A Survey of Graphical Languages for Monoidal Categories

By Peter Selinger. 2009

839 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/32b3c54e3b6d6d4661f849362b845f8e3923002c

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Citations in the corpus

Shadows and traces in bicategories (2009, methodology): “These diagrams were first used by Penrose [25,26], given a rigorous foundation by Joyal and Street [15,17], and since then have been adapted to many different contexts; a comprehensive overview can be found in [34].”
Mathematical Foundations for a Compositional Distributional Model of Meaning (2010, background): “Also the survey of graphical languages for monoidal categories [38] could be useful – note that Selinger refers to ‘non-commutative’ compact closed categories as (both left and right) planar autonomous categories.”
Buildings, spiders, and geometric Satake (2011, methodology): “The definitions used in this section are nicely summarized in a survey by Selinger [Sel11]; they are originally due to Freyd and Yetter [FY89] and Joyal and Street [JS93].”
The simplicial model of Univalent Foundations (after Voevodsky) (2012, neutral): “The situation is rather comparable to that of string diagrams, as used in monoidal and more elaborately structured categories [Sel11], or indeed of the traditional notations for diﬀerentiation and integration.”
Traces in symmetric monoidal categories (2011, background): “See [19, 21, 32, 37] for more about string diagram calculus.”
Categories for the practising physicist (2009, background): “Structures for Physics, which, among many other things, contain complementary tutorials on category theory and its graphical calculus [4, 12, 63 ].”
Quantum picturalism (2009, background): “But in fact, FHilb is very special as a dagger compact category, as the following beautiful result due to Selinger demonstrates.”
Category-theoretic quantitative compositional distributional models of natural language semantics (2013, methodology): “Fortunately, a graphical calculus, surveyed in [73], has been developed to provide both visual and practical support for these tasks.”