hypergeometric function

#Contents#
* table of contents
{:toc}

## Idea

The classical **hypergeometric series** (introduced by [[Gauss]]) are [[complex number|complex]] solutions of certain [[ordinary differential equations]] of second order, the hypergeometric differential equation.

Special cases appear in classical problems of [[mathematical physics]], as solutions to the [[wave equation]], Laplace equation or similar when attacked by Fourier method of separation of variables (cf. [[Legendre polynomial]], [[Hermite polynomial]]).

## Definition

The hypergeometric series is defined by the formula,

$$
{}_p F_q (a_1,\ldots,a_p; b_1,\ldots, b_q; x) = \sum_{n=0}^\infty
\frac{(a_1)_n (a_2)_n\cdots (a_p)_n}{(b_1)_n (b_2)_n\cdots (b_q)_n}\frac{x^n}{n!}
$$

where $(a)_0 = 1$ and, for $k = 1,2,3,\ldots$

$$
(a)_k := a (a+1) (a+2) \cdots (a+k-1)
$$

is the shifted [[factorial]]. In fact, let $\sum_{n = 0}^\infty c_n$ be any series of complex numbers such that $c_{n+1}/c_n$ is a rational function of $n$. Then we can find $x,p,q,a_1,\ldots,a_p,b_1,\ldots, b_q$ to write
$$
\frac{c_{n+1}}{c_n} = \frac{(n+a_1)(n+a_2)\cdots (n+a_p) x}{(n+b_1)(n+b_2)\cdots (n+b_q)(n+1)}
$$
and $\sum c_n = c_0\cdot {}_p F_q(a_1,\ldots,a_p; b_1,\ldots, b_q; x)$.

### Classical case, ${}_2 F_1$

Hypergeometric function $F = {}_2 F_1(a,b;c;x)$ satisfies the ordinary differential equation
$$
x(1-x)\frac{d^2 F}{d x^2} + [c-(a+b-1)x]\frac{d F}{d x}
- a b F = 0.
$$
For $Re(c)\gt 0$, $Re(b)\gt 0$ function ${}_2 F_1$ can be represented as the [[Euler integral]]
$$
{}_2 F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_0^1 t^b (1-t)^{c-b-1}(1-t x)^{-a} d t,\,\,\,\,x\notin [1,+\infty).
$$
The value of this function at origin is $1$. The second solution of the differential equation around $0$ is
 $x^{1-c} {}_2 F_1(a-c+1,b-c+1;2-c;x)$.
A basis of solutions around $\infty$ is given by $x^{-a}F(a,1-c+1;1-b+a;x^{-1})$ and $x^{-b}F(b,1-c+b;1-a+b;x^{-1})$.

### Confluent hypergeometric functions

If some singular points of the differential equation coalesce, in the limiting case we obtain confluent hypergeometric function (special case of which are e.g. [[Bessel functions]]).

## Examples

The classical orthogonal polynomials appear as special cases for particular choices of parameters, for example Jacobi polynomials and, as their special case, [[Legendre polynomial]]s.

## Generalizations and related equations

There are other variants like  $q$-hypergeometric functions and the basic hypergeometric series.

[[Heun equation]] is the second order [[Fuchsian differential equation|Fuchsian ODE]] with 4 regular singular points at $0,1,a$ and $\infty$; hypergeometric equation can be transformed into Heun equation by a change of variables.

There is a recent elliptic version of hypergeometric functions due Spiridonov, [Spiridonov2013](#Spiridonov2013).

There are now modern generalizations to many variables due Aomoto and another variant due [[Mikhail Kapranov]], [[Israel Gelfand]] and [[Andrei Zelevinsky]]. These multidimensional generalizations express pairings between representations of quantum groups at root of unity and representations of affine Lie algebras, which can be interpreted as "hypergeometric" pairings between certain kind of local homology and cohomology on configuration spaces, see [[hypergeometric construction of KZ solutions]]. This has been extensively studied by [[Alexander Varchenko]], [[Hiroaki Terao]], [[Peter Orlik]], [[Daniel Cohen]] and others; often in connection to the study of (complements of) [[arrangements of hyperplanes]] in $\mathbb{C}^n$. [[Selberg integral|Selberg-type integrals]] are involved.

## References

* G. E. Andrews, R. Askey, R. Roy, _Special functions_, Enc. of Math. and its Appl. __71__, Cambridge Univ. Press 1999

* G. Gasper, M. Rahman, _Basic hypergeometric series_ (1990)

* [[Israel Gelfand|I. M. Gelfand]], M. M. [[Kapranov]], [[Andrei Zelevinsky|A. Zelevinsky]], _Discriminants, resultants and multidimensional determinants_, Birkhäuser 1994, 523 pp.

*  [[Israel Gelfand|I. M. Gel’fand]], M. I. Graev, [[Vladimir Retakh|V. S. Retakh]], _General hypergeometric systems of equations and series of hypergeometric type_, Russian Math. Surveys __47__(4) (1992) 1--88 [doi](https://doi.org/10.1070/RM1992v047n04ABEH000915), transl. from _Общие гипергеометрические системы уравнений и ряды гипергеометрического типа_, УМН __47__:4(286) (1992) 3--82 [mathnet.ru](http://mi.mathnet.ru/umn4540); _General gamma functions, exponentials, and hypergeometric functions_, Russian Math. Surveys __53__:1 (1998) 1--55 [doi](https://doi.org/10.1070/RM1998v053n01ABEH000008), transl. from _Общие гамма-функции, экспоненты и гипергеометрические функции_, УМН, 1998, __53__:1 (319) 3--60 [doi](https://doi.org/10.4213/rm8)

* [[Ian G. Macdonald]], _Hypergeometric functions I_, 1987 ([arxiv/1309.4568](http://arxiv.org/abs/1309.4568))

* (dedicated chapter 2 of) Katsunori Iwasaki, Hironobu Kimura, Shun Shimomura, Masaaki Yoshida, _From Gauss to Painlevé, A modern theory of special functions_, 184 pp.

Lecture notes motivated by [[partial differential equations]] appearing in [[mathematical physics]]:

* [[Cliff P. Burgess]], §10 in: *Primer on Partial Differential Equations for Physicists*, lecture notes (1990) [[pdf](https://physics.mcmaster.ca/~cburgess/Notes/mathphys.pdf), [[Burgess-PDEs.pdf:file]]]

In relation to the [[Knizhnik-Zamolodchikov equation]] and [[quantum groups]]:

* [[Alexander Varchenko]], _Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups_, Adv. Ser. in Math. Phys. __21__, World Sci. Publ. 1995. x+371 pp. ([doi:10.1142/2467](https://doi.org/10.1142/2467))

* V. Tarasov, [[Alexander Varchenko]], _Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups_, Astérisque __246__ (1997), vi+135 pp. ([arXiv:q-alg/9703044](https://arxiv.org/abs/q-alg/9703044), [numdam:AST_1997__246__R1_0](http://www.numdam.org/item/AST_1997__246__R1_0))

Online entries/resources on hypergeometric function:

* at Wolframworld: [hypergeometric function](http://mathworld.wolfram.com/HypergeometricFunction.html), [confluent hypergeometric functon of the first kind](http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheFirstKind.html), [confluent hypergeometric functon of the second kind](http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheSecondKind.html), [generalized hypergeometric function](http://mathworld.wolfram.com/GeneralizedHypergeometricFunction.html), [$q$-hypergeometric function](http://mathworld.wolfram.com/q-HypergeometricFunction.html), [regularized hypergeometric function](http://mathworld.wolfram.com/RegularizedHypergeometricFunction.html)

* wikipedia: [hypergeometric series](http://en.wikipedia.org/wiki/Hypergeometric_series), [confluent hypergeometric function](http://en.wikipedia.org/wiki/Confluent_hypergeometric_function)

* [[Alexander Varchenko]]: [list](http://www.math.unc.edu/Faculty/av/complete.htm) of publications

There is also a far reaching elliptic generalization

* {#Spiridonov2013} V. P. Spiridonov, _Classical elliptic hypergeometric functions and their applications_, [pdf](http://www.math.kobe-u.ac.jp/publications/rlm18/17); _Aspects of elliptic hypergeometric functions_, [arxiv/1307.2876](http://arxiv.org/abs/1307.2876)

[[!redirects hypergeometric functions]]


hypergeometric KZ-solutions -- references

### Braid representations via twisted cohomology of configuration spaces
 {#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces}

The "[[hypergeometric construction of KZ solutions|hypergeometric integral]]" construction of [[conformal blocks]] for [[affine Lie algebra]]/[[WZW model]]-[[2d CFTs]] and of more general solutions to the [[Knizhnik-Zamolodchikov equation]], via [[twisted de Rham cohomology]] of [[configuration spaces of points]], originates with:

* [[Vadim Schechtman]], [[Alexander Varchenko]], *Integral representations of N-point conformal correlators in the WZW model*,  Max-Planck-Institut für Mathematik, (1989) Preprint MPI/89- $[$[cds:1044951](http://cds.cern.ch/record/1044951)$]$

* {#DateJimboMatsuoMiwa90} [[Etsuro Date]], [[Michio Jimbo]], [[Atsushi Matsuo]], [[Tetsuji Miwa]], *Hypergeometric-type integrals and the $\mathfrak{sl}(2,\mathbb{C})$-Knizhnik-Zamolodchikov equation*,  International Journal of Modern Physics B **04** 05 (1990) 1049-1057   $[$[doi:10.1142/S0217979290000528](https://doi.org/10.1142/S0217979290000528)$]$

  > (for [[sl(2)|$\mathfrak{sl}(2,\mathbb{C})$]])

* [[Atsushi Matsuo]], *An application of Aomoto-Gelfand hypergeometric functions to the $SU(n)$ Knizhnik-Zamolodchikov equation*, Communications in Mathematical Physics **134** (1990) 65–77 $[$[doi:10.1007/BF02102089](https://doi.org/10.1007/BF02102089)$]$

* [[Vadim Schechtman]], [[Alexander Varchenko]], *Hypergeometric solutions of Knizhnik-Zamolodchikov equations*, Lett. Math. Phys. **20** (1990) 279–283 $[$[doi:10.1007/BF00626523](https://doi.org/10.1007/BF00626523)$]$

* [[Vadim Schechtman]], [[Alexander Varchenko]], *Arrangements of hyperplanes and Lie algebra homology*, Inventiones mathematicae  **106** 1 (1991) 139-194 $[$[dml:143938](https://eudml.org/doc/143938), [[SchechtmanVarchenko_HyperplaneArrangements.pdf:file]]$]$

following precursor observations due to:

