HowTo

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Contents {: .clickToReveal}
### Contents {: .clickToHide tabindex="0"}
+-- {: .hide}
[[!include contents]]
=--
=--
=--

This page gives hints for how to edit [[nLab:HomePage|nLab]]-pages.

If you feel you can most easily start by modifying an  example, look at the *[[template page]]* and do experiments in the *[[Sandbox]]*.

The following provides more details. For technical help with   special features of Instiki (including peculiarities with itex), HTML and XML, and assorted specifics of the nLab, see also the [[FAQ]].

# Contents

* table of contents
{: toc}

## Getting Started

### How to edit the nLab

Hit "edit page" to see how pages are coded. Use the [[Sandbox|Sandbox]] to warm up.  The key point is that links to other pages are placed in <nowiki>[[double brackets]]</nowiki>.

There is no feature to preview your edits.  Instead, submit them and then edit again.  Two successive submissions with the same signature and made within 30 minutes of each other count as one.

### How to start a new page
{#newpage}

To create a new page on the $n$Lab first make an existing page request a link to it, and then click on that requested link.

In detail:

1. **Create a link request.**

   1. Identify any existing $n$Lab page which should eventually refer to your new page.

      (At least one such page ought to exist, if your new page is related to anything, but if you just cannot think of any, use the _[[Sandbox]]_ page).

   1. Hit "edit" on that existing page.

   1. In the resulting edit pane, add the string

      <nowiki>[[your new page name]]</nowiki>

      at some sensible point in the source code.

      (By default: in the "Related entries"-section.)

   1. Hit "submit" below the edit pane.

   1. See the existing page render again, now with your edit included. If <nowiki>your new page name</nowiki> did not exist yet as a page, you will see that string appear, in the existing entry, in gray with a clickable question mark behind it.

2. **Satisfy the link request.**

   1. Click on that question mark to open the edit pane for the new page.

   1. Edit.

   1. Once you hit "Submit" below that edit pane, the new page will appear.

   There is no preview mechanism: Just keep cycling through "Submit" and "Edit" until you are satisfied.

_Watch out_: the name of a page is case sensitive, so make your link lowercase if it comes at the beginning of a sentence. (<nowiki>[[like this|Like this]]</nowiki>.) We loosely agreed to try to follow that and some other naming conventions; see [below](#naming).

Use [[redirects]] to allow for other casing to be used when linking to the page elsewhere in the nLab.

### Signing an edit

When you edit a page, you should put your name (with normal capitalisation and spacing) in the box after 'Submit as'. If you don't, then your contribution will be credited to 'Anonymous' (formerly the [[AnonymousCoward]]).

Once you edit a page for the first time, your name will appear at the bottom, grayed out with a question mark since there is no page with your name yet.  You may take this as an invitation to create a user page and tell us about yourself.  But if you don't want to or don't have the time right now, you can forget about this. If you just want to show up on [category: people](http://www.ncatlab.org/nlab/list/people), then you make a page containing only 'category: people' (or someone else may do this for you).

To create your user page, simply click the question mark that appears next to your name at the bottom of the page after making a modification and add content to the edit box that appears. If you'd like to make a user page prior to modifying an existing page, you can do so by making some trivial modification to the [[Sandbox]], which will put your name at the bottom of the page where you can click the question mark. (Or hack the URL.)

### Naming conventions
{#naming}

The following naming conventions for new entries are not set in stone, but we\'re following them for now. Changing page titles results in unnecessary work for the [[lab elves]], so you should try to follow these if possible (or dispute them if not).  It is most important to follow these in links to pages that don\'t yet exist, so that the pages will be created at the correct title (and only once).

* Page titles should contain only ASCII characters.
  * Examples: `<nowiki>[[omega-category]]</nowiki>` instead of `<nowiki>[[$\omega$-category]]</nowiki>` or `<nowiki>[[∞-category]]</nowiki>`.
  * {#tricks} Tricks: To produce '$\omega$-[[omega-category|category]]', try `<nowiki>$\omega$-[[omega-category|category]]</nowiki>`. To produce '[[omega-category|∞-category]]', try `<nowiki>[[omega-category|∞-category]]</nowiki>`.  (The former has proper math-mode formatting, while the latter has proper link scope.)  Unfortunately, `<nowiki>[[omega-category|$\omega$-category]]</nowiki>` does not work.  You can also use [[redirects]] to make `<nowiki>[[∞-category]]</nowiki>` produce '[[∞-category]]', but only when the page already exists.
  * Reason: It's possible to put non-ASCII characters in a name, but not ones generated by iTeX or &...; character entities, so this can be difficult, depending on your browser. To keep things easy, therefore, use only ASCII characters in links. This also helps with creating automatic redirects (which are found by removing accents on letters).
  * Exception: We tend to use common European letters (such as 'ö', more or less the characters in Latin 1) in names, especially of people (such as [[Kurt Gödel]]), even though we're not really supposed to.

* Page titles should be singular nouns.

  * Examples: Use `<nowiki>[[category]]</nowiki>` instead of `<nowiki>[[categories]]</nowiki>`, `<nowiki>[[faithful functor]]</nowiki>` instead of `<nowiki>[[faithful]]</nowiki>`, and `<nowiki>[[categorification]]</nowiki>` instead of `<nowiki>[[categorify]]</nowiki>` or `<nowiki>[[categorified]]</nowiki>`.

  * Tricks: To produce '[[category|categories]]', try `<nowiki>[[category|categories]]</nowiki>`. To produce '[[faithful functor|faithful]] [[endofunctor]]', try `<nowiki>[[faithful functor|faithful]] [[endofunctor]] </nowiki>`.  Again, you can use redirects to make `<nowiki>[[categories]]</nowiki>` produce '[[categories]]', if the page already exists.

  * Reason: We want only one page on a given concept, regardless of variations in grammar.

  * Exception: Recently we have started to have page titles that are complete sentences (without capitalization or punctuation) that state a simple theorem (typically a universal affirmation in the Aristotelian sense) such as [[compact Hausdorff rings are profinite]] or [[derivations of smooth functions are vector fields]].

* {#PageTitlesShouldBeUncapitalized} Page titles should be uncapitalized, except for words that are always capitalized.

  * Examples: Use `<nowiki>[[homotopy theory]]</nowiki>` instead of `<nowiki>[[Homotopy Theory]]</nowiki>`, but use `<nowiki>[[Lie algebra]]</nowiki>`.

  * Tricks: To produce '[[homotopy theory|Homotopy theory]] is important!', try `<nowiki>[[homotopy theory|Homotopy theory]] is important!</nowiki>`.  If you do this a lot, then you can again create a redirect, if the page exists.

  * Reason: We want only one page on a given concept, regardless of variations in grammar.

* Except as contradicted above, use standard American English spelling conventions.

  * Examples: Use `<nowiki>[[center]]</nowiki>` instead of `<nowiki>[[centre]]</nowiki>` and hyphens as shown in the ASCII-only requirement.
  * Tricks: To produce '[[center|centre]]', try `<nowiki>[[center|centre]]</nowiki>`.  Or, again, make a redirect.

  * Reason: Much of this is written by [[Urs Schreiber]], who uses American English spelling (for the most part, at least).

* Regardless of the above, pages named after specific categories should use the capitalised singular abbreviated form.
  * Examples: Use `<nowiki>[[Set]]</nowiki>` instead `<nowiki>[[Sets]]</nowiki>` and `<nowiki>[[Cat]]</nowiki>` instead of `<nowiki>[[Category]]</nowiki>`.

  * Tricks: Although things like '$\Disc: \Set \to \Cat$' work best in math mode and even '$\Set$' alone looks most consistent that way (with proper math-mode formatting), you have to make the link '[[Set]]' outside of math mode.

### How to merge pages
{#merging}

It happens at times that you may wish the material in existing pages `A` and `B`, would be merged into a single page. (This occurs notably after any user accidentally created a new page unaware that another page on the topic was already in existence.)

In most cases it will be best to raise the issue on the [nForum](https://nforum.ncatlab.org/), check if regulars agree that the pages ought to be merged, and hope that one of them takes care of it.

If you do decide to merge on your own, here are instructions.

> The basic idea is evident: Merge the content, delete one of the entries and make its previous name redirect to the remaining page. But the redirecting takes some care; and a complication is that $n$Lab pages cannot be deleted (not by regular users, anyway) but only "orphaned".

Suppose that `A` is the name of the page to be kept and and that `B` is the name of the one to be deleted. Then:

1. Edit `A`

   1. so that it looks like the intended merged entry.

   1. Add the line "<nowiki>[[!redirects B]]</nowiki>" to the bottom.

      And in case the page `B` itself has <nowiki>[[!redirects X]]</nowiki>-lines in its source, copy them all over to `A`.

   1. Submit.

1. Edit `B`

   1. by first clearing its content (e.g. with a keystroke such as `Ctrl-A` `Del`).

   1. Click the checkbox 'Change page name.', which will bring up a new field 'New name:'.

   1.  In that field, append "`> history`" to the string "`B`", so that the new page name becomes "`B > history`".

       > (The idea is to change to any name that is neither coincides with an existing nor with a future entry.)

       Do *not* submit the edit *yet*.

   1. Before finishing we need to work around what the wiki software thinks is intended behaviour:

      1. Click in the big edit box -- this will automatically add the line "<nowiki>[[!redirects B]]</nowiki>".

         > (The wiki software is trying to be helpful, not knowing that we want to delete the entry for good.)

      1. Delete that line (to ensure that all requests for `B` will now take the reader to `A`),

      1. and instead add "`see` <nowiki>[[A]]</nowiki>".

         > (This just in case anyone ever comes across this page in the future -- which should not actually happen, due to the <nowiki>[[!redirects B]]</nowiki>-command meanwhile added to `A`.)

   1. Finally ready to submit this quasi-deletion of `B`.

      But since the page's name was changed in the process, the software will require announcing this edit in the box below the edit box. So enter there something like:

      "`As we discussed, I am orphaning this page and have merged its previous content into page "A".`"

   1. Submit.

### How to remove a page

Please make a request at the [nForum](https://nforum.ncatlab.org/).

## How to organize and write content
 {#HowToOrganizeAndWriteContent}

### Page organization
 {#PageOrganization}

We encourage authors to structure pages by sections and subsections
as indicated in the

* [[Template page]].

Every page starts out tiny and may not seem to need a structure by subsections. But it will grow and eventually the information it provides may be hard to _recognize_ without some global structure guiding the reader's attention. After all, the natural evolution of articles makes the $n$Lab function more like an encyclopedia than a novel.  Moreover, even with little material there in the beginning, a structure by subsections helps the collaborative process, as it makes it easer to decide for new authors where to add what.

### Definition-, Theorem-, Proof environments
 {#DefinitionTheoremProofEnvironments}

You should include actual mathematics in appropriate environments, as familiar from textbooks and research articles in mathematics.

#### LaTeX-style syntax
 {#LatexSyntax}

As of mid-2018, one can use environments exactly as in LaTeX.

For example the code

$$
  \array{
    \mathrlap{
      \backslash\text{begin}\{\text{theorem}\}
    }
    \\
    &
    \;\;
    \mathrlap{\backslash\text{label}\{\text{SomeTheorem}\}}
    \\
    &
    \;\;
    \mathrlap{\text{Some theorem.}}
    \\
    \mathrlap{\backslash\text{end}\{\text{theorem}\}}
  }
$$

produces:

> \begin{theorem} \label{SomeTheorem}
>   Some theorem.
> \end{theorem}

The available environments are listed below. Some environments can be specified in several ways.

Environment	Syntax

      Definition
      definition

      defn

      Theorem
      theorem

      thm

      Proof
      proof

      Proposition
      proposition

      prop

      prpn

      Remark
      remark

      rmk

      Corollary
      corollary

      cor

      Lemma
      lemma

      lem

      Notation
      notation

      notn

      Terminology
      terminology

      Conjecture
      conjecture

      conj

      Scholium
      scholium

      Assumption
      assumption

      assum

      Example
      example

      Exercise
      exercise

      Statement
      statement

Labels for the purpose of referencing work exactly as in LaTeX: put ```\label{SomeTheorem}``` anywhere inside the environment block. Later, one can reference it as Theorem \ref{SomeTheorem}. Again, see the [source](https://ncatlab.org/nlab/source/HowTo) of this page for how to produce this reference.

