hyperbolic functor

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## Idea

In the context of [[quadratic form|quadratic]]/[[Hermitian forms]] and in particular of [[Hermitian K-theory]], by the *hyperbolic functor* one refers to the construction which sends a [[module]] $\mathscr{V}$ over a given ([[star-algebra|star]]-)[[ground ring]] $R$ to its [[direct sum]]
$$
  \mathscr{V}
  \;\mapsto\;
  H(\mathscr{V})
  \;\coloneqq\;
  \mathscr{V}
  \oplus
  \mathscr{V}^\ast
$$
with its  ([[anti-dual linear space|anti]]-)[[linear dual]] and then equipped with the ([[sesquilinear form|sesqui]]- or) [[bilinear form]]

$$
  \array{
    H(\mathscr{V})
    \otimes
    H(\mathscr{V})
    &\longrightarrow&
    R
  }
$$
which on the mixed summands is (the [[complex conjugation|conjugation]] of) the [[evaluation]] map
$$
  \mathscr{V} \otimes \mathscr{V}^\ast
  \longrightarrow
  R
$$
and vanishing on the homogeneous summands.

This construction extends to [[functor]] from the [[core groupoid]] [[Mod|of $R$-modules]] to that of [[inner product spaces]] with [[isometries]] between them, by sending an [[isomorphism]] $\phi \colon \mathscr{V} \overset{\sim}{\to} \mathscr{W}$ to its [[direct sum]] with the [[dual linear map]] of its [[inverse morphism|inverse]]:

$$
  \phi \,\mapsto\, \phi \oplus (\phi^{-1})^\ast
  \,.
$$

Regarded with suitable $\mathbb{Z}/2 \times \mathbb{Z}/2$-[[equivariant homotopy theory|equivariance]], the hyperbolic functor establishes an equivalence between [[KR-theory]] and [[topological Hermitian K-theory]] (see the references [there](Hermitian+K-theory#ReferencesTopologicalHermitianKTheory)).

## Example

\begin{example}
  In the simple special case that the [[ground ring]] is the [[real numbers]] $\mathbb{R}$ and $\mathscr{V}$ is also $\mathbb{R}$, regared as the 1-dimensional [[vector space]] over itself, and finally identifying $\mathbb{R}^\ast \simeq \mathbb{R}$, canonically, the hyperbolic construction on $\mathbb{R}$ is the [[plane]] $\mathbb{R}^2 = \mathbb{R} \oplus \mathbb{R}$ equipped with its standard [[Minkowski metric]].

Here the subspaces of fixed non-vanishing [[norm]] are the usual [[hyperbolas]]. This gives the name "hyperbolic form" to all modules of the above form $\mathscr{V} \oplus \mathscr{V}^\ast$.
\end{example}

## References

The term "hyperbolic functor" seems to originate with:

* [[Hyman Bass]] (notes by Amit Roy), §5.2 in: *Lectures on Topics in Algebraic K-Theory*, Tata Institute of Fundamental Research (1965) [[[Bass-HyperbolicFunctor.pdf:file]]]

See also:

* {#Wall69} [[C. T. C. Wall]] (ed. [[Andrew Ranicki]]), p. 177 of: *Surgery on compact Manifolds*, Math. Surveys and Monographs **69** (1969) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/books/scm.pdf)]

Further discussion in the context of [[Hermitian K-theory]]:

* [[Max Karoubi]], [[Orlando Villamayor]], p. 60 of *K-théorie algébrique et K-théorie topologique II*, Math. Scand. **32** (1973) 57-86 [[jstor:24490565](https://www.jstor.org/stable/24490565)]

* [[Max Karoubi]], pp. 307 (7 of 111) in: *Périodicité de la K-théorie hermitienne*, in: [[Hyman Bass]] (ed.), *Algebraic K-Theory III -- Hermitian K-Theory and Geometric Applications*, Lecture Notes in Mathematics **343** (1973) 301-411 [[doi:10.1007/BFb0061366](https://doi.org/10.1007/BFb0061366)]

* [[Anthony Bak]], §3 in: *Grothendieck Groups of Modules and Forms Over Commutative Orders*, American Journal of Mathematics, **99** 1 (1977) 107-120 [[jstor:2374010](https://www.jstor.org/stable/2374010), [doi:10.2307/2374010](https://doi.org/10.2307/2374010)]

* {#Karoubi10} [[Max Karoubi]], §1.10 in: *Le théorème de périodicité en K-théorie hermitienne*, in *Quanta of Maths*, Clay Mathematics Proceesings **11**, AMS and Clay Math Institute Publications (2010) 257-282 [[arXiv:0810.4707](https://arxiv.org/abs/0810.4707), [pdf](https://www.claymath.org/library/proceedings/cmip011c.pdf)]

See also:

* [[Richard Elman]], [[Nikita Karpenko]], [[Alexander Merkurjev]], p. 4 of: *Algebraic and Geometric Theory of Quadratic Forms*, Colloquium Publication **56**, AMS (2008) [[ams:coll-56](https://bookstore.ams.org/coll-56), [pdf](https://www.math.ucla.edu/~rse/old_book/Kniga.pdf)]

* [[Matthew B. Young]], p. 15 of: *Self-Dual Hall modules*, PhD thesis, Stony Brook (2013) [[pdf](https://www.math.stonybrook.edu/alumni/2013-Matthew-Young.pdf), [[Young-HallModules.pdf:file]]]

* [[Marco Schlichting]], p. 8 of: *Higher $K$-Theory of Forms I. From Rings to Exact Categories*, Journal of the Institute of Mathematics of Jussieu **20** 4 (2021) 1205-1273 [[doi:10.1017/S1474748019000410](https://doi.org/10.1017/S1474748019000410)]

[[!redirects hyperbolic functors]]

[[!redirects hyperbolic form]]
[[!redirects hyperbolic forms]]


hyperbolic geometry

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## Idea

[[Riemannian geometry]] of [[Riemannian manifolds]] with [[constant function|constant]] [[negative number|negative]] [[sectional curvature]].