* [[Vladimir S. Dotsenko]], [[Vladimir A. Fateev]], *Conformal algebra and multipoint correlation functions in 2D statistical models*, Nuclear Physics B **240** 3 (1984) 312-348 $[$doi:10.1016/0550-3213(84)90269-4$]$

* [[Philippe Christe]], [[Rainald Flume]], *The four-point correlations of all primary operators of the $d = 2$ conformally invariant $SU(2)$ $\sigma$-model with Wess-Zumino term*, Nuclear Physics B
**282** (1987) 466-494 $[$doi:10.1016/0550-3213(87)90693-6$]$

The proof that for rational levels this construction indeed yields [[conformal blocks]] is due to:

* {#FeiginSchechtmanVarchenko90} [[Boris Feigin]], [[Vadim Schechtman]], [[Alexander Varchenko]], *On algebraic equations satisfied by correlators in Wess-Zumino-Witten models*,  Lett Math Phys **20** (1990)  291–297  $[$[doi:10.1007/BF00626525](https://doi.org/10.1007/BF00626525)$]$

* {#FeiginSchechtmanVarchenko94} [[Boris Feigin]], [[Vadim Schechtman]], [[Alexander Varchenko]], *On algebraic equations satisfied by hypergeometric correlators in WZW models. I*, Commun. Math. Phys. **163** (1994) 173–184 $[$[doi:10.1007/BF02101739](https://doi.org/10.1007/BF02101739)$]$

  > (for [[sl(2)|$\mathfrak{sl}(2, \mathbb{C})$]])

* [[Boris Feigin]], [[Vadim Schechtman]], [[Alexander Varchenko]], *On algebraic equations satisfied by hypergeometric correlators in WZW models. II*, Comm. Math. Phys. **170** 1 (1995) 219-247  $[$[euclid:cmp/1104272957](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-170/issue-1/On-algebraic-equations-satisfied-by-hypergeometric-correlators-in-WZW-models/cmp/1104272957.full)$]$

Review:

* {#Varchenko95} [[Alexander Varchenko]], *Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups*, Advanced Series in Mathematical Physics **21**, World Scientific 1995 ([doi:10.1142/2467](https://doi.org/10.1142/2467))

* [[Ivan Cherednik]], Section 8.2 of: *Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras*, Mathematical Society of Japan Memoirs **1998** (1998) 1-96 $[$[doi:10.2969/msjmemoirs/00101C010](https://doi.org/10.2969/msjmemoirs/00101C010)$]$

* {#EtingofFrenkelKirillov98} [[Pavel Etingof]], [[Igor Frenkel]], [[Alexander Kirillov]], Lecture 7 in: *Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations*, Mathematical surveys and monographs **58**, American Mathematical Society (1998) $[$[ISBN:978-1-4704-1285-2](https://bookstore.ams.org/surv-58), [review pdf](http://www.ams.org/journals/bull/2000-37-02/S0273-0979-00-00853-3/S0273-0979-00-00853-3.pdf)$]$

* [[Toshitake Kohno]], *Homological representations of braid groups and KZ connections*, Journal of Singularities **5** (2012) 94-108  $[$[doi:10.5427/jsing.2012.5g](http://dx.doi.org/10.5427/jsing.2012.5g),  [pdf](https://www.journalofsing.org/volume5/kohno.pdf)$]$

* [[Toshitake Kohno]], *Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks*, Acta Math Vietnam **39** (2014) 575–598 $[$[doi:10.1007%2Fs40306-014-0088-6](https://link.springer.com/article/10.1007%2Fs40306-014-0088-6), [pdf](https://www.ms.u-tokyo.ac.jp/~kohno/papers/kohno_config.pdf)$]$

* [[Toshitake Kohno]], *Introduction to representation theory of braid groups*, Peking 2018 $[$[pdf](https://www.math.pku.edu.cn/misc/puremath/summerschool/Peking_SummerSchool_kohno.pdf), [[KohnoBraidRepresentations.pdf:file]]$]$

  > (motivation from [[braid representations]])

See also:

* [[Alexander Varchenko]], *Asymptotic solutions to the Knizhnik-Zamolodchikov equation and crystal base*,  Comm. Math. Phys. **171** 1  (1995) 99-137 $[$[arXiv:hep-th/9403102](https://arxiv.org/abs/hep-th/9403102), [doi:10.1007/BF02103772](https://doi.org/10.1007/BF02103772)$]$

* [[Edward Frenkel]], [[David Ben-Zvi]], Section 14.3 in: *Vertex Algebras and Algebraic Curves*, Mathematical Surveys and Monographs **88**, AMS 2004 $[$[ISBN:978-1-4704-1315-6](https://bookstore.ams.org/surv-88-r), [web](https://math.berkeley.edu/~frenkel/BOOK/)$]$

This "hypergeometric" construction uses results on the [[twisted de Rham cohomology]] of [[configuration spaces of points]] due to:

* [[Peter Orlik]], [[Louis Solomon]], *Combinatorics and topology of complements of hyperplanes*, Invent Math **56** (1980) 167–189  $[$[doi:10.1007/BF01392549](https://doi.org/10.1007/BF01392549)$]$

* [[Kazuhiko Aomoto]], *Gauss-Manin connection of integral of difference products*,  J. Math. Soc. Japan **39** 2 (1987) 191-208  $[$[doi:10.2969/jmsj/03920191](https://projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-39/issue-2/Gauss-Manin-connection-of-integral-of-difference-products/10.2969/jmsj/03920191.full)$]$

* [[Hélène Esnault]], [[Vadim Schechtman]], [[Eckart Viehweg]], *Cohomology of local systems on the complement of hyperplanes*,  Inventiones mathematicae 109.1 (1992)  557-561 $[$[pdf](http://page.mi.fu-berlin.de/esnault/preprints/helene/26-es_sch_vi.pdf)$]$

* [[Vadim Schechtman]], H. Terao, [[Alexander Varchenko]], *Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors*, Journal of Pure and Applied Algebra **100** 1–3 (1995) 93-102 $[$[arXiv:hep-th/9411083](https://arxiv.org/abs/hep-th/9411083), doi:10.1016/0022-4049(95)00014-N$]$

also:

* [[Peter Orlik]], *Hypergeometric integrals and arrangements*, Journal of Computational and Applied Mathematics **105** (1999) 417–424 $[$doi:10.1016/S0377-0427(99)00036-9, [pdf](https://core.ac.uk/download/pdf/82631708.pdf)$]$

* [[Daniel C. Cohen]], [[Peter Orlik]], *Arrangements and local systems*, Math. Res. Lett. **7** (2000) 299-316 $[$[arXiv:math/9907117](https://arxiv.org/abs/math/9907117), [doi:10.4310/MRL.2000.v7.n3.a5](https://dx.doi.org/10.4310/MRL.2000.v7.n3.a5)$]$

reviewed in:

* Yukihito Kawahara, *The twisted de Rham cohomology for basic constructions of hyperplane arrangements and its applications*, Hokkaido Math. J. **34** 2 (2005) 489-505 $[$[doi:10.14492/hokmj/1285766233](https://projecteuclid.org/journals/hokkaido-mathematical-journal/volume-34/issue-2/The-twisted-de-Rham-cohomology-for-basic-constructionsof-hyperplane-arrangements/10.14492/hokmj/1285766233.full)$]$

{#HypergeometricReferencesForVanishingLevel} Discussion for the special case of [[level (Chern-Simons theory)|level]]$=0$ (cf. at *[logarithmic CFT -- Examples](logarithmic+CFT#Examples)*):

* [[Fedor A. Smirnov]], *Remarks on deformed and undeformed Knizhnik-Zamolodchikov equations*, $[$[arXiv:hep-th/9210051](https://arxiv.org/abs/hep-th/9210051)$]$

* [[Fedor A. Smirnov]], *Form factors, deformed Knizhnik-Zamolodchikov equations and finite-gap integration*, Communications in Mathematical Physics **155** (1993) 459–487 $[$[doi:10.1007/BF02096723](https://doi.org/10.1007/BF02096723), [arXiv:hep-th/9210052](https://arxiv.org/abs/hep-th/9210052)$]$

* S. Pakuliak, A. Perelomov, *Relation Between Hyperelliptic Integrals*, Mod. Phys. Lett. **9** 19 (1994) 1791-1798 $[$[doi:10.1142/S0217732394001647](https://doi.org/10.1142/S0217732394001647)$]$

Interpretation of the [[hypergeometric construction of KZ solutions|hypergeometric construction]] as happening in [[twisted equivariant differential K-theory]], showing that the [[K-theory classification of D-brane charge]] and the [[K-theory classification of topological phases of matter]] both reflect [[braid group representations]] as expected for [[defect branes]] and for [[anyons]]/[[topological order]], respectively:

* [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Anyonic defect branes in TED-K-theory]]* $[$[arXiv:2203.11838](https://arxiv.org/abs/2203.11838)$]$


hypergraph

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

# Contents
* table of contents
{:toc}

## Idea

In an ordinary undirected [[graph]], each edge $e$ links an [[unordered pair]] of vertices $x$ and $y$ (perhaps allowing for the possibility that $x = y$, as in the case of a loop).  Hypergraphs generalize this, allowing a "hyperedge" to link any set of "hypervertices".  Abstracting everything away but the _incidence_ relation between hypervertices and hyperedges, a hypergraph can be modelled as an arbitrary heterogenous [[relation]], or more generally as a [[span]].

## Categories of hypergraphs: Definition

As with "[[graph]]", the word "hypergraph" does not have an entirely standardized meaning.  We take as our starting point [(Schmidt & Ströhlein)](#SchmidtStroehlein), who define hypergraphs as heterogenous relations (usually presented as boolean-valued matrices) and give an appropriate notion of morphism.  We call these "simple" hypergraphs, since they are a special case of a more general definition given below.

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

The **category of simple hypergraphs** $SimpHGrph$ has objects consisting of a pair of sets $(V,E)$ equipped with a relation $R \subseteq V \times E$, and morphisms $(R \subseteq V \times E) \to (R' \subseteq V' \times E')$ consisting of pairs of functions $(f : V \to V', g : E \to E')$ which preserve the relation, i.e., such that for all $v\in V, e \in E$, if $(v,e) \in R$ then $(f v,g e) \in R'$.

=--

The category of simple hypergraphs could also be referred to more dryly as "the category of binary relations", although this has potential for confusion with [[Rel]] (whose _morphisms_ are binary relations).

