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This is a wiki for collaborative work on Mathematics, Physics, and Philosophy — especially (but far from exclusively) from the higher structures point of view: with a sympathy towards the tools and perspectives of homotopy theory/algebraic topology, homotopy type theory, higher category theory and higher categorical algebra.

Purpose

The nLab records and explores a wide range of mathematics, physics, and philosophy. Where mathematics, physics, and philosophy arise in other fields, computer science and linguistics for example, the nLab explores these too.

If you take a little time to find something in the literature, or to work through a proof or example, or to find an intuitive or new way to think about something, add a note on it to the nLab! Others will benefit, and you may well find that it proves useful to you too.

In doing so, follow common academic practice of substantiating your edit: Either justify it by citing relevant literature or else provide relevant proof (for claims) or other sanity checks (e.g. when introducing new definitions or terminology).

A fundamental idea behind the nLab is that the linking between pages of a wiki is a profound way to record knowledge, placing it in a rich context.

For more, see also Urs Schreiber’s thoughts at What is… the nLab?.

nForum, discussion, comments, and questions

Most nLab pages have a corresponding discussion thread at the nForum, linked to in the menus at the top and bottom of the page. For all but the most trivial edits (correcting spelling or punctuation, etc), we ask that you announce your changes at the nForum using the box at the bottom of the edit page. Typically this would be a short note summarising what you have done, but it can also include what you plan to do, or what you would like others to do! See Welcome to the nForum for more on the nForum.

If you have any questions about an nLab page, or would like to discuss or make a comment upon something in it, whether because you feel that something is wrong or missing or for any other reason, you are encouraged and very welcome to use the nForum discussion thread for the page for this purpose too. If the discussion thread does not yet exist, feel free to create it manually: use the ‘nLab - Latest changes’ category, and give it the same title (case-sensitive!) as the nLab page.

The pages of the nLab have almost always been edited by several people. In almost all cases, we recommend that you use the nForum rather than contact an author of an edit directly, but if you do have a reason to do this, the history of edits to the page with the details of who made them can be found by clicking on ‘History’ at the bottom of the page.

Contributing to the $n$Lab

If after looking around for a while you feel like contributing yourself, you are very welcome to do so! See below for more on how to actually edit, as well as the HowTo and writing in the nLab.

If you make any edits to the nLab, please follow the guidelines for announcing your changes at the nForum given above. If you are unsure whether what you would like to contribute is appropriate, please ask at the nForum before making the edit.

Viewing and editing the $n$Lab

The nLab should be viewable and editable in any modern web browser on any device. For technical reasons, modern browsers that are able to render MathML, such as Firefox 2+, Chrome/Chromium/Edge 109+, Safari 10+ may render very large pages quicker than older browsers, but these pages are few and far between, and most people will be able to use the browser of their choice.

The following custom themes are available for the nLab:

• Dark theme by BinaryQuantumSoul: https://userstyles.world/style/19674/nlab-full-redesign

• Dark theme by Paolo Perrone: https://userstyles.world/style/15519/nlab-dark

Editing the nLab these days can be done more or less as in LaTeX. Macros are not currently available, but mathematics can be entered within dollar signs as usual, theorem environments, sections and tables of contents can be defined as in LaTeX, commutative diagrams and figures can be created in Tikz or XyMatrix, and so on. For simple formatting, such as italics, and for some other things such as tables, Markdown is used.

See the HowTo and writing on the nLab for more on editing the nLab.

Making use of material from the $n$Lab

One goal of the $n$Lab is to help make information widely available and usefully related to other information. In this users and contributors are expected to follow traditional academic practice:

• Using and distributing content obtained from the $n$Lab is free and encouraged if you acknowledge the source, as usual in academia.

  (There is currently no consensus on a more formal license statement, but if it matters check if relevant individual contributors state such on their $n$Lab homepages.)

  If you cite a page you may want to point to a specific version of it, because $n$Lab pages can change. You can find a list of all the versions of a page by clicking on the History link at the bottom of the page itself.

• Conversely, any content contributed to the $n$Lab is publicly available and you should be aware that others may use your contributions (whatever you decide to do with their content elsewhere) and indeed may edit them. In the first case you trust that users will cite your contributions properly, in the second that they will respect and only improve on them. At the same time, you are expected to properly acknowledge sources of information for material entered into the $n$Lab.

Usually this works well. If there is need for discussion, the nForum is the forum to turn to. If serious problems arise, the steering committee might intervene.

Installation

The domain ncatlab.org is owned by Urs Schreiber.

The $n$Lab server is currently hosted at Carnegie Mellon University, kindly provided by Steve Awodey.

The nLab’s software and technical administrative matters are in the hands of the nLab’s technical board.

Much of the software behind the nLab has been written especially for the nLab: the latest source can be found at GitHub. It was originally an instance of Instiki.

The $n$Lab page style is due to Jake Bian, originating with his Kan browser extension

The $n$Lab logo is due to David Roberts, inspired by Matisse’s painting La Gerbe. Besides being an inside reference to higher structures known as gerbes, the logo represents maybe computational trinitarianism or the progression of modalities or generally the unity of diverse mathematical phenomena revealed by the nPOV.

Contact

The default venue for all communication regarding the nLab is the nForum. When posting there you get to choose a “category”-label for your message:

• Latest edit logs are to be posted under Latest changes.

• Organizational matters (such as concerning user accounts) are best posted under nLab Organisation.

• Technical bug reports or software feature requests are best raised in nLab Technical Matters

  or directly on github.com/ncatlab/nlab/issues.

Please do not try to contact the technical team on matters that are not purely technical: Policy decisions are made by the active nLab community or, if all fails, by the Steering Committee.

Steering Committee

The $n$Lab is a community undertaking. But for all matters that do require that the $n$Lab is represented to the outside by an official decision-taking body, we have the steering committee. Nobody “is in charge of the $n$Lab”. But the steering committee is the closest approximation to a body being in charge that we have.

category: meta