## Related concepts
 {#RelatedConcepts}

* [[Riemannian geometry]]

* [[Lorentzian geometry]]

| [[synthetic geometry]]  |  kind of [[space]]  |
|-------------------------|---------------------|
| [[Euclidean geometry]]  |  [[Euclidean space]]   |
| [[hyperbolic geometry]] |  [[hyperbolic space]]  |
| [[elliptic geometry]]   |    |

* [[hyperbolic manifold]]

  * [[hyperbolic 3-manifold]]

* [[hyperbolic link]]

* [[trigonometry]]

## References

### General

* [[James Cannon]], William Floyd, Richard Kenyon, Water Parry, _Hyperbolic geometry_, in _Flavors of geometry_, MSRI Publications Volume 31, 1997 ([pdf](http://www.math.ecnu.edu.cn/~lfzhou/others/cannon.pdf))

* {#Ratcliffe06} [[John Ratcliffe]], _Foundations of Hyperbolic Manifolds_, Graduate Texts in Mathematics 149, Springer 2006 ([doi:10.1007/978-0-387-47322-2](https://doi.org/10.1007/978-0-387-47322-2), pdf)

* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: §12 in: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]

* Brice Loustau, _Hyperbolic geometry_ ([arXiv:2003.11180](https://arxiv.org/abs/2003.11180))

See also

* Wikipedia, _[Hyperbolic geometry](http://en.wikipedia.org/wiki/Hyperbolic_geometry)_

For the special case of hyperbolic plane (but possibly over various fields) see

* Norman J. Wildberger, _Universal hyperbolic geometry I: Trigonometry_, Geometriae Dedicata 163(1), 2009 [arxiv/0909.1377](http://arxiv.org/abs/0909.1377) [doi](http://dx.doi.org/10.1007/s10711-012-9746-9); _Universal hyperbolic geometry II: A pictorial overview_, [arxiv/1012.0880](http://arxiv.org/abs/1012.0880)

### Relation to AdS3/CFT2

Relation of [[AdS3/CFT2]] to [[hyperbolic geometry]] and [[Arakelov geometry]] of [[algebraic curves]]:

* [[Yuri Manin]], [[Matilde Marcolli]], _Holography principle and arithmetic of algebraic curves_, Adv. Theor. Math. Phys. 5 (2002) 617-650 ([arXiv:hep-th/0201036](https://arxiv.org/abs/hep-th/0201036)

[[!redirects hyperbolic metric]]
[[!redirects hyperbolic metrics]]


hyperbolic link

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## Idea

A [[link]] $L \subset S^3$  in the [[3-sphere]] is called a _hyperbolic link_ if its [[knot complement]] $S^3 \setminus L$ admits the [[structure]] of a [[hyperbolic metric]], hence of a [[hyperbolic 3-manifold]], hence if it is [[isometry|isometric]] to a [[quotient space]] of [[hyperbolic 3-space]] by a [[torsion subgroup|torsion-free]] [[discrete group]] $\Gamma$ [[action|acting]] by [[isometries]]:

$$
  S^3 \setminus L
  \;\simeq\;
  \mathbb{H}^3/\Gamma
  \,.
$$

(e.g. [FKP 17, Def. 2.8](#FKP17))

If the hyperbolic link is in fact a [[knot]] (has a single [[connected component]]) it is called a _hyperbolic knot_.

## Related concepts

* [[hyperbolic 3-manifold]]

* [[volume conjecture]]

## References

* {#FKP17} David Futer, Efstratia Kalfagianni, Jessica S. Purcell, _A survey of hyperbolic knot theory_, Springer Proceedings in Mathematics & Statistics, vol. 284 (2019), 1-30 ([arXiv:1708.07201](https://arxiv.org/abs/1708.07201))

See also

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

[[!redirects hyperbolic links]]

[[!redirects hyperbolic knot]]
[[!redirects hyperbolic knots]]


hyperbolic manifold

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## Idea

A _hyperbolic space_ is the analog of a [[Euclidean space]] as one passes from [[Euclidean geometry]] to [[hyperbolic geometry]]. The generalization of the concept of [[hyperbolic plane]] to higher [[dimension]].

A _hyperbolic manifold_ is a [[geodesic completeness|geodesically complete]] [[Riemannian manifold]] $(X,g)$ of [[constant function|constant]] [[sectional curvature]] $-1$.

Of particular interest are [[hyperbolic 3-manifold|hyperbolic 3-manifolds]].

## Properties

### Conformal flatness
 {#ConformalFlatness}

Every hyperbolic manifold is a [[conformally flat manifold]].

(e.g. [Long-Reid 00, p. 4](#LongReid00))

### Zeta functions

There are canonical [[zeta functions]] associated with suitable (odd-dimensional) hyperbolic manifolds, see at _[[Selberg zeta function]]_ and _[[Ruelle zeta function]]_.

### Mostow rigidity theorem

The [[Mostow rigidity theorem]] states that every [[hyperbolic manifold]] of [[dimension]] $\geq 3$ and of [[finite number|finite]] [[volume]] is uniquely determined by its [[fundamental group]].

