Harvey Reall

* [webpage](http://www.damtp.cam.ac.uk/people/h.s.reall/)

## Selected writings

On [[D=5 supergravity]]:

* {#GGHPR02} [[Jerome Gauntlett]], [[Jan Gutowski]], [[Christopher Hull]], Stathis Pakis, [[Harvey Reall]], _All supersymmetric solutions of minimal supergravity in five dimensions_, Class. Quant. Grav. 20 (2003) 4587-4634 [[arXiv:hep-th/0209114](http://arxiv.org/abs/hep-th/0209114), [doi:10.1088/0264-9381/20/21/005](https://doi.org/10.1088/0264-9381/20/21/005)]

Finding [[black rings]] in [[D=5 supergravity]]:

* [[Henriette Elvang]], [[Roberto Emparan]], [[David Mateos]], [[Harvey Reall]], _Supersymmetric 4D Rotating Black Holes from 5D Black Rings_, JHEP0508:042 (2005) ([arXiv:hep-th/0504125](http://arxiv.org/abs/hep-th/0504125))

On reducing these to [[black holes]]:

* {#ElvangEmparanMateosReall05} [[Henriette Elvang]], [[Roberto Emparan]], [[David Mateos]], [[Harvey Reall]], _Supersymmetric 4D Rotating Black Holes from 5D Black Rings_, JHEP0508:042,2005 ([arXiv:hep-th/0504125](http://arxiv.org/abs/hep-th/0504125))

Review of [[black rings]]:

* {#EmparanReall06} [[Roberto Emparan]], [[Harvey Reall]], _Black Rings_, Class. Quant. Grav. 23:R169, 2006 ([arXiv:hep-th/0608012](https://arxiv.org/abs/hep-th/0608012))

## Related entries

* [[black ring]]

category: people


Harvey Wolff

* [MathGenealogy entry](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=4263)

## Selected writings

On the construction of [[action monads]] as an [[adjoint functor]] from [[monoids]] to [[strong monads]] (all in the generality of [[enriched category theory]]):

* {#Wolff73} [[Harvey Wolff]], *Monads and Monoids on Symmetric Monoidal Closed Categories*, Archiv der Mathematik **24** (1973) 113–120 [[doi:10.1007/BF01228184](https://doi.org/10.1007/BF01228184)]

Given a symmetric closed monoidal category $V$, a $V$-enriched category $A$ with underlying ordinary category $A_0$ and a subcategory $\Sigma$ of $A_0$
containing the identities of $A_0$, H. Wolff defines the corresponding theory of a [[localization of an enriched category]]:

* H. Wolff, _$V$-localizations and $V$-triples_, Dissertation, University of Illinois-Urbana, 1970.

* H. Wolff, _$V$-localizations and $V$-monads_, J. Alg. __24__, 405-438, 1973, [MR310041](http://mathscinet.ams.org/mathscinet-getitem?mr=310041), doi;

* H. Wolff, _V-localizations and $V$-monads. II_, Pacific J. Math. 63 (1976), no. 2, 579–589, [MR412253](http://www.ams.org/mathscinet-getitem?mr=412253), [euclid](http://projecteuclid.org/getRecord?id=euclid.pjm/1102867417);

* H. Wolff, _$V$-localizations and $V$-Kleisli algebras_, Manuscripta Math. __16__ (1975), no. 3, 203--228, [MR382383](https://mathscinet.ams.org/mathscinet-getitem?mr=382383), [doi](https://doi.org/10.1007/BF01164425)

On [[free monads]]:

* Harvey Wolff, *Free monads and the orthogonal subcategory problem*, Journal of Pure and Applied Algebra **13** 3 (1978) 233-242 [doi:10.1016/0022-4049(78)90010-5]

category: people

[[!redirects H. Wolff]]


Haskell

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

*Haskell* is a [[type theory|typed]] [[functional programming language]]. It is named after [[Haskell Brooks Curry]].

## Properties

### Category-theoretic properties

Viewed as a syntactic framework, we can identify a subset of Haskell called $\mathbf{Hask}$ that is often used to identify concepts used in basic category theory. One considers Haskell [[type|types]] as objects of a category whose morphisms are extensionally identified Haskell functions.

With $\to$ and $\times$ etc., we can work $\mathbf{Hask}$ almost as if it were even a Cartesian closed category. A particular problem is the polymorphic term $\text{undefined}$, which is defined to be term of every type. It prevents, for example, initial objects (i.e. there's no analog of the empty set). Or, when it comes to setting up the categorical product, the projections $\pi_1$ and $\pi_2$ couldn't distinguish between $(\text{undefined},\text{undefined})$ and just $\text{undefined}$, spoiling uniqueness. A related deficit is that when it comes to passing functions as arguments, Haskell sees more than just Hask morphisms.

The standard notion of functors in Haskell are not terms of a Hask type but operate on one level above those, the kind level. If '7' is of type 'Int' and 'Int' is of kind 'U', then a functor corresponds to a map 'U $\to$ U'. Functors form a class, in the sense used in computer science, i.e. a standardly employed abstraction or interface.

The object mappings of functors in Haskell are type formers (e.g. mapping a type of integers to the type of lists of such integers) and these are often used without any reference to the arrow mappings in the code. For each declaration of a functor, the user must code its action on the arrows (a function called 'fmap'), which here are Haskell function terms. It must be noted that all algebraic laws (e.g. compatibility of functors with function concatenation in the sense of the definition of functors) are not checked or enforced by the Haskell compiler. That is to say, in this language, its written code is only checked for the arrow mapping of a user defined functor is well-typed, while the user could, in principle, declare a "'Functor'" that doesn't actually fulfill all the defining properties of a functor.
Today, there are also software modules that implement category theoretical notions more abstractly, i.e. one may set up categories in which the arrows are not necessarily Haskell functions.

Haskell is famous for its use of [[monads (in computer science)]], a subclass of functors. Here, the unit (called 'return') and co-unit must be implemented. However, stemming from the way that monads are actually used by programmers, it is standard to implement the function 'return' and another function called 'bind', with infix '>>=', which is a composite of the functor and the two natural transformations and which can be derived from the others.

### Lifted products

Expanding on the caveat above about `undefined`, the built-in products in Haskell are "lifted", they are not exactly categorical products. For example, if we define
```
loop = loop
```
then the element `loop :: ((),())` is observably different from `(loop,loop) :: ((),())`.
To see this, note that
```
(\(_,_)->()) (loop,loop)
```
terminates but
```
(\(_,_)->()) loop
```
does not terminate.

As a result, the built-in [[currying]] is not strictly speaking a bijection in Haskell. For example,
```
(uncurry . curry) (\(_,_)->()) loop
```
terminates but
```
(\(_,_)->()) loop
```
does not terminate.

It _is_ consistent to have a cartesian closed category with a recursion operator, and indeed most semantic models of recursion in a call-by-name setting do actually form cartesian closed categories. In fact Haskell does provide "unboxed tuple" types, which are more like categorical products, but these are not so widely used.

### Similar and related software

Languages similar to Haskell but refining it from plain [[type theory]] to [[dependent type theory]] include

* [[Coq]], [[Agda]] and Idris

Coq and Agda is consistent with [[Homotopy type theory]], while Idris [is not](http://cstheory.stackexchange.com/questions/27979/formalizing-homotopy-type-theory-in-idris/28013#28013).

A wiki platform based on Haskell, running texmath

* [gitit](https://github.com/jgm/gitit)

## Related concepts

* [[Scala]]

* [[domain specific embedded programming language]]

  * [[Quipper]]

[[!include proof assistants and formalization projects -- list]]

## References

### General

* landing page: [www.haskell.org](https://www.haskell.org)

* [haskellwiki](http://www.haskell.org/haskellwiki/Haskell), [haskell platform](http://www.haskell.org/platform)

* [language and library specification](http://www.haskell.org/haskellwiki/Language_and_library_specification)

Joint introduction to [[functional programming]] in general and Haskell in particular:

* [[Simon Thompson]], *Haskell: the Craft of Functional Programming*, Addison-Wesley (1996) [[webpage](https://www.haskellcraft.com/craft3e/Home.html), [pdf ](https://simonjohnthompson.github.io/craft3e/craft3e.pdf)]

Discussion of the [[category]] of Haskell [[types]] (see at _[[relation between category theory and type theory]]_ and at _[[monad (in computer science)]]_) is in

* WikiBooks, _[Haskell/Category](http://en.wikibooks.org/wiki/Haskell/Category_theory)_

History of Haskell:

* {#HHPW07} [[Paul Hudak]], [[John Hughes]], [[Simon Peyton Jones]], [[Philip Wadler]], *A history of Haskell: being lazy with class*, History of Programming Languages (2007) 12-1 - 12-55 [[doi:10.1145/1238844.1238856](https://doi.org/10.1145/1238844.1238856)]

### Mathematics in Haskell
 {#ReferencesMathematicsInHaskell}

The use of Haskell in [[mathematics]] is discussed in the following references.

* haskellwiki, _[Haskell and mathematics](http://www.haskell.org/haskellwiki/Haskell_and_mathematics)_

* category theory, in haskellwiki, [wiki](http://www.haskell.org/haskellwiki/Category_theory)

* Jan van Eijck, _[The Haskell Road to Logic, Maths and Programming](http://homepages.cwi.nl/~jve/HR)_

* Dan Piponi, ["What Category do Haskell Types and Functions Live In?"](http://blog.sigfpe.com/2009/10/what-category-do-haskell-types-and.html), October 13, 2009.

The [[Kenzo]]-program for [[constructive mathematics|constructive]] [[algebraic topology]] ([[computational topology]]) re-written in [[Haskell]]:

* [[Mitchell Riley]]: [github.com/mvr/at](https://github.com/mvr/at)


Haskell Curry

Haskell Curry (b. September 12, 1900; d. September 1, 1982) was an American mathematician known for his work in developing [[combinatory logic]]. His name appears in concepts such as [[currying]] and the [[Curry-Howard correspondence]], as well as the computer programming language [[Haskell]].

* [Wikipedia article](https://en.wikipedia.org/wiki/Haskell_Curry)

category:people

[[!redirects Haskell Brooks Curry]]


Hasse diagram

# Hasse diagrams
* table of contents
{: toc}

+-- {: .standout}
Warning: This page is tentative and may contain errors.
=--

## Idea

Given a [[finite category|locally finite]] [[partially ordered set]] $C$, its _Hasse diagram_ encodes the minimal amount of information necessary to reproduce the ordering relation.

## Definition

A **Hasse diagram** $H$ is a [[directed graph]] (or [[quiver]]) such that the [[graph|adjacency relation]] equals the [[covering relation]].

In other words, a Hasse diagram is a directed graph in which for each edge $x\to y$ there is no other path from $x$ to $y$. There are no intermediate edges.

In particular, given a [[proset]] $C$, its Hasse diagram $H(C)$ is obtained by "forgetting all composite morphisms". The proset $C$ may then be recovered as the free poset on that Hasse diagram.

More formally, there is a [[forgetful functor]]

$$H: Ord \to Hasse,$$

where $Ord$ is the category of [[preordered sets]] and $Hasse$ is the category of Hasse diagrams, that forgets composite morphisms.

The corresponding [[free functor]]

$$F:Hasse\to Ord$$

allows us to identify a Hasse diagram with each proset.

## References

* See also [[Hasse n-graph]]
* [Wikipedia](http://en.wikipedia.org/wiki/Hasse_diagram)


Hasse n-graph

**Warning: This page is tentative.**

## Idea ##

Given a finite [[n-poset]] $C$, its _Hasse n-graph_ encodes the minimal amount of information necessary to reproduce the ordering relation.

## Definition ##

Given an [[n-poset]] $C$, a **Hasse $n$-graph** $H$ is a [[directed n-graph]] whose [[Hasse quiver]] $F(H)$ is [[equivalence|equivalent]] to $C$.

+-- {: .query}
As with [[Hasse diagram]], I *think* that declaring it to be smallest in unnecessary; the fact that it is merely an $n$-graph and not a $n$-category will do this.  But this should be an $n$-quiver, yes?  ---Toby

[[Eric]]: What should be an n-quiver? I thought I'd stick the "smallest" in there because if $x\lt y$, $y\lt z$ then the morphisms $x\to y$, $y\to z$, and $x\to z$ are in the quiver, but only $x\to y$ and $y\to z$ need to be in the Hasse diagram. However, we could have a graph including $x\to z$, but this would not be a Hasse diagram. Maybe I'm confused.

