holomorphic symplectic manifold

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#### Symplectic geometry
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## Idea

A _holomorphic symplectic manifold_ is a [[complex manifold]] equipped with a [[holomorphic differential form|holomorphic]] [[symplectic form]].

## References

### General

Survey:

* [[Arnaud Beauville]], _Holomorphic symplectic geometry_, 2011 ([pdf](https://math.unice.fr/~beauvill/conf/Lisbon.pdf), [[BeauvilleHolomorphicSymplectic.pdf:file]])

### Rozansky-Witten theory

That [[Rozansky-Witten invariants]], originally conceived only for [[hyperkähler manifolds]], are more generally defined for holomorphic symplectic manifolds is due to

* {#Kapranov99} [[Mikhail Kapranov]], _Rozansky–Witten invariants via Atiyah classes_,  Compositio Math.  __115__ (1999), no. 1, 71--113, [MR2000h:57056](http://www.ams.org/mathscinet-getitem?mr=2000h:57056), [doi](http://dx.doi.org/10.1023/A:1000664527238), [alg-geom/9704009](http://arxiv.org/abs/alg-geom/9704009)

See also

* {#RobertsWillerton10} [[Justin Roberts]], [[Simon Willerton]], _On the Rozansky-Witten weight systems_, Algebr. Geom. Topol. 10 (2010) 1455-1519 ([arXiv:math/0602653](https://arxiv.org/abs/math/0602653))

[[!redirects holomorphic symplectic manifolds]]

[[!redirects holomorphic symplectic structure]]
[[!redirects holomorphic symplectic structures]]

[[!redirects holomorphic symplectic geometry]]
[[!redirects holomorphic symplectic geometries]]


holomorphic vector bundle

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## Idea

A _holomorphic vector bundle_ is a [[complex vector bundle]] over a [[complex manifold]] such that it admits [[transition functions]] that are [[holomorphic functions]].

## Properties

### As complex algebraic vector bundles

By the [[GAGA]]-principle, holomorphic vector bundles and more generally analytic [[coherent sheaves]] over a [[projective variety|projective]] [[smooth variety|smooth]] [[complex variety]] coincide with complex [[algebraic vector bundles]]/[[coherent sheaves]].

### As complex vector bundles with holomorphically flat connection
 {#AsComplexVectorBundlesWithHolomorphicFlatConnection}

+-- {: .num_theorem #KoszulMalgrangeTheorem}
###### Theorem
**([[Koszul-Malgrange theorem]])**

Holomorphic vector bundles over a [[complex manifold]] are equivalently [[complex vector bundles]] which are equipped with a holomorphically [[flat connection]] (hence a [[connection on a vector bundle|connection]] whose holomorphic component vanishes). Under this identification the [[Dolbeault operator]] $\bar \partial$ acting on the [[sections]] of the holomorphic vector bundle is identified with the holomorphic component of the [[covariant derivative]] of the given connection.

The analogous statement is true for generalization of vector bundles to [[chain complexes]] of [[module sheaves]] with [[coherent cohomology]].

=--

For [[complex vector bundles]] over [[complex varieties]] this statement is due to [[Alexander Grothendieck]] and ([Koszul-Malgrange 58](#KoszulMalgrange58)), recalled for instance as ([Pali 06, theorem 1](#Pali06)). It may be understood as a special case of the [[Newlander-Nirenberg theorem]], see ([Delzant-Py 10, section 6](#DelzantPy10)), which also generalises the proof to [[infinite-dimensional manifold|infinite-dimensional]] vector bundles. Over [[Riemann surfaces]], see [below](#OverRiemannSurfaces), the statement was highlighted in ([Atiyah-Bott 83](#AtiyahBott83)) in the context of the [[Narasimhan-Seshadri theorem]].

The generalization from [[vector bundles]] to  [[coherent sheaves]] is due to ([Pali 06](#Pali06)). In the genrality of [[(∞,1)-categories of chain complexes]] ([[dg-categories]]) of holomorphic vector bundles the statement is discussed in ([Block 05](#Block05)).

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###### Remark

The equivalence in theorem \ref{KoszulMalgrangeTheorem} serves to relate a fair bit of [[differential geometry]]/[[differential cohomology]] with constructions in [[algebraic geometry]]. For instance [[intermediate Jacobians]] arise in [[differential geometry]] and [[quantum field theory]] as [[moduli spaces of flat connections]] equipped with [[symplectic structure]] and [[Kähler polarization]], all of which in terms of [[algebraic geometry]] directly comes down [[moduli spaces]] of [[abelian sheaf cohomology]] with [[coefficients]] in the [[structure sheaf]] (and/or some variants of that, under the exponential exact sequence).

=--

### Over Riemann surfaces
 {#OverRiemannSurfaces}

Over [[Riemann surfaces]] holomorphic vector bundles are a central part of the theory of the [[moduli space of flat connections]]. See at _[[Narasimhan-Seshadri theorem]]_.

A key observation here is ([Atiyah-Bott 83, section 7](#AtiyahBott83)), that a $U(n)$-[[principal connection]] induces a holomorphic structure on the [[associated bundle|associated]] [[complex vector bundle]] by taking the $(0,1)$-part of the connection 1-form as the [[Dolbeault operator]]. For review of the statement and its proof see ([Evans, lecture 10](#Evans)).

## Related concepts

* [[Kodaira vanishing theorem]]

* [[Chern connection]]

* [[Kähler polarization]]

* [[stable vector bundle]]

* [[moduli space of bundles]]

* [[Deligne line bundle]]

* [[Higgs bundle]]

* [[algebraic line bundle]]

* [[holomorphic line 2-bundle]], [[holomorphic line n-bundle]]

* [[complex analytic stack]]

* [[pseudoholomorphic vector bundle]]

* [[Oka principle]]

\
## References

### General

Textbook accounts:

* {#Forstneric11} [[Franc Forstnerič]], Section 1.5 in: _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))

  > (in the context of the [[Oka principle]])

See also

* Wikipedia, _[Holomorphic vector bundle](http://en.wikipedia.org/wiki/Holomorphic_vector_bundle)_

### Relation to complex vector bundles with flat holomorphic connection

The classical statement of theorem \ref{KoszulMalgrangeTheorem} is due to [[Alexander Grothendieck]] and

* {#KoszulMalgrange58} [[Jean-Louis Koszul]], [[Bernard Malgrange]], _Sur certaine structures fibrées complexes_, arch. mat, vol IX, 1958

Over [[Riemann surfaces]] and in the context of the [[moduli space of flat connections]]:

* {#AtiyahBott83} [[Michael Atiyah]], [[Raoul Bott]], _The Yang-Mills equations over Riemann surfaces_, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 ([jstor](http://www.jstor.org/stable/37156), [lighning summary](http://math.stackexchange.com/a/295505/58526))

* {#Evans} [[Jonathan Evans]], _[Aspects of Yang-Mills theory](http://www.homepages.ucl.ac.uk/~ucahjde/yangmills.htm)_, lecture notes, ([lecture 10](http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture10.pdf), [lecture 11](http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture11.pdf), [lecture 12](http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture12.pdf))

Generalization to [[coherent sheaves]] is due to

* {#Pali06} N. Pali, _Faisceaux $\bar \partial$-cohrents sur les variété complexes_ ( _$\bar \partial$-Coherent sheaves on complex manifolds) Math.
Ann. 336 (2006), no. 3, 571–615 ([arXiv:math/0305422](http://arxiv.org/abs/math/0305422))

Further Generalization to [[chain complexes]] of holomorphic vector bundles is discussed in

* {#Block05} [[Jonathan Block]], _Duality and equivalence of module categories in noncommutative geometry I_ ([arXiv:0509284](http://arxiv.org/abs/math/0509284))

in terms of [[Lie infinity-algebroid representations]] of the holomorphic [[tangent Lie algebroid]].

Generalization to [[infinite-dimensional manifold|infinite-dimensional]] vector bundles is in

* {#DelzantPy10} Thomas Delzant, Pierre Py, _Kähler groups, real hyperbolic spaces and the Cremona group_, Compositio Math. 148, no. 1 (2012), 153--184 ([arXiv:1012.1585](http://arxiv.org/abs/1012.1585))

[[!redirects holomorphic vector bundles]]

[[!redirects holomorphic line bundle]]
[[!redirects holomorphic line bundles]]

[[!redirects holomorphic section]]
[[!redirects holomorphic sections]]


holonomic D-module

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## Idea

(Solutions of) holonomic systems of [[differential equations]] are formalized in the notion of a _holonomic D-module_. A [[D-module]] $M$ on a smooth complex analytic variety $X$ of dimension $n$ is **holonomic** if its [[characteristic variety]] is of dimension $n$. It follows that the characteristic variety of a holonomic D-module is conic and [[lagrangian submanifold|lagrangian]].

Holonomicity of D-modules is important also in [[geometric representation theory]].

## Related concepts

* [[coherent D-module]]

* [[arithmetic D-module]]

## References

Lecture notes include

* [[Joseph Bernstein]], Lecture 2, section 11 of _Algebraic theory of D-modules_ ([[BernsteinDModule.pdf:file]], [ps](http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein-dmod.ps), [dvi](http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein-dmod.dvi))

See also

* [[Masaki Kashiwara]], _On the holonomic systems of linear differential equations. II_, Invent. Math. __49__ (1978), no. 2, 121--135, [doi](http://dx.doi.org/10.1007/BF01403082)

* Bernard Malgrange, _On irregular holonomic [[D-modules]]_, Séminaires et Congrès __8__, 2004, p. 391–410, [pdf](http://emis.matem.unam.mx/journals/SC/2004/8/pdf/smf_sem-cong_8_391-410.pdf); _Équations différentielles à coefficients polynomiaux_, Progress in Math. __96__, Birkhäuser 1991. vi+232 pp.

* P. Maisonobe, C. Sabbah, _D-modules cohérents et holonomes_, Hermann, Paris 1993.

* [[V. Ginzburg|V. Ginsburg]], _Characteristic varieties and [[vanishing cycle]]s_, Inv. Math. __84__, 327–402 (1986)

[[!redirects holonomic D-modules]]


holonomic quantum field

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## Idea

The Kyōto school of mathematical physics (Jimbo, Miwa, Sato etc.) in mid 1970-s discovered a nontrivial connection between the theory of isomonodromic deformations of differential equations (and closely related [[integrable systems]]) and the theory of a special class of [[quantum field theories]] (whose construction heavily relies upon [[Clifford algebras]] and [[Clifford groups]]). This subject lives under the title **holonomic quantum fields**. The work is also relevant to the study of [[Painleve transcendent|Painlevé transcendents]].

From [JMSM 2005](#JMSM05):

> Through recent study of the problems in mathematical physics, a deep, unexpected
link has emerged: a link between the monodromy preserving deformation theory for
linear (ordinary and partial) differential equations, and a class of quantum field
operators ([1] [2] [3]). The aim of this article is to give an overview to the
present stage of development in the theory (see also [4]).

