Centre of an algebra

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 (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.     

Skew Monoidal Monoids

Skew monoidal categories are monoidal categories with non-invertible “coherence” morphisms. As shown in a previous article, 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_R. This offers a new approach to bialgebroids and Hopf algebroids. Little is known about skew monoidal structures on general categories. In the present article, we study the one-object case: skew monoidal monoids (SMMs). We show that they possess a dual pair of bialgebroids describing the symmetries of the (co)module categories of the SMM. These bialgebroids are submonoids of their own base and are rank 1 free over the base on the source side. We give various equivalent definitions of SMM, study the structure of their (co)module categories, and discuss the possible closed and Hopf structures on a SMM.     

扭曲Smash积的辫Monoidal范畴与辫Monoidal范畴上的扭曲Smash积

得出了扭曲Ｓｍａｓｈ积模范畴Ａ＊ＨＭ是辫ｍｏｎｏｉｄａｌ范畴的一个充要条件;进一步讨论了与范畴Ａ—ＨＭ等价的范畴,引进了广义Ｙｅｔｔｅｒ－Ｄｒｉｎｆｅｌｄ范畴ＨＭＣ;最后,给出了辫Ｍｏｎｏｉｄａｌ范畴上扭曲Ｓｍａｓｈ积构成Ｈｏｐｆ代数充要的条件,这些结果统一了量子群中许多重要结论。     

Compatibility Conditions Between Rings and Corings

We introduce the notion of “bi-monoid” in general monoidal category, generalizing by this the notion of “bialgebra”. In the case of bimodules over a noncommutative algebra, we obtain compatibility conditions between rings and corings whenever both structures admit the same underlying bimodule. Several examples are expounded in this case. We also show that there is a class of right modules over a bi-monoid which is a monoidal category and the forgetful functor to the ground category is a strict monoidal functor.     

Cartesian Differential Categories as Skew Enriched Categories

We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

     

The Triangulated Hopf Category n +SL(2)

We discuss an example of a triangulated Hopf category related to SL(2). It is an equivariant derived category equipped with multiplication and comultiplication functors and structure isomorphisms. We prove some coherence equations for structure isomorphisms. In particular, the Hopf category is monoidal.     

Categories with sums and right distributive tensor product

Models for parallel and concurrent processes lead quite naturally to the study of monoidal categories (Inform. Comput. 88 (2) (1990) 105). In particular a category Tree of trees, equipped with a non-symmetric tensor product, interpreted as a concatenation, seems to be very useful to represent (local) behavior of non-deterministic agents able to communicate (Enriched Categories for Local and Interaction Calculi, Lecture Notes in Computer Science, Vol. 283, Springer, Berlin, 1987, pp. 57-70). The category Tree is also provided with a coproduct (corresponding to choice between behaviors) and the tensor product is only partially distributive w.r.t. it, in order to preserve non-determinism. Such a category can be properly defined as the category of the (finite) symmetric categories on a free monoid, when this free monoid is considered as a 2-category. The monoidal structure is inherited from the concatenation in the monoid. In this paper we prove that for every alphabet A, Tree(A), the category of finite A-labeled trees is equivalent to the free category which is generated by A and enjoys the afore-mentioned properties. The related category Beh(A), corresponding to global behaviors is also proven to be equivalent to the free category which is generated by A and enjoys a smaller set of properties.     

The Fundamental Pro-groupoid of an Affine 2-scheme

A natural question in the theory of Tannakian categories is: What if you don’t remember Forget? Working over an arbitrary commutative ring R, we prove that an answer to this question is given by the functor represented by the étale fundamental groupoid π ₁(spec(R)), i.e. the separable absolute Galois group of R when it is a field. This gives a new definition for étale π ₁(spec(R)) in terms of the category of R-modules rather than the category of étale covers. More generally, we introduce a new notion of “commutative 2-ring” that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of π ₁ for the corresponding “affine 2-schemes.” These results help to simplify and clarify some of the peculiarities of the étale fundamental group. For example, étale fundamental groups are not “true” groups but only profinite groups, and one cannot hope to recover more: the “Tannakian” functor represented by the étale fundamental group of a scheme preserves finite products but not all products.     

