Huan's inertia orbifold

[[!redirects Huan's inertia orbifold]]

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###Context###
#### Higher Geometry
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#### Higher Lie theory
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[[!include infinity-Lie theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The notion of an [[inertia orbifold]] is closely related to that of a [[free loop space]]/[[free loop stack]] of an orbifold. As such, one may expect there to be an [[action]] of the [[circle group]] $S^1$ on inertia orbifolds, corresponding to "rotation of loops". An explicit component-based modification of the inertia orbifold construction which builds in an $S^1$-action of sorts (akin to [[cyclic loop spaces]]/[[cyclic loop stacks]]) was highlighted in [Huan 18](#Huan18) (Def. \ref{HuanInertiaOrbifold} below), following [Ganter 07, Def. 2.3](#Ganter07) (and, apparently, following suggestions by [[Charles Rezk]]).

The [[orbifold K-theory]] of these modified inertia orbifolds ("quasi-elliptic cohomology") is closely related to [[equivariant elliptic cohomology]] at the [[Tate curve]] of the original orbifold (see also at *[[Tate K-theory]]*).

## Definition

In the following

* $G$ is a [[discrete group]];

* $\mathcal{X} \;\simeq\; X \!\sslash\! G$ is a [[good orbifold]] which is a [[global quotient orbifold]] of a [[smooth manifold]] $X$ by a [[smooth function|smooth]] [[proper action]] of $G$.

\begin{definition}\label{HuanCentralizer}
For $g \in G$ any element, with $C_g \subset G$ denoting its [[centralizer]], write
$$
  \Lambda_g
    \;\coloneqq\;
  \big(
    C_g \times \mathbb{R}
  \big)
  \,
  \big/
  \,
    \underset{
      \simeq \mathbb{Z}
    }{
    \underbrace{
      \langle (g^{-1},1) \rangle
    }}
$$

for the [[Lie group]] which is the [[quotient group]] of the [[direct product group]] of $G$ with the additive Lie group of [[real numbers]] by the [[subgroup]] ([[isomorphism|isomorphic]] to the [[natural numbers]]) which is generated from the [[pair]] $(g,-1) \in G \times \mathbb{R}$.

Hence this sits in a [[short exact sequence]] of [[Lie groups]] of this form:

\[
  1
  \to
  \mathbb{Z}
  \xhookrightarrow{ \; 1 \mapsto (g^{-1},1) \; }
  C_g \times \mathbb{R}
  \overset{\;p_g\;}{\twoheadrightarrow}
  \Lambda_g
  \to
  1
  \,.
\]

\end{definition}

\begin{remark}
  For $g \in G$ the [[group action]] of $G$ on $X$ restricts to an action of the [[centralizer]] $G$ on the [[fixed locus]] $X^g = X^{\langle g\rangle}$:

$$
  C_g \times X^g \xrightarrow{ (-) \cdot (-) } X^g
  \,.
$$

Moreover, since $g$ itself (a) commutes with all element in $C_g$ and (b) has trivial action on $X^g$, this [[lift|lifts]] to an action of $\Lambda_g$ (Def. \ref{HuanCentralizer})

\[
  \label{ActionOfHuanCentralizer}
  \array{
    \Lambda_g \times X^g
      &\xrightarrow{\;\;\;}&
    X^g
    \\
    ( [h,r], x ) &\mapsto& h \cdot x
    \,.
  }
\]

\end{remark}

The following definition modifies the [skeletal presentation of inertia orbifolds](inertia+orbifold#SkeletonForGlobalQuotientOrbifolds):

\begin{definition}\label{HuanInertiaOrbifold}
Write
  $$
    \Lambda_{S^1} \mathcal{X}
    \;\coloneqq\;
    \underset{[g] \in ConjCl(G)}{\coprod}
    X^g \!\sslash\! \Lambda_g
  $$
for the [[orbifold]] which is the [[disjoint union]] over the [[conjugacy classes]] $[g]$ of $G$ of the [[global quotient orbifolds]] of the [[fixed loci]] $X^g = X^{\langle g\rangle}$ by the [[group action]] (eq:ActionOfHuanCentralizer) of the group from Def. \ref{HuanCentralizer}.
\end{definition}

([Huan 18, Def. 2.14](#Huan18), review in [Dove 19, p. 62](#Dove19))

\begin{remark}\label{GanterOrbifold}
A similar definition is obtained by restricting [Ganter 07, Def. 2.3](#Ganter07) to constant loops and to $k = 1$, which yields
$$
  \simeq
  \underset{[g] \in ConjCl(G)}{\coprod}
  X^g \!\sslash\! (C_g/\langle g\rangle)
$$
\end{remark}

## Properties

### Relation to inertia orbifold

The canonical group homomorphism (via Def. \ref{HuanCentralizer})

$$
  \array{
    C_g
      &\xhookrightarrow{\;\;}&
    \Lambda_g
      &\xrightarrow{\;\;}&
    C_g/\langle g \rangle
    \\
    h
      &\mapsto&
    (h,0)
    \\
    &&
    [h,n]
    &\mapsto&
    [h]
  }
$$

induce canonical morphism from the plain [[inertia orbifold]] to Huan's (Def. \ref{HuanInertiaOrbifold}) and Ganter's orbifolds (Def. \ref{GanterOrbifold}):

$$
  \array{
    \Lambda\mathcal{X}
    &\xrightarrow{\;\;\;}&
    \Lambda_{S^1} \mathcal{X}
    &\xrightarrow{\;\;\;}&
    \cdots
    \,.
  }
$$

## Related concepts

* [[inertia orbifold]], [[free loop orbifold]]

* [[cyclic loop space]], [[cyclic loop stack]]

* [[Tate K-theory]], [[equivariant elliptic cohomology]]

* [[transchromatic character]]

## References

The notion is highlighted in:

* {#Huan18} [[Zhen Huan]], Def. 2.14 in: _Quasi-Elliptic Cohomology I_, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 ([arXiv:1805.06305](https://arxiv.org/abs/1805.06305), [doi:10.1016/j.aim.2018.08.007](https://doi.org/10.1016/j.aim.2018.08.007))

following

* [[Zhen Huan]], Exp. 2.1.14 in: _Quasi-elliptic cohomology_, 2017 ([hdl:2142/97268](http://hdl.handle.net/2142/97268))

following, in turn

* {#Rezk14} [[Charles Rezk]], *Quasi-Elliptic Cohomology* (2014) [[[Rezk-QuasiEllipticCohomology.pdf:file]]]

  > (publicly unavailable until upload here, in 2023)

which follows, finally, a similar construction in:

* {#Ganter07} [[Nora Ganter]], Def. 2.3 in: *Stringy power operations in Tate K-theory* ([arXiv:math/0701565](https://arxiv.org/abs/math/0701565))

and motivated (as made explicit on [p. 63](https://arxiv.org/pdf/1912.02374.pdf#page=69) of [Dove 19](#Dove19)) by the "[rotation condition](Tate+K-theory#RotationCondition)" on [[Tate K-theory]], due to

* {#Ganter13} [[Nora Ganter]], Def. 2.6 in: _Power operations in orbifold Tate K-theory_, Homology Homotopy Appl. Volume 15, Number 1 (2013), 313-342. ([arXiv:1301.2754](https://arxiv.org/abs/1301.2754), [euclid:hha/1383943680](https://projecteuclid.org/euclid.hha/1383943680))

following [Ganter 07, Def. 3.1](#Ganter07).

Streamlined review is in:

* {#Dove19} [[Thomas Dove]], p. 62 in: _Twisted Equivariant Tate K-Theory_ ([arXiv:1912.02374](https://arxiv.org/abs/1912.02374))

* [[Zhen Huan]], [[Matthew Spong]], Def. 2.1 in: *Twisted Quasi-elliptic cohomology and twisted equivariant elliptic cohomology* ([arXiv:2006.00554](https://arxiv.org/abs/2006.00554))

Characterization in [[cohesive homotopy theory]]:

* [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Cyclification of Orbifolds]]*, Comm. Math. Phys. (2023) [[arXiv:2212.13836](https://arxiv.org/abs/2212.13836), [[schreiber:cyclic loop spaces 2022|talk]]]

[[!redirects Huan's inertia orbifolds]]


Hubble's law

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### Context
#### Physics
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[[!include physicscontents]]
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#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[FLRW model]]

* [[dark energy]]

* [[big bang]]

* [[inhomogeneous cosmology]]

  [[KBC void]]

## References

### General
 {#ReferencesGeneral}

Hubble's law in relation to observation was arguably first discussed by

* [[Georges Lemaître]], *Un Univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques*, Annales de la Société Scientifique de Bruxelles, A **47** (1927) 49-59 [[adsabs:1927ASSB...47...49L](https://adsabs.harvard.edu/full/1927ASSB...47...49L)]

but gets its name from

* [[Edwin Hubble]], *A relation between distance and radial velocity among extra-galactic nebulae*, PNAS **15** (1929) 168-173 [[doi:10.1073/pnas.15.3.168](https://doi.org/10.1073/pnas.15.3.168)]

For more on the intricate history of the discovery of Hubble's law see:

* Harry Nussbaumer, Lydia Bieri, *Discovering the Expanding Universe*, Cambridge University Press (2009) [ISBN 978-0-521-51484-2]

* Harry Nussbaumer, Lydia Bieri, *Who discovered the expanding universe?* [[arXiv:1107.2281](https://arxiv.org/abs/1107.2281), [InSpire:917993](https://inspirehep.net/literature/917993)]

Review:

* R. Brent Tully, *The Hubble Constant: A Historical Review* [[arXiv:2305.11950](https://arxiv.org/abs/2305.11950)]

See also

* Wikipedia, _[Hubble's law](https://en.wikipedia.org/wiki/Hubble%27s_law)_

### Possible tension of model with observation
 {#H0TensionReferences}

Possible tensions in the [[standard model of cosmology]], regarding determination of the [[Hubble constant]] in the early and late universe, apparently observed by various groups, since the late 2010s:

review:

* L. Verde T. Treu, A.G. Riess, _Tensions between the Early and the Late Universe_ ([arXiv:1907.10625](https://arxiv.org/abs/1907.10625))

In

dication that the tension is not in the data but in systematic errors:

* Freedman et al. _The Carnegie-Chicago Hubble Program. VIII. An Independent Determination of the Hubble Constant Based on the Tip of the Red Giant Branch_ ([arxiv:1907.05922](https://arxiv.org/abs/1907.05922))

Indication that the tension is not in systematic errors:

* [[Adam Riess]], Gagandeep Anand, Wenlong Yuan, Stefano Casertano, Andrew Dolphin, Lucas Macri, Louise Breuval, Dan Scolnic, Marshall Perrin, Richard Anderson, *Crowded No More: The Accuracy of the Hubble Constant Tested\\with High Resolution Observations of Cepheids by JWST* ([arXiv:2307.15806](https://arxiv.org/abs/2307.15806))

Indication from new data from the [[James Webb Space Telescope]] that the old data does not have systematic errors and the tension is in the data:

* [[Wenlong Yuan]], [[Adam G. Riess]], [[Stefano Casertano]], [[Lucas M. Macri]], *A First Look at Cepheids in a SN Ia Host with JWST* ([arXiv:2209.09101](https://arxiv.org/abs/2209.09101))

Indication that the tension is not resolved by effects in [[inhomogeneous cosmology]] (such as the [[KBC void]]):

* Hao-Yi Wu, Dragan Huterer, _Sample variance in the local measurements of the Hubble constant_, Monthly Notices of the Royal Astronomical Society, Volume 471, Issue 4, November 2017, Pages 4946–4955 ([arXiv:1706.09723](https://arxiv.org/abs/1706.09723))

Indications that the Hubble constant is decreasing over time:

* Xuan-Dong Jia, Jian-Ping Hu, Fa-Yin Wang, *The evidence for a decreasing trend of Hubble constant*, ([arXiv:2212.00238](https://arxiv.org/abs/2212.00238))

On the status of the Hubble tension as of 25 January 2023

* Maria Dainotti, Biagio De Simone, Giovanni Montani, Tiziano Schiavone, Gaetano Lambiase, *The Hubble constant tension: current status and future perspectives through new cosmological probes* ([arXiv:2301.10572](https://arxiv.org/abs/2301.10572))

On the local value of the Hubble constant:

* [[Adam G. Riess]],  Louise Breuval, *The Local Value of $H_0$* ([arXiv:2308.10954](https://arxiv.org/abs/2308.10954))

On the status of various alternatives to the [[standard model of cosmology]] to resolve the Hubble tension:

* Ali Rida Khalife, Maryam Bahrami Zanjani, Silvia Galli, Sven Günther, Julien Lesgourgues, Karim Benabed, *Review of Hubble tension solutions with new SH0ES and SPT-3G data* ([arXiv:2312.09814](https://arxiv.org/abs/2312.09814))

That the Hubble tension might result from systematics in the [[cosmic microwave background]]:

* [[Stacy McGaugh]], *Early Galaxy Formation and the Hubble Tension* [[arXiv:2312.03127](https://arxiv.org/abs/2312.03127)]

[[!redirects Hubble constant]]

[[!redirects Hubble tension]]


Huber space

## Idea

one version of [[analytic spaces]] over [[nonarchimedean fields]].