> The fruit of the above link is multifold. On the one hand it enables one to
compute exactly the $n$ point correlation functions of the field in question in a
closed form, using solutions to certain non-linear differential equations of specific
type (such as the Painlevé equations). On the other hand it provides an effective
new tool of describing the deformation theory by means of quantum field operators.
Thus it stands as a good example of the fact that not only the pure mathematics is
applied to physical problems but also the converse is true.

## References

Introduction and review:

* [[Tetsuji Miwa]], [[Michio Jimbo]], *Introduction to holonomic quantum fields*, pp. 28--36 in: _The Riemann problem, complete integrability and arithmetic applications_, Lec. Notes in Math. **925**, Springer (1982) $[$[doi:10.1007/BFb0093497](http://dx.doi.org/10.1007/BFb0093497)$]$

* {#JMSM05} [[Michio Jimbo]], [[Tetsuji Miwa]], [[Mikio Sato]], Yasuko Môri, *Holonomic Quantum Fields — The unanticipated link between deformation theory of differential equations and quantum fields —*,  In: [[Konrad Osterwalder|K. Osterwalder]] (ed.), *Mathematical Problems in Theoretical Physics*, Lecture Notes in Physics, **116** Springer (2005) $[$[doi:10.1007/3-540-09964-6_310](https://doi.org/10.1007/3-540-09964-6_310)$]$

Original articles:

* [[Mikio Sato]], [[Tetsuji Miwa]], [[Michio Jimbo]], *Studies on holonomic quantum fields, I*, Proc. Japan Acad. Ser. A Math. Sci. **53** 1 (1977) 6-10 $[$[doi:10.3792/pjaa.53.6](https://projecteuclid.org/journals/proceedings-of-the-japan-academy-series-a-mathematical-sciences/volume-53/issue-1/Studies-on-holonomic-quantum-fields-I/10.3792/pjaa.53.6.full)$]$

* [[Mikio Sato]], [[Tetsuji Miwa]], [[Michio Jimbo]], *Holonomic quantum fields I*, Publ. RIMS __14__   1 (1978) 223--267 $[$[pdf](https://www.jstage.jst.go.jp/article/kyotoms1969/14/1/14_1_223/_pdf)$]$

  * _Holonomic quantum fields II — The Riemann-Hilbert Problem_, Publ. RIMS __15__ 1 (1979)   201--278 $[$[doi:10.2977/prims/1195188429](https://doi.org/10.2977/prims/1195188429), [pdf](https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=15&iss=1&rank=5)$]$

  * _Holonomic quantum fields III_, Publ. RIMS __15__ 2 (1979)  577-629, $[$[pdf](http://www.journalarchive.jst.go.jp/jnlpdf.php?cdjournal=kyotoms1969&cdvol=15&noissue=2&startpage=577&chr=ja), [doi:10.3792/pjaa.53.153](https://projecteuclid.org/journals/proceedings-of-the-japan-academy-series-a-mathematical-sciences/volume-53/issue-5/Studies-on-holonomic-quantum-fields-III/10.3792/pjaa.53.153.full)$]$

  * _Holonomic quantum fields IV_, Publ. RIMS __15__ (1979) No. 3  pp.871-972, [pdf](http://www.journalarchive.jst.go.jp/jnlpdf.php?cdjournal=kyotoms1969&cdvol=15&noissue=3&startpage=871&chr=ja);

  * _Holonomic quantum fields V_, Publ. RIMS __16__ (1980) No. 2  pp.531-584, [pdf](http://www.journalarchive.jst.go.jp/jnlpdf.php?cdjournal=kyotoms1969&cdvol=16&noissue=2&startpage=531&chr=ja)

  * _Supplement to Holonomic quantum fields IV_, Publ. RIMS 17 (1981) No. 1  pp.137-151 [pdf](http://www.journalarchive.jst.go.jp/jnlpdf.php?cdjournal=kyotoms1969&cdvol=17&noissue=1&startpage=137&chr=ja)

See also:

* Сато М., Дзимбо М., Мива Т.  _Голономные квантовые поля_ (a collection of the reprints in Russian of the articles of Kyōto school) vol. __30__ (1983) in the series Matematika -- novoe v zarubežnoj nauke ([description](http://lib.mexmat.ru/books/6928))

One of the primary ideas stems from an observation of

* L. Onsager, Phys. Rev. __65__ (1944), 117-149, "who discovered in effect that field operators on 2-dimensional Ising lattice are
elements of a Clifford group"

[[!redirects holonomic quantum fields]]


holonomy

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## Idea

Given [[connection on a bundle]] $\nabla$ over a [[space]] $X$, its [[parallel transport]] around some loop $\gamma : [0,1] \to X$, $\gamma(0) = \gamma(1) = x_0$ yields an element

$$
  hol_\nabla(\gamma) \in G
$$

in the [[automorphism group]] of the [[fiber]] $P_{x_0}$ of the bundle. This is the **holonomy** of $\nabla$ around $\gamma$.

Fixing a connection $\nabla$ and a point $x \in X$ the collection of all elements $hol_\nabla(\gamma)$ for all loops $\gamma$ based at $x$ forms a subgroup of $G$, called the [[holonomy group]].

If the [[Levi-Civita connection]] on a [[Riemannian manifold]] $(X,g)$ has a holonomy group that is a proper subgroup of the [[special orthogonal group]] one says that $(X,g)$ is a manifold with [[special holonomy]]. (More precise would be: "with special holonomy group for the Levi-Civita connection".)

## Properties

Proposition. If on a smooth principal bundle $P\to X$ there is a connection $\nabla$ whose holonomy group is $G$ then the structure group can be reduced to $G$.

(...)

+-- {: .un_theorem}
###### Theorem
**(Ambrose-Singer)**
[[Ambrose-Singer theorem]]: the [[Lie algebra]] of the holonomy group of a [[connection on a bundle]] $\nabla$ on $X$ at a point $x \in X$ is spanned by the [[parallel transport]] $Ad_{tra_\nabla(\gamma)}(F_A(v \vee w))$ of the [[curvature]] $F_A$ evaluated on any $v \vee w \in \wedge^2 T_y X$ at $y \in X$ along any path $\gamma$ from $x \to y$.
=--

We may think of $Id + \Ad_{tra_\nabla(\gamma)}(F_A(\phi))$ as being the holonomy around the loop obtained by

1. going along $\gamma$ from $x$ to $y$

1. going around the [[infinitesimal space|infinitesimal]] parallelogram spanned by $v$ and $w$;

1. coming back to $x$ along the reverse path $\gamma$.

## Applications

(...)

## Higher holonomy

The [[higher holonomy]] (see there) of [[circle n-bundles with connection]] is given by [[fiber integration in ordinary differential cohomology]].

## Related concepts

* [[connection on a bundle]], [[connection on a 2-bundle]], [[connection on an infinity-bundle]]

* [[parallel transport]], [[higher parallel transport]]

* **holonomy**

  * [[holonomy group]]

  * [[special holonomy]]

  * [[Wilson loop]]

  * [[nonabelian Stokes theorem]]

* [[fiber integration in differential cohomology]]

  * [[fiber integration in ordinary differential cohomology]]

## References

With an eye towards application in [[mathematical physics]]:

* [[Mikio Nakahara]], Section 10.2 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

[[!redirects holonomies]]

[[!redirects Ambrose-Singer theorem]]


holonomy group

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## Idea

For $X$ a [[space]] equipped with a $G$-[[connection on a bundle]] $\nabla$ (for some [[Lie group]] $G$) and for $x \in X$ any point, the [[parallel transport]] of $\nabla$ assigns to each [[curve]] $\Gamma : S^1 \to X$ in $X$ starting and ending at $x$ an element $ hol_\nabla(\gamma) \in  G$: the [[holonomy]] of $\nabla$ along that curve.

The **holonomy group** of $\nabla$ at $x$ is the subgroup of $G$ on these elements.

If $\nabla$ is the [[Levi-Civita connection]] on a [[Riemannian manifold]] and the holonomy group is a proper subgroup $H$ of the [[special orthogonal group]], one says that $(X,g)$ is a manifold of _[[special holonomy]]_ .

## Classification of holonomy groups of affine connections

Any [[closed Lie subgroup]] of $GL(V)$ occurs as the holonomy
group of some [[affine connection]] (with [[torsion]], in general).
See Hano–Ozeki \cite{HanoOzeki}.

Holonomy groups of locally symmetric connections can be classified using [[Élie Cartan]]'s [[classification of symmetric spaces]].

For [[Levi-Civita connections]], holonomy groups were classified by [[Marcel Berger]] \cite{Berger}.

The case of torsion-free [[affine connections]] that are not locally symmetric
and are not [[Levi-Civita connections]] was treated
by Merkulov and Schwachhöfer \cite{MerkulovSchwachhofer}.
A complete list of exotic holonomy groups (for the metric and nonmetric cases)
can be found in \cite{MerkulovSchwachhofer2}.

## Related concepts

* [[connection on a bundle]]

  * [[connection on a 2-bundle]], [[connection on an infinity-bundle]]

* [[parallel transport]], [[higher parallel transport]]

* [[holonomy]]

  * **holonomy group**

  * [[special holonomy]]

## References

* {#HanoOzeki} J. Hano, H. Ozeki, _On the holonomy groups of linear connections_, Nagoya Math. J. 10, 97-100 (1956).  [doi](https://doi.org/10.1017/S0027763000000106).

* {#Berger} [[Marcel Berger]], Sur les groupes d'holonomie homogènes de variétés à connexion affine et des variétés riemanniennes.  Bulletin de la Société mathématique de France 79:null (1955), 279-330.  [doi](http://dx.doi.org/10.24033/bsmf.1464).

* {#MerkulovSchwachhofer} [[Sergei Merkulov]], Lorenz Schwachhöfer.  Classification of Irreducible Holonomies of Torsion-Free Affine Connections.  Annals of Mathematics 150:1 (1999), 77–149.  [doi](http://dx.doi.org/10.2307/121098).

* {#MerkulovSchwachhofer2} [[Sergei Merkulov]], Lorenz Schwachhöfer.  Addendum to Classification of Irreducible Holonomies of Torsion-Free Affine Connections.  Annals of Mathematics 150:3 (1999), 1177–1179.  [doi](http://dx.doi.org/10.2307/121067).

[[!redirects holonomy groups]]


holonomy groupoid

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## Idea

A _holonomy groupoid_ is a ([[topological groupoid|topological]]/[[Lie groupoid|Lie]]-) [[groupoid]] naturally associated with a [[foliation]] $ \mathcal{F} $ of a [[manifold]] $ X $. It is, in some sense, the smallest de-singularization of the [[leaf space|leaf-space]] [[quotient]] $ X/\mathcal{F} $ of the foliation, which is, in general, not a manifold itself. Every [[foliation groupoid]] of $ \mathcal{F} $ has this de-singularization property, but the holonomy groupoid is, in some sense, minimal with respect to this property.

Explicitly, given a [[foliation]] $ \mathcal{F} $ on a manifold $ X $, the _holonomy groupoid_ of $ \mathcal{F} $ has as objects the points of $ X $. Given points $ x,y $ in the same leaf, a morphism between them is the equivalence class of a path in the leaf from $ x $ to $ y $, where two paths are identified if they induce the same [[germ]] of a holonomy transformation between small transversal sections through $ x $ and $ y $. If $ x $ and $ y $ are not in the same leaf, then there is no morphism between them.