Any character but closing brace is allowed in labels here (note that this is different from equation labels below).

#### Older syntax

An older syntax for this is described at [Instiki theorem environments](http://golem.ph.utexas.edu/wiki/instiki/show/Theorems):

A [[definition]] should be enclosed in

     +-- {: .num_defn}
     ###### Definition

      (Definition goes here)

     =--

A [[proposition]] or [[theorem]] should be enclosed in

     +-- {: .num_prop}
     ###### Proposition

      (Proposition goes here)

     =--

or

     +-- {: .num_theorem}
     ###### Theorem

      (Theorem goes here)

     =--

and their [[proof]] in

     +-- {: .proof}
     ###### Proof

       (Proof goes here.)

     =--

A corollary in

     +-- {: .num_cor}
     ###### Corollary

      (Corollary goes here)

     =--

and a remark in

     +-- {: .num_remark}
     ###### Remark

      (Remark goes here)

     =--

Beware:

1. Contrary to the behaviour with the LaTeX syntax described above, you do need to use the labels exactly as given above. That is, it needs to be `defn`, `thm`, and `cor`; neither `definition` nor `theorem` nor `corollary` (nor other abbreviations such as `def`) will be recognized by the software.

1. {#WhitespaceAfterMarukuEnvironment} There *must* be a whitespace after the closing `=--`. Otherwise the parser will not recognize the end of the environment and will in fact ignore its content and *all* following code (unless and until it happens to spot an unpaired `=--` further down the line).

Hence best to instead use the above [LaTeX syntax](#LatexSyntax), where possible.

### Cross-links
 {#CrossLinks}

A central point of the $n$Lab is that information is _linked_. Little value is added to the world when information is added to an $n$Lab page if it is not interlinked there with other $n$Lab entries such as to make sure that users who might look for it will actually eventually find it by following some links.

Therefore you should hyperlink the central keywords in what you write, by enclosing them in square brackets. Ideally, each and every keyword would be hyperlinked, certainly when it occurs first. Don't assume that the reader has the same background as you have. If you explain something by invoking other technical terms, at least provide links to $n$Lab entries where these other terms are defined.

Analogously, when you create a new page, make sure that it is _linked to_ from at least one relevant other page. An entry on a new kind of _manifold_ deserves to be linked to from a list of "related entries" at _[[manifold]]_, for instance.

Apart from guiding users to your entry this way, notably this will also help search engines to index the $n$Lab and make your new entry appear listed when people search for something related.

### Mathematical conventions
 {#MathematicalConventions}

There is really no such thing as a "global convention on the $n$Lab" for ambiguous meaning of mathematical terms and symbols. Unlike in a text book by a single or a handful of authors, there is just no way that we can even try to enforce global conventions. Typically each author decides on his or her own.

Therefore the general rule is: whatever you write in some entry, try to make sure that you provide enough context that the relevant conventions are clear. If you find yourself using some notation for the first time on a page and are worrying if readers will read it as intended, be sure to add a parenthetical remark (where we use here the convention that...) or similar.

If you feel that you need to refer to a specific convention repeatedly over several $n$Lab pages, just create another page that states the convention, and then simply point to that page.

One example of this that we have is the _[[implicit infinity-category theory convention]]_-page.

## Special Typesetting Features

### How to make links to subsections of a page ##
 {#HowToMakeLinksToSubsections}

#### Links to sections

When you create a section header, you can add an HTML anchor tag to it with the following syntax:

     <nowiki>## Heading {#anchorname}</nowiki>

__Note that the tag must be placed at the end.__ Then you can make a link to it using the syntax:

     <nowiki>[[some page#anchorname|link text]]</nowiki>

If the link is to the same page then the page name can be omitted by using the syntax:

     <nowiki>[link text](#anchorname)</nowiki>

__Note that a single bracket link with only an anchor doesn't function correctly when viewing a particular revision or a difference between revisions.__

{#strange}Strangely single bracket link texts allow math expressions while double bracket texts don't. For example

    <nowiki>[$Set^\to$](Sierpinski+topos)</nowiki> vs <nowiki>[[Sierpinski topos|$Set^\to$]]</nowiki> vs <nowiki>[$Set^\to$](https://ncatlab.org/nlab/show/Sierpinski+topos).</nowiki>

produces

+-- {: .standout}
[$Set^\to$](Sierpinski+topos) vs [[Sierpinski topos|$Set^\to$]] vs [$Set^\to$](https://ncatlab.org/nlab/show/Sierpinski+topos) .
=--

Of course, you can link to it from outside the nLab by adding `http://ncatlab.org` at the beginning of the link.

Note that if you skip the first step, it is still *possible* to create a link to a subsection, since (at least if the page has a table of contents) every section on the page is automatically assigned an HTML anchor by Instiki.  However, using such links is *not* encouraged, since the automatically generated anchor names will change whenever the page is rearranged and go away if a manual anchor name is added, which will cause such links to break.

{#thispgh}The same type of syntax enables one to link to arbitrary text blocks such as paragraphs or list elements, such as a [link](#thispgh) to this paragraph itself. __Note that these tags must be at the start of the text__. In particular, a bibliographic reference can be made by first by creating an anchor:

     * {#xyz} Reference

which can then be linked to from within the same page by writing

     [link text](#xyz).

#### Definition/Proposition/Theorem-numbering

When you write a numbered definition or proposition or theorem, you can also simultaneously create an anchor.

In the [LaTeX-style syntax](#LatexSyntax), use `\label` inside the theorem/etc. environment as usual.

In the older syntax, write:

     +-- {: .num_defn #definitionname}
     ###### Definition
     ...
     =--

     +-- {: .num_prop #propname}
     ###### Proposition
     ...
     =--

     +-- {: .num_theorem #theoremname}
     ###### Theorem
     ...
     =--

Then you can link to it in the same ways:

<nowiki>
     [[some page#theoremname|see this theorem]]
     [see this theorem](#theoremname)
</nowiki>

When you link to a theorem on the *same* page, however, it's better to use the `\ref` syntax (without the space between `\ref` and `{`, it is present here just to avoid it being parsed):

     see Theorem <nowiki>\ref {theoremname}</nowiki>

This inserts the number as well as creating a hyperlink, and will also work properly when the page is exported to LaTeX.

#### Equation numbering

To make an equation be automatically numbered use angular brackets instead of dollar signs

     \[
       \exp(\pi i) + 1 = 0
     \]

To refer to this numbered equation, add a label

     \[
       \label{SomeEquation}
       \exp(\pi i) + 1 = 0
     \]

and then refer to it later in the text by using `\eqref` just like in LaTeX:

     See equation <nowiki>\eqref {SomeEquation}</nowiki>.

(Leave out the space after `\eqref`, which is present here just to avoid it being parsed.)

There is also an older, non-TeX syntax for equation references (use an actual colon `:` in place of `-colon-`, which is used here to prevent the example from being parsed):

     See equation <nowiki>(eq-colon-SomeEquation)</nowiki>.

{#LabelNames} Only word characters are allowed in LaTeX display labels. In particular, the hyphen character `-` cannot be used. This is in contrast to labels for theorem environments. This is due to use of different parsers. Instead of <nowiki>\label{foo-bar}</nowiki>, consider <nowiki>\label{FooBar}</nowiki>.

#### Table of contents

Since 2019, the recommended way to make a table of contents for a page (and almost any page should have a table of contents) is to write the following at the top of the page, or wherever the table of contents is to appear, without the space between the backslash and `tableofcontents`.

```
     \ tableofcontents
```

An older method is to write

```
     * table of contents

     {: toc}

```

(including the line break!) at the position where the table of contents is to appear. You may want to label the table of contents by writing

     # Contents #
     * table of contents
     {: toc}

or (at the top of the page)

     # [page title] #
     * table of contents
     {: toc}

(which also gives the opportunity to use a more nicely formatted page title if special characters or mathematical formatting are wanted).

The items in the table of contents will be the section headlines marked by

     ## top level headline ##

     ### second level headline ###

     etc.

Instead of "table of contents" after the asterisk, you can write anything you like: this line will not be displayed but is still required to kick in the CSS that makes this possible.

It is important that the section headings not contain anything that shouldn't go in the table of contents.  Whilst formatting is allowed, wiki-links are not (since then the entry in the table of contents would have a link inside of a link).

### How to cite and record references
 {#HowToCiteAndRecordReferences}

At the end of each $n$Lab entry should be a section

      ## References

or sometimes

      ## Literature

which contains a bullet list of relevant references. A standard way to format an entry is this list is the following:

      * {#LastnameAnotherlastnameYear} <nowiki>[[Firstname Lastname]]</nowiki>, <nowiki>[[Anotherfirstname Anotherlastname]]</nowiki>, _Title_, Journal **volume** issue (year) firstpage-lastpage <nowiki>[[arXiv:xxxx.yyyyy](https://arxiv.org/abs/xxxx.yyyyy)</nowiki>, <nowiki>[doi:xyz](http://doi.xyz)]</nowiki>

which comes out as

>  * {#LastnameAnotherlastnameYear} [[HowTo|Firstname Lastname]], [[HowTo|Anotherfirstname Anotherlastname]], _Title_, Journal **volume** issue (year) firstpage-lastpage [[arXiv:xxxx.yyyyy](https://arxiv.org/abs/xxxx.yyyyy), [doi:xyz](http://doi.xyz)]

Given such an item in the list of references, a standard way to reference it from the main text of the entry is as follows:

      see (<nowiki>[Lastname & Anotherlastname Year, theorem 1.3](#LastnameAnotherlastnameYear)</nowiki>)

which comes out at

> see ([Lastname & Anotherlastname Year, theorem 1.3](#LastnameAnotherlastnameYear))

This produces a hyperlink, and in fact such that following it brings up the above bullet item highlighted by a gray box.

### How to make comments and ask questions {#query}

In general, the place to make comments and ask questions is at the [nForum](https://nforum.ncatlab.org/discussions/?CategoryID=0). Each page has a 'Discuss this page' link in the menu at the top. Clicking on this will take to the thread for discussion of the page in the 'Latest changes' category, if it already exists. Just add a comment to the thread. If it does not exist, you will be taken to the nForum home page. Just start a new discussion there, selecting 'Latest changes' as the category and with title the same as the nLab page, and make your comment/ask your question.

Whilst questions are answered and discussions are carried out at the nForum, the *conclusion* should often be recorded on an nLab page. If it's about a particular page, then of course put it on that page; if it's a general question then it could be recorded on [[HowTo|this How To]] or the [[FAQ]].  As this is a wiki, you can do this recording yourself when your question is answered.

That said, the wiki software does contain a mechanism for putting questions and comments on a wiki page itself: edit the page and put your comment or question in a __query block__ as shown in this example:

    +-- {: .query}
    How do I ask a question?
    =--

which produces

+-- {: .query}
How do I ask a question?
=--

In general, this is the wrong way to go about asking a question.  When you put a query box on a page, often none of the other contributors will notice it for a long time, so it will take a long time for the question to be answered, and in the meantime it makes the page cluttered and ugly for other readers.  However, *occasionally* it may be appropriate to record a brief question in a query box, if it has already been discussed at the nForum and the conclusion was not satisfactory.  In this case a link to the nForum discussion should be included in the query box in preference to any lengthy detail about the question and discussion.

(Before the nForum existed, it was much more common to ask questions using query boxes. Some such query boxes still remain undiscovered on various pages, but whenever we find one we generally migrate it to the n-Forum.)

### How to make a standout box
 {#standout}

If you want to make some text stand out (an important theorem, or slogan), you can do it using a __standout box__.  The mechanism is similar to a query box, although the purpose is different: while a query box is for a temporary question and should be removed once that question is resolved, a standout box is a permanent feature of a page that simply emphasizes a small amount of text.  To make a standout box, write:

    +-- {: .standout}
    First quantization is a mystery, but second quantization is a functor.
    =--

which produces

+-- {: .standout}
First quantization is a mystery, but second quantization is a functor.
=--

### How to include one page within another

If you have some material at a page called `foo` that you want to include directly in pages called `bar` and `baz`, then type <nowiki>[[!include foo]]</nowiki> in `bar` and `baz`.  For an example, see how [[contents]] is included at the tope of this page.  Also see how [[contents]] itself has been formatted so that it will appear as a sidebar when included.