_Toby_:  It would not be a Hasse diagram, but also its quiver would not be a poset, so that\'s taken care of.

However, the basic idea was wrong anyway, as shown by your counterxample [here](https://nforum.ncatlab.org/discussion/237/all-kdiagrams-commute/?Focus=1703#Comment_1703).
=--


Hasse quiver

**Warning: This page is tentative.**

# Idea #

A **Hasse $n$-quiver** is a free [[n-poset]] on a [[Hasse n-graph]] ...

A Hasse $1$-quiver is, as usual, referred to simply as a **Hasse quiver**.


Hasse-Schmidt derivation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of **Hasse-Schmidt derivation**, or **higher deriviation** is an extension of the notion of [[derivation]].

## Definition

Let $R$ be a [[commutative rig]] and $S$ a $R$-[[module]]. A Hasse-Schmidt derivation is a family of [[linear maps]] $(D^{n}:S \rightarrow S)_{n \ge 0}$ such that:

* $D^{0} = Id$
* $D^{n}(ab) = \underset{0 \le k \le n}{\sum}D^{k}(a)D^{n-k}(b)$

Note that $D^{1}$ is then a [[derivation]] in the usual sense.

## References

The notion is explicitely defined in:

* [[Morris Weifeld]]: *Purely Inseparable Extensions and Higher Derivations*, Transactions of the American Mathematical Society, **116** (1965) 435-449 [pdf](https://www.ams.org/journals/tran/1965-116-00/S0002-9947-1965-0191895-1/S0002-9947-1965-0191895-1.pdf)

where it is said that it is due to the paper:

* [[Helmut Hasse]]: *Noch eine Begründung der Theorie der höheren Differrentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten. (Nach einer brieflichen Mitteilung von [[Friedrich Karl Schmidt|F. K. Schmidt]] in Jena.)*, Reine Angew. Math.  **177** (1937) 215-223 [[paper](https://gdz.sub.uni-goettingen.de/id/PPN243919689_0177?tify={%22pages%22:[219],%22view%22:%22export%22})]

which defines the [[Hasse-Schmidt derivative]].


Hasse-Schmidt derivative

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In a [[field]] of [[positive characteristic]], the usual [[derivative]] of [[polynomials]] has bad properties. Let $\mathbb{K}$ be such a field of [[characteristic]] $p \gt 0$. Consider the [[polynomial algebra]] $\mathbb{K}[X]$. The usual derivative of polynomials in one indeterminate defines a [[derivation]] $\mathbb{K}[X] \rightarrow \mathbb{K}[X]$ which associates $P'$ to $P$.

We then have that $(X^{p})'=p.X^{p-1}=0$ but $X^{p} \neq 0$ because $1^{p} = 1 \neq 0$. We thus lack the property that $P' = 0$ iff $P$ is a constant.

The notion of Hasse-Schmidt derivative comes as a replacement of the usual derivative and enjoys better properties.

It seems to be very linked to [[differential linear logic]] and to the notion of [[graded bimonoid]].

## Definition for one indeterminate

Suppose that $R$ is a [[commutative rig]]. For every polynomial $P \in R[X]$ and $n \ge 0$, we define the $n-th$ Hasse-Schmidt derivative $P^{(n)}$ of $P$ by:

* $P^{(0)} = P$

and if $n \ge 1$:

* $(X^{n+p})^{(n)} = \left[\binom{n+p}{n}.1_{R}\right] X^{p}$ if $p \ge 0$
* $(X^{k})^{(n)} = 0$ if $n \gt k$
* $(aP+bQ)^{(n)} = aP^{(n)} + bQ^{(n)}$

## Definition for several indeterminates

Suppose that $R$ is a [[commutative rig]]. We look at the polynomials in $R[X_{1},...,X_{q}]$. For every multiset $Y_{1}...Y_{n} \in \{X_{1},...,X_{q}\}$ and $P \in R[X_{1},...,X_{q}]$ we define the derivative $D_{Y_{1}...Y_{n}}(P)$. Note that we treat in the same way all the higher-order derivatives and define them directly and not by induction. It is in this way directly clear that the order of the variables with respect to we derivate doesn't matter.

Suppose that $P$ is a monic monomial, that is a multiset of variables in $\{X_{1},...,X_{q}\}$.

* If $Y_{1}...Y_{n}$ doesn't divide $P$, then:
$$
D_{Y_{1}...Y_{n}}(P) = 0
$$

* If $Y_{1}...Y_{n}$ divides $P$, then:
$$
D_{Y_{1}...Y_{n}}(P) = \left[\underset{\substack{1 \le i \le q \\ X_{i} | Y_{1}...Y_{n}}}{\prod}\binom{nbr\;of\;time\;X_{i}\;in\;P}{nbr\;of\;time\;X_{i}\;in\;Y_{1}...Y_{n}}.1_{R}\right]\frac{P}{Y_{1}...Y_{n}}
$$

We then prolongate by linearity:

* $D_{Y_{1}...Y_{n}}(aP+bQ) = aD_{Y_{1}...Y_{n}}(P) + bD_{Y_{1}...Y_{n}}(Q)$

There is a very [[combinatorial]] interpretation which makes the link with the notion of [[graded bimonoid]]: if $P$ is a monic monomial, the derivative with respect to $Y_{1}...Y_{n}$ counts the number of ways to extract $Y_{1}...Y_{n}$ from $P$ and then multiplies it by $P$ where $Y_{1}...Y_{n}$ has been extracted. All the instances of a variable $Y_{i}$ in $P$ must be considered as different entities in this counting.

For example there is $\binom{3}{2}\binom{4}{1}$ ways to extract $X^{2}Y$ from $X^{3}Y^{4}$, we have $\binom{3}{2}$ choices of two instance of $X$ in $X^{3}$ and $\binom{4}{1}$ choices of two instances of $Y$ in $Y^{4}$, thus:
$$
D_{X^{2}Y}(X^{3}Y^{4}) = \binom{3}{2}\binom{4}{1}XY^{3} = 3.4.XY^{3} = 12XY^{3}
$$

## Related notions

* [[Hasse-Schmidt derivation]]

## References

The notion was introduced in:

* [[Helmut Hasse]]: *Noch eine Begründung der Theorie der höheren Differrentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten. (Nach einer brieflichen Mitteilung von [[Friedrich Karl Schmidt|F. K. Schmidt]] in Jena.)*, Reine Angew. Math.  **177** (1937) 215-223 [[paper](https://gdz.sub.uni-goettingen.de/id/PPN243919689_0177?tify={%22pages%22:[219],%22view%22:%22export%22})]

The definition and basic properties are reviewed on this blog page:

* _Hasse derivative_, Felix' Math Place (2009), [url](https://math.fontein.de/2009/08/12/the-hasse-derivative/)


Hasse-Weil L-function

#Contents#
* table of contents
{:toc}

## Idea

The _Hasse-Weil zeta function_ is a [[zeta function]]/[[L-function]] associated with [[algebraic varieties]] over a [[number field]] $K$.

Specifically on the [[spectrum of a commutative ring|spectrum]] $Spec(\mathcal{O}_K)$ of the [[ring of integers]] of $K$ it reduces to the [[Dedekind zeta function]] of $K$.

## Properties

### Special values

The [[Birch and Swinnerton-Dyer conjecture]] describes the first non-vanishing derivative of the Hasse-Weil zeta function at the [[special values of L-functions|special value]] $s= 1$ in analogy with the [[class number formula]].

### Function field analogy

[[!include function field analogy -- table]]

## Related concepts

* [[motivic L-function]]

## References

* Wikipedia, _[Hasse-Weil L-function](http://en.wikipedia.org/wiki/Hasse–Weil_L-function)_

* Wikipedia, _[Arithmetic zeta function](http://en.wikipedia.org/wiki/Arithmetic_zeta_function)_

[[!redirects Hasse-Weil L-functions]]

[[!redirects Hasse-Weil zeta function]]


Hassler Whitney

* [Wikipedia entry](http://en.wikipedia.org/wiki/Hassler_Whitney)

## Selected writings

Introducing the notion of *[[matroids]]*:

* [[Hassler Whitney]], _On the abstract properties of linear dependence_, American Journal of Mathematics (The Johns Hopkins University Press) **57** 3  (1935) 509-533 [[jstor:2371182](http://jstor.org/stable/2371182), [MR1507091](http://www.ams.org/mathscinet-getitem?mr=1507091)]

Introducing the term "[[cohomology]]":

* {#Whitney37} [[Hassler Whitney]], _On matrices of integers and combinatorial topology_.  Duke Mathematical Journal 3:1 (1937), 35–45 ([doi:10.1215/S0012-7094-37-00304-1](https://projecteuclid.org/journals/duke-mathematical-journal/volume-3/issue-1/On-matrices-of-integers-and-combinatorial-topology/10.1215/S0012-7094-37-00304-1.short))

See also:

* [[Hassler Whitney]], _On products in a complex_, Annals of Math. **39** (1938) 397–432 [[doi:10.2307/1968795](https://doi.org/10.2307/1968795)]

Early discussion of [[sphere bundles]] and introducing the general notion of [[fiber bundles]]:

* [[Hassler Whitney]], *On the theory of sphere-bundles*, Proceedings of the National Academy of Sciences **26** 2 (1940) 148-153 [[doi:10.1073/pnas.26.2.148](https://doi.org/10.1073/pnas.26.2.148)]

## Related entries

* [[Stiefel-Whitney class]], [[integral Stiefel-Whitney class]]

* [[Whitney sum]]

* [[Whitney embedding theorem]]

* [[Whitney extension theorem]]

* [[Alexander-Whitney map]]

category: people

[[!redirects Whitney]]


Hausdorff dimension

#Contents#
* table of contents
{:toc}

## Idea

Hausdorff dimension is a method of measuring the [[dimension]] of a [[metric space]].  It is always a non-negative [[real number]], but it need not be an [[integer]]; one way to define a [[fractal]] is a metric space with a fractional (non-integral) Hausdorff dimension. Hence Hausdorff dimension is an example of _[[fractal dimension]]_.

The Hausdorff dimension of the [[cartesian space]] $\mathbb{R}^n$ (or any [[inhabited set|inhabited]] [[open subset]] thereof) is $n$.  The Hausdorff dimension of a self-similar fractal which consists of $n$ copies of itself reduced in size by a factor of $m$ is $\log_m n$.

In general, Hausdorff dimension may be defined using [[Hausdorff measure]].

## References

* Wikipedia, [Hausdorff dimension](http://en.wikipedia.org/wiki/Hausdorff_dimension)

[[!redirects Hausdorff dimensions]]


Hausdorff implies sober

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

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

Assuming [[excluded middle]], then:

Every [[Hausdorff topological space]] is a [[sober topological space]].

More specifically, in a Hausdorff topological space
the [[irreducible closed subspaces]]
are precisely the [[singleton]] [[subspaces]].

=--

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

The second statement clearly implies the first. To see the second
statement, suppose that $F$ is an irreducible closed subspace which
contained two distinct points $x \neq y$. Then by the Hausdorff property
there are disjoint neighbourhoods $U_x, U_y$, and hence it would follow that the relative [[complements]] $F \setminus U_x$ and $F \setminus U_y$ were distinct proper closed subsets of
$F$ with

$$
  F = (F \setminus U_x) \cup (F \setminus U_y)
$$

in contradiction to the assumption that $F$ is irreducible.

This [[proof by contradiction|proves by contradiction]] that every irreducible closed subset is a singleton.
Conversely, generally the [[topological closure]] of every singleton is irreducible closed.

=--

## Strictness of implication

There are many examples of sober spaces which are not Hausdorff. For example, the spectrum of a ring which is not zero-dimensional is sober but not Hausdorff.

Any Hausdorff space is not only [[sober]], but also $T_1$. However, even the converse to this fails. [For example](https://en.wikipedia.org/wiki/Sober_space), let $X$ be the real line with a new point $p$ added, topologized such that any open set in the real line is open, and any cofinite set containing $p$ is open. Then $X$ is sober and $T_1$ but not Hausdorff.

