张量余单子与张量范畴

设是一个张量范畴,g和F均为上的张量余单子,p是一个余单子分配率.本文从FG的张量余单子结构和2-范畴的角度,描述了双余模范畴的张量结构,并给出了其做成张量范畴的一些充要条件.’     

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.     

Monoidal categories of corings

We introduce a monoidal category of corings using two different notions of corings morphisms. The first one is the (right) coring extensions recently introduced by T. Brzeziński in [2], and the other is the usual notion of morphisms defined in [6] by J. Gómez-Torrecillas.

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Sunto Introduciamo una categoria monoidale di coanelli usando due diverse nozioni di morfismi di coanelli. La prima è l'estensione (destra) di coanelli recentemente introdotta da Brzeziński in [2], mentre la seconda è la nozione usuale di morfismo definita in [6] da J. Gómez-Torrecillas.

     

Monoidal Categories and Multiextensions

We associate to a group-like monoidal grupoid a principal bundle E satisfying most of the axioms defining a biextension. The obstruction to the existence of a genuine biextension structure on E is exhibited. When this obstruction vanishes, the biextension E is alternating and a trivialization of E induces a trivialization of . The analogous theory for monoidal n-categories is also examined, as well as the appropriate generalization of these constructions in a sheaf-theoretic context. In the n-categorial situation, this produces a higher commutator calculus, in which some interesting generalizations of the notion of an alternating biextension occur. For n=2, the corresponding cocycles are constructed explicitly, by a partial symmetrization process, from the cocycle describing the n-category.     

Monoidal structures for N-complexes

We classify the monoidal structures for the category of N-complexes which respect the graded structure.     

Monoidal cut strengthening revisited

There are two distinct strengthening methods for disjunctive cuts with some integer variables; they differ in the way they modularize the coefficients. In this paper, we introduce a new variant of one of these methods, the monoidal cut strengthening procedure, based on the paradox that sometimes weakening a disjunction helps the strengthening procedure and results in sharper cuts. We first derive a general result that applies to cuts from disjunctions with any number of terms. It defines the coefficients of the cut in a way that takes advantage of the option of adding new terms to the disjunction. We then specialize this result to the case of split cuts for mixed integer programs with some binary variables, in particular Gomory mixed integer cuts, and to intersection cuts from multiple rows of a simplex tableau. In both instances we specify the conditions under which the new cuts have smaller coefficients than the cuts obtained by either of the two currently known strengthening procedures.     

Monoidal Uniqueness of Stable Homotopy Theory

We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, -spaces, orthogonal spectra, simplicial functors) with symmetric spectra. As an application we show that with an added assumption about underlying model structures Margolis' axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence.     

Cotensor Coalgebras in Monoidal Categories

We introduce the concept of cotensor coalgebra for a given bicomodule over a coalgebra in an Abelian monoidal category ?. If ? is also cocomplete, complete, and AB5, we show that such a cotensor coalgebra exists and satisfies a meaningful universal property which resembles the classical one. Here the lack of the coradical filtration is filled by considering a direct limit of a filtration consisting of wedge products. We prove that this coalgebra is formally smooth whenever the comodule is relative injective and the coalgebra itself is formally smooth.     

Weak Bialgebras and Monoidal Categories

We study monoidal structures on the category of (co)modules over a weak bialgebra. Results due to Nill and Szlachányi are unified and extended to infinite algebras. We discuss the coalgebra structure on the source and target space of a weak bialgebra.     

Militaru's D-Equation in Monoidal Categories

A FRT type construction is done in a minimal categorical context: the ambient monoidal category is only assumed to have coequalizers. The early motivation for this construction was G. Militaru's work on the D-equation. We get generalizations of Militaru's constructions and results. The D-equation is also studied using the classical FRT construction: this leads to a notion of D-bialgebra. New solutions of the D-equation are constructed.     

缠绕模的辫子范畴

该文给出了缠绕模范畴成为辫子范畴的充分必要条件.     

Operads within Monoidal Pseudo Algebras

A general notion of operad is given, which includes: (1) the operads that arose in algebraic topology in the 1970s to characterise loop spaces. (2) the higher operads of Michael Batanin [4] (3) braided and symmetric analogues of Batanin’s operads which are likely to be important in the study of weakly symmetric higher dimensional monoidal categories. The framework of this paper, links together two-dimensional monad theory, operads, and higher dimensional algebra, in a natural way.     

Functorial Calculus in Monoidal Bicategories

The definition and calculus of extraordinary natural transformations is extended to a context internal to any autonomous monoidal bicategory. The original calculus is recaptured from the geometry of the monoidal bicategory V-Mod whose objects are categories enriched in a cocomplete symmetric monoidal category V and whose morphisms are modules.     

A result on the composition of distributions

LetF be a distribution and letf be a locally summable function. The distributionF(f) is defined as the neutrix limit of the sequenceF _n (f), whereF _n (x) = F(x) * δ _n (x) andδ _n (x) is a certain sequence of infinitely differentiable functions converging to the Dirac delta-functionδ(x). The distribution (x^r)^−s is valuated forr, s = 1,2, ….     

Monoidal intervals of clones on infinite sets

Let X be an infinite set of cardinality κ. We show that if L is an algebraic and dually algebraic distributive lattice with at most 2^κ completely join irreducibles, then there exists a monoidal interval in the clone lattice on X which is isomorphic to the lattice 1+L obtained by adding a new smallest element to L. In particular, we find that if L is any chain which is an algebraic lattice, and if L does not have more than 2^κ completely join irreducibles, then 1+L appears as a monoidal interval; also, if λ?2^κ, then the power set of λ with an additional smallest element is a monoidal interval. Concerning cardinalities of monoidal intervals these results imply that there are monoidal intervals of all cardinalities not greater than 2^κ, as well as monoidal intervals of cardinality 2^λ, for all λ?2^κ.     

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。     

Free Monoid in Monoidal Abelian Categories

We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. Then 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 and free properad.      

K-Theory of Braided Monoidal Categories

We show that the commutativity constraint of a braided monoidal category gives rise to an algebraic structure on its K-theory known as a Gerstenhaber algebra. If, in addition, the braiding has a compatible balanced structure the Gerstenhaber bracket on the K-theory is generated by a Batalin–Vilkovisky differential. We use these algebraic structures to prove a generalization of the Anderson–Moore–Vafa theorem which says that the order of the twist, in a semi-simple balanced monoidal category with duals and finitely many simple objects, is finite.