Symmetric Bimonoidal Intermuting Categories and ω × ω Reduced Bar Constructions

A new, self-contained, proof of a coherence result for categories equipped with two symmetric monoidal structures bridged by a natural transformation is given. It is shown that this coherence result is sufficient for ω × ω-indexed family of iterated reduced bar constructions based on such a category.     

Labelled cospan categories and properads

We prove Steinebrunner's conjecture on the biequivalence between (coloured) properads and labelled cospan categories. The main part of the work is to establish a 1-categorical, strict version of the conjecture, showing that the category of properads is equivalent to a category of strict labelled cospan categories via the symmetric monoidal envelope functor.     

Weak bimonoids in duoidal categories

Weak bimonoids in duoidal categories are introduced. They provide a common generalization of bimonoids in duoidal categories and of weak bimonoids in braided monoidal categories. Under the assumption that idempotent morphisms in the base category split, they are shown to induce weak bimonads (in four symmetric ways). As a consequence, they have four separable Frobenius base (co)monoids, two in each of the underlying monoidal categories. Hopf modules over weak bimonoids are defined by weakly lifting the induced comonad to the Eilenberg–Moore category of the induced monad. Making appropriate assumptions on the duoidal category in question, the fundamental theorem of Hopf modules is proven which says that the category of modules over one of the base monoids is equivalent to the category of Hopf modules if and only if a Galois-type comonad morphism is an isomorphism.     

An internal language for autonomous categories

We present an internal language for symmetric monoidal closed (autonomous) categories analogous to the typed lambda calculus as an internal language for cartesian closed categories. The language we propose is the term assignment to the multiplicative fragment of Intuitionistic Linear Logic, which possesses exactly the right structure for an autonomous theory. We prove that this language is an internal language and show as an application the coherence theorem of Kelly and Mac Lane, which becomes straightforward to state and prove. Finally, we extend the language with the natural numbers and show that this corresponds to a weak Natural Numbers Object in an autonomous category.     

Making the Category of Doi-Hopf Modules into a Braided Monoidal Category

We investigate how the category of Doi-Hopf modules can be made into a monoidal category. It suffices that the algebra and coalgebra in question are both bialgebras with some extra compatibility relation. We study tensor identies for monoidal categories of Doi-Hopf modules. Finally, we construct braidings on a monoidal category of Doi-Hopf modules. Our construction unifies quasitriangular and coquasitriangular Hopf algebras, and Yetter-Drinfel'd modules.     

Center construction and duality of category of Hom-Yetter-Drinfeld modules over monoidal Hom-Hopf algebras

Let (H,α) be a monoidal Hom-Hopf algebra. In this paper, we will study the category of Hom-Yetter-Drinfeld modules. First, we show that the category of left-left Hom-Yetter-Drinfeld modules _H^HH Y D is isomorphic to the center of the category of left (H,α)-Hom-modules. Also, by the center construction, we get that the categories of left-left, left-right, right-left, and right-right Hom-Yetter-Drinfeld modules are isomorphic as braided monoidal categories. Second, we prove that the category of finitely generated projective left-left Hom-Yetter-Drinfeld modules has left and right duality.     

Quasi-bimonads and their representations

In this paper quasi-bimonads on a monoidal category are introduced and investigated. Quasi-bimonads generalize quasi-bialgebras to a non-braided setting. We discuss their representations and investigate the R-matrix of a quasi-bimonad. Such an R-matrix provides a new solution of the version of the Yang-Baxter equation adapted to the situation. We also introduce an equivalent relation on (quasitriangular) quasi-bimonads such that the categories of representations of two (quasitriangular) quasi-bimonads are (braided) monoidal equivalent. Finally, we discuss Drinfeld twists and Hom quasi-bialgebras.     