## Related entries

* [[schemes are sober]]

* [[compact Hausdorff spaces are normal]]

* [[a CW-complex is a Hausdorff space]]

* [[continuous images of compact spaces are compact]]

* [[maps from compact spaces to Hausdorff spaces are closed and proper]]

* [[closed subsets of compact spaces are compact]]

* [[compact subspaces of Hausdorff spaces are closed]]

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]

## References

* Topospaces, _[Hausdorff implies sober](https://topospaces.subwiki.org/wiki/Hausdorff_implies_sober)_

[[!redirects Hausdorff spaces are sober]]

[[!redirects every Hausdorff space is a sober space]]
[[!redirects every Hausdorff topological space is a sober topological space]]


hausdorff locally convex space > history

< [[hausdorff locally convex space]]


Hausdorff maximal principle

[[!redirects Hausdorff maximality principle]]

The __Hausdorff maximal principle__ is a version of [[Zorn's lemma]], equivalent to the usual version and thus (given [[excluded middle]]) equivalent to the [[axiom of choice]].

## Statement and proofs

Given a [[partial order|poset]] (or [[preorder|proset]]) $S$, let a _chain_ in $S$ be a [[subset]] $A$ of $S$ which, as a sub-proset, is [[total order|totally ordered]].  A chain $A$ is _maximal_ (as a chain) if the only chain that $A$ is contained in is $A$ itself.

+-- {: .un_theorem}
###### Theorem (Hausdorff maximal principle)

Every chain in a proset is contained in a maximal chain.
=--

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

We will use [[Zorn's lemma]]. Let $P$ be a proset and let $C \subseteq P$ be a chain. Consider the collection $\mathcal{C}$ of chains in $P$ that contain $C$, ordered by inclusion. If $\{C_\alpha\}_{\alpha \in A} \subseteq \mathcal{C}$ is a family totally ordered by inclusion, then the union $\bigcup_\alpha C_\alpha$, with the order coming from $P$, is also totally ordered: any two elements $x \in C_\alpha, y \in C_\beta$ are comparable in $max(C_\alpha, C_\beta)$. The hypotheses for Zorn's lemma therefore obtain on $\mathcal{C}$, and we conclude that $\mathcal{C}$ has a maximal element, which is clearly maximal in the collection of all chains.

=--

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

Conversely, suppose that the Hausdorff maximal principle holds; we will prove Zorn's lemma. Suppose given a poset (or preorder) $P$ such that every chain in $P$ has an upper bound. Since $\empty$ is a chain, the Hausdorff maximal principle implies that $P$ contains a maximal chain $C$; let $x$ be an upper bound of $C$. Then $x$ is maximal: if $x \leq y$, then $C = C \cup \{y\}$ by maximality of $C$; therefore $y \in C$ and hence $y \leq x$ since $x$ is an upper bound of $C$.

=--


Hausdorff metric

# The Hausdorff metric
* table of contents
{: toc}

## Idea

The *Hausdorff metric* is a [[metric space|metric]] on the [[power set]] of a given [[metric space]].

## Definition

Let $A$ be a [[metric space]], regarded as a [[category]] [[enriched category|enriched]] over $V = [0,\infty]$ (a [[Lawvere metric space]]).  The enriched [[functor category]] $[A,V]$ is, concretely, the set of [[short maps]] $A \to [0,\infty]$ with the [[supremum metric]], and the contravariant [[Yoneda embedding]] $A^{op} \to [A,V]$ sends $a$ to $d(a,-)$.

Now, for any [[subset]] $X \subseteq A$, each point $x \in X$ gives rise to the [[representable functor]] $d(x,-)$, and we can define the functor $d(X,-)\colon A \to V$ to be the [[coproduct]] of these representables over all $x \in X$.  Concretely, this means
$$ d(X,a)=inf_{x\in X} d(x,a) .$$
Note that if $X$ is [[closed subset|closed]], then it can be recovered from $d(X,-)$ as the set of points $x$ such that $d(X,x)=0$.  If $X$ is not closed, then in this way we recover its [[topological closure|closure]].

Finally, since $[A,V]$ is also a Lawvere metric space, we obtain an induced metric on the set of subspaces of $A$:
$$ d(X,Y) = d(d(X,-), d(Y,-)) .$$
This metric is not symmetric (so a [[quasimetric]]); its [[symmetrization of a metric|symmetrization]] is the **Hausdorff metric**.  Equivalently, we could start out by considering instead the functor category $[A, V_{sym}]$ where $V_{sym}$ is the symmetrization of $V = [0,\infty]$.

## References

* _Taking categories seriously_, Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1–24. ([pdf](http://www.emis.de/journals/TAC/reprints/articles/8/tr8.pdf))


Hausdorff series

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Magnus algebras and Magnus group

Let $X$ be a set of symbols. The [[free construction|free]] [[associative algebra|associative k-algebra]] $k\langle X \rangle$ on the [[set]] $X$ where $k$ is a commutative unital [[ring]], will be denoted $A(X)$. It is clearly [[graded algebra|graded]] (by the length of the word) as $A(X) = \oplus_n A^n(X)$. The product of $k$-modules $\hat{A}(X) = \prod_n A^n(X)$ has a natural multiplication

$$
(ab)_n = \sum_{i = 0}^n a_i b_{n-i}
$$

where $a = (a_n)_n$ and $b = (b_n)_n$. Furthermore, $\hat{A}(X)$ has the topology of the product of [[discrete topological space]]s. This makes $\hat{A}(X)$ a [[Hausdorff space|Hausdorff topological]] algebra, where
the ground field is considered discrete and $A(X)$ is dense in $\hat{A}(X)$. We say that $\hat{A}(X)$ is the __Magnus algebra__ with coefficients in $k$. (Bourbaki-Lie gr. II.5).

An element in $\hat{A}(X)$ is invertible (under multiplication) iff it's free term is invertible in $k$.

The __Magnus group__ is the (multiplicative) subgroup of the Magnus algebra consisting of all elements in the Magnus algebra with free
term $1$.

### Hausdorff group and Hausdorff series

The [[free Lie algebra]] $L(X)$ naturally embeds in (the Lie algebra corresponding to the associative algebra) $A(X)\hookrightarrow \hat{A}(X)$; one defines $\hat{L}(X)$ as the closure of $L(X)$ in $\hat{A}(X)$. The exponential series and the makes sense in $\hat{A}(X)$; when restricted to $\hat{L}(X)$ it gives a bijection between $\hat{L}(X)$ and a closed subgroup of the Magnus group which is sometimes called the Hausdorff group $exp(\hat{L}(X))$.

__Hausdorff series__ $H(U,V)$ is an element $log(exp(U)exp(V))$ in $\hat{L}(\{U,V\})$.

The formula $exp(X)exp(Y) = (exp(Y)exp(X))^{-1}$ implies the basic symmetry of the Hausdorff series: $H(-Y,-X) = -H(X,Y)$.

The specializations of the Hausdorff series
in Lie algebras which are not necessarily free are known as
the __Baker-Campbell-Hausdorff__ series and play the role in the corresponding BCH formula $exp(U)exp(V) = exp(H(U,V))$.

The BCH formula can be written in many ways, the most important which belong to Dynkin. The part which is linear in one of the variables involves [[Bernoulli number]]s.

There is a decomposition $H(X,Y) = \sum_{N=0}^\infty H_N(X,Y)$ where
__Dynkin's Lie polynomials__ $H_N = H_N(X,Y)$ are defined
recursively by $H_1 = X+Y$ and
$$
(N+1)H_{N+1} = \frac{1}{2}[X-Y,H_N] +
\sum_{r = 0}^{\lfloor N/2 -1\rfloor}
\frac{B_{2r}}{(2r)!}\sum_s
[H_{s_1},[H_{s_2},[ \ldots, [H_{s_{2r}},X+Y]\ldots]]]
$$

where the sum over $s$ is the sum over all
$2r$-tuples $s = (s_1,\ldots,s_{2r})$
of strictly positive integers whose sum $s_1 +\ldots+s_{2r} = N$.

Hausdorff series satisfies the symmetry $H(-Y,-X) = -H(X,Y)$.

First few terms of Hausdorff series are

$$
H(X,Y) = X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}([X,[X,Y]]+[Y,[Y,X]]) + \frac{1}{24}[Y,[X,[Y,X]]] + \ldots
$$

## Literature

Related $n$Lab entries include [[Lie theory]], [[Malcev completion]], [[exponential map]]

Here we list mostly references about the classical part of the subject. The references connecting Hausdorff series to [[Drinfeld associator]]s, [[Grothendieck-Teichmueller group]] and [[Kashiwara-Vergne conjecture]] see in the corresponding entries.

* [[N. Bourbaki]], _Lie groups and algebras_, chapter II
* [[M M Postnikov]], Lectures on geometry, Semester V, Lie groups and algebras
* E. B. Dynkin, _Calculation of the coefficents in the Campbell-Hausdorff formula_, Doklady Akad. Nauk SSSR (N.S.) 57, 323-326, (1947)

* [[Terence Tao]], 254A, Notes 1, _Lie groups, Lie algebras, and the Baker-Campbell-Hausdorff formula_, blog entry
* V. Kurlin, _Exponential Baker-Campbell-Hausdorff formula_, [http://arxiv.org/abs/math/0606330](http://arxiv.org/abs/math/0606330)
* Terry Tao's blog: [the-c11-baker-campbell-hausdorff-formula](https://terrytao.wordpress.com/2011/06/21/the-c11-baker-campbell-hausdorff-formula)
* [wikipedia](https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula)
* Kuo-Tsai Chen, _Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula_, Annals of Mathematics __65__:1 (1957) 163--178 [doi](https://doi.org/10.2307/1969671) [jstor](http://www.jstor.org/stable/1969671)
* [[Wilhelm Magnus]], _A connection between the Baker-Hausdorff formula and a problem of Burnside_, Ann. of Math. __52__ (1950) 111-126
* Federico Zadra et al. _The flow method for the Baker-Campbell-Hausdorff formula: exact results_, J. Phys. A: Math. Theor. __56__ (2023) 385206 [doi](https://doi.org/10.1088/1751-8121/acf102)

category: algebra
[[!redirects Baker-Campbell-Hausdorff series]]
[[!redirects BCH formula]]
[[!redirects Baker-Campbell-Hausdorff formula]]
[[!redirects BCH theorem]]
[[!redirects Campbell-Hausdorff formula]]
[[!redirects Hausdorff formula]]


Hausdorff space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A [[topological space]] (or more generally, a [[convergence space]]) is _Hausdorff_ if [[convergence]] is unique.  The concept can also be defined for [[locales]] (see Definition \ref{proper} below) and [[categorification|categorified]] (see [Beyond topological spaces](#BeyondTopologicalSpaces) below).  A Hausdorff space is often called $T_2$, since this condition came second in the original list of four [[separation axioms]] (there are more now) satisfied by [[metric spaces]].

[[!include main separation axioms -- table]]

Hausdorff spaces are a kind of [[nice topological space]]; they do not form a particularly [[nice category of spaces]] themselves, but many such nice categories consist of only Hausdorff spaces.  In fact, [[Felix Hausdorff]]\'s original definition of 'topological space' actually required the space to be Hausdorff, hence the name.  Certainly [[homotopy theory]] (up to [[weak homotopy equivalence]]) needs only Hausdorff spaces.  It is also common in analysis to assume that all spaces encountered are Hausdorff; if necessary, this can be arranged since every space has a Hausdorff quotient (in fact, the Hausdorff spaces form a [[reflective subcategory]] of [[Top]]), although usually an easier method is available than this sledgehammer.

## Definitions
 {#Definitions}

There are many equivalent ways of characterizing a space $S$ as __Hausdorff__. The traditional definition is this:

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

Given points $x$ and $y$ of $S$, if $x \neq y$, then there exist [[open neighbourhoods]] $U$ of $x$ and $V$ of $y$ in $S$ that are [[disjoint subsets|disjoint]]: such that their [[intersection]] $U \cap V$ is the [[empty set]] (or explicitly, such that $x' \ne y'$ whenever $x' \in U$ and $y' \in V$).
=--

That is, any two distinct points can be _separated_ by open neighbourhoods, and it is simply a mundane way of saying that $\ne$ is [[open subset|open]] in the [[product topology]] on $S \times S$.