## Examples

* [[hyperbolic plane]]

* [[hyperbolic 3-manifold]]

  * [[hyperbolic 3-space]]

  * [[hyperbolic solid torus]]

    * [[BTZ black hole]]

    * [[thermal AdS3]]

## Related concepts

A Riemannian manifold

* with zero sectional curvature is a [[Euclidean manifold]];

* with +1 sectional curvature is an [[elliptic manifold]]

See also

* [[hyperbolic geometry]]

* [[hyperbolic link]]

* [[volume conjecture]]

* [[Selberg zeta function]]

* [[prime geodesic]]

* [[Bloch group]]

* [[M-theory on hyperbolic manifolds]]

## References

Textbook accounts:

* {#Ratcliffe06} [[John Ratcliffe]], _Foundations of Hyperbolic Manifolds_, Graduate Texts in Mathematics 149, Springer 2006 ([doi:10.1007/978-0-387-47322-2](https://doi.org/10.1007/978-0-387-47322-2), pdf)

* [[Michael Kapovich]], _Hyperbolic Manifolds and Discrete Groups_,  Modern Birkhäuser Classics, Birkhäuser 2008 ([doi:10.1007/978-0-8176-4913-5](https://link.springer.com/book/10.1007/978-0-8176-4913-5))

See also

* {#LongReid00} Darren D. Long, A. W. Reid, _On the geometric boundaries of hyperbolic 4-manifolds_ ([arXiv:math/0007197](https://arxiv.org/abs/math/0007197))

* Wikipedia, _[Hyperbolic manifold](http://en.wikipedia.org/wiki/Hyperbolic_manifold)_

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

[[!redirects hyperbolic manifolds]]

[[!redirects hyperbolic space]]
[[!redirects hyperbolic spaces]]


hyperbolic plane

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## Idea

The _hyperbolic plane_ is the analog of the ordinary [[Euclidean space|Euclidean]] [[plane]] as one passes from [[Euclidean geometry]] to [[hyperbolic geometry]]. It is the 2-[[dimension|dimensional]] example of a [[hyperbolic space]].

## Definition

As a [[Riemannian manifold]], a **hyperbolic plane** is a [[complete space|complete]] [[contractible]] 2-[[dimension|dimensional]] [[smooth manifold]] of [[constant function|constant]] [[negative number|negative]] [[Riemann curvature]] (equivalently, Gaussian curvature).

## Properties

The intrinsic geometry of a hyperbolic plane is completely determined by its curvature; any two hyperbolic planes are related by a [[conformal transformation]], and a self-conformality of a hyperbolic plane is necessarily an [[isometry]].

Hyperbolic planes are also [[homogeneous spaces]]: there is a unique oriented [[isometry]] relating any two [[geodesic]] rays.

The [[incidence geometry]] of geodesics and points in the hyperbolic plane is equivalent to that of an open circular disc in the Euclidean plane (and several other obviously-equivalent things) — this is the Beltrami disc model.

## Related concepts

* [[hyperbolic tesselation]]

* [[hyperbolic 3-space]]

* [[hyperbolic 3-manifold]]

* [[hyperbolic manifold]]

[[!redirects hyperbolic planes]]


hyperbolic sine

## Definitions

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

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

   $$
     sinh'' = sinh
   $$

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

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

## Related concepts

* [[hyperbolic cosine]]

* [[hyerbolic tangent]]

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

## References

See also

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

[[!redirects sinh]]


hyperbolic solid torus

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## Idea

The [[solid torus]] of [[dimension]] 3 admits various [[mathematical structure|structures]] of a [[hyperbolic 3-manifold]]. Equipped with any such it is called a _hyperbolic solid torus_.

## Properties

### Knot complement

The [[solid torus]] is [[homeomorphism|homeomorphic]] to the [[knot complement]] of the [[unknot]] in the [[3-sphere]].

### Infinite volume

The [[volume]] of the [[hyperbolic solid torus]] is _not [[finite number|finite]]_.  Therefore this [[hyperbolic 3-manifold]] [[knot complement]] does not count as a "knot complement with hyperbolic structure" in the sense of [[William Thurston|Thurston]]'s [[classification of 3-manifolds]] (see also [this MO discussion](https://mathoverflow.net/q/302038/381)).

### As Euclidean BTZ black hole and thermal $AdS_3$

The [[Euclidean field theory|Euclidean]] [[BTZ black hole]] as well as [[thermal AdS3]] is a [[hyperbolic 3-manifold]] [[homeomorphism|homeomorphic]] to (the [[interior]] of) the [[hyperbolic solid torus|hyperbolic]] [[solid torus]], hence to the [[knot complement]] of the [[unknot]] in the [[3-sphere]].

(e.g [Gukov 03, appendix A](#Gukov03), [Kraus 06, around Fig. 1](#Kraus06), [BKR 07, 2.1](#BKR07))

## Related concepts

* [[hyperbolic 3-manifold]]

  * [[hyperbolic 3-space]]

## References

### General

(...)