## Related concepts

* [[Berkovich space]]

* [[perfectoid space]]

* [[global analytic geometry]]

## References

The original articles are

* {#Huber93} [[Roland Huber]], _Continuous valuations_, Math. Z., 212(3):455{477, 1993

* {#Huber93} [[Roland Huber]], _A generalization of formal schemes and rigid analytic varieties_. Math. Z., 217(4):513{551, 1994.

* {#Huber96} [[Roland Huber]],_ Etale cohomology of rigid analytic varieties and adic spaces_. Aspects of Mathematics, E30. Friedr. Vieweg & Sohn, Braunschweig, 1996.

Review with an eye towards [[perfectoid spaces]] is in

* [[Peter Scholze]], section 2 of _Perfectoid spaces_ ([arXiv:1111.4914](http://arxiv.org/abs/1111.4914))

[[!redirects Huber spaces]]

[[!redirects Huber adic space]]
[[!redirects Huber adic spaces]]

[[!redirects Huber's adic space]]
[[!redirects Huber's adic spaces]]


Hubert Goldschmidt

* [Institute webpage](https://www.ias.edu/scholars/hubert-goldschmidt)

* [Mathematica Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=10464)

## related $n$Lab entries

* [[differential equation]]

* [[formally integrable differential equation]]

category: people


Hubert Wagner

* [webpage](https://pub.ist.ac.at/~hwagner/)

## Selected writings

On [[persistent homology]] and [[Morse theory]]:

* [[Paweł Dłotko]], [[Hubert Wagner]], _Computing homology and persistent homology using iterated Morse decomposition_ $[$[arxiv/1210.1429](http://arxiv.org/abs/1210.1429)$]$

On [[persistent homology|persistent]] [[Cohomotopy]] and its improvement over [[well groups]]:

* [[Peter Franek]], [[Marek Krčál]], [[Hubert Wagner]], _Solving equations and optimization problems with uncertainty_, J Appl. and Comput. Topology (2018) 1: 297 ([arxiv:1607.06344](https://arxiv.org/abs/1607.06344), [doi:10.1007/s41468-017-0009-6](https://doi.org/10.1007/s41468-017-0009-6))

category: people


Hugh Everett III

* [Wikipedia entry](https://en.wikipedia.org/wiki/Hugh_Everett_III)

* A. L. Kuzemsky, *Biography of Hugh Everett (1930-1982)* [[pdf](http://theor.jinr.ru/~kuzemsky/Everett_Hugh.pdf), [web](http://theor.jinr.ru/~kuzemsky/everbio.html)]

## Selected writings

Introducing the *[[many-worlds interpretation of quantum mechanics]]*:

* {#Everett57a} [[Hugh Everett III]], *The Theory of the Universal Wavefunction* PhD thesis, Princeton (1957) [[pdf](https://www.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdf)] reprinted in: [De Witt & Graham 1973](#DeWittGraham73)

* {#Everett57b} [[Hugh Everett III]], *"Relative State" Formulation of Quantum Mechanics*, Rev. Mod. Phys. **29** (1957) 454-462 [[doi:10.1103/RevModPhys.29.454](https://doi.org/10.1103/RevModPhys.29.454)] (reprinted in [De Witt & Graham 1973](#DeWittGraham73))

* {#DeWittGraham73} [[Bryce S. DeWitt]], [[Neill Graham]] (eds.), *The Many-Worlds Interpretation of Quantum Mechanics*, Princeton Series in Physics, Princeton University Press (1973, 2016) [[ISBN:9780691645926](https://press.princeton.edu/books/hardcover/9780691645926/the-many-worlds-interpretation-of-quantum-mechanics), [jstor:j.ctt13x0wwk](https://www.jstor.org/stable/j.ctt13x0wwk)]

## Related entries

* [[interpretation of quantum mechanics]]

* [[deferred measurement principle]]

[[!redirects Hugh Everett]]


Hugh Osborn

* [webpage](http://www.damtp.cam.ac.uk/user/ho/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Hugh_Osborn)

## Selected writings

On [[D=4 N=1 super Yang-Mills theory]]:

* [[Hugh Osborn]], _$\mathcal{N}=1$ Superconformal Symmetry in Four Dimensional Quantum Field Theory_, Annals Phys. 272:243-294, 1999 ([arXiv:hep-th/9808041](https://arxiv.org/abs/hep-th/9808041))

category: people


Hugh Thomas

Interests: representation theory, algebraic geometry, algebraic combinatorics, discrete mathematics

[Homepage](http://www.math.unb.ca/~hugh)

[[!redirects hugh]]

category: people


Hugo Herbelin

* [webpage](http://pauillac.inria.fr/~herbelin/index-eng.html)

## Selected writings

On [[semi-simplicial types in homotopy type theory]]:

* {#Heberlin13} [[Hugo Herbelin]], *A dependently-typed construction of semi-simplicial types* (3/20/2013) [[pdf](https://ncatlab.org/ufias2012/files/semi-simplicial.pdf)]

category: people


Hugo Moeneclaey

* [website](https://www.hugo-moeneclaey.com/#publications)

## Publications

* [[Hugo Moeneclaey]], [[Peter Dybjer]], *Finitary higher inductive types in the groupoid model*, Electronic Notes in Theoretical Computer Science, Volume 336, pages 119-134, 2018 ([doi:10.1016/j.entcs.2018.03.019](https://doi.org/10.1016/j.entcs.2018.03.019))

## Talks

* [[Hugo Moeneclaey]], *Parametricity and cubes*, [[Homotopy Type Theory Electronic Seminar Talks]], 21 October 2021 ([video](https://www.youtube.com/watch?v=79Ik6KkoZ-w), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Moeneclaey-2021-10-21-HoTTEST.pdf))

category: people


Hugo Rossi

* [personal page](https://www.math.utah.edu/~rossi/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Hugo_Rossi)

## Selected writings

On [[analytic functions]] in [[complex analysis]]:

* {#GunningRossi} [[Robert C. Gunning]], [[Hugo Rossi]], *Analytic functions of several complex variables*, Prentice-Hall Inc., Englewood Cliffs (1965)

## Related entries

* [[Cartan theorem B]]

category: people


Hugo Steinhaus

Hugo Dyonizy Steinhaus was a mathematician at the University of Lwów and the University of Wrocław.

He get his PhD in 1911 from the University of Göttingen, advised by [[David Hilbert]].

His PhD students include [[Mark Kac]].

[[!redirects Hugo Dyonizy Steinhaus]]

category: people


Hurewicz cofibration

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#### Homotopy theory
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#Contents#
* table of contents
{:toc}

## Idea

In [[algebraic topology]] and [[homotopy theory]],
*Hurewicz cofibrations* are a kind of [[cofibration]] of [[topological spaces]], hence a kind of [[continuous function]] satisfying certain [[extension]] properties.

Specifically, a [[continuous function]] is a _Hurewicz cofibration_ ([Strøm 1966](#Strom66)) if it satisfies the [[homotopy extension property]] for all target spaces and with respect to the standard notion of [[left homotopy]] of topological spaces given by the standard topological [[interval object]]/[[cylinder object]].

A [[pointed topological space]] $(X,x)$ for which the base-point inclusion $\{x\} \xhookrightarrow{\;} X$ is a Hurewicz cofibration is called a *[[well-pointed topological space]]*. A [[simplicial topological space]] all whose degeneracy maps are Hurewicz cofibrations is called a *[[good simplicial topological space]]*.

In [[point-set topology]] Hurewicz cofibrations are often just called _cofibrations_,for short. If their [[image]] is a [[closed subspace]] they are called _[[closed cofibrations]]_.

Beware that there are other relevant classes of cofibrations between topological spaces, notably the *[[Serre cofibrations]]*, and (in [[homotopy theory]] and [[model category]]-theory, at least) also often just called "cofibrations". But "closed cofibration" always refers to closed Hurewicz cofibrations.

In fact, the notion of [[homotopy extension property]] makes sense in any [[category]] with a chosen [[cylinder object]]; and in this generality one also speaks of *[[h-cofibrations]]* (following [Mandell, May, Schwede & Shipley 2001, p. 16](#MandellMaySchwedeShipley01)).

## Definition
 {#Definition}

\begin{definition}\label{HurewiczCofibration}
**(Hurewicz cofibration)**
\linebreak
A map ([[continuous function]]) $i \colon A \xrightarrow{\;} X$ betwee [[topological spaces]] is a *Hurewicz cofibration* if it satisfies the [[homotopy extension property]] for all spaces, hence if

* for any map $f \colon X \longrightarrow Y$ and any [[left homotopy]] $\eta \,\colon\,  A \times [0,1] \longrightarrow Y$ such that $\eta(-,0) = f \circ i$, there exists a [[left homotopy]] $\widehat{\eta} \,\colon\,  X \times [0,1] \longrightarrow Y$ such that $\widehat{\eta} \circ (i \times id) = \eta$;

equivalently:

* given a [[commuting diagram]] in [[Top|TopSp]] of solid arrows as shown below, there exists a dashed arrow $\widehat \eta$ making all sub-diagrams commute:

\begin{tikzcd}[column sep={between origins, 40pt}]
    A
    \ar[
      dd,
      "{i}"{left}
    ]
    \ar[
      rr,
      "{(\mathrm{id},\, 0)}"
    ]
    &&
    A \times [0,1]
    \ar[
      dd,
      "{ i \times \mathrm{id} }"{right}
    ]
    \ar[
      dl,
      "{\eta}"{left, yshift=3pt}
    ]
    \\
    &
    Y
    \\
    X
    \ar[
      rr,
      "{ (\mathrm{id},\, 0) }"{below}
    ]
    \ar[
      ur,
      "{f}"
    ]
    &&
    X \times [0,1]
    \ar[
      ul,
      dashed,
      "{ \widehat \eta }"{description}
    ]
\end{tikzcd}
 \end{definition}

\begin{remark}
**(re-formulation in terms of right homotopies)**
\linebreak
In a [[convenient category of topological spaces|convenient]] [[cartesian closed category]] $TopSp$ of [[topological spaces]] (such as that of [[compactly generated topological spaces]]) the [[product]] $\dashv$ [[internal hom]]-[[adjunction]]

$$
  TopSp
    \underoverset
    {\underset{ (-)^{[0,1]} }{\longrightarrow}}
    {\overset{ (-) \times [0,1]  }{\longleftarrow}}
    {\;\;\;\;\;\bot\;\;\;\;\;}
  TopSp
$$

allows to pass to [[adjunct]] morphisms in Def. \ref{HurewiczCofibration}

$$
  \array{
    TopSp
    \big(
      A \times [0,1]
      ,\,
      Y
    \big)
    &\simeq&
    TopSp
    \big(
      A
      ,\,
      Y^{[0,1]}
    \big)
    \\
    \eta &\leftrightarrow& \eta'
  }
$$

and equivalently express the above diagram of [[left homotopies]] as the following (somewhat more transparent) diagram of [[right homotopies]], where $ev_0 \,\colon\, Y^{[0,1]}\to Y$ denotes [[evaluation]] at 0 $\gamma \mapsto \gamma(0)$:

\begin{tikzcd}
    A
    \ar[
      d,
      "{i}"{left}
    ]
    \ar[
      r,
      "{ \eta' }"{above}
    ]
    &
    Y^{[0,1]}
    \ar[
      d,
      "\mathrm{ev}_0"
    ]
    \\
    X
    \ar[
      r,
      "{f}"{below}
    ]
    \ar[
      ur,
      dashed,
      "\widehat \eta'"{description}
    ]
    &
    Y
\end{tikzcd}

In this equivalent formulation, the homotopy extension property is simply the [[right lifting property]] against the [[evaluation]] map $ev_0 \;\colon\; Y^{[0,1]} \xrightarrow{\;} Y$ out of the [[path space]] of $Y$.
\end{remark}

(e.g. [May 1999, p. 43 (51 of 251)](#May99))

\begin{definition}\label{ClosedHurewiczCofibration}
**(closed cofibrations)**
\linebreak
A Hurewicz cofibration $i \colon A\to X$ (Def. \ref{HurewiczCofibration}) is called a _[[closed cofibration]]_ if the [[image]] $i(A)$ is a [[closed subspace]] in $X$. In this case the pair $(A,X)$ is also called an *[[NDR-pair]]*.
\end{definition}

## Properties

### (Non-)Closed images
 {#Closedness}

\begin{prop}
Every Hurewicz cofibration $i \colon A \to X$ is [[injective function|injective]] and a [[homeomorphism]] onto its image. If $X$ is [[Hausdorff space|Hausdorff]], then $i$ is [[closed map|closed]].
\end{prop}

([tDKP 1970](#tDKP70) (1.17)).

\begin{prop}\label{HurewiczCofibrationsInCGWHSpacesAreClosed}
In the category of [[weakly Hausdorff space|weakly Hausdorff]] [[compactly generated spaces]], the image $i(A)$ of a Hurewicz cofibration is always closed.
The same holds in the category of all [[Hausdorff spaces]].
\end{prop}

(e.g. [May 1999, Sec. 6.2, p. 44 (52 of 251)](#May99))

But in the plain category [[Top]] of all topological spaces there are pathological counterexamples:

\begin{example}
Let $A =\{a\}$ and $X=\{a,b\}$ be the one and two element sets, both with the [[codiscrete topology]] (only $X$ and $\varnothing$ are [[open subsets]] of $X$), and $i:A\hookrightarrow X$ is the inclusion $a\mapsto a$. Then $i$ is a non-closed cofibration.
\end{example}
([Strøm 1966, p. 5](#Strom66))

### Characterization for closed subspace inclusions
 {#Inclusions}

When the [[image]] of a Hurewicz cofibration is a [[closed subspace]] -- which is automatically the case in [[weakly Hausdorff topological spaces]], by Prop. \ref{}HurewiczCofibrationsInCGWHSpacesAreClosed -- then there are a number of equivalent reformulations of the defining homotopy extension property (Def. \ref{HurewiczCofibration}).

