orientifold plane

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

#Contents#
* table of contents
{:toc}

## Idea

In [[type II string theory]] on [[orientifolds]] ([Dai-Leigh-Polchinski 89](#DaiLeighPolchinski89)), one says _O-plane_ for the [[fixed point]] locus of the $\mathbb{Z}_2$-[[involution]] (see at _[[real space]]_).

O-planes carry [[D-brane charges]] in [[KR-theory]] ([Witten 98](#Witten98)), see ([DMR 13](#DMR13)) for a mathematical account. They serve [[RR-field tadpole cancellation]] and as such play a key role in the construction of [[intersecting D-brane models]] for [[string phenomenology]].

## Properties

### T-Duality with type I string theory
 {#TDualityWithTypeIStringTheory}

Under [[T-duality]], [[type I string theory]] is [[duality in string theory|dual]] to [[type II string theory]] with orientifold planes (reviewed e.g. in [Ibanez-Uranga 12, section 5.3.2 - 5.3.4](#IbanezUranga12)):

### O-Plane charge
 {#OPlaneCharge}

O-planes carry effective negative [[RR-charge]] which may (must) cancel against the actual [[RR-charge|RR-]] [[D-brane charge]] via [[RR-field tadpole cancellation]].

#### For flat orientifolds
 {#OPlaneChargeForFlatOrientifolds}

The charge of the [[spacetime]]-filling $O9$-plane of plain [[type I string theory]] ([[type II string theory]] on the [[orientifold]] $\mathbb{R}^{9,1}\sslash \mathbb{Z}_2$ with [[trivial action|trival]] spacetime $\mathbb{Z}_2$-[[action]]) is found by [[worldsheet]]-computation to be $-32$ in [[physical unit|units]] of [[D9]]-[[D-brane charge|brane charge]]:

\[
  \label{O9PlaneCharge}
  q_{O9^-}
  \;=\;
  -32
  \,
  q_{D9}
\]

(e.g. [Blumenhagen-Lüst-Theisen 13 (9.83)](#BlumenhagenLustTheisen13)).

+-- {: .num_remark #ContingOfDBranesOnOrientifolds}
###### Remark
**counting of D-branes on orientifolds**

Beware that there is some convention involved in assigning an absolute value of unit D-brane charge $q_{D9}$. A common choice in the literature is to mean by "one D-brane" one of the two pre-images on the [[covering space]], in which case its obsolute charge is to be

\[
  \label{DBraneChargeOnOrbifold}
  q_{Dp}
  \;=\;
   1/2
\]

(e.g. [BDHKMMS 01, p. 46-47](#BDHKMMS01)). From [BLT 13, p. 318](#BlumenhagenLustTheisen13):

=--

This means that [[RR-field tadpole cancellation]] here requires the presence of 32 [[D-branes]] (or rather, by Remark \ref{ContingOfDBranesOnOrientifolds}: 16 and their $\mathbb{Z}_2$-mirror images), hence a space-filling [[D9-brane]] with [[Chan-Paton bundle]] of [[rank of a vector bundle|rank]] $32$, corresponding to a [[gauge group]] [[SO(32)]]. For more on this see at _[[type I string theory]] -- [Tadpole cancellation and SO(32)-GUT](type+I+string+theory#TadpoleCancellationAndSO32GUT)_.

From this the O$p^-$-brane charge for $p \leq n$ follows from [[T-duality]] (as [above](#TDualityWithTypeIStringTheory))
with respect to [[KK-compactification]] on a [[n-torus|d-torus]] $\mathbb{T}^d$ with $\mathbb{Z}_2$-[[action]] given by canonical [[coordinate function|coordinate]] [[reflection]]

$$
  \array{
    \mathbb{Z}_2 \times \mathbb{R}^{10-d-1,1} \times \mathbb{T}^d
    &\longrightarrow&
    \mathbb{R}^{10-d-1,1}
      \times
    \mathbb{T}^d
    \\
    (\sigma, (\vec x, \vec y))
    &\mapsto&
     (\vec x, - \vec y)
  }
  \,.
$$

This results in $O(9-d)^-$-planes with [[worldvolume]] $\mathbb{R}^{10-d-1,1}$. But since the [[orbifold]] $\mathbb{T}^d\sslash \mathbb{Z}_2$ now has $2^d$ [[singularities]] /[[fixed points]] ([this Example](Riemannian+orbifold#CoordinateReflectionOnNTorus)) there are now $2^d$ such $O(9-d)^-$-planes.

Since the number of [[D-branes]] does not change under [[T-duality]], the total O-plane charge should be the same as before

$$
  2^d \cdot q_{O(9-d)}
  \;=\;
  1 \cdot q_{O9}
  \;=\;
  -32 \cdot q_{D9}
  \;=\;
  - 2^5 \cdot q_{D9}
$$

which means that the $O(9-d)^-$-plane charge is

$$
  q_{O(9-d)^-}
  \;=\;
  - 2^{5-d} \cdot q_{D(p-d)}
$$

or equivalently

\[
  \label{OpPlaneCharge}
  q_{O p^-}
  \;=\;
  - 2^{ p - 4 } \cdot q_{D p}
\]

(e.g. [Ibáñez-Uranga 12 (5.52)](#IbanezUranga12), [Blumenhagen-Lüst-Theisen 13 (10.212)](#BlumenhagenLustTheisen13))

{#TableOfOPlaneCharges} In summary, we have the following table of O-plane charges on [[flat orbifolds]]:

| [[O-plane]]
 species | charge
 $q_{O p^-}/q_{D p}$| transverse
 [[n-torus|d-torus]] | [[fixed points]]
 $\left\vert\left( \mathbb{T}^d\right)^{\mathbb{Z}_2}\right\vert$ |
|------------|--------|-------|----|
| $O9^-$ | $-32$ | $\mathbb{T}^0$  | $\phantom{1}1$ |
| $O8^-$ | $-16$ | $\mathbb{T}^1$  | $\phantom{1}2$ |
| $O7^-$ | $-\phantom{1}8$ | $\mathbb{T}^2$  | $\phantom{1}4$ |
| $O6^-$ | $-\phantom{1}4$ | $\mathbb{T}^3$  |  $\phantom{1}8$ |
| $O5^-$ | $-\phantom{1}2$ | $\mathbb{T}^4$  |  $16$ |
| $O4^-$ | $-\phantom{1}1$ | $\mathbb{T}^5$  |  $32$ |

In particular the [[O4-plane]] has negative unit charge (in [[physical unit|units]] of [[D4]]-[[D-brane charge|brane charge]] $q_{D4}$), so that the total charge of $-32$ here comes entirely from the [[number]] $32 = 2^5$ of [[fixed points]] of the $\mathbb{Z}_2$-[[action]] on $\mathbb{T}^5$.

O-plane charges of different dimension may be present

\begin{center}

\end{center}

>  graphics grabbed from [Johnson 97](#Johnson97)

#### In the presence of discrete torsion
{#WithDiscreteTorsion}

In the presence of [[discrete torsion]] in the [[B-field]] and/or the [[RR-fields]], this charge structure of orientifold planes on [[flat orbifolds]] gets further modified ([Hanany-Kol 00, Sec. 2.1](#HananyKol00), see [Bergman-Gimon-Sugimoto 01, Sec. 1](#BergmanGimonSugimoto01)):

\begin{center}
\begin{imagefromfile}
  "file_name": "OPlaneChargeWithDiscreteTorsion.jpg",
  "width": 470
\end{imagefromfile}
\end{center}

> graphics grabbed from [Bergman-Gimon-Sugimoto 01](#BergmanGimonSugimoto01)

(In comparing this last table with the [above table](#TableOfOPlaneCharges),
notice that this shows the Op-plane charge in [[physical unit|units]] of $q_{Dq} \coloneqq 1/2$ as in (eq:DBraneChargeOnOrbifold).)

#### In differential equivariant KR-theory

A proposal for a formalization of a much more general formula for O-plane charge, regarded in [[differential K-theory|differential]] [[equivariant K-theory|equivariant]] [[KR-theory]] is briefly in [Distler-Freed-Moore 09, p. 6](#DistlerFreedMoore09).

### Duality with M-Theory
 {#DualityWithMTheory}

The possible O-planes in [[M-theory]] are $MO1$ ($\leftrightarrow$[[M-wave]]), [[MO5]] ($\leftrightarrow$[[M5-brane]]) and [[MO9]] ([Hanany-Kol 00 around (3.2)](#HananyKol00), [HSS 18, Prop. 4.7](#HSS18)).

Under the [[duality between M-theory and type IIA string theory]] the O8-plane is identified with the [[MO9]] of [[Horava-Witten theory]]:

\begin{center}

\end{center}

> graphics grabbed from [GKSTY 02, section 3](#GKSTY02)

while the [[O4-plane]] is [[duality between M-theory and type IIA string theory|dual]] to the [[MO5]] ([Hori 98](#Hori98), [Gimon 98, Sec. III](#Gimon98), [AKY 98, Sec. II B](#AKY98), [Hanany-Kol 00, 3.1.1](#HananyKol00))

\begin{center}

\end{center}

> graphics grabbed from [Gimon 98](#Gimon98)

and the $O0$ to the MO1 ([Hanany-Kol 00 3.3](#HananyKol00))

### Fractional branes at O-planes

By the discussion at _[D-branes ending on NS5 branes](NS5-brane#D6BranesEndingOnNS5Branes)__, a [[black brane|black]] [[D6-brane]] may end on a [[black brane|black]] [[NS5-brane]], and in fact a priori each [[black brane|brane]] [[NS5-brane]] has to be the junction of two [[black brane|black]] [[D6-branes]].

> from [GKSTY 02](#GKSTY02)

If in addition the [[black brane|black]] [[NS5-brane]] sits at an [[O8-plane]], hence at the [[orientifold]] [[fixed point]]-locus, then in the ordinary $\mathbb{Z}/2$-[[quotient]] it appears as a "[[half-brane]]" with only one copy of [[D6-branes]] ending on it:

> from [GKSTY 02](#GKSTY02)

(In [Hanany-Zaffaroni 99](#HananyZaffaroni99) this is interpreted in terms of the [['t Hooft-Polyakov monopole]].)

{#LiftToMTheortyOfNS5Halfbrane} The lift to [[M-theory]] of this situation is an [[M5-brane]] intersecting an [[M9-brane]]:

> from [GKSTY 02](#GKSTY02)

Alternatively the [[O8-plane]] may intersect the [[black brane|black]] [[D6-branes]] away from the [[black brane|black]] [[NS5-brane]]:

> from [HKLY 15](#HKLY15)

In general, some of the NS5 sit away from the [[O8-plane]], while some sit on top of it:

> from [Hanany-Zaffaroni 98](#HananyZaffaroni98)

See also at _[[intersecting D-brane models]]_ the section _[Intersection of D6s with O8s](intersecting+D-brane+model#IntersectionOfD6WithO8)_.

## Related concepts

* [[D-brane]]

* [[M-theory lift of gauge enhancement on D6-branes]]

* [NS5 half-branes](NS5-brane#NSHalfBranes)

## References

### General

The term "orientifold" originates around

* {#DaiLeighPolchinski89} Jin Dai, R.G. Leigh, [[Joseph Polchinski]], p. 12 of _New Connections Between String Theories_, Mod.Phys.Lett. A4 (1989) 2073-2083 ([spire:25758](http://inspirehep.net/record/25758))

Other early accounts include

* {#Johnson97} [[Clifford Johnson]], _Anatomy of a Duality_, Nucl.Phys. B521 (1998) 71-116 ([arXiv:hep-th/9711082](https://arxiv.org/abs/hep-th/9711082))

* {#HananyZaffaroni98} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Branes and Six Dimensional Supersymmetric Theories_, Nucl.Phys. B529 (1998) 180-206 ([arXiv:hep-th/9712145](https://arxiv.org/abs/hep-th/9712145))

* {#Witten98} [[Edward Witten]], section 5 of _D-branes and K-theory_, J. High Energy Phys., 1998(12):019, 1998 ([arXiv:hep-th/9810188](http://arxiv.org/abs/hep-th/9810188))

* [[Sunil Mukhi]], Nemani V. Suryanarayana, _Gravitational Couplings, Orientifolds and M-Planes_, JHEP 9909 (1999) 017 ([arXiv:hep-th/9907215](https://arxiv.org/abs/hep-th/9907215))

* Yoshifumi Hyakutake, Yosuke Imamura, [[Shigeki Sugimoto]], _Orientifold Planes, Type I Wilson Lines and Non-BPS D-branes_, JHEP 0008 (2000) 043 ([arXiv:hep-th/0007012](https://arxiv.org/abs/hep-th/0007012))

* {#BDHKMMS01} [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], section 3 of _Triples, Fluxes, and Strings_, Adv.Theor.Math.Phys. 4 (2002) 995-1186 ([arXiv:hep-th/0103170](https://arxiv.org/abs/hep-th/0103170))

Textbook accounts:

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 10 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge 2012

* {#BlumenhagenLustTheisen13} [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], Section 9.4 and 10.6 of _Basic Concepts of String Theory_ Part of the series Theoretical and Mathematical Physics, Springer 2013

See also

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

### With discrete torsion

O-Plane charge in the presence of [[discrete torsion]]:

* [Hanany-Kol 00](#HananyKol00)

* {#BergmanGimonSugimoto01} [[Oren Bergman]], [[Eric Gimon]], [[Shigeki Sugimoto]], _Orientifolds, RR Torsion, and K-theory_, JHEP 0105:047, 2001 ([arXiv:hep-th/0103183](https://arxiv.org/abs/hep-th/0103183))

* [[Atish Dabholkar]], [[Jaemo Park]], _Strings on Orientifolds_, Nucl. Phys. B477 (1996) 701-714 ([arXiv:hep-th/9604178](https://arxiv.org/abs/hep-th/9604178))

