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Free gs-monoidal categories and free Markov categories [article]

Tobias Fritz, Wendong Liang
2022 arXiv   pre-print
For free gs-monoidal categories, this comes in the form of an explicit combinatorial description of their morphisms as structured cospans of labeled hypergraphs.  ...  Closely related to Markov categories are gs-monoidal categories, also known as CD categories. These omit a condition that implements the normalization of probability.  ...  Special thanks are due to Fabio Gadducci for a long email exchange and copious help with the literature on gs-monoidal categories.  ... 
arXiv:2204.02284v2 fatcat:oook67drxfdvjao6bomgfe2kyy

Sheet diagrams for bimonoidal categories [article]

Cole Comfort, Antonin Delpeuch, Jules Hedges
2020 arXiv   pre-print
Bimonoidal categories (also known as rig categories) are categories with two monoidal structures, one of which distributes over the other.  ...  Our main result is a soundness and completeness theorem of the usual form for graphical calculi: we show that sheet diagrams form the free bimonoidal category on a signature.  ...  Jan Vaillant for his Javascript port of the GNU Linear Programming Kit.  ... 
arXiv:2010.13361v2 fatcat:yo72fdmpuzfnndf52f4sxb37ra

Quasistrict symmetric monoidal 2-categories via wire diagrams [article]

Bruce Bartlett
2014 arXiv   pre-print
In this paper we give an expository account of quasistrict symmetric monoidal 2-categories, as introduced by Schommer-Pries.  ...  We reformulate the definition using a graphical calculus called wire diagrams, which facilitates computations and emphasizes the central role played by the interchangor coherence isomorphisms.  ...  One can think of algebraic manipulations in a symmetric monoidal bicategory as being a form of stable 3 -dimensional algebra. Wire diagrams are one possible notation for this.  ... 
arXiv:1409.2148v1 fatcat:2hagc4d2y5eetkxxihkkrip7we

Belief propagation in monoidal categories

Jason Morton
2014 Electronic Proceedings in Theoretical Computer Science  
It also highlights the computational point of view in monoidal categories.  ...  Reverse-mode automatic differentiation is categorical belief propagation in the category of vectors and matrices over D.  ...  We also conjecture [6] that algorithms commonly used in quantum condensed matter physics such as DMRG [30] and its many extensions can be considered as an instance of the categorical belief propagation  ... 
doi:10.4204/eptcs.172.18 fatcat:fa37qdcblbfwpndjwedbzziowi

A Diagrammatic Axiomatisation for Qubit Entanglement [article]

Amar Hadzihasanovic
2015 arXiv   pre-print
Diagrammatic techniques for reasoning about monoidal categories provide an intuitive understanding of the symmetries and connections of interacting computational processes.  ...  In the context of categorical quantum mechanics, Coecke and Kissinger suggested that two 3-qubit states, GHZ and W, may be used as the building blocks of a new graphical calculus, aimed at a diagrammatic  ...  All diagrams were drawn with TikZiT [25] , whose developers also have the author's gratitude.  ... 
arXiv:1501.07082v1 fatcat:ps63nn7s2zgdvh5zlhzgoa2ywq

A diagrammatic calculus of fermionic quantum circuits [article]

Giovanni de Felice, Amar Hadzihasanovic, Kang Feng Ng
2019 arXiv   pre-print
We achieve this through a procedure that rewrites any diagram in a normal form.  ...  the parity of states, and the tensor product as monoidal product.  ...  We write Hilb Z 2 for the symmetric monoidal category of Z 2 -graded Hilbert spaces and pure maps, with the tensor product as monoidal product.  ... 
arXiv:1801.01231v4 fatcat:s5hq74tsqbeyfj7yjbs6dlvuly

Finite matrices are complete for (dagger-)hypergraph categories [article]

Aleks Kissinger
2015 arXiv   pre-print
Dagger-hypergraph categories are the same, but with dagger-symmetric monoidal categories and dagger-SCFAs.  ...  Hypergraph categories are symmetric monoidal categories where each object is equipped with a special commutative Frobenius algebra (SCFA).  ...  Furthermore they have extremely well-behaved normal forms, in that any connected diagram is equal.  ... 
arXiv:1406.5942v2 fatcat:fm2meueyobftjnzqsbu2xgowbm

A Diagrammatic Axiomatisation of Fermionic Quantum Circuits

Amar Hadzihasanovic, Giovanni De Felice, Kang Feng Ng, Michael Wagner
2018 International Conference on Rewriting Techniques and Applications  
We achieve this through a procedure that rewrites any diagram in a normal form.  ...  We introduce the fermionic ZW calculus, a string-diagrammatic language for fermionic quantum computing (FQC).  ...  We write Hilb Z2 for the symmetric monoidal category of Z 2 -graded Hilbert spaces and pure maps, with the tensor product as monoidal product. Remark 2.  ... 
doi:10.4230/lipics.fscd.2018.17 dblp:conf/rta/HadzihasanovicF18 fatcat:vndkk7qcufbnbfij4japvw56jm