#### Via retracts of the pushout-product with $0 \hookrightarrow [0,1]$

\begin{proposition}\label{CharacterizationViaRetractionOfPushoutProduct}
  A [[closed subset|closed]] [[topological subspace]]-inclusion $A \xhookrightarrow{\;i\;} X$ is a Hurewicz cofibration precisely if its [[pushout product]] with the endpoint inclusion $0 \,\colon\,  \ast \xhookrightarrow{\;} [0,1]$ into the [[topological interval]]
\[
  \label{PushoutProductMap}
  X \!\times\! \{0\}
  \,\cup\,
  A \!\times\! [0,1]
  \;
  \xrightarrow{\;\;\;\;\;\;}
  \;
  X \times [0,1]
\]

{#TheRetraction} admits a [[retraction]] $r$:

\begin{tikzcd}
        \mathrm{X} \!\times\! \{0\}
        \;\cup\;
        \mathrm{A} \!\times\! [0,1]
        \ar[
          r,
          "{ i \,\Box\, 0  }"
        ]
        \ar[
          rr,
          rounded corners,
          to path={
            -- ([yshift=-9pt]\tikztostart.south)
            --node[below]{
                \scalebox{.7}{$\mathrm{id}$}
              }
              ([yshift=-9pt]\tikztotarget.south)
            -- (\tikztotarget.south)}
        ]
        &
        \mathrm{X} \times [0,1]\;.
        \ar[
          r,
          dashed,
          "{r}"
        ]
        &
        \mathrm{X} \!\times\! \{0\}
        \;\cup\;
        \mathrm{A} \!\times\! [0,1]
\end{tikzcd}

\end{proposition}
([Strøm 1968 Thm. 2](#Strom68); review in [May 1999, Sec. 6.4, p. 45 (53 of 251)](#May99); [AGP 2002, Thm. 4.1.7](#AGP02); [Gutiérrez, Prop. 8.3](#Gutierrez))

\begin{example}\label{KifiedProductsOfCofibrationsWithCompactlyGeneratedSpaces}
  Let $A \xhookrightarrow{i} X$ be a [[closed subset|closed]] [[topological subspace]]-inclusion of [[compactly generated topological spaces]] which is a Hurewicz cofibration. Then for $Y$ any [[compactly generated topological space]], the [[k-ification|k-ified]] [[product space]]-construction $Y \times A \xhookrightarrow{ id_Y \times i  } Y \times X$ is itself a Hurewicz cofibration.
\end{example}
(e.g. [AGP 2002, Ex. 4.2.16](#AGP02))
\begin{proof}
  Since [[compactly generated spaces]] with the k-ified product space construction form a [[cartesian closed category]], the operation $Y \times (-)$ is a [[left adjoint]] and [[left adjoints preserve colimits|hence]] [[preserved colimit|preserves]] the [[pushout]] in (eq:PushoutProductMap). It follows that the retract which exhibits, according to Prop. \ref{CharacterizationViaRetractionOfPushoutProduct}, the cofibration property of $i$, may be extended as a [[constant function]] in $Y$ to yield a retract that exhibits the cofibration property of $Y \times i$.
\end{proof}

#### Via neighbourhood deformations

\begin{proposition}\label{CharactrerizationViaNeighbourhoodDeformation}
  A [[closed subset|closed]] [[topological subspace]]-inclusion $A \xhookrightarrow{\;i\;} X$ is a Hurewicz cofibration precisely if the following condition holds:

There exists:

(1) a [[neighbourhood]] $U$ of $A$ in $X$

   $$
     A \xhookrightarrow{\;} U \xhookrightarrow{\;} X
   $$

(2) a [[left homotopy]] $\eta$ shrinking this neighbourhood into $A$ relative to $A$, in that it makes the following [[commuting diagram|diagram commute]]:

\begin{tikzcd}[column sep=20pt]
    &
    A \times [0,1]
    \ar[
      dr,
      ->>
    ]
    \\[-25pt]
    U
    \ar[
      d,
      hook,
      "{
        (\mathrm{id}, 0)
      }"{left}
    ]
    \ar[
      drr,
      hook,
      "{
      }"
    ]
    &
    &
    A
    \ar[
      d,
      hook
    ]
    \\
    U \times [0,1]
    \ar[
      rr,
      dashed,
      "{\eta}"{description}
    ]
    \ar[
      from=uur,
      hook,
      crossing over
    ]
    &&
    X
    \\
    U
    \ar[
      u,
      hook,
      "{
        (\mathrm{id}, 1)
      }"{left}
    ]
    \ar[
      rr,
      dashed
    ]
    &&
    A
    \ar[
     u,
     hook
    ]
\end{tikzcd}

(3) another [[continuous function]] $\phi \colon X \xrightarrow{\;} [0,1]$ which makes the following [[commuting diagram|diagram commute]]:

\begin{tikzcd}[row sep=small]
    A
    \ar[
      d,
      hook
    ]
    \ar[rr]
    &&
    \{0\}
    \ar[
      d,
      hook
    ]
    \\
    X
    \ar[
      rr,
      dashed,
      "{\phi}"{description}
    ]
    &&
    {[0,1]}
    \\
    X \setminus U
    \ar[
      u,
      hook
    ]
    \ar[rr]
    &&
    \{1\}
    \ar[
      u,
      hook
    ]
\end{tikzcd}

and in fact such that only $A$ is the [[preimage]] of zero: $A \,=\, \phi^{-1}(\{0\})$.
\end{proposition}

(This is due to [Strøm 1966, Thm. 2](#Strom66); recalled, e.g., in [Bredon 1993, Thm. 1.5 on p. 431](#Bredon93))

### Assorted facts
 {#AssortedFacts}

\begin{prop}\label{hCofibrationIntoNormalSpaceIffSoIntoAnyOpenNeighbourhood}
  If $X$ is a [[normal topological space]] then any inclusion $A \xhookrightarrow{\;} X$ is an h-cofibration iff its factorization $A \hookrightarrow{\;} U_A$ through any [[open neighbourhood]] $U_A \subset X$ of $A$ is an h-cofibration.
\end{prop}
([AGP 2002, Prop. 4.1.9](#AGP02))

\begin{prop}
  The [[composition]] of two h-cofibrations is itself an h-cofibration.
\end{prop}
(e.g. [AGP 2002, Ex. 4.2.17](#AGP02))

### Interaction with (fiber) producs
 {#InteractionWithFiberProducts}

\begin{prop}\label{CrossedUnionOfhCofsIshCof}
  If

  $$
    \array{
      A_1
      & &
      A_2
      \\
      \big\downarrow {}^{\mathrlap{i_1}}
      & , &
      \big\downarrow {}^{\mathrlap{i_2}}
      \\
      X_1
      & &
      X_2
    }
  $$

  is a [[pair]] of Hurewicz cofibrations, with $i_1(A_1) \underset{clsd}{\subset} X_1$ [[closed subspace|closed]], then the inclusion

  $$
    \array{
      A_1 \times X_2
      \;\cup\;
      X_1 \times A_2
      \\
      \big\downarrow
      \\
      X_1 \times X_2
    }
  $$

  into the [[product space]] of the [[codomains]]
  is itself a Hurewicz cofibration.

\end{prop}
([Strøm 1968, Thm. 6](#Strom68))

\begin{proposition}\label{FiberProductPreserveshCofibrations}
**([[fiber product]] of [[Hurewicz fibrations]] preserves [[Hurewicz cofibrations]])**
\linebreak
Let

$$
  \array{
     X_0 &\hookrightarrow & X
     \\
     {}^{\mathllap{p_0}}\downarrow && \downarrow^{\mathrlap{p}}
     \\
     B_0 &\hookrightarrow& B
     \\
     \uparrow && \uparrow
     \\
     E_0 &\hookrightarrow& E
  }
$$

be a [[commuting diagram]] of [[topological spaces]] such that

* the horizontal morphisms are closed cofibrations;

* the morphisms $p_0$ and $p$ are [[Hurewicz fibrations]].

Then the induced morphism on [[fiber products]]

$$
  X_0 \times_{B_0} E_0 \hookrightarrow X \times_B E
$$

is also a closed cofibration.
\end{proposition}
([Kieboom 1987, Thm. 1](#Kieboom87)).

\begin{corollary}
**([[Cartesian product]] preserves [[Hurewicz cofibrations]])**
\linebreak
If

  $$
    \array{
      A_1
      & &
      A_2
      \\
      \big\downarrow {}^{\mathrlap{i_1}}
      & , &
      \big\downarrow {}^{\mathrlap{i_2}}
      \\
      X_1
      & &
      X_2
    }
  $$

is a [[pair]] of Hurewicz cofibrations, then their image under the [[product]] functor is, too:

$$
  \array{
    A_1 \times A_2
    \\
    \big\downarrow
    \\
    X_1 \times X_2
  }
$$

\end{corollary}
\begin{proof}
  This is the special case of Prop. \ref{FiberProductPreserveshCofibrations} for $B$ and $B_0$ being the [[point space]]
\end{proof}

### Strøm's model structure
 {#StromModelStructure}

The collections

* [[Hurewicz fibrations]],

* closed Hurewicz cofibrations (def. \ref{HurewiczCofibration}, def. \ref{ClosedHurewiczCofibration}),

* and [[homotopy equivalences]]

make one of the standard Quillen [[model category]] structures on the category [[Top]] of all topological spaces _[[Strøm's model category]].

The identity functor $id \colon Top \to Top$ is [[Quillen adjunction|left Quillen]] from the [[classical model structure on topological spaces]] (or the mixed model structure) to the Strøm model structure, and of course right Quillen in the other direction.

$$
  Top_{Strom}
    \stackrel{\overset{id}{\longleftarrow}}{\underset{id}{\longrightarrow}}
  Top_{Quillen}
  \,.
$$

This means in particular that any  [[retract]] of a  [[relative cell complex]] inclusion is a closed Hurewicz cofibration.

## Examples
 {#Examples}

### Relative cell complex inclusions

\begin{example}\label{RelativeCWComplexes}
  Any [[relative CW complex]]-inclusion is an h-cofibration.
\end{example}
Proofs may be found spelled out in: [Bredon 1993, Cor. I.4 on p. 431](#Bredon93), [Félix, Halperin & Thomas 2000, Prop. 1.9](rational+homotopy+theopry#FelixHalperinThomas00), [Mathew 2010b](#Mathew10b)

More generally, every [[retract]] of a [[relative cell complex]] inclusion is a closed Hurewicz cofibration.

This is part of the statement of the [[Quillen adjunction]] between then [[classical model structure on topological spaces]] and the [[Strøm model structure]] (see [below](#StromModelStructure)).

### Points in locally Euclidean Hausdorff spaces
 {#BasepointsInLocallyEuclideanSpaces}

\begin{proposition}
  Any [[point]]-inclusion into a ([[finite number|finite]]-[[dimension of a manifold|dimensional]]) [[locally Euclidean space|locally Euclidean]] [[Hausdorff space]] (e.g. a [[topological manifold]]) is an h-cofibration.
\end{proposition}
This is a simple special case of the general Prop. \ref{ClosedSubmanifoldsOfParacompactBanachManifoldsArehCof} below, but we give an explicit proof:
\begin{proof}
\label{ProofThatPointInclusionsInLocallyEuclideanHausdorffSpacesArehCofibs}
By definition of [[locally Euclidean spaces]], any point has a [[neighbourhood]] $U$ which is [[chart]], being a [[Euclidean space]] that may be identified with a [[vector space]] $\mathbb{R}^n$ with the given point being the origin $0 \,\in\, \mathbb{R}^n$:

$$
  \{0\}
    \;\in\;
  \mathbb{R}^n
    \;\simeq\;
  U
    \;\subset\;
  X
  \,.
$$

Now let:

* $\eta \colon U \times [0,1]\to X$ the homotopy given by $(\vec x, t) \mapsto (1-t)\cdot \vec x$;

* $K \;\coloneqq\; B_{\leq 1}(0)$ the [[closed ball]] of unit [[radius]] around the origin in $\mathbb{R}^n$ (hence [[compact space|compact]] by [[Heine-Borel theorem|Heine-Borel]]);

* $\phi \colon X \to[0,1]$ be given by:

  1. $x \mapsto min\big( \| x\|, 1 \big)$ on $U$ (the [[distance]] from the origin cut off at 1),

  1. $x \mapsto 1$ on the [[complement]] $X \setminus K$.