### In terms of KO-theory

O-Plane charge in [[differential K-theory|differential]] [[equivariant K-theory|equivariant]] [[KR-theory]]:

* {#DMR13} [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], _T-duality For Orientifolds and Twisted KR-theory_ ([arXiv:1306.1779](http://arxiv.org/abs/1306.1779))

* {#DistlerFreedMoore09} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf))

* {#DistlerFreedMooreII} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Spin structures and superstrings_ ([arXiv:1007.4581](http://arxiv.org/abs/1007.4581))

reviewed/surveyed in

* [[Daniel Freed]], _Dirac charge quantiation, K-theory, and orientifolds_, talk at a workshop _Mathematical methods in general relativity and quantum field theories_, Paris, November 2009 ([pdf](http://www.ma.utexas.edu/users/dafr/paris_nt.pdf), [[FreedK09.pdf:file]])

* {#Moore10} [[Greg Moore]], _The RR-charge of an orientifold_, Oberwolfach talk 2010 ([pdf](https://www.physics.rutgers.edu/~gmoore/Oberwolfach_June2010_FINAL.pdf), [[MooreOrientifold2010.pdf:file]], [ppt](http://www.physics.rutgers.edu/~gmoore/AnnArbor_Feb2010_FINAL.ppt))

* {#Freed} [[Daniel Freed]], _Lectures on twisted K-theory and orientifolds_, lecures at _[K-Theory and Quantum Fields](http://www.esi.ac.at/activities/events/2012/k-theory-and-quantum-fields)_, ESI 2012 ([[FreedESI2012.pdf:file]])

Actual construction of [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[KO-theory|orthogonal]] [[topological K-theory|K-theory]] and its relation to [[D-brane charge]] in [[type I string theory]] (on [[orientifolds]]):

* {#GradySati19b} [[Daniel Grady]], [[Hisham Sati]], _Twisted differential KO-theory_ ([arXiv:1905.09085](https://arxiv.org/abs/1905.09085))

### Examples / Models

The [[Witten-Sakai-Sugimoto model]] for [[QCD]] on O-planes:

* {#ImotoSakaiSugimoto09} Toshiya Imoto, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _$O(N)$ and $USp(N)$ QCD from String Theory_, Prog.Theor.Phys.122:1433-1453, 2010 ([arXiv:0907.2968](https://arxiv.org/abs/0907.2968))

### Lift to M-theory

Lift to [[M-theory]] ([[MO5]], [[MO9]]):

* {#Hori98} [[Kentaro Hori]], _Consistency Conditions for Fivebrane in M Theory on $\mathbb{R}^5/\mathbb{Z}_2$ Orbifold_, Nucl.Phys.B539:35-78, 1999 ([arXiv:hep-th/9805141](https://arxiv.org/abs/hep-th/9805141))

* {#Gimon98} [[Eric Gimon]], _On the M-theory Interpretation of Orientifold Planes_ ([arXiv:hep-th/9806226](https://arxiv.org/abs/hep-th/9806226), [spire:472499](http://inspirehep.net/record/472499))

* {#AKY98} Changhyun Ahn, Hoil Kim, Hyun Seok Yang, _$SO(2N)$ $(0,2)$ SCFT and M Theory on $AdS_7 \times \mathbb{R}P^4$_, Phys.Rev. D59 (1999) 106002 ([arXiv:hep-th/9808182](https://arxiv.org/abs/hep-th/9808182))

* {#GKSTY02} E. Gorbatov, V.S. Kaplunovsky, J. Sonnenschein, [[Stefan Theisen]], S. Yankielowicz, section 3 of _On Heterotic Orbifolds, M Theory and Type I' Brane Engineering_, JHEP 0205:015, 2002 ([arXiv:hep-th/0108135](https://arxiv.org/abs/hep-th/0108135))

* {#HananyKol00} [[Amihay Hanany]], [[Barak Kol]], _On Orientifolds, Discrete Torsion, Branes and M Theory_, JHEP 0006 (2000) 013 ([arXiv:hep-th/0003025](https://arxiv.org/abs/hep-th/0003025))

* {#HKLY15} Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Futoshi Yagi, _6d SCFTs, 5d Dualities and Tao Web Diagrams_ ([arXiv:1509.03300](https://arxiv.org/abs/1509.03300))

* {#HSS18} [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_ ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987))

The [[brane intersection|intersection]] with [[(p,q)5-brane webs]]:

* Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Masato Taki, Futoshi Yagi, _More on 5d descriptions of 6d SCFTs_, JHEP10 (2016) 126 ([arXiv:1512.08239](https://arxiv.org/abs/1512.08239))

* [[Amihay Hanany]], [[Alberto Zaffaroni]], _Issues on Orientifolds: On the brane construction of gauge theories with $SO(2n)$ global symmetry_, JHEP 9907 (1999) 009 ([arXiv:hep-th/9903242](https://arxiv.org/abs/hep-th/9903242))

* {#Zafrir2016} Gabi Zafrir, _Brane webs in the presence of an $O5^-$-plane and 4d class S theories of type D_, JHEP07 (2016) 035 ([arXiv:1602.00130](https://arxiv.org/abs/1602.00130))

[[!redirects orientifold planes]]

[[!redirects O-plane]]
[[!redirects O-planes]]

[[!redirects O6-plane]]
[[!redirects O6-planes]]

[[!redirects O7-plane]]
[[!redirects O7-planes]]

[[!redirects O8-plane]]
[[!redirects O8-planes]]

[[!redirects Op-plane]]
[[!redirects Op-planes]]

[[!redirects O-plane charge]]
[[!redirects O-plane charges]]


Orit Davidovich

* [arXiv page](https://arxiv.org/search/math?searchtype=author&query=Davidovich%2C+O)

## Selected writings

On [[number theory|number theoretic]] aspects of [[modular tensor categories]]:

* {#DavidovichHaggeWang2013} [[Orit Davidovich]], [[Tobias Hagge]], [[Zhenghan Wang]], _On Arithmetic Modular Categories_, arXiv preprit, 2013 [[arXiv:1305.2229](https://arxiv.org/abs/1305.2229)]

category: people


Orlando Alvarez

* [website](http://www.physics.miami.edu/~alvarez/)

* [inspire page](https://inspirehep.net/authors/1018545)

* [GoogleScholar page](https://scholar.google.com/citations?user=2Gn_5V8AAAAJ&hl=en)

## Selected writings

Interpreting [[elliptic genera]] as [[indices]] of [[Dirac-Ramond operators]] as would-be [[Dirac operators on smooth loop space]]:

* {#AlvarezKillingbackManganoWindey87} [[Orlando Alvarez]], T. P. Killingback, Michelangelo Mangano, [[Paul Windey]], _The Dirac-Ramond operator in string theory and loop space index theorems_, Nuclear Phys. B Proc. Suppl., 1A:189–215, 1987, in: _Nonperturbative methods in field theory_, 1987 (doi"10.1016/0920-5632(87)90110-1)

* [[Orlando Alvarez]], T. P. Killingback, Michelangelo Mangano, [[Paul Windey]], _String theory and loop space index theorems_, Comm. Math. Phys., 111(1):1–10, 1987 ([euclid:cmp/1104159462](https://projecteuclid.org/euclid.cmp/1104159462))

* [[Orlando Alvarez]], [[Paul Windey]], _Analytic index for a family of Dirac-Ramond operators_, Proc. Natl. Acad. Sci. USA, 107(11):4845–4850, 2010 ([arXiv:0904.4748](https://arxiv.org/abs/0904.4748))

On the [[Schwinger effect]]:

* {#AffleckAlvarezManton82} [[Ian Affleck]], [[Orlando Alvarez]], [[Nicholas Manton]], _Pair production at strong coupling in weak external fields_, Nuclear Physics B Volume 197, Issue 3 (1982) Pages 509-519 (doi:10.1016/0550-3213(82)90455-2)

  > (via [[worldline formalism]])

Formulating [[Dirac charge quantization]] via [[Čech cohomology]]:

* {#Alvarez85a} [[Orlando Alvarez]], *Cohomology and Field Theory*, talk at: *Symposium on Anomalies, Geometry, Topology*, Argonne IL (28-30 March 1985) [[inspire:965785](https://inspirehep.net/conferences/965785), [pdf](https://lib-extopc.kek.jp/preprints/PDF/1985/8507/8507262.pdf), [[Alvarez-CohomologyAndFieldTheory.pdf:file]]]

* [[Orlando Alvarez]], _Topological quantization and cohomology_, Comm. Math. Phys. **100** 2 (1985) 279-309 [[euclid:cmp/1103943448](https://projecteuclid.org/euclid.cmp/1103943448)]

category: people


Orlando Villamayor

* [GoogleScholar page](https://www.researchgate.net/profile/Orlando-Villamayor)

* [ResearchGate page](https://www.researchgate.net/profile/Orlando-Villamayor)

## Selected writings

Introducing [[Hermitian K-theory]]:

* [[Max Karoubi]], [[Orlando Villamayor]], *K-théorie algébrique et K-théorie topologique II*, Math. Scand. **32** (1973) 57-86 [[jstor:24490565](https://www.jstor.org/stable/24490565)]

## Related entries

* [[hyperbolic functor]]

category: people

[[!redirects Orlando E. Villamayor]]


Orlicz spaces

For Orlicz spaces see https://www.encyclopediaofmath.org/index.php/Orlicz_space


Orlov spectrum

## Idea

__Orlov's dimension spectrum__ (or simply Orlov spectrum) is an invariant of a triangulated category, introduced by [[Dmitri Orlov]]. When the triangulated category is of geometric origin (e.g. the bounded derived category of coherent sheaves on a projective variety, or the Fukaya category associated to a symplectic manifold) then the Orlov spectrum reflects some geometric information. The spectrum is
defined in terms of counting the extensions needed to generate all the objects from a fixed object, with some other operations, like competing under direct coproducts and summands not counted.

## Definitions

An object $E$ in a triangulated category $T$ defines the smallest triangulated subcategory $I_E\subset T$ which is closed under direct sum. Given two full triangulated subcategories $I_1$ and $I_2$ one defines the full triangulated category $I_1 \ast I_2$ consisting of all $M$ such that there exist $M_1$ in $I_1$ and $M_2$ in $I_2$ such that $M_1\to M \to M_2$ is a distinguished triangle, and by $\langle I_1 \ast I_2\rangle$ the smallest full subcategory of $T$ containing $I_1 \ast I_2$ and closed under finite coproducts, summands and shifts. Define by induction $\langle E\rangle_1 = I_E$ and $\langle E\rangle_{k+1} = \langle \langle E\rangle_k \ast I_E \rangle$, $k\gt 1$.

The __dimension of a triangulated category__ $T$ is the minimal integer $d\gt 0$ such that there is $E\in Ob T$ such that $\langle E\rangle_d = T$ or infinity otherwise. The __generation time__ $d_E$ of an object $E$ in $T$ such that $\langle E\rangle_{d+1} = T$ and $\langle E\rangle_d \neq T$. $E$ is a strong generator if the generation time $d_E$ is finite.

The __dimension spectrum__ of $T$ is the set $\sigma(T)$ of generation times of all strong generators of $T$.

## Basic results

By a result of Rouquier, the dimension of the triangulated category $D^b(Coh X)$ for a separated scheme $X$ of finite type over a perfect field is finite and, if $X$ is in addition reduced, it is equal or bigger than the Krull dimension of $X$ but smaller or equal the double dimension $2 dim(X)$. By a conjecture of Orlov, for any smooth variety it should be in fact equal to $dim(X)$.

## Literature

* [[Dmitri Orlov]], _Remarks on generators and dimensions of triangulated categories_, [arxiv/0804.1163](http://arxiv.org/abs/0804.1163)
* R. Rouquier, _Dimension of triangulated categories_, J. K-Theory 1 (2008), no.2, 193-256, [arXiv:math.CT/0310134](http://arxiv.org/abs/math/0310134)
* Matthew Ballard, David Favero, [[Ludmil Katzarkov]], _Orlov spectra: bounds and gaps_, [arxiv/1012.0864](http://arxiv.org/abs/1012.0864)
* [[A. Bondal]], M. van den Bergh, _Generators and representability of functors in commutative and non-commutative geometry_, Mosc. Math. J. __3__ (2003), no.1, 1-36, [math.AG/0204218](http://arxiv.org/abs/math/0204218)
* David Favero, _Dimensions of triangulated categories_, joint
work with M. Ballard and L. Katzarkov, slides, Jan 2010, [pdf](http://www-math.mit.edu/~auroux/frg/miami10-notes/D.%20Favero%20-%2021-01-10%20-%20The%20dimension%20spectrum%20of%20a%20triangulated%20category.pdf)
[[!redirects Orlov's spectrum]]
[[!redirects Orlov spectra]]
[[!redirects dimension spectrum]]


orthogonal basis

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

# Orthogonal and orthonormal subsets
* table of contents
{: toc}

## Definition

For $V$ an [[inner product space]], a collection of vectors $\{v_i\}$ of $V$ is **orthogonal** with respect to the inner $\langle{-, -}\rangle\colon V \times V \to k$ if for all $i \neq j$ we have $\langle{v_i, v_j}\rangle = 0$. It is __orthonormal__ if additionally we have $\langle{v_i, v_i}\rangle = 1$; that is, $\langle{v_i, v_j}\rangle = \delta_{i,j}$ (the [[Kronecker delta]]) for all $i, j$.  Note that an orthogonal set is necessarily [[linearly independent subset|linearly independent]], assuming (as one often does) that the inner product is nondegenerate.

## Bases

In speaking of the notion of orthonormal or orthogonal _[[basis]]_ of $V$, due attention must be paid to the meaning of 'basis'. When $V$ has finite [[dimension]], one can just use the [[Hamel basis|purely algebraic sense of basis]]: a linearly independent set whose span is all of $V$, or equivalently a [[maximal element|maximal]] linearly independent set. In this case, there is no restriction on the ground field $k$.

