nLab

nLab
Name	nLab
Type	Wiki
Language	English
Owner	Independent community
Launch date	2008

nLab nLab is an online collaborative wiki and knowledge base focusing on advanced topics in mathematics, physics, and related areas of philosophy and computer science. It serves as a repository of research-level expositions, definitions, conjectures, and references, functioning as a hub for practitioners working on category theory, homotopy theory, topology, algebraic geometry, and adjacent subjects. Contributors include researchers associated with institutions such as University of Cambridge, Massachusetts Institute of Technology, University of Oxford, IHÉS, CNRS, and Max Planck Institute for Mathematics.

Overview

nLab collects material on specialized topics like higher category theory, homotopy type theory, derived algebraic geometry, topological quantum field theory, and moduli space constructions. Articles interlink notions such as model category, ∞-category, sheaf, stack (mathematics), spectra (stable homotopy theory), and operad, often referencing results from figures like Grothendieck, Alexander Grothendieck, Quillen, Daniel Quillen, Vladimir Voevodsky, Jacob Lurie, John Baez, James Dolan, and Maxim Kontsevich. The site complements formal projects like Lean (proof assistant), Coq, Agda, and interfaces with formalization efforts at Univalent Foundations and Homotopy Type Theory.

History and Development

nLab emerged in the late 2000s from discussions among researchers frequenting forums such as Mathematics Stack Exchange, MathOverflow, and mailing lists connected to seminars at Perimeter Institute and Institute for Advanced Study. Early contributors drew on traditions from the nForum community and seminars inspired by works of Grothendieck, Pierre Deligne, Alexander Grothendieck, and the categorical program of Saunders Mac Lane. Development milestones include adoption of MediaWiki extensions, integration with GitHub-style workflows, and linkages to repositories at arXiv and seminar pages from Oxford University and Harvard University.

Content and Scope

Content spans expository articles, technical notes, examples, and research-level sketches across topics such as spectral sequence, Chern–Simons theory, mirror symmetry, Calabi–Yau manifold, Gromov–Witten invariant, K-theory, noncommutative geometry, von Neumann algebra, operator algebra, conformal field theory, string theory, brane, gauge theory, and loop quantum gravity. It documents constructions like groupoid, Lie groupoid, differential graded algebra, A∞-algebra, Batalin–Vilkovisky formalism, Koszul duality, Tannaka duality, Langlands program, motivic cohomology, and étale cohomology. The site cross-references works by authors such as Pierre Deligne, Jean-Pierre Serre, Alexander Grothendieck, André Weil, Michael Atiyah, Isadore Singer, Edward Witten, Nathan Seiberg, Cumrun Vafa, Maxwell Jacob, and Freeman Dyson.

Community and Governance

The contributing community comprises mathematicians and physicists affiliated with universities and institutes including Cambridge University, Princeton University, Yale University, ETH Zurich, Universität Bonn, University of California, Berkeley, Stanford University, Columbia University, and Imperial College London. Governance relies on communal norms, editorial practices influenced by peer review cultures at journals like Annals of Mathematics, Journal of the American Mathematical Society, Communications in Mathematical Physics, and Inventiones Mathematicae, and moderation inspired by models from Wikimedia Foundation projects. Contributors have organized around workshops and conferences at venues such as International Congress of Mathematicians, Strings Conference, APS March Meeting, and seminars at MPI for Mathematics.

Technical Infrastructure

nLab runs on a MediaWiki platform with customized templates, category systems, and linking conventions. It interfaces with external resources like arXiv, MathSciNet, Zentralblatt MATH, and bibliographic tools used by researchers at Institute Henri Poincaré and SISSA. Technical discussions include implementation of semantic features analogous to those in Semantic MediaWiki, citation handling inspired by BibTeX workflows, and archival strategies informed by digital preservation initiatives at libraries like Bodleian Library and Library of Congress. Tooling for diagrams often references packages used by contributors from LaTeX Project and diagrammatic languages employed in seminars at Perimeter Institute.

Reception and Influence

The project has been cited and consulted by researchers working on projects linked to Higher Topos Theory, Derived Categories, Derived Algebraic Geometry, Homotopy Type Theory book, and formalization initiatives at Microsoft Research and Google Research. It has influenced community knowledge-sharing practices alongside platforms like MathOverflow, arXiv, and collaborative efforts at n-Category Café and various thematic networks centered around figures such as Curtis T. McMullen, William Thurston, Michael Hopkins, and Jacob Lurie. Educational uses have appeared in graduate reading groups at Princeton, Cambridge, Harvard, and UCLA.

Category:Online encyclopedias