### As BTZ black hole and thermal $AdS_3$

* [[Kirill Krasnov]], _Holography and Riemann Surfaces_, Adv. Theor. Math. Phys. 4 (2000) 929-979 ([arXiv:hep-th/0005106](https://arxiv.org/abs/hep-th/0005106))

* [[Kirill Krasnov]], around Figure 6 of: _Analytic Continuation for Asymptotically AdS 3D Gravity_, Class. Quant. Grav. 19 (2002) 2399-2424 ([arXiv:gr-qc/0111049](https://arxiv.org/abs/gr-qc/0111049))

* {#Gukov03} [[Sergei Gukov]], Appendix A of: _Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial_, Commun. Math. Phys. 255: 577-627, 2005 ([arXiv:hep-th/0306165](https://arxiv.org/abs/hep-th/0306165))

* {#Kraus06} [[Per Kraus]], _Lectures on black holes and the $AdS_3/CFT_2$ correspondence_,  Lect. Notes Phys. 755: 193-247, 2008 ([arXiv:hep-th/0609074](https://arxiv.org/abs/hep-th/0609074))

* {#BKR07} [[Micha Berkooz]], Zohar Komargodski, Dori Reichmann, _Thermal $AdS_3$, BTZ and competing winding modes condensation_, JHEP 0712:020, 2007 ([arXiv:0706.0610](https://arxiv.org/abs/0706.0610))

* M. Cadoni, M. Melis, _Holographic entanglement entropy of the BTZ black hole_, Found. Phys. 40: 638-657, 2010 ([arXiv:0907.1559](https://arxiv.org/abs/0907.1559))


hyperbolic tangent

## Related concepts

* [[tangent]]

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

* [[rapidity]]

## References

* MathWorld, _[Hyperbolic tangent](http://mathworld.wolfram.com/HyperbolicTangent.html)_


hypercharge

## Idea

(...)

## Related concepts

* [[quantum hadrodynamics]]

* [[weak nuclear force]]

* [[electric charge]]

## References

* [[Howard Georgi]], §11 in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

See also

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


hypercohomology

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## Idea

Ordinary [[abelian sheaf cohomology]] is often  considered exclusively with coefficients being [[Eilenberg-MacLane objects]] $\mathbf{B}^n A$ given by [[chain complex]]es concentrated in degree $n$. If more generally any chain complex is allowed as a coefficient object, one speaks for emphasis of **hyper-cohomology**.

If abelian sheaf cohomology is thought of given by the [[derived functor]] of the [[global section]]s functor, then hypercohomology is given by the corresponding [[hyper-derived functor]].

In the literature, hypercohomology is typically denoted by blackboard bold.

In terms of the general [[nPOV]] on [[cohomology]] (as described there) this just means the following:

in a given [[(∞,1)-topos]] $\mathbf{H}$ (which if we are thinking of describing [[abelian sheaf cohomology]] over some [[site]] is the [[(∞,1)-category of (∞,1)-sheaves]] on that site) "ordinary" cohomology in degree $n$ with coefficients in a [[group object in an (∞,1)-category|group object]] A is just

$$
  H^n(X,A) := \pi_0 \mathbf{H}(X, K(A,n))
  \,,
$$

where $K(A,n) = \mathbf{B}^n A$ is the [[Eilenberg-MacLane object]] with $A$ in degree $n$. Typically this is the complex of sheaves $[\cdots \to 0 \to 0 \to A \to 0 \to 0 \to \cdots \to 0]$ turned into a [[simplicial presheaf|simplicial sheaf]] using the [[Dold-Kan correspondence]]

$$
  \Xi :  Ch_\bullet \to KanCplx
$$

and then interpreted as an [[∞-stack]] using the [[model structure on simplicial presheaves]].

As the discussion at [[cohomology]] amplifies, this definition of cohomology in terms of the [[derived hom-space]] $\mathbf{H}(-,-)$ depends in no way on the coefficient object being an [[Eilenberg-MacLane object]]. It could be any object. Only some familiar _properties_ of cohomology (related to the notion of degree) do depend the coefficients being Eilenberg-MacLane objects.

We could take a completely arbitrary coefficient object $K$ and consider

$$
  H(X,K) := \pi_0 \mathbf{H}(X,K)
  \,.
$$

This is then called [[nonabelian cohomology]]. The notion of **hypercohomology** lies in between Eilenberg-MacLane-type cohomology and fully general nonabelian cohomology. For hypercohomology we allow the coefficient object to be a _general sheaf of [[chain complex]]es_ $A_\bullet = [\cdots \to A_2 \to A_1 \to A_0]$, or rather the [[simplicial presheaf]] $\Xi A_\bullet$ represented by that. Then hypercohomology is

$$
  H(X,A_\bullet) := \pi_0\mathbf{H}(X,\Xi A_\bullet)
  \,.
$$

For a bit more on this see also the discussion at [[abelian sheaf cohomology]].

## Examples

* [[Deligne cohomology]] is hypercohomology for complexes of sheaves of [[differential form]]s of the form

  $$
    [
    \cdots
    \to
    0
    \to
    C^\infty(-,U(1))
    \stackrel{d_{dR} log}{\to}
    \Omega^1(-)
    \stackrel{d_{dR}}{\to}
    \Omega^2(-)
    \stackrel{d_{dR}}{\to}
    \cdots
    \Omega^{n-1}(-)
    \stackrel{d_{dR}}{\to}
    \Omega^n(-)
    ]
    \,.
  $$

* In the context of algebraic geometry, "de Rham cohomology" refers to the hypercohomology of the [[de Rham complex]] $\Omega^\bullet$. At least for nice varieties, say [[smooth morphism|smooth]] and [[proper morphism|proper]], this "de Rham cohomology" agrees with the classical de Rham cohomology of the [[GAGA|analytification]]. See [this discussion](http://mathoverflow.net/questions/17937/algebraic-de-rham-cohomology-vs-analytic-de-rham-cohomology) on MathOverflow. Note that, in the algebraic category, the cohomology of the constant sheaf $\underline{\mathbb{C}}$ is not the right thing to consider, for constant sheaves on [[irreducible space]]s are [[flasque]], hence [[acyclic]].