Often, however, 'basis' is supposed to mean the [[basis in functional analysis|sense of basis]] used for a [[topological vector space]] $V$, i.e., a linearly independent set whose span is [[dense subspace|dense]] in $V$. In the TVS case, it is usually understood that the ground field $k$ is a [[local field]], and the inner product is separately continuous with respect to the topology on $V$. A typical situation is where $k$ is $\mathbb{R}$ or $\mathbb{C}$ and the inner product is a positive definite inner product on $V$ which naturally induces a [[metric space|metric]] and from there a TVS structure on $V$.

We have this interesting pair of results:

* On one hand, no infinite-dimensional positive-definite inner-product space has an orthogonal (much less orthonormal) basis in the purely algebraic sense.
* On the other hand, any maximal orthogonal set (which must exist by an application of [[Zorn's lemma]]) is necessarily a basis in the topological sense.

## Gram–Schmidt process

For a finite-dimensional vector space equipped with a nondegenerate inner product, one may replace any [[well-ordered set|well-ordered basis]] by an well-ordered orthogonal basis using the [[Gram–Schmidt process]]. Assuming that we have $\langle v, v \rangle \neq 0$ for $v \neq 0$ (as for example in the case of a positive definite inner product space), and assuming square roots $\langle v, v \rangle^{1/2}$ exist in the ground field $k$, we can further produce a well-ordered orthonormal basis from a well-ordered basis.

In the TVS case, with the appropriate meaning of basis understood, the Gram-Schmidt process extends without difficulty to any well-ordered basis, because the process yields a linearly independent orthogonal set with the same span as the original basis.

## Examples

* [[Schur's lemma]] says that [[irreducible representations]] form an orthogonal basis of the [[representation ring]]. See [there](Schur's+lemma#InterpretationInCategoricalAlgebra) for more.

## Related concepts

* [[frame field]]

* [[orthogonality]]

* [[orthogonal structure]]

* [[normal vector]]

* [[mutually unbiased bases]]

## References

See also

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

[[!redirects orthogonal set]]
[[!redirects orthogonal sets]]
[[!redirects orthogonal subset]]
[[!redirects orthogonal subsets]]

[[!redirects orthonormal set]]
[[!redirects orthonormal sets]]
[[!redirects orthonormal subset]]
[[!redirects orthonormal subsets]]

[[!redirects orthogonal basis]]
[[!redirects orthogonal bases]]
[[!redirects orthogonal basises]]

[[!redirects orthogonal linear basis]]
[[!redirects orthogonal linear bases]]
[[!redirects orthogonal linear basises]]

[[!redirects orthonormal basis]]
[[!redirects orthonormal bases]]
[[!redirects orthonormal basises]]

[[!redirects orthonormal linear basis]]
[[!redirects orthonormal linear bases]]
[[!redirects orthonormal linear basises]]

[[!redirects orthonormal]]
[[!redirects orthonormal vector]]
[[!redirects orthonormal vectors]]

[[!redirects right angle]]


orthogonal calculus

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

#Contents#
* table of contents
{:toc}

## Idea

The _orthogonal calculus_ is a calculus of [[functors]] from [[vector spaces]] to [[homotopy types]],  similar to but different from the [[Goodwillie calculus]] of [[(infinity,1)-functors]].

The prime example is the assignment $V \mapsto B O(V)$ of the [[classifying space]] of the [[orthogonal group]] $O(V)$ to any [[inner product space|inner product]] [[vector space]] $V$. In this example, and in general, first derivatives in the orthogonal calculus reproduce and generalize much of the theory of [[Stiefel-Whitney
classes]]. Similarly, second derivatives in the orthogonal calculus reproduce
and generalize much of the theory of [[Pontryagin classes]]. (see [Weiss 95](#Weiss95))

The complex analog of the orthogonal calculus is known as the [[unitary calculus]].

## References

* {#Weiss95} [[Michael Weiss]], _Orthogonal calculus_, Trans. Amer. Math. Soc. 347 (1995), 3743-3796, [pdf](https://wwwmath.uni-muenster.de/u/michael.weiss/preprints/or.pdf)

* [[David Barnes]], [[Rosona Eldred]], _Comparing the orthogonal and homotopy functor calculi_, Journal of Pure and Applied Algebra **220** 11 (2016) 3650-3675 [[arXiv:1505.05458](http://arxiv.org/abs/1505.05458), [doi:10.1016/j.jpaa.2016.05.005](https://doi.org/10.1016/j.jpaa.2016.05.005)]


orthogonal factorization system

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

#Contents#
* table of contents
{: toc}

##Definition
 {#Definition}

### Orthogonal factorization systems

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

Let $C$ be a [[category]] and let $(E,M)$ be two [[classes]] of [[morphisms]] in
$C$. We say that $(E,M)$ is an **orthogonal factorization system** if $(E,M)$ is a [[weak factorization system]] in which solutions to [[lifting problems]] are unique.

=--

We spell out several equivalent explicit formulation of what this means.

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

$(E,M)$ is an **orthogonal factorization system** if
every morphism $f$ in $C$ factors $f = r\circ \ell$ as a morphism $\ell \in E$ followed by a morphism $r \in M$; and the following equivalent conditions hold

1. We have:

   a. $E$ is precisely the class of morphisms that are left [[orthogonality|orthogonal]] to every morphism in $M$;

   b. $M$ is precisely the class of morphisms that are right [[orthogonality|orthogonal]] to every morphism in $E$.

1. We have:

   a. The factorization is unique up to unique [[isomorphism]].

   b. $E$ and $M$ both contain all [[isomorphisms]] and are closed under composition.

1. We have:

   a. $E$ and $M$ are [[replete subcategories]] of the [[arrow category]] $C^I$.

   b. Every morphism in $E$ is left [[orthogonality|orthogonal]] to every morphism in $M$.

=--

OFS's are traditionally called just **factorization systems**.
See the _[[joyalscatlab:Factorization systems|Catlab]]_ for more of the theory.

An orthogonal factorization system is called **proper** if every morphism in $E$ is an [[epimorphism]] and every morphism in $M$ is a [[monomorphism]].

### Prefactorization systems

For any class $E$ of morphisms in $C$, we write $E^\perp$ for the class of all morphisms that are right orthogonal to every morphism in $E$.  Dually, given $M$ we write ${}^\perp M$ for the class of all morphisms that are  left orthogonal to every morphism in $M$.  The second condition in the definition of an OFS then says that $E= {}^\perp M$ and $M= E^\perp$.

In general, $(-)^\perp$ and ${}^\perp(-)$ form a [[Galois connection]] on the [[poset]] of classes of morphisms in $C$.  A pair $(E,M)$ such that $E= {}^\perp M$ and $M= E^\perp$ is sometimes called a **prefactorization system**.  Note that by generalities about Galois connections, for any class $A$ of maps we have prefactorization systems $({}^\perp(A^\perp),A^\perp)$ and $({}^\perp A, ({}^\perp A)^\perp)$.  We call these *generated* and *cogenerated* by $A$, respectively.

## Properties
 {#Properties}

### General

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

The different characterization in def. \ref{EquivalentDefinitions}
are indeed all equivalent.

=--

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

(...)

For the moment see ([Joyal](#Joyal)).

(...)

=--

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

A [[weak factorization system]] $(L,R)$ is an orthogonal factorization system precisely if $L \perp R$.

=--

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

(...)

For the moment see ([Joyal](#Joyal)).

(...)

=--

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

For $(L,R)$ an orthogonal factorization system
in a category $C$, the intersection $L \cap R$ is precisely the class of [[isomorphisms]] in $C$.

=--

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

If is clear that every isomorphism is in $L \cap R$.
Conversely, let $f : A \to B$ be a morphism in $L \cap R$. This implies that the two trivial factorizations

$$
  f = A \stackrel{id_A}{\to} A \stackrel{f}{\to} B
$$

and

$$
  f = A \stackrel{f}{\to} B \stackrel{id_B}{\to} B
$$

are both $(L,R)$-factorization. Therefore there is a unique morphism $\tilde f$ in the [[commuting diagram]]

$$
  \array{
    A &\stackrel{id_A}{\to}& A
    \\
    \downarrow^{\mathrlap{f}}
     &\nearrow_{\bar f}&
    \downarrow^{\mathrlap{f}}
    \\
    B &\stackrel{id_B}{\to}& B
  }
  \,.
$$

This says precisely that $\bar f$ is a left and right [[inverse]] of $f$.

=--

### Closure properties

A prefactorization system $(E,M)$ (and hence, also, a factorization system) satisfies the following closure properties.  We state them  for $M$, but $E$ of course satisfies the dual property.

* $M$ contains the isomorphisms and is closed under composition and [[pullback]] (insofar as pullbacks exist in $C$).
* If a composite $f g$ is in $M$, and $f$ is either in $M$ or a [[monomorphism]], then $g$ is in $M$.
* $M$ is closed under all [[limits]] in the [[arrow category]] $Arr(C)$.

If $C$ is a [[locally presentable category]], then for any *small set* of maps $A$, the prefactorization system $({}^\perp(A^\perp),A^\perp)$ is actually a factorization system.  The argument is by a transfinite construction similar to the [[small object argument]].

On the other hand, if $(E,M)$ is any prefactorization system for which $M$ consists of monomorphisms and $C$ is [[complete category|complete]] and [[well-powered category|well-powered]], then $(E,M)$ is actually a factorization system.  (Of course, there is a dual statement as well.)  In fact something slightly more general is true; see [[M-complete category]] for this and other related ways to construct factorization systems.

### Cancellation properties
 {#CancellationProperties}

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

For $(L,R)$ an orthogonal factorization system. Let

$$
  \array{
    && Y
    \\
    & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}}
    \\
    X
    &&
    \stackrel{g \circ f}{\to}
    &&
    Z
  }
$$

be two composable morphisms. Then

* If $f$ and $g \circ f$ are in $L$, then so is $g$.

* If $g$ and $g\circ f$ are in $R$, then so is $f$.

=--

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

Consider the first case. The second is directly analogous.

Choose an $(L,R)$-factorization of $g$

$$
  g : Y \stackrel{\ell}{\to} I \stackrel{r}{\to} Z
  \,.
$$

With this we have lifting diagrams of the form

$$
  \array{
    X &\stackrel{g \circ f}{\to}& Z
    \\
    \downarrow^{\mathrlap{f}} &&  \downarrow^{id_Z}
    \\
    Y & \nearrow_r&
    \\
    \downarrow^{\mathrlap{\ell}} && \downarrow^{id_Z}
    \\
    I &\stackrel{r}{\to}& Z
  }
  \;\;\;\;\;\;\;
  \;\;\;\;\;\;\;
  \array{
    X &\stackrel{f}{\to}& Y &\stackrel{\ell}{\to}& I
    \\
    {}^{\mathllap{g \circ f}}\downarrow
     & & \nearrow_{r^{-1}}& &
    \downarrow^{\mathrlap{r}}
    \\
    Z &\underset{id_Z}{\to}& &\underset{id_Z}{\to}& Z
  }
$$

exhibiting an inverse of $r$. Therefore $r$ is an isomorphism, hence is in $L$, by prop. \ref{IsomorphismsAreIntersection}, hence so is the composite $g = r \circ \ell$.

=--

### Characterization as Eilenberg-Moore algebras

Orthogonal factorization systems are equivalently described by the (appropriately defined) [[Eilenberg-Moore algebras]] with respect to the monad which belongs to the endofunctor $\mathcal{K} \mapsto \mathcal{K}^2$ of (the 2-category) [[Cat]] ([Korostenski-Tholen, Thrm B](#KorostenskiTholen)).

## Examples
 {#Examples}

Several classical examples of OFS $(E,M)$:

* in any [[topos]] or [[pretopos]], $E$ = class of all epis, $M$ = class of all monos: the [[(epi, mono) factorization system]];

* more generally, in any [[regular category]], $E$ = class of all [[regular epimorphisms]], $M$ = class of all monos

* in any [[quasitopos]], $E$ = all epimorphisms, $M$ = all [[strong monomorphisms]]

* In [[Cat]], $E$ = [[bo functors]], $M$ = [[fully faithful functors]]: the [[bo-ff factorization system]]

* (Street) in [[Cat]], $E$ = 0-[[final functors]], $M$ = [[discrete fibrations]] (This is called the [[comprehensive factorization system]].)

* (Street) in $\mathrm{Cat}$,  $E$ = 0-[[initial functors]], $M$ = [[discrete opfibration]]s

* in $\mathrm{Cat}$, $M$ = [[conservative functor]]s, $E$ = left orthogonal of $M$ ("iterated strict localizations" after A. Joyal)

* in the category of small categories where morphisms are functors which are [[exact functor|left exact]] and have [[right adjoint]]s, $E$ = class of all such functors which are also localizations, $M$ = class of all such functors which are also conservative

* if $F\to C$ is a [[fibered category]] in the sense of Grothendieck, then $F$ admits a factorization system $(E,M)$ where $E$ = arrows whose projection to $C$ is invertible, $M$ = cartesian arrows in $F$

* See the ([catlab](#Joyal)) for more examples.

## Generalizations

There is a [[categorification|categorified]] notion of a [[factorization system on a 2-category]], in which lifts are only required to exist and be unique up to isomorphism.  Some examples include:

* In [[Cat]], [[(eso, fully faithful) factorization system]]

* In [[Cat]], [[(eso+full, faithful) factorization system]]

Similarly, we can have a [[factorization system in an (∞,1)-category]], and so on; see the links below for other generalizations.

