NLab

Last updated

The nLab is a wiki for research-level notes, expositions and collaborative work, including original research, in mathematics, physics, and philosophy, with a focus on methods from category theory and homotopy theory. The nLab espouses the "n-categorical point of view" [1] (a deliberate pun on Wikipedia's "neutral point of view") that category theory and particularly higher category theory provide a useful unifying viewpoint for mathematics, physics and philosophy.

Contents

Overview

The nLab was originally conceived to provide a repository for ideas (and even new research) generated in the comments on posts at the n-Category Café, a group blog run (at the time) by John C. Baez, David Corfield and Urs Schreiber. Eventually the nLab developed into an independent project, which has since grown to include whole research projects and encyclopedic material. [2]

Associated to the nLab is the nForum, an online discussion forum for announcement and discussion of nLab edits (the analog of Wikipedia's "talk" pages) as well as for general discussion of the topics covered in the nLab. The preferred way of contacting the nLab steering committee is to post on the nForum. [3] An experimental sub-project of the nLab is the Publications of the nLab, intended as a journal for refereed research articles that are published online and cross-hyperlinked with the main wiki.

The nLab was set up on November 28, 2008 by Urs Schreiber using the Instiki software provided and maintained by Jacques Distler. Since May 2015 it runs on a server at Carnegie Mellon University that is funded in the context of Steve Awodey's Homotopy Type Theory MURI grant. [4] The system administrator is Adeel Khan Yusufzai. The domain ncatlab.org is owned by Urs Schreiber.

The nLab is listed on MathOverflow as a standard online mathematics reference to check before asking questions. [5] Many questions and answers link to the nLab for background material. [6] It is one of two wikis mentioned by the mathematical physicist John C. Baez in his review of math blogs for the American Mathematical Society. [7]

There is an informal steering committee, which "doesn't run the nLab", [8] but exists in order to resolve issues that would cause the whole project to run into trouble.

See also

Related Research Articles

Category theory Branch of mathematics

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows. A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.

PlanetMath is a free, collaborative, mathematics online encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be comprehensive, the project is currently hosted by the University of Waterloo. The site is owned by a US-based nonprofit corporation, "PlanetMath.org, Ltd".

John C. Baez American mathematician

John Carlos Baez is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, applications of higher categories to physics, and applied category theory.

Bogdanov affair French academic dispute

The Bogdanov affair was an academic dispute regarding the legitimacy of a series of theoretical physics papers written by French twins Igor and Grichka Bogdanov. These papers were published in reputable scientific journals, and were alleged by their authors to culminate in a proposed theory for describing what occurred at and before the Big Bang.

David Neil Corfield is a British philosopher specializing in philosophy of mathematics and philosophy of psychology. He is Senior Lecturer in Philosophy at the University of Kent.

In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology, where one studies algebraic invariants of spaces, such as their fundamental weak ∞-groupoid.

Jack Morava American mathematician

Jack Johnson Morava is an American homotopy theorist at Johns Hopkins University.

This is a timeline of category theory and related mathematics. Its scope is taken as:

MathOverflow is a mathematics question-and-answer (Q&A) website, which serves as an online community of mathematicians. It allows users to ask questions, submit answers, and rate both, all while getting merit points for their activities. It is a part of the Stack Exchange Network.

Ronald Brown (mathematician)

Ronald Brown is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor University, he has authored many books and more than 160 journal articles.

Stack Exchange is a network of question-and-answer (Q&A) websites on topics in diverse fields, each site covering a specific topic, where questions, answers, and users are subject to a reputation award process. The reputation system allows the sites to be self-moderating. As of August 2019, the three most actively-viewed sites in the network are Stack Overflow, Super User, and Ask Ubuntu.

In mathematics, more specifically category theory, a quasi-category is a generalization of the notion of a category. The study of such generalizations is known as higher category theory.

In topology, a branch of mathematics, a string group is an infinite-dimensional group introduced by Stolz (1996) as a -connected cover of a spin group. A string manifold is a manifold with a lifting of its frame bundle to a string group bundle. This means that in addition to being able to define holonomy along paths, one can also define holonomies for surfaces going between strings. There is a short exact sequence of topological groups

Homotopy type theory Type theory in logic and mathematics

In mathematical logic and computer science, homotopy type theory refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies.

Urs Schreiber is a mathematician specializing in the connection between mathematics and theoretical physics and currently working as a researcher at the Czech Academy of Sciences, Institute of Mathematics, Department for Algebra, Geometry and Mathematical Physics.

In mathematics, the cobordism hypothesis, due to John C. Baez and James Dolan, concerns the classification of extended topological quantum field theories (TQFTs). In 2008, Jacob Lurie proposed a broadly-accepted solution.

Univalent foundations are an approach to the foundations of mathematics in which mathematical structures are built out of objects called types. Types in univalent foundations do not correspond exactly to anything in set-theoretic foundations, but they may be thought of as spaces, with equal types corresponding to homotopy equivalent spaces and with equal elements of a type corresponding to points of a space connected by a path. Univalent foundations are inspired both by the old Platonic ideas of Hermann Grassmann and Georg Cantor and by "categorical" mathematics in the style of Alexander Grothendieck. Univalent foundations depart from the use of classical predicate logic as the underlying formal deduction system, replacing it, at the moment, with a version of Martin-Löf type theory. The development of univalent foundations is closely related to the development of homotopy type theory.

PhysicsOverflow is a physics website that serves as a post-publication open peer review platform for research papers in physics, as well as a collaborative blog and online community of physicists. It allows users to ask, answer and comment on graduate-level physics questions, post and review manuscripts from ArXiv and other sources, and vote on both forms of content.

Emily Riehl American mathematician

Emily Riehl is an American mathematician who has contributed to higher category theory and homotopy theory. Much of her work, including her PhD thesis, concerns model structures and more recently the foundations of infinity-categories. She is the author of two textbooks and serves on the editorial boards of three journals.

In mathematical physics higher gauge theory is the general study of counterparts of gauge theory that involve higher-degree differential forms instead of the traditional connection forms of gauge theories.

References

  1. nPOV in nLab
  2. Urs Schreiber, What is... the nLab?
  3. Steering committee in nLab meta
  4. Awodey, Steve (29 April 2014). "HoTT awarded a MURI". Homotopy Type Theory. Retrieved 8 August 2020.
  5. MathOverflow, 1.0 'How to ask' page. Archived on 2013-06-04.
  6. MathOverflow, Results for a search for 'nlab'. As of 2018-12-11 there are over 800 results.
  7. John C. Baez, "Math Blogs", Notices of the American Mathematical Society , March 2010
  8. Steering committee in nLab meta