octonionic Hopf fibration

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_Under construction_

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## Idea

The _octonionic Hopf fibration_ is the [[fibration]]

$$
  \array{
     S^7 &\hookrightarrow& S^15
     \\
     && \downarrow^{\mathrlap{p_{\mathbb{O}}}}
     \\
     && S^8
  }
$$

of the [[15-sphere]] over the [[8-sphere]] with [[fiber]] the [[7-sphere]]. This may be derived by the [[Hopf construction]] on the [[7-sphere]] $S^7$ with its [[Moufang loop]] structure.

Alternatively, we may construct a fibration by first decomposing $\mathbb{O}^2$ into the octonionic lines,

$l_m \coloneqq \{(x, m x)|x \in \mathbb{O}\}$ and $l_{\infty} := \{(0, y)|y \in \mathbb{O}\}$.

In this way the fibration $\mathbb{O}^2 \setminus (0, 0) \to S^8 = \{m \in \mathbb{O}\} \union \{\infty\}$ is obtained, with fibers $\mathbb{O} \setminus 0$, and the intersection with the [[unit sphere]] $S^{15} \subset \mathbb{O}^2$ provides the octonionic Hopf fibration (see [OPPV 12, p. 7](#OrneaPartonPiccinniVuletescu12)).

This second construction yields the standard parameterization of the octonionic Hopf fibration via $(x,y) \mapsto x y^{-1}$ (in one chart) and $(x,y ) \mapsto x^{-1} y$ (in the other), while the Hopf construction gives $(x,y) \mapsto x y$. The latter yields the generator $-1$ of $\pi_{15}(S^8) \cong \mathbb{Z}$, while the former yields $+1$.

## Properties

### $G_2$- and $Spin(9)$-equivariance

The [[automorphism group]] [[G2]] of the [[octonions]], as a [[normed algebra]], manifestly acts on the octonionic Hopf fibration, such that the latter is [[equivariant function|equivariant]].

(see also [Cook-Crabb 93](#CookCrabb93))

But the octonionic Hopf fibration is [[equivariant function|equivariant]] even with respect to the [[Spin(9)]]-[[action]], the one on $S^8 = S(\mathbb{R}^9)$ induced from the canonical action of $Spin(9)$ on $\mathbb{R}^9$, and on $S^{15} = S(\mathbb{R}^{16})$ induced from the canonical inclusion $Spin(9) \hookrightarrow Spin(16)$.

([Gluck-Warner-Ziller 86, Prop. 7.1](#GluckWarnerZiller86))

This equivariance is made fully manifest by realizing the octonionic Hopf fibration as a map of [[coset spaces]] as follows ([Ornea-Parton-Piccinni-Vuletescu 12, p. 7](#OrneaPartonPiccinniVuletescu12)):

$$
  \array{
    S^7
      &\overset{fib(h_{\mathbb{O}})}{\longrightarrow}&
    S^{15}
      &\overset{h_{\mathbb{O}}}{\longrightarrow}&
    S^8
    \\
     = && = && =
    \\
    \frac{Spin(8)}{Spin(7)}
    &\longrightarrow&
    \frac{Spin(9)}{Spin(7)}
    &\longrightarrow&
    \frac{Spin(9)}{Spin(8)}
  }
$$

### Subfibrations

*  The complex and quaternionic Hopf fibrations are not subfibrations of the octonionic one ([Parton & Piccinni 18, p. 4](#PartonPiccinni18)).

* The octonionic Hopf fibration does not admit any $S^1$ [[subfibration]] ([Parton & Piccinni 18, p. 13](#PartonPiccinni18)).

## Related concepts

* [[real Hopf fibration]], [[complex Hopf fibration]], [[quaternionic Hopf fibration]]

* [[octonionic projective space]]

## References

* {#GluckWarnerZiller86} [[Herman Gluck]], [[Frank Warner]], [[Wolfgang Ziller]], _The geometry of the Hopf fibrations_, L'Enseignement Mathématique, t.32 (1986), p. 173-198

* {#OrneaPartonPiccinniVuletescu12} Liviu Ornea, Maurizio Parton, [[Paolo Piccinni]], Victor Vuletescu, _Spin(9) geometry of the octonionic Hopf fibration_, Transformation Groups (2013) 18: 845 ([arXiv:1208.0899](http://arxiv.org/abs/1208.0899), [doi:10.1007/s00031-013-9233-x]( https://doi.org/10.1007/s00031-013-9233-x))

* {#PartonPiccinni18} Maurizio Parton, [[Paolo Piccinni]], _The role of $Spin(9)$ in octonionic geometry_, ([arXiv:1810.06288](https://arxiv.org/abs/1810.06288)).

* {#Miyaoka93} [[Reiko Miyaoka]], _The linear isotropy group of $G_2/SO(4)$, the Hopf fibering and isoparametric hypersurfaces_, Osaka J. Math. Volume 30, Number 2 (1993), 179-202. ([Euclid](http://projecteuclid.org/euclid.ojm/1200784357))

Discussion in [[parameterized homotopy theory]] is in

* {#CookCrabb93} A. L. Cook, M.C. Crabb, _Fiberwise Hopf structures on sphere bundles_, J. London Math. Soc. (2) 48 (1993) 365-384 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/crabbcook.pdf))

* {#Iriye95} [[Kouyemon Iriye]], _Equivariant Hopf structures on a sphere_, J. Math. Kyoto Univ. Volume 35, Number 3 (1995), 403-412 ([Euclid](http://projecteuclid.org/euclid.kjm/1250518704))


octonionic projective space

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## Idea

The notion of [[projective space]] $\mathbb{O}P^n$ over the [[octonions]] $\mathbb{O}$ makes sense for $n \,\in\, \{ 0,1,2 \}$ (but not beyond, see e.g. [Voelkel 16, Sec. 1.3](#Voelkel16)).
The [[octonionic projective plane]] $\mathbb{O}P^2$ is also called the *Cayley projective plane*.

## Properties

### Octonionic projective line and the 8-Sphere

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

We have a [[homeomorphism]]

$$
  \mathbb{O}P^1 \,\simeq\, S^8
$$

between the [[octonionic projective line]] and the [[8-sphere]].

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### Cell structure on octonionic projective plane

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###### Proposition

There is a [[homeomorphism]]

$$
  \mathbb{O}P^2 \,\simeq\, D^{16} \underset{h_{\mathbb{O}}}{\cup} \mathbb{O}P^1
$$

between the [[octonionic projective plane]] and the [[attaching space]] obtained from the [[octonionic projective line]] along the [[octonionic Hopf fibration]].

=--

([Lackmann 19, Lemma 3.4.](#Lackmann19))

([Mimura 67, page 166](#Mimura67))

See also at _[[cell structure of projective spaces]]_.

### Homotopy groups

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

The [[homotopy groups]] of octonionic projective plane are
\[
  \label{ListOfHomotopyGroupsOfOctonionicProjectivePlane}
    \pi_k
    \big(
      \mathbb{O}P^2
    \big)
    \;=\;
    \left\{
    \array{
      1 &\vert& k \leq 7
      \\
      \pi_k
      \big(
        S^8
      \big)
      &\vert&
      8 \leq k \leq 14
    }
    \right.
\]
Further homotopy groups are
$$\pi_{15}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}_{120}$$
$$\pi_{16}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}_2^3$$
$$\pi_{17}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}_2^4$$
$$\pi_{18}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}_{24}\times\mathbb{Z}_2$$
$$\pi_{19}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}_{504}\times\mathbb{Z}_2$$
$$\pi_{20}\big(\mathbb{O}P^2\big)
\cong 1$$
$$\pi_{21}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}_6$$
$$\pi_{22}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}_4$$
$$\pi_{23}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}\times\mathbb{Z}_{120}\times\mathbb{Z}_2^2\,.$$
(While $\pi_{15}\big(S^8\big)
\cong\mathbb{Z}\times\mathbb{Z}_{120}$, which includes the [[homotopy class]] of the [[octonionic Hopf fibration]].)

=--

([Lackmann 19, page 7](#Lackmann19))

([Mimura 67, Theorem 7.2.](#Mimura67))

### Cohomology

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

For $A \in $ [[Ab]] any [[abelian group]], then the [[ordinary cohomology]] [[cohomology groups|groups]] of octionionic projective plane $\mathbb{O}P^2$ with [[coefficients]] in $A$ are

$$
  H^k(\mathbb{O}P^2,A)
   \simeq
  \left\{
    \array{
       A & for \; k=0,8,16
       \\
       0 & otherwise
    }
  \right.
  \,.
$$

=--

([Lackmann 19, Corollary 4.1.](#Lackmann19))

## References

* {#Lackmann19} [[Malte Lackmann]], _The octonionic projective plane_, in MATRIX Book Series **4**, Springer (2021) [[doi:10.1007/978-3-030-62497-2_6](https://doi.org/10.1007/978-3-030-62497-2_6), [arXiv:1909.07047](https://arxiv.org/abs/1909.07047)]

* {#Voelkel16} [[Konrad Voelkel]], _Motivic cell structures for projective spaces over split quaternions_, 2016 [[freidok:11448](https://freidok.uni-freiburg.de/data/11448), [pdf](https://freidok.uni-freiburg.de/fedora/objects/freidok:11448/datastreams/FILE1/content)]

* Rowena Held, Iva Stavrov, Brian VanKoten, _(Semi-)Riemannian geometry of (para-)octonionic projective planes_, Diff. Geom. & its Appl. __27__ 4 (2009) 464-481 [[doi:/10.1016/j.difgeo.2009.01.007](https://doi.org/10.1016/j.difgeo.2009.01.007)]

* {#Mimura67} Mamoru Mimura, _The Homotopy groups of Lie groups of low rank_, J. Math. Kyoto Univ. **6** 2 (1967) 131-176 [[doi:10.1215/kjm/1250524375](https://doi.org/10.1215/kjm/1250524375)]

[[!redirects octonionic projective spaces]]

[[!redirects octonionic projective line]]
[[!redirects octonionic projective lines]]

[[!redirects octonion projective spaces]]
[[!redirects octonion projective spaces]]

[[!redirects octonion projective line]]
[[!redirects octonion projective lines]]


odd Chern character

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## Idea

Where the [[Chern character]] of a map to $\mathbf{B}U$ (for $U$ the [[stable unitary group]]) is a sum of [[characteristic classes]] corresponding to [[invariant polynomials]], the _odd Chern character_ $ch(g)$ of a map to $g \colon X \to U$ is a sum of the corresponding [[WZW term]] [[curvatures]]

$$
  ch(g) \coloneqq \underoverset{k= 0}{\infty}{\sum} (-1)^k \frac{k!}{(2k+1)!} tr[(g^{-1} d g)^{2k+1}]
  \,,
$$

where $g^{-1}d g \coloneqq g^\ast \theta$ is the [[pullback of differential forms]] along $g$ of the [[Maurer-Cartan form]] on $U$.

The odd Chern character appears in the index theory for [[Toeplitz operators]] and the [[eta invariant]] for even-dimensional manifolds ([Dai-Zhang 12](#DaiZhang12)).

## Related concepts

* [[K-theory]], [[K-homology]], [[index theory]]

## References

* [[Paul Baum]], R. G. Douglas, K-homology and index theory, in Proc. Sympos. Pure
and Appl. Math., Vol. 38, pp. 117-173, Amer. Math. Soc. Providence, 1982.

* [[Ezra Getzler]], _The odd Chern character in cyclic homology and spectral
flow_, Topology, 32(3):489–507, 1993.

Discussion of the relation to [[index theory]] of [[Toeplitz operators]] and [[eta invariants]] is in

* {#DaiZhang12} [[Xianzhe Dai]], [[Weiping Zhang]], _Eta invariant and holonomy, the even dimensional case_ ([arXiv:1205.0562](http://arxiv.org/abs/1205.0562))

Discussion of odd [[differential K-theory]] via the odd Chern character is in

* [[Thomas Tradler]], [[Scott Wilson]], [[Mahmoud Zeinalian]], _An Elementary Differential Extension of Odd K-theory_, J. of K-theory, K-theory and its Applications to Algebra, Geometry, Analysis
and Topology, ([arXiv:1211.4477](http://arxiv.org/abs/1211.4477))

* [[Scott Wilson]], _A loop group extension of the odd Chern character_ ([arXiv:1311.6393](http://arxiv.org/abs/1311.6393))

[[!redirects odd Chern characters]]


odd line

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## Definition

The **odd line** is the [[supermanifold]] $\mathbb{R}^{0|1}$ -- a [[super Cartesian space]] and in particular a [[superpoint]]  -- characterized by the fact that its $\mathbb{Z}_2$-graded algebra of functions is the algebra free on a single odd generator $\theta$: $C^\infty(\mathbb{R}^{0|1}) = \mathbb{R}[\theta] = \mathbb{R} \oplus \theta\cdot \mathbb{R}$.