Besides such sidebars that appear in many pages, you can also use inclusion to put in something that contains a bunch of ugly code (such as raw SVG, see [here](#IncludeSVG)) without mucking up the rest of the page.  That is, you put your messy code in `bar/foo` and then put <nowiki>[[!include bar/foo]]</nowiki> in `bar`.  Note that this is for something that, logically, should appear within `bar` itself, which is why `bar` appears in the name of the included page.

Note that the included page goes directly in where it is called with no surrounding whitespace.  This can mean that formatting rules are broken on the include.  For example, if the included file starts and ends with a `div` tag and is included with no surrounding blank lines then this breaks the rules and will generate an error.

### How to upload files
 {#HowToUploadFiles}

To upload a file, proceed as follows:

1. Type

   `<nowiki>[[FileName.xyz:file]]</nowiki>`

   into the edit page for some entry, or

   `<nowiki>[[FileName.xyz:pic]]</nowiki>`

   in the case of a picture;

1. hit "submit";

1. in the rendered entry, find a grayish link labeled "FileName.xyz" followed by a question mark;

1. click on the question mark to open an upload dialogue and follow the instructions there;

   notice that the dialogue field "Description" is asking for the text that will appear hyperlinked to your file. If you put no text here, no link to your file will appear by itself;

1. complete the dialogue by hitting "upload";

1. find in the rendered entry the previously grayish text "FileName.xyz" replaced by the text that you entered into the "Description"-box in the file upload dialogue, and hyperlinked to your file;

1. due to a bug, if you want that link to persist, you need to make any further edit to the entry (e.g. add a whitespace) and "submit" again.

After this, the file is now sitting at this URL:

      https://ncatlab.org/nlab/files/FileName.xyz

Hints:

1. If you uploaded a picture using the `:pic` syntax and do not wish to tweak the displayed image in any way, there is no need to do anything. If you uploaded it using the `:file` syntax, you will probably want to _remove_ the file upload code

           <nowiki>[[FileName.xyz:file]]</nowiki>

   (which renders to a link to the file) and instead add a line like

        \begin{imagefromfile}
            "file_name": "FileName.xyz"
        \end{imagefromfile}

   which makes the actual image show up in entry, There are a number of other posible parameters for an `imagefromfile` block, see {#ImageFiles}.

1. If you uploaded a pdf (or similar) for a reference, you will probably want to make sure that the "Description" text in the upload dialogue is "pdf" (or similar), so that the code

        A. Name, _A title_, A Journal, AYear ([[FileName.xyz:file]])

   renders to

   "A. Name, _A title_, A Journal, AYear (pdf)"

   with "pdf" hyperlinked to your file.

1. There is a size limit for files to upload. The functionality is meant to be used for uploading _ingredients_ of $n$Lab entries (such as pictures or references) which are not available for linking elsewhere on the web, or not reliably so.

   If you try to abuse the file upload for archiving your personal files, the [[steering committee]] will intervene.

### How to use redirects

See [[redirects]].

### How to put parentheses in external links

Since the mechanism for inserting links uses parentheses to delimit the link, it's not obvious how to put parentheses actually in the link itself.  Since Wikipedia uses them a fair bit, it's worth knowing how to put them in.  The trick is to use the URL codes rather than the actual characters.  URL codes are generally used to send "unsafe" characters in URLs (safe characters are <nowiki>`a`-`zA`-`Z0`-`9$-_.+!\*'(),`</nowiki>).  Although parentheses are actually "safe", due to their special meaning for the markdown filter, to put them in URLs here they need to be treated as "unsafe".  URL codes have the syntax `%hex` where `hex` is the index of the character in the ASCII character set represented as a 2-digit hexadecimal.  Wikipedia (among other places) has a [table](http://en.wikipedia.org/wiki/ASCII#ASCII_printable_characters) of the character set from which one can read off the required hexadecimal.  In particular, we see that `(` is `%28` and `)` is `%29`.  Thus

    [Monad (category theory)#Monads and adjunctions](http://en.wikipedia.org/wiki/Monad_%28category_theory%29#Monads_and_adjunctions)

produces

:  [Monad (category theory)#Monads and adjunctions](http://en.wikipedia.org/wiki/Monad_%28category_theory%29#Monads_and_adjunctions)

### How to add a floating table of contents

Many pages include a "floating table of contents" at the top right-hand side with links to other pages on similar topics.  The lists of related pages are separate pages with names such as [[category theory - contents]]; if you want to create a new one, look at the syntax of existing ones.

To add a floating TOC to a new page, in such a way that it will be automatically collapsed until moused over, use code such as the following at the top of the page:

~~~
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category Theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
~~~

If you want to include multiple contents pages, you can repeat the four lines from the one starting with `####` through the first `=--` line.

### How to draw commutative diagrams and pictures {#diagrams}

#### TikZ
 {#TikZ}

You can enter basic [`TikZ`](https://en.wikipedia.org/wiki/PGF/TikZ)-diagrams by (omitting the usual math delimiters such as `$$...$$' and instead)
directly opening a block of the form

<nowiki>
\begin{tikzpicture}
</nowiki>
<nowiki>
...
</nowiki>
<nowiki>
\end{tikzpicture}
</nowiki>

or

<nowiki>
\begin{tikzcd}
</nowiki>
<nowiki>
...
</nowiki>
<nowiki>
\end{tikzcd}
</nowiki>

and adding usual `TikZ`-code inside.

{#QuiverTool} A graphical user interface for generating TikZcd code is the [[quiver (editor)|quiver editor]] at [q.uiver.app](https://q.uiver.app/). (After drawing your diagram there, click on `LaTeX` at the bottom to copy-and-paste the code to be inserted into the above environments.)

{#TikZSupportIsAHack} Beware that this functionality is a hack: The `tikz`-code is compiled server-side and then included as an SVG in the page's HTML source.

In practice this means that `TikZ`-code does not interact with the ambient `instiki`-code, for instance it cannot be included inside bullet-items nor inside other math-environments. To prevent `instiki`'s indentation mechanism from clashing, be sure to align the outer block of the `TikZ`-code to the left of the edit window.

> (There used to be a claim here that `\usetikzlibrary` can be called inside the `tikz`-code block, but it does not seem to be the case.)

##### Examples

Picture:

\begin{centre}
    \begin{tikzpicture}
      \draw[blue, ->] (0,0) -- (1,0) -- (1,-1);
    \end{tikzpicture}
\end{centre}

Source:

```
<nowiki>\begin{centre}
    \begin{tikzpicture}
        \draw[blue, ->] (0,0) -- (1,0) -- (1,-1);
    \end{tikzpicture}
\end{centre}
</nowiki>
```

Single [[morphism]]:

\begin{centre}
    \begin{tikzcd}
        x \ar[r] & y
    \end{tikzcd}
\end{centre}

Source:

```
<nowiki>\begin{centre}
    \begin{tikzcd}
        x \ar[r] & y
    \end{tikzcd}
\end{centre}
</nowiki>
```

Diagram of [[adjoint functors]] {#TypesetAdjointFunctors}:

\begin{centre}
    \begin{tikzcd}
        \mathcal{D} \arrow[r, shift right=7pt, "R"', "\bot"] &
            \mathcal{C} \arrow[l, shift right=7pt, "L"']
    \end{tikzcd}
\end{centre}

Source:

```
<nowiki>\begin{centre}
    \begin{tikzcd}
        \mathcal{D} \arrow[r, shift right=7pt, "R"', "\bot"] &
            \mathcal{C} \arrow[l, shift right=7pt, "L"']
    \end{tikzcd}
\end{centre}
</nowiki>
```

#### Xymatrix

As of 2019 you can format commutative diagrams in the nLab using xymatrix command, almost as you would in a LaTeX document. The only significant differences are:

1. Use a `<nowiki>\begin{xymatrix} ... \end{xymatrix}</nowiki>` block rather than a `\xymatrix{...}` block.

1. Some global parameters can be passed inside the `<nowiki>\begin{xymatrix}</nowiki>` tag. For example, as usual, one can use `@=1.5em` to increase row and column separation, replacing `@` by `@C` (resp. `@R`) for column (resp. row=) separation. Thus one writes: `<nowiki>\begin{xymatrix@=1.5em}</nowiki>`. In addition, unlike in LaTeX, one can pass parameters in square brackets: `<nowiki>\begin{xymatrix[font = \large, border = 2em]}</nowiki>` or `<nowiki>\begin{xymatrix@=1.5em[font = \large, border = 2em]}</nowiki>`. Here the value of `font` controls the font size throughout the diagram, and `border` adds some whitespace around the figure, which is sometimes needed when using curved arrows. One can use only one of the two parameters instead of both.

Under the hood, this works by running pdflatex and then generating an SVG from the resulting pdf, which is included in the HTML source of the rendered page.

Use Tikz rather than Xymatrix if you do not have a strong preference or have not used either before. But it is fine to use Xymatrix if this is what you are used to.

##### Examples

Triangle:

\begin{centre}
    \begin{xymatrix}
        A \ar[r] \ar[dr] & B \ar[d] \\ & C
    \end{xymatrix}
\end{centre}

Source:

```
<nowiki>\begin{centre}
    \begin{xymatrix}
        A \ar[r] \ar[dr] & B \ar[d] \\ & C
    \end{xymatrix}
\end{centre}
</nowiki>
```

Bent arrows:

\begin{centre}
    \begin{xymatrix[border = 2pc]}
        A \ar@/^2.0pc/[r] \ar@/_2.0pc/[r] & B
    \end{xymatrix}
\end{centre}

Source:

```
<nowiki>\begin{centre}
    \begin{xymatrix[border = 2pc]}
        A \ar@/^2.0pc/[r] \ar@/_2.0pc/[r] & B
    \end{xymatrix}
\end{centre}
</nowiki>
```

Parallel arrows:

\begin{centre}
    \begin{xymatrix}
        A \ar@<.6ex>[r] \ar@<-.6ex>[r] & B
    \end{xymatrix}
\end{centre}

Source:

```
<nowiki>\begin{centre}
    \begin{xymatrix}
        A \ar@<.6ex>[r] \ar@<-.6ex>[r] & B
    \end{xymatrix}
\end{centre}
</nowiki>
```

2-cell:

\begin{centre}
    \begin{xymatrix[font = \large]@C+2pc}
        A \rtwocell^f_g{\alpha} & B
    \end{xymatrix}
\end{centre}

Source:

```
<nowiki>\begin{centre}
    \begin{xymatrix[font = \large]@C+2pc}
        A \rtwocell^f_g{\alpha} & B
    \end{xymatrix}
\end{centre}
</nowiki>
```

#### Older workaround for commutative diagrams

An older workaround is to use use arrays or matrices.  For example,

       $$
       \begin{matrix}
         (f/g)& \to & A \\
         \downarrow&\underset{\alpha}{\swarr}&\, \downarrow f\\
         B &\underset{g}{\to} & C
       \end{matrix}
       $$

   produces

   $$
   \begin{matrix}
     (f/g)& \to & A \\
     \downarrow&\underset{\alpha}{\swarr}&\, \downarrow f\\
     B &\underset{g}{\to} & C
   \end{matrix}
   $$

   In many cases, you may wish to tweak the alignments (say, of vertical arrows), using `\mathrlap{}` or `\mathllap{}`:

   $$
   \begin{matrix}
   B^{\mathrlap{A}} & \longrightarrow & 1^{\mathrlap{A}} \\
   \mathllap{\scriptsize{\sigma^A}}\downarrow & & \downarrow \mathrlap{\scriptsize{t^A}} \\
P(B)^{\mathrlap{A}} & \underset{\chi_\sigma^A}{\longrightarrow} &  P(1)^{\mathrlap{A}}
   \end{matrix}
   $$

   is produced by

        \[
        \begin{matrix}
        B^{\mathrlap{A}} & \longrightarrow & 1^{\mathrlap{A}} \\
        \mathllap{\scriptsize{\sigma^A}}\downarrow & & \downarrow               \mathrlap{\scriptsize{t^\alpha}} \\
        P(B)^{\mathrlap{A}} & \underset{\chi_\sigma^A}{\longrightarrow} & P(1)^{\mathrlap{A}}
        \end{matrix}
        \]

   Matrix and array environments also accept certain options. Here is an example using a "\rowlines{solid}" option to hack a sequent derivation (much more nicely than by resorting to \frac commands):

   $$
   \array{\arrayopts{\rowlines{solid}}
   X \times B \to P A\;\;\; in\; \mathbf{S} \\
   X \times B \nrightarrow A\;\;\; in\; Rel(\mathbf{S}) \\
   X \nrightarrow A \times B\;\;\; in\; Rel(\mathbf{S}) \\
   X \to P(A \times B)\;\;\; in \; \mathbf{S}
   }
   $$

   is produced by

       $$
       \array{
       \arrayopts{\rowlines{solid}}
       X \times B \to P A\;\;\; in\; \mathbf{S} \\
       X \times B \nrightarrow A\;\;\; in\; Rel(\mathbf{S}) \\
       X \nrightarrow A \times B\;\;\; in\; Rel(\mathbf{S}) \\
       X \to P(A \times B)\;\;\; in \; \mathbf{S}
       }$$

But use Tikz wherever possible.