Here is a classically equivalent definition that is more suitable for [[constructive mathematics]]:

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

Given points $x$ and $y$ of $S$, if every neighbourhood $U$ of $x$ in $S$ meets every neighbourhood $V$ of $y$ in $S$ (which means that $U \cap V$ is [[inhabited set|inhabited]]), then $x = y$.
=--

This is the mundane way of saying that $=$ is [[closed subset|closed]] in $S \times S$.

Another way of saying this, which makes sense also for [[locales]], is the following:

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

The [[diagonal]] embedding $S \to S \times S$ is a [[proper map]] (or equivalently a [[closed map]], since any closed subspace inclusion is proper).
=--

This way of stating the definition generalizes to [[topos theory]] and thus to many other contexts; but it is not always a faithful generalization of the classical notion for topological spaces.  See _[Beyond topological spaces](#BeyondTopologicalSpaces)_ below for more.

Here is an equivalent definition (constructively equivalent to Definition \ref{constructive}) that makes sense for arbitrary [[convergence spaces]] or [[preconvergence spaces]]:

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

Given a [[net]] (or equivalently, a proper [[filter]]) in $S$, if it converges to both $x$ and $y$, then $x = y$.
=--

That is, convergence in a Hausdorff space is unique.

## Examples

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

[[a CW-complex is a Hausdorff space]].

=--

## Properties
 {#Properties}

[[!include main separation axioms -- as lifting properties]]

### Hausdorff reflection
 {#HausdorffReflections}

+-- {: .num_prop #HausdorffReflection}
###### Proposition
**(Hausdorff reflection)**

For every [[topological space]] $X$ there exists
a [[Hausdorff topological space]] $H X$ and a [[continuous function]]

$$
  h_X
    \;\colon\;
  X \longrightarrow H X
$$

which is the  "closest approximation from the left" to $X$ by a Hausdorff topological space, in that
for $Y$ any [[Hausdorff topological space]], then [[continuous functions]] of the form

$$
  f \;\colon\; X \longrightarrow Y
$$

are in [[bijection]] with [[continuous function]] of the form

$$
  \tilde f \;\colon\; H X \longrightarrow Y
$$

and such that the bijection is constituted by

$$
  f = \tilde f \circ h_X
  \;\colon\;
    X
      \overset{h_X}{\longrightarrow}
    H X
      \overset{\tilde f}{\longrightarrow}
    Y
  \,.
$$

Here $H X$ (or more precisely $h_X$) is also called the _Hausdorff reflection_ (or _Hausdorffication_ or similar) of $X$.

Moreover, the operation $H(-)$ extends to [[continuous functions]] $f \colon X \to Y$

$$
  (X \overset{f}{\to} Y)
    \;\mapsto\;
  (H X \overset{H f}{\to} H Y)
$$

by setting

$$
  H f \;\colon\; [x] \mapsto [f(x)]
  \,,
$$

where $[x]$ denotes the [[equivalence class]] under $\sim_X$ of any $x \in X$.

Finally, the comparison map is compatible with this in that the follows [[commuting squares|squares commute]]:

$$
  \array{
      X &\overset{f}{\longrightarrow}& Y
      \\
      {}^{\mathllap{h_X}}\downarrow && \downarrow^{\mathrlap{h_Y}}
      \\
      H X &\underset{H f}{\longrightarrow}& H Y
  }
  \,.
$$

=--

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

There are various ways to construct $h_X$, see below prop. \ref{HausdorffReflectionViaHomsIntoAllHausdorffSpaces} and prop. \ref{HausdorffReflectionViaTransitiveClosureOfDiagonal}

=--

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

In the language of [[category theory]] the Hausdorff reflection of prop. \ref{HausdorffReflection}
says that

1. $H$ is a _[[functor]]_ $H \;\colon\; Top \longrightarrow Top_{Haus}$ from the [[category]] [[Top]] of [[topological spaces]] to the [[full subcategory]] $Top_{Haus} \overset{\iota}{\hookrightarrow} Top$ of Hausdorff topological spaces;

1. $h_X \colon X \to H X$ is a [[natural transformation]] from the [[identity functor]] on [[Top]] to the functor $\iota \circ H$

1. [[Hausdorff topological spaces]] form a [[reflective subcategory]] of all [[topological spaces]] in that $H$ is [[left adjoint]] to the inclusion functor $\iota$

   $$
     Top_{Haus}
       \underoverset{\underset{\iota}{\hookrightarrow}}{\overset{H}{\longleftarrow}}{\bot}
     Top
     \,.
   $$

=--

There are various ways to see the existence and to construct the Hausdorff reflection. The following is maybe the quickest way to see the existence, even though it leaves the actual construction rather implicit.

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

Let $(X,\tau)$ be a [[topological space]] and consider the [[equivalence relation]] $\sim$ on the underlying set $X$
for which $x \sim y$ precisely if for every [[surjective function|surjective]] [[continuous function]] $f \colon X \to Y$ into any
[[Hausdorff topological space]] $Y$ we have $f(x) = f(y)$.

Then the set of [[equivalence classes]]

$$
  H X \coloneqq X /{\sim}
$$

equipped with the [[quotient topology]] is a [[Hausdorff topological space]] and the quotient map $h_X \;\colon\; X \to X/{\sim}$ exhibits the Hausdorff reflection of $X$, according to prop. \ref{HausdorffReflection}.

=--

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

First observe that every continuous function $f \colon X \longrightarrow Y$ into a Hausdorff space $Y$
factors uniquely via $h_X$ through a continuous function $\tilde f$

$$
  f = \tilde f \circ h_X
$$

where

$$
  \tilde f \colon [x] \mapsto f(x)
  \,.
$$

That this is well defined and continuous follows directly from the definitions.

What remains to be seen is that $H X$ is indeed a Hausdorff space. Hence assume that $[x] \neq [y] \in H X$.
By construction of $H X$ this means that there exists a Hausdorff space $Y$ and a surjective continuous function
$f \colon X \longrightarrow Y$ such that $f(x) \neq f(y) \in Y$. Accordingly, since $Y$ is Hausdorff,
there exist disjoint open neighbourhoods $U_x, U_y \in \tau_Y$. Moreover, by the previous statement there
exists a continuous function $\tilde f \colon H X \to Y$ with $\tilde f([x]) = f(x)$ and $\tilde f([y]) = f(y)$.
Since, by the nature of continuous functions,
the pre-images $\tilde f^{-1}( U_x ), \tilde f^{-1}(U_y) \subset H X$ are still disjoint and open,
we have found disjoint neighbourhoods of $[x]$ and $[y]$. Hence $H X$ is Hausdorff.

=--

Some readers may find the following a more direct way of constructing the Hausdorff reflection:

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

For $(Y,\tau_Y)$ a [[topological space]], write $r_Y \subset Y \times Y$
for the [[transitive relation|transitive closure]] of the [[relation]] given by the [[topological closure]] $Cl(\Delta_Y)$ of the [[image]] of the [[diagonal]] $\Delta_Y \colon Y \hookrightarrow Y \times Y$.

$$
  r_Y \coloneqq Trans(Cl(Delta_Y))
  \,.
$$

Now for $(X,\tau_X)$ a [[topological space]], define by [[induction]] for each [[ordinal number]] $\alpha$ an [[equivalence relation]] $r^\alpha$ on $X$ as follows, where we write $q^\alpha \colon X \to H^\alpha(X)$ for the corresponding [[quotient topological space]] projection:

We start the induction with the trivial equivalence relation:

* $r^0_X \coloneqq \Delta_X$;

For a [[successor ordinal]] we set

* $r_X^{\alpha+1} \coloneqq \left\{ (a,b) \in X \times X  \,\vert\, (q^\alpha(a), q^\alpha(b)) \in r_{H^\alpha(X)} \right\}$

and for a [[limit ordinal]] $\alpha$ we set

* $r_X^\alpha \coloneqq \underset{\beta \lt \alpha}{\cup} r_X^\beta$.

Then:

1. there exists an ordinal $\alpha$ such that $r_X^\alpha = r_X^{\alpha+1}$

1. for this $\alpha$ then $H^\alpha(X) = H(X)$ is the Hausdorff reflection from prop. \ref{HausdorffReflectionViaHomsIntoAllHausdorffSpaces}.

=--

([vanMunster 14, section 4](#vanMunster14))

### Monadicity

The topology on a [[compact space|compact]] Hausdorff space is given precisely by the (existent because compact, unique because Hausdorff) limit of each [[ultrafilter]] on the space.  Accordingly, compact Hausdorff topological spaces are (perhaps surprisingly) described by a (large) [[algebraic theory]].  In fact, the category of compact Hausdorff spaces is [[monadic functor|monadic]] (over [[Set]]); the [[monad]] in question maps each set to the set ultrafilters on it.  (The results of this paragraph require the [[ultrafilter theorem]], a weak form of the [[axiom of choice]]; see [[ultrafilter monad]].)

A compact Hausdorff [[locale]] (or space) is necessarily [[regular locale|regular]]; a regular locale (or $T_0$ space) is necessarily Hausdorff.  Accordingly, [[locale]] theory usually speaks of 'compact regular' locales instead of 'compact Hausdorff' locales, since the definition of regularity is easier and more natural.  Then a version of the previous paragraph works for compact regular locales *without* the ultrafilter theorem, and indeed [[constructive mathematics|constructively]] over any [[topos]].

### Sobriety

Using [[classical logic]] (but not in [[constructive logic]]) every Hausdorff space is a [[sober topological space]]: _[[Hausdorff implies sober]]_.

### Relation to compact spaces

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

[[compact subspaces of Hausdorff spaces are closed]].

=--

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

[[maps from compact spaces to Hausdorff spaces are closed and proper]]

=--

### Dense subspaces and Epimorphisms

\begin{proposition}
 \label{DenseSubspaceInclusionsAreEpimorphismsAmongHausdorffSpaces}

In the [[category]] of [[Hausdorff topological spaces]] (with [[continuous functions]] between them), the inclusion of a [[dense subspace]]

$$
  A \overset{\;\;i\;\;}{\hookrightarrow} X
$$

is an [[epimorphism]].

\end{proposition}

Here is a proof in the language of [[category theory]]:

\begin{proof}

We have to show that for $(f,g)$ any [[pair]] of [[parallel morphisms]] out of $X$

$$
  A
  \overset{\;\;i\;\;}{\hookrightarrow}
  X
  \underoverset
  {\;\;g\;\;}
  {\;\;f\;\;}
  {\rightrightarrows}
  Y
$$

into a [[Hausdorff space]] $Y$, the [[equality]] $f \circ i = g \circ i$ implies $f = g$. Equivalently, that $f \circ i = g \circ i$ implies that $1_X \colon X \to X$ is the [[equalizer]] of the pair $(f, g)$. But the equalizer $E \to X$ is formed by taking a [[pullback]] of the [[diagonal map]] $\Delta \colon Y \to Y \times Y$ along $(f, g) \colon X \to Y \times Y$:

$$\array{
E & \to & Y \\
\downarrow & & \downarrow \mathrlap{\Delta} \\
X & \underset{(f, g)}{\to} & Y \times Y.
}$$

Since $Y$ is Hausdorff, the subset $\Delta: Y \to Y \times Y$ is closed, and the pullback $E \hookrightarrow X$ of this closed subset along the continuous map $h = (f, g)$, which is $E = h^{-1}(\Delta)$, is also closed. Since $E$ is a closed subset of $X$ and contains a dense subspace $i: A \hookrightarrow X$, it must be all of $X$ (as a subset of itself).
\end{proof}

Note, incidentally, that $X$ itself doesn't have to be Hausdorff for the argument to go through.

Alternatively, here is a proof in the language of [[general topology|basic topology]]:

\begin{proof}

We have to show that for $(f,g)$ any [[pair]] of [[parallel morphisms]] out of $X$

$$
  A
  \overset{\;\;i\;\;}{\hookrightarrow}
  X
  \underoverset
  {\;\;g\;\;}
  {\;\;f\;\;}
  {\rightrightarrows}
  Y
$$

into a [[Hausdorff space]] $Y$, the [[equality]] $f \circ i = g \circ i$ implies $f = g$. With [[classical logic]] we may equivalently show the [[contrapositive]]: That $f \neq g$ implies $f \circ i \neq  g \circ i$.