In a simple hypergraph, a hypervertex can be incident to a hyperedge at most once, but in some situations one wants to allow a hypervertex to be incident to a hyperedge multiple times.  To define hypergraphs in this more general sense, we keep the intuition of hypergraphs-as-heterogenous-relations, but generalize relations to spans.

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

The **category of hypergraphs** $HGrph$ has objects consisting of a pair of sets $(V,E)$ equipped with a span $V \overset{v}\leftarrow R \overset{e}\rightarrow E$, and morphisms $(V \overset{v}\leftarrow R \overset{e}\rightarrow E) \to (V' \overset{v'}\leftarrow R' \overset{e'}\rightarrow E')$ consisting of triples of functions $(f : V \to V', g : E \to E', h:R \to R')$ such that $v'\circ h = f\circ v$ and $e'\circ h = g\circ e$.

=--

Note that $HGrph$ is just the [[category of presheaves]] over the "[[walking structure|walking]] cospan" $\bullet \rightarrow \bullet \leftarrow \bullet$, and that $SimpHGrph$ is a [[full subcategory]] of $HGrph$.

## Hypergraphs from a topos-theoretic perspective

In Lawvere ([1986](#Law86) p.6, [1989](#Law89) pp.283-4) it is pointed out that $Set^{\bullet\leftarrow\bullet\rightarrow\bullet}$ is a [[spatial topos]] since it is equivalent to the topos of sheaves on the space $X=\{a,b,c\}$ with topology $\tau=\{\emptyset,\{a\},\{b\},\{a,b\},\{a,b,c\}\}$ i.e. $X$ has two isolated points $a,b$ and a [[focal point|focal]] one $c$ whose only neighborhood is the whole space.

The [[lattice of subtoposes]] of $Set^{\bullet\leftarrow\bullet\rightarrow\bullet}$ consists (besides the two obvious subtoposes) of one [[closed subtopos|closed]] and two [[open subtopos|open]] copies of $Set$, two closed copies of the [[Sierpinski topos]] complementing the open copies of $Set$ respectively and an open $Set\times Set=Sh(\{a,b\})=Sh_{\neg\neg}(Set^{\bullet\leftarrow\bullet\rightarrow\bullet})$ complementing the closed copy of $Set$. In particular, $Set^{\bullet\leftarrow\bullet\rightarrow\bullet}$ is a [[scattered topos]].

The complementary open-closed pairs of the subtopos lattice correspond precisely to analyses of $Set^{\bullet\leftarrow\bullet\rightarrow\bullet}$ by [[Artin gluing]].

For instance, take the product functor $\sqcap\colon Set\times Set\to Set$ with  $(X,Y)\mapsto X\times  Y$. $\sqcap$ is left exact since it forms an adjoint string with the [[diagonal functor]] and the coproduct functor: $\sqcup\dashv \triangle\dashv\sqcap$ .

Then $\mathbf{Gl}(\sqcap)$ has objects $((X,Y),Z, u\colon Z\to X\times Y)$ where $(X,Y)\in Set\times Set$ and $Z\in Set$ and $u$ is a morphism in $Set$. Since by the universal property of products $u\colon Z\overset{\langle u_0, u_1\rangle}{\rightarrow} X\times Y$ corresponds to a pair of maps $u_0$, $u_1$, one sees that the objects in $\mathbf{Gl}(\sqcap)$ really correspond to spans in $Set$.

The open inclusion of $Set\times Set$ into $\mathbf{Gl}(\sqcap)$ is given by the [[geometric morphism]]

$$ i_\ast \colon Set\times Set\to \mathbf{Gl}(\sqcap) \qquad (X,Y)\mapsto ((X,Y),X\times Y,id_{X\times Y})$$

$$ i^\ast\colon \mathbf{Gl}(\sqcap)\to Set\times Set \qquad ((X,Y),Z,u)\mapsto (X,Y)$$

$$ i_{!} \colon Set\times Set\to \mathbf{Gl}(\sqcap) \qquad (X,Y)\mapsto ((X,Y),0,0\to X\times Y) \quad ,$$

and the closed inclusion of $Set$ by

$$ j_\ast \colon Set\to \mathbf{Gl}(\sqcap) \qquad X\mapsto ((1,1),X,X\to 1\times 1)$$

$$ j^\ast\colon\mathbf{Gl}(\sqcap)\to Set \qquad ((X,Y),Z,u)\mapsto Z\quad .$$

Since $\triangle\dashv\sqcap$ the closed inclusion is [[essential geometric morphism|essential]]:

$$ j_!\colon Set \to \mathbf{Gl}(\sqcap)\qquad X\mapsto ((X,X), X,  X\overset{\langle id_X,id_X\rangle}{\to} X\times X)\quad .$$

Since $\sqcup\dashv \triangle$ there is a somewhat surprising further left adjoint:

$$ j^\circ\colon\mathbf{Gl}(\sqcap)\to Set \qquad ((X,Y),Z,u)\mapsto X\sqcup_u Y\quad .$$

Here $ X\sqcup_u Y$ denotes the [[pushout]] of $X\overset{u_0}{\leftarrow} Z\overset{u_1}{\rightarrow} Y$ where $u_0$, $u_1$ are the pair of maps provided by $u\colon Z\overset{\langle u_0, u_1\rangle}{\rightarrow} X\times Y$ . Since a map from $((X,Y),Z,u)$ to $j_!(W)$ is a triple $(f_0,f_1,f_2)$ such that the following diagram commutes:
$$
\array{
Z & \overset{\langle u_0, u_1\rangle}{\rightarrow} & X\times Y
\\
f_2\downarrow &  &f_0\downarrow \downarrow f_1
\\
W & \overset{\langle id_W, id_W\rangle}{\rightarrow} & W\times W
}
$$
$(f_0,f_1,f_2)$ has to satisfy the two conditions $f_0 u_0= f_2$ and $f_1 u_1=f_2$, or equivalently, $f_0 u_0=f_1 u_1$ . But this is the same as giving a map from $j^\circ ((X,Y),Z,u)= X\sqcup_u Y$ to $W$ by the universal property of the pushout.

Whence we get an adjoint string:

$$j^\circ\dashv j_!\dashv j^\ast \dashv j_\ast \colon Set\to \mathbb{Gl}(\sqcap)$$

with $j_!$, $j_\ast$ [[fully faithful]], exhibiting $\mathbf{Gl}(\sqcap)$ almost as a [[cohesive topos]] over $Set$. Of course, since $Set^{\bullet\leftarrow\bullet\rightarrow\bullet}$ is spatial it is not expected to satisfy all of Lawvere's axioms. In particular, the [[Nullstellensatz]] is violated since none of the copies of $Set$ is [[dense subtopos|dense]] in $Set^{\bullet\leftarrow\bullet\rightarrow\bullet}$.

Let $Q$ be the diagram category $N\overset{s}{\underset{t}{\rightrightarrows}} A$ underlying the topos $Set^{Q^{op}}$ of directed graphs or [[quiver|quivers]]. Consider the [[Yoneda embedding]] of the object $A$ into the presheaves: $Y(A)=Hom_Q(-,A)$. Viewed as a graph this gives the basic figure type of an _a_rrow $K_2=\bullet\to\bullet$ , the other basic figure being the _n_ode $\bullet$ given by $Y(N)$ .

The [[category of elements]] $\int_Q Y(A)$ is just the category $\bullet\rightarrow \bullet\leftarrow\bullet$ underlying the hypergraphs. Since $Y(A)$ is the representable presheaf corresponding to $A$ this is equivalent to the [[slice category]] $Q/A$. Then the following equivalence exhibits $Set^{Q^{op}}$ as an [[étendue|étendue topos]] using a general formula for [[slice topos|slices]] of [[presheaf topos|presheaf toposes]] and suggesting to view a quiver as a 'foliated' hypergraph:

$$Set^{Q^{op}}/Y(A)\simeq Set^{(Q/A)^{op}}\simeq Set^{(\int_Q Y(A))^{op}}\simeq Set^{\bullet\leftarrow\bullet\rightarrow\bullet}\quad .$$

This equivalence will be further explained in terms of graph colorings in the next section.

## Hypergraphs as 2-colored graphs

There is a classical equivalence between hypergraphs and [[vertex coloring|2-colored]] (hence [[bipartite graph|bipartite]]) graphs.  Given a hypergraph $H$, define a graph $G(H)$ whose vertex set is the [[disjoint union]] of the hypervertices and the hyperedges of $H$, and with an edge $x_v \sim x_e$ in $G(H)$ whenever the corresponding pair $(v,e)$ are incident in $H$ (creating multiple edges in the case of multiple incidence).  By coloring vertices corresponding to hypervertices in black and vertices corresponding to hyperedges in white, we satisfy the requirement that no pair of vertices sharing an edge have the same color.  Conversely, this construction is clearly reversible: any 2-colored graph $G$ determines a hypergraph $H(G)$ by interpreting black vertices as hypervertices and white vertices as hyperedges that connect all of their (black vertex) neighbors.

Giving a [[vertex coloring]] of a graph $G$ with $n$ colors is equivalent to building a graph homomorphism $G \to K_n$ into the complete graph on $n$ vertices, and so graphs equipped with a $2$-coloring are naturally organized as a [[slice]] category over $K_2$.  The classical equivalence between hypergraphs and 2-colored graphs can then be expressed formally as an [[equivalence of categories]]

$$ HGrph \cong Grph/K_2 $$

between the category of hypergraphs and the slice over $K_2$ of the category of graphs, where by the latter we mean "pseudographs" in the greatest level of generality, allowing for loops, multiple edges between vertices, and dangling half-edges (as described in [[graph#definition_in_terms_of_action_on_a_set_of_halfedges|this definition]]).  Moreover, this equivalence restricts to an equivalence

$$ SimpHGrph \cong LoopGrph/K_2 $$

where $LoopGraph$ is the category of "loop graphs", i.e., the full subcategory of $Grph$ whose objects are symmetric relations.

## Related concepts

* [[hypermap]]

* [[hyperstructure]]

* [[hypergraph category]]

* [[quiver]]

## References

* W. Dörfler and D. A. Waller, _A category-theoretical approach to hypergraphs_, Archiv der Mathematik **34** no.1 (1980) pp.185-192. DOI: 10.1007/BF01224952

* W. Grilliette, _A Functorial Link between Hypergraphs and Quivers_ , Electronic J. of Combinatorics **22** (2015). ([arXiv:1608:00058](http://arxiv.org/abs/1608.00058))

* {#Law86}[[F. William Lawvere]], _Categories of Spaces may not be Generalized Spaces as Exemplified by Directed Graphs_, Revista Colombiana de Matemáticas **XX** (1986) pp.179-186. Reprinted with commentary in TAC **9** (2005) pp.1-7. ([pdf](http://www.tac.mta.ca/tac/reprints/articles/9/tr9.pdf))

* {#Law89}[[F. William Lawvere]], _Qualitative Distinctions between some Toposes of Generalized Graphs_ , Cont. Math. **92** (1989) pp.261-299.

