Monoidal Categories

2019 ◽  
pp. 29-60
Author(s):  
Chris Heunen ◽  
Jamie Vicary
Keyword(s):  
Hilbert Spaces ◽  
Monoidal Category ◽  
Monoidal Categories ◽  
Graphical Calculus ◽  
The Core ◽  
Basic Language ◽  
Linear Maps ◽  
Visual Notation

A monoidal category is a category equipped with extra data, describing how objects and morphisms can be combined in parallel. This chapter introduces the theory of monoidal categories, including braidings, symmetries and coherence. They form the core of this book, as they provide the basic language with which the rest of the material will be developed. We introduce a visual notation called the graphical calculus, which provides an intuitive and powerful way to work with them. We also introduce the monoidal categories Hilb of Hilbert spaces and linear maps, Set of sets and functions and Rel of sets and relations, which will be used as running examples throughout the book.

Monoidal 2-Categories

2019 ◽  
pp. 265-312
Author(s):  
Chris Heunen ◽  
Jamie Vicary
Keyword(s):  
Hilbert Spaces ◽  
Two Dimensional ◽  
Monoidal Categories ◽  
Dense Coding ◽  
Graphical Calculus ◽  
Frobenius Algebras ◽  
Higher Dimensional ◽  
Modern Mathematics ◽  
Natural Way

Monoidal 2-categories are higher-dimensional versions of monoidal categories, allowing a more expressive syntax that plays an important role in modern mathematics. We explore their two-dimensional graphical calculus, and show how duality gives a language for oriented surfaces, from which Frobenius algebras emerge in a natural way. We describe 2-Hilbert spaces, categorifications of Hilbert spaces and explore the monoidal 2-category 2Hilb that they give rise to. We then show how we can use dualities in 2Hilb to give a concise and purely topological language to reason about teleportation, dense coding and complementarity.

Regular factorizations and perturbation analysis for the core inverse of linear operators in Hilbert spaces

2018 ◽  
Vol 96 (10) ◽  
pp. 1943-1956 ◽  
Author(s):  
Qianglian Huang ◽  
Saijie Chen ◽  
Zhirong Guo ◽  
Lanping Zhu
Keyword(s):  
Hilbert Spaces ◽  
Perturbation Analysis ◽  
Linear Operators ◽  
The Core ◽  
Core Inverse

Existence Theorems for Some Weak Abstract Variable Domain Hyperbolic Problems

1971 ◽  
Vol 23 (4) ◽  
pp. 611-626 ◽  
Author(s):  
Robert Carroll ◽  
Emile State
Keyword(s):  
Hilbert Spaces ◽  
Differential Operators ◽  
Separable Hilbert Space ◽  
Hyperbolic Equations ◽  
Existence Theorems ◽  
Hyperbolic Problems ◽  
Linear Maps ◽  
Variable Domains ◽  
Elliptic Differential Operators ◽  
Image Position

In this paper we prove some existence theorems for some weak problems with variable domains arising from hyperbolic equations of the type1.1where A = {A(t)} is, for example, a family of elliptic differential operators in space variables x = (x1, …, xn). Thus let H be a separable Hilbert space and let V(t) ⊂ H be a family of Hilbert spaces dense in H with continuous injections i(t): V(t) → H (0 ≦ t ≦ T < ∞). Let V’ (t) be the antidual of V(t) (i.e. the space of continuous conjugate linear maps V(t) → C) and using standard identifications one writes V(t) ⊂ H ⊂ V‘(t).

scholarly journals Operator Popoviciu’s inequality for superquadratic and convex functions of selfadjoint operators in Hilbert spaces

2019 ◽  
Vol 10 (4) ◽  
pp. 313-324
Author(s):  
Mohammad W. Alomari
Keyword(s):  
Hilbert Spaces ◽  
Convex Functions ◽  
Positive Operators ◽  
Selfadjoint Operators ◽  
Popoviciu’S Inequality ◽  
Linear Maps ◽  
Superquadratic Functions ◽  
Operator Version ◽  
Positive Linear Maps

AbstractIn this work, an operator version of Popoviciu’s inequality for positive operators on Hilbert spaces under positive linear maps for superquadratic functions is proved. Analogously, using the same technique, an operator version of Popoviciu’s inequality for convex functions is obtained. Some other related inequalities are also presented.

