https://arxiv.org/abs/1401.7220 Skip to main content Cornell University We gratefully acknowledge support from the Simons Foundation, member institutions, and all contributors. Donate arxiv logo > math > arXiv:1401.7220 [ ] Help | Advanced Search [All fields ] Search arXiv logo Cornell University Logo [ ] GO quick links * Login * Help Pages * About Mathematics > Category Theory arXiv:1401.7220 (math) [Submitted on 28 Jan 2014 (v1), last revised 9 Nov 2014 (this version, v2)] Title:Category Theory Using String Diagrams Authors:Daniel Marsden View a PDF of the paper titled Category Theory Using String Diagrams, by Daniel Marsden View PDF Abstract:In work of Fokkinga and Meertens a calculational approach to category theory is developed. The scheme has many merits, but sacrifices useful type information in the move to an equational style of reasoning. By contrast, traditional proofs by diagram pasting retain the vital type information, but poorly express the reasoning and development of categorical proofs. In order to combine the strengths of these two perspectives, we propose the use of string diagrams, common folklore in the category theory community, allowing us to retain the type information whilst pursuing a calculational form of proof. These graphical representations provide a topological perspective on categorical proofs, and silently handle functoriality and naturality conditions that require awkward bookkeeping in more traditional notation. Our approach is to proceed primarily by example, systematically applying graphical techniques to many aspects of category theory. We develop string diagrammatic formulations of many common notions, including adjunctions, monads, Kan extensions, limits and colimits. We describe representable functors graphically, and exploit these as a uniform source of graphical calculation rules for many category theoretic concepts. These graphical tools are then used to explicitly prove many standard results in our proposed diagrammatic style. Subjects: Category Theory (math.CT); Logic in Computer Science (cs.LO) Cite as: arXiv:1401.7220 [math.CT] (or arXiv:1401.7220v2 [math.CT] for this version) https://doi.org/10.48550/arXiv.1401.7220 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Daniel Marsden [view email] [v1] Tue, 28 Jan 2014 15:30:16 UTC (48 KB) [v2] Sun, 9 Nov 2014 21:13:55 UTC (51 KB) Full-text links: Access Paper: View a PDF of the paper titled Category Theory Using String Diagrams, by Daniel Marsden * View PDF * TeX Source * Other Formats view license Current browse context: math.CT < prev | next > new | recent | 2014-01 Change to browse by: cs cs.LO math References & Citations * NASA ADS * Google Scholar * Semantic Scholar 2 blog links (what is this?) a export BibTeX citation Loading... BibTeX formatted citation x [loading... ] Data provided by: Bookmark BibSonomy logo Reddit logo (*) Bibliographic Tools Bibliographic and Citation Tools [ ] Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) [ ] Litmaps Toggle Litmaps (What is Litmaps?) [ ] scite.ai Toggle scite Smart Citations (What are Smart Citations?) ( ) Code, Data, Media Code, Data and Media Associated with this Article [ ] Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) [ ] DagsHub Toggle DagsHub (What is DagsHub?) [ ] GotitPub Toggle Gotit.pub (What is GotitPub?) [ ] Links to Code Toggle Papers with Code (What is Papers with Code?) [ ] ScienceCast Toggle ScienceCast (What is ScienceCast?) ( ) Demos Demos [ ] Replicate Toggle Replicate (What is Replicate?) [ ] Spaces Toggle Hugging Face Spaces (What is Spaces?) [ ] Spaces Toggle TXYZ.AI (What is TXYZ.AI?) ( ) Related Papers Recommenders and Search Tools [ ] Link to Influence Flower Influence Flower (What are Influence Flowers?) [ ] Connected Papers Toggle Connected Papers (What is Connected Papers?) [ ] Core recommender toggle CORE Recommender (What is CORE?) * Author * Venue * Institution * Topic ( ) About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?) * About * Help * Click here to contact arXiv Contact * Click here to subscribe Subscribe * Copyright * Privacy Policy * Web Accessibility Assistance * arXiv Operational Status Get status notifications via email or slack