monoidal centre Z[A,V] of the convolution monoidal V-category [A,V]. Our paper extends these ideas somewhat. For general A, we construct a promonoidal V-category DA (which we suggest should be called the double of A) with an equivalence [DA,V] \simeq Tamb(A). When A is closed, we define strong (resp…
Motivated by algebraic structures appearing in Rational Conformal Field Theory we study a construction associating to an algebra in a monoidal category a commutative algebra ({\em full centre}) in the monoidal centre of the monoidal category. We establish Morita invariance of this construction by extending it to module categories. As an example we treat the case of group-theoretical categories.
In this work we develop some aspects of the theory of Hopf algebras to the context of autonomous map pseudomonoids. We concentrate in the Hopf modules and the Centre or Drinfel'd double. If $A$ is a map pseudomonoid in a monoidal bicategory \M, the analogue of the category of Hopf modules for $A$ is an Eilenberg-Moore construction for a certain monad in $\mathbf{Hom}(\M^{\mathrm{op}},\mathbf{Cat})$. We study the existence of the internalisation of this notion, called the Hopf module construction, by extending the completion under Eilenberg-Moore objects of a 2-category to a endo-homomorphism o…
This is a report on aspects of the theory and use of monoidal categories. The first section introduces the main concepts through the example of the category of vector spaces. String notation is explained and shown to lead naturally to a link between knot theory and monoidal categories. The second section reviews the light thrown on aspects of representation theory by the machinery of monoidal category theory, such as braidings and convolution. The category theory of Mackey functors is reviewed in the third section. Some recent material and a conjecture concerning monoidal centres is included. …
We study the biclosedness of the monoidal categories of modules and comodules over a (left or right) Hopf algebroid, along with the bimodule category centres of the respective opposite categories and a corresponding categorical equivalence to anti Yetter-Drinfel'd contramodules and anti Yetter-Drinfel'd modules, respectively. This is directly connected to the existence of a trace functor on the monoidal categories of modules and comodules in question, which in turn allows to recover (or define) cyclic operators enabling cyclic cohomology.
We present Trimble's definition of a tetracategory and prove that the spans in (strict) 2-categories with certain limits have the structure of a monoidal tricategory, defined as a one-object tetracategory. We recall some notions of limits in 2-categories for use in the construction of the monoidal tricategory of spans.
Brackets and products from centres in extension categories
该研究将Schwede关于结合代数双模上Ext群的Gerstenhaber代数构造推广至带幺半中心系数的一般幺半范畴，显式刻画了双代数胚左模范畴情形下的结构并给出operad证明，推广了经典结果，同时提出更一般情形的猜想。
Tricategorical Universal Properties Via Enriched Homotopy Theory
本文通过充实于Gray模型范畴的同伦良好极限与余极限，建立了三范畴极限与余极限的理论，并将其应用于伪单子Kleisli构造、中心构造及双范畴严格化等具体结构。
We introduce, for a symmetric fusion category $\mathcal{A}$ with Drinfeld centre $\mathcal{Z}(\mathcal{A})$, the notion of $\mathcal{Z}(\mathcal{A})$-crossed braided tensor category. These are categories that are enriched over $\mathcal{Z}(\mathcal{A})$ equipped with a symmetric tensor product, while being braided monoidal with respect to the usual tensor product on $\mathcal{Z}(\mathcal{A})$. In the Tannakian case where $\mathcal{A}=\mathbf{Rep}(G)$ for a finite group $G$, the 2-category of $\mathcal{Z}(\mathcal{A})$-crossed braided categories is shown to be equivalent to the 2-category of $G…
Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation
该研究从增强型2-范畴视角解释二维范畴论中结构的内在严格性，引入增强型2-范畴极限草图以弥补现有理论不足，并建立了这类结构的内化对称性。
Every small monoidal category with universal finite joins of central idempotents is monoidally equivalent to the category of global sections of a sheaf of local monoidal categories on a topological space. Every small stiff monoidal category monoidally embeds into such a category of global sections. An infinitary version of these theorems also holds in the spatial case. These representation results are functorial and subsume the Lambek-Moerdijk-Awodey sheaf representation for toposes, the Stone representation of Boolean algebras, and the Takahashi representation of Hilbert modules as continuous…
Pivotality, twisted centres and the anti-double of a Hopf monad
该研究将有限维Hopf代数中对合对、反Yetter–Drinfeld模与Drinfeld double同构的对应推广到刚性幺半范畴，构造研究Hopf单子的反Drinfeld double并给出单子论解释，还探讨了其与Drinfeld中心枢轴性的关联。
Grothendieck prelopologies: towards a closed monoidal sheaf category
本文推广 Grothendieck 预拓扑至半笛卡尔范畴，通过不保持有限极限的层化过程，构造闭幺半群层范畴，并受量子格上层概念的启发。
For a small category K enriched over a suitable monoidal category V, the free completion of K under colimits is the presheaf category [K*,V]. If K is large, its free completion under colimits is the V-category PK of small presheaves on K, where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed structures on PK.