## Related concepts

* [[hyper-derived functor]]


hypercomplete (infinity,1)-topos

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## Definition

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###### Definition

An [[(∞,1)-topos]] is **hypercomplete** if the [[Whitehead theorem]] is valid in it.

Equivalently: if all its object are [[hypercomplete objects]].

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## Properties

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

All [[geometric morphisms]] $X \to Y$ out of a hypercomplete $(\infty,1)$-topos $X$ factor through the hypercompletion $\hat Y$ of $Y$:

the inclusion $\hat Y \hookrightarrow Y$ induces an equivalence

$$
  Geom(X,\hat Y) \stackrel{\simeq}{\to}
  Geom(X,Y)
  \,.
$$

=--

This is [[Higher Topos Theory|HTT, prop. 6.5.2.13]].

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

Every [[(∞,1)-topos]] which is [[homotopy dimension|locally of homotopy dimension]] $\leq n$ for some finite $n\geq -1$ is hypercomplete.

=--

See the discussion at [[homotopy dimension]] for details and further implications.

## Examples

### Hypercompletions

Every [[(∞,1)-topos]] has a [[hypercompletion]], given by the full [[reflective sub-(∞,1)-category]] spanned by its [[hypercomplete objects]].

### Enough points
 {#EnoughPoints}

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

For $\mathcal{X}$ an [[(∞,1)-topos]], a **point** of $\mathcal{X}$ is a [[geometric morphism]]

$$
  (p^\ast \dashv p_*) \;\colon\;
  \infty Grpd
  \stackrel{\overset{p^\ast}{\leftarrow}}{\underset{p_\ast}{\to}}
  \mathcal{X}
$$

from [[∞Grpd]] $\simeq Sh_\infty(\ast)$ to $\mathcal{X}$. We say that $\mathcal{X}$ has **enough points** if a [[1-morphism]] $f \colon X \to Y$ in $\mathcal{X}$ is an [[equivalence in an (∞,1)-category|equivalence]] in $\mathcal{X}$ precisely if for all such points the [[inverse image]] $p^\ast (f)$ (the _[[stalk]]_ at the point) is an [[equivalence of ∞-groupoids|equivalence]] in [[∞Grpd]].

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

There exist [[1-site|1-sites]] $C$ such that the [[(n,1)-topos|(1,1)-topos]] of sheaves of sets on $C$ has [[point of a topos|enough points in the 1-topos sense]], but such that the corresponding [[n-localic (∞,1)-topos|1-localic (∞,1)-topos]] $\mathcal{X}$ does not have enough points in the sense of def. \ref{PointOfAndInfinityTopos}. An example is given by the site of open subsets of the topological space $\prod_{\mathbb{N}} \{x,z,y\}$ where the topology on $\{x,z,y\}$ is generated by the two open subsets $\{x,y\}$ and $\{x,z\}$. See [[Higher Topos Theory|HTT, Remark 6.5.4.7]].

However, the [[hypercompletion]] $\mathcal{X}^\wedge$ of $\mathcal{X}$ will have enough points. This follows from the fact that $\mathcal{X}^\wedge$ may be presented by the Jardine [[model structure on simplicial presheaves]] on the given site of definition, and that in the presence of enough 1-topos points the [[weak equivalences]] of that model structure are the [[stalk]]-wise weak equivalences in the [[model structure on simplicial sets]]. (See at _[[model structure on simplicial presheaves]]_ for details.)

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+-- {: .num_lemma}
###### Observation

An $(\infty,1)$-topos that has [[point of a topos|enough points]] is hypercomplete. WARNING: See the remark above.

=--

This is [[Higher Topos Theory|HTT, remark 6.5.4.7]]. See [SAG, A.4.0.5] for the converse: If an ∞-topos is hypercomplete and locally coherent, then it has enough points.

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

Recall from def. \ref{PointOfAndInfinityTopos} that a [[point of a topos|point of an (∞,1)-topos]] $\mathbf{H}$ is a [[geometric morphism]]  $p : \infty Grpd \stackrel{\overset{p^*}{\leftarrow}}{\underset{p_*}{\to}} \mathbf{H}$.

And by definition $\mathbf{H}$ has _enough points_ if a morphism $F : X \to Y$ in $\mathbf{H}$ is an [[equivalence in a quasi-category|equivalence]] if for all points $p$, the [[stalk]] $p^* f$ is an equivalence in [[∞Grpd]].

But if $f$ is $\infty$-[[connected]] in $\mathbf{H}$, then so is $p^* f$ in [[∞Grpd]], which is hypercomplete, by [[Whitehead's theorem]], so that $p^* f$ is an equivalence.

So in an $(\infty,1)$-topos with enough points all $\infty$-connected morphisms are equivalences.

=--

## Non-examples

### Goodwillie $n$-jet toposes

For $n \gt 1$ the [[Goodwillie n-jet (∞,1)-toposes]] are generically far from being hypercomplete.

## Models

### In terms of model category theory {#ModelCategory}

Hypercomplete [[∞-stack]] [[(∞,1)-topos]]es are precisely those that are [[presentable (∞,1)-category|presented]] by the local [[Andre Joyal|Joyal]]-Jardine [[model structure on simplicial presheaves]], where weak equivalences of simplicial presheaves are those morphisms that induce isomorphisms on homotopy sheaves. In these models the fibrant objects are those simplical presheaves that satisfy [[descent]] over all [[hypercover]]s.

