nLab
Template page

This page is meant to provide a general example and template for new nLab-pages. You can look at its source code to see how the various parts are done. A minimal template is given first which can be copy-and-pasted into newly created pages. See HowTo for more details.

category: meta

Minimal Template Code

Remove any sections that you don't use.


# Contents (or put a title here)
* the following line creates the automatic table of contents
{: toc}

## Idea ##

...


## Abstract ##

...


## Definition ##

...


## Properties ##

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## Examples ##

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## References ##

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[[!redirects <put page name here>]]
[[!redirects <put page name here>s]]

A longer example follows.


Contents

Idea

It is an old observation that xyz. One notices that from the nPOV this is just an abc. This leads to the definition of a uvw. It is useful for doing klm and provides the basis for the more general theory of äöü.

Abstract

A uvw is effectively a uv together with a w. Its main property is encoded in Somebody’s Theorem which says that it consists of precisely three letters. The archetypical example of a uvw is μνω\mu \nu \omega; details will be explained in the special examples paragraph.

Definition

As Jacques Distler said,

See more about definition/theorem/proof-environments.

Definition

(uvw)

A uvw is a UVW in which all letters are lower case.

This may be summed up in the slogan:

A uvw is just what it looks like.

Properties

Lemma

Every uvw contains at least one letter.

Proof

By inspection.

Proposition

Every uvw contains strictly more than one letter.

Proof

Use the above lemma and continue counting:

(1)1+1=2. 1 + 1 = 2 \,.
Theorem

Every uvw contains exactly three letters.

Proof

Along the lines of the above proposition, we use equation (1) and then conclude with

2+1=3. 2 + 1 = 3 \,.

Notice that this is indeed independent of in which order we sum up the letters, in that the diagram

×× Id×+ × +×Id + × + . \array{ \mathbb{N}\times \mathbb{N} \times \mathbb{N} &\stackrel{Id \times + }{\to}& \mathbb{N} \times \mathbb{N} \\ {}^{\mathllap{+ \times Id}}\downarrow && \downarrow^{\mathrlap{+}} \\ \mathbb{N} \times \mathbb{N} &\underset{+}{\to}& \mathbb{N} } \,.

commutes.

Corollary

No uvw contains more than three letters.

Examples

Special cases

  • First case

  • Second case

  • Third case

First person: I listed all of the special cases that I know above, but didn't Grothendieck study an important version too?

Second person: No, you're thinking of Lawvere. When I find the reference, I'll put it here.

Specific examples

For ease of reference, we will number the examples.

  1. The first example is obvious.

  2. The second example is a slight variation of (1).

  3. The third example is completely different from either (1) or (2).

References

The original definition appeared in section 3 of

  • FirstName LastName, Title, Journal (year) (arXiv, pdf).

Revised on September 15, 2012 19:50:25 by Urs Schreiber (82.113.99.10)