So assume that $f \neq g$. This means that there exists $x \in X$ with $f(x) \neq g(x)$. Since $Y$ is Hausdorff, there exist then [[disjoint subset|disjoint]] [[open neighbourhoods]] $O_{f(x)},\;O_{g(x)} \subset Y$, i.e. $f(x) \in O_{f(x)}$ and $g(x) \in O_{g(x)}$ with $O_{f(x)} \cap O_{g(x)} = \varnothing$.

But their [[preimages]] must intersect at least in $x \in f^{-1}\big( O_{f(x)} \big) \cap g^{-1}\big(  O_{g(x)} \big)$. Since this intersection is an [[open subset]] (as preimages of open subsets are open by definition of [[continuous functions]], and since finite [[intersections]] of open subsets are open by the definition of [[topological spaces]]) there exists a point $a \in A$ with $i(a) \in f^{-1}\big( O_{f(x)} \big) \cap g^{-1}\big(  O_{g(x)} \big)$ (by definition of [[dense subset]]). But since then $f(i(a)) \in O_{f(x)}$ and $g(i(a)) \in O_{g(x)}$ while $O_{f(x)}$ is [[disjoint subset|disjoint]] from $O_{g(x)}$, it follows that $f(i(a)) \neq g(i(a))$. This means that $f \circ i \neq g \circ i$.
\end{proof}

\linebreak

In fact the converse holds: any [[epimorphism]] in the category of Hausdorff spaces has dense image (e.g. [LL 18](#LL18)).

## Related notions

Arguably, the desire to make spaces Hausdorff ($T_2$) in analysis is really a desire to make them $T_0$; nearly every space that arises in analysis is at least [[regular space|regular]], and a regular $T_0$ space must be Hausdorff.  Forcing a space to be $T_0$ is like forcing a [[category]] to be [[skeletal category|skeletal]]; indeed, forcing a [[preorder]] to be a [[partial order]] is a special case of both (see [[specialisation topology]] for how).  It may be nice to assume, when working with a particular space, that it is $T_0$ but not to assume, when working with a particular underlying set, that every topology on it is $T_0$.

Whatever one thinks of that, there is a non-$T_0$ version of Hausdorff space, an __$R_1$ space__.  (The symbol here comes from being a weak version of a **r**egular space; in general a $T_i$ space is precisely both $R_{i-1}$ and $T_0$).  This is also called a __preregular space__ (in _[[HAF]]_) and a __reciprocal space__ (in convergence theory).

+-- {: .un_defn}
###### Definition (of $R_1$)
Given points $a$ and $b$, if every neighbourhood of $a$ meets every neighbourhood of $b$, then every neighbourhood of $a$ is a neighbourhood of $b$.  Equivalently, if any net (or proper filter) converges to both $a$ and $b$, then every net (or filter) that converges to $a$ also converges to $b$.
=--

There is also a notion of __sequentially Hausdorff space__:
+-- {: .un_defn}
###### Definition (of sequentially Hausdorff)
Whenever a [[sequence]] converges to both $x$ and $y$, then $x = y$.
=--
Some forms of [[predicative mathematics]] find this concept more useful.  Hausdorffness implies sequential Hausdorffness, but the converse is false even for [[sequential space]]s (although it is true for [[first-countable space]]s).

The reader can now easily define a _sequentially $R_1$ space_.

## Beyond topological spaces
 {#BeyondTopologicalSpaces}

### Hausdorff locales
 {#HausdorffLocale}

The most obvious definition for a [[locale]] $X$ to be **Hausdorff** is that its [[diagonal]] $X\to X\times X$ is a [[closed map|closed]] (and hence [[proper map|proper]]) inclusion.  However, if $X$ is a [[sober space]] regarded as a locale, this might not coincide with the condition for $X$ to be Hausdorff as a space, since the [[Cartesian product]] $X\times X$ in the [[category]] [[Loc]] of locales might not coincide with the product in the category [[Top]] of topological spaces (the [[Tychonoff product]]).  But it does coincide if $X$ is a [[locally compact locale]], so in that case the two notions of Hausdorff are the same.

### Separated toposes and schemes

This notion of a _Hausdorff locale_ is a special case of that of _[[Hausdorff topos]]_ in [[topos theory]]. This also is formally similar to notions such as a _[[separated scheme]]_ etc.  The corresponding relative notion (over an arbitrary [[base topos]]) is that of _[[separated geometric morphism]]_. For schemes see _[[separated morphism of schemes]]_.

## In constructive mathematics

In [[constructive mathematics]], the Hausdorff notion multifurcates further, due to the variety of possible meanings of [[closed subspace]].  If we ask the diagonal to be *weakly* [[closed subspace|closed]], then in the spatial case, this means that it contains all its limit points, giving Definition \ref{constructive} above.  But if we ask the diagonal to be *strongly* closed, i.e. the complement of an open set, then in the spatial case this means that there is a [[tight inequality]] $\ne$ (the [[exterior]] of $=$) relative to which Definition \ref{classical} holds.  (We use $\ne$ twice in that definition: in the hypothesis that $x \ne y$ and in the conclusion that $x' \ne y'$.)

It is natural to call these conditions *weakly Hausdorff* and *strongly Hausdorff*, but one should be aware of terminological clashes: in classical mathematics there is a different notion of a [[weak Hausdorff space]], whereas (strong) Hausdorffness for locales has by some authors been called "strongly Hausdorff" only to contrast it with Hausdorffness for spaces.

As a simple example, consider a [[discrete space]] $X$ regarded as a locale.  Since it is locally compact, the locale product $X\times X$ coincides with the space product (a theorem that is valid constructively); but nevertheless we have:

1. The diagonal $X\to X\times X$ always has an open complement.
2. Definition \ref{constructive} above always holds, since $\{x\}$ and $\{y\}$ are neighborhoods of $x$ and $y$, and if they intersect then $x=y$.  That is, $X$ is spatially weakly Hausdorff.
3. The diagonal $X\to X\times X$ is the complement of an open subset (i.e. $X$ is spatially strongly Hausdorff) if and only if equality in $X$ is stable under [[double negation]], in other words if $X$ admits a tight [[inequality relation]].
4. The locale diagonal $\Delta:X\to X\times X$ is a closed sublocale (i.e. $X$ is localically strongly Hausdorff) if and only if $X$ has [[decidable equality]].  For closedness of $\Delta$ means that $\Delta_\ast(U\cup \Delta^\ast(V)) \subseteq \Delta_\ast(U) \cup V$ for any $U\in O(X)$ and $V\in O(X\times X)$, where $x\in \Delta^\ast(V)$ means $(x,x)\in V$, while $(x,y)\in \Delta_\ast(U)$ means $(x=y)\to (x\in U)$.  Now let $U = \emptyset$ and $V = \{ (x,x) \mid x\in X \}$.  Then $(x,y) \in \Delta_\ast(U\cup \Delta^\ast(V))$ means $(x=y) \to (\bot \vee (x=x))$, which is a tautology; while $(x,y) \in \Delta_\ast(U) \cup V$ means $((x=y)\to \bot) \vee (x=y)$, i.e. $\neg(x=y) \vee (x=y)$.
5. I don't know what it means for $X$ to be localically weakly Hausdorff.  (Weak closure in locales is very inexplicit.)

In particular, the statement "all discrete locales are localically strongly Hausdorff" is equivalent to [[excluded middle]].

However, non-discrete spaces can constructively be localically strongly Hausdorff without having decidable equality.  For instance, any [[regular space]] is also regular as a locale, and hence localically strongly Hausdorff.  We can also say:

+-- {: .num_theorem #Apartness}
###### Theorem
In any topological space $X$, let $x\#y$ mean that there exist opens $U,V$ with $x\in U$ and $y\in V$ and $U\cap V = \emptyset$; then $\#$ is always an [[inequality relation]].  If the spatial product $X\times X$ coincides with the locale product (such as if $X$ is [[locally compact space|locally compact]]), then $X$ is localically strongly Hausdorff if and only if $\#$ is an [[apartness relation]] and every open set in $X$ is $\#$-open (i.e. for any $x\in U$ and $y\in X$ we have $y\in U \vee x\#y$).
=--
+-- {: .proof}
###### Proof
Note that $\#$ is, as a subset $W_\# \subseteq X\times X$, the exterior of the diagonal in the product topology, mentioned above.  If $X$ is localically strongly Hausdorff, then $W_\#$ *must* be the open set of which the diagonal is the complementary closed sublocale, since it is the largest open set disjoint from the diagonal.

To say that the diagonal *is* its complementary closed sublocale implies in particular that for any open set $U\subseteq X$, the open set $(U\times U) \cup W_\#$ is the largest open subset of $X\times X$ whose intersection with the diagonal is contained in $U\cap U = U$.  Specifically, therefore, $(U\times U) \cup W_\#$ contains $U\times X$ (since $U\times X$ is an open subset of $X\times X$ whose intersection with the diagonal is $U$).  That is, if $x\in U$ and $y\in X$, then either $(x,y)\in U\times U$ (i.e. $y\in U$) or $(x,y)\in W_\#$ (i.e. $x\#y$).  This shows that $U$ is $\#$-open.

To show that $\#$ is an apartness, note that for any $x$ the set $\{ z \mid x\# z \}$ is open, since it is the preimage of $W_\#$ under a section of the second projection $X\times X \to X$.  Thus, it is $\#$-open, which is to say that if $x\# z$ then for any $y$ either $x\#y$ or $y\#z$, which is the missing [[comparison]] axiom for $\#$ to be an apartness.

Conversely, suppose $\#$ is an apartness and every open set is $\#$-open (i.e. the apartness topology refines the given topology on $X$).  Let $A\subseteq X\times X$ be an open set; we must show that $A\cup W_\#$ is the largest open subset of $X\times X$ whose intersection with the diagonal is contained in $A\cap \Delta_X$.  In other words, suppose $U\times V$ is a basic open in $X\times X$ and $(U\times V)\cap \Delta_X$ (which is $U\cap V$) is contained in $A\cap \Delta_X$; we must show $U\times V\subseteq A\cup W_\#$.  In terms of elements, we assume that if $x\in U$ and $x\in V$ then $(x,x)\in A$, and we must show that if $x\in U$ and $y\in V$ then $(x,y)\in A \vee x\#y$.

Assuming $x\in U$ and $y\in V$, since $U$ and $V$ are $\#$-open we have either $y\in U$ or $x\# y$, and either $x\in V$ or $x\#y$.  Since we are done if $x\#y$, it suffices to assume $y\in U$ and $x\in V$.  Therefore, by assumption, $(x,x)\in A$ and $(y,y)\in A$.  Since $A$ is open in the product topology, we have opens $U',V'$ with $x\in U'$ and $y\in V'$ and $U'\times U'\subseteq A$ and $V'\times V' \subseteq A$.  But now $\#$-openness of $U'$ and $V'$ tells us again that either $x\#y$ (in which case we are done) or $y\in U'$ and $x\in V'$.  In the latter case, $(x,y)\in U'\times U'$ (and also $V'\times V'$), and hence is also in $A$.
=--

Note that the apartness $\#$ need not be [[tight relation|tight]], and in particular $X$ need not be spatially Hausdorff.  In particular, if $X$ might not even be $T_0$: since localic Hausdorffness is (of course) only a property of the open-set lattice, it only "sees" the [[sobrification]] and in particular the $T_0$ quotient ([[Kolmogorov quotient]]).  However, this is all that can go wrong: if $\neg(x\# y)$, then by $\#$-openness every open set containing $x$ must also contain $y$ and vice versa, so if $X$ is $T_0$ then $x=y$ and $\#$ is tight.