If the underlying ordinary [[category of sheaves|sheaf]] [[topos]] has [[point of a topos|enough points]], then this are equivalently the morphisms that induce [[stalk]]wise weak equivalences in the standard [[model structure on simplicial sets]].

Contrary to that one can consider local models by [[Bousfield localization of model categories|left Bousfield localization]] of the global [[model structure on simplicial presheaves]] only at [[Cech cover]]s. This yield in general a non-hypercomplete [[(∞,1)-categories of (∞,1)-sheaves]].

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

For $C$ [[site]], write $[C^{op}, sSet]_{inj,loc}$ for the Joyal-Jardine [[local model structure on simplicial presheaves]] (whose weak equivalence are morphisms that induce isomorphisms on homotopy-sheaves).

The $(\infty,1)$-topos that this presents is the [[hypercompletion]] of the [[(∞,1)-category of (∞,1)-sheaves]] on $C$:

$$
  ([C^{op}, sSet]_{inj,loc})^\circ
  \simeq
  \widehat{ Sh_{(\infty,1)}(C)}
  \,.
$$

=--

This is [[Higher Topos Theory|HTT, prop. 6.5.2.14]].

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

The strategy is to form the localization in a 2-step process, where
we first just form the Cech-localization, and then from that the
full hypercompletion. For that notice that among the weak equivalences in the Joyal-Jardine [[local model structure on simplicial presheaves]] are in particular the ordinary covering [[sieve]]s $S(\{U_i\}) \hookrightarrow j(U)$
(here $j$ is the [[Yoneda embedding]]) associated with a [[cover]]in family $\{U_i \to U\}$ in the [[site]] $C$:

since $C$ is an ordinary category, the simplicial presheaves $S(\{U_i\})$ and $j(U)$ have vanishing presheaves of homotopy groups in positive degree, while they coindide with their $\pi_0$-presheaves. Since the [[sheafification]] of $S(\{U_i\})$ is isomorphic to $j(U)$, by definition, it follows that the same holds for the $\pi_0$-presheaves and trivially for the $\pi_n$-presheaves. So $S(\{U_i\}) \to j(U)$ is a Joyal-Jardine weak equivalence.

We will now first localize with respect to these morphisms to obtain the Cech-localization whose fibrant objects are [[(∞,1)-sheaves]]. The point is that on these fibrant objects then, the Joyal-Jardine sheaves of homotopy groups can be seen to coincide with the [[categorical homotopy groups in an (∞,1)-topos|(∞,1)-categorical homotopy sheaves]] in terms of which [[hypercompletion]] is defined.

To start with,
as discussed at [[(∞,1)-category of (∞,1)-presheaves]]
we have that the global model structure presents the $(\infty,1)$-presheaves:

$$
  ([C^{op}, sSet]_{inj})^\circ \simeq PSh_{(\infty,1)}(C)
  \,.
$$

Observe that

1. every simplicially constant object is fibrant in $[C^{op}, sSet]_{inj}$ ;

1. hence since every object is cofibrant, the morphism $S(\{U_i\}) \to j(U)$
   is in $([C^{op}, sSet]_{inj})^\circ$;

1. under the abovee identification it is an [[monomorphism in an (∞,1)-category|(∞,1)-monomorphism]] in $PSh_{(\infty,1)}(C)$

   (discuss this bit in more detail...).

Therefore the [[topological localization]] of $PSh_{(\infty,1)}(C)$ at these monomorphisms, i.e. the [[(∞,1)-category of (∞,1)-sheaves]] on $C$ is presented by the [[Bousfield localization of model categories|left Bousfield localization]] $[C^{op}, sSet]_{inj,cov}$ of $[C^{op}, sSet]_{inj}$ at the covering subfunctors $S(\{U_i\}) \to j(U)$.

By the above remark, the Joyal-Jardine localization $[C^{op}, sSet]_{proj,loc}$ that we are after is a further localization of this Cech localization : we have the bottom row in the following diagram, and want to see that the top left corner is as indicated:

$$
  \array{
    \widehat{Sh_{(\infty,1)}(C)}
    &
     \stackrel{\overset{lex}{\leftarrow}}{\underset{}{\hookrightarrow}}
    &
    Sh_{(\infty,1)}(C)
    &
     \stackrel{\overset{lex}{\leftarrow}}{\underset{}{\hookrightarrow}}
    &
    PSh_{(\infty,1)}(C)
    \\
    \uparrow^{\mathrlap{\simeq}}
    &&
    \uparrow^{\mathrlap{\simeq}}
    &&
    \uparrow^{\mathrlap{\simeq}}
    \\
    ([C^{op}, sSet]_{inj,loc})^\circ
    &
    \stackrel{\overset{\mathbb{L} Id}{\leftarrow}}{\underset{\mathbb{R}Id}{\to}}
    &
    ([C^{op}, sSet]_{inj,cov})^\circ
    &
    \stackrel{\overset{\mathbb{L} Id}{\leftarrow}}{\underset{\mathbb{R}Id}{\to}}
    &
    ([C^{op}, sSet]_{inj})^\circ
  }
  \,.
$$

Now recall that the [[categorical homotopy groups in an (∞,1)-topos]] of an object $X$ are defined by first forming the [[power]]ing $X^{* \to S^n} : X^{S^n} \to X$ and then paassing to the 0-[[truncated|truncation]] $\tau_{\leq 0} ( X^{S^n} \to X)$ of this object in the [[over quasi-category|over (∞,1)-category]].