This is naturally a [[topological groupoid]] and a [[Lie groupoid]] if done right.

The **monodromy groupoid** of the foliation is obtained from this by further dividing out the [[homotopy]] between paths in a leaf.

## Definition

### Foliation Holonomy

Let $ (X,\mathcal{F}) $ be a foliated manifold with $ q = codim(\mathcal{F}) $. Let $ L $ be a leaf of $ \mathcal{F} $; let $ x,y \in L $ be two points; and let $ S $ and $ T $ be transversal sections through $ x $ and $ y $ respectively (i.e., $ S $ and $ T $ are sub-manifolds transversal to the leaves of $ \mathcal{F} $).

To a path $ \gamma: [0,1] \to L $ from $ x $ to $ y $, we assign the [[germ]] of a (partially defined) [[diffeomorphism]]

$$
{hol^{S,T}}(\gamma): (S,x) \to (T,y),
$$

called the **holonomy transformation** of the path $ \gamma $ with respect to $ S $ and $ T $, as follows:

If there exists a single foliation chart $ U $ of $ \mathcal{F} $ that contains the image of $ \gamma $, then there exists a sufficiently small open [[neighbourhood|neighborhood]] $ A $ of $ x $ in the space $ S \cap U $ for which there exists a unique smooth map $ f: A \to T $ satisfying the following conditions:

* $ f(x) = y $;

* For every $ a \in A $, the point $ f(a) $ lies in the same plaque in $ U $ as $ a $. Observe that $ f $ is a [[diffeomorphism]] onto its image.

Then define $ {hol^{S,T}}(\gamma) $ to be the [[germ]] of this diffeomorphism at $ x $:

$$
{hol^{S,T}}(\gamma) \stackrel{def}{=} {germ_{x}}(f).
$$

In general, the image of $ \gamma $ is not contained inside any single foliation chart $ U $, but as it is a compact subspace of $ X $, there exist finitely many foliation charts $ U_{1},\ldots,U_{n+1} $ and numbers $ t_{0},\ldots,t_{n+1} $ such that

* $ 0 = t_{0} \lt t_{1} \lt \cdots \lt t_{n} \lt t_{n+1} = 1 $ and

* $ \gamma([t_{i-1},t_{i}]) $ is contained in $ U_{i} $ for $ i = 1,\ldots,n+1 $.

Then arbitrarily choose transversal sections $ T_{i} $ through $ \gamma(t_{i}) $ for $ i = 1,\ldots,n $ and define

$$
{hol^{S,T}}(\gamma) \stackrel{def}{=}
{hol^{T_{n},T}} \left( \gamma|_{[t_{n},t_{n+1}]} \right) \circ
{hol^{T_{n-1},T_{n}}} \left( \gamma|_{[t_{n-1},t_{n}]} \right) \circ \cdots \circ
{hol^{T_{1},T_{2}}} \left( \gamma|_{[t_{1},t_{2}]} \right) \circ
{hol^{S,T_{1}}} \left( \gamma|_{[0,t_{1}]} \right).
$$

This definition is independent of the choice of $ U_{i} $'s, $ t_{i} $'s and $ T_{i} $'s. It only depends on the initial and final transversal sections $ S $ and $ T $.

**Proposition**

* Two homotopic paths with the same endpoints induce the same holonomy. (Note, however, that the converse is not true. Two paths with the same endpoints inducing the same holonomy may not be homotopic.)

* If $ S,S' $ are two transversal sections through $ x $ and $ T,T' $ two transversal sections through $ y $, then

$$
hol^{S',T'}(\gamma)
=
{hol^{T,T'}}(const_{y}) \circ {hol^{S,T}}(\gamma) \circ {hol^{S',S}}(const_{x}).
$$

### Holonomy Groupoid

Given a foliated manifold $ (X,\mathcal{F}) $, the **monodromy groupoid** is the [[disjoint union]] of the [[fundamental groupoids]] of the leaves of $ \mathcal{F} $, which is the groupoid having the following properties:

1. Its objects are the points of $ X $.

2. There are no morphisms between two points on different leaves.

3. The morphisms between two points on the same leaf are [[homotopy]]-classes of paths lying in the leaf joining those points.

The **holonomy groupoid** is defined analogously, where instead of identifying two paths if they are homotopic, they are identified if they induce the same holonomy as described above.

## References

The holonomy groupoid appears in

* [[Charles Ehresmann]], _Structures Feuilletées_ , Proc. 5th Canadian Math. Congress, Univ. of Toronto Press 1963, 1961, pp. 109–172.

and was studied extensively in

* H. E. Winkelnkemper, _The graph of a foliation_ , Ann. Global Anal. Geom. 1 (1983), no. 3, 51–75.

See also

* [[Marius Crainic]], [[Ieke Moerdijk]], _Foliation groupoids and their cyclic homology_, [arXiv:math/0003119](http://arxiv.org/abs/math/0003119)

* [[Ronnie Brown]], Osman Mucuk, _Foliations, locally Lie groupoids and holonomy_, [numdam](http://www.numdam.org/item?id=CTGDC_1996__37_1_61_0)

Applications to singular foliations and to Poisson manifolds in particular are in

* Iakovos Androulidakis, [[Georges Skandalis]],  _The holonomy groupoid of a singular foliation_, [arXiv:math/0612370](http://arxiv.org/abs/math/0612370)
* Iakovos Androulidakis, Marco Zambon, _Almost regular Poisson manifolds and their holonomy groupoids_, [arxiv/1606.09269](http://arxiv.org/abs/1606.09269)

[[!redirects holonomy groupoids]]

[[!redirects monodromy groupoid]]
[[!redirects monodromy groupoids]]


holonomy Lie algebra

Given a connection on a principal $G$-[[principal bundle|bundle]] $P\to M$ the __holonomy Lie algebra__ at a point $p$ in a total space principal bundle $P$ is the tangent Lie algebra of the [[holonomy group]] at $p$ of a principal $G$-bundle. By the __Ambrose-Singer theorem__ the holonomy Lie algebra is spanned as a vector subspace of the Lie algebra $\mathfrak{g}$ of $G$ spanned by all elements $\Omega_q(X,Y)$ where $q$ is a point in $P$ connected to $q$ by some horizontal path and $X,Y\in H_q P\subset T_q P$ are horizontal vectors at $q$.

## Literature

* M. M. Postnikov, _Lectures on differential geometry IV_

There are particular cases related to arrangements of hyperplanes like [[Drinfeld-Kohno Lie algebra]] and its generalizations.

* M. V. Feigin, A. P. Veselov, _$\vee$-systems, holonomy Lie algebras, and logarithmic vector fields_, Intern. Math. Res. Notices 2018, No. 7, pp. 2070-2098 [doi](https://doi.org/10.1093/imrn/rnw289)

For higher categorical analogue see

* [[Camilo Arias Abad]], [[Florian Schätz]], _Holonomies for connections with values in $L_\infty$-algebras_, Homology, Homotopy and Applications __16__:1 (2014) 89-118

category: geometry


hom type

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

\tableofcontents

## Idea

In [[dependent type theory]], a [[hom type]] on a type $A$ is a [[dependent type]] $\mathrm{hom}_A(x, y)$ which models the [[hom spaces]] of [[(infinity,1)-category|$\infty$-categories]].

## Definition

### In simplicial homotopy type theory

In [[simplicial homotopy type theory]], the hom type is defined as the [[extension type]]

$$\mathrm{hom}_A(x, y) \coloneqq \langle \Delta^1 \to A \vert_{[x,y]}^{\partial \Delta^1} \rangle$$

where $\Delta^1$ is the directed interval primitive $\mathbb{2}$ and $\partial \Delta^1$ is the [[boundary]] of $\Delta^1$.

## Related concepts

* [[simplicial homotopy type theory]]
* [[directed homotopy type theory]]
* [[identity type]]
* [[function type]]
* [[directed univalence]]
* [[Segal type]], [[Rezk type]]
* [[covariant type family]]
* [[dependent hom type]]

## References

* {#RiehlShulman17} [[Emily Riehl]], [[Michael Shulman]], *A type theory for synthetic $\infty$-categories* $[$[arXiv:1705.07442](https://arxiv.org/abs/1705.07442)$]$


hom-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Recall that in a [[category]] $C$, each pair of [[object]]s $x$ and $y$ determines a [[set]] $C(x,y)$, the [[hom-set]] of $x$ and $y$.  In a $2$-[[2-category|category]] $B$, each pair of objects determines a _category_ $B(x,y)$.  This category is the __hom-category__ of $x$ and $y$.

The [[objects]] in the hom-category $C(x,y)$ are the _[[1-morphisms]]_ in $C$ from $x$ to $y$, while the [[morphisms]] in the hom-category $C(x,y)$ are the _[[2-morphisms]]_ of $C$ that are _horizontally_ between $x$ and $y$.

As a $2$-category is [[enriched category|enriched]] over [[Cat]], a hom-category is a special case of a [[hom-object]].  But the hom-category makes sense also for the weakly enriched concept of [[bicategory]].

## Related concepts

* [[hom-object]]

* [[hom-set]]

* **hom-category**

* [[hom-space]], [[derived hom-space]]

[[!redirects hom-categories]]


hom-connection

hom-connections were introduced in

* [[Tomasz Brzeziński]], _Non-commutative connections of the second kind_,
J. Algebra Appl. __7__ (2008), no. 5, 557–-573, [MR2009m:58017](http://www.ams.org/mathscinet-getitem?mr=2459078), [arxiv/0802.0445](http://arxiv.org/abs/0802.0445), [doi](http://dx.doi.org/10.1142/S0219498808002977)

Later articles

* [[Tomasz Brzeziński]], Laiachi El Kaoutit, Christian Lomp. _Non-commutative integral forms and twisted multi-derivations_, J. Noncomm. Geom. __4__, n. 2, 2010, 281–-312, [doi](http://dx.doi.org/10.4171/JNCG/56) [arxiv/0901.2710](http://arxiv.org/abs/0901.2710) [MR2011g:58006](http://www.ams.org/mathscinet-getitem?mr=2595316)

* Tomasz Brzeziński, _Divergences on projective modules and
non-commutative integrals_, [arxiv/1010.1470](http://arxiv.org/abs/1010.1470)

* Serkan Karaçuha, Christian Lomp, _Integral calculus on quantum exterior algebras_, [arxiv/1302.5216](http://arxiv.org/abs/1302.5216)

[[!redirects hom-connections]]


hom-functor

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

#Contents#
* table of contents
{:toc}

## Definition

For $C$ a [[locally small category]], its **hom-functor** is the [[functor]]

$$
  hom : C^{op} \times C \to Set
$$

from the [[product category]] of the category $C$ with its [[opposite category]] to the category [[Set]] of [[sets]], which sends

* an [[object]] $(c, c') \in C^{op} \times C$, i.e. a pair of objects in $C$, to the [[hom-set]] $Hom_C(c,c')$ in Set, the set of [[morphisms]] $q : c \to c'$ in $C$;

* a morphism $(c,c') \stackrel{}{\to} (d,d')$, i.e. a pair of morphisms

  $$
    \array{
      c & c'
      \\
      \downarrow^{\mathrlap{f^{op}}} & \downarrow^{\mathrlap{g}}
      \\
      d & d'
    }
  $$

  in $C$ to the [[function]] $Hom_C(c,c') \to Hom_C(d,d')$ that sends

  $$
    (q : c \stackrel{}{\to} c')
    \;\;\;
    \mapsto
    \;\;\,
    \left(
    g \circ q \circ f \; : \;
    \array{
       c &\stackrel{q}{\to}& c'
       \\
       \uparrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}}
       \\
       d && d'
    }
    \right)
    \,.
  $$

+-- {: .query}
Note: when the symbol $\circ$ is used, it denotes traditional right-to-left order of composition. For those who prefer the left-to-right order, the symbol $;$ may be used in place of $\circ$. Further discussion of this should go to the nForum page [here](https://nforum.ncatlab.org/discussion/7239/realigned-arrows-for-clarification-and-fixed-composition-order/).
=--

More generally, for $V$ a [[closed monoidal category|closed]] [[symmetric monoidal category]] and $C$ a $V$-[[enriched category]], its _[[enriched hom-functor]]_ is the [[enriched functor]]

$$
  C(-,-) : C^{op} \times C \to V
$$

that sends objects $c,c' \in C$ to the [[hom-object]] $C(c,c') \in V$.