## Related concepts

* [[factorization system]]

  * [[weak factorization system]]

  * **orthogonal factorization system**

  * [[strict factorization system]]

  * [[reflective factorization system]]

  * [[stable factorization system]]

* [[factorization system in a 2-category]]

  * [[enhanced factorization system]]

* [[factorization system in an (∞,1)-category]]

  * [[orthogonal factorization system in an (∞,1)-category]]

  * [[orthogonal factorization system in a derivator]]

* [[factorization structure for sinks]]

## References

Factorisation systems appear to have been first studied by [[Mac Lane]] in the following paper under the term *bicategory* (not to be confused with [[bicategory]]), though this definition imposed extra conditions that are now not considered:

* [[Saunders Mac Lane]]. _Duality for groups_. Bulletin of the American Mathematical Society 56.6 (1950): 485-516.

The definition of orthogonal factorisation system essentially appears under the name "factorization" in:

* [[Peter Freyd]], [[Max Kelly]], *Categories of continuous functors*, J. Pure. Appl. Algebra **2** (1972) 169-191 [doi:10.1016/0022-4049(72)90001-1]

and under the name "factorization system" in:

* [[Aldridge Bousfield]], *Constructions of factorization systems in categories*, Journal of Pure and Applied Algebra **9** 2-3 (1977) 207-220 [doi:10.1016/0022-4049(77)90067-6]

See also:

* {#Joyal} [[André Joyal]], _[[joyalscatlab:Factorization systems]]_

* {#KorostenskiTholen} Mareli Korostenski, [[Walter Tholen]], _Factorization systems as Eilenberg-Moore algebras, ([doi](https://doi.org/10.1016/0022-4049%2893%2990171-O))

Introductory texts:

* [[Introduction to Homotopy Theory]]
* {#Riehl2008} [[Emily Riehl]], [_Factorization Systems_](https://math.jhu.edu/~eriehl/factorization.pdf), 2008
* The following dissertation section is entirely written after learning from Riehl's note above, but has complementary examples and may dive deeper into some proofs:
  * {#Nuyts2020} [[Andreas Nuyts]], _Contributions to Multimode and Presheaf Type Theory, section 2.4: Factorization Systems_, [PhD thesis](https://lirias.kuleuven.be/retrieve/581985), KU Leuven, Belgium, 2020

A connection to [[double categories]] may be found in:

* Miloslav Štěpán, _Factorization systems and double categories_, [arXiv:2305.06714](https://arxiv.org/abs/2305.06714).

[[!redirects orthogonal factorization system]]
[[!redirects orthogonal factorization systems]]
[[!redirects orthogonal factorisation system]]
[[!redirects orthogonal factorisation systems]]
[[!redirects unique factorization system]]
[[!redirects unique factorization systems]]
[[!redirects unique factorisation system]]
[[!redirects unique factorisation systems]]
[[!redirects prefactorization system]]
[[!redirects prefactorization systems]]


#Contents#
* table of contents
{:toc}

## Idea

The analogue of an [[orthogonal factorization system]] in a [[derivator]].

## Related concepts

* [[orthogonal factorization system in an (∞,1)-category]]
* [[homotopy factorization system in a model category]]

## References

* [[Fosco Loregian]], Simone Virili, *Factorization systems on (stable) derivators*, [arxiv](https://arxiv.org/abs/1705.08565)

[[!redirects orthogonal factorization system in a derivator]]
[[!redirects factorization system in a derivator]]
[[!redirects orthogonal factorization system on a derivator]]
[[!redirects factorization system on a derivator]]

[[!redirects orthogonal factorization systems in a derivator]]
[[!redirects factorization systems in a derivator]]
[[!redirects orthogonal factorization systems on a derivator]]
[[!redirects factorization systems on a derivator]]


orthogonal factorization system in an (infinity,1)-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

Let $C$ be an [[(∞,1)-category]] and $f : A \to B$ and $g : X \to Y$ two [[morphism]]s in $C$. Write $C_{A\sslash Y}$ for the [[over-(∞,1)-category|under-over-(∞,1)-category]].

We say that $f$ is **left orthogonal** to $g$ and that $g$ is **right orthogonal** to $f$ and write

$$
  f \perp g
$$

if for every [[commuting diagram]]

$$
  \array{
    A &\to& X
    \\
    {}^{\mathllap{f}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{g}}
    \\
    B &\to& Y
  }
$$

in $C$ we have that $C_{A\sslash Y}(B,X) \simeq *$ is [[contractible]].

=--

Note that the notation $C_{A\sslash Y}(B,X)$ subtly includes the given commuting diagram, since $C_{A\sslash Y}$ is only defined relative to a particular given morphism $A\to Y$.  Here we take that to be the common composite of the given commuting square, with $B$ and $X$ regarded as objects of $C_{A\sslash Y}$ via the resulting commuting triangles.

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

Let $C$ be an [[(∞,1)-category]]. An **orthogonal factorization system** on $C$ is a pair $(S_L, S_R)$ of [[class]]es of [[morphism]]s in $C$ that satisfy the following axioms.

1. Both classes are stable under [[retract]]s.

1. Every morphism in $S_L$ is left orthogonal to every morphism in $S_R$;

1. For every morphism $h : X \to Z$ in $C$ there exists a commuting triangle

   $$
     \array{
        && Y
        \\
        & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}}
        \\
       X &&\stackrel{h}{\to}&& Z
     }
   $$

   with $f \in S_L$ and $g \in S_R$.

=--

## Properties

### Closure properties
 {#ClosureProperties}

\begin{proposition}
\label{RightClassStableUnderLimitsInArrowCategory}
For $(L,R)$ a factorization system in an [[(∞,1)-category]] $\mathcal{C}$, the [[full sub-(∞,1)-category]] of the [[arrow category]] $Func(\Delta^1, \mathcal{C})$ on the morphisms in $R$ is closed under [[(∞,1)-limits]] of shapes that exist in $\mathcal{C}$. [[formal duality|Dually]], the full subcategory on $L$ is closed under [[(∞,1)-colimits]] that exist in $\mathcal{C}$.
\end{proposition}
([Lurie 2009, prop. 5.2.8.6 (7), (8)](#Lurie))

In fact:
\begin{proposition}
\label{RightClassReflectiveInArrowCategory}
The [[full sub-(infinity,1)-category|full sub-$\infty$-category]] of the [[arrow category]] on the right class is a [[reflective sub-(infinity,1)-category|reflective sub-$\infty$-category]]
$$
  \big(\mathcal{C}^{\Delta[1]}\big)_R
  \underoverset
    {\underset{}{\hookrightarrow}}
    {\overset{}{\longleftarrow}}
    {\;\; \bot \;\;}
  \mathcal{C}^{\Delta[1]}
$$

Moreover, the [[unit of an adjunction|adjunction units]]
$\eta_f \colon f \to \bar f$ are of the form

$$
  \array{
     X &\stackrel{\in L}{\longrightarrow}& \bar X
     \\
     {}^{\mathllap{f}}
     \big\downarrow
     &&
     \big\downarrow{}^{\mathrlap{\bar f \in R} }
     \\
     Y &\stackrel{\simeq}{\longrightarrow}& \bar Y
  }
  \,.
$$
(In words: the reflection into $\big(\mathcal{C}^{\Delta[1]}\big)_R$ is given by the factorization in $(L,R)$.)
\end{proposition}
([Lurie 2009, Lemma 5.2.8.19](#Lurie))

## Examples

* In an [[(∞,1)-topos]] the classe of [[n-connected]] and that of [[n-truncated]] morphisms form an orthogonal factorization system, for all $(-2) \leq n \leq \infty $.

  See [[(n-connected, n-truncated) factorization system]].

## Related concepts

* [[factorization system]]

  * [[weak factorization system]]

  * [[orthogonal factorization system]]

* [[factorization system in a 2-category]]

* [[factorization system in an (∞,1)-category]]

  * **orthogonal factorization system in an (∞,1)-category**

  * [[orthogonal factorization system in a derivator]].

  * [[homotopy factorization system in a model category]]

## References

Section 5.2.8 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_
 {#Lurie}

Formalization in [[homotopy type theory]] is discussed in

* [[Egbert Rijke]], _Orthogonal factorization in HoTT_, talk at IAS, January 24, 2013 ([video](http://video.ias.edu/1213/univalent/0124-EgbertRijke))

[[!redirects orthogonal factorization system in an (∞,1)-category]]

[[!redirects factorization system in an (∞,1)-category]]
[[!redirects factorization system in an (infinity,1)-category]]

[[!redirects factorization system in an infinity1-category]]
[[!redirects orthogonal factorization system in an infinity1-category]]


orthogonal group

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

#Contents#
* table of contents
{:toc}

## Definition

For $n \in \mathbb{N}$ the **orthogonal group** is the group of [[isometry|isometries]] of a real $n$-dimensional [[Hilbert space]]. This is naturally a [[Lie group]].
This is canonically isomorphic to the group of $n \times n$ orthogonal [[matrices]].

More generally there is a notion of _[[orthogonal group of an inner product space]]_.

The analog for complex Hilbert spaces is the [[unitary group]].

## Properties

### Compactness

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

The orthogonal group $O(n)$ is [[compact topological space]], hence in particular a [[compact Lie group]].

=--

### Homotopy groups
 {#HomotopyGroups}

+-- {: .num_prop #InclusionOfOnIntoOkIsnMinus1Equivalence}
###### Proposition

For $n, N \in \mathbb{N}$, $n \leq N$, then the canonical inclusion of orthogonal groups

$$
  O(n) \hookrightarrow O(N)
$$

is an [[n-equivalence|(n-1)-equivalence]], hence induces an [[isomorphism]] on [[homotopy groups]] in degrees $\lt n-1$ and a [[surjection]] in degree $n-1$.

=--

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

Consider the [[coset]] [[quotient]] [[projection]]

$$
  O(n)
    \longrightarrow
  O(n+1)
    \longrightarrow
  O(n+1)/O(n)
  \,.
$$

By prop. \ref{OrthogonalGroupIsCompact} and by [this corollary](coset#QuotientProjectionForCompactLieSubgroupIsPrincipal), the projection $O(n+1)\to O(n+1)/O(n)$ is a [[Serre fibration]]. Furthermore, example \ref{nSphereAsCosetSpace} identifies the [[coset]] with the [[n-sphere]]

$$
  S^{n}\simeq O(n+1)/O(n)
  \,.
$$

Therefore the [[long exact sequence of homotopy groups]] of the [[fiber sequence]] $O(n)\to O(n+1)\to S^n$ looks like

$$
  \cdots
    \to
  \pi_{\bullet+1}(S^n)
    \longrightarrow
  \pi_\bullet(O(n))
    \longrightarrow
  \pi_\bullet(O(n+1))
    \longrightarrow
  \pi_\bullet(S^n)
   \to
  \cdots
$$

Since $\pi_{\lt n}(S^n) = 0$, this implies that

$$
  \pi_{\lt n-1}(O(n))
    \overset{\simeq}{\longrightarrow}
  \pi_{\lt n-1}(O(n+1))
$$

is an isomorphism and that

$$
  \pi_{n-1}(O(n))
    \longrightarrow
  \pi_{n-1}(O(n+1))
$$

is surjective. Hence now the statement follows by induction over $N-n$.

=--

It follows that the [[homotopy groups]] $\pi_k(O(n))$ are independent of $n$ for $n \gt k + 1$ (the "stable range"). So if $O = \underset{\longrightarrow}{\lim}_n O(n)$, then $\pi_k(O(n)) = \pi_k(O)$. By [[Bott periodicity]] we have

$$
\array{
\pi_{8k+0}(O) & = \mathbb{Z}_2
\\
\pi_{8k+1}(O) & = \mathbb{Z}_2
\\
\pi_{8k+2}(O) & = 0
\\
\pi_{8k+3}(O) & = \mathbb{Z}
\\
\pi_{8k+4}(O) & = 0
\\
\pi_{8k+5}(O) & = 0
\\
\pi_{8k+6}(O) & = 0
\\
\pi_{8k+7}(O) & = \mathbb{Z}.
}
$$

