# 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} }

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…

## 9 Citations

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

- MathematicsNew Generation Computing
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!-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

- Computer ScienceQPL
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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

- Mathematics
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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

- Computer ScienceESOP
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### Categorical Quantum Mechanics I: Causal Quantum Processes

- Philosophy
- 2018

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.
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### A Compositional Approach to Parity Games

- Computer ScienceMFPS
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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

- MathematicsArXiv
- 2017

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

- MathematicsCALCO
- 2015

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…

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