A survey of graphical languages for monoidal categories

Abstract

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 to physicists, computer scientists, and others who use diagrammatic reasoning. We have opted for a somewhat informal treatment of topological notions, and have omitted most proofs. Nevertheless, the exposition is sufficiently detailed to make it clear what is presently known, and to serve as a starting place for more in-depth study. Where possible, we provide pointers to more rigorous treatments in the literature. Where we include results that have only been proved in special cases, we indicate this in the form of caveats.

Cited by in corpus (109)

• Quantum from principles
• Introduction to the book "Quantum Theory: Informational Foundations and Foils"
• The completeness of quantum theory for predicting measurement outcomes
• Quasi-quantization: classical statistical theories with an epistemic restriction
• Disintegration and Bayesian Inversion via String Diagrams
• Deligne Categories in Lattice Models and Quantum Field Theory, or Making Sense of $O(N)$ Symmetry with Non-integer $N$
• A compositional approach to quantum functions
• The Frobenius anatomy of word meanings II: possessive relative pronouns
• The blob complex
• Biunitary constructions in quantum information
• DisCoPy: Monoidal Categories in Python
• The Morita theory of quantum graph isomorphisms
• Dilation of states and processes in operational-probabilistic theories
• Strong Complementarity and Non-locality in Categorical Quantum Mechanics
• Grammar-aware sentence classification on quantum computers
• Finite dimensional Hilbert spaces are complete for dagger compact closed categories
• Bridging the gap between general probabilistic theories and the device-independent framework for nonlocality and contextuality
• Microcanonical thermodynamics in general physical theories
• Operational axioms for diagonalizing states
• Techniques to Reduce $\pi /4$-Parity-Phase Circuits, Motivated by the ZX Calculus
• Axiomatizing complete positivity
• Categories of relations as models of quantum theory
• Integral Categories and Calculus Categories
• Mixed quantum states in higher categories
• Completely positive classical structures and sequentializable quantum protocols
• A Finite Presentation of CNOT-Dihedral Operators
• Executions in (Semi-)Integer Petri Nets are Compact Closed Categories
• Schwinger's Picture of Quantum Mechanics IV: Composition and independence
• Agents, subsystems, and the conservation of information
• A Graphical Language for Proof Strategies
• Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus
• Autonomization of Monoidal Categories
• Parallel Self-Testing of the GHZ State with a Proof by Diagrams
• Towards Quantum Field Theory in Categorical Quantum Mechanics
• Recoupling coefficients and quantum entropies
• Compositional Models for Power Systems
• Reversible monadic computing
• A covariant Stinespring theorem
• Frobenius structures over Hilbert C*-modules
• The information loss of a stochastic map
• A Graph Theoretic Perspective on CPM(Rel)
• Abstract structure of unitary oracles for quantum algorithms
• Fundamental weight systems are quantum states
• On higher holonomy invariants in higher gauge theory II
• Modified traces for quasi-Hopf algebras
• Deconstructing the Calculus of Relations with Tape Diagrams
• Open Graphs and Computational Reasoning
• Categories of Quantum and Classical Channels (extended abstract)
• Picturing Indefinite Causal Structure
• A Compositional Approach to Parity Games
• Categorical composable cryptography
• Network Models from Petri Nets with Catalysts
• Purity through Factorisation
• A diagrammatic view of differential equations in physics
• Graphical Classification of Entangled Qutrits
• Monoidal categories, representation gap and cryptography
• Affinization of monoidal categories
• Proof Diagrams for Multiplicative Linear Logic: Syntax and Semantics
• Shaded Tangles for the Design and Verification of Quantum Programs (Extended Abstract)
• Towards a unified framework for decomposability of processes
• Functorial Question Answering
• United Monoids: Finding Simplicial Sets and Labelled Algebraic Graphs in Trees
• Inversion, Iteration, and the Art of Dual Wielding
• Towards Compositional Distributional Discourse Analysis
• Categorical composable cryptography: extended version
• Anomaly of $(2+1)$-Dimensional Symmetry-Enriched Topological Order from $(3+1)$-Dimensional Topological Quantum Field Theory
• Shaded tangles for the design and verification of quantum circuits
• Conditional Distributions for Quantum Systems
• Entanglement-symmetries of covariant channels
• A Structural and Nominal Syntax for Diagrams
• A Complete Language for Faceted Dataflow Programs
• Traced Monads and Hopf Monads
• Diagrammatics for real supergroups
• Functorial Language Games for Question Answering
• Language-independence of DisCoCirc's Text Circuits: English and Urdu
• A Complete Diagrammatic Calculus for Boolean Satisfiability
• Category-Graded Algebraic Theories and Effect Handlers
• Braids, twists, trace and duality in combinatory algebras
• On the Killing form of Lie Algebras in Symmetric Ribbon Categories
• Traced Monoidal Categories as Algebraic Structures in Prof
• Traced monoidal categories as algebraic structures in $\mathbf{Prof}$
• Categorifying the ZX-calculus
• Lower and Upper Conditioning in Quantum Bayesian Theory
• Monads on dagger categories
• String Diagrams for Layered Explanations
• Concurrent Process Histories and Resource Transducers
• The Word Problem for Braided Monoidal Categories is Unknot-Hard
• On the Structure of Abstract H*-Algebras
• Graphical Methods in Device-Independent Quantum Cryptography
• Diagrammatics for Comodule Monads
• Differential geometry and general relativity with algebraifolds
• A graphical calculus for semi-groupal categories
• ViCAR: Visualizing Categories with Automated Rewriting in Coq
• Intent Preserving Generation of Diverse and Idiomatic (Code-)Artifacts
• Zigzag normalisation for associative $n$-categories
• Operational Semantics with Hierarchical Abstract Syntax Graphs
• Why FHilb is Not an Interesting (Co)Differential Category
• Complexity of Grammar Induction for Quantum Types
• Belief propagation in monoidal categories
• The dagger lambda calculus
• Hopf-Frobenius Algebras and a Simpler Drinfeld Double
• Implicit automata in λ-calculi III: affine planar string-to-string functions
• Combs, Causality and Contractions in Atomic Markov Categories
• Completeness of Nominal PROPs
• Planar #CSP Equality Corresponds to Quantum Isomorphism -- A Holant Viewpoint
• On Traces in Categories of Contractions
• Composing Dinatural Transformations: Towards a Calculus of Substitution
• A Category of Surface-Embedded Graphs
• Information Flow in Pregroup Models of Natural Language