In the unstable range for [[low-dimensional topology|low degrees]] they instead start out as follows

| $G$ | $\pi_1$ | $\pi_2$ | $\pi_3$ | $\pi_4$ | $\pi_5$ | $\pi_6$ | $\pi_7$ | $\pi_8$ | $\pi_9$ | $\pi_10$ | $\pi_11$ | $\pi_12$ | $\pi_13$ | $\pi_14$ | $\pi_15$ |
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
| $SO(2)$ | $\mathbb{Z}$ |  0 | 0 |0 |0 |0 |0 |0 |0 |0 |0 |0 | 0 | 0 | 0 |
| $SO(3)$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{12}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{3}$ | $\mathbb{Z}_{15}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_2\oplus\mathbb{Z}_{12}$ | $\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}_{84}$ | $\mathbb{Z}_2^{\oplus 2}$ |
| $SO(4)$ | " | 0 | $\mathbb{Z}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{12}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{3}^{\oplus 2}$ | $\mathbb{Z}_{15}^{\oplus 2}$|$\mathbb{Z}_{2}^{\oplus 2}$ |$\mathbb{Z}_{2}^{\oplus 4}$  | $\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}_{12}^{\oplus 2}$ | $\mathbb{Z}_2^{\oplus 4}\oplus\mathbb{Z}_{84}^{\oplus 2}$ | $\mathbb{Z}_2^{\oplus 4}$ |
| $SO(5)$ | " | " | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 |$\mathbb{Z}_{120}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}^{\oplus 2}$| $\mathbb{Z}_2\oplus\mathbb{Z}_4$ | $\mathbb{Z}_{1680}$ | $\mathbb{Z}_2$ |
| $SO(6)$ | " | " | " | 0 | $\mathbb{Z}$ | 0 | $\mathbb{Z}$ | $\mathbb{Z}_{24}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2\oplus\mathbb{Z}_{120}$| $\mathbb{Z}_{4}$| $\mathbb{Z}_{60}$| $\mathbb{Z}_4$ | $\mathbb{Z}_2\oplus\mathbb{Z}_{1680}$ | $\mathbb{Z}_2\oplus\mathbb{Z}_{72}$ |
| $SO(7)$ | " | " | " | " | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{8}$| $\mathbb{Z}_2\oplus\mathbb{Z}$| 0 | $\mathbb{Z}_2$ | $\mathbb{Z}_2\oplus\mathbb{Z}_8\oplus\mathbb{Z}_{2520}$ | $\mathbb{Z}_2^{\oplus 4}$ |
| $SO(8)$ | " | " | " | " | " | 0 | $\mathbb{Z}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 3}$ | $\mathbb{Z}_{2}^{\oplus 3}$ | $\mathbb{Z}_{8} \oplus \mathbb{Z}_{24}$ | $\mathbb{Z}_2 \oplus \mathbb{Z}$ | 0 | $\mathbb{Z}^{\oplus 2}$ | $\mathbb{Z}_2\oplus\mathbb{Z}_8\oplus\mathbb{Z}_{120}\oplus\mathbb{Z}_{2520}$ | $\mathbb{Z}_2^{\oplus 7}$ |
| $SO(9)$ | " | " | " | " | " | " | $\mathbb{Z}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ |$\mathbb{Z}_{8}$ | $\mathbb{Z}_2\oplus \mathbb{Z}$ | 0 | $\mathbb{Z}_2$ | $\mathbb{Z}_2\oplus\mathbb{Z}_8$ | $\mathbb{Z}_2^{\oplus 3}\oplus\mathbb{Z}$ |
|$SO(10)$| " | " | " | " | " | " | " | $\mathbb{Z}_{2}$ | $\mathbb{Z}_2\oplus \mathbb{Z}$ | $\mathbb{Z}_{4}$| $\mathbb{Z}$ | $\mathbb{Z}_{12}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_8$ | $\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}$ |
|$SO(11)$| " | " | " | " | " | " | " | " |$\mathbb{Z}_{2}$ |$\mathbb{Z}_{2}$ | $\mathbb{Z}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_2^{\oplus 2}$ | $\mathbb{Z}_8$ | $\mathbb{Z}_2\oplus\mathbb{Z}$ |
|$SO(12)$| " | " | " | " | " | " | " | " | " | 0 | $\mathbb{Z}^{\oplus 2}$ | $\mathbb{Z}_{2}^{\oplus 2}$ | $\mathbb{Z}_2^{\oplus 2}$ | $\mathbb{Z}_4\oplus\mathbb{Z}_{24}$ | $\mathbb{Z}_2\oplus\mathbb{Z}$ |
|$SO(13)$| " | " | " | " | " | " | " | " | " | " | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z}_8$ | $\mathbb{Z}_2\oplus\mathbb{Z}$ |
|$SO(14)$| " | " | " | " | " | " | " | " | " | " | " | 0 | $\mathbb{Z}$ | $\mathbb{Z}_4$ | $\mathbb{Z}$ |
|$SO(15)$| " | " | " | " | " | " | " | " | " | " | " | " | 0 | $\mathbb{Z}_2$ | $\mathbb{Z}$ |
|$SO(16)$| " | " | " | " | " | " | " | " | " | " | " | " | " | 0 | $\mathbb{Z}^{\oplus 2}$ |
|$SO(17)$| " | " | " | " | " | " | " | " | " | " | " | " | " | " | $\mathbb{Z}$ |

The $SO(6)$ row can be found using [Mimura-Toda 63](#MimuraToda63), using $Spin(6) = SU(4)$, and that $Spin(6)$ is a $\mathbb{Z}_2$-[[covering space]] of $SO(6)$. The $SO(7)$ row can be derived from the homotopy groups of $Spin(7)$ as found in [Mimura 67](#Mimura67). Otherwise the table is given in columns $\pi_k$, $k=10,\ldots, 15$, and in rows $SO(n)$, $n=8,\ldots,17$, by the [[Encyclopedic Dictionary of Mathematics]], Table 6.VII in Appendix A.

Note that the maps
$$
  \array{
    \pi_3(SO(3)) \longrightarrow \pi_3(SO(4)) \longrightarrow \pi_3(SO(5))
    \\
     \mathbb{Z}\longrightarrow \mathbb{Z}\oplus \mathbb{Z} \longrightarrow\mathbb{Z}
  }
$$
are inclusion of the first summand followed by the map sending $(1,0)\mapsto 2$ and $(0,1)\mapsto 1$, so that stabilization from $SO(3)$ to $SO(5)$ induces multiplication by $2$ on $\pi_3$ (e.g. equations (2.1) and (2.2) in ([Tamura 57](#Tamura57)) and surrounding discussion). The same is also true for $\pi_7(SO(7)) \to \pi_7(SO(8)) \to \pi_7(SO(9))$.

### Homology and cohomology
 {#HomologyAndCohomology}

On the [[ordinary cohomology]] of the [[classifying spaces]] $B O(n)$ and $B SO(n)$

* with [[cyclic group of order 2|$\mathbb{Z}_2$]] [[coefficients]]:

  generated by the [[Stiefel-Whitney classes]]

  (e.g. [Milnor & Stasheff 74, Theorem 7.1. & Theorem 12.4.](#MilnorStasheff74))

* with [[integer]] [[coefficients]] ([[integral cohomology]]):

  generated by the [[integral Stiefel-Whitney classes]] and the [[Pontrjagin classes]]

  (e.g. [Brown 82](#Brown82), [Feshbach 83](#Feshbach83), [Pittie 91](#Pittie91), [Rudolph-Schmidt 17, Theorem 4.2.23](#RudolphSchmidt17))

### Whitehead tower and higher orientation structures

The [[Whitehead tower]] of the orthogonal group plays an important role in applications related to [[quantum physics]].

The first steps are

$$
  \cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n)
  \,.
$$

[[Fivebrane group]] to [[String group]] to [[Spin group]] to [[special orthogonal group]] to **orthogonal group**.

Given a [[manifold]] $X$, lifts of the structure map $X \to \mathcal{B}O(n)$ of the $O(n)$-[[principal bundle]] to which the [[tangent bundle]] is [[associated bundle|associated]] through this tower define, respectively

* [[orientation]]

* [[Spin structure]]

* [[String structure]]

* [[Fivebrane structure]]

on $X$.

### Coset spaces
 {#CosetSpaces}

+-- {: .num_example #nSphereAsCosetSpace}
###### Example

The [[n-spheres]] are [[coset spaces]] of orthogonal groups

$$
  S^n \simeq O(n+1)/O(n)
  \,.
$$

For fix a unit vector in $\mathbb{R}^{n+1}$. Then its [[orbit]] under the defining $O(n+1)$-[[action]] on $\mathbb{R}^{n+1}$ is clearly the canonical embedding $S^n \hookrightarrow \mathbb{R}^{n+1}$. But precisely the subgroup of $O(n+1)$ that consists of rotations around the axis formed by that unit vector [[stabilizer group|stabilizes]] it, and that subgroup is isomorphic to $O(n)$, hence $S^n \simeq O(n+1)/O(n)$.

=--

+-- {: .num_example #StiefelManifold}
###### Example

For $n \leq k$, the [[coset]]

$$
   V_n(\mathbb{R}^k) \coloneqq O(k)/O(k-n)
$$

is called the $n$th _real [[Stiefel manifold]]_ of $\mathbb{R}^k$.

=--

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

The [[Stiefel manifold]] $V_n(\mathbb{R}^k)$ (example \ref{StiefelManifold}) is [[n-connected topological space|(k-n-1)-connected]].

=--

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

Consider the [[coset]] [[quotient]] [[projection]]

$$
  O(k-n)
    \longrightarrow
  O(k)
    \longrightarrow
  O(k)/O(k-n)
    =
  V_n(\mathbb{R}^k)
  \,.
$$

By prop. \ref{OrthogonalGroupIsCompact} and by [this corollary](QuotientProjectionForCompactLieSubgroupIsPrincipal) the projection $O(k)\to O(k)/O(k-n)$ is a [[Serre fibration]]. Therefore there is induced the [[long exact sequence of homotopy groups]] of this [[fiber sequence]], and by prop. \ref{InclusionOfOnIntoOkIsnMinus1Equivalence} it has the following form in degrees bounded by $n$:

$$
  \cdots
    \to
  \pi_{\bullet \leq k-n-1}(O(k-n))
    \overset{epi}{\longrightarrow}
  \pi_{\bullet \leq k-n-1}(O(k))
    \overset{0}{\longrightarrow}
  \pi_{\bullet \leq k-n-1}(V_n(\mathbb{R}^k))
    \overset{0}{\longrightarrow}
  \pi_{\bullet-1 \lt k-n-1}(O(k))
    \overset{\simeq}{\longrightarrow}
  \pi_{\bullet-1 \lt k-n-1}(O(k-n))
    \to
  \cdots
  \,.
$$

This implies the claim. (Exactness of the sequence says that every element in $\pi_{\bullet \leq n-1}(V_n(\mathbb{R}^k))$ is in the kernel of zero, hence in the image of 0, hence is 0 itself.)

=--

## Related concepts

* [[orthogonal calculus]]

* [[Stiefel manifold]], [[Grassmannian]]

* [[stable orthogonal group]]

 $\cdots\to$ [[fivebrane group]] $\to$ [[string group]] $\to$ [[spin group]] $\to$ [[special orthogonal group]] $\to$ **orthogonal group**

[[!include table of orthogonal groups and related]]

## Examples

\linebreak

[[!include low dimensional rotation groups -- table]]

\linebreak

## References

Examples of sporadic (exceptional) [[isogenies]] from [[spin groups]] onto orthogonal groups are discussed in

* {#Garrett13} [[Paul Garrett]], _Sporadic isogenies to orthogonal groups_ (July 2013) [[pdf](http://www.math.umn.edu/~garrett/m/v/sporadic_isogenies.pdf), [[Garrett-SporadicIsogenies.pdf:file]] ]

The [[homotopy groups]] of $O(n)$ are listed for instance in

* {#Abanov09} Alexander Abanov, Homotopy groups of Lie groups 2009 ([pdf](http://felix.physics.sunysb.edu/~abanov/Teaching/Spring2009/Notes/abanov-cpA1-upload.pdf))

* {#MimuraToda63} M. Mimura and H. Toda, _Homotopy Groups of $SU(3)$, $SU(4)$ and $Sp(2)$_, J. Math. Kyoto Univ. Volume 3, Number 2 (1963), 217-250. ([Euclid](http://projecteuclid.org/euclid.kjm/1250524818))

* {#Mimura67} M. Mimura, _The Homotopy groups of Lie groups of low rank_, Math. Kyoto Univ. Volume 6, Number 2 (1967), 131-176. ([Euclid](http://projecteuclid.org/euclid.kjm/1250524375))

The [[ordinary cohomology]] and [[ordinary homology]] of the manifolds $SO(n)$ is discussed in

* {#MilnorStasheff74} [[John Milnor]], [[Jim Stasheff]], _Characteristic classes_, Princeton Univ. Press (1974) [[ISBN:9780691081229](https://press.princeton.edu/books/paperback/9780691081229/characteristic-classes-am-76-volume-76), [doi:10.1515/9781400881826](https://doi.org/10.1515/9781400881826), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/milnstas.pdf)]

* {#Brown82} [[Edgar H. Brown]], _The Cohomology of $B SO_n$ and $BO_n$ with Integer Coefficients_, Proceedings of the American Mathematical Society Vol. 85, No. 2 (Jun., 1982), pp. 283-288 ([jstor:2044298](https://www.jstor.org/stable/2044298))

* {#Feshbach83} Mark Feshbach, _The Integral Cohomology Rings of the Classifying Spaces of $\mathrm{O}(n)$ and $\mathrm{SO}(n)$,_ Indiana Univ. Math. J.  32 (1983), 511-516 ([doi:10.1512/iumj.1983.32.32036](https://doi.org/10.1512/iumj.1983.32.32036))

* {#Pittie91} Harsh V. Pittie, _The integral homology and cohomology rings of SO(n) and Spin(n)_, Journal of Pure and Applied Algebra Volume 73, Issue 2, 19 August 1991, Pages 105–153 (doi:10.1016/0022-4049(91)90108-E))

* {#RudolphSchmidt17} Gerd Rudolph, Matthias Schmidt, around Theorem 4.2.23 of _Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields_, Theoretical and Mathematical Physics series, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

See also

* {#Tamura57} Itiro Tamura, _On Pontrjagin classes of homotopy types of manifolds_, Journal of the mathematical society of Japan, Vol. 9 No. 2 , 1957 [pdf](http://www.maths.ed.ac.uk/~aar/papers/tamura3.pdf)

[[!redirects orthogonal groups]]

[[!redirects O(n)]]

[[!redirects O(2)]]


orthogonal group of an inner product space

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

#Contents#
* table of contents
{:toc}

## Idea

Given an [[inner product space]] $V = (V,\langle-,-\rangle)$, the *[[orthogonal group]]* of $V$ is the subgroup of the [[general linear group]] $GL(V)$ which leaves invariant the inner product.

## Definition

Given an element $A$ of $GL(V)$ we say it _preserves the inner product_ $\langle-,-\rangle$ if $\langle A v ,A w \rangle = \langle v,w \rangle$ for all $v,w\in V$.

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

 If $A$ and $B$ preserve the inner product on $V$, then so do $AB$ and $A^{-1}$.
=--

We thus see that we have a group.

+-- {: .num_defn #def_of_O_n}
###### Definition
The **orthogonal group** $O(V,\langle-,-\rangle)$ is the subgroup of $GL(V)$ consisting of those elements that preserve the inner product.
=--

(... more detail ...)

