Abstract Tensor Systems as Monoidal Categories

@inproceedings{Kissinger2014AbstractTS,
  title={Abstract Tensor Systems as Monoidal Categories},
  author={Aleks Kissinger},
  booktitle={Categories and Types in Logic, Language, and Physics},
  year={2014}
}
  • A. Kissinger
  • Published in
    Categories and Types in Logic…
    16 August 2013
  • Mathematics
The primary contribution of this paper is to give a formal, categorical treatment to Penrose’s abstract tensor notation, in the context of traced symmetric monoidal categories. To do so, we introduce a typed, sum-free version of an abstract tensor system and demonstrate the construction of its associated category. We then show that the associated category of the free abstract tensor system is in fact the free traced symmetric monoidal category on a monoidal signature. A notable consequence of… 

Tensors, !-graphs, and Non-commutative Quantum Structures

Abstract!-graphs provide a means of reasoning about infinite families of string diagrams and have proven useful in manipulation of (co)algebraic structures like Hopf algebras, Frobenius algebras, and

Tensors, !-graphs, and non-commutative quantum structures (extended version)

!-graphs provide a means of reasoning about infinite families of string diagrams and have proven useful in manipulation of (co)algebraic structures like Hopf algebras, Frobenius algebras, and

Encoding !-tensors as !-graphs with neighbourhood orders

Diagrammatic reasoning using string diagrams provides an intuitive language for reasoning about morphisms in a symmetric monoidal category. To allow working with infinite families of string diagrams,

!-Logic : first order reasoning for families of non-commutative string diagrams

Equational reasoning with string diagrams provides an intuitive method for proving equations between morphisms in various forms of monoidal category. !-Graphs were introduced with the intention of

Computing with Semirings and Weak Rig Groupoids

This work proposes a variant of the Curry---Howard correspondence which is inspired by conservation of information and recent homotopy theoretic approaches to type theory, and naturally relates semirings to reversible programming languages.

Categorical Quantum Mechanics I: Causal Quantum Processes

We derive the category-theoretic backbone of quantum theory from a process ontology. More specifically, we treat quantum theory as a theory of systems, processes and their interactions. In this

A Compositional Approach to Parity Games

The category of open parity games is introduced, which is defined using standard definitions for graph games, and a suitable semantic category inspired by the work by Grellois and Melliès on the semantics of higher-order model checking is introduced.

Disintegration and Bayesian Inversion, Both Abstractly and Concretely

The notions of disintegration and Bayesian inversion are presented here in abstract graphical formulations, and the resulting abstract descriptions are used for proving basic results in conditional probability theory.

A First-order Logic for String Diagrams

Equational reasoning with string diagrams provides an intuitive means of proving equations between morphisms in a symmetric monoidal category. This can be extended to proofs of infinite families of

References

SHOWING 1-10 OF 27 REFERENCES

Functors between tensored categories

to be coherent with respect to coherent commutative and associative tensor products in d and ~. This type of coherence is necessary when introducing cup products and Steenrod operations into the

Compact Monoidal Categories from Linguistics to Physics

This is largely an expository paper, revisiting some ideas about compact 2-categories, in which each 1-cell has both a left and a right adjoint. In the special case with only one 0-cell (where the

Finite Dimensional Vector Spaces Are Complete for Traced Symmetric Monoidal Categories

We show that the category FinVectk of finite dimensional vector spaces and linear maps over any field k is (collectively) complete for the traced symmetric monoidal category freely generated from a

Tortile tensor categories

A Survey of Graphical Languages for Monoidal Categories

This article is intended as a reference guide to various notions of monoidal categories and their associated string diagrams. It is hoped that this will be useful not just to mathematicians, but also

Interacting Quantum Observables: Categorical Algebra and Diagrammatics

The ZX-calculus is introduced, an intuitive and universal graphical calculus for multi-qubit systems, which greatly simplifies derivations in the area of quantum computation and information and axiomatize phase shifts within this framework.

Coherence for compact closed categories

Quantum Invariants of Knots and 3-Manifolds

This monograph, now in its second revised edition, provides a systematic treatment of topological quantum field theories in three dimensions, inspired by the discovery of the Jones polynomial of

Pictures of processes : automated graph rewriting for monoidal categories and applications to quantum computing

The introduction of a discretised version of a string diagram called a string graph is introduced, and it is shown how string graphs modulo a rewrite system can be used to construct free symmetric traced and compact closed categories on a monoidal signature.

Two-spinor calculus and relativistic fields

Preface 1. The geometry of world-vectors and spin-vectors 2. Abstract indices and spinor algebra 3. Spinors and world-tensors 4. Differentiation and curvature 5. Fields in space-time Appendix