Pictures of Processes: Automated Graph Rewriting for Monoidal Categories and Applications to Quantum Computing [article]

Aleks Kissinger
2012 arXiv   pre-print
If we allow plugging input and output wires together, we can intuitively represent complex compositions of processes, formalised as morphisms in a monoidal category. [...]  ...  String diagrams are used to represent a collection of processes, depicted as "boxes" with multiple (typed) inputs and outputs, depicted as "wires".  ...  Repeating the process for each of the traces, we obtain a diagram in normal form.  ... 
arXiv:1203.0202v2 fatcat:zx2s4qevsre53ecyigq6bnhjhe

Propification and the Scalable Comonad [article]

Titouan Carette
2022 arXiv   pre-print
String diagrams can nicely express numerous computations in symmetric strict monoidal categories (SSMC). To be entirely exact, this is only true for props: the SSMCs whose monoid of objects are free.  ...  In this paper, we show a propification theorem asserting that any SSMC is monoidally equivalent to a coloured prop. As a consequence, all SSMCs are within reach of diagrammatical methods.  ...  The left adjoint functor, called propification, turns any SSMC into a monoidally equivalent prop, allowing for manipulation of arrows in this category as string diagrams via bureaucracy isomorphisms.  ... 
arXiv:2205.07760v1 fatcat:ilwthcqpd5fuzdct7yzawsgkdy

DisCoPy: Monoidal Categories in Python [article]

Giovanni de Felice, Alexis Toumi, Bob Coecke
2020 arXiv   pre-print
We introduce DisCoPy, an open source toolbox for computing with monoidal categories. The library provides an intuitive syntax for defining string diagrams and monoidal functors.  ...  Its modularity allows the efficient implementation of computational experiments in the various applications of category theory where diagrams have become a lingua franca.  ...  Two diagrams are equal in the free monoidal category if they are related by a series of interchangers a boundary-connected diagram with n boxes, a normal form can be reached in at most O(n 3 ) steps  ... 
arXiv:2005.02975v2 fatcat:g7ukimlfrjbytgiph7m6wmyln4

Nominal String Diagrams

Samuel Balco, Alexander Kurz, Michael Wagner
2019 Conference on Algebra and Coalgebra in Computer Science  
To this end, we develop the beginnings of a theory of monoidal categories internal in a symmetric monoidal category.  ...  We introduce nominal string diagrams as string diagrams internal in the category of nominal sets.  ...  the same general lines as the well-known proofs for SMTs (see eg Lafont [14] ) and proceed by showing that each diagram f : A → B can be rewritten to one in normal form, with the normal form being a  ... 
doi:10.4230/lipics.calco.2019.18 dblp:conf/calco/Balco019 fatcat:fh32n4ewnnez3kdfrhqrnz5fne

String Diagram Rewrite Theory I: Rewriting with Frobenius Structure [article]

Filippo Bonchi, Fabio Gadducci, Aleks Kissinger, Pawel Sobocinski, Fabio Zanasi
2022 arXiv   pre-print
String diagrams are a powerful and intuitive graphical syntax, originated in the study of symmetric monoidal categories.  ...  In the last few years, they have found application in the modelling of various computational structures, in fields as diverse as Computer Science, Physics, Control Theory, Linguistics, and Biology.  ...  We are thankful to the anonymous referees for their helpful comments; Fabio  ... 
arXiv:2012.01847v2 fatcat:5defrofqqrhm7aiqtzb4pkywfi

Synthesising Graphical Theories [article]

Aleks Kissinger
2012 arXiv   pre-print
In recent years, diagrammatic languages have been shown to be a powerful and expressive tool for reasoning about physical, logical, and semantic processes represented as morphisms in a monoidal category  ...  Naively, we could automate this procedure by enumerating all diagrams up to a given size and check for equalities, but this is intractable in practice because it produces far too many equations.  ...  In [14] , Kissinger showed that string graphs could be used to form the free symmetric traced category over a monoidal signature.  ... 
arXiv:1202.6079v2 fatcat:bemtn6mwvvcgzp4py4zr2ggepy

Graphical Piecewise-Linear Algebra [article]

Guillaume Boisseau, Robin Piedeleu
2021 arXiv   pre-print
In this paper, we introduce to the family its most expressive member to date: Graphical Piecewise-Linear Algebra, a new language to specify piecewise-linear subsets of vector spaces.  ...  Reid Barton in particular contributed significantly to the proof of completeness. The first author is funded by the EPSRC under grant OUCS/GB/1034913.  ...  Acknowledgements The authors would like to thank the various Twitter and Zulip users who contributed to the genesis and development of the theory contained in this paper, notably Jules Hedges, Cole Comfort  ... 
arXiv:2111.03956v1 fatcat:kt4bypqbjndmla2a3ctuceyto4
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