* T. R. S. Walsh, _Hypermaps Versus Bipartite Maps_,  Journal of Combinatorial Theory (B) **18** (1975) pp.155-163.

* Martin Schmidt, _Functorial Approach to Graph and Hypergraph Theory_, ([arXiv:1907.02574](https://arxiv.org/abs/1907.02574))

[[!redirects hypergraphs]]


hypergraph category

+-- {: .rightHandSide}
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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Graph theory
+-- {: .hide}
[[!include graph theory - contents]]
=--
=--
=--

# Hypergraph categories

* table of contents
{: toc}

## Idea

A *hypergraph category* is a [[monoidal category]] whose [[string diagrams]] are [[hypergraphs]].  Recall that in general the [[vertices]] of a string diagram correspond to [[morphisms]] in a [[category]], and its [[edges]] to [[objects]].  An ordinary string diagram is a [[directed graph]], where the inputs and outputs of a vertex describe objects appearing in a [[tensor product]]-decomposition of the [[domain]] and [[codomain]] of a morphism; each edge is connected to only one vertex as input and one vertex as output because of how morphisms in a category are composed.  A hypergraph category allows edges to connect to many vertices as input and many vertices as output, which [[category theory|category theoretically]] means that we may [[composition|compose]] many morphisms containing an object in their [[codomain]] with many morphisms containing that object in their [[domain]].

Hypergraph categories have been reinvented many times and given many different names, such as "well-supported compact closed categories" ([Carboni](#Carboni) and [RSW](#RSW)), "dgs-monoidal categories" ([GH](#GH)), and "spidered/dungeon categories" ([Morton](#Morton)).  The name "hypergraph category" is more recent ([Kissinger](#Kissinger) and [Fong](#FongDC)).

## Definition

A **hypergraph category** is:

* a [[symmetric monoidal category]] in which
* each object is equipped with the structure of a special commutative [[Frobenius algebra]], such that
* the Frobenius algebra structure of any tensor product $X\otimes Y$ is induced in the canonical way from those of $X$ and $Y$.

Note in particular that we do *not* require the morphisms of the category to be Frobenius algebra morphisms.

## Examples

* The category [[Rel]] of sets and relations (with cartesian product of sets as the monoidal product) is a hypergraph category, with the Frobenius algebra on $X$ given by the "deleting and copying" [[comonoid]] $\{ (x, *) \mid x \in X \}$, $\{ (x, (x, x)) | x \in X \}$ together with its [[opposite relation|opposite]].

* More generally, categories of [[span|spans]], [[cospan|cospans]], [[relation|relations]] and [[corelation|corelations]] in any category (with the appropriate structure) can be made into hypergraph categories by choosing the correct monoidal structure.  For example, the category [[FinRel]] is hypergraph when this category is given the $\times$ monoidal structure (beware: this is not the categorical product in $FinRel$; it comes from cartesian product in [[FinSet]]).  The same is probably true of relations in any [[regular category]].    The category $FinRel$ is not hypergraph when given the $+$ monoidal structure.   The category [[FinCorel]] is hypergraph when this category is given the $+$ monoidal structure (coming from disjoint unions).

* Categories of [[decorated cospan|decorated cospans]] and decorated corelations are hypergraph categories.

## Remarks

* The reason for the definition is that if $X$ is a special commutative Frobenius algebra, then there is a unique morphism $X^{\otimes m}\to X^{\otimes n} $ induced by the Frobenius algebra structure.  It can of course be defined as the $m$-ary multiplication followed by the $n$-ary comultiplication; the real point is that the special commutative Frobenius axioms ensure that any composite of two such morphisms is again another such morphism.  This is what enables the hypergraph string diagrams described informally above. (Some authors refer to these morphisms as "spiders" due to their appearance in [[string diagram|string diagrams]], as a black node with $m + n$ legs.)

* The free hypergraph category on one object is the category of finite sets and isomorphism classes of cospans.  This is a decategorification of the fact that the free monoidal category containing a (non-special) commutative Frobenius algebra is the category of 1-dimensional [[manifolds]] and isomorphism classes of 2-dimensional [[cobordisms]].  More general free hypergraph categories can be constructed using labeled cospans.

* Note that the special commutative Frobenius algebras are not required to be "extra-special", meaning that the morphism $I = X^{\otimes 0} \to X^{\otimes 0} = I$ need not be the identity.  Thus, the relevant sort of "hypergraphs" can contain "edges not incident to any vertices".  If we add the extra-special condition, the cospans are replaced by "co-relations", i.e. jointly surjective cospans.

* A hypergraph category can be equivalently defined as a symmetric monoidal category that [[supply in a monoidal category|supplies]] special commutative Frobenius algebras.

## Properties

### As monoids in a presheaf category

Let $\mathbf{Cospan}_\Delta$ be the free hypergraph category on [[generators and relations|generators]] $\Delta$. The objects are [[lists]] of [[elements]] of $\Delta$, or equivalently [[pairs]] $(m,l)$ of a [[natural number]] $m$ and a labelling $l\colon \big[ m \big] \to \Delta$. The morphisms are compatible [[cospans]] of [[functions]] (up to [[isomorphism]]).

+-- {: .un_prop}
###### Theorem
[Fong, Spivak](#FongSpivak).
To give a hypergraph category with chosen objects $\Delta$ that generate it is to give a lax monoidal functor $\mathbf{Cospan}_\Delta\to \mathbf{Set}$.
=--
(Formally, this can be made into an equivalence between the category of objectwise-free hypergraph categories and the lax monoidal functors; and every hypergraph category is in some sense equivalent to an objectwise-free one.)

A hypergraph category is thus a [presheaf](https://ncatlab.org/nlab/show/presheaf) $H\colon \mathbf{Cospan}_\Delta\to \mathbf{Set}$ that has lax monoidal structure
\[
1\to H()\qquad H(\vec a)\times H(\vec b)\to H(\vec a,\vec b)
\]
which is strongly reminiscent of the [mix rule](https://ncatlab.org/nlab/show/mix%20rule).
A lax monoidal functor is the same thing as a [monoid for the Day convolution](https://ncatlab.org/nlab/show/Day+convolution#Monoids). Thus hypergraph categories are monoids in the presheaf category $[\mathbf{Cospan}_\Delta, \mathbf{Set}]$.

## References

* {#Kissinger} [[Aleks Kissinger]], *Finite matrices are complete for (dagger-)hypergraph categories*. ([arxiv:1406.5942](https://arxiv.org/abs/1406.5942))

* {#FongDC} [[Brendan Fong]], *Decorated cospans*, Theory and Applications of Categories, Vol. 30, 2015, No. 33, 1096-1120. ([arxiv:1502.00872](https://arxiv.org/abs/1502.00872))

* {#FongThesis} [[Brendan Fong]], *The Algebra of Open and Interconnected Systems*, Ph.D. Thesis, Department of Computer Science, University of Oxford, 2016. ([arxiv:1609.05382](https://arxiv.org/abs/1609.05382))

* {#Carboni} [[Aurelio Carboni]], _Matrices, relations and group representations_ J. Algebra, 138(2):497–529, 1991.

* {#RSW} [[Robert Rosebrugh]], N. Sabadini, and R. F. C. Walters. *Generic commutative separable algebras and cospans of graphs*. Th. App. Cat. 15(6):164–177, 2005. [online](http://www.tac.mta.ca/tac/volumes/15/6/15-06abs.html).

* {#GH} F. Gadducci, R. Heckel. *An inductive view of graph transformation*. In "Recent Trends in Algebraic Development Techniques", Lecture Notes in
Computer Science 1376:223–237. Springer–Verlag, Berlin Heidelberg, 1998.

* {#Morton} [[Jason Morton]], _Belief propagation in monoidal categories_ In [[Bob Coecke]], I. Hasuo, P. Panangaden (eds.) _Quantum Physics and Logic_ 2014 (QPL 2014), EPTCS 172:262–269.

* {#FongSpivak} [[Brendan Fong]] and [[David Spivak]], *Hypergraph categories*. ([arxiv:1806.08304](https://arxiv.org/abs/1806.08304))

[[!redirects hypergraph categories]]
[[!redirects well-supported compact closed category]]
[[!redirects well-supported compact closed categories]]
[[!redirects dgs-monoidal category]]
[[!redirects dgs-monoidal categories]]
[[!redirects spidered category]]
[[!redirects spidered categories]]
[[!redirects dungeon category]]
[[!redirects dungeon categories]]


hypergroupoid

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###Context###
#### Higher category theory
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[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _$n$-hypergroupoid_ is a model for an [[n-groupoid]]: it is an [[Kan complex]] that is like the [[nerve]] of a [[groupoid]] ($n= 1$), [[bigroupoid]] ($n = 2$) etc.

## Definition

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

An **$n$-hypergroupoid** is a [[Kan complex]] $K$ in which the [[horn]]-fillers are _unique_ in dimension greater than $n$:

$$
  (k \gt n) \Rightarrow
  \left(
     \array{
       \Lambda^i[k] &\to&  K
       \\
       \downarrow & \nearrow_{\exists !}
       \\
       \Delta[k]
     }
  \right)
  \,.
$$

=--

(The lower dimensional horn fillers of course also exist, but are not in general unique.)

This is due to ([Duskin 79](#Duskin79), [Glenn 82](#Glenn82)), however their definition does not ask $K$ has lower dimensional horn fillers. In [Beke 04](#Beke04) these are called _exact $n$-types_ instead. For a review on the definition see ([Pridham 09, section 2](#Pridham09)).

Equivalently, this are those [[Kan complex]]es which are $(n+1)$-[[coskeletal]] and such that the $(n+1)$-horns and $(n+2)$-horns have unique fillers.