## As a Lie group

When the base field is $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ (the last being a [[skewfield]], but this is ok), and the vector space is finite-dimensional then we can put the structure of a finite-dimensional  [[manifold]] on $O(V,\langle-,-\rangle)$. We shall denote the group in this case by $O(n,\mathbb{K})$ where $V = \mathbb{K}^n$, $\mathbb{K} \in \{ \mathbb{R},\mathbb{C},\mathbb{H}\}$ and we take the 'standard' inner products on these spaces.

A relatively easy way to see that $O(n,\mathbb{K})$ is a manifold is that it is a smooth [[affine variety]] in [[Euclidean space]], but this requires some machinery. We can however construct explicit [[charts]] for $O(n,\mathbb{K})$ as a real manifold.

### Charts on the underlying manifold

One can use the algebra structure on matrices over the base [[field]] (or base [[division ring]]) of $V$ to define [[charts]] which are different to the charts one gets via the [exponential map](/nlab/show/exponential+map#exp_of_Lie_groups).

For $A \in O(n,\mathbb{K})$ Define the sets
$$
  \Omega(A) := \{ X\in O(n,\mathbb{K}) | A + X \in GL(V) \}
$$
and
$$
  T_{A^\dagger} := \{ Y\in M_n(\mathbb{K}) | AY + (AY)^\dagger = 0 \}.
$$
Here $T_{A^\dagger}$ is a vector space, and will turn out to be the tangent space to $O(n,\mathbb{K})$ at $A^\dagger$.

+-- {: .num_prop #isomorphism_of_opens_with_tangent_spaces}
###### Proposition
There is an isomorphism $c_A\colon \Omega(A) \stackrel{\sim}{\to} T_{A^\dagger}$.
=--
+-- {: .proof}
###### Proof
The map $c_A$ is defined as
$$
  X \mapsto (I - A^\dagger X)(A + X)^{-1} = (I - A^\dagger X)(I + A^\dagger X)^{-1}A^\dagger
$$
the latter equality using that $A^{-1} = A^\dagger$. We need to check that this is indeed a map to $T_{A^\dagger}$ and that it is an isomorphism.

Since
$$
   (A c_A(X))^\dagger = A c_I(A^\dagger X)^\dagger A^\dagger,
$$
if we show that $c_I(A^\dagger X)^\dagger = - c_I(A^\dagger X)$ we are done. Let $Z = X^\dagger A$, which is again in $O(n,\mathbb{K})$. Then
$$
\begin{aligned}
    c_I(Z)^\dagger & = \left( (I - Z)(I + Z)^{-1} \right)^\dagger\\
    & = \left( (I - Z)Z^\dagger Z^{\dagger -1}(I + Z)^{-1} \right)^\dagger\\
    & = \left( -(I - Z^\dagger)(I + Z^\dagger)^{-1} \right)^\dagger \\
    & = -(I + Z)^{-1}(I - Z)
\end{aligned}
$$
(tbc...)

=--

## Examples

* [[Euclidean space]] $\mathbb{R}^n$ with the standard inner product $\langle\mathbf{x},\mathbf{y}\rangle = \mathbf{x}\cdot\mathbf{y}$ has as orthogonal group the standard [[orthogonal group]] $O(n)$.

* The finite-dimensional [[Hilbert space]] $\mathbb{C}^n$ with the standard inner product has as orthogonal group $U(n)$, the [[unitary group]]

* The space $\mathbb{H}^n$, for $\mathbb{H}$ the [[quaternions]], has an inner product ... such that the corresponding orthogonal group is the [[compact symplectic group]] $Sp(n)$.

* (...Examples of indefinite signature go here...)


orthogonal Lie algebra

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

#Contents#
* table of contents
{:toc}

## Idea

The _orthogonal Lie algebra_ $\mathfrak{o}$ is the [[Lie algebra]] of the [[orthogonal group]] $O$.

The _special orthogonal Lie algebra_ is the Lie algebra $\mathfrak{so}$ of the [[special orthogonal group]] $SO$.

Since the two [[Lie groups]] differ by an discrete group $\mathbb{Z}_2$, these two Lie algebras coincide; we traditionally write $\mathfrak{so}$ instead of $\mathfrak{o}$.

## Properties

### In the classification of Lie algebras

For $n \gt 2$, except for $n = 4$, where $\mathfrak{so}(4)\cong\mathfrak{so}(3)\oplus\mathfrak{so}(3)$, $\mathfrak{so}(n)$ is a [[simple Lie algebra]], either $\mathfrak{d}_{n/2}$ when $n$ is even or $\mathfrak{b}_{(n-1)/2}$ when $n$ is odd.  For $n = 2$, $\mathfrak{so}(n)$ is the [[line]], an [[abelian Lie algebra]], which is also a [[simple object]] in [[LieAlg]] but is not counted as a simple Lie algebra.  For $n \lt 2$, $\mathfrak{so}(n)$ is the [[trivial Lie algebra]] (which is [[too simple to be simple]] by any standard).

### Relation to Clifford algebra

For $V$ an [[inner product space]],
the special orthogonal Lie algebra on $V$ is naturally isomorphism to the algebra of [[bivectors]] in the [[Clifford algebra]] $Cl(V)$ under the Clifford [[commutator]] bracket.

## Related concepts

* **special orthogonal Lie algebra** , [[unitary Lie algebra]], [[Poincare Lie algebra]]

* [[special linear Lie algebra]]

* [[symplectic Lie algebra]]

* [[string Lie 2-algebra]]

* [[fivebrane Lie 6-algebra]]

* [[orthosymplectic super Lie algebra]]

[[!redirects orthogonal Lie algebra]]
[[!redirects orthogonal Lie algebras]]
[[!redirects special orthogonal Lie algebra]]
[[!redirects special orthogonal Lie algebras]]

[[!redirects Lorentz Lie algebra]]
[[!redirects Lorentz Lie algebras]]

[[!redirects so(n)]]


orthogonal matrix

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

#Contents#
* table of contents
{:toc}

## Definition

An _orthogonal matrix_ is a [[square matrix]] $A$ whose [[transpose matrix]]  equals its [[inverse matrix]] $A^T = A^{-1}$, hence such that $A^T A = 1$ under [[matrix multiplication]].

Orthogonal matrices form a [[subgroup]] of the [[general linear group]], namely the [[orthogonal group]].

For a generalization see _[[J-orthogonal matrix]]_.

## Properties

Every real matrix $A$ can be factorized $A = Q R$ where $Q$ is orthogonal and $R$ is   a (say, upper) triangular matrix (wikipedia/[QR decomposition](https://en.wikipedia.org/wiki/QR_decomposition) which is a special case of [[Iwasawa decomposition]] for semisimple Lie groups). This is consequence of Gram-Schmidt orthogonalization. Similarly, every complex matrix can be factorized into a unitary and a complex upper triangular (complex) matrix.

Over the reals, the [[Cayley transform]] is a [[diffeomorphism]] between the linear space skewsymmetric matrices and an open subset of the Lie group of orthogonal matrices ($A$ such that $I+A$ is invertible) - a chart which is often having an advantage over using the exponential map.

## Related concepts

* [[J-orthogonal matrix]]

* [[orthogonal group]], [[Gram-Schmidt orthogonalization]]

### References

* wikipedia/[orthogonal matrix](https://en.wikipedia.org/wiki/Orthogonal_matrix)

[[!redirects orthogonal matrices]]


orthogonal ring spectrum

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _orthogonal ring spectrum_ is a [[ring spectrum]] modeled as an [[orthogonal spectrum]]. Equivalently this is a [[monoid object]] with respect to the [[symmetric monoidal smash product of spectra]], for orthogonal spectra.
Morever, if we regard orthogonal spectra as $\mathbb{S}_{orth}$-[[modules]], as discussed at _[[model structure on orthogonal spectra]]_, then this, in turn, is equivalent to a $\mathbb{S}_{orth}$-algebra, where $\mathbb{S}_{orth}$ is the standard model of the [[sphere spectrum]] as an orthogonal spectrum.

There is a [[model structure for ring spectra|model structure for orthogonal ring spectra]] ([MMSS 00](#MandellMaySchwedeShipley01)) under which orthogonal ring spectra represent genuine [[A-infinity rings]], and commutative orthogonal ring spectra represent genuine [[E-infinity rings]]. This fact is one of the key motivation for passing from [[sequential spectra]] to the richer model of [[orthogonal spectra]].

Despite all this, the component expression of the structure in an orthgonal ring spectrum, in the fashion of _[[functors with smash product]]_, is rather straightforward, and directly analogous to the structure in a [[dg-algebra]]: essentially there is for all pairs of indices $n_1, n_2$ a pairing between the component spaces of the spectrum in these degrees

$$
  E_{n_1}\wedge E_{n_2}\longrightarrow E_{n_1 + n_2}
$$

such that this respects the given action of the [[orthogonal groups]] and of [[suspensions]], and such that it is is [[associativity|associative]] and [[unitality|unital]] in the evident way.

## Definition

The definition is directly analogous to that of _[[symmetric ring spectrum]]_, just with the [[symmetric groups]] replaced by [[orthogonal groups]] throughout.

## References

* {#MandellMaySchwedeShipley01} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], _[[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))

[[!redirects orthogonal ring spectra]]


orthogonal space

\tableofcontents

## Definition

The [[category]] of __orthogonal spaces__ is the category
of [[topological functors]] from the [[topological category]] $\mathbf{L}$
of finite-dimensional real inner product spaces (with linear isometric
embeddings) to the [[topological category]] of topological spaces (with continuous maps).

Here a __topological space__, or simply a __space__,
refers to an object in a [[convenient category of spaces]],
namely, [[compactly generated weakly Hausdorff space]].

Orthogonal spaces are turned into a [[relative category]]
using the notion of a __global equivalence__,
which are introduced via an equivariant version of the [[Whitehead theorem]].
A morphism $f\colon X\to Y$ is a __global equivalence__
if for any [[compact Lie group]] $G$,
any continuous orthogonal representation $V$ of $G$
on a finite-dimensional real inner product space,
any $k\ge0$,
and any continuous maps $\alpha\colon\partial D^k\to X(V)^G$
and $\beta\colon D^k\to Y(V)^G$ such that $\beta|_{\partial D^k}=f(V)^G\circ\alpha$,
we can find a continuous orthogonal representation $W$ of $G$,
a $G$-equivariant linear isometric embedding $\phi\colon V\to W$
and a continuous map $\lambda\colon D^k\to X(W)^G$
such that $\lambda|_{\partial D^k}=X(\phi)^G\circ\alpha$
and $f(W)^G\circ\lambda$ is homotopic to $Y(\phi)^G\circ\beta$ relative $\partial D^k$.

In particular, if $X$ and $Y$ are [[constant functors]],
then $f\colon X\to Y$ is a global equivalence
if and only if $f(0)\colon X(0)\to Y(0)$ is a [[weak homotopy equivalence]]
of topological spaces.

Global equivalences can also be characterized as morphisms $f\colon X\to Y$
such that for any [[compact Lie group]] $G$
and for some (hence any) complete $G$-[[universe]] $U$,
the canonical map of __underlying $G$-spaces__
$$hocolim_V f(V)\colon hocolim_V X(V)\to hocolim_V Y(V)$$
is a $G$-[[equivariant weak equivalence]] (i.e., a [[weak homotopy equivalence]]
on $H$-fixed points for any closed subgroup $H$ of $G$),
where $V$ runs over all finite-dimensional subrepresentations of $U$.

## Example: the global classifying space of a compact Lie group

The __global classifying space of a compact Lie group__ is defined as
$$\mathrm{B}_{gl} G = \mathbf{L}_{G,V} = \mathbf{L}(V,-)/G,$$
where $V$ is any faithful representation of $G$.
Here $\mathbf{L}(V,-)$ denotes the [[enriched hom]]
in the [[topological category]] $\mathbf{L}$.

If $V'$ is another such faithful representation, then the canonical maps
$$\mathbf{L}_{G,V} \gets \mathbf{L}_{G,V\oplus V'} \to \mathbf{L}_{G,V'}$$
are global equivalences.

For example, if $G=Z/2Z$, then the underlying nonequivariant space
of $\mathrm{B}_{gl} G$ is $\mathbf{RP}^\infty$.

## References

* [[Stefan Schwede]], _[[Global homotopy theory]]_.


orthogonal spectrum

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Orthogonal spectra are one version of [[highly structured spectra]] that support a [[symmetric monoidal smash product of spectra]]. An orthogonal spectrum is a sequence of [[pointed topological spaces]] $\{X_n\}_{n \in \mathbb{N}}$ equipped with maps $X_n \wedge S^1 \longrightarrow X_{n+1}$ from the [[suspension]] of one into the next, but such that the $n$th [[topological space]] is equipped with an [[action]] of the [[orthogonal group]] $O(n)$ and such that the induced structure maps.
$X_n \wedge S^k \longrightarrow X_{n+k}$
are all $O(n)\times O(k)$-[[equivariance|equivariant]], hence are [[action]] [[homomorphisms]]. There is a natural [[homotopy theory]] of such orthogonal spectra and it is equivalent to the standard [[stable homotopy theory]] ([MMSS 98](#MMSS08)).

The [[category]] of _orthogonal spectra_  is a [[presentable (∞,1)-category|presentation]] of the [[symmetric monoidal (∞,1)-category]] [[stable (infinity,1)-category of spectra|of spectra]], with the special property that it implements the [[smash product of spectra]] such as to yield itself a [[symmetric monoidal category|symmetric]] [[monoidal model category|monoidal]] [[model category of spectra]]: the _[[model structure on orthogonal spectra]]_.

This implies in particular that with respect to this [[symmetric smash product of spectra]] an [[E-∞ ring]] is presented simply as a plain [[commutative monoid]] [[internalization|in]] orthogonal spectra ("[[highly structured ring spectrum]]"). See at _[[orthogonal ring spectrum]]_.