If the locale product $X\times X$ does not coincide with the spatial product, then the "only if" direction of the above proof still works, if we define $W_\#$ to be the open part of the locale product $X\times X$ given by $W_\# = \bigvee \{ U\otimes V \mid U\cap V = \emptyset \}$.  A different proof is to recall that by [this theorem](/nlab/show/apartness+relation#ClosedLocalicEquivalenceRelation), an apartness relation is the same as a (strongly) closed equivalence relation on a discrete locale, and the quotient of such an equivalence relation is the $\#$-topology.  Thus, if $X$ is localically strongly Hausdorff, its diagonal is a closed equivalence relation, which yields by pullback a closed equivalence relation on the discrete locale $X_d$ on the same set of points.  This is the [[kernel pair]] of the canonical surjection $X_d \to X$, and hence its quotient (the $\#$-topology) maps to $X$, i.e. refines the topology of $X$.

## History

First introduced by [[Hausdorff]] in [[Grundzüge der Mengenlehre]], Hausdorff spaces were the original concept of a [[topological space]].  Later the Hausdorffness condition was dropped, apparently first by [[Kuratowski]] in 1920s.

## Related concepts

* [[non-Hausdorff topological space]]

* [[locally Hausdorff topological space]]

* [[Kolmogorov topological space]]

* [[nice topological space]]

## References

The notion is due to

* [[Felix Hausdorff]], *[[Grundzüge der Mengenlehre]]*, Leipzig: Veit (1914) [ISBN:978-0-8284-0061-9] Reprinted by Chelsea Publishing Company (1944, 1949, 1965) [[ark:/13960/t2891gn8g](https://archive.org/details/grundzgedermen00hausuoft/page/n5/mode/2up)]

See also

* Wikipedia, _[Hausdorff space](https://en.wikipedia.org/wiki/Hausdorff_space)_

and the references at *[[topology]]*.

A detailed discussion of Hausdorff reflection is in

* {#vanMunster14} Bart van Munster, _The Hausdorff quotient_, 2014 ([pdf](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.728.5810))

On epimorphisms of Hausdorff spaces:

* {#LL18} Jérôme Lapuyade-Lahorgue, _The epimorphisms of the category Haus are exactly the image-dense morphisms_ ([arXiv:1810.00778](https://arxiv.org/abs/1810.00778))

Comments on the relation to [[topos theory]]:

* S. Niefield, _A note on the locally Hausdorff property_, Cahiers TGDC (1983) ([numdam](http://www.numdam.org/item?id=CTGDC_1983__24_1_87_0))

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Hawaiian earring space

[[!redirects Shrinking wedge of circles]]

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#Contents#
* table of contents
{:toc}

## Idea

The [[topological space]] which is the [[union]] of an infinite number of [[circles]] in the [[plane]], of decreasing radius and all sharing exactly one point, has commonly been known as the *Hawaiian earring space* (apparently following [Dudley 61](#Dudley61), though the space appears earlier in [Griffiths 56](#Griffiths56)). Other terminology is “clamshell” (also [Dudley 61](#Dudley61)) or, more recently, “infinite earring” or “shrinking wedge of circles”.

In [[algebraic topology]] this space has been of interest mainly as a counterexample, showing the need for care in hypotheses for a good theory of [[covering spaces]] -- see at *[Properties](#Properties)* below.

## Definition

The (Hawaiian) earring space is the [[topological space]] defined to be the set

$$
  \bigcup_{n \in \mathbb{N}} \{(x, y) \in \mathbb{R}^2: (x - 1/2^n)^2 + y^2 = 1/2^{2n}\}
$$

endowed with [[subspace topology]] inherited from the [[Euclidean topology]] of the [[plane]] $\mathbb{R}^2$.

More abstractly: the earring space can be described up to [[homeomorphism]] as the [[one-point compactification]] of a [[coproduct]] (in [[Top]]; a [[disjoint union space]]) of [[countably infinite set|countably]] many open [[intervals]] (e.g. section 2.1 of ([Cannon-Conner 2000](#CannonConner00))). This shows that the specific radii converging to zero (which was taken to be $1/2^n$ above) don't actually matter. Equivalently, the earring space is [[homeomorphic]] to the [[one-point compactification]] of the [[product space]] $\mathbb{N}\times \mathbb{R}$, where the space of [[real numbers]] $\mathbb{R}$ has the [[Euclidean space|Euclidean topology]] and that of [[natural numbers]] $\mathbb{N}$ has the [[discrete topology]].

Another topological characterisation is as the [[reduced suspension]] of the [[one-point compactification]] of $\mathbb{N}$ with the [[discrete topology]] (that is, the "free convergent sequence"). The basepoint is taken to be the added limit point "at infinity".

## Properties
 {#Properties}

Every [[neighborhood]] of $(0, 0)$ in the Hawaiian earring space has non-contractible [[loops]] inside it (this would not be the case under the CW topology, i.e., the evident [[quotient space]] of countably many circles with the [[coproduct]] or [[disjoint union topological space|disjoint sum topology]], making the result a countable bouquet of circles). This means that the space is not
[[semi-locally simply connected space|semi-locally simply connected]].

Viewed in terms of [[general topology]], it would be hard to sell the earring space as a genuinely "pathological space": as it is a [[compactum|compact, Hausdorff]], [[connected space|connected and locally path-connected]] [[metric space]], etc. It's really through the lens of [[algebraic topology]] and particularly the classical theory of [[covering spaces]] that it seems strange: it is not a [[CW complex]], it is not
[[semi-locally simply connected space|semi-locally simply connected]] and its [[fundamental group]] is hard to compute.

## Related concepts

* [[Warsaw circle]]

## References

An early paper discussing the concept without giving it any specific name is:

* {#Griffiths56} H. B. Griffiths, _Infinite Products of Semi-Groups and Local Connectivity_, Proceedings of the London Mathematical Society, s3-6 (1956) 455-480 ([doi:10.1112/plms/s3-6.3.455](https://doi.org/10.1112/plms/s3-6.3.455))

The name _Hawaiian earring_ seems to be due to:

* {#Dudley61} R. M. Dudley, _Continuity of homomorphisms_, Duke Math. J. 28:4 (1961), 587-594 ([doi:10.1215/S0012-7094-61-02859-9](https://doi.org/10.1215/S0012-7094-61-02859-9))

See also:

* Wikipedia, *[Hawaiian earring](https://en.wikipedia.org/wiki/Hawaiian_earring)*

The following discusses the combinatorial structure of the fundamental group of the earring space.

* {#CannonConner00}  J. W. Cannon and G. R. Conner, _The combinatorial structure of the Hawaiian earring group_, Topology and its Applications **106** (2000) pp 225–271 (doi:[10.1016/S0166-8641(99)00103-0](https://doi.org/10.1016/S0166-8641%2899%2900103-0))

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Hawaiian earring space > history

[[!redirects Hawaiian earring space]]


Haynes Miller

* [website](http://www-math.mit.edu/~hrm/)

* [German Wikipedia entry](http://de.wikipedia.org/wiki/Haynes_Miller)

## Selected writings

On the [[elliptic Chern character]] and the [[elliptic genus]]/[[Witten genus]] as [[fiber integration]] in [[elliptic cohomology]]:

* [[Haynes Miller]], _The elliptic character and the Witten genus_, in: [[Mark Mahowald]], [[Stewart Priddy]] (eds.), _Algebraic topology_ (Evanston, IL, 1988), Contemp. Math. 96, Amer. Math. Soc., Providence, RI (1989) 281–289 ([pdf](http://dedekind.mit.edu/~hrm/papers/ell-char.pdf), [[MillerEllipticGenus.pdf:file]], [doi:10.1090/conm/096](http://dx.doi.org/10.1090/conm/096))

category: people


Hayo Thielecke

[webpage](http://www.cs.bham.ac.uk/~hxt/)

## Related entries

* [[continuation-passing style]]

category: people

category: computer science


Hazewinkel, Witt vectors

This entry provides pointers related to the text

* [[Michael Hazewinkel]],

  _Witt vectors Part 1_,

  ([arXiv:0804.3888](http://arxiv.org/abs/0804.3888))

on [[Lambda-rings]] of [[Witt vectors]].

* Hazewinkel talks about both the 'orthodox' and 'heterodox' approaches to $\lambda$-rings.  Indeed he starts out with a lot of material on [[Witt vectors]] and their relation to $p$-adics.   The [[Lambda-rings]] only make their debut on page 87, where the operation of 'taking the Witt vectors' of a [[commutative ring]] is revealed to be the [[right adjoint]] to the [[forgetful functor]] from $\lambda$-rings to commutative rings.

* He then goes ahead and defines $\lambda$-rings on page 88. At first his definition looks a bit frustrating, because Hazewinkel defines '$\lambda$-ring' using the concept of 'morphism of $\lambda$-rings'!  But it's not actually circular; it's really just a trick to spare us certain ugly equations that appear in the usual definition.

* Just for fun, note the unusual remark in footnote 62 on page 88: warning the reader to 'steer clear' of a certain book by two famous authors.

* On page 92, Hazewinkel proves the Wilkerson theorem getting $\lambda$-rings from rings equipped with [[Adams operations]] $\psi_p$.  And then, at the bottom of page 94, he goes heterodox and defines '$\psi$-rings' to be commutative rings equipped with Adams operations --- and notes that over a
field of characteristic zero, $\lambda$-rings are the same as $\psi$-rings.

* On page 97, he describes 'taking the Witt vectors' as a [[comonad]] on the category of commutative rings.  This is just another way of talking about the [[right adjoint|right adjointness]] property he mentioned on page 87; now he's proving it.

* On page 98 he goes orthodox again, and shows that Symm, the ring of [[symmetric functions]] in countably many [[variables]], is the [[free construction|free]] [[Lambda ring]] on one generator.

* On page 102 starts explaining '[[plethysm]]'.

category: reference


He Wang

* [personal page](https://web.northeastern.edu/he.wang/)

## Selected writings

A [[spectral sequence]] for higher [[twisted de Rham cohomology]] (analgous to the [[Atiyah-Hirzebruch spectral sequence]] for [[twisted K-theory]] from [Atiyah & Segal 2005](Atiyah–Hirzebruch+spectral+sequence#AtiyahSegal05)):

* {#LiLiuWang09} Weiping Li, Xiugui Liu, [[He Wang]], _On a spectral sequence for twisted cohomologies_,  Chin. Ann. Math. Ser. B 35, 633–658 (2014) ([arXiv:0911.1417](https://arxiv.org/abs/0911.1417), [doi:10.1007/s11401-014-0842-z](https://doi.org/10.1007/s11401-014-0842-z)).

category: people


heap

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# Heaps
* table of contents
{: toc}

## Idea
 {#Idea}

In [[algebra]], a _heap_ is an algebraic [[structure]] which is almost that of a [[group]] structure but without the specification of a [[neutral element]].

While notions such as *[[affine spaces]]*, *[[principal homogeneous spaces]]* are very similar,
the concrete definition of heaps is more direct and simpler in the sense that it postulates just a single ternary operation satisfying just two [[axioms]].

Concretely, for heaps [[underlying]] a [[group]] [[structure]], this ternary operation is defined from the binary group operation $(\text{-})\cdot(\text{-})$ by

\[
  \label{HeapOperationInAGroup}
  t
  \,\colon\,
  (a,b,c)
  \;\mapsto\;
  a \cdot b^{-1} \cdot c
\]

and every heap arises this way, up to [[isomorphism]], from a group called its *automorphism group* (Prop. \ref{PrinIsEssentiallySurjective} below).

{#HowToThinkAboutT} The discussion there (eq:BinaryOperationInSecondModel) shows that one may usefully think of this heap operation (eq:HeapOperationInAGroup) as bracketed to the right

$$
  t(a,b,c) \,=\, a \cdot \big( b^{-1} \cdot c \big)
$$

and thought of as the action of group elements $a$ on elements of its [[underlying]] [[torsor]], which reduces to the group action on itself (eq:BinaryOperationInFirstModel) once the otherwise arbitrary origin "$b$" of this torsor is (re-)identified with the [[neutral element]].

There is also a dual version of this concept, see at *[[quantum heap]]*.

\begin{remark}\label{Disambiguation}
**(disambiguation)**
\linebreak
Heaps in the sense of algebra should not be confused with wither

* [heaps](http://en.wikipedia.org/wiki/Heap) in the sense of theoretical [[computer science]],

nor with

* *heaps of pieces* in the [[combinatorics|combinatorial]] sense of [Viennot 2006](https://link.springer.com/chapter/10.1007/BFb0072524).

