scholarly journals Some remarks on symmetry for a monoidal category

1981 ◽  
Vol 23 (2) ◽  
pp. 209-214
Author(s):  
Stefano Kasangian ◽  
Fabio Rossi
Keyword(s):  
Tensor Product ◽  
Monoidal Category

It is shown that, for a monoidal category V, not every commutation is a symmetry and also that a commutation does not suffice to define the tensor product A ⊗ B of V-categorles A and B. Moreover, it is shown that every symmetry can be transported along a monoidal equivalence.

Hom-L-R-smash biproduct and the category of Hom–Yetter–Drinfel’d–Long bimodules

2018 ◽  
Vol 17 (07) ◽  
pp. 1850133 ◽  
Author(s):  
Daowei Lu ◽  
Xiaohui Zhang
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Monoidal Category ◽  
Admissible Pair ◽  
Smash Product ◽  
Monoidal Categories ◽  
Finite Dimensional

Let [Formula: see text] be a Hom-bialgebra. In this paper, we firstly introduce the notion of Hom-L-R smash coproduct [Formula: see text], where [Formula: see text] is a Hom-coalgebra. Then for a Hom-algebra and Hom-coalgebra [Formula: see text], we introduce the notion of Hom-L-R-admissible pair [Formula: see text]. We prove that [Formula: see text] becomes a Hom-bialgebra under Hom-L-R smash product and Hom-L-R smash coproduct. Next, we will introduce a prebraided monoidal category [Formula: see text] of Hom–Yetter–Drinfel’d–Long bimodules and show that Hom-L-R-admissible pair [Formula: see text] actually corresponds to a bialgebra in the category [Formula: see text], when [Formula: see text] and [Formula: see text] are involutions. Finally, we prove that when [Formula: see text] is finite dimensional Hom-Hopf algebra, [Formula: see text] is isomorphic to the Yetter–Drinfel’d category [Formula: see text] as braid monoidal categories where [Formula: see text] is the tensor product Hom–Hopf algebra.

scholarly journals Abstract bivariant Cuntz semigroups II

2020 ◽  
Vol 32 (1) ◽  
pp. 45-62 ◽  
Author(s):  
Ramon Antoine ◽  
Francesc Perera ◽  
Hannes Thiel
Keyword(s):  
Tensor Product ◽  
Monoidal Category ◽  
Additional Structure ◽  
Mild Conditions ◽  
Symmetric Monoidal Category

AbstractWe previously showed that abstract Cuntz semigroups form a closed symmetric monoidal category. This automatically provides additional structure in the category, such as a composition and an external tensor product, for which we give concrete constructions in order to be used in applications. We further analyze the structure of not necessarily commutative {\mathrm{Cu}}-semirings, and we obtain, under mild conditions, a new characterization of solid {\mathrm{Cu}}-semirings R by the condition that {R\cong\llbracket R,R\rrbracket}.

scholarly journals The Chromatic Brauer Category and Its Linear Representations

2020 ◽  
Author(s):  
L. Felipe Müller ◽  
Dominik J. Wrazidlo
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Topological Field Theories ◽  
Monoidal Category ◽  
Field Theories ◽  
Vector Spaces ◽  
Real Vector ◽  
Linear Representations ◽  
Linear Maps ◽  
Topological Field

AbstractThe Brauer category is a symmetric strict monoidal category that arises as a (horizontal) categorification of the Brauer algebras in the context of Banagl’s framework of positive topological field theories (TFTs). We introduce the chromatic Brauer category as an enrichment of the Brauer category in which the morphisms are component-wise labeled. Linear representations of the (chromatic) Brauer category are symmetric strict monoidal functors into the category of real vector spaces and linear maps equipped with the Schauenburg tensor product. We study representation theory of the (chromatic) Brauer category, and classify all its faithful linear representations. As an application, we use indices of fold lines to construct a refinement of Banagl’s concrete positive TFT based on fold maps into the plane.

scholarly journals Comodules over weak multiplier bialgebras

2014 ◽  
Vol 25 (05) ◽  
pp. 1450037 ◽  
Author(s):  
Gabriella Böhm
Keyword(s):  
Tensor Product ◽  
Hopf Algebras ◽  
Monoidal Category ◽  
Base Algebra ◽  
Finite Dimensional ◽  
Linear Maps ◽  
Multiplier Hopf Algebras ◽  
Hopf Modules ◽  
Total Algebra ◽  
Module Structures