Other presentations sharing this property are _[[symmetric spectra]]_ and _[[S-modules]]_. In contrast to symmetric spectra, orthogonal spectra need to consist of [[topological spaces]] instead of [[simplicial sets]].

One advantages of orthogonal spectra over symmetric spectra is that for them the naive definition of [[homotopy groups]] comes out homotopically correct, while for symmetric spectra an intransparent replacement is needed first (see [symmetric spectrum -- Homotopy groups](symmetric+spectrum#HomotopyGroups)).

Another advantage is that orthogonal spectra support a similarly good model for [[equivariant stable homotopy theory]] with equivariance by [[compact Lie groups]], while [[symmetric spectra]] share this property only for equivariance under [[finite groups]].

## Definition
 {#Definition}

### Orthogonal spectra

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

An _orthogonal spectrum_ $X$ consists of for each $n \in \mathbb{N}$

1. a sequence of [[pointed topological spaces]] $X_n$ (the _$n$th level_);

1. a base-point preserving [[continuous function|continuous]] [[action]] of the [[topological group|topological]] [[orthogonal group]] $O(n)$ on $X_n$;

1. based-point preserving [[continuous functions]] $\sigma_n \colon X_n \wedge S^1 \longrightarrow X_{n+1}$ from the [[smash product]] with the [[1-sphere]] (the _$n$th structure map_)

such that for all $n,k \in \mathbb{N}$ with $k \geq 1$

* the [[continuous functions]] $\sigma^k \colon X_n \wedge S^k \longrightarrow X_{n+k}$ given as the [[compositions]]

  $$
    \sigma^k \colon
    X_n \wedge S^k
     \stackrel{\sigma_n \wedge S^{k-1}}{\longrightarrow}
    X_{n+1} \wedge S^{k-1}
     \stackrel{\sigma_{n-1} \wedge S^{k-2}}{\longrightarrow}
    X_{n+2} \wedge S^{k-2}
     \stackrel{\sigma_{n-2} \wedge S^{k-3}}{\longrightarrow}
     \cdots
     \stackrel{\sigma_{n+k-2} \wedge S^{1}}{\longrightarrow}
   X_{n+k-1} \wedge S^1
     \stackrel{\sigma_{n+k-1} }{\longrightarrow}
    X_{n+k}
  $$

  is $O(n) \times O(k)$-equivariant

  (with respect to the $O(k)$-[[action]] on $S^k$ regarded as the [[representation sphere]] of the defining action on $\mathbb{R}^k$ and via the diagonal embedding $O(n)\times O(k) \hookrightarrow O(n+k)$).

A [[homomorphism]] $f \colon X \longrightarrow Y$ of orthogonal spectra is a sequence of $O(n)$-equivariant based continuous functions $f_n \colon X_n \longrightarrow Y_n$ [[commuting diagram|commuting]] with the structure maps

$$
  \array{
    X_n \wedge S^1 & \stackrel{\sigma_n^X}{\longrightarrow} & X_{n+1}
    \\
    \downarrow^{\mathrlap{f_n}} && \downarrow^{\mathrlap{f_{n+1}}}
    \\
    Y_n \wedge S^1 & \stackrel{\sigma_n^Y}{\longrightarrow} & Y_{n+1}
  }
  \,.
$$

We write $OrthSpectra$ for the [[category]] of orthogonal spectra with homomorphisms between them.

=--

### Homotopy groups and Weak homotopy equivalences
 {#HomotopyGroups}

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

Given an orthogonal spectrum $X$, def. \ref{OrthogonalSpectrum},
then for $n,k \in \mathbb{N}$ the _stabilization map_ $\iota_{n,k}$ on [[homotopy groups]] $\pi_\bullet(X_\bullet)$ of the level spaces $X_\bullet$ is

$$
  \iota_{n,k}
    \;\colon\;
  \pi_{n+k} X_n
    \stackrel{(-)\wedge S^1}{\longrightarrow}
  \pi_{n+k+1}(X_n \wedge S^1)
    \stackrel{(\sigma_n)_\ast}{\longrightarrow}
  \pi_{n+k+1} X_{n+1}
  \,.
$$

=--

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

Given an orthogonal spectrum $X$, def. \ref{OrthogonalSpectrum}, then for $k \in \mathbb{Z}$ its $k$th _stable [[homotopy group]]_ is the [[colimit]]

$$
  \pi_k X
  \;\coloneqq\;
  \underset{\longrightarrow}{\lim}_n
  \pi_{n+k} X_n
$$

of the [[homotopy groups]] of the level spaces, taken with respect to the stabilization maps, def. \ref{StabilizationMap}.

=--

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

A homomorphism $f\colon X \longrightarrow Y$ of orthogonal spectra, def. \ref{OrthogonalSpectrum}, is a _[[weak homotopy equivalence]]_ if it induces [[isomorphisms]] (of [[abelian groups]])

$$
  \pi_\bullet(f) \;\colon\;  \pi_\bullet(X) \longrightarrow \pi_\bullet(Y)
$$

on all stable homotopy groups, def. \ref{StableHomotopyGroup}.

=--

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

The [[simplicial localization]] of the category of orthogonal spectra, def. \ref{OrthogonalSpectrum}, at the weak homotopy equivalences, def. \ref{WeakHomotopyEquivalences}, is [[equivalence of (infinity,1)-categories|equivalent]] to the [[(infinity,1)-category of spectra]]:

$$
  L_{whe} OrthSpectra \simeq Spectra
  \,.
$$

See at _[[model structure on orthogonal spectra]]_.

=--

### Smash product
 {#SmashProduct}

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

Given two orthogonal spectra $X,Y\in OrthSpectra$, def. \ref{OrthogonalSpectrum}, their _[[smash product of spectra]]_ is the orthogonal spectrum

$$
  X \wedge Y \in OrthSpectra
$$

whose $n$th level space is the [[coequalizer]]

$$
  \left(
    \underset{p+1+q = n}{\bigvee}
     O(n)_+  \underset{O(p)\times 1 \times O(q)}{\wedge} X_p \wedge S^1 \wedge Y_q
  \right)
  \stackrel{\overset{}{\longrightarrow}}{\underset{}{\longrightarrow}}
  \left(
   \underset{p+q = n}{\bigvee}
    O(n)_+ \underset{O(p)\times O(q)}{\wedge} X_p \wedge Y_q
   \right)
   \longrightarrow
   \left(X\wedge Y\right)_{n}
$$

of the two maps whose components are $\sigma_p^X \wedge Y_q$ and $X_p \wedge \sigma_q^Y \circ X_p \wedge braid_{S^1, Y_q}$, respectively, and whose structure maps are induced, under the coequalizer, by the component maps $X_p\wedge \sigma_q^Y$.

=--

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

The [[smash product of spectra]] from def. \ref{SmashProduct} naturally extends to a [[functor]]

$$
  (-)\wedge (-)
  \;\colon\;
  OrthSpectra \times OrthSpectra
  \longrightarrow
  OrthSpectra
$$

which makes $OrthSpectra$ into a [[symmetric monoidal category]] with [[unit]] the orthogonal [[sphere spectrum]] $\mathbb{S}$, example \ref{OrthogonalSphereSpectrum}.

=--

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

For $X,Y,Z \in OrthSpectra$, def. \ref{OrthogonalSpectrum}, a _[[bilinear map|bilinear]]-homomorphism_
$$
  b
    \;\colon\;
  (X,Y)
   \longrightarrow
  Z
  \,,
$$

is a collection of, for each $p,q\in \mathbb{N}$, base-point preserving $O(p) \times O(q)$-equivariant [[continuous functions]]

$$
  b_{p,q}
   \;\colon\;
  X_p \wedge Y_q
    \longrightarrow
  Z_{p+q}
$$

(out of the [[smash product]] of [[pointed topological spaces]]) which are _[[bilinear map|bilinear]]_ in that the following [[diagrams]] [[commuting diagram|commutes]]:

$$
  \array{
    X_p \wedge Y_q \wedge S^1
     &\stackrel{b_{p,q} \wedge S^1}{\longrightarrow}&
    Z_{p+q} \wedge S^1
    \\
    \downarrow^{\mathrlap{X_p \wedge \sigma_q}}
      &&
    \downarrow^{\mathrlap{\sigma_{p+q}}}
    \\
    X_p \wedge Y_{q+1}
     &\stackrel{b_{p,q+1}}{\longrightarrow}&
    Z_{p+q+1}
  }
  \;\;\;\;,\;\;\;\;\;
  \array{
    X_p \wedge Y_q \wedge S^1
     &\stackrel{b_{p,q} \wedge S^1}{\longrightarrow}&
    Z_{p+q} \wedge S^1
    \\
    \downarrow^{\mathrlap{X_p \wedge braid_{Y_q, S^1}}}
     &&
    \downarrow^{\mathrlap{\sigma_{p+q}}}
    \\
    X_p \wedge S^1 \wedge Y_q
     &&
    Z_{p+q+1}
    \\
    \downarrow^{\mathrlap{\sigma_{p} \wedge Y_q}}
      &&
    \downarrow^{\mathrlap{id \times \chi_{1,q}}}
    \\
    X_{p+1} \wedge Y_{q}
     &\stackrel{b_{p+1,q}}{\longrightarrow}&
    Z_{p+1+q}
  }
  \,.
$$

=--

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

The smash product of orthogonal spectra $X \wedge Y$, def. \ref{SmashProduct}, is the [[universal construction|universal]] recipient in $OrthSpectra$ of bilinear maps, def. \ref{BilinearHomomorphisms}, out of $(X,Y)$.

=--

## Examples

+-- {: .num_example #OrthogonalSphereSpectrum}
###### Example

The canonical incarnation of the [[sphere spectrum]] $\mathbb{S}$ as an orthogonal spectrum, def. \ref{OrthogonalSpectrum}, has $n$th level space

$$
  \mathbb{S}_n = S^n
$$

the [[representation sphere]] of the defining [[linear representation]] of $O(n)$ on $\mathbb{R}^n$, and as structure maps the canonical [[smash product]] [[isomorphisms]] ([[homeomorphisms]])

$$
  S^p \wedge S^1  \longrightarrow S^{p+1}
  \,.
$$

=--

## Properties

### Relation to the J-homomorphism

relation to the [[J-homomorphism]]:

see ([Schwede 15, example 4.22](#Schwede15))

### Relation to the cobordism hypothesis

> check

A [[connective spectrum]] is equivalently a grouplike [[E-∞ space]], hence a [[Picard ∞-groupoid]]. As such it is an [[(∞,0)-category]] of [[fully dualizable objects]]. By the [[cobordism hypothesis]] this means that it is equipped with an $O(n)$-[[∞-action]] for all $n$, coming from the action $O(n)$ on the [[framed manifold|n-framings]] of the point in the [[(∞,n)-category of cobordisms]]. This $O(n)$-action is that which is encoded by the definition of orthogonal spectrum ([Lurie 09, example 2.4.15](#Lurie09)).

### Relation to global equivariant stable homotopy theory

Since orthogonal spectra are by definition equipped with orthogonal group [[actions]], they serve as models for [[equivariant homotopy theory]] "for all groups at once", called _[[global stable homotopy theory]]_.

## Related concepts

* [[free orthogonal spectrum]]

* [[orthogonal ring spectrum]]

[[model structure on spectra]], [[symmetric monoidal smash product of spectra]]

* [[symmetric spectrum]], [[model structure on symmetric spectra]]

* **orthogonal spectrum**, [[model structure on orthogonal spectra]]

* [[S-module]], [[model structure on S-modules]]

* [[global equivariant stable homotopy theory]]

## References

Reviews include

* Knut Berg, _Orthogonal spectra_ ([pdf](http://folk.uio.no/rognes/theses/orthogonal.pdf))

* {#Malkiewich14} [[Cary Malkiewich]], section 2.3 of _The stable homotopy category_, 2014 ([pdf](https://people.math.binghamton.edu/malkiewich/stable.pdf), [[MalkiewichStableHomotopyCategory.pdf:file]])

Lecture notes include

* {#Schwede12} [[Stefan Schwede]], chapter I, section 7.1 of _[[Symmetric spectra]]_ (2012)

* {#Schwede14} [[Stefan Schwede]], section 1 of _[[Lectures on Equivariant Stable Homotopy Theory]]_, 2014 ([pdf](http://www.math.uni-bonn.de/people/schwede/equivariant.pdf))

and ([Schwede 15](#Schwede15)) (take throughout $\mathcal{F} = \{1\}$ there to be the trivial family to restrict to the non-equivariant case).

Orthogonal spectra were introduced around

* {#MMSS98a} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], _Diagram spaces, diagram spectra, and FSP's_, 1998 ([KTheory:0319](http://www.math.uiuc.edu/K-theory/0319/))

and their [[homotopy theory]] and [[Quillen equivalences]] of [[model categories of spectra]] were discussed in

* [[Michael Mandell]], [[Peter May]], _Orthogonal spectra and $S$-modules_ ([K-theory:0318](http://www.math.uiuc.edu/K-theory/0318/), [pdf](http://www.math.uchicago.edu/~may/PAPERS/mmLMSDec30.pdf))

* {#MMSS98b} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]],  _[[Model categories of diagram spectra]]_, 1998 ([KTheory:0320](http://www.math.uiuc.edu/K-theory/0320/))

and for [[equivariant spectra]] in

* {#MandellMay00} [[Michael Mandell]], [[Peter May]], _Equivariant orthogonal spectra and $S$-modules_, 2000 [KTheory:0408](http://www.math.uiuc.edu/K-theory/0408/)

and for [[operads]] enriched over orthogonal spectra in

* [[Tore Kro]], _Model structure on operads in orthogonal spectra_, Homology Homotopy Appl. Volume 9, Number 2 (2007), 397-412.([Euclid](http://projecteuclid.org/euclid.hha/1201127343))

and in the context of [[equivariant stable homotopy theory]] in ([Schwede 14](#Schwede14)) and in

* {#MMSS00} [[Michael Mandell]], [[Peter May]], _Equivariant orthogonal spectra and S-modules_. Preprint, April 29, 2000, ([KTheory:0408](http://www.math.uiuc.edu/K-theory/0408/))

and in [[global equivariant stable homotopy theory]]:

* {#Schwede15} [[Stefan Schwede]], _[[Global homotopy theory]]_, 2015 ([pdf](http://www.math.uni-bonn.de/people/schwede/global.pdf))

See also

* {#Lurie09} [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_, Current Developments in Mathematics Volume 2008 (2009), 129-280 ([arXiv:0905.0465](http://arxiv.org/abs/0905.0465))

[[!redirects orthogonal spectra]]


orthogonal structure

{:classqn: style="border:solid #cccccc;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

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

#Contents#
* table of contents
{:toc}

## Idea ##

The idea of an **orthogonal structure** is a generalisation to arbitrary [[vector bundles]] of the distinguishing ingredient of a [[Riemannian manifold]]: the [[Riemannian metric]].  In short, an orthogonal structure on a vector bundle is a smooth choice of [[inner product]] on the fibres.