We axiomatize the extended operators in topological orders (possibly gravitationally anomalous, possibly with degenerate ground states) in terms of monoidal Karoubi-complete $n$-categories which are mildly dualizable and have trivial centre. Dualizability encodes the word "topological," and we take it as the definition of "(separable) multifusion $n$-category"; triviality of the centre implements the physical principle of "remote detectability." We show that such $n$-categorical algebras are Morita-invertible (in the appropriate higher Morita category), thereby identifying topological orders w…
We collate information about the fusion categories with $A_n$ fusion rules. This note includes the classification of these categories, a realisation via the Temperley-Lieb categories, the auto-equivalence groups (both braided and tensor), identifications of the subcategories of invertible objects, and explicit descriptions of the Drinfeld centres. The first section describes the classification of these categories (as monoidal, dagger, pivotal, and braided categories). The second section describes the properties of these categories.
Hopf bimodules for bialgebroids
本文在双代数胚框架下定义四类Hopf模及双边双余边Hopf模（Hopf双模），通过Hopf-Galois余模证明基本定理，并构造两种辫子幺半范畴等价于Yetter-Drinfeld模范畴。
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。
Tape Diagrams for Monoidal Monads
本文扩展磁带图表示法，使其适用于由代数理论给出的对称幺半群单子的克莱斯利范畴，突破了原有仅适用于双积范畴的限制。
The Grouplike Functor and Monoidal Adjunctions
本文在辫子幺半范畴中范畴化Hopf代数的群样元素函子，通过有限极限构造群对象，证明其函子性并给出自由Hopf幺半群函子的右伴随，推广经典对应。
The main objective of the paper is to define the category of monoids as a weighted limit. We also define the category of actions of monoids along the action of a monoidal category as a weighted limit.
We give several criteria to decide whether a given tensor category is the abelian envelope of a fixed symmetric monoidal category. As a main result we prove that the category of finite-dimensional representations of a semisimple simply connected algebraic group is the abelian envelope of the category of tilting modules. Benson and Etingof conjectured that a certain limit of finite symmetric tensor categories is tensor equivalent to the finite dimensional representations of $SL_2$ in characteristic $2$. We use our results on the abelian envelopes to prove this conjecture and its variants for an…
Accessible aspects of 2-category theory
该研究针对范畴结构及其伪映射难构成局部可展示2-范畴的问题，证明足够弱的范畴结构对应的2-范畴具有可达性，并探究了这类2-范畴的柔性极限及其与滤过余极限的相互作用，丰富了2-范畴论的可达性相关结论。
The construction of the cotensor coalgebra for an "abelian monoidal" category $\M$ which is also cocomplete, complete and AB5, was performed in [A. Ardizzoni, C. Menini and D. Ştefan, \emph{Cotensor Coalgebras in Monoidal Categories}, Comm. Algebra, to appear]. It was also proved that this coalgebra satisfies a meaningful universal property which resembles the classical one. Here the lack of the coradical filtration for a coalgebra $E$ in $\M$ is filled by considering a direct limit $\widetilde{D}$ of a filtration consisting of wedge products of a subcoalgebra $D$ of $E$. The main aim of this …
Monoidal closure of Grothendieck constructions via $Σ$-tractable monoidal structures and Dialectica formulas
本文通过引入 $Σ$-可处理幺半结构，刻画 Grothendieck 构造的纤维极限与余极限，建立全范畴幺半闭的充分条件，推广 Gödel 的 Dialectica 解释，并讨论闭结构的纤维性。
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。
We construct free monoids in a monoidal category with finite limits and countable colimits, in which tensoring on either side preserves reflexive coequalizers and colimits of countable chains.
We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We find concrete generators for the categories of coalgebras in these monoidal categories, and explicitly construct cofree coalgebras, products and limits of coalgebras in each case. This answers an open question of A. Agore on the existence of a cofree coring, and constructs the cofree (co)module coalgebra on a $B$-(co)module, for a bialgebra $B$.