Character sheaves on unipotent groups in positive characteristic: foundations

In this article we formulate and prove the main theorems of the theory of character sheaves on unipotent groups over an algebraically closed field of characteristic $p>0$ . In particular, we show that every admissible pair for such a group $G$ gives rise to an $\mathbb{L }$ -packet of character sheaves on $G$ and that conversely, every $\mathbb{L }$ -packet of character sheaves on $G$ arises from a (nonunique) admissible pair. In the Appendices we discuss two abstract category theory patterns related to the study of character sheaves. The first Appendix sketches a theory of duality for monoidal categories, which generalizes the notion of a rigid monoidal category and is close in spirit to the Grothendieck–Verdier duality theory. In the second one we use a topological field theory approach to define the canonical braided monoidal structure and twist on the equivariant derived category of constructible sheaves on an algebraic group; moreover, we show that this category carries an action of the surface operad. The third Appendix proves that the “naive” definition of the equivariant $\ell $ -adic derived category with respect to a unipotent algebraic group is equivalent to the “correct” one.     

Coherence for closed categories with biproducts

A coherence result for symmetric monoidal closed categories with biproducts is shown in this paper. It is also explained how to prove coherence for compact closed categories with biproducts and for dagger compact closed categories with dagger biproducts by using the same technique.     

Module categories,weak Hopf algebras and modular invariants

We develop a theory of module categories over monoidal categories (this is a straightforward categorization of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects is equivalent to the category of representations of a weak Hopf algebra (theorem of T. Hayashi) and we classify module categories over the fusion category of sl(2) at a positive integer level where we meet once again the ADE classification pattern.     

Intermutation

This paper proves coherence results for categories with a natural transformation called intermutation made of arrows from (A ∧ B) ∨ (C ∧ D) to (A ∨ C) ∧ (B ∨ D), for ∧ and ∨ being two biendofunctors. Intermutation occurs in iterated, or n-fold, monoidal categories, which were introduced in connection with n-fold loop spaces, and for which a related, but different, coherence result was obtained previously by Balteanu, Fiedorowicz, Schw?nzl and Vogt. The results of the present paper strengthen up to a point this previous result, and show that two-fold loop spaces arise in the manner envisaged by these authors out of categories of a more general kind, which are not two-fold monoidal in their sense. In particular, some categories with finite products and coproducts are such. Coherence in Mac Lane’s “all diagrams commute” sense is proved here first for categories where for ∧ and ∨ one assumes only intermutation, and next for categories where one also assumes natural associativity isomorphisms. Coherence in the sense of coherence for symmetric monoidal categories is proved when one assumes moreover natural commutativity isomorphisms for ∧ and ∨. A restricted coherence result, involving a proviso of the kind found in coherence for symmetric monoidal closed categories, is proved in the presence of two nonisomorphic unit objects. The coherence conditions for intermutation and for the unit objects are derived from a unifying principle, which roughly speaking is about preservation of structures involving one endofunctor by another endofunctor, up to a natural transformation that is not an isomorphism. This is related to weakening the notion of monoidal functor. A similar, but less symmetric, justification for intermutation was envisaged in connection with iterated monoidal categories. Unlike the assumptions previously introduced for two-fold monoidal categories, the assumptions for the unit objects of the categories of this paper, which are more general, allow an interpretation in logic.     

Multiplier Hopf Monoids

The notion of multiplier Hopf monoid in any braided monoidal category is introduced as a multiplier bimonoid whose constituent fusion morphisms are isomorphisms. In the category of vector spaces over the complex numbers, Van Daele’s definition of multiplier Hopf algebra is re-obtained. It is shown that the key features of multiplier Hopf algebras (over fields) remain valid in this more general context. Namely, for a multiplier Hopf monoid A, the existence of a unique antipode is proved — in an appropriate, multiplier-valued sense — which is shown to be a morphism of multiplier bimonoids from a twisted version of A to A. For a regular multiplier Hopf monoid (whose twisted versions are multiplier Hopf monoids as well) the antipode is proved to factorize through a proper automorphism of the object A. Under mild further assumptions, duals in the base category are shown to lift to the monoidal categories of modules and of comodules over a regular multiplier Hopf monoid. Finally, the so-called Fundamental Theorem of Hopf modules is proved — which states an equivalence between the base category and the category of Hopf modules over a multiplier Hopf monoid.