\end{remark}

\begin{remark}\label{Synonyms}
**(synonyms)**
\linebreak
There are also a number of synonyms for the term 'heap'; [below](#HeapsAndTorsors) we consider '[[torsor]]' in this light.  In Russian one term for a heap is 'груда' ('gruda') meaning a heap of soil; this is a pun as it is parallel to the russian word 'группа' ('gruppa') meaning a group: forgetting the unit element is sort of creating an amorphous version.  This term also appears in English as 'groud'. In universal algebra the standard name is __associative Malcev algebra__ (in various spellings, including Mal'cev, Mal'tsev and Maltsev), other names include __herd__.
\end{remark}

## Definition

\begin{definition}
A __heap__ $(H,t)$ is a [[inhabited set|nonempty]] [[set]] $H$ equipped with a ternary operation $t \colon H \times H \times H\to H$ satisfying the following two relations

1. $ t(b,b,c) = c = t(c,b,b)$,

1. $ t\big(a,b,t(c,d,e)\big) = t\big(t(a,b,c),d,e\big) $.

\end{definition}

More generally, a ternary operation in some [[variety of algebras]] satisfying the first pair of equations is called a [[Mal'cev operation]].  A Mal'cev operation is  called _associative_ if it also satisfies the latter equation (i.e. it makes its domain into a heap).

A heap is **abelian** if it additionally satisfies the relation
$$
  t(a,b,c) = t(c,b,a)
$$

A **heap [[homomorphism]]**, of course, is a function that preserves the ternary operations.  This defines a [[category]] $Heap$ of heaps.

The [[hom-sets]] of the [[full subcategory]] $AbHeap$ of abelian heaps inherit an abelian heap structure from the pointwise operation in the codomain: given $f,g,h\colon H \to G$, the function $a\mapsto t_G(f(a),g(a),h(a))$ is again a heap homomorphism.

## Properties

### Automorphism groups of heaps
 {#AutomorphismsGroupsOfHeaps}

As indicated in (eq:HeapOperationInAGroup) a [[group]] $G$ becomes a heap by setting

\[
  a,b,c \,\in\, G
  \;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;
  t(a,b,c) \,\coloneqq\, a \cdot b^{-1} \cdot c
  \,.
\]

\begin{proposition}
\label{PrinIsEssentiallySurjective}
This construction (eq:HeapOperationInAGroup) defines a [[functor]]

$$
  Prin
    \,\colon\,
  Grp \longrightarrow Heap
$$

from the category [[Grp]] of [[groups]] to that of heaps,
which is [[essentially surjective functor|essentially surjective]], meaning that, up to [[isomorphism]] all heaps arise in this way.

In fact, there is a converse functor
$$
  Aut
   \,\colon\,
  Heap \longrightarrow Grp
$$

such that

1. there exist [[natural isomorphisms|natural]] group isomorphisms of the form

   $Aut\big(Prin(G)\big) \,\cong\, G$

1. there exist heap isomorphisms of the form

   $Prin\big(Aut(H)\big) \,\cong\, H$

   which however are *not* [[natural transformation|natural]] (whence we do *not* have an [[equivalence of categories]]).

\end{proposition}
This group $Aut(H)$ is called the *automorphism group* of the heap $H$.
\begin{proof}
Given a heap $H$, the claimed automorphism group  $Aut(H)$ is described, up to [[isomorphism]], by any of the following constructions:

1. Choosing any [[element]] $\mathrm{e} \in H$, then the [[binary operation]]

   \[
     \label{BinaryOperationInFirstModel}
     a \cdot b \,\coloneqq\, t(a, \mathrm{e} ,b)
     \,.
   \]

   constitutes a [[group]] [[structure]] on $H$, with [[neutral element]] $\mathrm{e}$.

   This serves as the required automorphism group: $Aut(H) \,\coloneqq\, (H, \cdot)$.

2. Take the [[underlying set]] of $Aut(H)$ to be that of [[equivalence classes]] of [[pairs]] $(a,b) \in H\times H$, subject to the [[equivalence relation]]

   $$
     (a,b) \,\sim\, (a',b')
     \;\;\;\;\text{iff}\;\;\;\;
     t(a,a',b') \,=\, b
   $$

   (the idea is to think of the pair $(a,b)$ as the representative of $a^{-1} \cdot b$)

   and take the [[binary operation]] on the group to be given on representatives by

   \[
     \label{BinaryOperationInSecondModel}
     (c,d) \cdot (a,b)
     \,\coloneqq\,
     \big( c, t(d,a,b) \big)
     \,.
   \]

   This again defines a group $Aut(H)$.

   Notice that:

   * the [[inverse element|inverse]] of (the [[equivalence class]] of) $(a,b)$ is (the equivalence class of) $(b,a)$

   * the [[neutral element]] is (the equivalence class of) $(a,a)$ (for any $a$).

3. Finally, $Aut(H)$ is realized also as an actual [[subgroup]] of the [[symmetric group]] on the [[underlying set]] of  $H$, analogously to [[Cayley's theorem]] for groups.  We take the elements of $Aut(H)$ to be set [[bijections]] of the form

   \[
      \label{GroupActionInThirdModel}
      t(-,a,b) \,\colon\, H \rightarrow H
      \,,
   \]

    where $a,b \in H$, with [[composition]] as the group's [[binary operation]].

   Notice here that

   \[
      t(-,c,d) \cdot_{Aut(H)} t(-, a,b)
      \;=\;
      t\big(t(-,c,d),a,b\big) \;=\; t\big(-,c,t(d,a,b)\big)
      \,,
   \]

   so that $Aut(H)$ is closed under this operation.

   The first axiom of a  heap shows that $Aut(H)$ contains the [[neutral element]] $t(-,x,x)$, for any  $x$), and the [[inverse element]] of $t(\cdot,a,b)$ is $t(\cdot,b,a)$; thus  $Aut(H)$ is a subgroup of the symmetric group of $H$.

It remains to see that these constructions all agree and are functorial.

First to see that the [[equivalence relations]] used in the second and third construction agree, notice that
the following are equivalent:

* (i) the bijections $t(\cdot,a,b)$ and $t(\cdot,a',b')$ coincide

* (ii) $t(a,a',b') = b$,

* (iii) $t(b,b',a') = a$.

Namely:

(ii) follows from (i) and $t(a,a,b) = b$.

(iii) follows from (ii) by applying $t(\cdot,b',a')$ on the
right. Similarly (ii) follows from (iii).

(i) follows from (ii) by the calculation:
\[ t(x,a',b') = t(t(x,a,a),a',b')= t(x,a,t(a,a',b'))
 = t(x,a,b).\]

Since the composition laws are also easily seen to agree, we have that the second two constructions of $Aut(H)$ are canonically isomorphic.

To compare them to the first construction, observe that for a fixed $\mathrm{e} \in H$, any equivalence class contains a unique pair of the form $(\mathrm{e},a)$.  (If $(b,c)$ is in the equivalence class, then $a$ is determined by $a = t(\mathrm{e},b,c)$.)  This sets up a bijection between the first two constructions, which we can easily show is an isomorphism.

Finally, the second constructions is manifestly functorial.
\end{proof}

### Relation to torsors
 {#HeapsAndTorsors}

The automorphism group $Aut(H)$ in (eq:PrinIsEssentiallySurjective) comes equipped with a canonical [[group action]] on $H$, as is most clear from the third definition (eq:GroupActionInThirdModel).

This action is

* [[transitive action|transitive]]

  by $t(a,a,b) = b$

and

* [[free action|free]]

  since if $t(a,b,c) = a$ then by the previous statement $t(x,b,c) = x$ for each $x$, and in particular $t(b,b,c) = b$ and also $t(b,b,c) = c$.

Therefore, $H$ is an $Aut(H)$-[[torsor]] (over a point).

Conversely, any [[torsor]] $H$ over a group $G$ becomes a heap, by defining

$$
  t(a,b,c) \,\coloneqq\, g \cdot c
  \,,
$$

where $g\in G$ is the unique group element such that $g\cdot b = a$.

In fact, the category $Heap$ is [[equaivalence of categories|equivalent]] to the following category $Tors$: its objects are pairs $(G,H)$ consisting of a group $G$ and a $G$-torsor $H$, and its morphisms are pairs $(\phi,f):(G,H)\to (G',H')$ consisting of a group homomorphism $\phi:G\to G'$ and a $\phi$-equivariant map $f:H\to H'$.

## The empty heap {#empty}

If we wish $Heap$ to be an [[algebraic category]], then we must remove the clause that the underlying set of a heap must be nonempty (see also at *[[pseudo-torsor]]*).  Then the [[empty set]] becomes a heap in a unique way.

However, in this case, the various theorems relating heaps to groups above all break down.  For this reason, one usually requires a heap to be inhabited.

On the other hand, we could generalize the notion of [[group]] to allow for an empty group.  This even remains a purely algebraic notion: we can define a group as an [[inhabited]] [[invertible semigroup]] $(S,\cdot,(-)^{-1})$ or an inhabited [[associative quasigroup]]. Then any [[invertible semigroup]] or [[associative quasigroup]] is a possibly-empty-heap, and every possibly-empty-heap arises in this way from its automorphism invertible semigroup or automorphism associative quasigroup (defined by either method (2) or (3)); the category of possibly-empty-heaps is equivalent to the category of invertible semigroups equipped with torsors over the point, and the category of associative quasigroups equipped with torsors over the point; etc.

This is even [[constructive mathematics|constructive]]; the theorems can be proved uniformly, rather than by treating the empty and inhabited cases separately. (This rather trivial method is obvious to a classical mathematician, but it\'s not constructively valid, since a invertible semigroup/associative quasigroup/heap as defined here can\'t be constructively proved empty or inhabited; it can only be proved empty iff not inhabited.  Indeed, taking any group $G$ and any [[truth value]] $P$, the invertible subsemigroup or associative subquasigroup $\{x \in G \;|\; P\}$ is empty or inhabited iff $P$ is false or true.)

## Examples in homotopy theory

The map $[0,1]\to[0,1]$ that sends $0\mapsto 0$, $1/3\mapsto 1$, $2/3\mapsto 0$, $1\mapsto 1$,
and interpolates linearly between these points
yields an associative Malcev operation on $[\Sigma X,Y]$,
where $X$ and $Y$ are (unpointed) spaces, $\Sigma$ is the [[suspension]] functor, and $[-,-]$ denotes the set of morphisms in the [[homotopy category]].

Thus, $[\Sigma X,Y]$ is a (nonabelian) heap.
Likewise, the full mapping space $Map(\Sigma X,Y)$ can be turned into an (∞,1)-heap, defined as an (∞,1)-algebra (in spaces) over the algebraic theory of heaps.

See Vokřínek {#Vokrinek} for more information.

## References and remarks

Related notions: [[torsor]], [[affine space]], [[Mal'cev variety]], [[truss]]

* [[Christopher D. Hollings]], [[Mark V. Lawson]], _Wagner’s theory of generalised heaps_, 2017, Springer.  [doi](https://doi.org/10.1007/978-3-319-63621-4)

* [[A. K. Sushkevich]], _Theory of generalised groups_, DNTVU, Kharkov-Kiev (1937) (Russian original: [[А. К. Сушкевич]], _Теория обобщенных групп_, Государственное научно-техническое издательство Украины, 1937).

* G.M. Bergman, A.O. Hausknecht,
_Cogroups and co-rings in categories of associative rings_, Ch.IV, paragraph 22, p.95ff, AMS 1996.

* Z. Škoda, _Quantum heaps, cops and heapy categories_, Mathematical Communications 12, No. 1, pp. 1--9 (2007); arXiv:[math.QA/0701749](http://arxiv.org/abs/math.QA/0701749)

* Thomas Booker, [[Ross Street]], _Torsors, herds and flocks_, J. Algebra 330 (2011) 346--374 [pdf](https://core.ac.uk/download/pdf/81154582.pdf) [arXiv:0912.4551](https://arxiv.org/abs/0912.4551)

* [[Peter T. Johnstone]], _The ‘closed subgroup theorem’ for localic herds and pregroupoids_, Journal of Pure and Applied Algebra 70 (1991) 97-106.  [doi](http://dx.doi.org/10.1016/0022-4049(91)90010-y).