By the discussion at Tensoring and cotensoring with an ∞-groupoid we have that this powering operation is on fibrant objects modeled by the powering in the [[sSet]]-[[enriched model category]] $[C^{op}, sSet]_{inj,cov}$. But the powering of [[simplicial presheaves]] by [[simplicial set]] is just objectwise the [[internal hom]] of simplicial sets. In terms of this are defined the objectwise [[simplicial homotopy group]]s and hence the Joyal-Jardine homotopy-presheaves.

Furthermore, if $X \in [C^{op}, sSet]_{inj,cov}$ is fibrant, it satisfies [[descent for simplicial presheaves]] at [[Cech cover]]s. Since powering is a [[Quillen bifunctor]], the same is then true for $X^S$, formed in the model category, so $X^S$ is an $\infty$-stack. But that means its 0-[[truncated|truncation]] $\tau_{\leq 0}(X^{S^n})$ is an ordinary [[sheaf]].
(Observe that truncation commutes with localization, as discussed here.)

In total this shows that _on fibrant objects_ $X$ in $[C^{op}, sSet]_{inj,cov}$, the Joyal-Jardine homotopy sheaves coincide with the $(\infty,1)$-categorical homotopy sheaves of the object $X$.

It remains to observe that under left [[Bousfield localization of model categories|Bousfield lcoalization]], the new fibrant objects are precisely those old fibrant objects that are also [[local object]]s with respect to the morphisms at which one localizes. With the above this implies that the left Bousfield localization $[C^{op}, sSet]_{inj,cov} \to [C^{op}, sSet]_{inj,loc}$ does model the [[hypercompletion]] $Sh_{(\infty,1)}(C) \to \widehat {Sh_{(\infty,1)}}(C)$.

=--

### In classical topos theory

In classical [[topos theory]] literature frequently [[simplicial object]]s in an ordinary [[topos]] are considered, with acyclic fibrations taken to be those morphisms $Y_\bullet \to X_\bullet$ such that for all [[horn]] inclusions the induced morphism

$$
  (X^{\Delta[n]} \to X^{\partial \Delta[n]} \times_{Y^{\partial \Delta[n]}} Y^{\Delta[n]})
$$

is an [[epimorphism]] in the [[topos]].

See for instance page 17 of

* [[Ieke Moerdijk]], _Classifying spaces and classifying topoi_ (LNM 1616)

> (and it looks like this is the discussion planned for part E of the [[Elephant]]).

For [[Grothendieck topos|sheaf toposes]] epimorphism means [[stalk]]-wise epimorphism. Therefore this amounts to using on simplicial sheaves the structure of a [[category of fibrant objects]] as defined in [[BrownAHT]], where acyclic fibrations are the stalkwise acyclic [[Kan fibration]]s.

The [[homotopy category]] of this [[homotopical category]] is the same as that of the Joyal-Jardine [[model structure on simplicial presheaves]] in the presence of enough points (since in both cases weak equivalences are the stalkwise weak equivalences), hence is the same as the [[homotopy category of an (infinity,1)-category|homotopy category of the hypercomplete (∞,1)-topos]].

For more discussion of how this classical definition interplays with other definitions see also [[homotopy groups in an (∞,1)-topos]].

## References

The notion of hypercompleteness appears as **$t$-completeness** in

* [[Bertrand Toen]] and [[Gabriele Vezzosi]], _Segal topoi and stacks over Segal categories_ , Proceedings of the
Program Stacks, Intersection theory and Non-abelian Hodge Theory, MSRI, Berkeley, January-May 2002 ([arxiv:math/0212330](http://arxiv.org/abs/math/0212330)).

The notion of hypercomplete $(\infty,1)$-toposes is the topic of  section 6.5 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

[[!redirects hypercomplete (∞,1)-topos]]
[[!redirects hypercomplete (∞,1)-topoi]]
[[!redirects hypercomplete (∞,1)-toposes]]


hypercomplete object

<div class="rightHandSide toc">
[[!include (infinity,1)-topos - contents]]
</div>

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

## Idea

An object in an [[(∞,1)-topos]] $\mathbf{H}$ is _hypercomplete_ if it regards the [[Whitehead theorem]] to be true in $\mathbf{H}$, i.e. if homming [[weak homotopy equivalence]]s into it produces an equivalence.

## Definition

Let $\mathbf{H}$ be an [[(∞,1)-topos]].

An [[object]] $A \in \mathbf{H}$ is **hypercomplete** if it is a [[local object]] with respect to all $\infty$-[[connected]] [[morphism]]s.

This means: if for every morphism $f : X \to Y$ which is $\infty$-[[connected]] as an object of the [[over quasi-category|over category]] $\mathbf{H}_{/Y}$ (roughly: all its [[homotopy fiber]]s have vanishing [[homotopy group in an (∞,1)-topos|homotopy groups]]), then the induced morphism

$$
  \mathbf{H}(f,A) : \mathbf{H}(Y,A) \to \mathbf{H}(X,A)
$$

is an [[equivalence in a quasi-category]] in [[∞Grpd]].

The $(\infty,1)$-topos $\mathbf{H}$ itself is a
  **[[hypercomplete (∞,1)-topos]]** if all its objects are hyercomplete. See there for more details.

## Remarks

* Hypercompleteness is a notion that appears only due to the possible unboundedness of the degree of [[homotopy groups in an (∞,1)-topos]]. The notion is empty in an [[(n,1)-topos]] for finite $n$.