Some categories $C$ are equipped with an operation that behaves like a hom-functor, but takes values in $C$ itself

$$
  [-,-] : C^{op} \times C \to C
  \,.
$$

Such an operation is called an [[internal hom]] functor, and categories carrying this are called [[closed categories]].

### In homotopy type theory

Note: the [[HoTT book]] calls a [[category]] a "precategory" and a [[univalent category]] a "category", but here we shall refer to the standard terminology of "category" and "univalent category" respectively.

For any [[category]] $A$, we have a hom-functor
$$hom_A : A^{op} \times A \to \mathit{Set}$$
It takes a pair $(a,b):(A^{op})_0 \times A_0 \equiv A_0\times A_0$ to the set $hom_A(a,b)$. For a morphism $(f,f') : hom_{A^{op} \times A}((a,b),(a',b'))$, by definition we have $f:hom_A(a',a)$ and $f': hom_A(b,b')$, so we can define

$$(hom_A)_{(a,b),(a',b')}(f,f') \equiv (g\mapsto f' g f) : hom_A(a,b) \to hom_A(a',b')$$

## Properties

### Representable functors

Given a hom-functor $hom:C^{op}\times C\to Set$, for any object $c \in C$ one obtains a functor

$$
  h^c: C \to Set
$$

given by $h^c\coloneqq hom(c,-)$ and a functor

$$
  h_c : C^{op} \to Set
$$

given by $h_c\coloneqq hom(-,c)$, i.e. by fixing one of the arguments of $hom: C^{op} \times C \to Set$ to be $c$.

Formally this is

$$
  hom(c,-) : C \stackrel{\simeq}{\to} * \times C  \stackrel{(c,Id)}{\to} C^{op} \times C \stackrel{hom(-,-)}{\to} Set
$$

and

$$
  hom(-,c) : C \stackrel{\simeq}{\to} C^{op} \times *  \stackrel{(Id,c)}{\to} C^{op} \times C \stackrel{hom(-,-)}{\to} Set
  \,.
$$

Functors of the form $C^{op} \to Set$ are called [[presheaves]] on $C$, and functors [[natural isomorphism|naturally isomorphic]] to $hom(-,c)$ are called [[representable functors]] or **representable presheaves** on $C$.

Functors of the form $C \to Set$ are called [[copresheaves]] on $C$, and functors naturally isomorphic to $hom(c,-)$ are called co[[representable functors]] or **representable copresheaves** on $C$.

### Preservation of limits

The [[hom-functor preserves limits]] in both arguments separately. This means:

* for fixed object $c \in C$ the functor $hom(c,-) : C \to Set$ sends limit [[diagrams]] in $C$ to limit diagrams in $Set$;

* for fixed object $c' \in C$ the functor $hom(-,c') : C^{op} \to Set$ sends limit diagrams in $C^{op}$ -- which are [[colimit]] [[diagrams]] in $C$! -- to limit diagrams in $Set$.

For instance for

$$
  \array{
     y \times_x z &\to& y
     \\
     \downarrow && \downarrow
     \\
     z &\to& x
  }
$$

a [[pullback]] [[diagram]] in $C$ and for $c \in C$ any object, the induced diagram

$$
  \array{
     Hom_C(c,y) \times_{Hom_C(c,x)} Hom_C(c,z)\simeq & Hom_C(c,y \times_x z)  &\to& Hom_C(c,y)
     \\
     & \downarrow && \downarrow
     \\
     & Hom_C(c,z) &\to& Hom_C(c,x)
  }
$$

in [[Set]] is again a pullback diagram. A moment of reflection shows that this statement is _equivalent_ to the very definition of limit.

### Relation to profunctors

The hom-functor $hom : C^{op}\times C\to Set$ is also the identity [[profunctor]] $1_C: C ⇸ C$.

One way to see this is to notice that its [[adjunct]]

$$
  C \to [C^{op}, Set]
$$

under the [[internal hom]] [[adjunction]] in the 1-category [[Cat]] is the functor

$$
  C \stackrel{id}{\to} C \stackrel{j}{\to} [C^{op}, Set]
  \,,
$$

where $j$ is the [[Yoneda embedding]]. Profunctors $\mathbf{F} : C^{op} \times C \to Set$ whose hom-adjunct is of the form $C \stackrel{F}{\to} C \stackrel{j}{\to} [C^{op}, Set]$ for $F$ an ordinary functor are those in the inclusion of these ordinary functors into profunctors. So the hom-functor is the image of the identity functor under this inclusion.

## Examples

...

## Related concepts

* [[hom-set]], [[hom-object]]

* [[hom-functor]]

* [[enriched hom-functor]], [[internal hom-functor]]

* [[derived hom-functor]]

[[!include homotopy-homology-cohomology]]

* [[internal hom]], [[enriched hom]], [[derived hom space]], [[Ext]]

[[!redirects hom-functors]]

[[!redirects hom functor]]
[[!redirects hom functors]]

[[!redirects hom]]
[[!redirects Hom]]


hom-functor preserves limits

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

#Contents#
* table of contents
{:toc}

## Idea

One of the basic facts of [[category theory]] is that the [[hom-functor]] on a [[category]] $\mathcal{C}$ [[preserved limit|preserve]] [[limits]] in both [[variables]] separately (remembering that a limit in the first variable, due to contravariance, is actually a [[colimit]] in $\mathcal{C}$).

## Statement

### Ordinary hom-functor

+-- {: .num_prop #HomFunctorPreservesLimits}
###### Proposition
**([[hom-functor]] [[preserved limit|preserves]] [[limits]])**

Let $\mathcal{C}$ be a [[category]] and write

$$

  Hom_{\mathcal{C}}
   \;\colon\;
  \mathcal{C}^{op} \times \mathcal{C}
   \longrightarrow
  Set
$$

for its [[hom-functor]]. This [[preserved limit|preserves]] [[limits]] in both its arguments (recalling that a limit in the [[opposite category]] $\mathcal{C}^{op}$ is a [[colimit]] in $\mathcal{C}$).

More in detail, let $X_\bullet \colon \mathcal{I} \longrightarrow \mathcal{C}$ be a [[diagram]]. Then:

1. If the [[limit]] $\underset{\longleftarrow}{\lim}_i X_i$ exists in $\mathcal{C}$ then for all $Y \in \mathcal{C}$ there is a [[natural isomorphism]]

   $$
     Hom_{\mathcal{C}}\left(Y, \underset{\longleftarrow}{\lim}_i X_i \right)
     \simeq
     \underset{\longleftarrow}{\lim}_i \left( Hom_{\mathcal{C}}\left( Y, X_i \right) \right)
      \,,
   $$

   where on the right we have the limit over the diagram of [[hom-sets]] given by

   $$
     Hom_{\mathcal{C}}(Y,-) \circ X \;\colon\; \mathcal{I} \overset{X}{\longrightarrow} \mathcal{C} \overset{Hom_{\mathcal{C}}(Y,-)  }{\longrightarrow} Set\,.
   $$

1. If the [[colimit]] $\underset{\longrightarrow}{\lim}_i X_i$ exists in $\mathcal{C}$ then for all $Y \in \mathcal{C}$ there is a [[natural isomorphism]]

   $$
     Hom_{\mathcal{C}}\left(\underset{\longrightarrow}{\lim}_i X_i ,Y\right)
     \simeq
     \underset{\longleftarrow}{\lim}_i \left( Hom_{\mathcal{C}}\left( X_i , Y\right) \right)
      \,,
   $$

   where on the right we have the limit over the diagram of [[hom-sets]] given by

   $$
     Hom_{\mathcal{C}}(-,Y) \circ X \;\colon\; \mathcal{I}^{op} \overset{X}{\longrightarrow} \mathcal{C}^{op} \overset{Hom_{\mathcal{C}}(-,Y)  }{\longrightarrow} Set\,.
   $$

=--

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

We give the proof of the first statement. The proof of the second statement is [[formal dual|formally dual]].

First observe that, by the very definition of [[limit|limiting]] [[cones]], maps out of some $Y$ into them are in natural bijection
with the set $Cones\left(Y, X_\bullet \right)$ of cones over the diagram $X_\bullet$ with tip $Y$:

$$
  Hom\left(
    Y,
    \underset{\longleftarrow}{\lim}_{i} X_i
  \right)
    \;\simeq\;
  Cones\left(
    Y, X_\bullet
  \right)
  \,.
$$

Hence it remains to show that there is also a natural bijection like so:

$$
  Cones\left(
    Y,
    X_\bullet
  \right)
    \;\simeq\;
  \underset{\longleftarrow}{\lim}_{i} \left( Hom(Y,X_i) \right)
  \,.
$$

Now, again by the very definition of limiting cones, a single element in the limit on the right is equivalently a cone of the form

$$
  \left\{
  \array{
    && \ast
    \\
    & {}^{\mathllap{const_{p_i}}}\swarrow && \searrow^{\mathrlap{const_{p_j}}}
    \\
    Hom(Y,X_i)
     && \underset{X_\alpha \circ (-)}{\longrightarrow} &&
    Hom(Y,X_j)
  }
  \right\}_{i, j \in Obj(\mathcal{I}), \alpha \in Hom_{\mathcal{I}}(i,j) }
  \,.
$$

This is equivalently for each object $i \in \mathcal{I}$ a choice of morphism $p_i \colon Y \to X_i$ , such that  for each pair of objects $i,j \in \mathcal{I}$
and each $\alpha \in Hom_{\mathcal{I}}(i,j)$ we have $X_\alpha \circ p_i = p_j$. And indeed, this is precisely the characterization of an element in the
set $  Cones\left( Y, X_{\bullet} \right)$.