A monoidal Dold-Kan correspondence for comodules
本文构造了余模范畴中基于 Postnikov 塔的归纳纤维性替换，证明单纯余模与连通微分分次余模 Quillen 等价，并推出单连通空间的有理 A-理论等价于相应余模结构完美链复形的 K-理论。
We construct for every $\infty$-operad $\mathcal{O}^\otimes$ with certain finite limits new $\infty$-operads of spectrum objects and of commutative group objects in $\mathcal{O}$. We show that these are the universal stable resp. additive $\infty$-operads obtained from $\mathcal{O}^\otimes$. We deduce that for a stably (resp. additively) symmetric monoidal $\infty$-category $\mathcal{C}$ the Yoneda embedding factors through the $\infty$-category of exact, contravariant functors from $\mathcal{C}$ to the $\infty$-category of spectra (resp. connective spectra) and admits a certain multiplicative…
Categorical orthodoxy has it that collections of ordinary mathematical structures such as groups, rings, or spaces, form categories (such as the category of groups); collections of 1-dimensional categorical structures, such as categories, monoidal categories, or categories with finite limits, form 2-categories; and collections of 2-dimensional categorical structures, such as 2-categories or bicategories, form 3-categories. We describe a useful way in which to regard bicategories as objects of a 2-category. This is a bit surprising both for technical and for conceptual reasons. The 2-cells of t…
We prove 2-categorical conservativity for any {0,T}-free fragment of MALL over its corresponding intuitionistic version: that is, that the universal map from a closed symmetric monoidal category to the *-autonomous category that it freely generates is fully faithful, and similarly for other doctrines. This implies that linear logics and graphical calculi for *-autonomous categories can also be interpreted canonically in closed symmetric monoidal categories. In particular, every closed symmetric monoidal category can be fully embedded in a *-autonomous category, preserving both tensor products …
Hopf 2-algebras and Braided Monoidal 2-Categories
本文提出Hopf 2-代数的同伦范畴化，构造2-量子双并证明其对偶性，同时定义2-R矩阵并探讨其在4维扩展缺陷编织中的应用。
We develop a theory of categories which are simultaneously (1) indexed over a base category S with finite products, and (2) enriched over an S-indexed monoidal category V. This includes classical enriched categories, indexed and fibered categories, and internal categories as special cases. We then describe the appropriate notion of "limit" for such enriched indexed categories, and show that they admit "free cocompletions" constructed as usual with a Yoneda embedding.
We axiomatically define (pre-)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small pre-Hilbert category whose monoidal unit is a simple generator embeds (weakly) monoidally into the category of pre-Hilbert spaces and adjointable maps, preserving adjoint morphisms and all finite (co)limits. An intermediate result that is important in its own right is that the scalars in such a category necessarily form an involutive field. In case of a Hilbert category, the embedding extends …
Finite Temperley-Lieb (TL) algebras are diagram-algebra quotients of (the group algebra of) the famous Artin's braid group $B_N$, while the affine TL algebras arise as diagram algebras from a generalized version of the braid group. We study asymptotic `$N\to\infty$' representation theory of these quotients (parametrized by $q\in\mathbb{C}^{\times}$) from a perspective of braided monoidal categories. Using certain idempotent subalgebras in the finite and affine algebras, we construct infinite `arc' towers of the diagram algebras and the corresponding direct system of representation categories, …
This paper presents some basic facts about the so-called connectivity spaces. In particular, it studies the generation of connectivity structures, the existence of limits and colimits in the main categories of connectivity spaces, the closed monoidal category structure given by the so-called tensor product on integral connectivity spaces; it defines homotopy for connectivity spaces and mention briefly related difficulties; it defines smash product of pointed integral connectivity spaces and shows that this operation results in a closed monoidal category with such spaces as objects. Then, it st…
We develop a homotopy theory of categories enriched in a monoidal model category V. In particular, we deal with homotopy weighted limits and colimits, and homotopy local presentability. The main result, which was known for simplicially-enriched categories, links homotopy locally presentable V-categories with combinatorial model V-categories, in the case where has all objects of V are cofibrant.
We define for each $n \geq 1$ a symmetric monoidal $(\infty, n+1)$-category $n\mathrm{Pr}^L$ whose objects we call presentable $(\infty,n)$-categories, generalizing the usual theory of presentable $(\infty,1)$-categories. We show that each object $\mathcal{C}$ in $n\mathrm{Pr}^L$ has an underlying $(\infty,n)$-category $ψ_n(\mathcal{C})$ which admits all conical colimits, and that conical colimits of right adjointable diagrams in $ψ_n(\mathcal{C})$ can be computed in terms of conical limits after passage to right adjoints.
Higher categories of push-pull spans, I: Construction and applications
本文构造了一族对称幺半$(\infty,3)$-范畴，包含配备局部系统的对应关系，并应用此构造为Rozansky-Witten模型的$3$-范畴提供近似，验证相关猜想。
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