  It is manifest that this is a well-defined function and that the [[restrictions]] $\phi|_{U}$ and $\phi|_{X\setminus K}$ are [[continuous functions]]. Moreover, since [[compact subspaces of Hausdorff spaces are closed]], $X \setminus K$ is [[open subset|open]] and $\big\{ U ,\, X \setminus K \big\}$ is an [[open cover]] of $X$. Therefore (by the [[sheaf]]-property of continuous functions), $\phi$ is continuous on all of $X$.

It is immediate to see that this data satisfies the conditions discussed in Prop. \ref{CharactrerizationViaNeighbourhoodDeformation}. Since [[Hausdorff spaces]] are [[T1|$T_1$]],  so that all of their points are [[closed point|closed]], that proposition applies and implies the claim.
\end{proof}

### Closed submanifolds of Banach manifolds
 {#ClosedSubmanifoldsOfBanachManifolds}

\begin{proposition}\label{ClosedInclusionIntoANRIsANRIffhCofibration}
  Let $X$ be an [[absolute neighbourhood retract]] (ANR) and $A \xhookrightarrow{i} X$ a [[closed subspace]]-inclusion. Then $i$ is a [[Hurewicz cofibration]] iff $A$ is itself an ANR.
\end{proposition}
([Aguilar, Gitler & Prieto 2002, Thm. 4.2.15](#AGP02))

\begin{proposition}\label{ClosedSubmanifoldsOfParacompactBanachManifoldsArehCof}
  Let $X$ be a [[paracompact topological space|paracompact]] [[Banach manifold]]. Then the inclusion $A \hookrightarrow X$ of any [[closed subspace|closed]] sub-[[Banach manifold]] is a [[Hurewicz cofibration]].
\end{proposition}
\begin{proof}
  Being a closed subspace of a paracompact space, $A$ is itself paracompact (by [this Prop.](paracompact+topological+space#ClosedSubspacesOfParacompactsAreParacompact)). But paracompact Banach manifolds are [[absolute neighbourhood retracts]] ([this Prop.](absolute+retract#ParacompactBanachManifoldsAreANRs))
Therefore the statement follows with Prop. \ref{ClosedInclusionIntoANRIsANRIffhCofibration}.
\end{proof}

### Further examples

\begin{example}\label{PointInclusionIntoPUH}
**(point inclusion into [[PU(ℋ)]])**
\linebreak
  The [[projective unitary group]] [[PU(ℋ)]] on an infinite-dimensional [[separable Hilbert space|separable]] [[Hilbert space]] is

* a [[Banach Lie group]] in its [[norm topology]], and as such [[well-pointed topological group|well-pointed]] by Prop. \ref{ClosedSubmanifoldsOfParacompactBanachManifoldsArehCof};

* no longer a Banach space in its weak/strong [[operator topology]], but nevertheless still [[well-pointed topological group|well-pointed]] in this case, by [this Prop.](projective+unitary+group+on+a+Hilbert+space#WellPointedInOperatorTopology).

\end{example}

## Related concepts

* [[Hurewicz fibration]]

* [[Strøm model structure]]

* [[Serre cofibration]]

## References

Named after:

* [[Witold Hurewicz]], _On the concept of fiber space_, Proc. Nat. Acad. Sci. USA __41__ (1955) 956--961; ([doi:10.1073%2Fpnas.41.11.956](https://dx.doi.org/10.1073%2Fpnas.41.11.956), [jstor:89187](https://www.jstor.org/stable/89187), [pdf](http://www.pnas.org/content/41/11/956.full.pdf). MR0073987)

Original articles:

* {#Strom66} [[Arne Strøm]], _Note on cofibrations_,  Math. Scand.  **19** (1966) 11-14 ([jstor:24490229](https://www.jstor.org/stable/24490229), [dml:165952](https://eudml.org/doc/165952), MR0211403)

* {#Strom68} [[Arne Strøm]], _Note on cofibrations II_,  Math. Scand.  **22** (1968) 130--142 ([jstor:24489730](https://www.jstor.org/stable/24489730), [dml:166037](https://eudml.org/doc/166037), MR0243525)

* [[Dieter Puppe]], _Bemerkungen über die Erweiterung von Homotopien_, Arch. Math. (Basel) 18 1967 81--88 ([doi:10.1007/BF01899475](http://dx.doi.org/10.1007/BF01899475), MR0206954)

Textbook accounts:

* {#tDKP70} [[Tammo tom Dieck]], [[Klaus Heiner Kamps]], [[Dieter Puppe]], Chapter I of: *Homotopietheorie*, Lecture Notes in Mathematics **157** Springer 1970 ([doi:10.1007/BFb0059721](https://link.springer.com/book/10.1007/BFb0059721))

* {#Bredon93} [[Glen Bredon]], Section VII.1 of: _Topology and Geometry_, Graduate texts in mathematics **139**, Springer 1993 ([doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0),  [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf))

* {#May99} [[Peter May]], Chapter 6 of: *[[A concise course in algebraic topology]]*, University of Chicago Press 1999 ([ISBN: 9780226511832](https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3777031.html), [pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf))

* {#AGP02} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, Section 4.1 in: _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([doi:10.1007/b97586](https://link.springer.com/book/10.1007/b97586), [toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))

Lecture notes:

* {#Gutierrez} [[Javier Gutiérrez]], *Cofibrations*, ([pdf](https://www.math.ru.nl/~gutierrez/files/Lecture08.pdf), [[Gutierrez_Cofibrations.pdf:file]])

Exposition:

* {#Mathew10a} [[Akhil Mathew]], *Cofibrations*, 2010 ([web](https://amathew.wordpress.com/2010/10/07/cofibrations/))

* {#Mathew10b} [[Akhil Mathew]], *Examples of cofibrations*, 2010 ([web](https://amathew.wordpress.com/2010/10/08/examples-of-cofibrations/))

The fact that morphisms of fibrant pullback diagrams along closed cofibrations induce closed cofibrations is in

* {#Kieboom87} [[Rudger Kieboom]], _A pullback theorem for cofibrations_, Manuscripta Math **58** (1987) pp 381–384 ([doi:10.1007/BF01165895](https://doi.org/10.1007/BF01165895))

The terminology "h-cofibration" is due to:

* {#MandellMaySchwedeShipley01} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], p. 16 in: _[[Model categories of diagram spectra]]_, Proceedings of the London Mathematical Society, 82 (2001), 441-512 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf), [doi:10.1112/S0024611501012692]( https://doi.org/10.1112/S0024611501012692))

[[!redirects Hurewicz cofibrations]]

[[!redirects closed cofibration]]
[[!redirects closed cofibrations]]

[[!redirects closed Hurewicz cofibration]]
[[!redirects closed Hurewicz cofibrations]]


Hurewicz connection

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

#Contents#
* table of contents
{:toc}

## Setup and definition ##

Given a [[continuous map]] $\pi : E\to B$ of [[topological spaces]], one constructs the __[[mapping cocylinder]]__ $Cocyl(\pi)$ as the [[pullback]]

$$
  \array{
    Cocyl(\pi) &\overset{pr_{\mathcal{P}(B)}}\to & \mathcal{P}(B)
    \\
    \pr_E\downarrow && \downarrow
    \\
    E & \stackrel{\pi}\to & B
  }
$$

where $\mathcal{P}(B)$ is the [[path object|path space]] in [[Top]], the space of continuous [[interval object|path]]s
$u:[0,1]\to B$ in $B$, and where $\mathcal{P}(B)\to B$ is the map sending a path $u$ to its value $u(0)$.
The cocylinder can be realized as a subspace of $E\times \mathcal{P}(B)$ consisting of pairs $(e,u)$ where $e\in E$ and $u:[0,1]\to B$ are such that $\pi(e)=u(0)$.

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

A __Hurewicz connection__ is any continuous [[section]]
$$s:Cocyl(\pi)\to \mathcal{P}(E)$$
of the map $\pi_!:\mathcal{P}(E)\to Cocyl(\pi)$ given by $\pi_!(u)=(u(0),\pi\circ u)$.

=--

## Characterization of Hurewicz fibrations ##

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

A map $\pi:E\to B$ is a __[[Hurewicz fibration]]__ iff there exists at least one Hurewicz connection for $\pi_!$.

=--

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

To see that consider the following diagram
$$\begin{matrix}
Y& \stackrel{\theta}\to & Cocyl(\pi) &\overset{pr_E}\to & E\\
\sigma_0\downarrow&&\sigma_0\downarrow&&\downarrow \pi\\
Y\times I&\stackrel{\theta\times I}\underset{}{\to}& Cocyl(\pi)\times I& \underset{ev}\to &B
\end{matrix}$$
where $pr_E: Cocyl(\pi)\to E$ is the restriction of the projection $E\times B^I\to E$ to the factor $E$ and the map $Cocyl(\pi)\times I\to B$ is the evaluation $(e,u,t)\mapsto u(t)$ for $(e,u)\in Cocyl(\pi)$. The right-hand square is commutative and this square defines a homotopy lifting problem. If $\pi$ is a fibration this **universal** homotopy lifting problem has a solution, say $\tilde{s}:Cocyl(p)\times I\to E$. By the hom-mapping space adjunction (exponential law) this map corresponds to some map $s:Cocyl(\pi)\to \mathcal{P}(E)$. One can easily check that this map is a section of $\pi_!$.

Conversely, let a Hurewicz connection $s$ exist, and fill the right-hand square of the diagram with diagonal $\tilde{s}$ obtained by hom-mapping space adjunction. Let the data for the general homotopy lifting problem be given: $\tilde{f}:Y\to E$, $F:Y\times I\to B$ with $F_0 = p\circ \tilde{f}:Y\to E$; let furthermore $F':Y\to \mathcal{P}(B)$ be the map obtained from $F$ by the hom-mapping space adjunction. By the universal property of the cocylinder (as a pullback), there is a unique mapping $\theta: Y\to Cocyl(\pi)$ such that $pr_{\mathcal{P}(B)}\circ\theta=F':Y\to \mathcal{P}(B)$ and $pr_E\circ\theta =\tilde{f}:Y\to E$. Now notice that the by composing the horizontal lines we obtain $\tilde{f}$ upstairs and $F$ downstairs, hence the external square is the square giving the homotopy lifting problem for this pair. The lifting is then given by $\tilde{s}\circ (\theta\times id_I):Y\times I\to E$. Simple checking finishes the proof.

=--

Of course there are many other equivalent characterizations of Hurewicz fibrations.

## Special cases and properties ##

If $\pi:E\to B$ is a [[covering space]] where $B$ is [[Hausdorff space|Hausdorff]], then $\pi_!$ is a [[homeomorphism]]; thus in that case the Hurewicz connection is unique.

If $\pi$ is a smooth principal bundle equipped with a distribution of horizontal spaces forming an [[Ehresmann connection]], then one can define a corresponding "smooth" Hurewicz connection in the sense that the Ehresmann connection provides a continuous choice of smooth path lifting, with prescribed initial point, of a smooth path in the base. This can be expressed in terms as a continuous section of $\pi_!^{smooth}:\mathcal{P}^{smooth}(E)\to Cocyl^{smooth}(\pi)$ where the subspaces of smooth paths are used.

## References

The original article is

* [[Witold Hurewicz]], _On the concept of fiber space_, Proc. Nat. Acad. Sci. USA __41__ (1955) 956--961; MR0073987 (17,519e) [PNAS,pdf](http://www.pnas.org/content/41/11/956.full.pdf).

A review is for instance in

* [[James Eells]], Jr., _Fibring spaces of maps_, in Richard Anderson (ed.) _Symposium on infinite-dimensional topology_

[[!redirects Hurewicz connections]]


Hurewicz fibration

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

[[Serre fibration]] $\Leftarrow$ **Hurewicz fibration** $\Rightarrow$ [[Dold fibration]] $\Leftarrow$ [[shrinkable map]]

***

# Contents
* table of contents
{:toc}

## Definition

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

A [[continuous map]] $p \;\colon\; E\longrightarrow B$ of [[topological space]] is called a **Hurewicz fibration**  if it satisfies the [[right lifting property]] against all [[continuous functions]]

$$
  \sigma_0
  \;\colon\;
  X \cong X \times\{0\}
    \xhookrightarrow{\phantom{---}}
  X \times I
$$

including any [[topological space]] $X$ as one end of the [[cylinder]] over it (where $I \coloneqq [0,1]$ denotes the [[topological interval]]).

=--

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

In this context, the defining [[right lifting property]] is called the *[[homotopy lifting property]]*, because the maps from $X\times I$ are understood as [[homotopies]]. In more detail, for every space $X$, any homotopy $F:X\times I\to B$, and a continuous map $f:X\to E$, there is a homotopy $\tilde{F}:X\times I\to E$ such that $f =\tilde{F}
\circ\sigma_0 :=\tilde{F}_0$ and $F=p\circ\tilde{F}$:

$$
  \array{
    X &\stackrel{f}\longrightarrow& E
    \\
    {}^{\mathllap{\sigma_0}}
    \big\downarrow
      &{}^{\tilde{F}}\nearrow&
    \big\downarrow {}^{\mathrlap{p}}
    \\
    X\times I &\stackrel{F}{\to}& B
  }
  \,.
$$

=--

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

Strictly speaking, the "all" in this context should be interpreted to refer to all spaces in whatever ambient [[category]] of [[TopologicalSpaces]] one is working in, since frequently this is a [[convenient category of spaces]].  In theory, therefore, a map in such a category could be a Hurewicz fibration in that category without necessarily being a Hurewicz fibration in the category of *all* topological spaces, but in practice this usually doesn't make a whole lot of difference.