=--

### Internal hom-functor
 {#InternalHomFunctor}

+-- {: .num_prop #InternalHomPreservesLimits}
###### Proposition
**(internal hom-functor preserves limits)**

Let $\mathcal{C}$ be a [[symmetric monoidal category|symmetric]] [[closed monoidal category]] with [[internal hom]]-[[bifunctor]] $[-,-]$. Then this bifunctor preserves [[limits]] in the second variable, and sends colimits in the first variable to limits:

$$
  [X, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Y(j)]
  \;\simeq\;
  \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [X, Y(j)]
$$

and

$$
  [\underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j),X]
  \;\simeq\;
  \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j),X]
$$

=--

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

For $X \in \mathcal{C}$ any object, $[X,-]$ is a [[right adjoint]] by definition, and hence preserves limits as _[[adjoints preserve (co-)limits]]_.

For the other case, let $Y \;\colon\; \mathcal{L} \to \mathcal{C}$ be a [[diagram]] in $\mathcal{C}$, and let $C \in \mathcal{C}$ be any object. Then there are isomorphisms

$$
  \begin{aligned}
    Hom_{\mathcal{C}}(C, [ \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X ] )
    & \simeq
    Hom_{\mathcal{C}}( C \otimes \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X )
    \\
    & \simeq
    Hom_{\mathcal{C}}( \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} (C \otimes Y(j)), X )
    \\
    & \simeq
    \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim}
    Hom_{\mathcal{C}}( (C \otimes Y(j)), X )
    \\
    & \simeq
    \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim}
    Hom_{\mathcal{C}}( C, [Y(j), X] )
    \\
    & \simeq
    Hom_{\mathcal{C}}( C, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] )
  \end{aligned}
$$

which are [[natural isomorphism|natural]] in $C \in \mathcal{C}$, where we used that the ordinary [[hom-functor]] respects (co)limits as shown (see at _[[hom-functor preserves limits]]_), and that the [[left adjoint]] $C \otimes (-)$ preserves colimits (see at _[[adjoints preserve (co-)limits]]_).

Hence by the [[fully faithful functor|fully faithfulness]] of the [[Yoneda embedding]], there is an isomorphism

$$
  \left[ \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X \right]
  \overset{\simeq}{\longrightarrow}
  \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X]
  \,.
$$

=--

## Related statements

* [[limits preserve limits]]

* [[adjoints preserve (co-)limits]]

* [[limits commute with limits]]

* [[limits of presheaves are computed objectwise]]

[[!redirects hom-functors preserve limits]]

[[!redirects internal hom-functor preserves limits]]
[[!redirects internal hom-functors preserve limits]]


hom-group

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

#Contents#
* table of contents
{:toc}

## Idea

A [[hom-object]] in the case of [[enriched category|enrichment]] over [[groups]]
(often: [[abelian groups]] for an [[Ab-enriched category]]).

## Examples

* The abelian hom-groups of the [[stable homotopy category]] are [[cohomology groups]] in [[Whitehead-generalized cohomology theories]].

## Related concepts

* [[hom-set]]

* [[additive category]]

[[!redirects hom-groups]]

[[!redirects hom group]]
[[!redirects hom groups]]


hom-groupoid

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

# Contents
* table of contents
{:toc}

## Idea

A $V$-[[enriched category]] $C$, for $V = $ [[Grpd]] (the [[category]] of [[groupoids]]), has for every ordered pair $x,y$ of [[object]]s a [[groupoid]] $C(x,y)$ of [[morphism]]s between $x$ and $y$. This [[hom-object]] is hence a **hom-groupoid** in this case.

For this reason such a category may be thought of as a [[locally groupoidal 2-category]], or [[(2,1)-category]].

## Examples

* For $C = $ [[Grpd]] itself, the hom-groupoids are the [[functor categories]] between two groupoids.

* For $C$ any [[small category]], the [[(2,1)-presheaf]]-category $[C^{op},Grpd]$ has as hom-groupoid $[C^{op}, Grpd](A,B)$  the groupoid of [[pseudonatural transformation]]s and [[modification]]s between the [[pseudo-functor]]s $A, B$.

[[!redirects hom-groupoid]]
[[!redirects hom-groupoids]]
[[!redirects hom groupoid]]
[[!redirects hom groupoids]]


hom-object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An ordinary [[locally small category]] $\mathcal{C}$ has for any ordered pair of  [[object]]s $x,y$ a [[hom-set]] $\mathcal{C}(x,y)$—an object in the category [[Set]].

For $\mathcal{C}$ more generally an [[enriched category]] over a [[closed monoidal category]] $\mathcal{V}$, there is --  by definition -- for all $x,y$ an _object_ $\mathcal{C}(x,y) \in obj V$ that plays the role of the "collection of morphisms" from $x$ to $y$.

Accordingly, if here $\mathcal{V}$ is a category of "spaces" of sorts, then one also speaks of a _hom space_.
For instance if $\mathcal{V}$ =[[vector spaces]] such that $\mathcal{C}$ is a [[linear category]], then hom-spaces are vector spaces.

Or if $\mathcal{V}$=[[topological spaces]] such that $\mathcal{C}$ is [[topologically enriched category]], then hom-spaces are [[topological spaces]]. If this last example is regarded in the context of [[homotopy theory]], then one may consider just the [[homotopy type]] of these topological spaces, which is equivalently modeled as a [[simplicial set]] ([[Kan complex]]) and thought of as an [[infinity-groupoid]]. In all these cases people still often speak of "hom spaces", but see at _[[derived hom space]]_.

## Examples

* The category Prop of [[proposition]]s is enriched over itself. Hence for any two proposition $a,b$, there is a hom-proposition $Prop(A,B)$. This is the [[implication]] proposition $a \implies b$.

* The category [[Set]] of [[set]]s is enriched over itself. Hence for any two sets $A,B$, there is a hom-set $Set(A,B)$. This is the [[function set]] $A \to B$.

* The category [[Grpd]] of [[groupoid]]s is enriched over itself. Hence for any two groupoids $A,B$, there is a [[hom-groupoid]] $Grpd(A,B)$. This is the [[functor category]] $Func(A,B)$.

* The category [[Pos]] of [[poset]]s is enriched over itself. Hence for any two posets $A,B$, there is a hom-poset $Pos(A,B)$, the poset of [[monotone]]s.

## Related concepts

* [[enriched hom-functor]]

* **hom-object**

* [[hom-set]], [[hom-group]], [[hom-category]]

* [[hom-functor]]

* [[derived hom-space]]

* [[hom-functor]]

* [[enriched hom-functor]], [[internal hom-functor]]

* [[derived hom-functor]]

[[!include homotopy-homology-cohomology]]

[[!redirects hom-objects]]
[[!redirects hom object]]
[[!redirects hom objects]]

[[!redirects enriched hom]]

[[!redirects hom space]]
[[!redirects hom spaces]]

[[!redirects hom-space]]
[[!redirects hom-spaces]]

[[!redirects object of morphisms]]
[[!redirects objects of morphisms]]


hom-object in a quasi-category

<div class="rightHandSide toc">
[[!include quasi-category theory contents]]
</div>

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

## Idea

Between any two [[objects]] $x,y$ in an [[(∞,1)-category]] $C$ there is an
[[∞-groupoid]] of morphisms. It is well-defined up to equivalence.
When the $(\infty,1)$-category is incarnated as a [[quasi-category]], there are
several explicit ways to extract representatives of this [[hom-object]].

## Definition

Let $C$ be a [[quasi-category]] (or any [[simplicial set]]), and $x,y \in C_0$
any two objects. Then write

$$
  Hom_C(x,y) := [\tau(C)(x,y)] \in Ho(sSet_{Quillen}) \simeq Ho(\infty Grpd)
  \,,
$$

where

* $\tau$ is the [[left adjoint]] to the [[homotopy coherent nerve]];

* $\tau(C)$ is accordingly the [[simplicially enriched category]] incarnation of $C$,

* $\tau(C)(x,y)$ is the [[sSet]]-[[hom-object]] of that $sSet$-[[enriched category]];

* $[\tau(C)(x,y)]$ is the equivalence class of this object.

This defines $Hom_C(x,y)$ as an equivalence class of $\infty$-groupoids, but
at the same time defines a particular representative: if $C$ is a [[quasi-category]] then
$\tau(C)(x,y)$ is a [[Kan complex]] that represents this class.

This is useful for many purposes, but $\tau(C)$ is usually hard to compute explicitly.
The following three other definitions of representatives of $Hom_C(x,y)$ are often useful.

**Definition**

For $C$ and $x,y$ as above, write

$$
  Hom_C^{L R}(x,y) := \{x\} \times_C C^{\Delta[1]} \times_C \{y\}
$$

for the [[pullback]]

$$
  \array{
    \{x\} \times_C C^{\Delta[1]} \times_C \{y\} &\to& C^{\Delta[1]}
    \\
    \downarrow && \downarrow^{\mathrlap{d_1 \times d_0}}
    \\
    \{x\} \times \{y\}
    &\to&
    C \times C
  }
$$

in [[sSet]] of the [[path object]] $C^{\Delta[1]}$ (the
cartesian [[internal hom]] in [[sSet]] with the 1-[[simplex]] $\Delta[1]$) .

Write

$$
  Hom^R_C(x,y) \in sSet
$$

for the simplicial set whose $n$-[[simplex|simplices]] are defined to be those morphisms
$\sigma : \Delta[n+1] \to C$ such that the restriction to $\Delta\{0, \cdots, n\}$
is the constant map to $x$ and the restriction to $\Delta\{n+1\}$ is the map to $y$.

Analogously, write

$$
  Hom^L_C(x,y) \in sSet
$$

for the simplicial set whose $n$-simplices are morphisms $\Delta[n+1] \to X$ which restrict to
$x$ on $\{0\}$ and are constant on $y$ when restricted to $\{1, \cdots, n+1\}$.

**Remark** The 1-cells in $Hom_C^R(x,y)$, $Hom_C^L(x,y)$ and $Hom_C^{L R}(x,y)$
are 2-[[globes]] in $C$. The 2-cells are commuting squares of vertical composites
of 2-globes forming a 3-globe.

Equivalently this may be understood in terms of fibers of [[over quasi-categories]].

Recall that for $p : K \to C$ a morphism, we have the [[over quasi-category]] $C_{/p}$
defined by

$$
  (C_{/p})_n := Hom(\Delta[n],C^{/p}) :=
  Hom_p(\Delta[n] \star K, C)
  \,,
$$

where on the right we have the set of morphisms in $sSet$ out of the
[[join of simplicial sets]] that restrict on $K$ to $p$.

This comes with the canonical projection $C^{p/} \to C$, which sends
$(\Delta[n] \star K \to C)$ to the restriction
$(\Delta[n] \to \Delta[n] \star K \to C)$.

There is also the other, equivalent, definition $C^{/p}$ of [[over quasi-category]],
defined using the other, equivalent, definition $\diamondsuit$
of [[join of quasi-categories]] by

$$
  (C^{/p})_n := Hom_{K/sSet}( \Delta[n] \diamondsuit K, C)
  \,.
$$

**Obervation**

We have

$$
  Hom^R_C(x,y) = C_{/{y}} \times_C \{x\}
  \,,
$$

where on the right we have the [[pullback]] in [[sSet]] in the diagram

$$
  \array{
    C_{/{y}} \times_C \{x\} &\to& C_{/y}
    \\
    \downarrow && \downarrow
    \\
    \{x\} &\to& C
  }
$$

and the equality sign denotes an [[isomorphism]] in [[sSet]].