This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals [The Larson–Sweedler theorem for weak multiplier Hopf algebras, in preparation] are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite-dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.

scholarly journals A MONOIDAL STRUCTURE ON THE CATEGORY OF RELATIVE HOPF MODULES

2012 ◽  
Vol 11 (02) ◽  
pp. 1250026
Author(s):  
DANIEL BULACU ◽  
STEFAAN CAENEPEEL
Keyword(s):  
Tensor Product ◽  
Monoidal Category ◽  
Drinfeld Modules ◽  
Braided Monoidal Category ◽  
Monoidal Structure ◽  
Comodule Algebra ◽  
Hopf Modules

Let B be a bialgebra, and A be a left B-comodule algebra in a braided monoidal category [Formula: see text], and assume that A is also a coalgebra, with a not-necessarily associative or unital left B-action. Then we can define a right A-action on the tensor product of two relative Hopf modules, and this defines a monoidal structure on the category of relative Hopf modules if and only if A is a bialgebra in the category of left Yetter–Drinfeld modules over B. Some examples are given.

Monoids and Comonoids

2019 ◽  
pp. 127-146
Author(s):  
Chris Heunen ◽  
Jamie Vicary
Keyword(s):  
Quantum Information ◽  
Tensor Product ◽  
Monoidal Category ◽  
Classical Information ◽  
Abstract Notion

The tensor product of a monoidal category allows us to consider multiplications on its objects, leading to an abstract notion of monoid. This chapter investigates monoids and their relation to dual objects, as well as comonoids, which axiomatize copying and deleting operations. These play a major role in the categorical description of classical information, since classical information can be copied and deleted, while quantum information cannot. We prove categorical no-deleting and no-cloning theorems, showing that if these structures are able to copy and delete every state of the system, then the category collapses. We also characterize when a tensor product is a categorical product.

scholarly journals An explicit formula for the free exponential modality of linear logic

2017 ◽  
Vol 28 (7) ◽  
pp. 1253-1286 ◽  
Author(s):  
PAUL-ANDRÉ MELLIÈS ◽  
NICOLAS TABAREAU ◽  
CHRISTINE TASSON
Keyword(s):  
Tensor Product ◽  
Explicit Formula ◽  
Monoidal Category ◽  
Linear Logic ◽  
Double Negation ◽  
Symmetric Monoidal Category ◽  
Algebraic Description ◽  
Classical Duality ◽  
First Time ◽  
Coherence Spaces

The exponential modality of linear logic associates to every formula A a commutative comonoid !A which can be duplicated in the course of reasoning. Here, we explain how to compute the free commutative comonoid !A as a sequential limit of equalizers in any symmetric monoidal category where this sequential limit exists and commutes with the tensor product. We apply this general recipe to a series of models of linear logic, typically based on coherence spaces, Conway games and finiteness spaces. This algebraic description unifies for the first time a number of apparently different constructions of the exponential modality in spaces and games. It also sheds light on the duplication policy of linear logic, and its interaction with classical duality and double negation completion.

OPERADS AND JET MODULES

2005 ◽  
Vol 02 (06) ◽  
pp. 1133-1186 ◽  
Author(s):  
MARC A. NIEPER-WISSKIRCHEN
Keyword(s):  
Tensor Product ◽  
Commutative Algebra ◽  
Monoidal Category ◽  
Model Categories ◽  
Commutative Algebras ◽  
Symmetric Monoidal Category ◽  
Atiyah Class ◽  
Algebra Over An Operad ◽  
The Right ◽  
Model Structures

Let A be an algebra over an operad in a cocomplete closed symmetric monoidal category. We study the category of A-modules. We define certain symmetric product functors of such modules generalizing the tensor product of modules over commutative algebras, which we use to define the notion of a jet module. This in turn generalizes the notion of a jet module over a module over a classical commutative algebra. We are able to define Atiyah classes (i.e., obstructions to the existence of connections) in this generalized context. We use certain model structures on the category of A-modules to study the properties of these Atiyah classes. The purpose of the paper is not to present any really deep theorem. It is more about the right concepts when dealing with modules over an algebra that is defined over an arbitrary operad, i.e., the aim is to show how to generalize various classical constructions, including modules of jets, the Atiyah class and the curvature, to the operadic context. For convenience of the reader and for the purpose of defining the notations, the basic definitions of the theory of operads and model categories are included.

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