## Properties

\begin{example}
\label{OneHypergroupoids}
1-Hypergroupoids are precisely the [[nerves]] of [[groupoids]], see also the example [here](simplicial+skeleton#CoskeletalityOfSimplicialNervesOfCategories).
\end{example}

\begin{example}
2-Hypergroupoids are precisely the [[Duskin nerves]] of [[bigroupoids]].
\end{example}

## References
 {#References}

The term _hypergroupoid_ is due to

* {#Duskin79} [[John Duskin]], _Higher-dimensional torsors and the cohomology of topoi: the abelian theory_, p. 255-279 in: _Applications of sheaves_,  Lecture Notes in Mathematics **753**, Springer (1979) [[doi:10.1007/BFb0061822](https://doi.org/10.1007/BFb0061822)]

and

* {#Glenn82} [[Paul G. Glenn]], *Realization of cohomology classes in arbitrary exact categories*, J. Pure Appl. Algebra **25** 1 (1982) 33-105 [doi:10.1016/0022-4049(82)90094-9]

The term _exact $n$-type_ is used in

* {#Beke04} [[Tibor Beke]], *Higher Čech theory*, K-Theory **32** 4 (2004) 293-322 [[K:0646](https://faculty.math.illinois.edu/K-theory/0646)]

On presentation of [[infinity-stack|higher stacks]] (higher [[geometric stacks]]) by hypergroupoid objects:

* {#Pridham09} [[Jonathan Pridham]], *Presenting higher stacks as simplicial schemes*,  Adv. Math. **238** (2013) 184-245 [[arXiv:0905.4044](http://arxiv.org/abs/0905.4044), [doi:10.1016/j.aim.2013.01.009](https://doi.org/10.1016/j.aim.2013.01.009)]

[[!redirects hypergroupoids]]

[[!redirects n-hypergroupoid]]
[[!redirects 1-hypergroupoid]]
[[!redirects 2-hypergroupoid]]

[[!redirects n-hypergroupoids]]
[[!redirects 1-hypergroupoids]]
[[!redirects 2-hypergroupoids]]


hyperimaginary element

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### Context
#### Model theory
+-- {: .hide}
[[!include model theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition {#sec:orgcc9169c}

An
[[imaginary element]] of a (finitary)
[[first-order theory]] is an equivalence
class $X/E$ of a (finitary first-order) [[definable
set]] $X$ quotiented by a (finitary first-order)
[[definable groupoid|definable equivalence
relation]] $E$. A **hyperimaginary** is an equivalence
class $X/E$ of a [[definable set#type-definable-set |
type-definable]] $X$ quotiented by a
[[definable set#type-definable-set | type-definable]] equivalence relation $E$.

## Examples {#sec:orgdc24af7}

The prototypical example of a hyperimaginary is an equivalence class of
the quotient of the [[hyperreal numbers]] by
the equivalence relation "$x$ is infinitesimally close to $y$", which is
definable by the countable conjunction
$\left( \left| x - y \right| < \frac{1}{n} \right)_{n \in \mathbb{N}}$.

## Elimination of hyperimaginaries {#sec:org59c5525}

Let $X/E$ be a hyperimaginary. $T$ is said to **eliminate** the
hyperimaginary $X/E$ (c.f. [[elimination of
imaginaries]]) if $X/E$ is interdefinable with a sequence
of imaginaries, i.e. if every type-definable equivalence relation is, in
fact, a conjunction of definable equivalence relations.

[[stable theory|Stable theories]] eliminate
hyperimaginaries. This is related to a result of Hrushovski's that in a stable theory, (pro-definable) groupoids are pro-(definable groupoids).

## Properties {#sec:org623f6fa}

-   In the same way that the [[automorphism
    group]] $\operatorname{Aut}(M)$ of a model acts on
    [[imaginary element|imaginaries]],
    $\operatorname{Aut}(M)$ acts on hyperimaginaries.

-   Any hyperimaginary $X/E$ in the sense of
    the previous paragraph can be extended to a quotient of the entire
    model $M^{\alpha}$ (in the (possibly infinitary)
    [[type|sort]] $\alpha$ of $E$) by a
    type-definable equivalence relation $E'$ obtained by preserving $E$ on $X$ and extending it to be the discrete equivalence relation on $M^{\alpha} \backslash X$. In symbols, $E' \overset{\operatorname{df}}{=} (E(x,y) \wedge (x,y) \in X \times X) \vee (x = y)$. (This is a special case of the fact that finite disjunctions of type-definable sets are again type-definable. It is not true that arbitrary disjunctions of type-definable sets are again type-definable, since $\operatorname{Aut}(\mathbb{M})$-invariant subsets of $\mathbb{M}$ the [[monster model]] have this form.)

-   The process of taking points of hyperimaginaries in models
    ($M \mapsto M(X/E)$) does not commute
    with [[ultraproducts]]. For example, if
    $E$ is the equivalence relation in the reals of being
    infinitesimally close together, it's easy to see that
    $\mathbb{R}^{\mathcal{U}}/E \not \simeq \left(\mathbb{R}/E \right)^{\mathcal{U}}$,
    since $\mathbb{R}/E \simeq \mathbb{R}$ (after all, the real numbers
    are Hausdorff) so that the right hand side is the nonstandard reals
    $^* \mathbb{R}$ again (so every number has a [[halo|cloud]] of infinitesimals
    around it) while the left hand side has no points which are
    infinitesimally close together.

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

This is actually an excellent example of why we shouldn't expect in
general the process of taking points of hyperimaginaries in a model to
commute with ultraproducts: on one hand, if $E$ is a type-definable
equivalence relation on $X$, and $\mathcal{U}$ an
[[ultrafilter]], then computing
$\left(\prod_{i \in I} M_i/\mathcal{U} \right)(X/E)$ yields
$E$-equivalence classes of [[germs]]
$[m_i]_{\mathcal{U}}$. That is, for every condition $\varphi(x_1, x_2)$
in $E$ that needs to be checked to conclude that two tuples are
$E$-equivalent, every pair of $E$-equivalent germs
$[m_i]_{\mathcal{U}}$ and $[n_i]_{\mathcal{U}}$ agree on $\varphi$ on
some subset $P \subseteq I$ for $P$ in $\mathcal{U}$.But these $P$s
depend on $\varphi$. They aren't uniform, and there's possibly no set
$P'$ in the ultrafilter (the intersection of all the $P$'s might be
empty) on which one could decide if $[m_i]$ and $[n_i]$ are
$E$-equivalent for every condition $\varphi \in E$.

On the other hand, one could try computing
$\left( \prod_{i \in I} M_i(X/E) \right) /\mathcal{U}$. One ends up
instead with germs of hyperimaginaries, as in
$[[m_i]_E]_{\mathcal{U}}$, but this is a tighter
equivalence relation: $([x_i]_E)_i$ and $([y_i]_E)_i)$ are
$\mathcal{U}$-equivalent if and only if there is a single $P \in \mathcal{U}$,
uniform for every $i \in P$ and for every $\varphi \in E$ so that
$M_i \models \varphi(x_i, y_i)$. This means that the commutation of
these two procedures should fail precisely when the $P$'s on the left
hand side intersect to something outside of the ultrafilter.

Indeed, in the example we first mentioned, consider the sequences
$(1, 2, 3, \dots)$ and $(1 + 1/2, 2 + 1/3, 3 + 1/4, \dots)$. If we take
them mod-$\mathcal{U}$, then mod-$E$, then they are equivalent as
hyperimaginaries in $^*\mathbb{R}$, since for each condition "x is less
than $1/n$ away from y", these two sequences agree cofinitely many times
(and every ultrafilter contains the cofinite filter). But these cofinite
sets of indices are getting smaller and smaller and intersect to
$\emptyset$. And if we take them mod-$E$, then mod-$\mathcal{U}$, they
remain distinct.

=--

## Related concepts {#sec:org1bb2b22}

-   [[imaginary element]]

-   [[infinitesimal]]

-   [[simple theory]]

## References {#sec:org00f5b3c}

-   Frank Wagner, *Simple Theories* (2000)

-   Byunghan Kim, *Simplicity Theory* (2014)

-   Enrique Casanovas, *Simple theories and hyperimaginaries* (2011)

[[!redirects hyperimaginary]]
[[!redirects hyperimaginaries]]
[[!redirects hyperimaginary elements]]
[[!redirects elimination of
hyperimaginaries]]


hyperkähler manifold

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#### Riemannian geometry
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $n \in \mathbb{N}$ a [[natural number]], a $4n$-[[dimension|dimensional]] [[Riemannian manifold]] is a _hyperkähler manifold_ if its [[holonomy group]] is (a [[subgroup]] of) the [[quaternionic unitary group]] [[Sp(n)]]. Regarded as a subgroup of the [[central product group]] [[Sp(n).Sp(1)]] this means that hyper-Kähler manifolds are special cases of [[quaternion-Kähler manifolds]], though the latter are often taken to be only those [[Riemannian manifolds]] with full [[Sp(n).Sp(1)]]-[[holonomy]].

Equivalently, a _hyperkähler manifold_ is a [[Riemannian manifold]] $(M,g)$ with three [[complex structure|complex structures]] $I, J, K$ which are [[Kähler manifold|Kähler]] with respect to the metric $g$ and satisfy the [[quaternion|quaternionic]] identities
$$I^2=J^2=K^2=I J K=-1.$$

## Properties

### As part of the Berger classification

[[!include special holonomy table]]

### As special $\mathbb{H}$-Riemannian manifolds

[[!include normed division algebra Riemannian geometry -- table]]

### Rozansky-Witten weight systems

Every [[hyperkähler manifold]] induces a [[Rozansky-Witten weight system]] with [[coefficients]] in certain [[Dolbeault cohomology]]-groups. For [[compact hyperkähler manifolds]] there are induced [[Rozansky-Witten weight system]] with values in the [[ground field]], hence actual [[weight systems]].

## Examples

### Compact hyperkähler manifolds

The only known examples of [[compact hyperkähler manifolds]] are [[Hilbert schemes of points]] $X^{[n+1]}$ (for $n \in \mathbb{N}$) for $X$ either

1. a [[K3-surface]]

1. a [[4-torus]] (in which case the [[compact hyperkähler manifolds]] is really the [[fiber]] of $(\mathbb{T}^4)^{[n]} \to \mathbb{T}^4$)

([Beauville 83](#Beauville83))
and two exceptional examples ([O'Grady 99](#OGrady99), [O'Grady 03](#OGrady03)
), see [Sawon 04, Sec. 5.3](#Sawon04).

### Coulomb- and Higgs-branches of $D=3$ $\mathcal{N} =4 $ SYM

Both the [[Coulomb branch]] and the [[Higgs branch]] of [[D=3 N=4 super Yang-Mills theories]] are [[hyperkähler manifolds]] ([Seiberg-Witten 96](D%3D3+N%3D4+super+Yang-Mills+theory#SeibergWitten96), see e.g. [dBHOO 96](D=3+N=4+super+Yang-Mills+theory#dBHOO96)).
In special cases they are [[compact hyperkähler manifolds]]
([Intriligator 99](D=3+N=4+super+Yang-Mills+theory#Intriligator99)).