In finite dimensions, that is pretty much all that can be said about the basic definition.  For [[infinite dimensional manifolds]], however, there is considerably more to be said because the _linear_ notion of an inner product is much, much richer.

## Definition ##

Although the definition makes sense in the topological category, we shall confine ourselves to the smooth category for the time being.

+-- {: .num_defn #orthstr}
###### Definition
Let $\pi \colon E \to M$ be a smooth [[vector bundle]] over a [[smooth manifold]].  An **orthogonal structure** on $E$ is a smooth choice of [[inner product]] on $E$.  That is to say, an orthogonal structure $g$ on $E$ is a section of $E^* \otimes E^*$ which restricts to an inner product on each fibre.
=--

As for [[Riemannian metrics]], a standard [[partition of unity]] argument yields the following existence result.

+-- {: .num_proposition #exist}
###### Proposition
Let $\pi \colon E \to M$ be a smooth vector bundle with finite dimensional fibres which admits a trivialisation over a numerable cover.  Then $E$ admits an orthogonal structure.
=--

Moreover, in finite dimensions all orthogonal structures are equivalent in that there is a bundle isomorphism between any two.

## Properties

### Reduction of structure Groups

#### In finite dimensions

In finite dimensions, to put an orthogonal structure on a vector bundle is to give a [[reduction of structure groups|reduction of its structure group]] from the [[general linear group]] to the corresponding [[orthogonal group|orthogonal]] subgroup along the defining inclusion $O(n) \hookrightarrow GL(n)$.  Such a reduction is also known as a choice of _[[vielbein]]_.

Thus the existence in general is equivalent to the fact that the inclusions $O(n) \to GL(n)$ and $U(n) \to GL(n)$ are [[homotopy equivalences]], as for the inclusion of any [[maximal compact subgroup]].  A similar statement can be made about the fact that any two orthogonal structures are equivalent.

#### In infinite Dimensions

The situation in infinite dimensions is more complicated due to the fact that there are more situations to consider.  In finite dimensions, any two orthogonal structures are equivalent, which stems from the fact that any two inner products define equivalent norms in finite dimensions.  In infinite dimensions, it is no longer true even that all [[locally convex topological vector spaces]] admit inner products, and when one does then it is highly unlikely that it will admit only one (up to equivalence).

What gives Riemannian geometry much of its power is the induced isomorphism between the [[tangent bundle|tangent]] and [[cotangent bundles]].  Thus a natural first question to ask of an orthogonal structure in infinite dimensions is whether or not this is still true.

+-- {: classqn}
**Question:** Does the orthogonal structure induce an isomorphism $E \cong E^*$?

**Yes:** It is _strong_.

**No:** It is _weak_.
=--

The condition that an orthogonal structure be strong is very restrictive.  The bundle must be modelled on Hilbert spaces (with the correct topology) and the inner product must be the right choice.  Many orthogonal structures that occur "in the wild" do not fit those two conditions and are thus relegated to the "weak" bin.  However, all is not lost and it is still possible to work with such orthogonal structures.  Although a weak orthogonal structure means that the bundle is not modelled on Hilbert spaces, it does mean that there are Hilbert spaces nearby and it is possible to further refine the notion of a "weak orthogonal structure" by asking how "nice" are those associated Hilbert spaces.

The idea is quite simple.  To put an orthogonal structure on a vector _space_, say $V$, is to give an injection $V \to H$ in to some Hilbert space.  We can assume that $V$ has dense image, but this is not strictly necessary.  Thus an inner product is a diagram:
$$
\iota \colon V \to H
$$
and the classification of weak orthogonal structures is essentially a study of how well that diagram extends over the base space.

## Related concepts

* [[orthogonality]], [[orthogonal basis]], [[normal vector]]

* [[pseudo-orthogonal structure]]

* A choice of orthogonal structure on a [[manifold]] is also equivalently called a choice of _[[vielbein]]_. See there for more details.

* [integrability of G-structures -- Examples -- Orthogonal structure](integrability+of+G-structures#ExamplesOrthogonalStructure)

## References

* _How to Construct a Dirac Operator in Infinite Dimensions_ [[Andrew Stacey]] [0809.3104](http://arxiv.org/abs/0809.3104)

* _The Co-Riemannian Structure of Smooth Loop Spaces_ [[Andrew Stacey]] [0809.3108](http://arxiv.org/abs/0809.3108)

[[!redirects orthogonal structures]]


orthogonal subcategory problem

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

The **orthogonal subcategory problem** for a [[class]] of [[morphism]]s $\Sigma$ in a [[category]] $C$ asks whether the [[full subcategory]] $\Sigma^\perp$ of [[object]]s [[orthogonal]] to $\Sigma$ is a [[reflective subcategory]].

Here, an object $X$ is said to be orthogonal to a map $f:A\to B$ if every morphism $A\to X$ factors uniquely through $f$. For example, in a category of presheaves, $X:C^{op}\to Set$ is orthogonal to the canonical map $colim_i C(-,a_i)\to C(-,\colim_i a_i)$ if it preserves the limit $\lim_i a_i$ in $C^{op}$.

This problem is related to the problem of [[localization]]. Suppose $\Sigma^\perp$ is indeed a [[reflective subcategory]]; let $r: C \to \Sigma^\perp$ be the reflector (the [[left adjoint]] to the inclusion $i: \Sigma^\perp \to C$).

## Properties

+-- {: .un_lemma}
###### Lemma

If $f: a \to b$ belongs to $\Sigma$, then $r(f): r(a) \to r(b)$ is an [[isomorphism]] in $\Sigma^\perp$.

=--

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

By definition of [[orthogonality]], for every object $X$ in $\Sigma^\perp$, $f$ induces an isomorphism of [[hom-set]]s

$$\hom_C(f, i(X)): \hom_C(B, i(X)) \stackrel{\sim}{\to} \hom_C(A, i(X))$$

Since $r \dashv i$, this means that for all $X$ in $\Sigma^\perp$ the map

$$\hom_{\Sigma^\perp}(r(f), X): \hom_{\Sigma^\perp}(r(B), X) \to \hom_{\Sigma^\perp}(r(A), X)$$

is an [[isomorphism]], so that $\hom_{\Sigma^\perp}(r(f), -)$ is a [[natural isomorphism]] between [[representable functor|representables]]. By the [[Yoneda lemma]], this means $r(f)$ is an isomorphism.

=--

This lemma can be sharpened. First, given a category $C$, there is a [[Galois connection]] between classes of morphisms $\Sigma$ and classes of objects $K$, induced by the predicate that $\hom_C(\sigma, k)$ is a bijection for all $\sigma \in \Sigma$ and all $k \in K$. The induced monad on the collection of morphism classes $\Sigma$ (partially ordered by inclusion) may be called "saturation", denoted $sat(\Sigma)$. An easy argument due to Gabriel-Zisman is that if $C$ is finitely [[cocomplete category|cocomplete]], then $(C, sat(\Sigma))$ satisfies the axioms for a [[calculus of fractions]], so that the [[localization]] $C[(sat(\Sigma))^{-1}]$ can be constructed. An easy argument then establishes the following sharpened lemma:

+-- {: .un_lemma}
###### Lemma

If $\Sigma^\perp \hookrightarrow C$ is [[reflective subcategory|reflective]], then the reflection $r: C \to \Sigma^\perp$ exhibits a canonical [[equivalence of categories|equivalence]]
$$C[(sat(\Sigma))^{-1}] \simeq \Sigma^\perp$$

=--

+-- {: .un_lemma}
###### Lemma

If $\Sigma^\perp \hookrightarrow C$ is [[reflective subcategory|reflective]], then the reflection $r: C \to \Sigma^\perp$ exhibits a canonical [[equivalence of categories|equivalence]]
$$C[(sat(\Sigma))^{-1}] \simeq \Sigma^\perp$$

=--

+--{.query}

To be connected with things like [[small object argument]], [[Bousfield localization]], and others...

=--

## Theorem in the setting of locally presentable categories

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

If $\Sigma$ is a set of morphisms in a [[locally presentable]] category $\mathcal{A}$ then the orthogonality subcategory $\Sigma^\top$ is reflective in $\mathcal{A}$.

=--

## References ##

* [[Gabriel-Zisman]]

* [[Peter Freyd]], [[Max Kelly]], _Categories of continuous functors I_, Jour. Pure Appl. Alg. 2 (1972), 169-191. ([web](http://www.sciencedirect.com/science/article/pii/0022404972900011))

* {#AdamekRosicky} [[Jiří Adámek]], [[Jiří Rosický]], _[[Locally presentable and accessible categories]]_, Cambridge University Press, (1994). But see the [errata](https://www.tu-braunschweig.de/Medien-DB/iti/cor.pdf).


orthogonality

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

#Contents#
* table of contents
{:toc}

(This entry describes two distinct notions, one in the theory of [[inner product spaces]], and the second in a more purely [[category theory|category theoretic]] context.)

## In inner product spaces

Two elements $x,y$ in an [[inner product space]], $(V, \langle -,-\rangle)$, are **orthogonal** or [[normal vectors]], denoted $x \perp y,$ if $\langle x,y\rangle = 0$.

## In category theory

### Definition

Two [[morphism]]s $e:A\to B$ and $m:C\to D$ in a [[category]] are said to be **orthogonal**, written $e\perp m$, if $e$ has the [[left lifting property]] with respect to $m$, i.e. if in any [[commutative diagram|commutative square]]

$$
\array{ A & \overset{e}{\to} & B\\
  \downarrow && \downarrow \\
  C & \underset{m}{\to} & D}
$$

there exists a unique diagonal [[filler]] making both triangles commute:

$$
\array{ A & \overset{e}{\to} & B\\
  \downarrow & \swarrow & \downarrow \\
  C & \underset{m}{\to} & D}
$$

Given a [[class]] of maps $E$, the class $\{m | e\perp m \;\forall e\in E\}$ is denoted $E^{\downarrow}$ or $E^\perp$.  Likewise, given $M$, the class $\{e | e\perp m \;\forall m\in M\}$ is denoted $M^{\uparrow}$ or ${}^\perp M$.  These operations form a [[Galois connection]] on the poset of classes of morphisms in the ambient category.  In particular, we have $({}^\perp(E^\perp))^\perp = E^\perp$ and ${}^\perp(({}^\perp M)^\perp) = {}^\perp M$.

A pair $(E,M)$ such that $E^\perp = M$ and $E = {}^\perp M$ is sometimes called a **prefactorization system**.  If in addition every morphism factors as an $E$-morphism followed by an $M$-morphism, it is an (orthogonal) [[orthogonal factorization system|factorization system]].

### Examples

* Of course, any [[orthogonal factorization system]] gives plenty of examples.  The ur-example is that $e\perp m$ in [[Set]] (or actually, any [[pretopos]]) for any surjection $e$ and injection $m$.

* A [[strong epimorphism]] in any category is, by definition, an epimorphism in ${}^\perp(Mono)$, where $Mono$ is the class of monomorphisms.  (If the category has equalizers, then every map in ${}^\perp(Mono)$ is epic.)  Dually, a [[strong monomorphism]] is a monomorphism in $(Epi)^\perp$.

* The [[orthogonal subcategory problem]] for a [[class]] of [[morphism]]s $\Sigma$ in a [[category]] $C$ asks whether the [[full subcategory]] $\Sigma^\perp$ of [[object]]s $X$ [[orthogonal]] to $\Sigma$ is a [[reflective subcategory]]. Here we define $f \perp X$ to mean $f \perp !: X \to 1$.

  The orthogonal subcategory problem is related to [[localization]]. Suppose $\Sigma^\perp$ is indeed a [[reflective subcategory]]; let $r: C \to \Sigma^\perp$ be the reflector (the [[left adjoint]] to the inclusion $i: \Sigma^\perp \to C$). Certainly $r$ sends arrows in $\Sigma$ to isomorphisms in $\Sigma^\perp$. Indeed, if $f: A \to B$ belongs to $\Sigma$, then the inverse to $r(f): r(A) \to r(B)$ is the unique arrow extending $1_{r(A)}$ along $r(f): r(A) \to r(B)$ to an arrow $g: r(B) \to r(A)$, using the fact that $r(A)$ belongs to $\Sigma^\perp$.

## Related concepts

* [[injective object]]

[[!include (co)isotropic subspaces - table]]

* [[orthogonal structure]], [[orthogonal basis]]

* [[Grassmannian]], [[Stiefel manifold]]

* [[orthogonal spectrum]], [[coordinate-free spectrum]], [[equivariant spectrum]]

[[!redirects orthogonal]]

[[!redirects orthogonal complement]]
[[!redirects orthogonal complements]]

[[!redirects perpendicular]]