And we have

$$
  Hom_C^{L R}(x,y) = C^{/y} \times_C \{x\}
$$

**Proof** For the first statement
use the identification of $\Delta[n+1]$ with the [[join of simplicial sets]]
$\Delta[n] \star \Delta[0]$, as described there.

For the second statement use that $\Delta[n] \diamondsuit \Delta[0]$
is the colimit $\Delta[n+1]$ in

$$
  \array{
  && \Delta[n] \times \Delta[0] &\to & \Delta[0]
  \\
  && \downarrow &\downarrow&
  \\
  \Delta[n] \times \Delta[0] &\to& C \times \Delta[0] \times \Delta[1] &\to& \Delta[n+1]
  \\
  \downarrow && \downarrow && \downarrow
  \\
  \Delta[n] &\to& C \times \Delta[1] &\to& \Delta[n+1]
  }
  \,,
$$

so that

$$
  C^{/y} = C^{\Delta[1]} \times_C \{y\}
$$

because

$$
  (C^{/y})_n = Hom_{\Delta[0]/sSet}(
    \Delta[n]\times \Delta[1] \coprod_{\Delta[1] \times \Delta[0]} \Delta[0], C)
    =
    Hom(\Delta[n] \times \Delta[1], C) \times_{Hom(\Delta[1],C)} \{y\}
    = (C^{\Delta[1]} \times \{y\})_n
    \,.
$$

**Remark** One advantage of the representative $\tau(C)(c,y)$ of $Hom_C(x,y)$ is that,
by the fact that $\tau(C)$ is an [[sSet]]-[[enriched category]], there is a strict
composition operation

$$
  \tau(C)(x,y) \times \tau(C)(y,z) \to \tau(C)(x,z)
  \,.
$$

This is not available for the $Hom_C^R(x,y)$ and $Hom_C^L(x,y)$

## Properties

**Proposition**

If the simplicial set $C$  is a [[quasi-category]], then $Hom_C^R(x,y)$ is a
[[Kan complex]].

**Proof**

This is [[Higher Topos Theory|HTT, prop 1.2.2.3]].

From the definition it is clear that $Hom_C^R(x,y)$ has fillers for all inner and
right outer horns $\Lambda[n]_{1 \leq i \leq n}$, because these yield inner horns in
$\Delta[n+1] = \Delta[n] \star \Delta[0]$. The claim follows then with the fact that
every [[right fibration]] over the point is a Kan complex, as described there.

**Proposition**

If $C$ is a [[quasi-category]] then the canonical inclusions

$$
  Hom_C^R(x,y) \to Hom_C^{L R}(x,y) \leftarrow Hom_C^L(x,y)
$$

are [[homotopy equivalences]] of [[Kan complexes]].

**Proof**

This is [[Higher Topos Theory|HTT, cor. 4.2.1.7]].

As described at [[join of quasi-categories]] the canonical morphism
$C_{/y} \to C^{/y}$ is an equivalence of quasi-categories. So
for the statement for $Hom_C^R(x,y)$ it suffices to
show that this induces an equivalence of fibers over $C$.
This follows from the fact that both $C_{/y} \to C$ and $C^{/y} \to C$
are [[Cartesian fibrations]].

See [[Cartesian fibrations]] for these statements. This is
[[Higher Topos Theory|HTT, prop. 3.3.1.5. (2)]].

The statement for $Hom_C^L(x,y)$ follows dually.


hom-set

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

#Contents#
* table of contents
{:toc}

## Definition

### For locally small categories

Given [[object|objects]] $x$ and $y$ in a [[locally small category]], the __hom-set__ $hom(x,y)$ is the collection of all morphisms from $x$ to $y$.  In a [[closed category]], the hom-set may also be called the __external hom__ to distinguish it from the [[internal hom]].

* We say a category is [[locally small category|locally small]] if this collection is always actually a set instead of a proper class, or in the context of a [[Grothendieck universe]]: if this set is a $U$-small set.  This holds for many examples, but not for all. For example, if $C$ and $D$ are not [[small category|small categories]], then the [[functor category]] $C^D$ (whose morphisms are [[natural transformation]]s) will not be locally small.

### For enriched categories

For a category $C$ [[enriched category theory|enriched over]] a category $V$, the "hom-set" $C(x,y)$ is an object of $V$, the [[hom-object]].

### For internal categories

For $C = (C_0, C_1, s,t,e, c)$ an [[internal category]], the [[generalized element|generalized]] objects of $C$ are morphisms $x: X \to C_0$ and $y: Y \to C_0$, and the "hom-set" becomes the [[pullback]] $C(x,y)$ in

$$ \array{
C(x,y)     & \to          & Y            \\
\downarrow & \searrow     &              & \searrow^{y}    \\
X          &              & C_1          & \stackrel{t}\to & C_0 \\
           & \searrow^{x} & \downarrow_s \\
           &              & C_0
} $$

In particular, in a category with a [[terminal object|terminal]] [[generator]] $*$, we may identify morphisms $x,y: * \to C_0$ with [[global element|global]] objects of $C$ and form $C(x,y)$ as above.

## Related concepts

* [[hom-object]]

* **hom-set**, [[hom-group]]

  [[heteromorphism]]

* [[hom-category]]

* [[hom-space]]

* [[derived hom-space]]

* [[hom-functor]]

* [[enriched hom-functor]], [[internal hom-functor]]

* [[derived hom-functor]]

## References

Textbook accounts:

* [[Saunders MacLane]], §I.8 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

[[!redirects homsets]]
[[!redirects homset]]
[[!redirects hom-sets]]
[[!redirects hom sets]]
[[!redirects external hom]]

[[!redirects hom set]]

[[!redirects set of morphisms]]
[[!redirects sets of morphisms]]


hom-space > history

> see _[[hom-object]]_


homeomorphism

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

#Contents#
* table of contents
{:toc}

## Definition

A **homeomorphism** (also spelt 'homoeomorphism' and 'homœomorphism' but *not* '[[homomorphism]]') is an [[isomorphism]] in the [[category]] [[Top]] of [[topological spaces]].

That is, a homeomorphism $f : X \to Y$ is a [[continuous map]] of [[topological spaces]] such that there is an [[inverse]] $f^{-1}: Y \to X $ that is also a continuous map of topological spaces.  Equivalently, $f$ is a [[bijection]] between the underlying sets such that both $f$ and its inverse are continuous.

For more see at _[[Introduction to Topology -- 1]]_ the section _[Homeomorphisms](Introduction+to+Topology -- 1#Homeomorphisms)_.

The term 'homeomorphism' is also applied to isomorphisms in categories that generalise $Top$, such as the category of [[convergence spaces]] and the category of [[locales]].

## Counter-examples

Beware that a continous [[bijection]] is not necessarily a homeomorphism; that is, $Top$ is not a [[balanced category]].

For example $\exp(2\pi i(-)) [0,1) \to S^1$ is continuous and the underlying function of sets is a [[bijection]], but the inverse function at the level of sets is not continuous.

But see prop. \ref{HomeoContinuousOpenBijection}.

## Properties

+-- {: .num_prop #HomeoContinuousOpenBijection}
###### Proposition
**([[homeomorphisms]] are the [[continuous function|continuous]] and [[open map|open]] [[bijections]])

Let $f \;\colon\; (X, \tau_X) \longrightarrow (Y,\tau_Y)$ be a [[continuous function]]
between [[topological spaces]]. Then the following are equivalence:

1. $f$ is a [[homeomorphism]];

1. $f$ is a [[bijection]] and an [[open map]];

1. $f$ is a [[bijection]] and a [[closed map]].

=--

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

It is clear from the definition that a homeomorphism in particular has to be a bijection.
The condition that the [[inverse function]] $Y \leftarrow X \colon g$ be continuous means
that the [[pre-image]] function of $g$ sends open subsets to open subsets. But by $g$ being the inverse to
$f$, that pre-image function is equal to $f$, regarded as a function on subsets:

$$
  g^{-1} = f \;\colon\; P(X) \to P(Y)
  \,.
$$

Hence $g^{-1}$ sends opens to opens precisely if $f$ does, which is the case precisely if $f$ is an open map,
by definition.
This shows the equivalence of the first two items. The equivalence between the first and the third follows
similarly via prop. \ref{ClosedSubsetContinuity}.

=--

## Related concepts

* [[homeomorphism invariant]]

* [[homotopy equivalence]]

* [[weak homotopy equivalence]]

## References

The terminology "homeomorphism" is due to:

* [[Henri Poincaré]], *[[Analysis Situs]]*, Journal de l'École Polytechnique. (2). 1: 1–123 (1895) ([gallica:12148/bpt6k4337198/f7](https://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7), Engl: [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf), [[Stillwell_AnalysisSitus.pdf:file]])

[[!redirects homeomorphisms]]
[[!redirects homœomorphism]]
[[!redirects homœomorphisms]]
[[!redirects homoeomorphism]]
[[!redirects homoeomorphisms]]

[[!redirects homeomorphic]]
[[!redirects homœomorphic]]
[[!redirects homoeomorphic]]

[[!redirects homeomorphism class]]
[[!redirects homeomorphism classes]]


HomePage

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

+-- {: .query}
$\,$

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\tableofcontents

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{#purpose}

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[[!redirects HomePage]]
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[[!redirects homepage]]
[[!redirects home page]]
[[!redirects nLab]]


HOMFLY-PT polynomial

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

# The HOMFLY-PT Polynomial
* tic
{: toc}

## Idea

The HOMFLY-PT polynomial is a [[knot]] and [[link]] [[knot invariants|invariant]].  Confusingly, there are several variants depending on exactly which relationships are used to define it.  All are related by simple substitutions.

## Definition

To compute the HOMFLY-PT polynomial, one starts from an [[oriented link diagram]] and uses the following rules:

1. $P$ is an isotopy invariant (thus, unchanged by [[Reidemeister moves]]).
1. $P(\text{unknot}) = 1$
1. Let $L_+$, $L_-$, and $L_0$ be links which are the same except for one part where they differ according to the diagrams below.  Then, depending on the choice of variables:

   1. $l \cdot P(L_+) + l^{-1} \cdot P(L_-) + m \cdot P(L_0) = 0$.
   1. $a \cdot P(L_+) - a^{-1} \cdot P(L_-) = z \cdot P(L_0)$.  (Sometimes $\nu$ is used instead of $a$)
   1. $\alpha^{-1} \cdot P(L_+) - \alpha \cdot P(L_-) = z \cdot P(L_0)$.
   1. Using *three* variables: $x \cdot P(L_+) + y \cdot P(L_-) + z \cdot P(L_0) = 0$.

   $$
   \begin{array}{ccc}
   \begin{svg}[[!include SVG skein positive crossing]]\end{svg} &
   \begin{svg}[[!include SVG skein negative crossing]]\end{svg} &
   \begin{svg}[[!include SVG skein no crossing]]\end{svg} \\
   L_+ & L_- & L_0
   \end{array}
   $$

From the rules, one can read off the relationships between the different formulations:

1. $y = \alpha = a^{-1}$
1. $x = - \alpha^{-1} = -a$
1. $a = - i l$, $l = i a$
1. $z = i m$, $m = - i z$.