=--

Instead of checking the homotopy lifting property, one can instead solve a universal problem:

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

A map is a Hurewicz fibration precisely if it admits a [[Hurewicz connection]]. (See there for details.)

=--

## Properties

### Appearance in a model structure

There is a [[Quillen model category]] structure on [[Top]] where fibrations are Hurewicz fibrations, cofibrations are closed [[Hurewicz cofibrations]] and weak equivalences are [[homotopy equivalences]] -- see at *[[model structure on topological spaces]]* and specifically at *[[Strøm's model category]]*.

There is a version of Hurewicz fibrations for [[pointed spaces]], as well as in the [[slice category]] $Top/B_0$ where $B_0$ is a fixed base (the [[slice model structure]]).

There is also a model category whose fibrations are the Hurewicz fibrations and whose weak equivalences are the [[weak homotopy equivalences]], obtained by [[mixed model structure|mixing]] the above model structure with the [[classical model structure on topological spaces]].

### Relation to Serre fibrations

Every Hurewicz fibration is a [[Serre fibration]].  Conversely, [[a Serre fibration between CW-complexes is a Hurewicz fibration]].

### Local recognition over numerable open covers
 {#LocalRecognition}

The following proposition says that being a Hurewicz fibration is a *local property* with respect to the [[codomain]] space, at least as long as the local [[open cover]] used is [[numerable open cover|numerable]].

\begin{prop}\label{MapIsHurewiczFibrationIfPullbackToNumerableCoverIsSo}
**(Local recognition of Hurewicz fibrations over numerable open covers)**\linebreak

Let $p \colon E \to B$ be a continuous function such that there exists a [[numerable open cover]] $U \overset{\phi}{\longrightarrow} B$ over which $E$ is a Hurewicz fibration, hence such that the [[pullback]] $\phi^\ast p$ is a Hurewicz fibration.

Then $p$ is a Hurewicz fibration

\end{prop}

A proof may be found spelled out in e.g. [May 99, Sec. 7.4](#May99)

## Abstract Hurewicz fibrations
 {#Abstractly}

The concept of Hurewicz fibrations makes sense also more generally in the presence of a (well behaved) [[interval object]], see for instance the early example of such in ([Kamps 72](#Kamps72)) and see ([Williamson 13](#Williamson13)) for review and further developments. Discussion with a view towards [[homotopy type theory]] is in ([Warren 08](#Warren08)).

## Examples
 {#Examples}

### Empty bundles

\begin{example}\label{EmptyBundlesAreHurewiczFibrations}
**([[empty bundles]] are [[Hurewicz fibrations]])** \linebreak
 All [[empty bundles]] $\varnothing \longrightarrow B$ are Hurewicz fibrations,  because none of the [[commuting squares]] that one would have to non-trivially [[lifting property|lift in]] actually exist:

$$
  \array{
    X &\overset{ \not \exists }{\longrightarrow}& \varnothing
    \\
    \big\downarrow && \big\downarrow
    \\
    X \times I &\longrightarrow& B
    \mathrlap{\,,}
  }
$$

since the [[empty topological space]] is a [[strict initial object]]: There is *no* morphism to it from any [[inhabited space]] $X$

(If $X$ itself is [[empty topological space|empty]], then so is $X \times I$ and then a square only exists if $B$ in turn is empty. This square, consisting entirely of empty spaces, does have a unique lift, trivially, and hence also the empty bundle over the empty space is a Hurewicz fibration -- for what it's worth.)

\end{example}

### Covering spaces

+-- {: .num_example #CoveringSpaceIsHurewiczFibration}
###### Example
**([[covering spaces]] are Hurewicz fibrations)**

Every [[covering space]] projection is a Hurewicz fibration, by [this prop.](covering#space#HomotopyLiftingPropertyOfCoveringSpaces).

=--

### Fiber bundles
 {#FiberBundles}

\begin{prop}\label{NumerableFiberBundlesAreSerreFibrations}
**([[numerable fiber bundles]] are Hurewicz fibrations)** \linebreak

Every [[numerable fiber bundle]], hence in particular every [[fiber bundle]] over a [[paracompact topological space]], is a Serre fibration.

\end{prop}
\begin{proof}
  By definition of local triviality,
  for a [[numerable fiber bundle]] $p \colon E \to B$ -- such as a [[fiber bundle]] over a [[paracompact topological space]] -- there exists a [[numerable open cover]] $\phi \colon U \to B$ such that the [[pullback bundle]] $\phi^\ast p$ is trivial, i.e. is the [[Cartesian product]] [[projection]]

$$
  \phi^\ast p \;\colon\; U \times F \longrightarrow U
$$

(for $F$ the [[typical fiber]]). Since such a projection is clearly a Hurewicz fibration the claim follows by the local recognition of Hurewicz fibrations over numerable open covers, from Prop. \ref{MapIsHurewiczFibrationIfPullbackToNumerableCoverIsSo}.
\end{proof}

Notice that the example of [[empty bundles]] (Example \ref{EmptyBundlesAreHurewiczFibrations}) is a special case of a [[fiber bundle]]: With [[typical fiber]] the [[empty topological space]].

## Related concept

* [[Hurewicz cofibration]]

* [[Strøm model structure]]

* [[Serre fibration]]

## References

The original article:

* [[Witold Hurewicz]], _On the concept of fiber space_, Proc. Nat. Acad. Sci. USA __41__ (1955) 956--961; ([doi:10.1073%2Fpnas.41.11.956](https://dx.doi.org/10.1073%2Fpnas.41.11.956), [jstor:89187](https://www.jstor.org/stable/89187), [pdf](http://www.pnas.org/content/41/11/956.full.pdf). MR0073987)

A decent review of Hurewicz fibrations, [[Hurewicz connections]] and related issues is in

* [[James Eells]], Jr., _Fibring spaces of maps_, in Richard Anderson (ed.) _Symposium on infinite-dimensional topology_

Textbook accounts:

* {#tDKP70} [[Tammo tom Dieck]], [[Klaus Heiner Kamps]], [[Dieter Puppe]], Chapter II of: *Homotopietheorie*, Lecture Notes in Mathematics **157** Springer 1970 ([doi:10.1007/BFb0059721](https://link.springer.com/book/10.1007/BFb0059721))

* [[Alan Hatcher]], from p. 61 on in _Algebraic Topology_ ([web](http://www.math.cornell.edu/~hatcher/AT/ATpage.html))

See also

* [[R. Schwänzl]], [[R. Vogt]], _Strong cofibrations and fibrations in enriched categories_, 2002.

* the textbooks on algebraic topology by Whitehead and Spanier.

* {#May99} [[Peter May]], _[[A concise course in algebraic topology]]_,   University of Chicago Press 1999 ([ISBN: 9780226511832](https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3777031.html), [pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf))

* {#Cutler20} [[Tyrone Cutler]], _Fibrations III -- Locally trivial maps and bundles_, 2020 ([pdf](https://www.math.uni-bielefeld.de/~tcutler/pdf/Fibrations%20III.pdf), [[CutlerFibrationsIII.pdf:file]])

Abstract analogues of Hurewicz fibrations can be found in

* {#Kamps72} [[K.H.Kamps]], _Kan-Bedingungen und abstrakte Homotopietheorie_, Math. Z. 124,1972, 215 -236

summarised in

* [[K. H. Kamps]], [[Tim Porter]], _Abstract Homotopy and Simple Homotopy Theory_ ([GoogleBooks](http://books.google.de/books?id=7JYKxInRMdAC&dq=Porter+Kamps&printsec=frontcover&source=bl&ots=uuyl_tIjs4&sig=Lt8I92xQBZ4DNKVXD0x76WkcxCE&hl=de&sa=X&oi=book_result&resnum=3&ct=result#PPP1,M1))

and further developed in

* {#Williamson13} [[Richard Williamson]], _Cylindrical model structures_ ([arXiv:1304.0867](http://arxiv.org/abs/1304.0867), [web](http://rwilliamson-mathematics.info/cylindrical_model_structures.html))

Discussion with an eye towards [[homotopy type theory]] is in

* {#Warren08} [[Michael Warren]], _Homotopy theoretic aspects of constructive type theory_, 2008 ([PDF](http://mawarren.net/papers/phd.pdf)) {#Warren}

[[!redirects Hurewicz fibration]]
[[!redirects Hurewicz fibrations]]


Hurewicz model structure on chain complexes

The analogue of the [[Strom model structure]] but for [[chain complexes]] instead of [[topological spaces]].  Its weak equivalences are [[chain homotopy equivalences]] rather than [[quasi-isomorphisms]], and its [[homotopy category]] is the [[homotopy category of chain complexes]].


Hurewicz theorem

> Not to be confused with the [[Hurwitz theorem]].

***

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#### Homotopy theory
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=--
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# Contents
* table of contents
{: toc}

## Idea

The first nonzero [[homotopy group]] and [[ordinary homology|ordinary]]/[[singular homology]] [[homology group|group]] of a [[simply-connected topological space]] occur in the same dimension and are [[isomorphism|isomorphic]].

## Hurewicz homomorphism
 {#HurewiczHomomorphismSection}

### For topological spaces

+-- {: .num_defn #HurewiczHomomorphism}
###### Definition
**(Hurewicz homomorphism)**

For $(X,x)$ a [[pointed object|pointed]] [[topological space]], the **Hurewicz homomorphism** is the [[function]]

$$
  \Phi : \pi_k(X,x) \to H_k(X)
$$

from the $k$th [[homotopy group]] of $(X,x)$ to the $k$th [[singular homology]] group defined by sending

$$
  \Phi
  \;\colon\;
  (f \colon S^k \to X)_{\sim} \mapsto f_*[S_k]
$$

any representative singular $k$-sphere $f$ in $X$ to the push-forward along $f$ of the [[fundamental class]] $[S_k] \in H_k(S^k) \simeq \mathbb{Z}$.

=--

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

The Hurewicz homomorphism is a [[natural transformation]]

$$
  \Phi \;\colon\; \pi_k(-) \to H_k(-)
$$

between [[functors]] [[pointed topological spaces|$Top^{\ast/}$]] $\to$ [[Ab]].

=--

### For spectra
 {#ForSpectra}

The above construction has an immediate analog in [[stable homotopy theory]]:

For $R$ a [[ring]], its [[Eilenberg-MacLane spectrum]] is an [[E-infinity ring]] and hence receives a canonical [[unit]] homomorphism $\mathbb{S} \longrightarrow H R$ from the [[sphere spectrum]].

Under [[smash product]] and passing to [[stable homotopy group]], this induces a [[natural transformation]] from [[stable homotopy groups]] of $X$ (its [[stable homotopy homology theory]]) to [[ordinary homology]] of $X$ with [[coefficients]] in $R$:

$$
  \pi^{st}_\bullet(X)
  \;\simeq\;
  \pi_\bullet( \mathbb{S} \wedge X_+ )
  \longrightarrow
  \pi_\bullet( H R \wedge X_+ )
  \simeq
  H_\bullet(X,R)
  \,.
$$

If here the [[Eilenberg-MacLane spectrum]] $H R$ is replaced by any other [[E-infinity ring spectrum]] the analogous construction is called the _[[Boardman homomorphism]]_.

## Hurewicz theorem
 {#HurewiczTheorem}

In general, [[homology theory|homology]] is a coarser invariant than [[homotopy type|homotopy]], and [[ordinary homology]] is the coarsest of all [[generalized homology]]-invariants. Therefore the Hurewicz homomorphism (Def. \ref{HurewiczHomomorphism}) is bound to lose information, in general.

Indeed, the Hurewicz homomorphism exhibits a kind of abelianization of the [[homotopy type]] (in the sense of [[stable homotopy theory]], see at *[[Boardman homomorphism]]* for more on this), a statement that in low degrees is true in the plain sense of *[[abelianization]]*: this is the content of Prop. \ref{HurewiczTheoremInDegreeZero} and Prop. \ref{HurewiczTheoremInDegreeOne} below.

While in higher degrees the Hurewicz homomorphism is in general far from being an isomorphism, the thrust of the Hurewicz theorem is to show that high connectivity is a sufficient condition to ensure that it is. This is Theorem \ref{HurewiczTheoremInDegreeTwoAndHigher} below.

\begin{proposition}\label{HurewiczTheoremInDegreeZero}
**(in degree 0)**
\linebreak
For $X$ a topological space, the Hurewicz homomorphism (Def. \ref{HurewiczHomomorphism}) in degree 0 exhibits an [[isomorphism]] between the [[free abelian group]] $\mathbb{Z}[\pi_0(X)]$ on the set of [[connected components]] of $X$ and the degree-0 singular homlogy:

$$
  \mathbb{Z}[\pi_0(X)]
   \simeq
  H_0(X)
  \,.
$$

\end{proposition}

Since a [[homotopy group]] in [[positive number|positive]] degree depends on the [[homotopy type]] of the [[connected component]] of the base point, while the [[ordinary homology]] does not depend on a basepoint, it is interesting to compare these groups only for the case that $X$ is connected:

\begin{proposition}\label{HurewiczTheoremInDegreeOne}
**(in degree 1)**
\linebreak
For $X$ a [[connected topological space]] the [[Hurewicz homomorphism]] (Def. \ref{HurewiczHomomorphism})
in degree 1

$$
  \Phi \colon \pi_1(X,x) \longrightarrow H_1(X)
$$

is [[surjection|surjective]]. Its [[kernel]] is the [[commutator subgroup]]
of $\pi_1(X,x)$. Therefore it induces an [[isomorphism]] from the [[abelianization]] $\pi_1(X,x)^{ab} \coloneqq \pi_1(X,x)/[\pi_1,\pi_1]$:

$$
  \pi_1(X,x)^{ab}
  \overset{\simeq}{\longrightarrow}
  H_1(X)
  \,.
$$

\end{proposition}

\begin{theorem}\label{HurewiczTheoremInDegreeTwoAndHigher}
**(in degree $\geq 2$)**
\linebreak
If a [[topological space]] (or [[infinity-groupoid]]) $X$ is [[n-connected object in an (infinity,1)-topos|(n-1)-connected]] for $n \geq 2$ then the [[Hurewicz homomorphism]], Def. \ref{HurewiczHomomorphism}

$$
  \Phi
  \;\colon\;
  \pi_n(X,x)
    \longrightarrow
  H_n(X)
$$

is an [[isomorphism]].

