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Free gs-monoidal categories and free Markov categories
[article]
2022
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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]
2014
arXiv
pre-print
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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
2014
Electronic Proceedings in Theoretical Computer Science
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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]
2015
arXiv
pre-print
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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]
2019
arXiv
pre-print
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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]
2015
arXiv
pre-print
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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
2018
International Conference on Rewriting Techniques and Applications
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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]
2012
arXiv
pre-print
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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]
2022
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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
2019
Conference on Algebra and Coalgebra in Computer Science
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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]
2022
arXiv
pre-print
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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]
2012
arXiv
pre-print
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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]
2021
arXiv
pre-print
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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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