This algebra is essentially the [[ring of dual numbers]], but with the single generator in odd degree.

## Properties

### The automorphism super-group
 {#TheAutomorphismSuperGroup}

The internal [[automorphism group]] of the odd line in the [[topos]] of [[smooth super infinity-groupoids|smooth super spaces]] is the [[supergroup]]

$$
  \mathbf{Aut}(\mathbb{A}^{0|1})
  \simeq
  \mathbb{G}_m \ltimes (\Pi \mathbb{G}_{ad})
$$

which is the [[semidirect product group]] of the [[multiplicative group]] (the [[group of units]], hence $\mathbb{R}^\times$ when working over the real numbers) with the [[additive group]] shifted into odd degree. (See at _[References -- Automorphism group](#ReferencesAutomorphismGroup)_ for the origin of this observation.)

In the [[topos]] over [[superpoints]] $\mathbb{R}^{0|q}$ this is seen over the test space $\mathbb{R}^{0|1}$ itself with canonical odd coordinate $\theta$ by taking the canonical odd coordinate of the odd line that we are taking automorphism of to be $\epsilon$ and observing that maps

$$
  \mathbb{R}^{0|1} \to \mathbf{Aut}(\mathbb{R}^{0|1})
$$

are then given, under the [[evaluation map]]-isomorphism and via the [[Yoneda lemma]] by [[Grassmann algebra]] [[homomorphisms]] of the form

$$
  \langle \epsilon, \theta\rangle
  \leftarrow
   \langle \epsilon\rangle
$$

that send

$$
  \epsilon \mapsto x \epsilon + y \theta
$$

with $x \neq 0$ in even degree and arbitrary $y$ in odd degree. Notice that the [[inverse]] of this map exists and is given by

$$
  \epsilon \mapsto x^{-1}\epsilon - y x^{-1} \theta
  \,.
$$

Moreover, observe that an [[action]] of $\mathbf{Aut}(\mathbb{R}^{0|1})$ on a [[supermanifold]] corresponds to a choice of [[graded object|grading]] and a choice of [[differential]]. A clean account of this statement is in ([Carchedi-Roytenberg 12](#CarchediRoytenberg12)). (At least for the grading this is essentially the classical statement about graded rings, see at _[[affine line]]_ the section  _[Properties -- Grading](affine+line#Grading)_).

## Related concepts

* [[Cartesian space]]

* [[infinitesimal space]]

* [[synthetic differential supergeometry]]

## References

### Automorphism group
 {#ReferencesAutomorphismGroup}

That an action of the endomorphism/automorphism supergroup of the odd line on a supermanifold is equivalent to a choice of grading and a [[differential]] first observed in

* {#Kontsevich97} [[Maxim Kontsevich]], _Deformation quantization of Poisson manifolds, I_, Lett. Math. Phys. 66:157-216,2003 ([arXiv:q-alg/9709040](http://arxiv.org/abs/q-alg/9709040))

It was later amplified in section 3.2 of

* {#KochanSevera03} Denis Kochan, [[Pavol Ševera]], _Differential gorms, differential worms_ ([arXiv:math/0307303](http://arxiv.org/abs/math/0307303)),

where it is used to exhibit the canonical [[de Rham differential]] action on the [[odd tangent bundle]] $Maps(\mathbb{R}^{0|1}, X)$ of a [[supermanifold]] $X$.

The same mechanism is amplified further in the discussion of [[derived differential geometry]] in

* [[David Carchedi]], [[Dmitry Roytenberg]], _Homological Algebra for Superalgebras of Differentiable Functions_ ([arXiv:1212.3745](http://arxiv.org/abs/1212.3745))
 {#CarchediRoytenberg12}

An interpretation of an $\mathbb{G}_m \ltimes \Pi \mathbb{G}_{ad}$-[[action]] on a [[supermanifold]] of local [[quantum observables]] of a [[supersymmetry|supersymmetric]] [[field theory]] as the formalization of the concept of [[topologically twisted super Yang-Mills theory]] is in section 15 of

* {#Costello11} [[Kevin Costello]], *Notes on supersymmetric and holomorphic field theories in dimension 2 and 4*, talk at *[Geometric and Algebraic Structures in Mathematics](https://www.math.stonybrook.edu/dennisfest/)*, Stony Brook (2011) [[arXiv:1111.4234](https://arxiv.org/abs/1111.4234)]

For more on this see at _[topologically twisted super Yang-Mills theory -- Formalization](topologically+twisted+D%3D4+super+Yang-Mills+theory#Formalization)_.


Oded Regev

* [research group page](https://cims.nyu.edu/~regev/)

* Wikipedia entry

## Selected writings

On [[adiabatic quantum computation]]:

* [[Dorit Aharonov]], [[Wim van Dam]], [[Julia Kempe]], [[Zeph Landau]], [[Seth Lloyd]], [[Oded Regev]], *Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation*, SIAM Journal of Computing **37** 1 (2007) 166-194 [[arXiv:quant-ph/0405098](https://arxiv.org/abs/quant-ph/0405098), [jstor:20454175](https://www.jstor.org/stable/20454175), [doi:10.1109/FOCS.2004.8](https://doi.org/10.1109/FOCS.2004.8), [doi:10.1137/080734479](https://doi.org/10.1137/080734479)]

category: people


Oded Yacobi

is a lecturer at the University of Sydney.  Here is his [webpage](http://www.maths.usyd.edu.au/u/oyacobi/).


Odysseus

## References

See also

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


Ofer Aharony

* [webpage](http://www.weizmann.ac.il/physics/staff/aharony.htm)

## Selected writings

On the [[AdS-CFT correspondence]]:

* {#AharonyGubserMaldacenaOoguriOz99} [[Ofer Aharony]], [[Steven Gubser]], [[Juan Maldacena]], [[Hirosi Ooguri]], [[Yaron Oz]], _Large $N$ Field Theories, String Theory and Gravity_, Phys.Rept.323:183-386,2000 ([arXiv:hep-th/9905111](http://arxiv.org/abs/hep-th/9905111))

and specifically for [[higher spin gauge theory]]:

* [[Ofer Aharony]], [[Shai Chester]], [[Erez Urbach]], _A Derivation of AdS/CFT for Vector Models_ ([arXiv:2011.06328](https://arxiv.org/abs/2011.06328))

On [[(p,q)5-brane webs]]:

* [[Ofer Aharony]], [[Amihay Hanany]], [[Barak Kol]], _Webs of $(p,q)$ 5-branes, Five Dimensional Field Theories and Grid Diagrams_, JHEP 9801:002,1998 ([arXiv:hep-th/9710116](http://arxiv.org/abs/hep-th/9710116))

* [[Ofer Aharony]], [[Amihay Hanany]], _Branes, Superpotentials and Superconformal Fixed Points_, Nucl. Phys. B504:239-271, 1997 ([arXiv:hep-th/9704170](https://arxiv.org/abs/hep-th/9704170))

On [[D-brane polarization]] in a context of [[AdS/QCD]]:

* [[Ofer Aharony]], [[Arvind Rajaraman]], _String Theory Duals for Mass-deformed $SO(N)$ and $USp(2N)$ $\mathcal{N}=4$ SYM Theories_, Phys. Rev. D62:106002, 2000 ([arXiv:hep-th/0004151](https://arxiv.org/abs/hep-th/0004151))

On [[multi trace operators]] in [[AdS/CFT]]:

* [[Ofer Aharony]], [[Micha Berkooz]], [[Eva Silverstein]], _Multiple-Trace Operators and Non-Local String Theories_, JHEP 0108:006, 2001 ([arXiv:hep-th/0105309](https://arxiv.org/abs/hep-th/0105309))

* [[Ofer Aharony]], [[Micha Berkooz]], [[Eva Silverstein]], _Non-local string theories on $AdS_3$ times $S^3$ and stable non-supersymmetric backgrounds_, Phys. Rev. D65:106007, 2002 ([arXiv:hep-th/0112178](https://arxiv.org/abs/hep-th/0112178))

On [[D=2 Yang-Mills theory]] as the [[effective field theory]] of a [[string theory]]:

* [[Ofer Aharony]], Suman Kundu, Tal Sheaffer, *A String Theory for Two Dimensional Yang-Mills Theory I* [[arXiv:2312.12266](https://arxiv.org/abs/2312.12266)]

## Related entries

* [[ABJM theory]]

category: people


Ofer Gabber

**Ofer Gabber** is an Israeli-French mathematician (with thesis under Barry [Mazur](http://en.wikipedia.org/wiki/Barry_Mazur)), famous for his technical excellence in a number of mathematical subjects, now a CNRS member [working](http://seven.ihes.fr/IHES-A/People/per-cnrsA.html) at [IHES](http://www.ihes.fr). Gabber has produced very many results from foundational results to counterexamples, mainly in [[algebraic geometry]], [[arithmetic geometry]], number theory, [[representation theory]] and [[algebra]], only small part of which are published, and many are known only indirectly via works of other mathematicians who were acquainted with his results.

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

## Selected writings

On [[almost ring theory]]

* [[Ofer Gabber]], [[Lorenzo Ramero]], _Almost ring theory_, Springer 2003 ([arXiv:math/0002064](https://arxiv.org/abs/math/0002064), [doi:10.1007/b10047](https://link.springer.com/book/10.1007/b10047))

and its foundations (including [[2-category]] theory):

* [[Ofer Gabber]], [[Lorenzo Ramero]], *Foundations for almost ring theory* ([arXiv:math/0409584](https://arxiv.org/abs/math/0409584))

## Related nLab entries

* [[BBDG decomposition theorem]]

* [[almost scheme]]

category: people


off-shell Poisson bracket

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## Idea

Given a [[symplectic manifold]] $(X,\omega)$ and given a [[Hamiltonian]] [[function]] $H \colon X \longrightarrow \mathbb{R}$, there is a [[Poisson bracket]] on an [[algebra of functions]] on the [[smooth space|smooth]] [[path space]] $[I,X]$  -- the "space of histories" or "space of [[trajectories]]" -- for $I = [0,1]$ the closed [[interval]], which is such that its [[symplectic leaves]] are each a copy of $X$, but regarded as the space of initial conditions for evolution with respect to $H$ with a [[source]] term added.

The first statement was first observed for the [[Peierls bracket]] of [[local prequantum field theory]] in ([Marolf 93, section II](#Marolf93)), but the construction there is not specific to the [[Peierls bracket]]. That the construction provides a [[foliation]] of trajectory space by [[symplectic leaves]] labeled by [[level sets]] of the [[Euler-Lagrange equation|Euler-Lagrange function]] was explicitly pointed out at ([Brunetti-Fredenhagen-Ribeiro 12, top of p. 4](#BrunettiFredenhagenRibeiro12)). (Again for the [[Peierls bracket]], but the statement holds more generally.) These references and that this means a symplectic foliation by [[source]] terms was highlighted out by ([Khavkine 13](#Khavkine13)).

## On paths in a symplectic manifold

We describe here the [[off-shell]] Poisson bracket in the context of [[mechanics]], hence for [[mechanical systems]] with [[finite set|finite]]-[[dimension|dimensional]] [[phase space]]. The basic idea is that sketched in ([Marolf 93, section II](#Marolf93)), but we try to make it precise. Then we similarly look into the [[foliation]] by [[symplectic leaves]] as suggested by ([Khavkine 13](#Khavkine13)).

### The trajectory space of a symplectic manifold

Let $(X,\omega)$ be a [[symplectic manifold]]. We write

$$
  \{-,-\} \;\colon\; C^\infty(X)\otimes C^\infty(X) \longrightarrow C^\infty(X)
$$

for the [[Poisson bracket]] induced by the [[symplectic form]] $\omega$, hence by the [[Poisson bivector]] $\pi \coloneqq \omega^{-1}$.

For notational simplicity we will restrict attention to the special case that

$$
  X = \mathbb{R}^2 \simeq T^\ast \mathbb{R}
$$

with canonical [[coordinates]]

$$
  q,p \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R}
$$

and symplectic form

$$
  \omega = \mathbf{d}q \wedge \mathbf{d}p
  \,.
$$

The general case of the following discussion is a straightforward generalization of this, which is just notationally more inconvenient.