\end{theorem}

A proof is spelled out for instance with theorem 2.1 in ([Hutchings](#Hutchings)).

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

With the [[universal coefficient theorem]] a corresponding statement follows for the [[cohomology group]] $H^n(X,A)$.

=--

## Related concepts

* [[Boardman homomorphism]]

* [[Hopf degree theorem]]

* The _[[Adams spectral sequence]]_ is a vast generalization of the computation of [[homotopy groups]] from [[cohomology groups]] via the Hurewicz theorem.

## References

The original reference is:

* [[Witold Hurewicz]], Koninklijke Akademie van Wetenschappen: Proceedings of the Section of Sciences 38 (1935), 112–119.  Mathematics. — Beiträge zur Topologie der Deformationen (I. Höherdimensionale Homotopiegruppen).  [PDF](https://dwc.knaw.nl/DL/publications/PU00016667.pdf)

This article is reproduced on pages 341–348 of

* [Collected Works of Witold Hurewicz](https://bookstore.ams.org/cworks-4).  (Edited by Krystyna Kuperberg.)  American Mathematical Society, 1995.  ISBN 0-8218-0011-6

The [[simplicial homotopy theory|simplicial]] version is due to

* [[Daniel M. Kan]],  _The Hurewicz theorem_, 1958 Symposium internacional de topología algebraica (International symposium on algebraic topology), pp. 225–231 Universidad Nacional Autónoma de México and UNESCO, Mexico City.

The basic statement is for instance in

* [[Allen Hatcher]], _Algebraic Topology_ ([web](http://www.math.cornell.edu/~hatcher/AT/ATpage.html))

Lecture notes:

* {#Hutchings} [[Michael Hutchings]], _Introduction to higher homotopy groups and obstruction theory_ (2011) ([pdf](http://math.berkeley.edu/~hutching/teach/215b-2011/homotopy.pdf))

* {#Kobin16} Andrew Kobin, Section 7.3 of: _Algebraic Topology_, 2016 ([[KobinAT2016.pdf:file]])

For discussion in [[stable homotopy theory]] modeled on [[symmetric spectra]] is in

* {#Schwede12} [[Stefan Schwede]], part II, prop. 6.30 of _[[Symmetric spectra]]_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

See also

* wikipedia, _[Hurewicz theorem](http://en.wikipedia.org/wiki/Hurewicz_theorem)_

In the generality of the [[Boardman homomorphism]]:

* [[Frank Adams]], Part II.6 of _[[Stable homotopy and generalised homology]]_, 1974

Discussion of the stable Hurewicz homomorphism includes

* [[Akhil Mathew]], _Torsion exponents in stable homotopy and the Hurewicz homomorphism_, Algebr. Geom. Topol. 16 (2016) 1025-1041 ([arXiv:1501.07561](https://arxiv.org/abs/1501.07561))

Proof of the [[Hurewicz theorem]] in [[homotopy type theory]], hence in general [[(∞,1)-toposes]]:

* [[Daniel Christensen]], [[Luis Scoccola]], _The Hurewicz theorem in Homotopy Type Theory_ ([arXiv:2007.05833](https://arxiv.org/abs/2007.05833))

Discussion of [[persistent homotopy]] with focus on the [[van Kampen theorem]], [[excision]] and the [[Hurewicz theorem]]:

* Mehmet Ali Batan, Mehmetcik Pamuk, Hanife Varli, *Persistent Homotopy* $[$[arXiv:1909.08865](https://arxiv.org/abs/1909.08865)$]$

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Hurwitz theorem

> not to be confused with the [[Hurewicz theorem]].

***

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# Contents
* table of contents
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## Statement

The only Euclidean [[composition algebras]] / [[division algebra|division]] [[composition algebras]] over the [[real numbers]] (i.e. real [[normed division algebras]]) are, up to [[isomorphism]], the algebras of

1. [[real numbers]]

2. [[complex numbers]]

3. [[quaternions]]

4. [[octonions]].

For more see at _[composition algebra -- Hurwitz theorem](composition+algebra#HurwitzTheoremSection)_.

## Implications

### Real spinor representations and supersymmetry

This classification turns out to closely connect to various other systems of exceptional structures in [[mathematics]] and [[physics]]. Notably patterns related to [[Majorana spinors]] in [[spin geometry]] are intimately related to the four normed division algebras, and, induced by this, so is the classification of [[super Poincaré Lie algebras]] and [[super Minkowski spacetimes]] (which are built from these real spin representations). For more on this see at _[[supersymmetry and division algebras]]_ and _[[geometry of physics -- supersymmetry]]_.

[[!include exceptional spinors and division algebras -- table]]

### Hopf invariant one maps

The classification matches that of [[Hopf invariant one]] maps. See there for more.

See also at _[[cross product]]_.

## Generalized Hurwitz Theorem

As describe in [Elduque (2021)](#Elduque21), the Hurwitz Theorem can be extended to more general [[composition algebras]] that are not necessarily unital, but that are *para-unital* or even *non-unital* (see [here](https://ncatlab.org/nlab/show/composition+algebra#definition) for the relevant definitions).

The Generalized Hurwitz Theorem then becomes the statement that the only Euclidean composition algebras over the real numbers are, up to isomorphism:

1. Seven unital algebras: real numbers, complex numbers, quaternions, and octonions (along with their [split](https://ncatlab.org/nlab/show/composition+algebra#split_composition_algebras) companions).

2. Seven para-unital algebras, corresponding to the para- counterparts of the unital algebras above.

3. Two eight-dimensional non-unital algebras: the [[Okubo algebra]] $\mathcal{O}$ and its split counterpart $\mathcal{O}_s$.

## References

Due to [[Adolf Hurwitz]] (1859–1919), published posthumously in 1923.

* {#Hurwitz1898} [[Adolf Hurwitz]], _Über die Composition der quadratischen Formen von beliebig vielen Variabeln_, Nachr. Ges. Wiss. Göttingen (1898) pp 309--316 ([GDZ pdf](http://gdz.sub.uni-goettingen.de/download/PPN252457811_1898/LOG_0034.pdf), [EuDML entry](https://eudml.org/doc/58420))

* [[Adolf Hurwitz]],  _Über die Komposition der quadratischen Formen_, Math. Ann. 88 (1923) pp 1--25, doi:[10.1007/bf01448439](http://dx.doi.org/10.1007/bf01448439) ([GDZ pdf](http://gdz.sub.uni-goettingen.de/download/PPN235181684_0088/LOG_0005.pdf), [EuDML entry](https://eudml.org/doc/158975))

* Wikipedia, _[Hurwitz theorem (composition algebras)](http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_%28composition_algebras%29)_

* {#Elduque21} [[Alberto Elduque]], Composition algebras: “Algebra and Applications I: Non-associative Algebras and Categories”, Chapter 2, pp. 27-57, edited by Abdenacer Makhlouf, Sciences-Mathematics, ISTEWiley, London 2021.

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Huzihiro Araki

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

## Selected writings

On the [[time tube theorem]]:

* [[Huzihiro Araki]], *A generalization of Borchers' theorem*, Helvetica Physica Acta **36** (1963) 132-139

On [[relative entropy]] of [[state on a star-algebra|states]] on  [[von Neumann algebras]]:

* {#Araki} [[Huzihiro Araki]], _Relative Entropy of States of von Neumann Algebras_, Publications of the Research Institute for Mathematical Sciences, **11** 3 (1976) 809-833 ([pdf](https://ems.press/content/serial-article-files/2833), [doi:10.2977/prims/1195191148]( https://doi.org/10.2977/prims/1195191148))

## Related entries

* [[AQFT]], [[Haag-Kastler axioms]]

* [[relative entropy]]

category: people


Hvedri Inassaridze

**Hvedri Inassaridze** is a Georgian mathematician, a member of the Georgian Academy of Sciences. His fields of interest include homological and homotopical algebra, K-theory, algebraic topology, category theory, and algebraic geometry. In addition to numerous articles, he has authored the monographs

* _Nonabelian homological algebra and its applications_, Kluwer Acad. Publishers, Amsterdam, 1997, 270 pages.

* _Algebraic K-theory_, Kluwer Acad. Publishers, Amsterdam, 1995, 440 pages.

He is the founding editor of the following journals.

* [[Homology, Homotopy and Applications]] ([HHA](https://web.archive.org/web/20000830082304/http://www.rmi.acnet.ge/hha/geninfo.html#edlist), founded in 1998)

* [[Journal of Homotopy and Related Structures]] ([JHRS](http://tcms.org.ge/Journals/JHRS/))

* Tbilisi Mathematical Journal, founded in 2008, renamed in September 2021 to Advanced Studies: Euro-Tbilisi Mathematical Journal.

In the early 1990s, Hvedri Inassaridze was active in politics.

His son [[Niko Inassaridze|Niko (Nick) Inassaridze]]  is also a mathematician (who has studied extensively a notion of [nonabelian tensor product](http://www.bangor.ac.uk/~mas010/nonabtens.html) useful in nonabelian homological algebra).

*  [homepage](http://www.rmi.ge/~hvedri)

category: people


hydrodynamics

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

__Hydrodynamics__ is the dynamics of liquids, i.e. of low compressible fluids. It is part of _rheology_, a general science of fluids. For a while this entry will have a redirect fluid dynamics.

## Related entries

* [[Navier-Stokes equation]]

* [[magnetohydrodynamics]]

* [[phase of matter]]

* [[fluid/gravity correspondence]]

## References

* L. D. Landau, E. M. Lifschitz, _Fluid mechanics_ (Russian original is _Hydrodynamics_), vol. 6 of the [[Landau-Lifschitz|Course of Theoretical Physics]], Pergamon Press

* [[Vladimir Arnold]], [[Boris Khesin]], _Topological methods in hydrodynamics_, Applied Mathematical Sciences __125__, Springer 1998 ([doi:10.1007/b97593](https://link.springer.com/book/10.1007/b97593))

  (using methods of [[topology]] and [[homotopy theory]])

* [[Dennis Sullivan]], _Algebra, topology and algebraic topology of 3D ideal fluids_, [arxiv/1010.2721](http://arxiv.org/abs/1010.2721)

  (some ideas on need for [[higher geometry]] at the end)

* Benjamin Doyon, _Lecture notes on Generalised Hydrodynamics_ ([arXiv:1912.08496](https://arxiv.org/abs/1912.08496))

See also

* Wikipedia, _[Fluid dynamics](http://en.wikipedia.org/wiki/Fluid_dynamics)_

[[!redirects fluid dynamics]]
[[!redirects hydrodynamic]]


hydrogen atom

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#Contents#
* table of contents
{:toc}

## Idea

In [[physics]], the [[atom (physics)|atom]] whose [[nucleus (physics)|nucleus]] consists of a single [[proton]] and whose shell contains a single [[electron]] is called the _hydrogen atom_ (hence a [[bound state]] of an [[electron]] and a [[proton]], see also _[[positronium]]_).

## Related concepts

* [[Lamb shift]]

* [[hydrogen line]]

* [[harmonic oscillator]]

* [[quantum lattice system]]

## References

* {#Nanni15} Luca Nanni, _The hydrogen atom: a review of the Birth of Modern Quantum Mechanics_ ([arXiv:1501.05894](https://arxiv.org/abs/1501.05894))

See also

* Wikipedia, _[Hydrogen atom](https://en.wikipedia.org/wiki/Hydrogen_atom)_

[[!redirects hydrogen atoms]]

[[!redirects hydrogen]]


hydrogen line

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## Idea

What is called the _21cm hydrogen line_ or just _hydrogen line_ is the signature of the [[electromagnetic radiation]] of [[wavelength]] about 21[[cm]] which is emitted by [[hydrogen]] [[atoms]] when the [[electrons]] in their shell change [[quantum state]] from the "triplet" to the singlet [[ground state]].

## In astrononmy

In [[astronomy]], the _[[Tully-Fisher relation]]_ is a relation between the asymptotic behaviour of [[galaxy rotation curves]], witnessed by the width of their [[21cm hydrogen line]], and the total visible [[baryon|baryonic]] [[mass]] of the [[galaxies]].