### Image files
  {#ImageFiles}

One can upload an image to the nLab as follows.

1. Add `<nowiki>[[some_picture.jpg:pic]]</nowiki>` or `<nowiki>[[some text that you wish to display|some_picture.jpg:pic]]</nowiki>` to the page to which you wish to add the image. Save your edit.

2. Click on the `?` symbol that appears after 'some_picture.jpg' or 'some text that you wish to display' on the page.

3. You will be sent to a page where you can upload the image.

4. After uploading the image, you will need to make some edit to the page to refresh it. After this, the picture should appear on the page.

The `:pic` functionality does not allow much configuration.Once it is uploaded, one can replace it by an `imagefromfile` block, such as the following.

<nowiki>
        \begin{imagefromfile}
            "file_name": "FileName.xyz"
        \end{imagefromfile}
</nowiki>

The contents of the block must be a valid JSON, but without curly opening and closing braces. There are several possible parameters. A common case is the following, which sets the width of the image (the height is automatically adjusted correspondingly).

<nowiki>
        \begin{imagefromfile}
            "file_name": "FileName.xyz",
            "width": 300
        \end{imagefromfile}
</nowiki>

The default unit is pixels; to specify a different one, or to make it explicit, one uses the 'unit' parameter. Currently the only permitted values are "px" and "em", which correspond to pixels and 'em' respectively.

<nowiki>
        \begin{imagefromfile}
            "file_name": "FileName.xyz",
            "width": 300,
            "unit": "px"
        \end{imagefromfile}
</nowiki>

Another common case is the following, which puts the image to the right of the page with the other text fitting around it, with a nice margin in between. The 'unit' inside the 'margin' block behaves in the same way as the overall width/height, in particular defaulting to pixels, but applies only to the margin.

<nowiki>
        \begin{imagefromfile}
            "file_name": "FileName.xyz",
            "float": "right",
            "margin": {
                "top": -40,
                "right": 0,
                "bottom": 20,
                "left": 20
            }
        \end{imagefromfile}
</nowiki>

The full list of options is shown below. Except for "file_name", these are all either optional or have defaults.

<nowiki>
        \begin{imagefromfile}
            "file_name": "FileName.xyz",
            "web": "nlab",
            "width": 300,
            "height": 300,
            "unit": "px",
            "float": "right",
            "margin": {
                "top": -40,
                "right": 0,
                "bottom": 20,
                "left": 20,
                "unit": "px"
            },
            "alt": "Some picture",
            "caption": "This is some picture"
        \end{imagefromfile}
</nowiki>

These should be fairly self-explanatory. The `alt` parameter corresponds to the `<img>` attribute of the same name.

Note in particular that a caption can be added. If one is floating the graphic and one uses a negative value for `top` in the values for `margin`, which is quite often a good idea, then one typically will need a positive value for `bottom` in order for text wrapping around the image not to obscure the caption.

Be judicious when floating graphics. It may be easier to read a page, especially on a mobile phone, if the graphic is simply placed below a paragraph rather than floating to the side of it.

### How to centre

Text or figures can be centred exactly as in LaTeX. For instance,

\begin{centre}
    Hello!
\end{centre}

is produced by

```
<nowiki>
    \begin{centre}
        Hello!
    \end{centre}
</nowiki>
```

or by the following (i.e. either UK or US spelling can be used).

```
<nowiki>
    \begin{center}
        Hello!
    \end{center}
</nowiki>
```

## Other Sources of Information

### Instiki HowTo

For general information and help with Instiki, see the [Instiki](http://golem.ph.utexas.edu/instiki/show/HomePage) wiki.

Here are some useful specifics:

* [Use basic Markdown syntax](http://daringfireball.net/projects/markdown/syntax)
* [Make tables, footnotes, etc](http://maruku.rubyforge.org/maruku.html#extra)
* [Add definitions and theorems](http://golem.ph.utexas.edu/instiki/show/Theorems)
* [Add metadata to your markup](https://golem.ph.utexas.edu/~distler/maruku/proposal.html)
* [Type itex equations](http://golem.ph.utexas.edu/~distler/blog/itex2MMLcommands.html)
* [Use wiki syntax](http://golem.ph.utexas.edu/instiki/show/Syntax#wiki)
* [Embed SVG in equations](http://golem.ph.utexas.edu/instiki/show/SVG)
* [Upload files](http://golem.ph.utexas.edu/instiki/show/File+Uploads)
* [Use keyboard shortcuts](http://golem.ph.utexas.edu/instiki/show/AccessKeys)

### Survey of available math typesetting commands

* [itex2MML Command Summary](http://golem.ph.utexas.edu/~distler/blog/itex2MMLcommands.html)
* [webTeX documentation](https://golem.ph.utexas.edu/~distler/WebTeX/docs/wtxsec5.html)

This is worth perusing carefully;
 for example, iTeX does not provide a command such as `\prescript` to generate left subscripts.
 Instead it contains the more general `\multiscripts` command to typeset subscripts and superscripts
 both to the left and right of an expression. For instance,
 the math mode text `\multiscripts{_0^2_1}{R}{_i^j_k}` produces

$$\multiscripts{_0^2_1}{R}{_i^j_k}$$

## Technical

### Software required to use the $n$Lab
{#software}

The $n$Lab serves mathematical symbols as [MathML](http://en.wikipedia.org/wiki/MathML). Presently the browsers with _native_ MathML support, and hence immediate rendering of the formulas, are those based on the Gecko, Blink, and WebKit rendering engines, e.g., Firefox, Chrome/Chromium, Safari, etc.

Older unmaintained browsers like Internet Explorer, as well as a few obscure mobile and desktop browsers,
fall back to rendering MathML formulas using the [MathJax](https://www.mathjax.org/) [polyfill](https://en.wikipedia.org/wiki/Polyfill_(programming%29). This works, but is slow.  On small pages it is fine, on larger pages with many formulas MathJax may take up to several minutes for rendering.

### How to search the nLab & nForum from firefox

(Firefox - and clones - specific)

Here are some search plugins for firefox that will let you search the nLab from the firefox search bar.  **EDIT:** These are old and may not work with recent versions of Firefox.  Instead you can use the [search engine creator plugin](https://addons.mozilla.org/en-US/firefox/addon/search-engine-creator/).  Someone should update these files to conform to [OpenSearch](https://developer.mozilla.org/en-US/Add-ons/Creating_OpenSearch_plugins_for_Firefox); then they might work with other browsers too.

* [[nlab-search.xml:file]]: searches the nLab (like the search box at the top of every page).
* [[nlab-goto.xml:file]]: takes you directly to the page with a given exact title (if it exists; otherwise it takes you to an edit box to create such a page).
* [[nlab-edit.xml:file]]: takes you directly to the "edit" page for a given title.
* [[nforum-topic-search.xml:file]]: searches the nForum Topics.
* [[nforum-comment-search.xml:file]]: searches the nForum Comments.

It would be nice if these had different icons.  To use one or more of these, drop them in the 'searchplugins' directory of your firefox profile.

Alternatively, you can try the [add to search bar](https://addons.mozilla.org/en-US/firefox/addon/add-to-search-bar/) add-on for Firefox.  This will let you right-click in the search box on any page and add it to your search bar.  (You can't duplicate the "goto" and "edit" plugins this way, though.)

If DuckDuckGo is your search engine, you can type "!nl" to search nLab, e.g. "!nl morphism".

### How to edit nLab pages in your favorite text editor

(Firefox - and clones - specific)

One way to do this is to install one of several firefox extensions, e.g. [for Emacs](https://addons.mozilla.org/en-US/firefox/addon/edit-with-emacs1/?src=search) or [for Sublime Text, Atom, VS Code, and Vim](https://addons.mozilla.org/en-US/firefox/addon/ghosttext/?src=search). There are also extensions which add some basic editor features to html text areas, for instance [Find & Replace for Text Editing](https://addons.mozilla.org/en-US/firefox/addon/find-replace-for-text-editing/?src=search) allows you to use regular expressions and text templates.

If your favorite editor is [Emacs](http://www.gnu.org/software/emacs/) with [AucTeX](http://www.gnu.org/software/auctex/), you may find the following snippet useful to put in your `.emacs` file:

    (add-to-list 'auto-mode-alist '("/\\(www.\\)?ncatlab.org" . latex-mode))
    (add-to-list 'auto-mode-alist '("/golem.ph.utexas.edu" . latex-mode))
    (defun nlab-latex-fixes ()
      (when (or (string-match "/\\(www.\\)?ncatlab.org" buffer-file-name)
                (string-match "/golem.ph.utexas.edu" buffer-file-name))
      (longlines-mode t)
      (set (make-local-variable 'TeX-open-quote) "\"")
      (set (make-local-variable 'TeX-close-quote) "\"")))
    (add-hook 'LaTeX-mode-hook 'nlab-latex-fixes)

This will tell Emacs to automatically edit nLab pages (and nCafe comments as well, for good measure) in LaTeX mode, with long lines wrapped using soft returns, and ordinary double-quotes rather than LaTeX ones.

### How to print a page from the nLab

Use the 'Print' link at the bottom of the page.  As a shortcut in most browsers on most operating systems, hit Alt-Shift-P while the focus is in the page to go to the printable page.

### How to customize the nLab

(Firefox - and clones - specific)

You may wish to customize the font scheme (both for math or text) on the nLab, as well as tweak things such as the small edit box for comments. Try using the Stylish add-on if you are using Firefox; Stylish is a plug-in for firefox enabling you to customize websites; it is available [here](https://addons.mozilla.org/en-US/firefox/addon/2108).

Currently, the following stylish themes are available:

* [nLab Stylish theme](http://userstyles.org/styles/17934) by [[Bruce Bartlett]].  This nLab theme changes the fonts on the nLab to a serif-style, and makes the edit box much bigger for an overall more pleasant experience!
* [nLab -- nCafe style](http://userstyles.org/styles/22800) by [[Daniel Schäppi]].  This is based on Bruce Bartlett's theme but changes the overall colour scheme somewhat to something a little more like the n-Cafe.

### How to download a local copy of the nLab
  {#download}

A current git repository of the source code for nLab pages is available at <https://github.com/ncatlab/nlab-content> and a current git repository of the compiled HTML code for nLab pages is available at <https://github.com/ncatlab/nlab-content-html>.

category: meta


Hoàng Xuân Sính

**Hoàng Xuân Sính** is a Vietnamese mathematician who was a student of [[Grothendieck]]. Her thesis was on *Gr-categories* (now more often called (weak) [[2-groups]]) and *[[Picard groupoids|Picard category]]* (also called *[[symmetric 2-groups]]*, i.e. [[symmetric monoidal categories]] with all objects [[invertible object|invertible]] and all morphisms [[invertible morphism|invertible]]).

