mathematics

NLab

NLab is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand NLab rather than just read about it. In short: The nLab is a wiki for research-level notes, expositions and collaborative work. It includes original research in mathematics, physics, and philosophy with a focus on methods from type theory, category theory, and homotopy theory.

Key takeaways

NLab belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect NLab to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of NLab from memory before moving on to harder problems.

Reference excerpt

The nLab is a wiki for research-level notes, expositions and collaborative work. It includes original research in mathematics, physics, and philosophy with a focus on methods from type theory, category theory, and homotopy theory. The nLab espouses the "n-point of view" (a deliberate pun on Wikipedia's "neutral point of view"), which is that type theory, homotopy theory, category theory, and higher category theory provide a useful, unifying viewpoint for mathematics, physics and philosophy. The n in n-point of view could refer to either n-categories as found in higher category theory, n-groupoids as found in both homotopy theory and higher category theory, or n-types as found in homotopy type theory.

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. Associated with 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. 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. This sub-project appears to be inactive as of 2014. 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. 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. Many questions and answers link to the nLab for background material. 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. There is an informal steering committee, which doesn't run the nLab, but exists in order to resolve issues that could cause the whole project to run into trouble. The content of this wiki is not placed under a specific copyright license.

Worked examples

Example 1 — a first encounter with NLab

Start with the simplest possible case. Write down what NLab claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to NLab before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about NLab ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of NLab

In research: NLab appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses NLab in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: NLab is common in secondary-school and first-year university syllabi. It links to neighbouring topics Higher category theory, Homotopy theory, Mathematics websites, so understanding it makes those chapters shorter.
In everyday life: Look for NLab outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study NLab in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what NLab means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain NLab out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is NLab in simple terms?

Why does NLab matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study NLab?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on NLab.