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A correctness proof establishes a strictification-based construction, yielding a novel graphical proof of Mac Lane’s strictness theorem and an elementary proof of the coherence theorem via canonical isomorphisms.",{"@graph":14,"@context":71},[15,34,54],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & 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diagrams?","Question",{"text":61,"@type":62},"It addresses how to expand the graphical language of string diagrams so it can express non-strict tensors in monoidal categories, while keeping the notation manageable and intuitive.","Answer",{"name":64,"@type":59,"acceptedAnswer":65},"How does the proposed construction handle non-strict tensors?",{"text":66,"@type":62},"It introduces operations that pack pairs of wires into structured single wires and unpack structured wires into pairs labeled by tensor component objects, plus additional operations for hiding or revealing wires labeled with the tensor unit.",{"name":68,"@type":59,"acceptedAnswer":69},"What are the main theoretical results proved in the paper?",{"text":70,"@type":62},"The paper proves correctness by showing monoidal equivalence between the strictified category used by the diagrams and the original non-strict category, leading to novel graphical proofs of Mac Lane’s strictness theorem and an elementary graphical proof of the coherence theorem.","https://schema.org",{"og:url":32,"og:type":73,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":75,"canonical":32},"index,follow",{"doc_id":77,"site_id":7},665542,1793304830,{"code":4,"msg":80,"data":81},"success",[82,86,90,94,99,104,109,113,118,121,125],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":83,"show_sort_weight":84,"slug":85},"Story & 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\nUniversity of Birmingham, UK Fabio Zanasi \\# Ñ University College London, UK  \n~~ Abstract ~~  \nWhereas string diagrams for strict monoidal categories are well understood, and have found application in several fields of Computer Science, graphical formalisms for non-strict monoidal categories are far less studied. In this paper, we provide a presentation by generators and relations of string diagrams for non-strict monoidal categories, and show how this construction can handle applications in domains such as digital circuits and programming languages. We prove the correctness of our construction, which yields a novel proof of Mac Lane’s strictness theorem. This in turn leads to an elementary graphical proof of Mac Lane’s coherence theorem, and in particular allows for the inductive construction of the canonical isomorphisms in a monoidal category.  \n2012 ACM Subject Classification Theory of computation Keywords and phrases String Diagrams, Strictness, Coherence Digital Object Identifier 10.4230/LIPIcs.CSL.2023.37  \nRelated Version Extended Version: [https://arxiv.org/abs/2201.11738](https://arxiv.org/abs/2201.11738)  \n 1  Introduction  \nString diagrams are a rigorous graphical notation for category theory that is proving useful in a broad variety of application domains, such as quantum systems [9], computational linguistics [8], digital circuits [12], or signal flow analysis [5] . What the majority of string diagrammatic notations have in common is that they are devised for monoidal categories in which the tensor is strict, i.e. the associator and unitor morphisms are identities. As Joyaland Street explain in their seminal Geometry of Tensor Calculus [16], the choice of using a strict monoidal category was motivated by convenience (“simplicity of exposition”) and by a wish to focus on “aspects other than the associativity of tensor product”. Furthermore, they believed that “most results obtained with the hypothesis that a tensor category is strict can be reformulated and proved without this condition.”  \nIndeed, in terms of mathematical power, this statement is true. However, string diagrams have been used increasingly as a convenient syntax for languages with models in (strict) monoidal categories. And, when used as syntax, the distinction between strict and nonstrict tensor becomes relevant, if not in terms of mathematical expressiveness then at least as a mechanism of abstraction. This is why modern programming languages, and even some modern hardware design languages such SystemVerilog [22], use non-strict features such as tuples and structs which can nest in non-trivial ways. These non-strict structures could be manually “strictified” by the programmer by flattening them into arrays. Using such programmer conventions instead of native syntactic support does not entail a loss of expressiveness, but a loss of code readability, convenience, and general programmer effectiveness.  \n© Paul Wilson, Dan Ghica, and Fabio Zanasi;  \nlicensed under Creative Commons License CC-BY 4.0  \n31st EACSL Annual Conference on Computer Science Logic (CSL 2023) .  \nEditors: Bartek Klin and Elaine Pimentel; Article No. 37; pp. 37:1–37:19  \nLeibniz International Proceedings in Informatics  \n Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany  \n37:2 String Diagrams for Non-Strict Monoidal Categories  \nIn this paper, we address the problem of expanding the graphical language of string diagrams with the required features that allow the expression of non-strict tensors. What makes the language of strict tensors convenient for the graphical representation is that objects are naturally represented as lists of wires. This suggests that string diagrams make use of strictness in an essential way and, indeed, naive attempts to define string diagram languages for non-strict monoidal categories can render the notation","cbCaik0UOBWd76HJ","https://ap.wps.com/l/cbCaik0UOBWd76HJ","pdf",861611,"English","# Introduction\n## Contributions","[{\"question\":\"What problem does the paper address regarding string diagrams?\",\"answer\":\"It addresses how to expand the graphical language of string diagrams so it can express non-strict tensors in monoidal categories, while keeping the notation manageable and intuitive.\"},{\"question\":\"How does the proposed construction handle non-strict tensors?\",\"answer\":\"It introduces operations that pack pairs of wires into structured single wires and unpack structured wires into pairs labeled by tensor component objects, plus additional operations for hiding or revealing wires labeled with the tensor unit.\"},{\"question\":\"What are the main theoretical results proved in the paper?\",\"answer\":\"The paper proves correctness by showing monoidal equivalence between the strictified category used by the diagrams and the original non-strict category, leading to novel graphical proofs of Mac Lane’s strictness theorem and an elementary graphical proof of the coherence theorem.\"}]","String Diagrams for Non-Strict Monoidal Categories | PDF",48]