## Properties

The HOMFLY polynomial generalises both the [[Jones polynomial]] and the [[Alexander polynomial]] (equivalently, the [[Conway polynomial]]).

* To get the [[Jones polynomial]], make one of the following substitutions:

  1. $a = q^{-1}$ and $z = q^{1/2} - q^{-1/2}$
  1. $\alpha = q$ and $z = q^{1/2} - q^{-1/2}$
  1. $l = i q^{-1}$ and $m = i (q^{-1/2} - q^{1/2})$

* To get the [[Conway polynomial]], make one of the following substitutions:

  1. $a = 1$
  1. $\alpha = 1$
  1. $l = i$, $m = -i z$

* To get the [[Alexander polynomial]], make one of the following substitutions:

  1. $a = 1$, $z = q^{1/2} - q^{-1/2}$
  1. $\alpha = 1$, $z = q^{1/2} - q^{-1/2}$
  1. $l = i$, $m = i (q^{-1/2} - q^{1/2})$

## References

See the [wikipedia page](http://en.wikipedia.org/wiki/HOMFLY_polynomial) for the origin of the name.

Some fairly elementary discussion of the HOMFLY polynomial is given in introductory texts such as

* [[N. D. Gilbert]] and [[T. Porter]], Knots and Surfaces, Oxford U.P., 1994.

The original work was published as

*   [[P. Freyd]], [[D. Yetter]], J.  Hoste,  W.B.R. Lickorish, K. Millett, and [[A. Ocneanu]]. (1985). _A New Polynomial Invariant of Knots and Links_ Bulletin of the American Mathematical Society 12 (2): 239–246.

More recent work includes:

* A.Mironov, A.Morozov, An.Morozov, _Character expansion for HOMFLY polynomials. I. Integrability and difference equations_, [arxiv/1112.5754](http://arxiv.org/abs/1112.5754)
* Hugh Morton, Peter Samuelson, _The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra_, [arxiv/1410.0859](http://arxiv.org/abs/1410.0859)

category: knot theory

[[!redirects HOMFLY-PT polynomial]]
[[!redirects homfly-pt polynomial]]
[[!redirects HOMFLY polynomial]]
[[!redirects homfly polynomial]]
[[!redirects HOMFLY-PT]]
[[!redirects homfly-pt]]
[[!redirects HOMFLY]]
[[!redirects homfly]]


homogeneous C-star-algebra

## Idea

A [[C*-algebra]] is called __$n$-homogeneous__ (or n-dimensionally homogeneous)
if each irreducible $C^\star$-representation of the algebra is $n$-dimensional.

## Related entries

* [[Azumaya algebra]]
* [[Fell bundle]]

## References

* Jun Tomiyama, Masamichi Takesaki, _Applications of fibre bundles to the certain class of $C^\ast$-algebras_, Tohoku Math. J. (2) 13:3 (1961) 498-522 [euclid](http://projecteuclid.org/euclid.tmj/1178244253) [doi](http://dx.doi.org/10.2748/tmj/1178244253)

The comparison of the Artin's theorem on characterization of [[Azumaya algebra]]s and Tomiyama-Takesaki's theorem on $n$-homogeneous $C^\ast$-algebras is in chapter 9 of

* Edward Formanek, _Noncommutative invariant theory_, in: Group actions on rings (Brunswick, Maine, 1984), 87–119, Contemp. Math. 43, Amer. Math. Soc. 1985 [doi](http://dx.doi.org/10.1090/conm/043)

category: operator algebras

[[!redirects homogeneous C*-algebra]]
[[!redirects homogeneous C*-algebras]]


homogeneous function

#Contents#
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## Definition

For $R$ some [[ring]] and $N,N_1,\dots,N_n$ [[modules]] over this ring, a [[function]]

$$
  f \;\colon\; N_1 \times \cdots \times N_n \longrightarrow N
$$

is _homogeneous_ of degree $k$ if for all $r \in R$ we have

$$
  f(r a_1, \cdots, r a_n) = r^k f(a_1, \cdots, a_n)
  \,.
$$

## Examples

* [[quadratic form]]

## References

See also

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

[[!redirects homogeneous functions]]


homogeneous polynomial

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## Idea

A polynomial all of whose monomial terms have the same [[degree of a polynomial|degree]]. See also [[polynomial]].

## Definition

If $R$ is a [[commutative rig]], then the set $R_{n}[X_{1},...,X_{q}]$ of homogeneous polynomials of degree $n$ with variables $X_{1},...,X_{q}$ is given by all the expressions of the form $\underset{\substack{0\le i_{1},...,i_{q} \le n\\ i_{1}+...+i_{q} = n}}{\sum} a_{i_{1},...,i_{q}}X_{1}^{i_{1}}...X_{q}^{i_{q}}$.

## Euler identity

A specific property of homogeneous polynomials is the Euler identity:

\begin{theorem}
If $R$ is a commutative rig and if $P \in R_{n}[X_{1},...,X_{q}]$ then $\underset{1 \le k \le q}{\sum} \frac{\partial P}{\partial X_{k}}X_{k} = n.P$. Reciprocally, if $R$ is a commutative [[multiplicatively cancellable rig]], if $P \in R[X_{1},...,X_{q}]$ and $\underset{1 \le k \le q}{\sum} \frac{\partial P}{\partial X_{k}}X_{k} = n.P$, then $P \in R_{n}[X_{1},...,X_{q}]$.
\end{theorem}

\begin{proof}
Suppose that $P \in R_{n}[X_{1},...,X_{q}]$ and write $P=\underset{\substack{0\le i_{1},...,i_{q} \le n\\ i_{1}+...+i_{q} = n}}{\sum} a_{i_{1},...,i_{q}}X_{1}^{i_{1}}...X_{q}^{i_{q}}$. Then $\underset{1 \le k \le q}{\sum} \frac{\partial P}{\partial X_{k}}X_{k} = \underset{1 \le k \le q}{\sum}\underset{\substack{0\le i_{1},...,i_{q} \le n\\ i_{1}+...+i_{q} = n}}{\sum} a_{i_{1},...,i_{q}}.i_{k}.X_{1}^{i_{1}}...X_{q}^{i_{q}}$

$ = \underset{\substack{0\le i_{1},...,i_{q} \le n\\ i_{1}+...+i_{q} = n}}{\sum} a_{i_{1},...,i_{q}}.(i_{1}+...+i_{q}).X_{1}^{i_{1}}...X_{q}^{i_{q}}$

$ = \underset{\substack{0\le i_{1},...,i_{q} \le n\\ i_{1}+...+i_{q} = n}}{\sum} a_{i_{1},...,i_{q}}.n.X_{1}^{i_{1}}...X_{q}^{i_{q}}$

$=n.P$

Reciprocally, suppose that $R$ is multiplicatively cancellable, that $P \in R[X_{1},...,X_{q}]$ and that $\underset{1 \le k \le q}{\sum} \frac{\partial P}{\partial X_{k}}X_{k} = n.P$. Write $P=\underset{0 \le i_{1},...,i_{q} \le n}{\sum} a_{i_{1},...,i_{q}}X_{1}^{i_{1}}...X_{q}^{i_{q}}$.

Then $\underset{1 \le k \le q}{\sum} \frac{\partial P}{\partial X_{k}}X_{k} = \underset{1 \le k \le q}{\sum}\underset{0 \le i_{1},...,i_{q} \le q}{\sum} a_{i_{1},...,i_{q}}.i_{k}.X_{1}^{i_{1}}...X_{q}^{i_{q}}$

$ = \underset{0 \le i_{1},...,i_{q} \le n}{\sum} a_{i_{1},...,i_{q}}.(i_{1}+...+i_{q}).X_{1}^{i_{1}}...X_{q}^{i_{q}}$.

Thus, if $\underset{1 \le k \le q}{\sum} \frac{\partial P}{\partial X_{k}}X_{k} = n.P$, then

$\underset{0 \le i_{1},...,i_{q} \le n}{\sum} a_{i_{1},...,i_{q}}.n.X_{1}^{i_{1}}...X_{q}^{i_{q}}  = \underset{0 \le i_{1},...,i_{q} \le n}{\sum} a_{i_{1},...,i_{q}}.(i_{1}+...+i_{q}).X_{1}^{i_{1}}...X_{q}^{i_{q}}$

which is equivalent to $a_{i_{1},...,i_{q}}.(i_{1}+...+i_{q}) = a_{i_{1},...,i_{q}}.n$ for all $0 \le i_{1},...,i_{q} \le n$ which is equivalent to $a_{i_{1},...,i_{q}}.(i_{1}+...+i_{q}) = a_{i_{1},...,i_{q}}.n$ for all $0 \le i_{1},...,i_{q} \le n$ such that $a_{i_{1},...,i_{q}} \neq 0$. $R$ is multiplicatively cancellable, thus it is equivalent to $i_{1}+...+i_{q} = n$ for all $0 \le i_{1},...,i_{q} \le n$ such that $a_{i_{1},...,i_{q}} \neq 0$ which is equivalent to $P\in R_{n}[X_{1},...,X_{q}]$.
\end{proof}

[[!redirects homogeneous polynomials]]
[[!redirects Euler identity]]


Homogeneous polynomial codifferential category

## Idea

An homogeneous polynomial codifferential category is a $\mathbb{N}$-[[graded codifferential category]] which verifies a kind of [[Euler identity]]. It is conjectured that the only homogeneous polynomial codifferential categories are those given by actual [[homogeneous polynomials]] (in a sense precised below).


homogeneous space

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## Definition

Given a [[topological group]] or [[algebraic group]] or [[Lie group]], etc.,  $G$, a __homogeneous $G$-space__ is a [[topological space]] or [[scheme]], or [[smooth manifold]] etc. with [[transitive action|transitive]] $G$-[[action]].

A __principal homogeneous $G$-space__ is the total space of a $G$-[[torsor]] over a point.

There are generalizations, e.g. the [[quantum homogeneous space]] for the case of quantum groups.

## Examples

* A special case of homogeneous spaces are [[coset spaces]] arising from the [[quotient]] $G/H$ of a [[group]] $G$ by a subgroup. For the case of [[Lie groups]] this is also called [[Klein geometry]].

* Specifically for $G$ a [[compact Lie group]] and $T\hookrightarrow G$ a [[maximal torus]], then the coset $G/T$ play a central role in [[representation theory]] and [[cohomology]], for instance in the [[splitting principle]].

* In [[analysis]] and [[number theory]], certain functions on certain coset spaces play a role as _[[automorphic forms]]_ (e.g. [[modular forms]]). See there for more.

## Properties

### Relation to coset spaces
 {#RelationToCosetSpaces}

Under weak topological conditions (cf. [Helgason](#Helgason)), every topological homogeneous space $M$ is isomorphic to a __[[coset]] space__ $G/H$ for a [[closed subspace|closed]] subgroup $H\subset G$ (the [[stabilizer]] of a fixed point in $X$).