* wikipedia:[heap](https://en.wikipedia.org/wiki/Heap_%28mathematics%29) (mathematics)

* [[John Baez]] on [Heaps and torsors](http://www.math.ucr.edu/home/baez/torsors.html)

* [[Lukáš Vokřínek]], _Heaps and unpointed stable homotopy theory_, [arXiv](https://arxiv.org/abs/1312.1709v1).

There is an oidification ([[horizontal categorification]]) of a heap, sometimes called a _heapoid_.

category: algebra

[[!redirects groud]]
[[!redirects grouds]]

[[!redirects heap]]
[[!redirects heaps]]
[[!redirects Heap]]
[[!redirects herd]]
[[!redirects associative Malcev algebra]]


hearing the shape of a drum

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{:toc}

## Idea

Physical oscillations (of [[elasticity|elastic]] bodies) are described via a [[wave equation]] and hence by the [[spectrum of an operator|spectrum]] of a [[Laplace operator]] on a [[Riemannian manifold]]. The phrase _hearing the shape of a drum_ in this context refers to the issue of recovering a Riemannian manifold from just the spectrum of the Laplace operator on it ("[[spectral geometry]]").

In plain Riemannian geometry there are counter-examples to the possibility of recovering Riemannian manifolds from just the spectrum of their Laplace operator. But from a little bit more of information, enhancing the Laplace operator to a _[[spectral triple]]_ it becomes possible in general. This theorem due to [[Alain Connes]] is the motivation for the formulation of all of metric geometry via [[spectral triples]], a program which is known as Connes' _[[noncommutative geometry]]_.

## Related concepts

* [[rubber-sheet analogy of gravity]]

## References

* Wikipedia, _[Hearing the shape of a drum](http://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum)_

* Pierre Martinetti, _[Designing the sound of a cut-off drum](https://arxiv.org/pdf/1502.02720.pdf)_


heat equation

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#### Differential geometry
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#### Equality and Equivalence
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#Contents#
* table of contents
{:toc}

## Idea

The _heat equation_ is a [[partial differential equation]]:

$$\frac{\partial u}{\partial t}(t,x)=\frac{\alpha}{2}\Delta u(t,x)$$

where $\Delta$ is a [[Laplace operator]].

Its [[fundamental solution]] is called the _[[heat kernel]]_.

## Related concepts

* [[wave equation]], [[Klein-Gordon equation]]

* [[Schrödinger equation]]

* [[Dirac equation]]

## References

See also

* Wikipedia, _[Heat equation](https://en.wikipedia.org/wiki/Heat_equation)_

[[!redirects heat equations]]


heat kernel

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* table of contents
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## Idea

### Heat kernel as a fundamental solution

One of the simplest linear partial differential equations of parabolic type is the heat (conductivity) equation. Recall that a [[fundamental solution]] of a linear partial differential operator $P$ is a solution of the PDE $P f = \delta$ where the inhomogeneous term $\delta$ is a [[delta function]] (in appropriate boundary conditions).

The [[fundamental solution]] of a [[heat equation]] is called the **heat kernel**.

### Role in index theory

The study of heat kernel led to a new simpler proof of the [[index theorem]] by [[Michael Atiyah|Atiyah]], [[Raoul Bott|Bott]] and Patodi.

### Heat kernel for operators over Riemannian manifolds

Let $E\to X$ be a smooth vector bundle over a [[Riemannian manifold]] $X$, $\Gamma(E)$ the space of the smooth sections of $E$ and $P:\Gamma(E)\to\Gamma(E)$ a positive self-adjoint elliptic differential operator. The **heat operator** symbolically denoted by $e^{-tP}:\Gamma(E)\to\Gamma(E)$ is an infinitely smoothening operator characterized by the property that

$$
\frac{d}{dt} (e^{-tP}u) = -Pe^{-tP}u
$$

for all $u\in\Gamma(E)$. The **heat kernel** $K$ for $P$ is then the kernel of an [[integral operator]] representing the heat operator:

$$
(e^{-tP}u)(x) = \int_X K_t(x,y) u(y) dy
$$

$K_t(x,y):E_y\to E_x$ is a linear map for all $x,y$ and $t$. Of course, one needs to justify this definition by the proof of the existence.

### Heat kernel and path integrals

The [[Schrödinger equation]] without potential term is similar to the
[[heat equation]] (there is an additional $\sqrt{-1}$); hence its fundamental solution is similar. The heat equation on the other hand can describe [[diffusion]]. Therefore also the similarity in the [[path integral]] description: the [[Wiener measure]] [[integral]] describes diffusion using [[Brownian motion]], similarly the Feynman path integral (for a finite-dimensional system) describes [[quantum mechanics]]; many points in the standard calculations are parallel.

## Related concepts

* [[Laplace operator]]

* [[heat kernel]]

## References

A standard textbook account is

* [[Nicole Berline]], [[Ezra Getzler]], [[Michele Vergne]], _Heat kernels and Dirac operators_, Grundlehren __298__, Springer 1992, "Text Edition" 2003.

For the relation to the [[index theorem]] see also

* [[Michael Atiyah]], [[Raoul Bott]], V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330.

Discussion in the context of [[renormalization]] in [[quantum field theory]] is around section 6.5 of

* [[Kevin Costello]], _Renormalisation and the Batalin-Vilkovisky formalism_ ([arXiv:0706.1533](http://arxiv.org/abs/0706.1533))

See also

* H. Blaine Lawson, Jr. , Marie-Louise Michelson, _Spin geometry_, Princeton Univ. Press 1989.

* Wikipedia, _[Heat kernel](http://en.wikipedia.org/wiki/Heat_kernel)_

[[!redirects heat kernels]]


Heath Emerson


heavy flavor

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## Idea

In the [[standard model of particle physics]], the [[masses]] of the various [[flavour (particle physics)|flavors]] of [[quarks]], and thus of the [[hadron]] [[bound states]] built from them, differ by orders of magnitude ("mass hierarchy"):

* The [[up quark]] and [[down quark]] are both very light and of comparable mass.

* The [[strange quark]] is an order of magnitude heavier than the [[up quark|up]] and [[down quark|down]].

* The [[charm quark]] and [[bottom quark]] are two orders of magnitude heavier than the [[strange quark]].

* The [[top quark]] is another two orders of magnitude heavier than the bottom quark.

Since in the formulation of [[quantum hadrodynamics]] as [[chiral perturbation theory]] one needs to decide which quarks count as "approximately massless" and which count as "approximately of infinite mass" for the purpose of [[effective field theory]], it is very common to group the quark species into "light" and "heavy".

There are two versions of this grouping in use:

1. In "2-flavor"-[[chiral perturbation theory]] with [[SU(2)]] [[chiral symmetry]]

* the [[up quark]] and [[down quark]] are regarded as light (i.e. the [[generation of fundamental particles|first generation quarks]])

* the [[strange quark]], [[charm quark]], [[bottom quark]] and [[top quark]] are regarded as heavy.

1. In "3-flavor"-[[chiral perturbation theory]] with [[SU(3)]] [[chiral symmetry]]

* the [[up quark]], [[down quark]] and also the [[strange quark]] are regarded as light,

* the [[charm quark]], [[bottom quark]] and [[top quark]] are regarded as heavy.

Either choice gives rise to a corresponding notion of light and heavy [[hadrons]].

For [[mesons]] one has:

1. In "2-flavor"-[[chiral perturbation theory]] with [[SU(2)]] [[chiral symmetry]]

* the [[pion]], [[omega-meson]] and [[rho meson]] are light (being [[bound states]] of [[up quarks]] and [[down quarks]]);

* the heavy mesons are the [[eta-meson]], [[phi-meson]] and [[K-mesons]] (involving a [[strange quark]]); the [[D-mesons]] (involving a [[charm quark]]) and the [[B-mesons]] (involving a [[bottom quark]])

1. In "3-flavor"-[[chiral perturbation theory]] with [[SU(3)]] [[chiral symmetry]]

* the [[pion]], [[omega-meson]], [[rho meson]] and [[K-mesons]] are light (being [[bound states]] of [[up quarks]], [[down quarks]] and [[strange quarks]]);

* the heavy mesons are the [[D-mesons]] (involving a [[charm quark]]) and the [[B-mesons]] (involving a [[bottom quark]])

For [[baryons]] one has:

(...)

## References

### General

* Benjamin Grinstein, _An Introduction to Heavy Mesons_ ([arXiv:hep-ph/9508227](https://arxiv.org/abs/hep-ph/9508227), [cds:285792](http://cds.cern.ch/record/285792))

* Daniel Kikoła, _Five things you should remember about heavy flavor measurements_,  Journal of Physics: Conference Series, Volume 612, 2014 ([doi:10.1088/1742-6596/612/1/012023](https://iopscience.iop.org/article/10.1088/1742-6596/612/1/012023))

* {#Brambilla14} Brambilla et al., Sections 3-4 in: _[[QCD and strongly coupled gauge theories - challenges and perspectives]]_, Eur Phys J C Part Fields. 2014; 74(10): 2981  ([arXiv:1404.3723](https://arxiv.org/abs/1404.3723), [doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5))

* D.R. Entem, P.G. Ortega, J. Segovia, F. Fernández, _Heavy mesons in the Quark Model_, EPJ Web of Conferences 199, 01012 (2019) ([arXiv:1809.06985](https://arxiv.org/abs/1809.06985), [doi:10.1051/epjconf/201919901012](https://doi.org/10.1051/epjconf/201919901012))

[[!include Skyrme hadrodynamics with heavy mesons -- references]]

[[!include heavy flavor hadrodynamics via holographic QCD -- references]]

[[!redirects light flavor]]
[[!redirects heavy flavors]]
[[!redirects light flavors]]
[[!redirects light and heavy flavor]]
[[!redirects light and heavy flavors]]

[[!redirects heavy quark]]
[[!redirects light quark]]
[[!redirects heavy quarks]]
[[!redirects light quarks]]
[[!redirects light and heavy quark]]
[[!redirects light and heavy quarks]]

[[!redirects heavy meson]]
[[!redirects light meson]]
[[!redirects heavy mesons]]
[[!redirects light mesons]]
[[!redirects light and heavy meson]]
[[!redirects light and heavy mesons]]

[[!redirects heavy vector meson]]
[[!redirects light vector meson]]
[[!redirects heavy vector mesons]]
[[!redirects light vector mesons]]
[[!redirects light and heavy vector meson]]
[[!redirects light and heavy vector mesons]]

[[!redirects heavy hadron]]
[[!redirects light hadron]]
[[!redirects heavy hadrons]]
[[!redirects light hadrons]]
[[!redirects light and heavy hadron]]
[[!redirects light and heavy hadrons]]

[[!redirects heavy baryon]]
[[!redirects light baryon]]
[[!redirects heavy baryons]]
[[!redirects light baryons]]
[[!redirects light and heavy baryon]]
[[!redirects light and heavy baryons]]

[[!redirects light lepton]]
[[!redirects light leptons]]


heavy flavor hadrodynamics via holographic QCD -- references

### Heavy flavor in holographic QCD
 {#HeavyFlavorHadrodynamicsViaHolographicQCDReferennces}

Discussion of [[heavy flavor]] in [[holographic QCD]]:

* Yizhuang Liu, [[Ismail Zahed]], _Heavy and Strange Holographic Baryons_, Phys. Rev. D 96, 056027 (2017) ([arXiv:1705.01397](https://arxiv.org/abs/1705.01397))

* Si-wen Li, _A holographic description of heavy-flavoured baryonic matter decay involving glueball_, Phys. Rev. D 99, 046013 (2019) ([arXiv:1812.03482](https://arxiv.org/abs/1812.03482))

  (via additional [[flavor branes]])

* Nicolas Kovensky, Andreas Schmitt, _Heavy Holographic QCD_ ([arxiv:1911.08433](https://arxiv.org/abs/1911.08433))

* Daisuke Fujii, [[Atsushi Hosaka]], _Heavy baryons in holographic QCD with higher dimensional degrees of freedom_ ([arXiv:2003.13415](https://arxiv.org/abs/2003.13415))

  (via higher [[KK-compactification|KK-modes]])

* {#ChenHuang21} Yidian Chen, Mei Huang, *Holographic QCD model for $N_f = 4$* ([arXiv:2110.08215](https://arxiv.org/abs/2110.08215))