## Related concepts

* [[Kähler manifold]], **hyper-Kähler manifold**, [[quaternionic Kähler manifold]]

* [[compact hyperkähler manifold]]

* [[ADE singularity]]

* [[Rozansky-Witten theory]]

## References

### General

See also

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

### Compact hyperkähler manifolds

* {#Beauville83} [[Arnaud Beauville]], _Variétés Kähleriennes dont la premiere classe de Chern est nulle_, Jour.
Diff. Geom. 18 (1983), 755–782 ([euclid.jdg/1214438181](https://projecteuclid.org/euclid.jdg/1214438181))

* {#OGrady99} [[Kieran O’Grady]], _Desingularized moduli spaces of sheaves on a K3_, J. Reine Angew. Math. 512 (1999), 49–117 ([arXiv:alg-geom/9708009](https://arxiv.org/abs/alg-geom/9708009), [arXiv:math/9805099](https://arxiv.org/abs/math/9805099))

* {#OGrady03} [[Kieran O’Grady]], _A new six dimensional irreducible symplectic variety_, J. Algebraic Geom. 12 (2003), 435-505 ([arXiv:math/0010187](https://arxiv.org/abs/math/0010187))

* [[Daniel Huybrechts]], _Compact Hyperkähler Manifolds_, In: Ellingsrud G., Ranestad K., Olson L., Strømme S.A. (eds.) _Calabi-Yau Manifolds and Related Geometries_, Universitext. Springer, Berlin, Heidelberg 2003 ([doi:10.1007/978-3-642-19004-9_3](https://doi.org/10.1007/978-3-642-19004-9_3),
[arXiv:alg-geom/9705025](https://arxiv.org/abs/alg-geom/9705025))

### Rozansky-Witten invariants

With an eye towards [[Rozansky-Witten theory]] ([[ground field]]-valued [[Rozansky-Witten weight systems]]):

* {#RobertsWillerton06} [[Justin Roberts]], [[Simon Willerton]], p. 17 of: _On the Rozansky-Witten weight systems_, Algebr. Geom. Topol. 10 (2010) 1455-1519 ([arXiv:math/0602653](https://arxiv.org/abs/math/0602653))

* {#Sawon04} [[Justin Sawon]], Section 5.3 of: _Rozansky-Witten invariants of hyperkähler manifold_, Cambridge 2000 ([arXiv:math/0404360](https://arxiv.org/abs/math/0404360))

### Examples

The example of [[Coulomb branches]] of [[D=3 N=4 super Yang-Mills theory]] originates with

* {#SeibergWitten96} [[Nathan Seiberg]], [[Edward Witten]], _Gauge Dynamics And Compactification To Three Dimensions_, In: J.M. Drouffe, J.B. Zuber (eds.) _The mathematical beauty of physics: A memorial volume for Claude Itzykson_ Proceedings, Conference, Saclay, France, June 5-7, 1996 ([arXiv:hep-th/9607163](http://arxiv.org/abs/hep-th/9607163), [spire:420925](http://inspirehep.net/record/420925))

[[!redirects hyperkähler manifolds]]

[[!redirects hyper-Kähler manifold]]
[[!redirects hyper-Kähler manifolds]]

[[!redirects hyper-Kähler geometry]]
[[!redirects hyper-Kähler geometries]]

[[!redirects hyper-Kaehler manifold]]
[[!redirects hyper-Kaehler manifolds]]

[[!redirects Hyperkähler manifold]]
[[!redirects Hyperkähler manifolds]]

[[!redirects hyper-Kähler structure]]
[[!redirects hyper-Kähler structures]]

[[!redirects hyper-Kähler structure]]
[[!redirects hyper-Kähler structures]]

[[!redirects hyper-Kaehler structure]]
[[!redirects hyper-Kaehler structures]]

[[!redirects Hyperkähler structure]]
[[!redirects Hyperkähler structures]]


hypermagma

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### Context
#### Algebra
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=--
=--
=--

# Contents
* table of contents
{:toc}

[[!redirects hypergroup]]
[[!redirects hypermagmas]]
[[!redirects hypergroups]]
[[!redirects comagma]]

## Idea

A **hypermagma** $X$ relates to a [[magma]] like a [[hypergraph]] relates to an ordinary [[graph]] i. e. the binary operation on $X$ becomes multi-valued by taking value in $2^X$ instead of $X$.

By imposing further axioms one obtains the concept of a **hypergroup** corresponding to the generalization of the [[group|group concept]] in this context.

With the commutative variant of a [[canonical hypergroup]] this "_multi-valued hyperalgebra_" has recently gained prominence in [[number theory]] and algebraic geometry over $\mathbb{F}_1$ (cf. [[hyperring]]).

## Definition

A set $X$ together with a function $X\times X\to 2^X$ is called a _hypermagma_. The function is normally denoted by $\cdot$ and called the _hyperlaw_ (, _hyperoperation_ or _hyperproduct_) of $X$. A _morphism of hypermagmas_ from $X$ to $Y$ is a function $f:X\to Y$ such that $f(x\cdot y)\subset f(x)\cdot f(y)$ for all $x,y\in X$. The morphism is called _good_ if $f(x\cdot y) = f(x)\cdot f(y)$.

A hypermagma $X$ that satisfies furthermore:

* $(x\cdot y)\cdot z = x\cdot (y\cdot z)$ for all $x,y,z\in X$ ( _associativity_ ) and

* $x\cdot X= X = X\cdot x$ for all $x\in X$ ( _reproduction_ )

is called a _hypergroup_.

## Remarks

Given a hypermagma $X$ the hyperlaw is extended to $A,B\in 2^X$ by $A \cdot B=\cup_{a\in A,b\in B} a\cdot b$. Hence, $x\cdot Y$  is understood as $\{x\}\cdot Y$ etc.

Suppose $X$ is an associative hypermagma and $x\cdot y=\emptyset$ then $x\cdot y \cdot X=\emptyset$ and, accordingly, $y\cdot X\neq\X$ or $x\cdot X\neq X$ whence $X$ can't be a hypergroup. So we see that the hyperlaws of hypergroups are in fact valued in _non-empty_ subsets - hypermagmas with this property are sometimes called _hypergroupoids_.

But note that $\emptyset$ together with the empty map is nevertheless a hypergroup, in fact, the initial hypergroup in the obvious _category of hypergroups_ $\mathbf{HypGrp}$.

By imposing commutativity one arrives at the notion of a [[canonical hypergroup]] that enters into the definition of a [[hyperring]].

Hypergroups whose hyperlaw is valued in singleton subsets correspond to groups.

## Example

Let $G$ be a compact [[Lie group]] and $\hat{G}$ the set of its irreducible representations. Given $a,b\in\hat{G}$ define $a\cdot b$ as the set of irreducible representations $\mu_1,\dots,\mu_k$ occurring in the decomposition $a\otimes b=\Sum m_i \mu_i$. This endows $\hat{G}$ with the structure of a hypergroup. (For further details and the connection to operator algebra see [Litvinov (2011)](#Lit11)).

## Related entries

* [[magma]]

* [[hypermonoid]]

* [[canonical hypergroup]]

* [[hyperring]]

* [[hypergraph]]

* [[historical notes on quasigroups]]

## Reference

* S. D. Comer, _Polygroups derived from cogroups_ , J. Algebra **89** no.2 (1984) pp.397-405.

* P. Corsini, V. Leoreanu-Fotea, _Applications of Hyperstructure Theory_ , Kluwer Dordrecht
2003.

* {#HS90}L. Haddad, Y. Sureau, _Les cogroupes et la construction de Utumi_ , Pacific J. Math. **145** no.1 (1990) pp.17-58. ([abstract](http://projecteuclid.org/euclid.pjm/1102645606))

* L. Haddad, Y. Sureau, _Les groupes, les hypergroupes et l'énigme des Murngin_ , JPAA **87** (1993) pp.221-235.

* J. Jantosciak, _Transposition Hypergroups: Noncommutative Join Spaces_ , J. Algebra **187** (1997) pp.97-119.

* {#Lit11}G. L. Litvinov, _Hypergroups and Hypergroup Algebras_ , arXiv:1109.6596 (2011). ([abstract](http://arxiv.org/abs/1109.6596))

* F. Marty,  _Sur une généralization de la notion de groupe_ , IV Congrès des Mathématiciens Scandinaves, Stockholm 1934.


hypermatrix

Another term for _[[tensor]]_, mostly used when speaking of [[hyperdeterminants]]. See there.

[[!redirects hypermatrices]]


hypermonoid

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

## Definition

By removing structure and properties from the notion of [[canonical hypergroup|hypergroup]], a **hypermonoid** is a set equipped with an associative multivalued multiplication. There are several equivalent ways of making this more precise:

* A _hypermonoid_ is a monoid $M$ in the monoidal [[Pos]]-enriched category of sets and total ([[entire relation|entire]]) relations, where the tensor product of sets is cartesian product and the tensor product of relations $R: A \to B$, $S: C \to D$ is defined by the truth-value assignment
$$(R \times S)((a, c), (b, d)) = R(a, b) \wedge S(c, d)$$
The unit $u: 1 \to M$ is assumed to be a well-defined relation (therefore a function).

* A _hypermonoid_ is a $\mathbf{2}$-enriched [[promonoidal category]] structure on a discrete poset $S$, whose prounit is a singleton and whose promultiplication $S \times S \to P S$ takes values in $P_+(S)$ (nonempty subsets);

* A _hypermonoid_ is a structure of [[quantale]] on a power set $P(S)$, whose unit is a singleton and where the quantale multiplication $\cdot$ satisfies $T \cdot U \neq \bot$ whenever $T, U \neq \bot$.

(In the opinion of one of the article authors ([[Todd Trimble]]), one may legitimately ask what is the mathematical gain in restricting to total relations and in the condition that the unit is functional. Lifting these restrictions, one obtains the simpler notion of quantale.)

## Examples

### Special types

Analogous to how ordinary [[monoid]]s refine to [[group]]s and to [[ring]]s, hypermonoids refine to

* [[hypergroup]]s

* and [[hyperring]]s.

See there for more examples.

### Specific examples

#### Boolean Hecke algebra of a Coxeter group

Let $W$ be a finite Coxeter group. A [[Coxeter group]] is _by definition_ a group presentation, generated by elements $s_1, \ldots, s_n$ and subject to relations of the form

$$(s_i s_j)^{m_{i j}} = 1$$

where $m_{i j}$ is a collection of integers specified by a Coxeter diagram or Coxeter matrix, assumed symmetric and with $m_{i i} = 1$. We may rewrite the presentation as

$$s_{i}^{2} = 1, \qquad s_i s_j \ldots = s_j s_i \ldots$$

where the words on either side of the second equation alternate between $s_i$ and $s_j$ and have length $m_{i j}$.