For more on Hoàng Xuân Sính see:

* [Wikipedia entry](https://en.wikipedia.org/wiki/Ho%C3%A0ng_Xu%C3%A2n_S%C3%ADnh)

* [[John Baez]], *[Hoàng Xuân Sính](https://golem.ph.utexas.edu/category/2022/06/hong_xun_snh.html)* $n$-Category Café (June 20, 2022)

## Selected writings:

Early discussion of [[2-groups]]:

* {#Sinh73} Hoàng Xuân Sính, _Gr_-catégories, PhD thesis, Hanoi (1973, 1975) [[web](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh.html), [pdf](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh_I.pdf), [[Sinh_Gr-Categories.pdf:file]]]

The version linked to above is a handwritten draft copy, with some annotations by someone (perhaps [[Grothendieck]]). A typed version does exist which does not contain the preliminary material in the first file. The date of the official thesis defense was 22 May 1975, with [[Jean-Louis Verdier]], [[Henri Cartan]], [[Laurent Schwartz]] and [[Michel Zisman]] as well as [[Alexander Grothendieck]] as the 'jury'. It was a 'doctorat d'état'. The link to her thesis contains other material, including a collection of remarks by Grothendieck on her work as well as two papers by Sính related to the subject of her thesis:

* Hoàng Xuân Sính,  Gr-catégories strictes, *Acta Mathematica Vietnamica* **3** 2 (1978) 47-59 [[pdf](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/Hoang_Xuan_Sinh_Gr_categories_strictes.pdf)]

* Hoàng Xuân Sính, Catégories de Picard restreintes, *Acta Mathematica Vietnamica* **7** 1 (1983) 117-122  [[pdf](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/Hoang_Xuan_Sinh_Categories_de_Picard_restreintes.pdf)]

The first appears to prove that every 2-group is equivalent to a strict 2-group arising from a [[crossed module]].  The second calls a symmetric 2-group a *Picard category*, and calls a Picard category *restrained* if the [[braiding]] $B_{x,x} \colon x \otimes x \to x \otimes x$ is the identity for all objects $x$.  It then proves that every Picard category is equivalent to one arising from a 2-term chain complex of abelian groups.

## Related entries

* [[2-group]], [[symmetric 2-group]]

* [[crossed module]]

* [[Picard groupoid]]

category:people

[[!redirects H.X. Sinh]]
[[!redirects Hoang Xuan Sinh]]


Hořava-Witten theory

> This entry is about the conjectured relation ([[duality in string theory|duality]]) between [[M-theory]] at [[MO9]]-planes and [[heterotic string theory]] on these. For the relation of [[M-theory]] [[KK-compactification|KK-compactified]] on a [[K3-surface]] and [[heterotic string theory]] on a [[3-torus]] see instead at _[[duality between M/F-theory and heterotic string theory]]_.

***

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

There is an observation by [Hořava--Witten 95](#HoravaWitten95), [Hořava--Witten 96](#HoravaWitten96) which suggests that [[M-theory]] on an  [[cyclic group of order 2|Z/2]]-[[orbifold]] (actually a [[higher orientifold]]) of the form $X_{10} \times (S^1 \sslash \mathbb{Z}_2))$ is "[[duality in string theory|dual]]" to [[heterotic string theory]] on its [[boundary]] ([[fixed point]]) "[[M9-brane]]". Therefore one also speaks of "heterotic [[M-theory]]" ([Ovrut 02](#Ovrut02)).

> from [Kashima 00](#Kashima00)

In the above the circle factor is taken to be the [[circle]] [[fiber]] over the 10d [[type IIA string theory|type IIA]] [[spacetime]].

If instead one considers another of the spatial dimensions to be compactified on $S^1\sslash \mathbb{Z}_2$, then, after [[T-duality]] ([[F-theory]]) result is supposed to be [[type I string theory]]. In this case the intersection of the [[M2-brane]] with the [[M9-brane]] (the latter now wrapping the M-theory circle fiber) is called the _[[E-string]]_.

More in detail:

One considers the [[KK-compactification]] of [[M-theory]] on a [[cyclic group of order 2|Z/2]]-[[orbifold]] of a [[torus]], hence of the [[Cartesian product]] of two [[circles]]

$$
  \array{
    & S^1_A &\times& S^1_B
    \\
    \text{radius}: &  R_{11} && R_{10}
  }
$$

such that the reduction on the first factor $S^1_A$ corresponds to the [[duality between M-theory and type IIA string theory]], hence so that subsequent [[T-duality]] along the second factor yields [[type IIB string theory]] (in its [[F-theory]]-incarnation). Now the diffeomorphism which exchanges the two circle factors and hence should be a symmetry of M-theory is interpreted as [[S-duality]] in [[type II string theory]]:

$$
  IIB \overset{S}{\leftrightarrow} IIB
$$

> graphics taken from [Horava-Witten 95, p. 15](#HoravaWitten95)

If one considers this situation additionally with a $\mathbb{Z}/2\mathbb{Z}$-[[orbifold]] quotient of the first circle factor, one obtains the [[duality between M-theory and heterotic string theory]] ([[Horava-Witten theory]]). If instead one performs it on the second circle factor, one obtains [[type I string theory]].

Here in both cases the [[involution]] [[action]] is by [[reflection]] of the circle at a line through its center. Hence if we identify $S^1 \simeq \mathbb{R} / \mathbb{Z}$ then the action is by multiplication by /1 on the [[real line]].

In summary:

M-theory on

* $(S^1_A \sslash \mathbb{Z}_2 ) \times S^1_B$ yields [[heterotic string theory]]

* $S^1_A \times \left( S^1_B \sslash \mathbb{Z}_2 \right)$ yields [[type I' string theory]]

Hence the [[S-duality]] that swaps the two circle factors corresponds to a duality between heterotic E and type I' string theory. And T-dualizing turns this into a _[[duality between type I and heterotic string theory]]_.

$$
  \array{
     HE
       &\overset{KK/\mathbb{Z}^A_2}{\leftrightarrow}&
     M
       &\overset{KK/\mathbb{Z}^B_2}{\leftrightarrow}&
     I'
     \\
     \mathllap{T}\updownarrow
     &&  &&
     \updownarrow \mathrlap{T}
     \\
     HO
     && \underset{\phantom{A}S\phantom{A}}{\leftrightarrow} &&
     I
  }
$$

> graphics taken from [Horava-Witten 95, p. 16](#HoravaWitten95)

### Duality between M-theory and heterotic string theory

Here each of the two copies of the heterotic gauge theory is a "[[hidden sector]]" with respect to the other.

The orbifold equivariance condition of the [[supergravity C-field]] is that discussed at _[[orientifold]]_ (there for the [[B-field]]). Therefore it has to vanish at the two fixed fixed points of the $\mathbb{Z}_2$-action. Thereby the quantization condition

$$
  [2G_4] = 2 [c_2] - [\frac{1}{2} p_1]
$$

on the [[supergravity C-field]] becomes the condition for the [[Green-Schwarz mechanism]] of the [[heterotic string theory]] on the "boundary" (the orbifold fixed points).

### Duality between M-theory and type I string theory

[[duality between M-theory and type I string theory]]

originally suggested in ([Horava-Witten 95, section 3](#HoravaWitten95))

Further evidence is reviewed in [APT 98](#APT98)

### Duality between heterotic and type I string theory

[[duality between heterotic and type I string theory]]

via [[S-duality]] in the [[F-theory]]-picture...

## Related concepts

* [[duality between F-theory and heterotic string theory]]

[[!include KK-compactifications of M-theory -- table]]

## Properties

### Boundary conditions
 {#BoundaryConditions}

The [[supergravity C-field]] $\hat G_4$ is supposed to vanish, and _differentially_ vanish at the boundary in the HW model, meaning that also the local connection 3-form $C_3$ vanishes there. The argument is roughly as follows (similar for as in [Falkowski, section 3.1](#Falkowski)).

The [[higher dimensional Chern-Simons theory|higher Chern-Simons term]]

$$
  C_3 \mapsto C_3 \wedge G_4 \wedge G_4
$$

in the [[Lagrangian]] of [[11-dimensional supergravity]] is supposed to be well-defined on fields on the [[orbifold]] and hence is to be $\mathbb{Z}_2$-invariant.

Let $\iota_{11}$ be the canonical vector field along the circle factor. Then the component of $G \wedge G$ which is annihilated by the contraction $\iota_{11}$ is necessarily even, so the component $d x^{11}\wedge \iota_11 C_3$ is also even. It follows that also $d x^{11}\wedge \iota_11 G_4$ is even.

Moreover, the kinetic term

$$
  C \mapsto G \wedge \star G
$$

is to be invariant. With the above this now implies that the components of $G$ annihiliated by $\iota_{11}$ is odd, because so is the mixed component of the [[metric]] tensor.

This finally implies that the restriction of $C_3$ to the orbifold fixed points has to be closed.

## Related concepts

* [[duality in string theory]]

  * [[duality between F-theory and heterotic string theory]]

* [[11-dimensional supergravity]]

* [[M-theory]], [[heterotic string theory]]

* [[M9-brane]]

## References

### General

The original articles are

* {#HoravaWitten95} [[Petr Hořava]], [[Edward Witten]], _Heterotic and Type I string dynamics from eleven dimensions_, Nucl. Phys. B **460** (1996) 506 [[arXiv:hep-th/9510209](http://arxiv.org/abs/hep-th/9510209), doi:10.1016/0550-3213(95)00621-4]

* [[Edward Witten]], _Strong Coupling Expansion Of Calabi-Yau Compactification_, Nucl. Phys.B 471:135-158, 1996 ([arXiv:hep-th/9602070](https://arxiv.org/abs/hep-th/9602070))

* {#HoravaWitten96} [[Petr Hořava]], [[Edward Witten]],  _Eleven dimensional supergravity on a manifold with boundary_, Nucl. Phys. B475 (1996) 94 ([arXiv:hep-th/9603142](http://arxiv.org/abs/hep-th/9603142))

Review is in

* [[Piyush Kumar]], _Hořava-Witten theory_ (2004) ([pdf](http://theory.uchicago.edu/~sethi/Teaching/P484-W2004/hwitten.pdf))

* [[Paul Townsend]], _Four Lectures on M-Theory_ ([arXiv:hep-th/9612121](http://arxiv.org/abs/hep-th/9612121)).

* {#Ovrut02} [[Burt Ovrut]], _Lectures on Heterotic M-Theory_, in *Strings, Branes and Extra Dimensions*, TASI 2001, World Scientific (2004) 359-407  [[arXiv:hep-th/0201032](http://arxiv.org/abs/hep-th/0201032), [doi:10.1142/9789812702821_0007](https://doi.org/10.1142/9789812702821_0007)]

* {#Falkowski} [[Adam Falkowski]], section 3 of _Five dimensional locally supersymmetric theories with branes_, Master Thesis, Warsaw ([[FalkowskiLecture.pdf:file]])

The [[black brane|black]] [[M2-brane]] solution in HW-theory, supposedly yielding the [[black brane|black]] [[heterotic string]] at the intersection with the [[M9-brane]] is discussed in

* {#LalakLukasOvrut97} Zygmunt Lalak, André Lukas, [[Burt Ovrut]], _Soliton Solutions of M-theory on an Orbifold_, Phys. Lett. B425 (1998) 59-70 ([arXiv:hep-th/9709214](https://arxiv.org/abs/hep-th/9709214))

* {#Kashima00} Ken Kashima, _The M2-brane Solution of Heterotic M-theory with the Gauss-Bonnet $R^2$ terms_, Prog.Theor.Phys. 105 (2001) 301-321 ([arXiv:hep-th/0010286](https://arxiv.org/abs/hep-th/0010286))

More on the [[Green-Schwarz mechanism]] in [[Hořava-Witten theory]]:

* E. Dudas, J. Mourad, _On the strongly coupled heterotic string_, Phys. Lett. B400 (1997) 71-79 ([arXiv:hep-th/9701048](https://arxiv.org/abs/hep-th/9701048))

* [[Adel Bilal]], Jean-Pierre Derendinger, Roger Sauser, _M-Theory on $S^1/\mathbb{Z}_2$: New Facts from a Careful Analysis_, Nucl. Phys. B576 (2000) 347-374 ([arXiv:hep-th/9912150](https://arxiv.org/abs/hep-th/9912150))

* Ian G Moss, _A new look at anomaly cancellation in heterotic M-theory_, Phys. Lett. B637 (2006) 93-96 ([arXiv:hep-th/0508227](https://arxiv.org/abs/hep-th/0508227))

* Sergio Lukic, [[Gregory Moore]], _Flux corrections to anomaly cancellation in M-theory on a manifold with boundary_ ([arXiv:hep-th/0702160](https://arxiv.org/abs/hep-th/0702160))

* Ian G Moss, _Higher order terms in an improved heterotic M theory_, JHEP 0811:067, 2008 ([arXiv:0810.1662](https://arxiv.org/abs/0810.1662))

* [[Fei Han]], [[Kefeng Liu]], [[Weiping Zhang]], _Anomaly Cancellation and Modularity. II: $E_8 \times E_8$ case_, Sci. China Math. 60, 985–994 (2017) ([arXiv:1209.4540](https://arxiv.org/abs/1209.4540), [doi:10.1007/s11425-016-9034-1](https://doi.org/10.1007/s11425-016-9034-1))

Explicit discussion of [[worldvolume]] [[CFT]] of the [[M2-branes]] ending on the HW fixed points and becoming [[heterotic strings]] is discussed, via the [[BLG model]], in

* {#Lambert15} [[Neil Lambert]], _Heterotic M2-branes_, Physics Letters B Volume 749, 7 October 2015, Pages 363–367 ([arXiv:1507.07931](http://arxiv.org/abs/1507.07931))

After [[KK-reduction]] to [[5d supergravity]] there is a corresponding 5d mechanism, see the references [there](5-dimensional+supergravity#ReferencesHWCompactification).