Write $I \coloneqq [0,1]$ for the standard [[interval]] regarded as a [[smooth manifold]] [[manifold with boundary|with boundary]]. The [[mapping space]]

$$
  P X \coloneqq [I, X]
$$

canonically exists as a [[smooth space]], but since $I$ is [[compact topological space|compact]] this structure canonically refines to that of a [[Fréchet manifold]] (see at _[[manifold structure of mapping spaces]]_). This implies that there is a good notion of [[tangent space]] $T P X$. The task now is to construct a certain [[Poisson bivector]] as a [[section]] $\pi \in \Gamma^{\wedge 2}(T P X)$.

Among the [[smooth functions]] on $P X$ are the [[evaluation maps]]

$$
  ev \;\colon\; P X \times I = [I,X] \times I \stackrel{}{\longrightarrow} X
$$

whose components we denote, as usual, for $t \in I$ by

$$
  q(t) \coloneqq q \circ ev_t \;\colon\; P X \longrightarrow \mathbb{R}
$$

and

$$
  p(t) \coloneqq p \circ ev_t \;\colon\; P X \longrightarrow \mathbb{R}
  \,.
$$

Generally for $f \colon X \to \mathbb{R}$ any [[smooth function]], we write $f(t) \coloneqq f \circ ev_t \in C^\infty(P X)$. This defines an embedding

$$
  C^\infty(X) \times I \hookrightarrow C^\infty(P X)
  \,.
$$

Similarly we have

$$
  \dot q(t) \;\colon\; P X \longrightarrow \mathbb{R}
$$

and

$$
  \dot q(t) \;\colon\; P X \longrightarrow \mathbb{R}
$$

obtained by [[differentiation]] of $t \mapsto q(t)$ and $t \mapsto p(t)$.

### Hamiltonian evolution

Let now

$$
  H \;\colon\; X \times I \longrightarrow \mathbb{R}
$$

be a [[smooth function]], to be regarded as a time-dependent [[Hamiltonian]]. This induces a time-dependent function on trajectory space, which we denote by the same symbol

$$
  H
  \;\colon\;
  P X \times I
   \stackrel{(ev,id)}{\longrightarrow}
  X \times X
   \stackrel{H}{\longrightarrow}
  \mathbb{R}
  \,.
$$

Hence for $t \in I$ we write

$$
  H(t)
  \;\colon\;
  P X \times \{t\}
  \stackrel{(ev, id)}{\longrightarrow}
  X \times \{t\}
  \stackrel{H}{\longrightarrow}
  \mathbb{R}
$$

for the function that assigns to a trajectors $(q(-),p(-)) \colon I \longrightarrow X$ its [[energy]] at (time) parameter value $t$.

Define then the [[Euler-Lagrange equation|Euler-Lagrange density]] induced by $H$ to be the functions

$$
  EL(t)
  \;\colon\;
  P X
  \longrightarrow
  \mathbb{R}^2
$$

with components

$$
  EL(t)
  =
  \left(
    \array{
      \dot q(t) - \frac{\partial H}{\partial p}(t)
      \\
      \dot p(t) + \frac{\partial H}{\partial p}(t)
    }
  \right)
  \,.
$$

The [[trajectories]] $\gamma \colon I \to X$ on which $EL(t)$ vanishes for all $t \in I$ are equivalently those

* for which the [[tangent vector]] $\dot \gamma \in T_{\gamma}X$ is a [[Hamiltonian vector field|Hamiltonian vector]] for $H$;

* which satisfy [[Hamilton's equations]] [[equations of motion|of motion]] for $H$.

Since the [[differential equations]] $EL = 0$ have a unique solution for given initial data $(q(0), p(0))$, the evaluation map

$$
  \left\{
    \gamma \in P X | \forall_{t \in I}\; EL_\gamma(t) = 0
  \right\}
  \stackrel{\gamma \mapsto \gamma(0)}{\longrightarrow}
  X
$$

is an [[equivalence]] (an [[isomorphism]] of [[smooth spaces]]).

### The off-shell Poisson bracket
 {#TheOffShellPoissonBracketOnSpaceOfPathsInSymplecticManifold}

Write

$$
  Poly(P X) \hookrightarrow C^\infty(P X)
$$

for the [[subalgebra]] of [[smooth functions]] on [[path space]] which are

* [[polynomials]]

* of [[integrals]] over $I$

* of the smooth functions in the image of $C^\infty(X) \times I \hookrightarrow C^\infty(P X)$

* and all their [[derivatives]] along $I$.

Define a [[bilinear function]]

$$
  \{-,-\}
  \;\colon\;
  Poly(P X) \otimes Poly(P X)
  \longrightarrow Poly(P X)
$$

as the unique function which is a [[derivation]] in both arguments and moreover is a solution to the [[differential equations]]

$$
  \frac{\partial}{\partial t_2}
  \left\{f(t_1), q(t_2)\right\}
  =
  \left\{
    f(t_1), \frac{\partial H}{\partial p}(t_2)
  \right\}
$$

$$
  \frac{\partial}{\partial t_2}
  \left\{f(t_1), p(t_2)\right\}
  =
  -
  \left\{
    f(t_1), \frac{\partial H}{\partial q}(t_2)
  \right\}
$$

subject to the initial conditions

$$
  \{f(t), q(t)\} = \{f,q\}
$$

$$
  \{f(t), p(t)\} = \{f,p\}
$$

for all $t \in I$, where on the right we have the original [[Poisson bracket]] on $X$.

This bracket directly inherits skew-symmetry and the [[Jacobi identity]] from the [[Poisson bracket]] of $(X, \omega)$, hence equips the [[vector space]] $Poly(P X)$ with the structure of a [[Lie bracket]]. Since it is by construction also a [[derivation]] of  $Poly(P X)$ as an [[associative algebra]], we have that

$$
  \left(
    Poly\left(P X\right),
    \;
    \left\{
      -,-
    \right\}
  \right)
  \;\;\;
  \in P_1 Alg
$$

is a [[Poisson algebra]]. This is the "off-shell Poisson algebra" on the space of [[trajectories]] in $(X,\omega)$.

### The symplectic leaves

Observe that by construction of the off-shell Poisson bracket,
specifically by the [[differential equations]] defining it, the
[[Euler-Lagrange equation|Euler-Lagrange function]] $EL$
generate a [[Poisson reduction|Poisson ideal]].

For instance

$$
 \left(
  \array{
  \frac{\partial}{\partial t_2}
  \left\{f(t_1), q(t_2)\right\}
  &=&
  \left\{
    f(t_1), \frac{\partial H}{\partial p}(t_2)
  \right\}
  \\
  \frac{\partial}{\partial t_2}
  \left\{f(t_1), p(t_2)\right\}
  &=&
  -
  \left\{
    f(t_1), \frac{\partial H}{\partial q}(t_2)
  \right\}
  }
  \right)
  \;\;\;
  \Leftrightarrow
  \;\;\;
  \left(
    \left\{
      f(t_1), \;
      EL(t)
    \right\}
    =
    0
  \right)
  \,.
$$

Moreover, since $\{EL(t) = 0\}$ are  [[equations of motion]] the [[Poisson reduction]] defined by this Poisson idea is the subspace of those [[trajectories]] which are solutions of [[Hamilton's equations]], hence the "on-shell trajectories".

As remarked above, the initial value map canonically identifies this on-shell trajectory space with the original [[phase space]] manifold $X$. Moreover, by the very construction of the off-shell Poisson bracket as being the original Poisson bracket at equal times, hence in particular at time $t = 0$, it follows that restricted to the [[zero locus]] $EL = 0$ the off-shell Poisson bracket becomes [[symplectic manifold|symplectic]].

All this clearly remains true with the function $EL$ replaced by the function $EL - J$, for  $J \in C^\infty(I)$ any function of the (time) parameter (since $\{J,-\} = 0$). Any such choice of $J$ hence defines a symplectic subspace

$$
  \left\{
    \gamma \in P X
    \;|\;
    \forall_{t \in I}\; EL_\gamma(t) = J
  \right\}
$$

of the off-shell Poisson structure on trajectory space. Hence $\left(O X, \left\{-,-\right\}\right)$ has a [[foliation]] by [[symplectic leaves]] with the [[leaf space]] being the [[smooth space]] $C^\infty(I)$ of [[smooth functions]] on the interval.

Notice that changing $EL \mapsto EL - J$ corresponds changing the time-dependent [[Hamiltonian]] $H$ as

$$
  H \mapsto H - J q
  \,.
$$

Such a term linear in the [[canonical coordinates]] (the [[field (physics)|fields]]) is a _[[source]] term_. (The [[action functionals]] with such [[source]] terms added serve as [[integrands]] of [[generating functions]] for [[correlators]] in [[statistical mechanics]] and in [[quantum mechanics]].)

### Boundary field theory interpretation
 {#BoundaryFieldTheoryInterpretation}

Hence in conclusion we find the following statement:

The [[trajectory]] space (history space) of a [[mechanical system]] carries a natural [[Poisson manifold|Poisson structure]] whose [[symplectic leaves]] are the subspaces of those trajectories which satisfy the [[equations of motion]] with a fixed [[source]] term and hence whose symplectic [[leaf space]] is the space of possible sources.

Notice what becomes of this statement as we consider the the [[2d Chern-Simons theory]] induced by the off-shell Poisson bracket (the [[non-perturbative field theory|non-perturbative]] [[Poisson sigma-model]]) whose [[moduli stack]] of [[field (physics)|fields]] is the [[symplectic groupoid]] $SG\left(P X, \left\{-,-\right\}\right)$ induced by the Poisson structure.

By the discussion at [[motivic quantization]] in the section _[The Poisson manifold at the boundary of the 2d Chern-Simons theory](motivic%20quantization#PoissonManifoldAtTheBoundaryOf2dChernSimonsTheory)_, the Poisson space $\left(P X, \left\{-,-\right\}\right)$ defines a  [[boundary field theory]] (in the sense of [[local prequantum field theory]]) for this [[2d Chern-Simons theory]], exhibited by a boundary [[correspondence]] of the form

$$
  \array{
    && P X
    \\
    & \swarrow && \searrow
    \\
    \ast && \swArrow_{\xi} && SG\left(P X, \left\{-,-\right\}\right)
    \\
    & \searrow && \swarrow_{\mathrlap{\chi}}
    \\
    && \mathbf{B}^2 U(1)
    \\
    && \downarrow^{\mathrlap{\rho}}
    \\
    && KU Mod
  }
  \,.
$$

Notice that the [[symplectic groupoid]] is a version of the [[symplectic leaf|symplectic]] [[leaf space]] of the given [[Poisson manifold]] (its [[0-truncation]] is exactly the leaf space). Hence in the case of the off-shell Poisson bracket, the [[symplectic groupoid]] is the space of _[[source field|sources]]_ of a mechanical system. At the same time it is the [[moduli space]] of [[field (physics)|fields]] of the [[2d Chern-Simons theory]] of which the mechanical system is the [[boundary field theory]].
Hence the [[field (physics)|fields]] of the [[bulk field theory]] are identified with the [[sources]] of the [[boundary field theory]]. Hence conceptually the above boundary correspondence diagram is of the following form

$$
  \array{
    && Fields
    \\
    & \swarrow && \searrow
    \\
    \ast && \swArrow_{} && Sources
    \\
    & \searrow && \swarrow_{\mathrlap{}}
    \\
    && Phases
  }
  \,.
$$

Such a relation

|  [[bulk field theory]]  |  [[boundary field theory]]  |
|---|----|
| [[field (physics)|field]] | [[source]] |

between bulk fields and boundary sources is the characteristic feature of what is called the _[[holographic principle]]_ in its realization as the [[AdS-CFT correspondence]].