## Related concepts

* [[Klein geometry]]

* [[conjugacy class]]

* [[orbit category]]

* [[Grassmannian]], [[flag variety]]

* [[Schubert calculus]]

## References

Textbook accounts:

* {#Bredon72} [[Glen Bredon]], Section I.4 of: _[[Introduction to compact transformation groups]]_, Academic Press  1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

* {#Helgason} Sigurdur Helgason, _Differential geometry, Lie groups and symmetric spaces_

On [[homogeneous spaces]] with the same [[rational cohomology]] as a [[product space|product]] of [[n-spheres]]:

* [[Linus Kramer]], _Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurface_,  Memoirs of the American Mathematical Society number 752 ([arXiv:math/0109133] (http://arxiv.org/abs/math/0109133), [doi:10.1090/memo/0752](http://dx.doi.org/10.1090/memo/0752), [GoogleBooks](http://books.google.com/books?id=SA8O6ihrDFkC&printsec=frontcover&hl=de&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false))

The following article has categorical analysis of relation between the total space of a principal bundle and of the corresponding quotient space both for the classical case and for noncommutative generalizations:

* [[Tomasz Brzeziński]], _On synthetic interpretation of quantum principal bundles_, AJSE D - Mathematics 35(1D): 13-27, 2010 [arxiv:0912.0213](http://uk.arxiv.org/abs/0912.0213); _Quantum group differentials, bundles and gauge theory_, Encyclopedia of Mathematical Physics, Acad. Press. 2006, pp. 236–244
[doi](http://dx.doi.org/10.1016/B0-12-512666-2/00050-X)

[[!redirects homogeneous spaces]]


homoiconicity

In computer programming, homoiconicity (from the Greek words _homo_ meaning "the same" and _icon_ meaning "representation") is a property of some [[programming languages]] in which the [[program]] [[structure]] is similar to its [[syntax]], and therefore the program's internal representation can be inferred by reading the text's layout. If a language has homoiconicity, it means that the language text has the same structure as its abstract syntax tree (i.e. the AST and the syntax are [[isomorphic]]).

In a homoiconic language the primary representation of programs is also a data structure in a primitive type of the language itself. This makes metaprogramming easier than in a language without this property, since code can be treated as [[data]]: [[reflection]] in the language (examining the program's entities at runtime) depends on a single, homogeneous structure, and it does not have to handle several different structures that would appear in a complex syntax. To put that another way, homoiconicity is where a program's source code is written as a basic data structure that the programming language knows how to access.

A typical, commonly cited example is the programming language Lisp, which was created to be easy for lists manipulation and where the structure is given by S-expressions that take the form of nested [[lists]].
Lisp programs are written in the form of lists; the result is that the program can access its own functions and procedures while running, and programmatically reprogram itself on the fly. Homoiconic languages typically include full support of syntactic macros allowing the programmer to express program transformations in a concise way. A more recent example is the programming language Clojure, which is a contemporary dialect of Lisp.

* From the [Wikipedia article on homoiconicity](https://en.wikipedia.org/wiki/Homoiconicity)


Homological Algebra

A foundational book on [[homological algebra]]:

* [[Henri Cartan]] and [[Samuel Eilenberg]]

  \linebreak

  **Homological Algebra**

  \linebreak

  Princeton University Press (1956)

  Princeton Mathematical Series **19** (1999)

  \linebreak

  [doi:10.1515/9781400883844](https://doi.org/10.1515/9781400883844)

  [ISBN:9780691049915](https://press.princeton.edu/books/paperback/9780691049915/homological-algebra-pms-19-volume-19)

  [pdf](http://www.math.stonybrook.edu/~mmovshev/BOOKS/homologicalalgeb033541mbp.pdf)

\linebreak

{#FromCollins06} From [Collins 2006](homological+algebra#Collins06):

> In 1942 came the first move forward towards [[homological algebra]] as we know it today, with the arrival of a [paper by Samuel Eilenberg and Saunders MacLane](homological+algebra#EilenbergMacLane42). In it we find [[Hom]] and [[Ext]] defined for the very first time, and along with it the notions of a [[functor]] and [[natural isomorphism]]. These were needed to provide a precise language for talking about the properties of $Hom(A,B)$; in particular the fact that it varies naturally, contravariantly in $A$ and covariantly in $B$.

> Only three years later this language [was expanded](category#EilenbergMacLane45) to include [[category]] and [[natural equivalence]]. However, this terminology was not widely accepted by the mathematical community until the appearance of [Cartan and Eilenberg's book](homological+algebra#CartanEilenberg56).

> Cartan and Eilenberg's book was truly a revolution in the subject, and in fact it was here that the term "Homological Algebra" was first coined. The book used [[derived functors]] in a systematic way which united all the previous [[homology theories]], which in the past ten years had arisen in [[group theory]], [[Lie algebras]] and [[algebraic geometry]]. The sheer list of terms that were first defined in this book may give the reader an idea of how much of this project is due to the existence of that one book! They defined what it means for an object to be [[projective object|projective]] or [[injective object|injective]], and defined the notions of [[projective resolution|projective]] and [[injective resolutions]]. It is here that we find the first mention of $Hom$ being [[left exact functor|left exact]] and the first occurrence of $Ext^n$ as the [[right derived functors]] of $Hom$.

## Contents
## Preface

## Chapter I. Rings and Modules 3
### 1. Preliminaries 3
### 2. Projective modules 6
### 3. Injective modules 8
### 4. Semi-simple rings 11
### 5. Hereditary rings 12
### 6. Semi-hereditary rings 14
### 7. Noetherian rings 15
### Exercises 16

## Chapter II. Additive Functors 18
### 1. Definitions 18
### 2. Examples 20
### 3. Operators 22
### 4. Preservation of exactness 23
### 5. Composite functors 27
### 6. Change of rings 28
### Exercises 31

## Chapter III. Satellites 33
### 1. Definition of satellites 33
### 2. Connecting homomorphisms 37
### 3. Half exact functors 39
### 4. Connected sequence of functors 43
### 5. Axiomatic description of satellites 45
### 6. Composite functors 48
### 7. Several variables 49
### Exercises 51

## Chapter IV. Homology 53
### 1. Modules with differentiation 53
### 2. The ring of dual numbers 56
### 3. Graded modules, complexes 58
### 4. Double gradings and complexes 60
### 5. Functors of complexes 62
### 6. The homomorphism a 64
### 7. The homomorphism a (continuation) 66
### 8. Künneth relations 71
### Exercises 72

## Chapter V. Derived Functors 75
### 1. Complexes over modules; resolutions 75
### 2. Resolutions of sequences 78
### 3. Definition of "derived functors 82
### 4. Connecting homomorphisms 84
### 5. The functors R?T and L^T 89
### 6. Comparison with satellites 90
### 7. Computational devices 91
### 8. Partial derived functors 94
### 9. Sums, products, limits 97
### 10. The sequence of a map 101
### Exercises 104

## Chapter VI. Derived Functors of ⊗ and Hom 106
### 1. The functors Tor and Ext 106
### 2. Dimension of modules and rings 109
### 3. Künneth relations 112
### 4. Change of rings 116
### 5. Duality homomorphisms 119
### Exercises 122

## Chapter VII. Integral Domains 127
### 1. Generalities 127
### 2. The field of quotients 129
### 3. Inversible ideals 132
### 4. Prufer rings 133
### 5. Dedekind rings 134
### 6. Abelian groups 135
### 7. A description of Torj (A,C) 137
### Exercises 139

## Chapter VIII. Augmented Rings 143
### 1. Homology and cohomology of an augmented ring 143
### 2. Examples 146
### 3. Change of rings 149
### 4. Dimension 150
### 5. Faithful systems 154
### 6. Applications to graded and local rings 156
### Exercises 158

## Chapter IX. Associative Algebras 162
### 1. Algebras and their tensor products 162
### 2. Associativity formulae 165
### 3. The enveloping algebra A* 167
### 4. Homology and cohomology of algebras 169
### 5. The Hochschild groups as functors of A 171
### 6. Standard complexes 174
### 7. Dimension 176
### Exercises 180

## Chapter X. Supplemented Algebras 182
### 1. Homology of supplemented algebras 182
### 2. Comparison with Hochschild groups 185
### 3. Augmented monoids 187
### 4. Groups 189
### 5. Examples of resolutions 192
### 6. The inverse process 193
### 7. Subalgebras and subgroups 196
### 8. Weakly injective and projective modules 197
### Exercises 201

## Chapter XI. Products 202
### 1. External products 202
### 2. Formal properties of the products 206
### 3. Isomorphisms 209
### 4. Internal products 211
### 5. Computation of products 213
### 6. Products in the Hochschild theory 216
### 7. Products for supplemented algebras 219
### 8. Associativity formulae 222
### 9. Reduction theorems 225
### Exercises 228

## Chapter XII. Finite Groups 232
### 1. Norms 232
### 2. The complete derived sequence 235
### 3. Complete resolutions 237
### 4. Products for finite groups 242
### 5. The uniqueness theorem 244
### 6. Duality 247
### 7. Examples 250
### 8. Relations with subgroups 254
### 9. Double cosets 256
### 10. p-groups and Sylow groups 258
### 11. Periodicity 260
### Exercises 263

## Chapter XIII. Lie Algebras 266
### 1. Lie algebras and their enveloping algebras 266
### 2. Homology and cohomology of Lie algebras 270
### 3. The Poincare-Witt theorem 271
### 4. Subalgebras and ideals 274
### 5. The diagonal map and its applications 275
### 6. A relation in the standard complex 277
### 7. The complex F(g) 279
### 8. Applications of the complex K(g) 282
### Exercises 284

## Chapter XIV. Extensions 289
### 1. Extensions of modules 289
### 2. Extensions of associative algebras 293
### 3. Extensions of supplemented algebras 295
### 4. Extensions of groups 299
### 5. Extensions of Lie algebras 304
### Exercises 308

## Chapter XV. Spectral Sequences 315
### 1. Filtrations and spectral sequences 315
### 2. Convergence 319
### 3. Maps and homotopies 321
### 4. The graded case 323
### 5. Induced homomorphisms and exact sequences 325
### 6. Application to double complexes 330
### 7. A generalization 333
### Exercises 336

## Chapter XVI. Applications of Spectral Sequences 340
### 1. Partial derived functors 340
### 2. Functors of complexes 342
### 3. Composite functors 343
### 4. Associativity formulae 345
### 5. Applications to the change of rings 347
### 6. Normal subalgebras 349
### 7. Associativity formulae using diagonal maps 351
### 8. Complexes over algebras 352
### 9. Topological applications 355
### 10. The almost zero theory 358
### Exercises 360

## Chapter XVII. Hyperhomology 362
### 1. Resolutions of complexes 362
### 2. The invariants 366
### 3. Regularity conditions 368
### 4. Mapping theorems 371
### 5. Künneth relations 372
### 6. Balanced functors 374
### 7. Composite functors 376

## Appendix: Exact categories, by David A. Buchsbaum 379

## List of Symbols 387

## Index of Terminology 389

[[!redirects Homological algebra]]

category: reference