Discussion of the [[duality between heterotic and type I string theory]] includes

* {#APT98} I. Antoniadis, H. Partouche, T.R. Taylor, _Lectures on Heterotic-Type I Duality_, Nucl.Phys.Proc.Suppl. 61A (1998) 58-71; Nucl.Phys.Proc.Suppl. 67 (1998) 3-1

Discussion of [[string phenomenology]] in Horava-Witten theory:

* {#Ovrut18} [[Burt Ovrut]], _Vacuum Constraints for Realistic Heterotic M-Theories_, Symmetry 2018, 10(12), 723 ([arXiv:1811.08892](https://arxiv.org/abs/1811.08892), [doi:10.3390/sym10120723](https://doi.org/10.3390/sym10120723))

### Phenomenology

Discussion of [[phenomenology]]/[[string phenomenology]] for the [[heterotic M-theory]], relating to the [[standard model of particle physics]]:

* {#DOPW99} [[Ron Donagi]], [[Burt Ovrut]], [[Tony Pantev]], [[Daniel Waldram]], _Standard Models from Heterotic M-theory_, Adv. Theor. Math. Phys. 5 (2002) 93-137 ([arXiv:hep-th/9912208](https://arxiv.org/abs/hep-th/9912208))

* {#DOPW00} [[Ron Donagi]], [[Burt Ovrut]], [[Tony Pantev]], [[Daniel Waldram]], _Standard Model Vacua in Heterotic M-Theory_, talk at [Strings '99](http://strings99.aei.mpg.de/), Potsdam, Germany, 19 - 24 Jul 1999 ([arXiv:hep-th/0001101](https://arxiv.org/abs/hep-th/0001101))

and with emphasis on [[heterotic line bundle]]-models:

* [[Sebastian Dumitru]], *The strongly coupled $E_8 \times E_8$ heterotic string: Geometry & Phenomenology* [[arXiv:2206.12310](https://arxiv.org/abs/2206.12310), [spire:2100628](https://inspirehep.net/literature/2100628)]

See also:

* Cédric Deffayet, [[Burt Ovrut]], [[Paul Steinhardt]], *Stable Vacua with Realistic Phenomenology and Cosmology in Heterotic M-theory Satisfying Swampland Conjectures* [[arXiv:2401.04828](https://arxiv.org/abs/2401.04828)]

### Gaugino condensation and supersymmetry breaking
 {#ReferencesGauginoCondensation}

Discussion of [[gaugino condensation]] and [[supersymmetry breaking]] in Horava-Witten theory, where the two heterotic [[MO9-branes]] provide a natural mechanism for [[supersymmetry breaking]] on one (the physical brane) by gaugino condensation on the other (the "dark sector" brane):

lecture notes:

* {#Nilles04} [[Hans-Peter Nilles]], _Gaugino Condensation and SUSY Breakdown_, Lectures at Cargese School of Physics and Cosmology, Cargese, France, August 2003 ([arXiv:hep-th/0402022](https://arxiv.org/abs/hep-th/0402022))

original articles:

* I. Antoniadis, M. Quiros, _On the M-theory description of gaugino condensation_, Phys.Lett. B416 (1998) 327-333 ([arXiv:hep-th/9707208](https://arxiv.org/abs/hep-th/9707208))

* I. Antoniadis, M. Quiros, _Supersymmetry breaking in M-theory and gaugino condensation_, Nucl.Phys. B505 (1997) 109-122 ([arXiv:hep-th/9705037](https://arxiv.org/abs/hep-th/9705037))

* [[André Lukas]], [[Burt Ovrut]], [[Daniel Waldram]], _Gaugino condensation in M theory on $S^1/Z_2$_, Phys. Rev. D 57, 7529 (1998) ([arXiv:hep-th/9711197](https://arxiv.org/abs/hep-th/9711197), [doi:10.1103/PhysRevD.57.7529](https://doi.org/10.1103/PhysRevD.57.7529))

* [[André Lukas]], [[Burt Ovrut]], [[Daniel Waldram]], _Five-Branes and Supersymmetry Breaking in M-Theory_, JHEP 9904:009, 1999 ([arXiv:hep-th/9901017](https://arxiv.org/abs/hep-th/9901017))

specifically for [[M-theory on S1/G_HW times H/G_ADE]]:

* Zygmunt Lalak, Steven Thomas, _Gaugino Condensation, Moduli Potentials and Supersymmetry Breaking in M-Theory Models_, Nuclear Physics B
Volume 515, Issues 1–2, 30 March 1998, Pages 55-72
Nuclear Physics B ([hep-th/9707223](https://arxiv.org/abs/hep-th/9707223), doi:10.1016/S0550-3213(97)00784-0)

With emphasis on [[heterotic line bundles]] in the [[hidden sector]]:

* Anthony Ashmore, Sebastian Dumitru, [[Burt Ovrut]], Section 4.2 of: _Line Bundle Hidden Sectors for Strongly Coupled Heterotic Standard Models_ ([arXiv:2003.05455](https://arxiv.org/abs/2003.05455))

* Anthony Ashmore, Sebastian Dumitru, [[Burt Ovrut]], _Explicit Soft Supersymmetry Breaking in the Heterotic M-Theory B−L MSSM_ ([arXiv:2012.11029](https://arxiv.org/abs/2012.11029))

### Generalization to M-theory on $S^1/G_{HW} \times \mathbb{H}/G_{ADE}$

Generalization to [[M-theory on S1/G_HW times H/G_ADE]]:

* Zygmunt Lalak, Steven Thomas, _Gaugino Condensation, Moduli Potentials and Supersymmetry Breaking in M-Theory Models_ ([hep-th/9707223](https://arxiv.org/abs/hep-th/9707223))

* V. Kaplunovsky, [[Jacob Sonnenschein]],[[Stefan Theisen]], S. Yankielowicz, _On the Duality between Perturbative Heterotic Orbifolds and M-Theory on $T^4/Z_N$_ ([arXiv:hep-th/9912144](https://arxiv.org/abs/hep-th/9912144))

* {#GKSTY02} E. Gorbatov, V.S. Kaplunovsky, J. Sonnenschein, [[Stefan Theisen]], S. Yankielowicz, _On Heterotic Orbifolds, M Theory and Type I' Brane Engineering_, JHEP 0205:015, 2002 ([arXiv:hep-th/0108135](https://arxiv.org/abs/hep-th/0108135))

* {#GaiottoTomasiello14} [[Davide Gaiotto]], [[Alessandro Tomasiello]], _Holography for $(1,0)$ theories in six dimensions_, JHEP12(2014)003 ([arXiv:1404.0711](https://arxiv.org/abs/1404.0711))

* {#DHTV14} Michele Del Zotto, [[Jonathan Heckman]], [[Alessandro Tomasiello]], [[Cumrun Vafa]], Section 6 of: _6d Conformal Matter_, JHEP02(2015)054 ([arXiv:1407.6359](https://arxiv.org/abs/1407.6359))

* {#HSS18} [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], Example 2.2.7 of: _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_, CMP (2019) ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987), [doi:10.1007/s00220-019-03442-3](http://link.springer.com/article/10.1007/s00220-019-03442-3))

### For 7d supergravity

Discussion for [[7d supergravity]]:

* {#GherghettaKehagias02} Tony Gherghetta, [[Alex Kehagias]], _Anomaly Cancellation in Seven-Dimensional Supergravity with a Boundary_, Phys.Rev. **D68** (2003), 065019, ([arXiv:hep-th/0212060](http://arxiv.org/abs/hep-th/0212060))

* Spyros D. Avramis, [[Alex Kehagias]], _Gauged $D=7$ Supergravity on the $S^1/\mathbb{Z}_2$ Orbifold_ ([arXiv:hep-th/0407221](https://arxiv.org/abs/hep-th/0407221))

* T.G. Pugh, [[Ergin Sezgin]], [[Kellogg Stelle]], _$D=7$ / $D=6$ Heterotic Supergravity with Gauged R-Symmetry_ ([arXiv:1008.0726](https://arxiv.org/abs/1008.0726))

[[!redirects Hořava-Witten theory]]
[[!redirects Hořava–Witten theory]]
[[!redirects Hořava--Witten theory]]
[[!redirects Horava-Witten theory]]
[[!redirects Horava–Witten theory]]
[[!redirects Horava--Witten theory]]

[[!redirects heterotic M-theory]]

[[!redirects duality between M-theory and heterotic string theory]]
[[!redirects duality between heterotic string theory and M-theory]]

[[!redirects duality heterotic string theory and M-theory]]

[[!redirects duality between M-theory and type I string theory]]
[[!redirects duality between type I string theory and M-theory]]

[[!redirects HET - M]]
[[!redirects HET - M duality]]

[[!redirects M - HET]]
[[!redirects M - HET duality]]


HQFT

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### General idea

What is called a  **homotopy quantum field theory** is a [[TQFT]] defined on [[cobordisms]] that are equipped with the extra [[structure]] of a [[continuous function]] into a fixed [[topological space]] $B$.

Hence if $Bord_n(B)$ denotes a [[category of cobordisms]] suitably equipped with maps into $B$, then an HQFT is a [[monoidal functor]]
$$
  Z \;\colon\; Bord_n(B)^{\coprod} \longrightarrow Vect^{\otimes}
  \,.
$$

### History

HQFTs were first defined (under a different name) by [[Graeme Segal]] as early as 1988 in ([Segal 88](#Segal88)).

Starting from 1991, several papers by [[Dan Freed]],
including [FreedQuinn1991](#FreedQuinn1991), further developed the notion of a [[functorial field theory]], where bordisms in the domain category are equipped with a map
to a target space (such as the [[Eilenberg–MacLane space]] $K(G,1)$ for a finite group $G$).
See, in particular, Theorem 1.7 and the preceding discussion on page 6 of [FreedQuinn1991](#FreedQuinn1991).

From 1999, HQFTs were studied systematically by [[Vladimir Turaev]] ([Turaev 99](#Turaev99)) for 2-dimensional [[manifolds]]/[[cobordisms]] and extended to 3-dimensional ones in ([Turaev 00](#Turaev00)).
Turaev also introduced the term “homotopy quantum field theory”.

At about the same time, ([Brightwell-Turner 00](#BrightwellTurner00)) looked at what they called the [[homotopy]] surface category and its [[representation]]s. There are two viewpoints which interact and complement each other. Turaev's seems to be to see HQFTs as an extension of the tool kit for studying [[manifolds]] already given by [[TQFT]]s, whilst in Brightwell and Turner's,  it is the 'background space', which is probed by the surfaces in the sense of [[sigma-models]].