## References

The off-shell extension of the [[Peierls bracket]] is observed in section II of

* [[Don Marolf]],  _Poisson Brackets on the Space of Histories_ Annals of Physics Volume 236, Issue 2, December 1994, Pages 374-391 ([arXiv:hep-th/9308141](http://arxiv.org/abs/hep-th/9308141))
  {#Marolf93}

The observation that the off-shell bracket has a symplectic foliation by the level sets of the Euler-Lagrange functions appears on the top of p. 4 of

* [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Pedro Ribeiro]], _Algebraic Structure of Classical Field Theory I: Kinematics and Linearized Dynamics for Real Scalar Fields_ ([arXiv:1209.2148](http://arxiv.org/abs/1209.2148))
  {#BrunettiFredenhagenRibeiro12}

All this and the interpretation of the resulting symplectic foliation as a foliation by source terms has been highlighted by

* [[Igor Khavkine]], personal communication
  {#Khavkine13}

[[!redirects off-shell Poisson brackets]]


oidification - table

| [[algebra|algebraic]] [[mathematical structure|structure]] | [[oidification]] |
|----------------|----------|
|[[magma]]|[[magmoid]]|
|[[pointed set|pointed]] [[magma]] with an [[endofunction]]|[[setoid]]/[[Bishop set]]|
|[[unital magma]]|[[unital magmoid]]|
|[[quasigroup]]|[[quasigroupoid]]|
|[[loop (algebra)|loop]]|[[loopoid]]|
|[[semigroup]]|[[semicategory]]|
|[[monoid]]|[[category]]|
|[[anti-involution|anti-involutive]] [[monoid]]|[[dagger category]]|
|[[associative quasigroup]]|[[associative quasigroupoid]]|
|[[group]]|[[groupoid]]|
|[[flexible magma]]|[[flexible magmoid]]|
[[alternative magma]]|[[alternative magmoid]]|
|[[absorption monoid]]|[[absorption category]]|
|[[cancellative monoid]]|[[cancellative category]]|
|[[rig]]|[[CMon]]-[[enriched category]]|
|[[nonunital ring]]|[[Ab]]-[[enriched magmoid|enriched]] [[semicategory]]|
|[[nonassociative ring]]|[[Ab]]-[[enriched magmoid|enriched]] [[unital magmoid]]|
|[[ring]]|[[ringoid]]|
|[[nonassociative algebra]]|[[linear magmoid]]|
|[[nonassociative algebra|nonassociative unital algebra]]|[[unital magmoid|unital]] [[linear magmoid]]|
|[[nonunital algebra]]|[[linear magmoid|linear]] [[semicategory]]|
|[[associative unital algebra]]|[[linear category]]|
|[[C-star algebra]]|[[C-star category]]|
|[[differential algebra]]|[[differential algebroid]]|
|[[flexible algebra]]|[[flexible magmoid|flexible]] [[linear magmoid]]|
|[[alternative algebra]]|[[alternative magmoid|alternative]] [[linear magmoid]]|
|[[Lie algebra]]|[[Lie algebroid]]|
|[[monoidal poset]]|[[2-poset]]|
|[[strict monoidal groupoid]]|[[strict (2,1)-category]]|
|[[strict 2-group]]|[[strict 2-groupoid]]|
|[[strict monoidal category]]|[[strict 2-category]]|
|[[monoidal groupoid]]|[[(2,1)-category]]|
|[[2-group]]|[[2-groupoid]]/[[bigroupoid]]|
|[[monoidal category]]|[[2-category]]/[[bicategory]]|


Oidification > history

< [[Oidification]]
[[!redirects Oidification -- history]]


Oka manifold

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

#Contents#
* table of contents
{:toc}

## Idea

*Oka manifolds* (the term is due to [Forstnerič 2009a](#Forstneric09a)) comprise the class of [[complex analytic manifolds]] which, when used as [[classifying spaces]], satisfy the weak homotopy equivalence form of the *[[Oka principle]]*.

The relative notion are *Oka maps* ([Forstnerič 2009a](#Forstneric09a)), these are [[fibrations]] in the Jardine-Lárusson model structure on the category of [[simplicial presheaves]] on the simplicial Stein site. Those cofibrant objects which are representable by complex manifolds are in fact [[Stein manifolds]].

## Examples

\begin{prop}\label{ComplexProjectiveSpaceIsOkaManifold}
**([[complex projective spaces]] are [[Oka manifolds]])** \linebreak
Every [[complex projective space]] $\mathbb{C}P^n$, $n \in \mathbb{N}$, is an [[Oka manifold]]. More generally every [[Grassmannian]] over the [[complex numbers]] is an Oka manifold.
\end{prop}
(review in [Forstnerič & Lárusson 11, p. 9](#ForstnericLarusson11), [Forstnerič 2013, Ex. 2.7](#Forstneric13))

Yet more generally:
\begin{prop}\label{CosetSpacesOfComplexLieGroupsAreOkaManifolds}
**([[coset spaces]] of [[complex Lie groups]] are [[Oka manifolds]])** \linebreak
Every [[complex Lie group]] and every [[coset space]] ([[homogeneous space]]) of complex Lie groups is an [[Oka manifold]].
\end{prop}
(review in [Forstnerič & Lárusson 11, p. 9](#ForstnericLarusson11), [Forstnerič 2013, Thm. 2.6](#Forstneric13))

\begin{example}\label{ComplementsOfCompactPolynomiallyConvextSubsets}
The [[complement]] of any compact polynomially convex subset of $\mathbf{C}^n$ ($n\gt1$) is an Oka manifold.
\end{example}
([Kusakabe 20, Theorem 1.2, Corollary 1.3](#Kusakabe20))

## References

Introduction and review:

* {#Larusson10} [[Finnur Lárusson]], _What is an Oka manifold?_, Notices AMS Volume 57, Number 1, 2010 ([pdf](http://www.ams.org/notices/201001/rtx100100050p.pdf), [[Larussen_OkaManifolds.pdf:file]])

* {#ForstnericLarusson11} [[Franc Forstnerič]], [[Finnur Lárusson]], *Survey of Oka theory*, New York J. Math., 17a (2011), 1-28 ([arXiv:1009.1934](https://arxiv.org/abs/1009.1934), [eudml:232963](https://eudml.org/doc/232963))

* {#Forstneric11} [[Franc Forstnerič]], Section 5 of: _Stein manifolds and holomorphic mappings -- The homotopy principle in complex analysis_, Springer 2011 ([doi:10.1007/978-3-642-22250-4](https://link.springer.com/book/10.1007/978-3-642-22250-4))

* {#Forstneric13} [[Franc Forstnerič]] (appendix by [[Finnur Lárusson]]), *Oka manifolds: From Oka to Stein and back*, Annales de la Faculté des sciences de Toulouse, Mathématiques, Série 6, Tome 22 (2013) no. 4, pp. 747-809 ([numdam:AFST_2013_6_22_4_747_0](https://afst.centre-mersenne.org/item/?id=AFST_2013_6_22_4_747_0))

* [[Franc Forstnerič]], *Developments in Oka theory since 2017* ([arXiv:2006.07888](https://arxiv.org/abs/2006.07888))

See also:

* [[Finnur Lárusson]], *Affine simplices in Oka manifolds*, Documenta Mathematica (2009) Volume: 14, page 691-697 ([eudml:228723](https://eudml.org/doc/228723))

* {#Kusakabe20} [[Yuta Kusakabe]], _Oka properties of complements of holomorphically convex sets_ ([arXiv:2005.08247](https://arxiv.org/abs/2005.08247)).

Proof of the [[homotopy theory|homotopy-theoretic]] [[Oka principle]]:

* {#Forstneric09a} [[Franc Forstnerič]], _Oka manifolds_, Comptes Rendus Mathematique, Acad. Sci. Paris 347 (2009), 1017–20 ([arXiv:0906.2421](https://arxiv.org/abs/0906.2421), [doi:10.1016/j.crma.2009.07.005](https://doi.org/10.1016/j.crma.2009.07.005))

* {#Forstneric09b} [[Franc Forstnerič]], _Oka maps_, Comptes Rendus Mathematique, Acad. Sci. Paris, Ser. I 348 (2010) 145-148 ([arxiv/0911.3439](http://arxiv.org/abs/0911.3439), [doi:10.1016/j.crma.2009.12.004](https://doi.org/10.1016/j.crma.2009.12.004))

[[!redirects Oka manifolds]]
[[!redirects Oka map]]
[[!redirects Oka maps]]


Oka principle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In [[complex geometry]], the _Oka-Grauert principle_ states that over [[complex manifolds]] $S$ which are *[[Stein manifolds]]*, the [[non-abelian cohomology]]-classification of [[holomorphic vector bundles]] coincides with that of [[topological vector bundles]],

$$
  S \;\text{is Stein}
  \;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;
  H^1
  \Big(
    S;
    \,
    \underline{GL(n, \mathbb{C})}_{hol}
  \Big)
  \,\simeq\,
  H^1
  \Big(
    S;
    \,
    \underline{GL(n, \mathbb{C})}_{top}
  \Big)
  \,.
$$

This was originally proven for holomorphic [[line bundles]] in [Oka 1939](#Oka39) (in which case it says that [[holomorphic line bundles]] over [[Stein manifolds]] are fully classified by their [[first Chern class]]) and generalized in [Grauert 1958](#Grauert58) to holomorphic [[vector bundles]] and further to holomorphic [[principal bundles]] with [[structure group]] any [[complex Lie group]].

As a principle, this *Oka-Grauert principle* is sometimes stated as ([Forstnerič 12](#Forstneric12)):

> Analytic problems on Stein spaces which can be cohomologically formulated have only topological [[obstructions]].

More generally, for suitable [[complex manifolds]] $A$ now called *[[Oka manifolds]]* ([Forstnerič 2009a](#Forstneric09a)) -- including (see [here](Oka+manifold#ComplexProjectiveSpaceIsOkaManifold)) the complex [[Grassmannians]] that serve as [[classifying spaces]] for [[complex vector bundles]] --, the inclusion into the [[compact-open topology|space of continuous maps]] $S \to A$, out of a [[Stein manifold]] $S$, of the [[topological subspace|subspace]] of [[holomorphic functions]] is a [[weak homotopy equivalence]]:

$$
  Maps_{hol}(S, \, A)
    \xhookrightarrow{ \simeq_{whe} }
  Maps(S, \, A)
  \;\;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;\;
  \underset{n \in \mathbb{N}}{\forall}
  \;\;
  \pi_n
  \big(
    Maps_{hol}(S, \, A)
  \big)
    \xrightarrow{\sim}
  \pi_n
  \big(
    Maps(S, \, A)
  \big)
  \,.
$$

(review in [Forstnerič & Lárusson 2011, Cor. 3.5](#ForstnericLarusson11), [Forstnerič 2013, (1.1)](#Forstneric13))

More generally, an analogous statement applies to suitable [[fiber bundles]] of [[Oka manifolds]] over [[Stein manifolds]] and their [[spaces of sections]] ([Forstnerič 2009b](#Forstneric09b)).

This [[homotopy theory|homotopy theoretic]] *weak homotopy equivalence Oka principle* goes back to results of [Gromov 89](#Gromov89), where (?) it is viewed an an example of the [[h-principle]].

The [[duality]] between [[Stein manifolds]] and [[Oka manifolds]] in this [[homotopy theory|homotopy]]-theoretic Oka principle is fully brought out by the existence of a [[model category]] for [[complex analytic ∞-groupoids]] in which a [[complex manifold]] is [[cofibrant object|cofibrant]]/[[fibrant object]] if it is [[Stein manifold|Stein]]/[[Oka manifold|Oka]], respectively ([Lárusson 2001](#Larusson01), [03](#Larusson03)).

## Homotopical Oka principle
 {#HomotopicalOkaPrinciple}

\begin{prop}\label{WeakHomotopyEquivalenceOkaPrinciples}
**(weak homotopy equivalence Oka principle)**
For

* $S$ a [[Stein manifold]],

* $X$ an [[Oka manifold]],

the inclusion
$$
  Maps_{hol}
  \big(
    S, \, X
  \big)
  \xhookrightarrow{\;\simeq_{whe}\;}
  Maps
  \big(
    S ,\, X
  \big)
$$
of the [[subspace]] of [[holomorphic functions]]
into the [[mapping space]] of their [[underlying]] [[topological spaces]] (with the [[compact-open topology]])
is a [[weak homotopy equivalence]].
\end{prop}
(review in [Forstnerič & Lárusson 2011, Cor. 3.5](#ForstnericLarusson11))

More generally, for $Z \xrightarrow{\;} S$ a stratified holomorphic [[fiber bundle]] of [[Oka manifolds]], the corresponding inclusion of [[spaces of sections]]

$$
  \Gamma_{hol}\big(S, \, Z \big)
  \xhookrightarrow{\;\simeq_{whe}\;}
  \Gamma\big(S, \, Z \big)
$$

is a [[weak homotopy equivalence]].

([Forstnerič 2011, Cor. 5.4.8](#Forstneric11))

## In higher complex analytic geometry
 {#StatementInComplexAnalyticGeometry}

In ([Lárusson 2001](#Larusson01), [Lárusson 2003](#Larusson03)) this is formulated in terms of [[higher complex analytic geometry]] of [[complex analytic infinity-groupoids]].