The Boolean [[Hecke algebra]] of a Coxeter group $W$ is a quantale whose underlying sup-lattice is $P W$, and whose quantalic multiplication is uniquely determined by a hypermonoid structure which satisfies

$$\{s_i\} \cdot \{s_i\} = \{s_i, 1\}, \qquad s_i s_j \ldots = s_j s_i \ldots$$

and described as follows:

* If $w$ is a reduced word $s_{i_1} s_{i_2} \ldots s_{i_k}$, then the hypermonoid product $\prod_j s_{i_j}$ is $\{[w]\}$ where $[w]$ is the value of $w$ in $W$;

* Given a generator $s$ and a reduced word $w = w' s$ in the Coxeter group presentation, define $\{[w]\} \cdot \{s\} = \{[w' s], [w']\}$.

It may be shown that this recipe gives a well-defined map $W \times W \to P(W)$ that gives a hypermonoid structure.

This description appears to depend on combinatorial technicalities of Coxeter groups, but it can be made more conceptual if the Coxeter group $W$ is seen as arising from a BN-pair. A [[BN-pair]] (see also references below) is given by a simple [[algebraic group]] $G$, a Borel subgroup $B$, and a normalizer $N$ of a maximal torus $T$ for which $T = B \cap N$ and $G = B N$, satisfying some conditions. The corresponding Boolean Hecke algebra may be defined to be the quantale whose elements are $G$-equivariant sup-preserving maps

$$P(G/B) \to P(G/B)$$

where the quantale multiplication is composition. As a sup-lattice, this quantale is isomorphic to $P(B\backslash G/B)$, the power set of the set of [[double cosets]] $B g B$. According to the general theory of BN-pairs, the double cosets are in bijection with the elements of a Coxeter group $W$ given by $N/T$; the bijection is given by the well-defined map

$$W \to B \backslash G/B: w \mapsto B w B$$

Hence $P(W) \cong P(B\backslash G/B)$ as sup-lattices, but the algebra structure on $P(B \backslash G/B)$ can be regarded as a deformation of the group algebra structure. The unit is the singleton $\{B e B\}$.

#### Other examples

Hecke algebras are a [[deformation theory|deformation]] of the Coxeter group algebra. Under certain conditions, other deformations of group or monoid algebras of monoids $M$ can be exploited to yield hypermonoids.

For example, if the multiplication table of such a deformation is given by equations

$$g_i g_j = \sum_k a_{i j}^{k} g_k$$

where all the $a_{i j}^k$ are nonnegative rational numbers of which at least one is nonzero for each pair $i, j$, then we may obtain a hypermonoid by replacing each positive $a_{i j}^{k}$ by $1$ and interpreting the sum as a join in the Boolean algebra $P(M)$. This procedure may be termed "Booleanization". For example, in the case of Hecke algebras, we have

$$s_{i}^{2} = \frac{q-1}{q} s_i + \frac1{q}$$

and if the parameter $q$ is specialized to positive integer values (e.g., powers of primes), one may apply Booleanization to obtain the Boolean Hecke algebra.

In general the principle is this: if the structure coefficients $a_{i j}^{k}$ belong to a [[rig]] $A$, so that we have a deformation $A[M]_d$ of a monoid rig, and if one is provided with a rig homomorphism $p: A \to\mathbf{2} = (\{0, 1\}, + = \vee, \cdot = \wedge)$ then there is a Booleanized hypermonoid structure defined by the composite

$$M \times M \stackrel{mult}{\to} A[M]_d \cong A \otimes_A A[M]_d \stackrel{p \otimes_A 1}{\to} \mathbf{2} \otimes_A A[M]_d \cong \mathbf{2}[M]_d \hookrightarrow P(M)$$

For instance, there is a unique rig homomorphism $p: \mathbb{Q}_+ \to \mathbf{2}$ from the rig of nonnegative rational numbers.

## References

* [[Todd Trimble]],  _[Notes on BN-pairs and Boolean Hecke algebras](http://math.ucr.edu/home/baez/trimble/BN-pairs_and_Boolean_Hecke_algebras.html)_  (2005)

[[!redirects hypermonoids]]


hyperplane

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## Idea

By a *hyperplane* one usually means an affine or [[linear subspace]] of an [[affine space]] or [[linear space]], respectively, typically required to be a [[positive number|positive]] [[dimension of a vector space|dimension]] and [[codimension]] and often required to have [[codimension]]  = 1.

## Examples

In the archetypical example the ambient space is the time-honored [[Euclidean space|Euclidean]]/[[Cartesian space]] $\mathbb{R}^3$ and a hyperplane of [[codimension]] 1 is an ordinary [[plane]].

More sophisticated examples arise for instance in the discussion of [[configuration spaces of points]] in a [[Cartesian space]] $E = \mathbb{R}^d$, where hyperplanes in $E^n$ given by $\big\{ (\vec x_1, \cdots, \vec x_n) \,\in\, E^n \;\big\vert\; \vec x_i = \vec x_j \big\}$ reflect the subspaces where positions of a given [[pair]] of points coincide. This leads to discussion of [[arrangements of hyperplanes]] and their [[complements]], see also at *[[Knizhnik-Zamolodchikov equation]]* the references on *[twisted cohomology of configuration spaces of points](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces)*.

## Related concepts

* [[plane]]

* [[hypercube]]

* [[hyperplane line bundle]]

* [[arrangements of hyperplanes]]

* [[distribution of subspaces]]

## References

See also:

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

[[!redirects hyperplanes]]


hyperplane line bundle

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## Idea

Let $\mathbb{P}^n = \mathbb{P}^n_k$ be the $n$-dimensional [[projective space]] over a [[field]] $k$, whose points are the equivalence classes $[z_0,\ldots,z_n]$ of $(n+1)$-tuples $(z_0,\ldots,z_n)\in k^{n+1}\backslash \{0\}$; the (co)domains of usual open charts in the sense of [[manifolds]], which are [[Zariski topology|Zariski-open]] subsets for general $k$, have $U_i = \{[z_0,\ldots,z_n] | z_i\neq 0\}$.

The __hyperplane line bundle__ $\mathcal{O}(1)$ on $\mathbb{P}^n$ is a [[line bundle]] given by the [[transition functions]] $g_{i j}([z_0,\ldots,z_n])=z_j/z_i$ on $U_i\cap U_j$. Its dual bundle $\mathcal{O}(1)^*$ is the __tautological line bundle__ (or universal bundle) usually denoted by $\mathcal{O}(-1)$ and the tensor powers are $\mathcal{O}(n)=\mathcal{O}(1)^{\otimes n}$, $\mathcal{O}(-n)=\mathcal{O}(-1)^{\otimes n}$ for $n\geq 0$. The total space of the tautological line bundle can be identified with $\mathbb{C}^{n+1}$ and the projection is exactly $(z_0,\ldots,z_m)\mapsto[z_0,\ldots,z_m]$, i.e. the fiber over $[z_0,\ldots,z_m]$ is the line $\{(\lambda z_0,\ldots,\lambda z_n), 0 \neq \lambda\in \mathbb{C}\}$. The __[[canonical line bundle]]__ $K = \Lambda^n T^* \mathbb{P}^n$ equals $\mathcal{O}(-n-1)$.

The bundles $\mathcal{O}(n)$ are holomorphic if $k=\mathbb{C}$. The sheaves of (regular or holomorphic) sections are also denoted as $\mathcal{O}(n)$ and are said to be the _twists_ of the structure sheaf $\mathcal{O}$; they restrict to the equally denoted sheaves on any projective subvariety and these restrictions up to an isomorphism do not depend on a particular embedding into a particular projective space.

## Related concepts

* [[hyperplane]]

* [[classifying space]]

* [[canonical line bundle]]

* [[tautological line bundle]]

[[!redirects hyperplane line bundles]]


hyperplane section theorem

The Lefschetz hyperplane section theorem says that [[cohomology|cohomologically]] a nonsingular [[complex variety]] looks like its [[hyperplane]] sections. More precisely,

+-- {: .un_theorem}
###### Hyperplane section theorem
Let $X$ be an [[algebraic variety|algebraic subvariety]] of complex projective space and $H$ a generic hyperplane in $\mathbb{C}^n$.
Then the $i$-th [[relative cohomology]] $H_i(X,X\cap H) = 0$ for $i\lt n$.
=--

There is a related deeper theorem, also due to Lefschetz, the [[hard Lefschetz theorem]].

There is also a version of the quantum hyperplane section theorem due to Y.-P. Lee, where the cohomology is replaced by the quantum cohomology.

* Goresky, Mac Pherson, _Stratified Morse theory_

* Kyle Hofmann, _The Lefschetz hyperplane section theorem_, [pdf](http://www-personal.umich.edu/~krhofman/lefschetz-hyperplane.pdf)

* [wikipedia](http://en.wikipedia.org/wiki/Lefschetz_hyperplane_theorem)

* Y-P. Lee, _Quantum Lefschetz hyperplane theorem_, Inventiones Math. __145__, 1, 121--149, 2001 ([doi](http://dx.doi.org/10.1007/s002220100145))

* Mark Andrea A. de Cataldo, Luca Migliorini, _The perverse filtration and the Lefschetz hyperplane theorem_, accepted to Annals of Math. [pdf](http://pjm.math.berkeley.edu/editorial/uploads/annals/accepted/090120-Cataldo/090120-Cataldo-v1.pdf)

[[!redirects Lefschetz hyperplane theorem]]
[[!redirects Lefschetz hyperplane section theorem]]


hyperring

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

## Idea

A **hyperring** is like a [[ring]] not with an underlying [[abelian group]] but an underlying [[canonical hypergroup]].

It is a [[hypermonoid]] with additional ring-like [[stuff, structure, property|structure and properties]].

This means that in a hyperring $R$ addition is a multi-valued operation.

## Definition

A **hyperring** is a non-empty [[set]] $R$ equipped with a hyper-addition $+ : R\times R \to P^*(R)$ (where $P^*(R)$ is the set of non-empty subsets) and a multiplication $\cdot : R \times R \to R$ and with elements $0,1 \in R$ such that

1. $(R,+)$ is a [[canonical hypergroup]];

1. $(R,\cdot)$ is a [[monoid]] with identity element $1$;

1. $\forall r,s \in R : r(s+t) = r s + r t$ and $(s + t) r = s r + t r$;

1. $\forall r \in R : r \cdot 0 = 0 \cdot r = 0$;

1. $0 \neq 1$.