In the proof of the [[cobordism hypothesis]] in ([[On the Classification of Topological Field Theories|Lurie 09]]) the concept of HQFTs was refined to [[extended TQFT]] by considering an [[(∞,n)-category of cobordisms]] $Bord_n(X)$ with maps to a given [[homotopy type]] $X$. For these the [[cobordism hypothesis]] essentially says (see at [For framed cobordisms in a topological space](cobordism+hypothesis#StatementForCobordismsInAManifold)) that $Bord_n(X)^\coprod$ is the [[free construction]] [[(∞,n)-category with duals]] on the [[fundamental ∞-groupoid]] $\Pi(X)$ of $X$.

## Definition

An HQFT is going to be defined as assigning data to $B$-cobordisms. We first introduce these

* [B-cobordisms](#BCobordism)

and then define

* [HQFTs](#HQFTs)

themselves in terms of these.

### $B$-Cobordisms
 {#BCobordism}

Let $B$ be a [[pointed topological space]].

+-- {: .num_defn }
###### Definition

A **$B$-manifold** is a pair $(X, g)$, where $X$ is a closed oriented $n$-[[manifold]] (with a choice of base point $m_i$ in each connected component $X_i$ of $X$), and $g$ is a [[continuous function]] $g : X \to B$, called the **characteristic map**, such that $g(m_i) = \ast$ for each base point $m_i$.

=--

+-- {: .num_defn }
###### Definition

A **$B$-isomorphism** between $B$-manifolds, $\phi : ( X, g) \to ( Y, h)$ is an [[isomorphism]] $\phi : X \to Y$ of the manifolds, preserving the orientation, taking base points into base points and such that $h\phi = g$.

=--

+-- {: .num_remark }
###### Remark

If as  is often the case, the manifolds under consideration will be [[smooth manifolds]]  and then '[[isomorphism]]' is interpreted as '[[diffeomorphism]]', but equally well we can position the theory in the category of PL-manifolds or triangulable topological manifolds with the obvious changes.  In fact for some of the time it is convenient to  develop constructions for simplicial complexes rather than manifolds, as it is triangulations that provide the basis for the combinatorial descriptions of the structures that we will be using.

=--

+-- {: .num_defn }
###### Definition

Denote by $\mathbf{Man}(n,B)$ the [[category]] of $n$-dimensional $B$-manifolds and $B$-isomorphisms.  We define a 'sum' operation on this category using [[disjoint union]].  The disjoint union of $B$-manifolds is defined by
$$( X, g) \amalg ( Y, h) := ( X\amalg Y, g\amalg h),$$
with the obvious characteristic map, $g\amalg h : X \amalg Y \to B$.  With this 'sum' operation, $\mathbf{Man}(n,B)$ becomes a symmetric monoidal category with the unit being given by the empty $B$-manifold, $\emptyset$, with the empty characteristic map.  Of course, this is an $n$-manifold by default.

=--

These $B$-manifolds are the objects of interest, but they have to be related by the analogue of [[cobordisms]] for this setting.

+-- {: .num_defn }
###### Definition

A **$B$-cobordism**, $(W,F)$, from $(X_0,g)$ to $(X_1,h)$ is a [[cobordism]] $W : X_0 \to X_1$ endowed with a [[homotopy class]] relative to the boundary of a [[continuous function]] $F : W \to B$  such that

$$
  F|_{X_0} =  g \,, \;\;\;\;  F|_{X_1} = h
$$

=--

+-- {: .num_remark }
###### Remark

Generally, unless necessary in this entry, we will not make a notational distinction between the [[homotopy class]] $F$ and any of its representatives.

=--

+-- {: .num_defn }
###### Definition

A **$B$-isomorphism of $B$-cobordisms**, $\psi : (W,F) \to (W^\prime, F^\prime)$, is an isomorphism $\psi : W \to W^\prime$ such that

$$\psi (\partial_+W) = \partial_+W^\prime, \quad \psi (\partial_-W) = \partial_-W^\prime,$$

and $F^\prime \psi = F$, in the obvious sense of homotopy classes relative to the boundary.

=--

We can glue $B$-cobordisms along their boundaries, or more generally, along a $B$-isomorphism between their boundaries, in the usual way. This gives rise to a [[symmetric monoidal category]] $\mathbf{HCobord}(n,B)$ of $B$-cobordisms

The detailed structure of $B$-[[cobordisms]] and the resulting category $\mathbf{HCobord}(n,B)$ is given in ([Rodrigues 03, appendix](#Rodrigues03)), at least in the important case of [[smooth manifold|smooth]] $B$-manifolds.  This category is a [[monoidal category]] with strict [[dual objects]].

### Homotopy Quantum Field Theories
 {#HQFTs}

The general absract definition of an HQFT is now the following.

Fix an integer $n \geq 0$ and a [[field]], $K$. All vector spaces will be tacitly assumed to be finite dimensional.  In general $K$ can be replaced by a [[commutative ring]] merely by replacing finite dimensional vector spaces by projective $K$-[[modules]] of finite type, but we will not do this here.

+-- {: .num_defn }
###### Definition

A **homotopy quantum field theory** is  a [[symmetric monoidal functor]] from $\mathbf{HCobord}(n,B)$  to the category, [[Vect]], of finite dimensional vector spaces over the [[field]] $K$.

=--

This definiting unwinds to the following structure in components

+-- {: .num_defn }
###### Definition

 A **$(n + 1)$-dimensional homotopy quantum field theory**, $\tau$, with background $B$  assigns

* to any $n$-dimensional $B$-manifold, $(X,g)$, a vector space, $\tau{(X,g)}$,

* to any $B$-isomorphism, $\phi : (X, g) \to ( Y, h)$, of $n$-dimensional $B$-manifolds, a $K$-linear isomorphism $\tau(\phi)  : \tau{(X, g)} \to \tau{( Y, h)}$,

 and

* to any $B$-cobordism, $(W,F) : (X_0,g_0) \to (X_1,g_1)$, a $K$-linear transformation, $\tau(W) : \tau{(X_0,g_0)} \to \tau{(X_1,g_1)}$.

These assignments are to satisfy the following axioms:

1. $\tau$ is functorial in $\mathbf{Man}(n,B)$, i.e., for two $B$-isomorphisms, $\psi: (X, g) \to ( Y, h)$ and $\phi : ( Y, h) \to (P,j)$, we have $\tau(\phi\psi) = \tau(\phi)\tau(\psi),$
and if $1_{(X,g)}$ is the identity $B$-isomorphism on $(X,g)$, then $\tau(1_{(X,g)}) = 1_{\tau{(X,g)}}$

 1.  There are natural isomorphisms
$$c_{(X,g),(Y,h)} : \tau((X,g)\amalg (Y,h)) \cong \tau(X,g)\otimes \tau(Y,h),$$
and an isomorphism, $u : \tau(\emptyset) \cong K$, that satisfy the usual axioms for a [[symmetric monoidal functor]].

1.  For $B$-cobordisms, $(W,F) : (X,g) \to (Y,h)$ and $(V,G): (Y^\prime, h^\prime) \to (P,j)$ glued along a $B$-isomorphism $\psi :(Y,h) \to (Y^\prime,h^\prime)$, we have
$\tau((W,F)\amalg_\psi (V,G))= \tau(V,G)\tau(\psi)\tau(W,F).$

1.  For the identity $B$-cobordism, $1_{(X,g)} = (I\times X, 1_g)$, we have
$\tau( 1_{(X,g)}) = 1_{\tau(X,g)}.$

1.  For $B$-cobordisms $(W,F) : (X,g) \to (Y,h)$ and $(V,G) : (X^\prime,g^\prime) \to (Y^\prime,h^\prime)$ and $(P,J): \emptyset \to \emptyset$, some fairly obvious diagrams are commutative.

=--

+-- {: .num_remark }
###### Remark

These axioms are slightly different from those given in the original paper of Turaev in 1999.  The really significant difference is in axiom 4, which is weaker than as originally formulated, where any $B$-cobordism structure on $I \times X$ was considered as trivial.  The effect of this change is important as it is now the case that the HQFT is determined by the $(n+1)$-type of $B$, cf. ([Rodrigues 03](#Rodrigues03)).

With the revised version of the axioms, it becomes possible to attempt to classify HQFTs with a given $n$ and $B$. Turaev did this in the original paper with $n = 2$ and $B$ an [[Eilenberg-MacLane space]], $K(G,1)$.  The results of Brightwell and Turner essentially gave the solution for $B$ a $K(A,2)$.

=--

## Examples

### 1+1 dimensional HQFTs with background a $K(G,1)$

If we look at the case $n= 1$ and with background an [[Eilenberg-Mac Lane space]] $K(G,1)$ for a [[discrete group]] $G$, then HQFTs correspond to [[crossed G-algebra|crossed G-algebras]], in much the same way that commutative [[Frobenius algebras]] correspond to [[2d TQFT]]s.  There the correspondence is given by a [[2d TQFT]], $Z$, corresponds to the [[Frobenius algebra]], $Z(S^1)$. This is because the circle $S^1$ is a [[Frobenius algebra]], sometimes called a  Frobenius object, in the category $Bord_2$ of 2d-[[cobordism]]s between 1-manifolds.

In the case of HQFTs, the role of the circle is replaced by the family of  circles with characteristic maps to $B$.  Each one gives, combinatorially, a circle together with a labelling of the boundary by an element of $G$. (It does not seem to be known how to get a $G$-graded version of an abstract Frobenius object that will correspond to this situation, although this is probably not too hard to do.)

### Equivariant

In ([Moore-Segal 06](#MooreSegal06)) are discussed $G$-[[equivariant TFT]]s and it is shown that they naturally correspond to a simple case of Turaev's HQFTs.  They relate (1+1) equivariant TFTs to Turaev's [[crossed G-algebras]] (which they call Turaev algebras).

## History

The original definition is due to [[Graeme Segal]], who introduced them under the term _[[elliptic objects]]_.
Specifically, in [Segal 88](#Segal88), we read in §6:

> 6\. Speculation about the definition of elliptic cohomology

> For any space $X$ let $\mathcal{P}_X$ be the category
whose objects are the points of $X$ and whose morphisms from $x_0$ to $x_1$ are the paths in $X$ from $x_0$ to $x_1$,
two such paths being identified if they differ only by reparametrization.
A functor from $\mathcal{P}_X$ to finite dimensional vector spaces is essentially the same thing as a vector bundle on $X$ with a connection.
(The functor must be continuous in a suitable sense.)
It is well known how K-theory is constructed from such objects.

> I have described elsewhere [23] a category $\mathcal{C}$
whose objects are all compact oriented one-dimensional manifolds,
and whose morphisms from $S_0$ to $S_1$ are pairs $(\Sigma,\alpha)$,
where $\Sigma$ is a Riemann surface with boundary $\partial\Sigma$,
and $\alpha$ is an isomorphism between $\partial X$ and $S_1-S_0$.
Two pairs $(\Sigma,\alpha)$, $(\Sigma',\alpha')$ are identified if they are isomorphic.
For any space $X$ one can now define a category $\mathcal{C}_X$.
Its objects are pairs $(S,s)$, where $S$ is an object of $\mathcal{C}$ and $s \colon S \to X$ is a map.
Its morphisms from $(S_0,s_0)$ to $(S_1,s_1)$ are triples $(\Sigma,\alpha,\sigma)$,
where $(\Sigma,\alpha)\colon S_0 \to S_1$ is a morphism in $\mathcal{C}$,
and $\sigma\colon\Sigma\to X$ is a map compatible with $(s_O,s_1)$.
The category $\mathcal{C}_X$ is a natural analogue of the category $\mathcal{P}_X$ which gives rise to vector bundles.

> It is appropriate to consider functors from $\mathcal{C}$ to the category $\mathcal{V}$ of topological vector spaces and trace-class maps.
If such a functor $E$ is holomorphic in the natural sense then $E(S^1)$ is a positive energy representation of $Diff(S^1)$ of finite type.
More precisely, as is familiar in the representation theory of $Diff(S^1)$,
one must consider projective representations of $\mathcal{C}$ of some definite positive integral level $k$.
Imposing a further condition — the contraction condition below — on the functor $E$ ensures that the character of $E(S^1)$ is a modular form of weight $k$.