Say that a [[complex manifold]] $X$ is an _[[Oka manifold]]_ if for every [[Stein manifold]] $\Sigma$ the canonical morphism

$$
  Maps_{hol}(\Sigma, X) \longrightarrow Maps_{top}(\Sigma, X)
$$

from the [[mapping space]] of [[holomorphic functions]] to that of [[continuous functions]] (both equipped with the [[compact-open topology]]) is a [[weak homotopy equivalence]].

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

This is the case precisely if $Maps_{hol}(-,X) \in Psh_\infty(SteinSp)$ satisfies [[descent]] with respect to [[finite set|finite]] [[covers]].

=--

([Larusson 01, theorem 2.1](#Larusson01))

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

The category of [[complex manifolds]] and [[holomorphic maps]] can be embedded into a Quillen [[model category]] such that:

* a holomorphic map is a [[weak equivalence]] in the ambient model category if and only if it is a [[homotopy equivalence]] in the usual topological sense;

* a holomorphic map is a [[fibration]] if and only if it is an Oka map. In particular, a complex manifold is [[fibrant object|fibrant]] if and only if it is an [[Oka manifold]];

* a [[complex manifold]] is [[cofibrant]] if and only if it is [[Stein manifold|Stein]];

* a Stein inclusion is a [[cofibration]].

=--

([Larusson 03](#Larusson03) -- apparently this follows an observation due to [[J. F. Jardine]] and uses his [[intermediate model structure on simplicial presheaves]])

## Related concepts

* [[Oka manifold]]

* [[homotopy of rational maps]]

* [[smooth Oka principle]]

## References

Introduction and review:

* {#Larusson10} [[Finnur Lárusson]], _What is an Oka manifold?_, Notices AMS Volume 57, Number 1, 2010 ([pdf](http://www.ams.org/notices/201001/rtx100100050p.pdf), [[Larussen_OkaManifolds.pdf:file]])

* {#ForstnericLarusson11} [[Franc Forstnerič]], [[Finnur Lárusson]], *Survey of Oka theory*, New York J. Math., 17a (2011), 1-28 ([arXiv:1009.1934](https://arxiv.org/abs/1009.1934), [eudml:232963](https://eudml.org/doc/232963))

* {#Forstneric11} [[Franc Forstnerič]], Section 5.3 of: _Stein manifolds and holomorphic mappings -- The homotopy principle in complex analysis_, Springer 2011 ([doi:10.1007/978-3-642-22250-4](https://link.springer.com/book/10.1007/978-3-642-22250-4))

* {#Forstneric13} [[Franc Forstnerič]] (appendix by [[Finnur Lárusson]]), *Oka manifolds: From Oka to Stein and back*, Annales de la Faculté des sciences de Toulouse, Mathématiques, Série 6, Tome 22 (2013) no. 4, pp. 747-809 ([numdam:AFST_2013_6_22_4_747_0](https://afst.centre-mersenne.org/item/?id=AFST_2013_6_22_4_747_0))

* [[Franc Forstnerič]], *Developments in Oka theory since 2017* ([arXiv:2006.07888](https://arxiv.org/abs/2006.07888))

* {#Forstneric12} [[Franc Forstnerič]], *Gromov's contribution to the Oka principle*, 2012 ([pdf](https://www.fmf.uni-lj.si/~forstneric/papers/2012Gromov%27scontribution.pdf), [[Forstneric_GromovContributionToOka.pdf:file]])

See also:

* Georges Elencwajg, [MO:a/60053](http://mathoverflow.net/a/60053/381)

Original articles:

* {#Oka39} [[Kiyoshi Oka]], _Sur les fonctions des plusieurs variables. III: Deuxième problème de Cousin_, J. Sc. Hiroshima Univ. __9__, 7–19 (1939) ([doi:10.32917/hmj/1558490525](https://projecteuclid.org/journals/hiroshima-mathematical-journal/volume-9/issue-none/Sur-les-fonctions-analytiques-de-plusieurs-variables-IIIDeuxi%c3%a8me-probl%c3%a8me-de/10.32917/hmj/1558490525.full))

* {#Grauert58} [[Hans Grauert]], _Analytische Faserungen über holomorph-vollständigen Räumen_, Math. Ann. __135__, 263–-273 (1958) ([doi:10.1007/BF01351803](http://dx.doi.org/10.1007/BF01351803))

* {#Gromov89} [[Mikhail Gromov]], _Oka's principle for holomorphic sections of elliptic bundles_, J. Amer. Math. Soc. __2__ (1989), 851–-897 ([doi:10.2307/1990897](https://doi.org/10.2307/1990897))

* [[Franc Forstnerič]], _The Oka principle for sections of stratified fiber bundles_, Pure Appl. Math. Quarterly (Special Issue in honor of Joseph J. Kohn), 6 (2010), no. 3, 843--874 ([arxiv/0705.0591](http://arxiv.org/abs/0705.0591), [doi:10.4310/PAMQ.2010.v6.n3.a11](https://dx.doi.org/10.4310/PAMQ.2010.v6.n3.a11))

Proof of the [[homotopy theory|homotopy-theoretic]] Oka principle:

* {#Forstneric09a} [[Franc Forstnerič]], _Oka manifolds_, Comptes Rendus Mathematique, Acad. Sci. Paris 347 (2009), 1017–20 ([arXiv:0906.2421](https://arxiv.org/abs/0906.2421), [doi:10.1016/j.crma.2009.07.005](https://doi.org/10.1016/j.crma.2009.07.005))

* {#Forstneric09b} [[Franc Forstnerič]], _Oka maps_, Comptes Rendus Mathematique, Acad. Sci. Paris, Ser. I 348 (2010) 145-148 ([arxiv/0911.3439](http://arxiv.org/abs/0911.3439), [doi:10.1016/j.crma.2009.12.004](https://doi.org/10.1016/j.crma.2009.12.004))

See also:

* [[Finnur Lárusson]], *Applications of a Parametric Oka Principle for Liftings*, In: *Complex Analysis*. Trends in Mathematics. Birkhäuser (2010) ([doi:10.1007/978-3-0346-0009-5_12](https://doi.org/10.1007/978-3-0346-0009-5_12))

Discussion in terms of [[higher complex analytic geometry]] and [[complex analytic ∞-groupoids]]:

* {#Larusson01} [[Finnur Lárusson]], _Excision for simplicial sheaves on the Stein site and Gromov's Oka principle_,  International Journal of Mathematics Vol. 14, No. 02, pp. 191-209 (2003)  ([arXiv:math/0101103](http://arxiv.org/abs/math/0101103), [doi:10.1142/S0129167X03001727](https://doi.org/10.1142/S0129167X03001727))

* {#Larusson03} [[Finnur Lárusson]], _Model structures and the Oka principle_, Journal of Pure and Applied Algebra Volume 192, Issues 1–3, 1 September 2004, Pages 203-223 ([math.CV/0303355](http://arxiv.org/abs/math/0303355), [doi:10.1016/j.jpaa.2004.02.005](https://doi.org/10.1016/j.jpaa.2004.02.005))

Application to [[minimal surfaces]]:

* [[Antonio Alarcón]], [[Franc Forstnerič]], *New complex analytic methods in the theory of minimal surfaces: a survey*,  Journal of the Australian Mathematical Society, 106(3), 287-341 ([arXiv:1711.08024](https://arxiv.org/abs/1711.08024), [doi:10.1017/S1446788718000125](https://doi.org/10.1017/S1446788718000125))

See also:

* Tyson Ritter, _A strong Oka principle for embeddings of some planar domains into $C\times C^*$_, [arxiv/1011.4116](http://arxiv.org/abs/1011.4116)

[[!redirects Oka theory]]
[[!redirects Oka's theory]]
[[!redirects Oka's principle]]

[[!redirects Oka-Grauert principle]]

[[!redirects homotopical Oka principle]]


Ola Bratteli

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

* *Ola Bratteli and His Diagrams*, Notices of the AMS **67** 5 (May 2020) 665-676 [[pdf](https://www.ams.org/notices/202005/rnoti-p665.pdf), [doi:10.1090/noti2085](https://dx.doi.org/10.1090/noti2085), full issue:[pdf](https://www.ams.org/notices/202005/202005FullIssue.pdf)]

## Selected writings

On [[operator algebras]] and [[AQFT|algebraic]] [[quantum statistical mechanics]]  ([[C-star algebras|$C^\ast$-algebras]], [[W-star algebras|$W^\ast algebras$]]):

* {#BratteliRobinson79} [[Ola Bratteli]], [[Derek W. Robinson]], *Operator Algebras and Quantum Statistical Mechanics* -- vol 1: *$C^\ast$- and $W^\ast$-Algebras. Symmetry Groups. Decomposition of States.*, Springer (1979, 1987, 2002) [[doi:10.1007/978-3-662-02520-8](https://doi.org/10.1007/978-3-662-02520-8)]

category: people


Olaf Hohm

* [webpage](http://www.theorie.physik.uni-muenchen.de/cosmology/members/former_mem/hohm/index.html)

## Selected writings

On [[D=6 supergravity]] with [[number of supersymmetries|$\mathcal{N} = (4,0)$]]:

* [[Yannick Bertrand]], [[Stefan Hohenegger]], [[Olaf Hohm]], [[Henning Samtleben]], *Toward Exotic 6D Supergravities*, Phys. Rev. D **103** (2021) 046002 [[arXiv:2007.11644](https://arxiv.org/abs/2007.11644), [doi:10.1103/PhysRevD.103.046002](https://doi.org/10.1103/PhysRevD.103.046002)]

* [[Yannick Bertrand]], [[Stefan Hohenegger]], [[Olaf Hohm]], [[Henning Samtleben]], *Supersymmetric action for 6D $(4,0)$ supergravity*, JHEP (2022) [[arXiv:2206.04100](https://arxiv.org/abs/2206.04100)]

## Related entries

* [[exceptional generalized geometry]]

* [[exceptional field theory]]

category: people


Olaf Lechtenfeld

* [webpage](http://olaf-lechtenfeld.de)

## Selected writings

On the [[Nicolai map]]:

* [[Olaf Lechtenfeld]], *Konstruktion der Nicolai-Abbildung in supersymmetrischen Feldtheorien*, PhD thesis, Bonn (1984) [[pdf](https://lib-extopc.kek.jp/preprints/PDF/2000/0030/0030157.pdf), [[Lechtenfeld-NicolaiAbbildung.pdf:file]]]

* [[Klaus Dietz]], [[Olaf Lechtenfeld]], *Nicolai maps and stochastic observables from a coupling constant flow*, Nuclear Physics B **255** (1985) 149-173 [doi:10.1016/0550-3213(85)90132-4]

* [[Klaus Dietz]], [[Olaf Lechtenfeld]], *Ghost-free quantisation of non-abelian gauge theories via the Nicolai transformation of their supersymmetric extensions*, Nuclear Physics B **259** 2–3 (1985) 397-411 [doi:10.1016/0550-3213(85)90642-X]

* [[Olaf Lechtenfeld]], *Stochastic variables in ten dimensions?*, Nuclear Physics B **274** 3–4 (1986) 633-652 [doi:10.1016/0550-3213(86)90531-6]

* [[Rainald Flume]], [[Olaf Lechtenfeld]], *On the stochastic structure of globally supersymmetric field theories*, Physics Letters B **135** 1–3 (1984) 91-95 [doi:10.1016/0370-2693(84)90459-3]

* [[Olaf Lechtenfeld]], [[Maximilian Rupprecht]], *Universal form of the Nicolai map*, Phys. Rev. D **104** 021701 (2021) [[arXiv:2104.00012](https://arxiv.org/abs/2104.00012), [doi:10.1103/PhysRevD.104.L021701](https://doi.org/10.1103/PhysRevD.104.L021701)]

* [[Olaf Lechtenfeld]], [[Maximilian Rupprecht]], *Is the Nicolai map unique?*,  J. High Energ. Phys. **2022** 139 (2022) [[arXiv:2207.09471](https://arxiv.org/abs/2207.09471), doi:10.1007/JHEP09(2022)139]

* [[Olaf Lechtenfeld]], *Supersymmetric large-order perturbation with the Nicolai map*, Physics Letters B
**835** (2022) 137507 [[arXiv:2208.06420](https://arxiv.org/abs/2208.06420), [doi:10.1016/j.physletb.2022.137507](https://doi.org/10.1016/j.physletb.2022.137507)]

* [[Olaf Lechtenfeld]], *The Nicolai-map approach to supersymmetry* [[arXiv:2309.00481](https://arxiv.org/abs/2309.00481)]

On [[Sasakian manifolds]] and [[quiver gauge theory]]:

* [[Olaf Lechtenfeld]], Alexander Popov, [[Richard Szabo]], _Sasakian quiver gauge theories and instantons on Calabi-Yau cones_ ([arXiv:1412.4409](http://arxiv.org/abs/1412.4409))

A derivation of 4d [[Skyrmions]] by [[KK-compactification]] of [[D=5 Yang-Mills theory]] on a [[closed interval]], motivated by [[M5-branes]] instead of by [[D4/D8-brane intersections]] as in the [[Sakai-Sugimoto model]]:

* [[Tatiana Ivanova]], [[Olaf Lechtenfeld]], [[Alexander Popov]], _Skyrme model from 6d $\mathcal{N}= (2,0)$ theory_, Physics Letters B Volume 783, 10 August 2018, Pages 222-226 ([arXiv:1805.07241](https://arxiv.org/abs/1805.07241))

following

* [[Tatiana Ivanova]], [[Olaf Lechtenfeld]], [[Alexander Popov]], _Non-Abelian sigma models from Yang-Mills theory compactified on a circle_, Physics Letters B Volume 781, 10 June 2018, Pages 322-326 ([arXiv:1803.07322](https://arxiv.org/abs/1803.07322))

On a [[perturbation series]] for the [[M2-brane]] [[Green-Schwarz sigma-model]] quantized to the [[BFSS matrix model]]:

* [[Olaf Lechtenfeld]], [[Hermann Nicolai]], *A perturbative expansion scheme for supermembrane and matrix theory* ([arXiv:2109.00346](https://arxiv.org/abs/2109.00346))

On the [[Nicolai map]] for the [[CP^N sigma-model|$\mathbb{C}P^n$ sigma-model]]:

* [[Lorenzo Casarin]], [[Olaf Lechtenfeld]], [[Maximilian Rupprecht]], *Nicolai maps with four-fermion interactions* [[arXiv:2310.19946](https://arxiv.org/abs/2310.19946)]

category: people


Oleg Zaboronsky

* [webpage](https://warwick.ac.uk/fac/sci/maths/people/staff/oleg_zaboronski)

## Selected writings

Introducing the [[AKSZ sigma-model]]:

* {#AKSZ} M. Alexandrov, [[Maxim Kontsevich]], [[Albert Schwarz]], [[Oleg Zaboronsky]], _The geometry of the master equation and topological quantum field theory_, Int. J. Modern Phys. A 12(7):1405--1429, 1997 ([arXiv:hep-th/9502010](http://arxiv.org/abs/hep-th/9502010))

On [[supersymmetry]] and [[equivariant localization]]:

* [[Albert Schwarz]], [[Oleg Zaboronsky]], _Supersymmetry and localization_, Comm. Math. Phys. __183__, 2 (1997), 463-476 ([euclid](http://projecteuclid.org/euclid.cmp/1158328185))

category: people


Oleksii Kotov

__Oleksii Kotov__ (also signed in older papers as Alexei Kotov) is an Ukrainian mathematical physicist at the University of Hradec Králové, Czech Republic.

* [[Martin Bojowald]], Alexei Kotov, Thomas Strobl, _Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries_,  J. Geom. Phys.  __54__:4  (2005) 400--426, [doi](https://doi.org/10.1016/j.geomphys.2004.11.002)
* Maxim Grigoriev, Alexei Kotov, _Presymplectic AKSZ formulation of Einstein gravity_, JHEP09(2021)181, (doi)

category: people

[[!redirects Alexei Kotov]]


Oliver Goertsches

* [personal page](https://www.uni-marburg.de/de/fb12/kooperationen/diffgeoana/prof-dr-oliver-goertsches)

## Selected writings

On [[equivariant de Rham cohomology]]:

* [[Oliver Goertsches]], [[Leopold Zoller]], _Equivariant de Rham Cohomology: Theory and Applications_, São Paulo J. Math. Sci. (2019) ([arXiv:1812.09511](https://arxiv.org/abs/1812.09511), [doi:10.1007/s40863-019-00129-4](https://doi.org/10.1007/s40863-019-00129-4))

category: people


Oliver Lorscheid

* [personal page](http://w3.impa.br/~lorschei/)

## Selected writings

On [[K-theory]] over the [[field with one element]] and its relation to [[stable Cohomotopy]] and [[equivariant stable Cohomotopy]]:

* {#ChuLorscheidSanthanam10} Chenghao Chu, [[Oliver Lorscheid]], Rekha Santhanam, *Sheaves and K-theory for $\mathbb{F}_1$-schemes*, Advances in Mathematics, **229** 4, (2012) 2239-2286 [[arxiv:1010.2896](https://arxiv.org/abs/1010.2896), [doi:10.1016/j.aim.2011.12.023](https://doi.org/10.1016/j.aim.2011.12.023)]

## Related $n$Lab entries

* [[F1]]

category: people


Oliver Röndigs

* [web page](http://www.mathematik.uni-osnabrueck.de/forschung/ag_topologie_und_geometrie/roendigs_oliver.html)

* [MathSciNet](http://ams.math.uni-bielefeld.de/mathscinet/search/publications.html?pg4=AUCN&s4=R%22ondigs%2C+O.&co4=OR&pg5=ICN&s5=R%22ondigs%2C+Oliver)

* [MathGenealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=86729)

## Selected writings

On [[Goodwillie calculus]] via [[model structures on excisive functors]] and [[model structure for n-excisive functors]]:

* {#BiedermannChornyRondings06} [[Georg Biedermann]], [[Boris Chorny]], [[Oliver Röndigs]], _Calculus of functors and model categories_, Advances in Mathematics **214** (2007) 92-115 [[arXiv:math/0601221](http://arxiv.org/abs/math/0601221), [doi:10.1016/j.aim.2006.10.009](https://doi.org/10.1016/j.aim.2006.10.009)]

* [[Georg Biedermann]], [[Oliver Röndigs]], _Calculus of functors and model categories II_, Algebr. Geom. Topol. **14** (2014) 2853-2913 [[arXiv:1305.2834v2](http://arxiv.org/abs/1305.2834), [doi:10.2140/agt.2014.14.2853](https://doi.org/10.2140/agt.2014.14.2853)]

## Related entries

* [[motivic homotopy theory]]

* [[algebraic cobordism]]


Oliver Schlotterer

* [webpage Uppsala](http://www.physics.uu.se/research/theoretical-physics/people/oliver-schlotterer)

* [webpage Potsdam](https://www.aei.mpg.de/person/23635/2168)

## Selected writings

On [[string scattering amplitudes]] for the [[Berkovits superstring|Berkovits-formulation]] of the [[superstring]]:

* {#MafraSchlotterer18a} [[Carlos Mafra]], [[Oliver Schlotterer]], _Towards the $n$-point one-loop superstring amplitude I: Pure spinors and superfield kinematics_ ([arXiv:1812.10969](https://arxiv.org/abs/1812.10969))

* {#MafraSchlotterer18b} [[Carlos Mafra]], [[Oliver Schlotterer]], _Towards the $n$-point one-loop superstring amplitude II: Worldsheet functions and their duality to kinematics_ ([arXiv:1812.10970](https://arxiv.org/abs/1812.10970))

* {#MafraSchlotterer18c} [[Carlos Mafra]], [[Oliver Schlotterer]], _Towards the $n$-point one-loop superstring amplitude III: One-loop correlators and their double-copy structure_ ([arXiv:1812.10971](https://arxiv.org/abs/1812.10971))

* [[Carlos R. Mafra]], [[Oliver Schlotterer]], *Tree-level amplitudes from the pure spinor superstring* [[arXiv:2210.14241](https://arxiv.org/abs/2210.14241)]

On [[2-loop]] [[string scattering amplitudes]] of the [[superstring]]:

* [[Eric D'Hoker]], [[Carlos Mafra]], [[Boris Pioline]], [[Oliver Schlotterer]], _Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors_ ([arXiv:2006.05270](https://arxiv.org/abs/2006.05270))

* [[Eric D'Hoker]], [[Carlos Mafra]], [[Boris Pioline]], [[Oliver Schlotterer]], _Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality_ ([doi:2008.08687](https://arxiv.org/abs/2008.08687))

## Related $n$Lab entries

* [[superstring scattering amplitudes]]

* [[KLT relations]]

* [[pure spinor]]

category: people


Olivia Caramello

Olivia Caramello is an Italian mathematician who works as an Associate Professor at the University of Insubria in Como and also holds the Gelfand Chair at IHES. She has pioneered the use of [[toposes]] as unifying bridges in mathematics.

* [webpage](http://www.oliviacaramello.com/)

###Recent papers and preprints:

* Olivia Caramello, [_Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic ‘bridges’_](https://global.oup.com/academic/product/theories-sites-toposes-9780198758914), Oxford University Press (2017).

* Olivia Caramello, [_Grothendieck toposes as unifying ‘bridges’ in Mathematics_] (https://www.oliviacaramello.com/Unification/HDROliviaCaramello.pdf), habilitation thesis, University of Paris 7 (2016).

* Olivia Caramello, _Topos theoretic background_, [PDF](http://www.oliviacaramello.com/Unification/ToposTheoreticPreliminariesOliviaCaramello.pdf)(2014)

* Olivia Caramello, _Topological Galois theories_, ([arXiv: 1301.0300](http://arxiv.org/abs/1301.0300)) (2013)

* Olivia Caramello, _The unification of Mathematics via Topos Theory_, ([arXiv: 1006.3930](http://arxiv.org/abs/1006.3930)) (2010)

category: people


Olivier Pons

* [personal page](http://www-sop.inria.fr/croap/personnel/Olivier.Pons/MAIN/summary-eng.html)

* [personal page](http://www-sop.inria.fr/lemme/Olivier.Pons/htdocs/accueil-eng.html)

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

## Selected writings

On [[type equivalence]]:

* [[Gilles Barthe]], [[Olivier Pons]], *Type Isomorphisms and Proof Reuse in Dependent Type Theory*, in *Foundations of Software Science and Computation Structures. FoSSaCS 2001*, Lecture Notes in Computer Science **2030**, Springer (2001) [[doi:10.1007/3-540-45315-6_4](https://doi.org/10.1007/3-540-45315-6_4)]

On [[setoids]] in [[type theory]]:

* [[Gilles Barthe]], Venanzio Capretta, [[Olivier Pons]], *Setoids in type theory*, Journal of Functional Programming **13** 2 (2003) 261-293 [[doi:10.1017/S0956796802004501](https://doi.org/10.1017/S0956796802004501)]

category: people


Olivier Renaudin

* [Mathematics Genealogy Page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=56210)

## Selected writingd

* {#Renaudin06} [[Olivier Renaudin]], *Plongement de certaines théories homotopiques de Quillen dans les dérivateurs*, Journal of Pure and Applied Algebra Volume 213, Issue 10, October 2009, Pages 1916-1935
([arXiv:math/0603339](https://arxiv.org/abs/math/0603339), [doi:10.1016/j.jpaa.2009.02.014](https://doi.org/10.1016/j.jpaa.2009.02.014))

## Related $n$Lab entries

* [[2-localization of a 2-category]]

* [[Ho(CombModCat)]]

category: people


Olivier Schiffmann

* [personal page](https://sites.google.com/site/olivierschiffmann/)

* [GoogleScholar page](https://scholar.google.fr/citations?user=5JuidGsAAAAJ&hl=fr)

## Selected writings

On [[Hall algebras]]:

* [[Olivier Schiffmann]], *Lectures on Hall algebras* [[arXiv:math/0611617](https://arXiv.org/abs/math/0611617)]

category: people


Olivier Verdier

* [webpage](http://www.olivierverdier.com)

## related $n$Lab entries

* [[function monad]]

category: people


Olof Hanner

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

## Selected writings

On [[absolute neighbourhood retracts]]:

* [[Olof Hanner]], _Some theorems on absolute neighbourhood retracts_, Arkiv För Matematik Band 1 nr 30 (1950) ([doi:10.1007/BF02591376](https://doi.org/10.1007/BF02591376))

category: people


Olov Wilander

* [MathGenelogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=161630)

# Selected writings

On [[setoids]]:

* {#Wilander12} [[Olov Wilander]], *Constructing a small category of setoids*, Mathematical Structures in Computer Science **22** 1 (2012) 103-121 [[doi:10.1017/S0960129511000478](https://doi.org/10.1017/S0960129511000478), [pdf](http://www.diva-portal.org/smash/get/diva2:399799/FULLTEXT01.pdf)]

* {#PalmgrenWilander14} [[Erik Palmgren]], [[Olov Wilander]], *Constructing categories and setoids of setoids in type theory*, Logical Methods in Computer Science, **10** 3 (2014) lmcs:964 [[arXiv:1408.1364](https://arxiv.org/abs/1408.1364), doi:10.2168/LMCS-10(3:25)2014]

category: people


Omar Khayyam

* [St. Andrews History of Mathematics entry](http://www-history.mcs.st-and.ac.uk/Biographies/Khayyam.html)

category:people
[[!redirects Khayyam]]


omega-categorical structure

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

#Contents#
* table of contents
{:toc}

## Idea
$\omega$-categorical structures are "highly symmetric" first-order [[structure|structures]].
## Definition
A countable structure $A$ is $\omega$-categorical if any other countable structure $B \models \mathbf{Th}(A)$ with the same [[first-order theory]] as $A$ is isomorphic to $A$.

