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


# Contents (or put a title here)
* this block creates the table of contents, leave as is
{: toc}

## Idea

(...)

## Definition

(...)

## Properties

(...)

## Examples

(...)

## References

(...)



A more detailed example follows. Check out the source code here to see how it’s coded:

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 (Def. ) 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 (Def. ) 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 following diagram commutes:

$\array{ \mathbb{N}\times \mathbb{N} \times \mathbb{N} & \overset{Id \times + }{\longrightarrow} & \mathbb{N} \times \mathbb{N} \\ {}^{\mathllap{+ \times Id}} \big\downarrow && \big\downarrow^{\mathrlap{+}} \\ \mathbb{N} \times \mathbb{N} & \underset{+}{\longrightarrow} & \mathbb{N} } \,.$

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.

Example

The first example is obvious.

Example

The second example is a slight variation of Exp. .

Example

The third example is completely different from both Exp. .

References

The original definition appeared in section 3 of