We can form many examples of hyperrings by quotienting a ring $R$ by a subgroup $G \subset R^{\times}$ of its multiplicative group.

A **morphism of hyperrings** is a map $f : R_1 \to R_2$ such that

1. $\forall a,b \in R_1 : f(a + b) \subset f(a) + f(b)$;

1. $\forall a,b\in R_1 : f(a \cdot b) = f(a) \cdot f(b)$.

A **hyperfield** is a hyperring for which $(R - \{0\}, \cdot)$ is a [[group]].

## Examples

### Hyperfield extension of field with one element

The **hyperfield extension of the [[field with one element]]** is

$$
  \mathbf{K} := (\{0,1\}, +, \cdot)
$$

with additive neutral element $0$ and the hyper-addition rule

$$
  1 + 1 = \{0,1\}
  \,.
$$

This is to be thought of as the hyperring of [[integer]]s modulo the relation "is 0 or not 0": think of $0 \in \mathbf{K}$ as being the integer 0 and of $1 \in \mathbf{K}$ as being _any_ non-zero integer, then the addition rule says that 0 plus any non-zero integer is non-zero, and that the sum of a non-zero integer with another non-zero integer is either zero or non-zero.

### The signature hyperfield $\mathbf{S}$

Let $\mathbf{S} = \{0,1,-1\}$ be the hyperfield with multiplication induced from $\mathbb{Z}$ and with addition given by 0 being the additive unit and the laws

* $1+1 = \{1\}$;

* $-1 + -1 = \{-1\}$

* $1 + -1 = \{-1, 0, 1\}$.

This we may think of as being the hyperring of [[integer]]s modulo the relation "is positive or negative or 0": think of $1$ as being any positive integer, $0$ as being the integer $0$ and $-1$ as being any negative integer. Then the hyper-addition law above encodes how the signature of integers behaves under addition.

**Proposition**

To each element, $\phi$, of $Hom(\mathbb{Z}[X], \mathbf{S})$ there corresponds an extended real number, $Re(\phi) \in [-\infty, \infty]$ given as a Dedekind cut. This is a surjective mapping. The inverse image of each real algebraic number contains three elements, while that of a nonalgebraic number is a singleton. For real algebraic $\alpha$, the three homomorphisms from $\mathbb{Z}[X]$ to $\mathbf{S}$ are

$$
P(T) \mapsto \underset{\epsilon \to 0+} {lim} sign P(\alpha + t \epsilon), t \in \{-1, 0, 1\}.
$$

## Related entries

* [[hypergroup]]

* [[canonical hypergroup]]

* [[hypermonoid]]

## Book

* B. Davvaz and V. Leoreanu-Fotea, Hyperring Theory and Applications, International Ackademic Press, USA, 2007.

## References

The notion of hyperring and hyperfield is due to Marc Krasner:

* M. Krasner, _Approximation des corps valués complets de caractéristique $p\neq 0$ par ceux de caractéristique
0_ , pp.126-201 in _Colloque d' Algébre Supérieure (Bruxelles), 1956_ , Ceuterick Louvain 1957.

Another early reference is

* D. Stratigopoulos, _Hyperanneaux non commutatifs: Hyperanneaux, hypercorps, hypermodules, hyperespaces vectoriels et leurs propriétés élémentaires_ (French) C. R. Acad. Sci. Paris Sér. A-B 269 (1969) A489–A492.

Modern applications in connection to the [[field with one element]] are discussed in

* [[Alain Connes]], [[Caterina Consani]], _The hyperring of adèle classes_ ([arXiv:1001.4260](http://arxiv.org/abs/1001.4260))

* [[Alain Connes]], [[Caterina Consani]], _From monoids to hyperstructures: in search of an absolute arithmetic_ ([arXiv:1006.4810](http://arxiv.org/abs/1006.4810))

An overview is in

* Jaiung Jun, _Algebraic geometry over hyperrings_, [arXiv:1512.04837](http://arxiv.org/abs/1512.04837)

category: algebra

[[!redirects hyperring]]
[[!redirects hyperrings]]
[[!redirects hyper-ring]]
[[!redirects hyper-rings]]

[[!redirects hyperfield]]
[[!redirects hyperfields]]
[[!redirects hyper-field]]
[[!redirects hyper-fields]]


hypersequent calculus

#Contents#

## Idea

(...)

## Related entries

* [[sequent calculus]]

* [[Gödel logic]]

## References

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

[[!redirects hypersequent calculi]]

[[!redirects hypersequent]]
[[!redirects hypersequents]]


hyperset

* Peter Aczel, _Non-well-founded sets_, CSLI Publications, Stanford, 1988. [MR89j:03039](http://www.ams.org/mathscinet-getitem?mr=940014)

* [[Jacob Lurie]], _Anti-admissible sets_, J. Symbolic Logic __64__ (1999), no. 2, 407–435, [MR2002g:03086](http://www.ams.org/mathscinet-getitem?mr=1777761), [doi](http://dx.doi.org/10.2307/2586475)

[[!redirects hypersets]]


hypersheaf

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## Idea

A **hypersheaf** is a [[presheaf]] satisfying [[descent]] with respect to all [[hypercovers]].


hypersimplex

#Contents#
* table of contents
{:toc}

## Idea

The $(k,n)$-**hypersimplex** is a convex [[polytope]] $\Delta(k, n)$ (or $\Delta_{k,n}$) in $\mathbb{R}^n$ which is the [[convex hull]] of the $\left(\array{n \\ k}\right)$ points $e_{i_1}+ e_{i_2}+ \cdots + e_{i_k}$, $1\leq i_1\lt i_2\lt \ldots \lt i_k\leq n$, where the $e_{j}$ are the standard basis vectors. $\Delta(k,n)$ may be seen as a convex hull of the [[barycenter]]s of the $(k-1)$-dimensional faces of $a (n-1)$-dimensional [[simplex]]. Special cases, $\Delta(1,n)$ and $\Delta(n-1,n)$ are themselves simplices of dimension $(n-1)$.

The combinatorics of hypersimplices extends the combinatorics of distinguished triangles and octahedra in the standard [[triangulated categories]]; in fact they are postulated in [[Georges Maltsiniotis|Maltsiniotis]]'s strong version of a triangulated category. The [[octahedron]] is a $(2,4)$-hypersimplex. Higher hypersimplices were indeed obtained from $A_\infty$-enrichments by [[Volodymyr Lyubashenko]], so one could expect that they can also be obtained from the homotopy category of a [[stable (∞,1)-category]]. There are also connections between distinguished hypersimplices and [[Postnikov system in a triangulated category|Postnikov towers in triangulated categories]].

## References

* [[Israel Gelfand|I.M.Gelfand]], R.D.MacPherson, _Geometry in Grassmannians and a generalization of the dilogarithm_, Adv. Math., 44 A982), 279--312; Coll. pap. of I.M. Gelfand v. __3__, Springer 1989, p. 492--525.

* I. M. Gelfand, [[Mikhail Kapranov|M. Kapranov]], [[Andrei Zelevinsky|A. Zelevinsky]], _Discriminants, resultants, and multidimensional determinants_, 1994.

* [[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.

* Fred J. Rispoli, _The graph of the hypersimplex_, [arxiv/0811.2981](http://arxiv.org/abs/0811.2981)

* T. Lam, A. Postnikov, _Alcoved polytopes I_, [math.CO/0501246](http://arxiv.org/abs/math/0501246)

* Yu. Bespalov, [[Volodymyr Lyubashenko]], O. Manzyuk, _Pretriangulated $A_\infty$-categories_, Proceedings of the Institute of Mathematics of NAS of Ukraine, vol. 76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008, 598 ([ps.gz](http://www.math.ksu.edu/~lub/cmcMonad.gz)) (chapter 13)

* A. [[Beilinson]], J. Bernstein, P. Deligne, _Faisceaux pervers_, Astérisque __100__ (1982). MR 86g:32015 (diagrams in Rem. 1.1.14 are hypersimplices)

* [[Georges Maltsiniotis]], _Catégories triangulées supérieures_, Pré-preprint [ps](http://people.math.jussieu.fr/~maltsin/ps/triansup.ps) (2005)

* [[Jacob Lurie]], _Derived Algebraic Geometry I: Stable $\infty$-Categories_, ([arXiv](http://www.arXiv.org/abs/math.CT/0608228), [pdf](http://www.math.harvard.edu/~lurie/papers/DAG-I.pdf))


hyperstonean space

#Contents#
* table of contents
{:toc}

## Definition

A __hyperstonean space__ (French: espace hyperstonien)
is a [[Stonean space]] such that the union of supports
of all normal measures is [[everywhere dense]].

A __normal measure__ is a [[Radon measure]] that vanishes
on [[nowhere dense]] subsets.

## Properties

The support of a normal measure is a [[clopen]] subset.

In a hyperstonean space, all [[meager subsets]] are [[nowhere dense]]
and the support of any normal measure is a [[clopen subset]].

Every [[Stonean space]] decomposes as a [[coproduct]] of three [[clopen subsets]] $E_1$, $E_2$, and $E_3$ with the following properties:

* $E_1$ has an [[everywhere dense]] [[meager subset]].  The only normal measure on $E_1$ is the zero measure.

* $E_2$ is a hyperstonean spaces.

* Every [[meager subset]] of $E_3$ is [[nowhere dense]] and every [[Radon measure]] has a [[nowhere dense]] support.  The only normal measure on $E_3$ is the zero measure.

By [[Gelfand-type duality for commutative von Neumann algebras]],
the category of hyperstonean spaces and open continuous maps
is equivalent to the opposite category of [[localizable Boolean algebras]],
the category of [[measurable locales]],
and the opposite category of commutative [[von Neumann algebras]].

## Hyperstonean locales

Hyperstonean locales can be defined as [[Stonean locales]]
that admit sufficiently many [[normal valuations]].

Assuming the [[axiom of choice]], [[Stonean locales]] are
[[spatial]], so the category of hyperstonean locales
is equivalent to the category of hyperstonean spaces.

## References

Hyperstonean spaces were introduced and studied by [[Jacques Dixmier]]:

* [[Jacques Dixmier]], _Sur certains espaces considérés par M. H. Stone_. Summa Brasiliensis Mathematicae 2, (1951), 151–182.  [PDF](https://dmitripavlov.org/scans/dixmier.pdf)

An expository account is given by [[Masamichi Takesaki]] in

* [[Masamichi Takesaki]], _Theory of Operator Algebras_, Chapter III, Section 1.

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