## Examples
- The [[DLO|countable dense linear orders without endpoints]] are $\omega$-categorical.

- The [[countable random graph]] is $\omega$-categorical.

- Many [[Fraisse limit|Fraisse limits]] are $\omega$-categorical.

- The canonical (orbit) structure induced by an oligomorphic (finitely many orbits in each power) permutation group on a countable set is $\omega$-categorical.

## Remarks
- Conversely, the action $\Aut(A) \curvearrowright A$ is oligomorphic.

- The automorphism group of an $\omega$-categorical structure $A$ equipped with the topology of pointwise convergence comprises a complete set of invariants for $\mathbf{Th}(A)$ up to bi-interpretability: this is the Coquand-Ahlbrandt-Ziegler theorem.

## References

- Silvia Barbina, [Automorphism groups of $\omega$-categorical structures](https://pdfs.semanticscholar.org/d936/e201d29053b749d0459b2b23dd62fa0bbc79.pdf)

[[!redirects aleph_0-categorical structure]]


omega-category

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

#Contents#
* table of contents
{:toc}

## Idea

The term _$\omega$-category_ is used more or less synonymously with _[[∞-category]]_, though the two terms appear with different frequency in different parts of the literature.

Originally, $\omega$-categories were introduced as

* [[strict ∞-categories]].

The general concept is that of

* [[weak ∞-categories]].

## Related concepts

* [[∞-groupoid]]

* [[globular set]], [[globular operad]], [[globular theory]]

* [[cellular set]]

[[!redirects ω-category]]
[[!redirects ω-categories]]
[[!redirects omega-categories]]
[[!redirects (∞,∞)-categories]]


omega-complete poset

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An *$\omega$-complete poset*, in the following sense, is a [[poset]] with [[countable set|countable]] [[joins]], hence a countably [[cocomplete category|cocomplete]] poset.

## Definition ##

A __$\omega$-complete poset__ is a [[poset]] $(P, \leq)$ with

* an [[initial object|initial]] [[element]] $\bot \in P$ ([[bottom]]), hence such that for every element $a \in P$ we have $\bot \leq a$;

* a [[function]]

  $$
    \Vee_{n \colon \mathbb{N}} (-)(n) \;\colon\;
    (\mathbb{N} \to P) \to L
  $$

  [[universal property|exhibiting]] the existence of [[countable set|denumerable/countable]] [[joins]] in the poset, namely such that

  1. for every [[natural number]] $n \in \mathbb{N}$ and every [[sequence]] $s \colon \mathbb{N} \to P$ we have

     $$
       s(n)
         \leq
       \Vee_{n \colon \mathbb{N}} s(n)
       \,;
     $$

  1. for every element $a \in P$ and sequence $s \colon \mathbb{N} \to P$ of elements $\leq a$,

     $$
       \prod_{n \colon \mathbb{N}}
       (s(n) \leq a)
       \,,
     $$

     we have

     $$
       \Vee_{n \colon \mathbb{N}}
       s(n) \leq a
       \,.
     $$

## See also ##

* [[poset]]

* [[join-semilattice]]

* [[sigma-complete lattice]]

* [[partial function classifier]]

* [[directed-complete poset]]

## References ##

* *Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type* ([arXiv:1610.09254](https://arxiv.org/abs/1610.09254))


omega-graph > history

< [[omega-graph]]


Omega-group

# $\Omega$-groups
* table of contents
{: toc}

## Idea

An $\Omega$-group is a [[group]] equipped with additional algebraic operations (of [[signature]] $\Omega$) that distribute over the group operations.

## Definition

We take an [[algebraic structure]] in a traditional sense as a set $\Omega$ with a (not necessarily finite) number of operations $\alpha$ of various arity and satisfying some axioms (not necessarily of first order).  That is, we are discussing objects of an [[equationally presentable category|equationally presentable]] or [[algebraic category]].

An __$\Omega$-group__ is an algebraic structure which amounts to a [[group]] (usually written additively but not necessarily commutative) together with a set $\Omega$ of operations of any arity, such that for each $n$-ary operation $\alpha \in \Omega$, distributivity holds in each variable over the group operations:

$$ \alpha(x_1,\ldots,x_j + y_j,\ldots,x_n)=
\alpha(x_1,\ldots,x_j,\ldots,x_n) + \alpha(x_1,\ldots,y_j,\ldots,x_n) .$$

(This states only distributivity over addition; however, distributivity over all other group operations follows.)

## Examples

The classical examples are of course groups (where $\Omega$ is [[empty set|empty]]), [[rngs]] (where $\Omega$ consists of only multiplication), and [[rings]] (where $\Omega$ consists of multiplication and the nullary operator that gives the multiplicative identity).  Given a fixed [[ground ring]] $k$, the [[modules]] over $k$ form another example: each element of $k$ gives a unary multiplication operation.

The older term __group with operators__ is traditionally used for $\Omega$-groups when only unary operations are considered (as in the case of modules).

[[Philip Higgins]] discussed a more general notion of $\Omega$-group: a group $G$ endowed with a set $\Omega$ of finitary operations satisfying the condition that the neutral element of the group should form a one-element subalgebra.
$\Omega$-groups in the sense of Higgins form a [[protomodular category]], which is not in general strongly protomodular. A counterexample is provided by the category of digroups, i.e. sets with two group group structures that share the same neutral element.

Grace Orzech introduced a notion of extension in a special related setting known, somewhat opaguely, as a _category of interest_. (In the n-Lab we are trying out a perhaps more informative term_ [[category of group-based universal algebras]]_

## Remarks

The general theory of $\Omega$-groups is similar to the basics of group and ring theory, including [[normal subgroups]] / [[ideals]] / [[submodules]], [[quotient algebra|quotient]] $\Omega$-groups, Noether's [[isomorphism theorem]]s, etc. For example, the [[Jordan-Holder theorem|Jordan?Holder theorem]] holds: if there is a [[composition series]], then every two composition series are equivalent up to permutation of factors. An obvious [[horizontal categorification]] of $\Omega$-groups is also interesting.

Note that $\Omega$ is a capital Greek letter; $\omega$-[[infinity-group|group]] is rather a synonym for (for some people strict) $\infty$-[[infinity-groupoid|groupoid]] with a single object, hence nothing to do with $\Omega$-groups.

## References

* Wikipedia, [Group with operators](http://en.wikipedia.org/wiki/Group_with_operators).

* N. Bourbaki, _Algebra I_, ch. 1-3.

* [[P. J. Higgins]], "Groups with multiple operators", Proceedings of the London Mathematical Society, 1956

* E. I. Khukhro, _Local nilpotency in varieties of groups with operators_, Russ. Acad. Sci. Sbornik Mat. 78 379, 1994.
([doi](http://dx.doi.org/10.1070/SM1994v078n02ABEH003475))

* Grace Orzech, _Obstruction theory in algebraic categories I, II_, J. Pure Appl. Algebra __2__ (1972) 287-340, 315–340.

[[!redirects Omega-group]]
[[!redirects Omega-groups]]
[[!redirects $\Omega$-group]]
[[!redirects $\Omega$-groups]]

[[!redirects group with operators]]
[[!redirects groups with operators]]


omega-groupoid

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## Idea

An _$\omega$-groupoid_ is an [[∞-category]] (see there for more details) in which all [[k-morphisms]] for all $k \in \mathbb{N}$ are [[equivalences]].

This is also called an _[[∞-groupoid]]_. In the literature the term "$\omega$-groupoid" is usually reserved for [[algebraic model of higher categories|algebraic]] models instead of [[geometric model of higher categories|geometric models]].

How strict the $\omega$-category and the inverses must be can vary. Somes authors that use the term '$\omega$-groupoid' mean [[strict ∞-groupoid]] by default and speak of _weak $\omega$-groupoid_ otherwise.

## Related concepts

* [[groupoid]], [[2-groupoid]], [[3-groupoid]], [[n-groupoid]]

* [[Grothendieck-Maltsiniotis omega-groupoid]]

* [[opetopic type theory]]

[[!redirects omega-groupoids]]

[[!redirects globular omega-groupoid]]


omega-nerve

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## Definition

The **$\omega$-nerve** functor

$$
  N : \omega Cat \to sSet
$$

$$
  C \mapsto N(C)
$$

from [[∞-categories]] to [[simplicial sets]] is the functor induced by the general logic of [[nerve and realization]] from the [[orientals]]: the [[cosimplicial object|cosimplicial]] $\omega$-category

$$
  O : \Delta \to \omega Cat
  \,.
$$

So for $C$ an $\omega$-category, its $\omega$-nerve is the [[simplicial set]] whose $k$-[[simplex|simplices]] are precisely all possible images of the $k$-[[oriental]] in $C$:

$$
  N(C)_k := \omega Cat(O[k], C)
  \,.
$$

Where the $\omega$-category itself provided rules for how exactly to compose [[k-morphism]]s, its $\omega$-nerve just records all possible ways of how $(k+1)$-morphisms connect pasting diagrams of $k$-morphisms in $C$. This is however precisely the same information.

## Characterization of $\omega$-categories by their $\omega$-nerves

Accordingly, omega-nerves may be used to _define_ and identify $\omega$-categories. For instance

* the $\omega$-nerve $N(C)$ is a simplicial set in which _all_  [[horn]]s have _unique_ fillers precisely if $C$ is a 1-[[groupoid]];

* the $\omega$-nerve $N(C)$ is a simplicial set in which all _inner_ [[horn]]s have _unique_ fillers precisely if $C$ is an ordinary [[category]];

* the $\omega$-nerve $N(C)$ is a simplicial set in which _all_ [[horn]]s have any fillers precisely if $C$ is an [[∞-groupoid]] (see [[Kan complex]] for more on this);

* the $\omega$-nerve $N(C)$ is a simplicial set in which all _inner_ [[horn]]s have any fillers precisely if $C$ is an [[(∞,1)-category]].

* in full generality, a simplicial set is the $\omega$-nerve of an $\omega$-category if it is a [[weak complicial set]].

## Related concepts

* [[nerve]]

* [[Duskin nerve]]

* **$\omega$-nerve**

* [[homotopy coherent nerve]]

[[!redirects omega-nerves]]
[[!redirects ∞-nerve]]
[[!redirects ∞-nerves]]


Omega-spectrum

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## Idea

Among [[sequential spectrum|sequential]] [[pre-spectra]] $X$, the $\Omega$-spectra are those for which each structure map $X_n \to \Omega X_{n+1}$ (from each component space to the [[based loop space]] of the next component space) is a [[weak homotopy equivalence]].

(Beware that some authors require a [[homeomorphism]] instead and say "weak $\Omega$-spectrum", for the more general case).

Omega-spectra are particularly good representatives among [[pre-spectra]] of the objects of the [[stable (∞,1)-category of spectra]], hence of the [[stable homotopy category]]. For instance they are (after [[geometric realization]]) the [[fibrant objects]] of the [[Bousfield-Friedlander model structure]].

## Definition

With $\Omega$ the notation for the [[loop space]] construction (whence the name), an $\Omega$-spectrum is a sequence $E_\bullet = (E_n)_{n \in \mathbb{N}}$ of [[pointed object|pointed]] [[∞-groupoids]] ([[homotopy types]]) equipped for each $n \in \mathbb{N}$ with an [[equivalence of ∞-groupoids]]

$$
  E_n \stackrel{\simeq}{\longrightarrow} \Omega E_{n+1}
  \,.
$$

Remark: In terms of [[model category]] [[presentable (∞,1)-category|presentation]] one may also consider sequences of [[topological spaces]] equipped with _[[homeomorphisms]]_ $E_n \longrightarrow \Omega E_{n+1}$  See at _[[spectrum]]_ the section _[Omega-spectra](spectrum#OmegaSpectrum)_.