## In cosmology

In the context of [[cosmology]] it has been argued ([Higan-Rees 79](#HiganRees79)) that observation of a absorption at the 21cm hydrogen line in the [[cosmic microwave background]] would be a signal from the time when the first [[stars]] formed and ignited, a cosmological epoch not otherwise accessible to observation. A claim of detection of this signal by the [[EDGES]] [[experiment]] appeared in [BRMMM 18](#BRMMM18).

## Related entries

* [[observable universe]]

## References

### General

* Wikipedia, _[Hydrogen line](https://en.wikipedia.org/wiki/Hydrogen_line)_

* Hyperphysics, _[The Hydrogen 21-cm line](http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/h21.html)_

### In astrophysics and cosmology
 {#ReferencesInAstrophysicsAndCosmology}

The suggestion that the 21cm hydrogen absorption line in the [[cosmic microwave background]] should be an interesting probe for early universe [[cosmology]] goes back to

* {#HiganRees79} C. J. Hogan, [[Martin Rees]], _Spectral appearance of non-uniform gas at high $z$_, Monthly Notices of the Royal Astronomical Society, Volume 188, Issue 4, 1 October 1979, Pages 791–798 ([doi:10.1093/mnras/188.4.791]( https://doi.org/10.1093/mnras/188.4.791))

Review includes

* {#PritchardLoeb11} Jonathan Pritchard, A. Loeb,  _21 cm cosmology in the 21st century_, Rep. Prog. Phys. 75, 086901 (2012) ([arXiv:1109.6012](https://arxiv.org/abs/1109.6012))

* Jonathan Pritchard, _Cosmological probes of inflation and the 21 cm line_ ([video recording](http://www.perimeterinstitute.ca/videos/cosmological-probes-inflation-and-21-cm-line))

* {#Zaroubi12} Saleem Zaroubi, _The Epoch of Reionization_, in Wiklind T., Mobasher B., Bromm V. (eds) _The First Galaxies_, Springer ([arXiv:1206.0267](https://arxiv.org/abs/1206.0267))

A claim of detection of the cosmological signal by the [[EDGES]] [[experiment]] is due to

* {#BRMMM18} Judd D. Bowman, Alan E. E. Rogers, Raul A. Monsalve, Thomas J. Mozdzen,  Nivedita Mahesh, _An absorption profile centred at 78 megahertz in the sky-averaged spectrum_, Nature volume 555, pages 67–70 (01 March 2018) ([doi:10.1038/nature25792](https://www.nature.com/articles/nature25792))

* Raul A. Monsalve, Bradley Greig, Judd D. Bowman, Andrei Mesinger, Alan E. E. Rogers, Thomas J. Mozdzen, Nicholas S. Kern, Nivedita Mahesh, _Results from EDGES High-Band: II. Constraints on Parameters of Early Galaxies_,  The Astrophysical Journal, Volume 863, Number 1 ([arXiv:1806.07774](https://arxiv.org/abs/1806.07774))

Discussion incluces

* {#Falkowski18} [[Adam Falkowski]], _[21cm to dark matter](http://resonaances.blogspot.fr/2018/03/21cm-to-dark-matter.html)_, March 2018

Calling this argument into question:

* Richard Hills, Girish Kulkarni, P. Daniel Meerburg, Ewald Puchwein, _Concerns about Modelling of the EDGES Data_ ([arXiv:1805.01421](https://arxiv.org/abs/1805.01421))

Arguments that this detection of the 21cm line rules out substantial contributions of [[fuzzy dark matter]]:

* {#Nebrin17} Olof Nebrin, _Cosmic Dawn in a Fuzzy Universe: Constraining the nature of Dark Matterwith 21 cm Cosmology_, Stockholm 2017 ([diva2:1195402](http://www.diva-portal.org/smash/record.jsf?pid=diva2%3A1195402&dswid=-8037), [urn:nbn:se:su:diva-154861](http://su.diva-portal.org/smash/record.jsf?pid=diva2%3A1195402&dswid=6755))

* {#NebrinGharaMellema18} Olof Nebrin, Raghunath Ghara, Garrelt Mellema, _Fuzzy Dark Matter at Cosmic Dawn: New 21-cm Constraints_ ([arXiv:1812.09760](https://arxiv.org/abs/1812.09760), [reddit](https://www.reddit.com/r/Physics/comments/a9zeeb/my_first_paper_fuzzy_dark_matter_at_cosmic_dawn/))

[[!redirects hydrogen lines]]

[[!redirects 21cm hydrogen line]]
[[!redirects 21cm hydrogen lines]]


Hyman Bass

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

## Selected writings

On the [[Eilenberg swindle]] (and introducing that terminology):

* {#Bass63} [[Hyman Bass]], _Big projective modules are free_, Illinois Journal of Mathematics **7** (1963) 24–31  [[doi:10.1215/ijm/1255637479](https://doi.org/10.1215%2Fijm%2F1255637479)]

On [[Hermitian K-theory]]:

* [[Hyman Bass]] (ed.), *Algebraic K-Theory III -- Hermitian K-Theory and Geometric Applications*, Lecture Notes in Mathematics **343** (1973) [[doi:10.1007/BFb0061366](https://doi.org/10.1007/BFb0061366)]

category:people

[[!redirects H. Bass]]


hyper-derived functor

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## Idea

In the context of [[homological algebra]] [[derived functors]] are traditionally considered on a [[model structure on chain complexes]] and often they are evaluated only on chain complexes that are concentrated in a single degree. If instead they are evaluated on general chain complexes, one sometimes speaks of **hyper-derived functors** for emphasis.

For more see at _[[derived functor in homological algebra]]_.

## Examples

If [[abelian sheaf cohomology]] is considered in terms of the [[derived functor]] of the [[global section]] functor, then the corresponding hyper-derived functor is [[hypercohomology]]. This, too, is really just the basic definition of (abelian) [[cohomology]], but not restricted to [[Eilenberg-MacLane object]]s concentrated in a single degree.

## Properties

There is a certain [[spectral sequence]] that can help to compute values of hyper-derived functors. See the section Spectral sequences for hyper-derived functors.

## Related concepts

* [[hypercohomology]]

[[!redirects hyper-derived functors]]

[[!redirects hyperderived functor]]
[[!redirects hyperderived functors]]


hyper-envelope of a Lie algebra

Let $L$ be a finite dimensional Lie algebra over real or complex numbers with basis $\hat{x}_1,\ldots,\hat{x}_n$ and let $x_1,\ldots, x_n$ be the corresponding basis of the symmetric algebra $S(L)$ via the identity and $U(L)$ the [[universal enveloping algebra]] of $L$. Given any $y\in L\hookrightarrow U(L)$ by $\hat{y}$ we denote the same element understood in $L\hookrightarrow S(L)$. This correspondence extends to a coalgebra isomorphism $e: S(L)\to U(L)$, the [[coexponential map]] (or symmetrization map)

$$
e : y_1 \ldots y_k \mapsto \frac{1}{k!}\sum_{\sigma\in \Sigma(k)} \hat{y}_{\sigma 1}\cdots \hat{y}_{\sigma k}
$$

for any $y_1,\ldots, y_n\in L\hookrightarrow S(L)$ (not necessarily basis elements). The coexponential map is the only functorial in $L$ isomorphism of coalgebras and it fixes $k\otimes L$ where $k$ is the ground field. It is also characterized by $y^n\mapsto (\hat{y})^n$, where $y\in L\hookrightarrow S(L)$.

The Raševskii [[hyper-envelope of a Lie algebra]] $L$ is a completion of $U(L)$ by means of a countable family of norms $\hat{f}\mapsto \|\hat{f}\|_{\epsilon}$ for all $\epsilon$ in an arbitrary fixed family of positive numbers having $0$ as  an accumulation points, where

$$
\| \hat{f}\|_{\epsilon} = max_{s_1,\ldots, s_n}\epsilon^{-(s_1+s_2+\ldots+s_n)} |f_{s_1\ldots s_n}|,
$$

for $s_1+\cdots + s_n = s$, and where   $f_{s_1,\ldots,s_n}$ is the Taylor coefficient in front of $x_1^{s_1}\cdots x_n^{s_n}$ of the commutative polynomial $f = e^{-1}(\hat{f})$, i.e.
$$
\hat{f} = e \left(\sum_{s_1\ldots s_n} f_{s_1\ldots s_n}\frac{x_1^{s_1}\cdots x_n^{s_n}}{s_1! s_2!\cdots s_n!}  \right)
$$
Here $x_1,\ldots, x_n$ is a fixed basis of $L$, viewed as commutative coordinates.

It is nontrivial and proved by Raševskii that the algebra multiplication in $U(L)$ is continuous in this topology and hence that the completion of the $U(L)$ as a countably normed vector space carries the unique structure of a topological algebra extending the algebra operations on $U(L)$.

* P. K. Raševskii, _Associative hyper-envelopes of Lie algebras, their regular representations and ideals_, Trudy Mosk. Mat. Ob.

It may be tried to use the same definition with $e$ replaced by another coalgebra isomorphism $S(L)\to U(L)$.

Remark: It is announced (by author of these lines [[Zoran Škoda|ZŠ]], June 11, 2011), that under mild conditions this modified definition results in the isomorphic completion of $U(L)$ as a topological algebra.

There is also an non-Archimedean version of the notion in the literature.

[[!redirects hyperenvelope of a Lie algebra]]


hyper-Kähler manifold > history

see _[[hyperkähler manifold]]_


hyperalgebra

## Overview

__Hyperalgebra__ of an [[affine algebraic group]] $G$ is the [[finite dual]]s of the Hopf algebra of [[representative function]]s of (irreducible component containing the unit element in) $G$. It can be interpreted as (and is sometimes called) the algebra of [[distribution on an affine algebraic group|distribution]]s supported at unit. This algebra comes with a natural filtration. In characteristic $0$ it coincides (by L. Schwarz's theorem) with the [[universal enveloping algebra]] of the Lie algebra of $G$, but it is much bigger in positive characteristic. It can also be obtained by base change from the Kostant's integral form of the universal enveloping algebra of the complex Lie algebra associated to $G$.

## Literature

See also closely related entry [[distribution on an affine algebraic group]].

Some books on algebraic groups and on [[Hopf algebra]]s have chapters dedicated to this topic e.g.

* M. Sweedler, _Hopf algebras, Benjamin, NY 1969
* C. Jantzen, _Representations of algebraic groups_, chapter 7
* Η. Yanagihara, _Theory of Hopf algebras attached to group schemes_, Springer Lecture
Notes in Mathematics 614 (1977)

Articles:

* J. Sullivan, _Simply connected groups, the hyperalgebra, and Verma's conjecture_, Amer. J. Math. 100 (1978) 1015-1019.
* [[Mitsuhiro Takeuchi]], _Tangent coalgebras and hyperalgebras I_, Japan. J. Math. __42__ (1974) 1-143 [pdf](https://www.jstage.jst.go.jp/article/jjm1924/42/0/42_0_1/_pdf); _On coverings and hyperalgebras of affine algebraic groups_, Trans. AMS
211 (1975), 249-275; _A hyperalgebraic proof of the isomorphism and isogeny theorems for
reductive groups_, J. Algebra 85 (1983), 179-196; _Generators and relations for the hyperalgebras of reductive groups_, doi
* W. J. Haboush, _Central differential operators of split semisimple groups over fields of positive characteristic_, In: Séminaire d'Algèbre Paul Dubreil et Marie-Paule Malliavin, Springer Lecture Notes in Mathematics __795__, pp 35-85 [doi](https://doi.org/10.1007/BFb0090113)
* Edward Cline, Brian Parshall, Leonard Scott, _Cohomology, hyperalgebras, and representations_,  Journal of Algebra __63__:1, 98-123 (1980) doi

MathOverflow: [which-is-the-correct-universal-enveloping-algebra-in-positive-characteristic](http://mathoverflow.net/questions/7112/which-is-the-correct-universal-enveloping-algebra-in-positive-characteristic)

A quantum version at root of unity is proposed in

* Iván Angiono, _A quantum version of the algebra of distributions of $SL(2)$_, [arxiv/1607.04869](https://arxiv.org/abs/1607.04869)

and another approach is in

* W. Chin, L. Krop, _Quantized hyperalgebra of rank 1_, [pdf](http://condor.depaul.edu/~wchin/qhyperalgebra.pdf); _Spectra of quantized hyperalgebras_, [pdf](http://condor.depaul.edu/wchin/Spectra.pdf)

[[!redirects hyperalgebras]]
[[!redirects hyperalgebra of an affine algebraic group]]


hyperbola

## Related concepts

* [[conic section]]

  * [[ellipse]], [[parabola]]

* [[hyperbolic geometry]]

* [[hyperbolic functor]]

## References

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

[[!redirects hyperbolas]]


hyperbolic 3-manifold

[[!redirects hyperbolic 3-manifolds]]

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## Idea

A [[Riemannian manifold]] which is both a [[3-manifold]] and a [[hyperbolic manifold]] is a _hyperbolic 3-manifold_.

Equivalently this is a [[Riemannian manifold]] which is [[isometry|isometric]] to the [[quotient space]] $\mathbb{H}^3/ \Gamma$ of [[hyperbolic 3-space]] by the [[action]] of a [[torsion subgroup|torsion-free]] [[discrete group]] $\Gamma$ acting by [[isometries]].