scholarly journals Monoidal Categories, 2-Traces, and Cyclic Cohomology

2019 ◽  
Vol 62 (02) ◽  
pp. 293-312 ◽  
Author(s):  
Mohammad Hassanzadeh ◽  
Masoud Khalkhali ◽  
Ilya Shapiro
Keyword(s):  
Associative Algebra ◽  
Monoidal Category ◽  
Cyclic Homology ◽  
Cyclic Cohomology ◽  
Monoidal Categories ◽  
Central Pair ◽  
Cyclic Type ◽  
Unital Associative Algebra

AbstractIn this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category ( $\mathscr{C},\otimes$ ) endowed with a symmetric 2-trace, i.e., an $F\in \text{Fun}(\mathscr{C},\text{Vec})$ satisfying some natural trace-like conditions, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra “with coefficients in $F$ ”. Furthermore, we observe that if $\mathscr{M}$ is a $\mathscr{C}$ -bimodule category and $(F,M)$ is a stable central pair, i.e., $F\in \text{Fun}(\mathscr{M},\text{Vec})$ and $M\in \mathscr{M}$ satisfy certain conditions, then $\mathscr{C}$ acquires a symmetric 2-trace. The dual notions of symmetric 2-contratraces and stable central contrapairs are derived as well. As an application we can recover all Hopf cyclic type (co)homology theories.

scholarly journals Rank one preserving R-linear maps on spaces of self-adjoint operators on complex Hilbert spaces

2006 ◽  
Vol 416 (2-3) ◽  
pp. 568-579 ◽  
Author(s):  
Hong You ◽  
Shaowu Liu ◽  
Guodong Zhang
Keyword(s):  
Hilbert Spaces ◽  
Linear Maps ◽  
Rank One ◽  
Adjoint Operators

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 Autonomization of Monoidal Categories

2017 ◽  
Author(s):  
Antonin Delpeuch
Keyword(s):  
Monoidal Category ◽  
Point Of View ◽  
Monoidal Categories ◽  
Categorical Point

We define the free autonomous category generated by a monoidal category and study some of its properties. From a linguistic perspective, this expands the range of possible models of meaning within the distributional compositional framework, by allowing nonlinearities in maps. From a categorical point of view, this provides a factorization of the construction in [Preller and Lambek, 2007] of the free autonomous category generated by a category. ; Comment: Under review. Comments welcome!

scholarly journals Dagger linear logic for categorical quantum mechanics

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Robin Cockett ◽  
Cole Comfort ◽  
Priyaa Srinivasan
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Quantum Physics ◽  
Linear Logic ◽  
Infinite Dimensional ◽  
Graphical Calculus ◽  
Finite Dimensional ◽  
Categorical Quantum Mechanics ◽  
Definition Of ◽  
Categorical Semantics

Categorical quantum mechanics exploits the dagger compact closed structure of finite dimensional Hilbert spaces, and uses the graphical calculus of string diagrams to facilitate reasoning about finite dimensional processes. A significant portion of quantum physics, however, involves reasoning about infinite dimensional processes, and it is well-known that the category of all Hilbert spaces is not compact closed. Thus, a limitation of using dagger compact closed categories is that one cannot directly accommodate reasoning about infinite dimensional processes. A natural categorical generalization of compact closed categories, in which infinite dimensional spaces can be modelled, is *-autonomous categories and, more generally, linearly distributive categories. This article starts the development of this direction of generalizing categorical quantum mechanics. An important first step is to establish the behaviour of the dagger in these more general settings. Thus, these notes simultaneously develop the categorical semantics of multiplicative dagger linear logic. The notes end with the definition of a mixed unitary category. It is this structure which is subsequently used to extend the key features of categorical quantum mechanics.

scholarly journals Linear Maps Preserving Rank-additivity and Rank-sum-minimal on Tensor Products of Matrix Spaces

2019 ◽  
pp. 1-10
Author(s):  
Lele Gao ◽  
Yang Zhang ◽  
Jinli Xu
Keyword(s):  
Matrix Theory ◽  
Tensor Products ◽  
Linear Mapping ◽  
Linear Map ◽  
The Core ◽  
Research Areas ◽  
Linear Maps ◽  
Matrix Spaces

The problems of characterizing maps that preserve certain invariant on given sets are called the preserving problems, which have become one of the core research areas in matrix theory. If for any a linear map, , as established, there is we say that  preserves the rank-additivity. If for any , and a linear map,  established, there is  we say that rank-sum-miminal. In this paper, we characterize the form of linear mapping .

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