## Properties

### $\Omega$-spectrification

The inclusion of $\Omega$-spectra into all sequential pre-spectra has a [[left adjoint]], [[spectrification]]. See there for more.

## Examples

### $\Omega$-spectrification of suspension spectra
 {#CompletionOfSuspensionSpectra}

Given a [[pointed topological space]] $X$, its [[suspension spectrum]] $\Sigma^\infty X$ is far from being an $\Omega$-spectrum. The $\Omega$-spectrum that it induces (its [[spectrification]]) is given by [[free infinite loop space]] constructions:

write

$$
  Q\;\colon\; Top^{\ast/} \longrightarrow Top^{*/}
$$

for the [[free infinite loop space]] [[functor]] given as the [[colimit]]

$$
  Q X \coloneqq \underset{\longrightarrow}{\lim}_k \Omega^k \Sigma^k X
$$

over iterated [[suspension]] and [[loop space]] construction.

Then $(Q \Sigma^\infty X)_n \coloneqq Q(\Sigma^n X)$ is the $\Omega$-spectrum corresponding to the suspension spectrum of $X$.

### K-theory spectrum

The standard incarnation of the spectrum [[Brown representability theorem|representing]] complex and real [[topological K-theory]] $K U$ and $K O$ is already an $\Omega$-spectrum, due to [[Bott periodicity]]

$$
  \Omega^2(B U \times \mathbb{Z}) \simeq B U \times \mathbb{Z}
$$

and

$$
  \Omega^8 B O \simeq B O \times \mathbb{Z}
  \,.
$$

## Related concepts

* [[coordinate-free spectrum]]

* [[excisive functor]]

## References

* {#Adams74} [[Frank Adams]], part III, section 2 of _[[Stable homotopy and generalised homology]]_, 1974

* {#ElmendorfKrizMay95} [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], section 1 of  _[[Modern foundations for stable homotopy theory]]_, in [[Ioan Mackenzie James]], _[[Handbook of Algebraic Topology]]_, Amsterdam: North-Holland (1995) pp. 213–253,  ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

* {#Kochmann96} [[Stanley Kochmann]], section 3.3 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Malkiewich14} [[Cary Malkiewich]], section 2.2 of _The stable homotopy category_, 2014 ([pdf](http://math.stanford.edu/~carym/stable.pdf))

[[!redirects Omega-spectra]]

[[!redirects Omega spectrum]]
[[!redirects Omega spectra]]


Omega-spectrum type

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## Idea

This entry is about the formulation of [[Omega-spectra]] [[spectrum object|in]] [[homotopy type theory]], i.e. [[sequential spectrum types]] that are [[stable homotopy types]].

## Definition ##

In [[homotopy type theory]], a **spectrum** or $\Omega$-**spectrum** is a [[sequential spectrum type]] $E$ in which each pointed map $e_n$ is an [[equivalence in homotopy type theory|equivalence]].

$$\Spectrum \equiv \sum_{E : \PreSpectrum} \prod_{n : \mathbb{Z}} \IsEquiv (e_n)$$

## See also ##

* [[pointed type]]

* [[sequential spectrum type]]

* [[spectrum]]

* [[Omega-spectrum]] (in contrast to an [[sequential spectrum]])

* [[spectrification of a sequential spectrum type]]

* [[cohomology in homotopy type theory]]

## References ##

* [[Evan Cavallo]], §3.2 of: *Synthetic Cohomology in Homotopy Type Theory* (2015) [[pdf](https://staff.math.su.se/evan.cavallo/works/thesis15.pdf), [[Cavallo-CohomologyInHoTT.pdf:file]]]

* [[Floris van Doorn]], §5.3 in: _On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory_ (2018) [[arXiv:1808.10690](https://arxiv.org/abs/1808.10690), [web](http://florisvandoorn.com/papers/dissertation.pdf)]

[[!redirects Omega-spectrum types]]

[[!redirects spectrum type]]
[[!redirects spectrum types]]


This page has material related to the article

* {#BousfieldGugenheim76} [[Aldridge Bousfield]], [[Victor Gugenheim]]:

  _On PL deRham theory and rational homotopy type_,

  Memoirs of the AMS, vol. 179 (1976)

  [ams:memo-8-179](https://bookstore.ams.org/memo-8-179)

on [[rational homotopy theory]] via a [[model structure on dg-algebras]] (hence on objects [[formal duality|dual]] to the [[dg-Lie algebras]] originally considered in [Quillen 69](rational+homotopy+theory#Quillen69)).

\linebreak

Related references:

* [[Sergei Gelfand]], [[Yuri Manin]], _[[Methods of homological algebra]]_,  transl. from the 1988 Russian (Nauka Publ.) original, Springer 1996. xviii+372 pp. 2nd corrected ed. 2002 ([doi:10.1007/978-3-662-12492-5](https://doi.org/10.1007/978-3-662-12492-5))

Review:

* {#Hess06} [[Kathryn Hess]], _Rational homotopy theory: a brief introduction_, contribution to _[Summer School on Interactions between Homotopy Theory and Algebra](https://jdc.math.uwo.ca/summerschool/)_, University of Chicago, July 26-August 6, 2004, Chicago ([arXiv:math.AT/0604626](http://arxiv.org/abs/math.AT/0604626)), chapter in Luchezar Lavramov, [[Dan Christensen]], [[William Dwyer]], [[Michael Mandell]], [[Brooke Shipley]] (eds.), _Interactions between Homotopy Theory and Algebra_, Contemporary Mathematics 436, AMS 2007 ([doi:10.1090/conm/436](http://dx.doi.org/10.1090/conm/436))

* {#Moerman15} [[Joshua Moerman]], _Rational Homotopy Theory_, 2015 ([pdf](https://www.ru.nl/publish/pages/813282/rational_homotopy_theory.pdf), [[MoermanRationalHomotopyTheory.pdf:file]])

Generalization to [[equivariant rational homotopy theory]] ([[fundamental theorem of equivariant rational homotopy theory]]):

* {#Scull08} [[Laura Scull]], _A model category structure for equivariant algebraic models_, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 ([doi:10.1090/S0002-9947-07-04421-2](https://doi.org/10.1090/S0002-9947-07-04421-2))

\linebreak

#Contents#
* table of contents
{:toc}

## The simplicial Algebra $\nabla$

* [[polynomial differential forms on simplices]]

* [[PL de Rham complex]]

## The polynomial de Rham Theory

* [[PL de Rham theorem]]

## Multiplicative structure and the Eilenberg-Moore theorem

## A Quillen Homotopy Theory for DG Algebras

* [[model structure on connective dgc-algebras]]

## Function Spaces for DG Algebras

## The Homotopy Relation for DG Algebras

## Minimal DG Algebras

* [[minimal Sullivan model]]

## The de Rham functors and their Adjoints

* [[Quillen adjunction between simplicial sets and connective dgc-algebras]]

## The Sullivan-de Rham Equivalence Therem

* [[fundamental theorem of dg-algebraic rational homotopy theory]]

## Proof of the Sullivan-de Rham Equivalence Therem

## The Sullivan-de Rham Localization and Homotopy Theorems

* [[rationalization]] via the [[fundamental theorem of dg-algebraic rational homotopy theory]]

\linebreak

***

category: reference


On Scientific Contributions to the nLab > history


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This entry is about the article

* [[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)

  [doi:10.4310/CDM.2008.v2008.n1.a3](https://dx.doi.org/10.4310/CDM.2008.v2008.n1.a3)

  [euclid:cdm/1254748657](https://projecteuclid.org/euclid.cdm/1254748657)

whose main content is a formalization and proof of the

* [[cobordism hypothesis]].

# Contents

## Topological Field Theories and Higher Categories

* [[higher category theory]]

* [[(infinity,n)-category]]

* [[quantum field theory]]

  * [[FQFT]]

* [[n-categorical physics]]

## Formulation of the Cobordism Hypothesis

* [[complete Segal space]]

* [[cobordism]]

* [[(infinity,n)-category of cobordisms]]

* [[cobordism hypothesis]]

  * [[generalized tangle hypothesis]]

## Proof of the Cobordism Hypothesis

## Beyond the Cobordism Hypothesis

* [[factorization algebra]]

* [[Calabi-Yau object]]

* [[TCFT]]

category: reference

[[!redirects On the CLassification of Topological Field Theories]]


On the CLassification of Topological Field Theories > history

< [[On the CLassification of Topological Field Theories]]
[[!redirects On the CLassification of Topological Field Theories -- histo]]
[[!redirects On the Classification of Topological Field Theories -- history]]
[[!redirects On the CLassification of Topological Field Theories -- history]]


* [[Guillaume Brunerie]]

  **On the homotopy groups of spheres in homotopy type theory**

  [arxiv:1606.05916](https://arxiv.org/abs/1606.05916)

on [[homotopy groups of spheres in HoTT]].

> **Abstract.** The goal of this thesis is to prove that [[first stable homotopy group of spheres|$\pi_4(S^3)\simeq \mathbb{Z}/2\mathbb{Z}$]] in [[homotopy type theory]]. In particular it is a [[constructive mathematics|constructive]] and purely [[homotopy theory|homotopy-theoretic]] proof. We first recall the basic concepts of homotopy type theory, and we prove some well-known results about the [[homotopy groups of spheres]]: the computation of the homotopy groups of the [[circle]], the triviality of those of the form $\pi_k(S^n)$ with $k\lt n$, and the construction of the [[Hopf fibration]]. We then move to more advanced tools. In particular, we define the [[James construction]] which allows us to prove the [[Freudenthal suspension theorem]] and the fact that there exists a natural number $n$ such that $\pi_4(S^3)\simeq\mathbb{Z}/n\mathbb{Z}$. Then we study the [[smash product]] of spheres, we construct the [[cohomology]] ring of a space, and we introduce the [[Hopf invariant]], allowing us to narrow down the $n$ to either $1$ or $2$. The Hopf invariant also allows us to prove that all the [[groups]] of the form ${\pi_{4n-1}}(S^{2n})$ are infinite. Finally we construct the [[Gysin exact sequence]], allowing us to compute the [[cohomology]] of $\mathbb{C}P^2$ and to prove that $\pi_4(S^3)\simeq \mathbb{Z}/2\mathbb{Z}$ and that more generally $\pi_{n+1}(S^n)\simeq \mathbb{Z}/2\mathbb{Z}$ for every $n\ge 3$.

category: reference


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A classical text regarded as envisioning the modern notion of ([[differentiable manifold|differentiable]], [[smooth manifold|smooth]]) [[manifolds]] and of what came to be called [[Riemannian geometry]].

\linebreak

* [[Bernhard Riemann]]:

  \linebreak

  **Über die Hypothesen, welche der Geometrie zu Grunde liegen**

  talk  before the Göttingen Faculty (including [[Carl Gauss|Gauss]], [[Richard Dedekind|Dedekind]] & Weber)

  June 10, 1854

  \linebreak

  Reprinted with historical commentary as:

  [[Jürgen Jost]] (ed.)

  Klassische Texte der Wissenschaft

  Springer (2013)

  [doi:10.1007/978-3-642-35121-1](https://doi.org/10.1007/978-3-642-35121-1)

  \linebreak

  English translation by [[William Clifford]]:

  **On the hypotheses which underlie geometry**

  Nature **VIII** (1873) 183-184

  [pdf](http://www.cs.jhu.edu/~misha/ReadingSeminar/Papers/Riemann54.pdf), [pdf](https://www.emis.de/classics/Riemann/WKCGeom.pdf) [[Riemann-OnTheHypotheses.pdf:file]]

  \linebreak

  Reprinted with historical commentary in:

  [[Jürgen Jost]] (ed.)

  **On the Hypotheses Which Lie at the Bases of Geometry**

  Classic Texts in the Sciences

  Springer (2016)

  [doi:10.1007/978-3-319-26042-6](https://doi.org/10.1007/978-3-319-26042-6)

\linebreak

Quotes:

* On the potential breakdown of differential-geometric space at microscopic distances, reconsidered much later in the context of [[quantum gravity]]:

  > [[§III.3](https://ncatlab.org/nlab/files/Riemann-OnTheHypotheses.pdf#page=11)] *Now it seems that the empirical notions on which the metric determinations of Space are based, the concept of a solid body and a light ray, lose their validity in the infinitely small; it is therefore quite definitely conceivable that the metric relations of Space in the infinitely small do not conform to the hypotheses of geometry; and in fact, one ought to assume this as soon as it permits a simpler way of explaining phenomena.*

[[!redirects Über die Hypothesen, welche der Geometrie zu Grunde liegen]]