# 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)
{: toc}

## Idea ##

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## Abstract ##

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## Definition ##

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## Properties ##

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## Examples ##

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## References ##

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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 $\mu \nu \omega$; details will be explained in the special examples paragraph.

## Definition

As Jacques Distler said,

###### 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.

By inspection.

###### Proposition

Every uvw contains strictly more than one letter.

###### Proof

Use the above lemma and continue counting:

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

Every uvw contains exactly three letters.

###### Proof

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

$2 + 1 = 3 \,.$

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

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

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).

The original definition appeared in section 3 of

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