> Now let us define an elliptic object of level $k$ on $X$
as a projective functor $E\colon\mathcal{C}_X\to \mathcal{V}$ of level $k$ which is holomorphic and satisfies the contraction condition.
Such an object consists of an infinite dimensional vector bundle on the loop space $\mathcal{L}X$, equivariant under $Diff(S^1)$,
together with some additional data amounting to a kind of connection.
The primary example is the spin bundle of $\mathcal{L}X$, which is defined when $X$ is a spin manifold with $p_1=0$.

> I have nothing precise to say about elliptic objects,
but it seems to me quite likely that the objects of each level lead to an interesting cohomology theory,
and that the theories for different levels are related by “Bott maps”.
That would fit in well with Theorem (5.3), for just as elements of $K(B G)$ are
elated to flat bundles, so elements of $Ell^*(B G)$ seem to be related to flat elliptic objects,
i.e. ones such that the operator associated to $(\Sigma,\alpha,\sigma)$ depends on $\sigma$ only up to homotopy, and is therefore a homomorphism $\pi_1(\Sigma)\to G$.

> I should mention that the category $\mathcal{C}$ can be modified by equipping the Riemann surfaces $\Sigma$ with chosen spin structures.
That is certainly needed to obtain genuine elliptic cohomology.

> Finally I return to the “contraction property”.
This is motivated by the path-integral point of view.
If a surface $\Sigma$ is a morphism from $S$ to itself
then the trace of the operator $E(\Sigma)\colon E(S)\to E(S)$ associated to $\Sigma$
must depend only on the _closed_ surface $\check\Sigma$ obtained by attaching the two boundary pieces of $\Sigma$ to each other.
Thus if $\Sigma_\tau$ is the annulus $\{z \in \mathbf{C} \mid |e^{i\tau}|\le |z|\le 1\}$ then $\check\Sigma_\tau\cong \check\Sigma_{\tau'}$, when $\tau'=-1/\tau$,
and therefore the trace of $E(\Sigma_\tau)$ is invariant under $\tau\mapsto -1/\tau$.

> Brylinski [9] has proposed a similar approach to elliptic cohomology.

> Postscript.
After giving this talk, I learnt of the work [29] 1 which gives a good account of the Dirac operator on loop space from the path integral point of view.

##  References

The original definition is due to [[Graeme Segal]] (who introduced them under the name (flat) _[[elliptic objects]]_), see §6 of

* {#Segal88} [[Graeme Segal]], _Elliptic cohomology (after Landweber–Stong, Ochanine, Witten, and others)_, Séminaire Bourbaki, 40e année, 1987–88, No. 695 ([numdam:SB_1987-1988__30__187_0](http://www.numdam.org/item/?id=SB_1987-1988__30__187_0))

These ideas were further developed in

* {#FreedQuinn1991} [[Daniel S. Freed]], [[Frank Quinn]], _Chern-Simons Theory with Finite Gauge Group_, [arXiv:hep-th/9111004](https://arxiv.org/abs/hep-th/9111004).

* {#Quinn95} [[Frank Quinn]], *Lectures on axiomatic topological quantum field theory*, in [[Dan Freed]], [[Karen Uhlenbeck]] (eds.) *Geometry and Quantum Field Theory* **1** (1995) [[doi:10.1090/pcms/001](https://doi.org/10.1090/pcms/001)]

The theory of HQFTs was developed in

*  {#Turaev99} [[Vladimir Turaev]], _Homotopy field theory in dimension 2 and group-algebras_ ([arXiv:math.QA/9910010](http://arxiv.org/abs/math.QA/9910010))

* {#Turaev00} [[Vladimir Turaev]], _Homotopy field theory in dimension 3 and crossed group-categories_ ([arXiv:math.GT/0005291](http://arxiv.org/abs/math.GT/0005291)).

* [[Vladimir Turaev]], _Homotopy Quantum Field Theory_, Tracts in Mathematics 10, (with Appendices by Michael Muger and Alexis Vurelizier), European Mathematical Society, June 2010.

* {#BrightwellTurner00} M. Brightwell and P. Turner, _Representations of the homotopy surface category of a simply connected space_, J. Knot Theory and its Ramifications, 9 (2000), 855--864.

* {#Rodrigues03} G. Rodrigues, _Homotopy Quantum Field Theories and the Homotopy Cobordism Category  in Dimension 1 + 1_, J. Knot Theory and its Ramifications, 12 (2003) 287--317 ([arXiv:math.QA/0105018](http://arxiv.org/abs/math.QA/0105018)).

* [[T. Porter]] and [[Vladimir Turaev|V. Turaev]], _Formal Homotopy Quantum Field Theories, I: Formal Maps and Crossed $C$-algebras_, Journal of Homotopy and Related Structures 3(1), 2008, 113--159. ([arXiv:math.QA/0512032](http://arxiv.org/abs/math.QA/0512032)).

* [[Tim Porter]], _Formal Homotopy Quantum Field Theories II: Simplicial Formal Maps_, Cont. Math. 431, p. 375 - 404 (Streetfest volume: Categories in Algebra, Geometry and Mathematical Physics - edited by A. Davydov, M. Batanin, and M. Johnson, S. Lack, and A. Neeman) ([arXiv:math.QA/0512034](http://arxiv.org/abs/math.QA/0512034))

A treatment of HQFTs that includes some details of the links with TQFTs is given in [[HQFT-XMenagerie.pdf|HQFTs meet the Menagerie:file]], which is a set of notes prepared by [[Tim Porter]] for a school and workshop in Lisbon, Feb. 2011.

Related ideas are discussed in

* {#MooreSegal06} [[Greg Moore]], [[Graeme Segal]], _D-branes and K-theory in 2D topological field theory_ ([arXiv:hep-th/0609042](http://arxiv.org/abs/hep-th/0609042))

See also:

* [[João Faria Martins]], [[Timothy Porter]], *A categorification of Quinn's finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids* [[arXiv:2301.02491](https://arxiv.org/abs/2301.02491)]

* [[Christoph Schweigert]], [[Lukas Woike]], *Extended Homotopy Quantum Field Theories and their Orbifoldization*, Journal of Pure and Applied Algebra **224** 4 (2020) 106213 [[arXiv:1802.08512](https://arxiv.org/abs/1802.08512), doi:10.1016/j.jpaa.2019.106213]

Understanding [[higher parallel transport]] of [[circle n-bundles with connection]] as an [[extended functorial field theory|extended]] [[homotopy field theory]]:

* [[Lukas Müller]], [[Lukas Woike]], *Parallel Transport of Higher Flat Gerbes as an Extended Homotopy Quantum Field Theory*,  J. Homotopy Relat. Struct. **15** (2020) 113–142  [[arXiv:1802.10455](https://arxiv.org/abs/1802.10455), [doi:10.1007/s40062-019-00242-3](https://doi.org/10.1007/s40062-019-00242-3)]

[[!redirects HQFTs]]

[[!redirects homotopy quantum field theory]]
[[!redirects homotopy quantum field theories]]

[[!redirects homotopy field theory]]
[[!redirects homotopy field theories]]


HR manifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### Basic definition

We state the definition below in Def. \ref{HRManifold}. First we need the following preliminaries:

Denote by $\mathbf{D}$ the duplex (sometimes called _paracomplex_ or _hyperbolic_) numbers, which is the [[associative algebra]] over the [[real numbers]] $\mathbf{R}$ generated by the elements $1, \mathbf{k}$ s.t. $\mathbf{k}^2 = 1 $, in other words the real [[Clifford algebra]] $C\ell _{1, 0} (\mathbf{R})$. A [[PDE]] theory analogous to complex holomorphy may be developed based on this algebra; for a function $\psi = (\psi_1 , \psi_2) : \mathbf{D} \rightarrow \mathbf{D}$ (under an identification of $\mathbf{D}$ with $\mathbf{R}^2$), paracomplex linearity of $d \psi$ means the real components of $\psi$ must satisfy the equations $\partial_1 \psi_2 = \partial_2 \psi_1$ and $\partial_1 \psi_1 = \partial_2 \psi_2$. These are the hyperbolic analogue of the [[Cauchy-Riemann equations]], although clearly not defining an [[elliptic differntial equation|elliptic system]] since the components of $\psi$ therefore satisfy the [[wave equations]] $\Box \psi_i =0$. In the context of [[differential geometry]] over $\mathbf{D}$, such functions are sometimes called _paraholomorphic_.

As with [[CR geometry]], one can study real hypersurfaces of manifolds carrying such hyperbolic structure (discussed [below](#hypmanifold)):

+-- {: .num_defn #HRManifold}
###### Definition
**(HR manifold)**

An _HR manifold_ (for "hyperbolic-real") is a [[differentiable manifold]] $M$ together with a [[sub-bundle]] $H$ of the hyperbolified [[tangent bundle]], $H \subset TM \otimes_\mathbf{R} \mathbf{D}$ such that $[H, H ] \subset H$ and $H \cap H^{\dagger} =\{ 0 \} $, where $\dagger$ is the bundle [[involution]] s.t. $\mathbf{k} \mapsto - \mathbf{k}$.

=--

### As $G$-structure
  {#hypmanifold}

[[G-structures]] of this type only exist on even-dimensional differentiable manifolds, and have been known since the classical contributions of Libermann. Explicitly, an _almost-hyperbolic_ structure on a real $2n$-manifold $M$ is determined by a [[reduction of the structure group]] $\text{GL}(n, \mathbf{D}) \hookrightarrow \text{GL}(2n, \mathbf{R})$, defining a bundle [[automorphism]] $K \in \text{End}(TM)$ s.t. $K^2 = \text{id}_{TM}$. Locally this means that $K$, when integrable, is of the form:
$$
\left( \begin{matrix}
0 & I_n \\
I_n & 0
\end{matrix} \right)
$$
on fibers, so that the [[transition functions]] of $M$ satisfy the [[wave equations]] just discussed. One can also give various integrability conditions of $K$, although as a Dirac structure the simplest to state is the vanishing of the [[Nijenhuis tensor]] $N_K (X, Y) = [KX, KY] + [X, Y] - K ([KX, Y] + [X, KY]) $, a sign away from its complex analogue.

## Examples

(...)

## Other Properties

(...)

## Other Clifford-type Hypersurfaces

* [[BR manifold]]

* [[CR manifold]]

## References

The classical articles are:

* P. Libermann, _Sur le probleme d’equivalence de certaines structures infinitesimales_, Ann. Mat. Pura Appl., 36 (1954), 27-120.

* P. Libermann, _Sur les structures presque paracomplexes_, C.R. Acad. Sci. Paris, 234 (1952), 2517-2519.

A convenient modern survey appears in::

* V. Cruceanu, P. Fortuny and P. M. Gadea, _A Survey on Paracomplex Geometry_ , Rocky Mountain J. Math. Volume 26, Number 1 (1996), 83-115.

And a more recent article done in the style of [[generalized complex geometry]] is:

* Aïssa Wade, _Dirac structures and paracomplex manifolds_, C. R. Acad. Sci. Paris, Ser. I 338 (2004) 889–894.

[[!redirects HR manifolds]]

[[!redirects HR-manifold]]
[[!redirects HR-manifolds]]

[[!redirects HR geometry]]
[[!redirects HR geometries]]


Hrvoje Nikolić

* [webpage](http://www.irb.hr/eng/Research/Divisions/Division-of-Theoretical-Physics/Particle-Physics-and-Cosmology-Group/Hrvoje-Nikolic)

## related $n$Lab entries

* [[vacuum stability]]

* [[Casimir effect]]

category: people

[[!redirects Hrvoje Nikolic]]


Hsian-Hua Tseng

* [website](https://people.math.osu.edu/tseng.109/)

## Selected writings

On [[Lie integration]] of [[Lie algebroids]] to [[Lie groupoids]] ([[smooth stacks]], cf. [[Lie's third theorem]]):

* [[Hsian-Hua Tseng]], [[Chenchang Zhu]], *Integrating Lie algebroids via stacks*, Compositio Mathematica **142** (2006) 251–270 [[doi:10.1112/S0010437X05001752](https://doi.org/10.1112/S0010437X05001752)]

category: people