## Properties

### Volume conjecture

For the moment see at _[[volume conjecture]]_.

## Examples

* [[hyperbolic 3-space]]

* [[hyperbolic solid torus]]

  * [[BTZ black hole]]

  * [[thermal AdS3]]

* [[knot complement]] of [[hyperbolic knot]]

## Related concepts

* [[low dimensional topology]]

* [[SnapPy]]

## References

### General

* [[William Thurston]], _Hyperbolic Structures on 3-manifolds, I: Deformation of acylindrical manifolds_, Annals of Math, 124 (1986), 203--246 ([jstor:1971277](https://www.jstor.org/stable/1971277), [arXiv:math/9801019](https://arxiv.org/abs/math/9801019))

* [[William Thurston]], _Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle_ ([arXiv:math/9801045](https://arxiv.org/abs/math/9801045))

* {#Thurston82} [[William Thurston]], _Three dimensional manifolds, Kleinian groups and hyperbolic geometry_, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 3 (1982), 357-381 ([euclid.bams/1183548782](https://projecteuclid.org/euclid.bams/1183548782))

See also

* [[Manifold Atlas]], _[Hyperbolic 3-manifolds](http://www.map.mpim-bonn.mpg.de/index.php/Hyperbolic_3-manifolds)_

* Wikipedia, _[Hyperbolic 3-manifold](https://en.wikipedia.org/wiki/Hyperbolic_3-manifold)_

### M5-branes wrapped on hyperbolic 3-manifolds

Discussion of [[black brane|black]] [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-manifolds]]:

* [[Jerome Gauntlett]], [[Nakwoo Kim]], [[Daniel Waldram]], Section 3.1 of: _M-Fivebranes Wrapped on Supersymmetric Cycles_, Phys. Rev. D63 (2001) 126001 ([arXiv:hep-th/0012195](https://arxiv.org/abs/hep-th/0012195))

* Aristomenis Donos, [[Jerome Gauntlett]], [[Nakwoo Kim]], Oscar Varelam, _Wrapped M5-branes, consistent truncations and AdS/CMT_, JHEP 1012:003, 2010 ([arXiv:1009.3805](https://arxiv.org/abs/1009.3805))

* {#GangKimLee14a} Dongmin Gang, [[Nakwoo Kim]], Sangmin Lee, _Holography of Wrapped M5-branes and Chern-Simons theory_, Physics Letters B
Volume 733, 2 June 2014, Pages 316-319 ([arXiv:1401.3595](https://arxiv.org/abs/1401.3595))

* {#GangKimLee14b} Dongmin Gang, [[Nakwoo Kim]], Sangmin Lee, _Holography of 3d-3d correspondence at Large $N$_, JHEP04(2015) 091 ([arXiv:1409.6206](https://arxiv.org/abs/1409.6206))

* {#GangKim18} Dongmin Gang, [[Nakwoo Kim]], _Large $N$ twisted partition functions in 3d-3d correspondence and Holography_, Phys. Rev. D 99, 021901 (2019) ([arXiv:1808.02797](https://arxiv.org/abs/1808.02797))

* {#GangKimPandoZayas19} Dongmin Gang, [[Nakwoo Kim]], Leopoldo A. Pando Zayas, _Precision Microstate Counting for the Entropy of Wrapped M5-branes_ ([arXiv:1905.01559](https://arxiv.org/abs/1905.01559))

Suggestion that the statement of the [[volume conjecture]] is reall8 [[AdS-CFT duality]] combined with the [[3d-3d correspondence]] for [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-manifolds]]:

* [Gang-Kim-Lee 14b, Section 3.2](#GangKimLee14b)

* [Gang-Kim 18, around (21)](#GangKim18)


hyperbolic 3-space

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## Idea

Hyperbolic 3-space $\mathbb{H}^3$ is [[generalized the|the]] [[simply connected topological space|simply connected]] [[geodesically complete]] [[hyperbolic 3-manifold]].

Every [[hyperbolic 3-manifold]] is [[isometry|isometric]] to the [[quotient space]] $\mathbb{H}^3/\Gamma$ of hyperbolic 3-space by the [[action]] of a [[torsion subgroup|torsion-free]] [[discrete group]] acting via [[isometries]].

## Related concepts

* [[hyperbolic solid torus]]

  * [[BTZ black hole]]

  * [[thermal AdS3]]

* [[hyperbolic plane]]

## References

* John Parker, Chapter 5 of _Hyperbolic spaces_ ([pdf](http://maths.dur.ac.uk/~dma0jrp/img/HSjyvaskyla.pdf))

* William Abikoff, _The bounded model for hyperbolic 3-space and a quaternionic uniformization theorem_, Mathematica Scandinavica
Vol. 54, No. 1 (August 23, 1984), pp. 5-16 ([jstor:24491416](https://www.jstor.org/stable/24491416))

See also

* [[Manifold Atlas]], _[Hyperbolic 3-manifolds](http://www.map.mpim-bonn.mpg.de/index.php/Hyperbolic_3-manifolds)_

[[!redirects hyperbolic 3-spaces]]


hyperbolic cosine

## Definitions

The _hyperbolic cosine function_ is the [[function]] $\cosh \;\colon\; \mathbb{R} \to \mathbb{R}$ from the [[real numbers]] to themselves which is characterized by the following condition:

1. $\cosh $ is the unique [[solution]] among [[smooth functions]] to the [[differential equation]]/[[initial value problem]]

   $$
     cosh '' = cosh
   $$

   (where a prime indicates the [[derivative]]) subject to the initial conditions

   $$
     \begin{aligned}
       cosh(0) &= 1
       \\
       cosh'(0) & = 0
       \,.
     \end{aligned}
   $$

## Related concepts

* [[hyperbolic sine]]

* [[sine]], [[cosine]]

## References

See also

* Wikipedia, _[Hyperbolic function](https://en.wikipedia.org/wiki/Hyperbolic_function#Cosh)_

[[!redirects cosh]]


hyperbolic curves

Let $k$ be an algebraically closed field. Let $X$ be a smooth one-dimensional variety over $k$. Such a curve has a unique projective model $\overline X$, and $X=\overline X\smallsetminus \{x_1,\dots,x_r\}$ for some finite collection $x_1,\dots,x_r$ of points in $\overline X$. Let $g$ be the genus of $\overline X$. We say that $X$ is _hyperbolic_ if $2 g-2+r \gt 0$[^moc].

[^moc]: Mochizuki, Shinichi. [_Correspondences on hyperbolic curves_](http://www.kurims.kyoto-u.ac.jp/~motizuki/Correspondences%20on%20Hyperbolic%20Curves.pdf)


hyperbolic differential operator

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## Idea

A [[differential operator]] whose [[principal symbol]] is hyperbolic (...)

Over a [[Lorentzian manifold]] a _normally hyperbolic differential operator_ is a hyperbolic differential operator whose [[principal symbol]] coincides with the given [[pseudo-Riemannian metric]].

A _[[Green hyperbolic differential operator]]_ is a hyperbolic differential operator that behaves like a [[normally hyperbolic differential operator]] on a [[globally hyperbolic Lorentzian manifold]] in that it has unique [[advanced and retarded Green functions]].

## Examples

* [[wave operator]]

* [[Klein-Gordon operator]]

## Related concepts

* [[elliptic differential operator]]

## References

* {#BaerGinouxPfaeffle07} [[Christian Bär]], [[Nicolas Ginoux]], [[Frank Pfäffle]], _Wave Equations on Lorentzian Manifolds and Quantization_, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover ([arXiv:0806.1036](https://arxiv.org/abs/0806.1036))

[[!redirects hyperbolic differential operators]]

[[!redirects hyperbolic differential equation]]
[[!redirects hyperbolic differential equations]]

[[!redirects normally hyperbolic differential operator]]
[[!redirects normally hyperbolic differential operators]]

[[!redirects normally hyperbolic differential equation]]
[[!redirects normally hyperbolic differential equations]]


hyperbolic function

[[!redirects hyperbolic functions]]

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## Idea

The hyperbolic functions (or sometimes *hyperbolic trigonometric functions*) are analogues of the usual [[trigonometric functions]], but adapted to the [[hyperbola]] $x^2 - y^2 = 1$ rather than the [[circle]] $x^2 + y^2 = 1$.

## Definition

There are multiple ways of introducing the hyperbolic functions. Probably the most straightforward is to use the following definitions based on the [[exponential function]] $\exp$:

* $\cosh(x) \coloneqq \frac1{2} (\exp(x) + \exp(-x))$ (*hyperbolic cosine*, sometimes pronounced as "kosh"). This can be interpreted as a function $\mathbb{R} \to \mathbb{R}$, or as a function $\mathbb{C} \to \mathbb{C}$.

* $\sinh(x) \coloneqq \frac1{2} (\exp(x) - \exp(-x))$ (*hyperbolic sine*, sometimes pronounced as "cinch"). This also can be interpreted as a function $\mathbb{R} \to \mathbb{R}$, or as a function $\mathbb{C} \to \mathbb{C}$.

* The remaining hyperbolic functions are defined by $\tanh \coloneqq \frac{\sinh}{\cosh}$, $\coth = \frac{\cosh}{\sinh}$, $\sech = \frac1{\cosh}$, $\csch = \frac1{\sinh}$.

Note that $(\cosh(t), \sinh(t))$ (as a pair of real-valued functions) lies on the hyperbola $x^2 - y^2 = 1$, and in fact this is a parametrization of the hyperbola, much as $(\cos(t), \sin(t))$ parametrizes the unit circle $x^2 + y^2 = 1$.

It is straightforward to establish the following further properties by exploiting properties of the exponential function:

* $\cosh(x + y) = \cosh(x)\cosh(y) + \sinh(x)\sinh(y)$ and $\sinh(x+y) = \sinh(x)\cosh(y) - \cosh(x)\sinh(y)$ ("addition formulas"),

* $(\cosh)' = \sinh$ and $(\sinh)' = \cosh$,

* $\cosh(x + 2\pi i) = \cosh(x)$ and $\sinh(x + 2\pi i) = \sinh(x)$ for all $x \in \mathbb{C}$,

* $\cosh(x) = \sum_{n \geq 0} \frac{x^{2 n}}{(2 n)!}$ and $\sinh(x) = \sum_{n \geq 0} \frac{x^{2 n + 1}}{(2 n + 1)!}$.

### Alternative presentation

Another avenue is first to introduce the _inverse_ hyperbolic functions as [[integrals]] of suitable [[algebraic functions]], e.g.,

$$\sinh^{-1}(t) = \int_0^t \; \frac{d x}{y}$$

where $y = \sqrt{x^2 + 1}$. Or, what is essentially the same, construct a solution $p(t)$ to the [[differential equation]]

$$(p')^2 = p^2 + 1, \qquad p(0) = 0$$

so that $(p'(t), p(t))$ parametrizes the curve $x^2 - y^2 = 1$; the approach by invoking an integral is in accordance with an elementary method in differential equations that goes by the name "separation of variables".

This particular approach is similar to the way that [[elliptic functions]] were introduced historically, by studying integrals of algebraic functions

$$\int \; \frac{d x}{y}$$

where $y$ is related to $x$ via an algebraic relation such as $y^2 = x^3 + a x + b$. For suitable such relations these give the various so-called _elliptic integrals_; for more on what this has to do with ellipses, see Wikipedia, e.g., [here](http://en.wikipedia.org/wiki/Elliptic_integral#Incomplete_elliptic_integral_of_the_first_kind). Elliptic functions are then suitable inverses of elliptic integrals, following Jacobi, Abel, and others throughout the nineteenth century (e.g., Weierstrass; see also [[Weierstrass elliptic function]]).

### Area parametrization

The way that hyperbolic functions $(\cosh t, \sinh t)$ parametrize the hyperbola $x^2 - y^2 = 1$ is not through an arc length parametrization (the usual way that circular functions $(\cos t, \sin t)$ parametrize the circle $x^2 + y^2 = 1$), but rather through an area parametrization. In fact, both circular and hyperbolic functions may be parametrized in unison through area, as follows.

Consider, for a parameter $t \in [0, \infty)$, the area $A$ of the planar region which is bounded on three sides by

* The line segment from $(0, 0)$ to $(1, 0)$,

* The line segment from $(0, 0)$ to $(\cosh t, \sinh t)$,

* The arc along the hyperbola $x^2 - y^2 = 1$ that is between the points $(1, 0)$ and $(\cosh t, \sinh t)$.

Then $t = 2A$ for positive $t$. (Negative $t$ can also be accommodated under this scheme if we consider oriented area.)

Likewise, consider for a parameter $t \in [0, \pi)$ the area $A$ of the planar region which is bounded on three sides by

* The line segment from $(0, 0)$ to $(1, 0)$,

* The line segment from $(0, 0)$ to $(\cos t, \sin t)$,

* The arc along the circle $x^2 + y^2 = 1$ that is between the points $(1, 0)$ and $(\cos t, \sin t)$.

Then $t = 2A$. (Other values of $t$ can be accommodated by considering [[orientation]], as well as areas swept out multiple times if need be.)

## Related entries

* [[trigonometry]]

[[!redirects hyperbolic functions]]

[[!redirects hyperbolic trigonometric function]]
[[!redirects hyperbolic trigonometric functions]]