* An object being hypercomplete in $\mathbf{H}$ means that it regards the [[Whitehead theorem]] to be true in $\mathbf{H}$. If $\mathbf{H}$ itself is hypercomplete, then the Whitehead theorem _is_ true in $\mathbf{H}$.

## References

This is the topic of section 6.5.2 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

The definition appears before lemma 6.5.2.9

[[!redirects hypercomplete]]
[[!redirects hypercomplete objects]]


hypercomplete site

[[!redirects hypercomplete sites]]
## Definition

A [[site]] is __hypercomplete__ if the [[model structure]] on [[simplicial presheaves]] implementing [[Čech descent]] yields a [[hypercomplete (∞,1)-topos]].

## Examples

* [[Zariski site]], [[Nisnevich site]]

* [[Cartesian site]]

* [[Stein site]]

## Related concepts

* [[Čech descent]], [[hyperdescent]]

* [[hypercomplete (∞,1)-topos]]


hypercompletion

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

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

## Idea

The _hypercompletion_ ([Lurie](#Lurie)) or _$t$-completion_ ([Rezk](#Rezk), [ToënVezzosi](#ToenVezzosi)) of an [[(∞,1)-topos]] of [[(infinity,1)-category of (infinity,1)-sheaves|(∞,1)-sheaves]] is a further localization/[[(infinity,1)-category of (infinity,1)-sheaves|(∞,1)-sheafification]] which corresponds to retaining only those [[(∞,1)-sheaves]] which satisfy [[descent]] with respect to all [[hypercover]]s.

## Definition

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

An [[(∞,1)-topos]] of [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaves]] is a **[[hypercomplete (∞,1)-topos]]**  if every $\infty$-[[n-connective|connective]] morphism is an [[equivalence in an (∞,1)-category|equivalence]].

=--

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

This may be read as saying that the [[Whitehead theorem]] is valid in the [[(∞,1)-topos]].

=--

## Examples

* The $(\infty,1)$-category defined by the Joyal-Jardine [[local model structure on simplicial presheaves]] with weak equivalences those morphisms that induce isomorphisms on homotopy sheaves is the hypercompletion of the [[localization]] of simplicial presheaves at ordinary covers. For more details see [[descent for simplicial presheaves]].

## Related concepts

* [[topological localization]]
* [[cotopological localization]]

## References

Section 10 of

* [[Charles Rezk]], _Toposes and homotopy toposes_ ([pdf](http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf))
{#Rezk}

Section 6.5 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

* [[Bertrand Toën]], [[Gabriele Vezzosi]], ([pdf](http://poincare.dma.unifi.it/~vezzosi/papers/msri.pdf))
{#ToenVezzosi}

[[!redirects hypercompletions]]


hypercomplex manifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[complex manifold]]

* [[quaternion-Kähler manifold]]

## References

In terms of [[G-structure]]:

* [[Dominic Joyce]], _Manifolds with many complex structures_, The Quarterly Journal of Mathematics, Volume 46, Issue 2, June 1995, Pages 169–184 ([doi:10.1093/qmath/46.2.169](https://doi.org/10.1093/qmath/46.2.169), [pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.596.1560&rep=rep1&type=pdf))

[[!redirects hypercomplex manifolds]]


hypercomplex number

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

# Hypercomplex numbers
* table of contents
{: toc}

## Idea

A __hypercomplex number system__ is a [[finite-dimensional vector space|finite-dimensional]] unital [[nonassociative algebra|algebra]] over the [[field]] $\mathbb{R}$ of [[real numbers]].  A __hypercomplex number__ is an [[element]] of one of these algebras.

## Examples by dimension

*  There is [[the|only one]] hypercomplex number system of [[dimension]] $0$, of course.

*  There is also only one hypercomplex number system of dimension $1$, which is $\mathbb{R}$ itself.

*  Up to [[isomorphism]], there are three hypercomplex number systems of dimension $2$, each of which is a [[commutative algebra]]. Given the rule

   $$ e^2 = a + b e $$

   (for $a, b$ fixed real numbers), the algebra $\mathbb{R}[e]$ may be identified as one of these three cases by the sign of $b^2 - 4 a$.  They are:

   *  the [[complex numbers]] (elliptic case, $b^2 - 4 a \gt 0$),

   *  the [[dual numbers]] (parabolic case, $b^2 - 4 a = 0$),

   *  the [[perplex numbers]] (hyperbolic case, $b^2 - 4 a \lt 0$).

   Notice that the [[complex numbers]] have 2 distinct [[continuous function|continuous]] [[automorphisms]], given, over $\mathbb{R}$, by $e \mapsto \pm e$ (see at _[[automorphism of the complex numbers]]_).

   On the other hand, the [[ring of dual numbers]] has a continuous automorphism $e \mapsto k e$ for each $k \in \mathbb{R} \setminus \{0\}$. But the  latter space is [[homotopy equivalence|homotopy equivalent]] to two points, corresponding to the [[connected components]] of, again, $e \mapsto \pm e$.

   Finally, $e \mapsto \pm e$ are the only two non-trivial continuous automorphisms of the [[perplex numbers]]. (For let $\phi$ be an automorphism of the perplex numbers. Observing that $\phi(e)^2 = \phi(e^2) = \phi(1) = 1$, it follows that $\phi(e) \in \{1,-1,e,-e\}$. But if $\phi(e) \in \{+1, -1\}$ then $\phi$ is not injective. So therefore $\phi(e) \in \{e, -e\}$.)

   Therefore, the [[Euler characteristic]] of the space of continuous automorphisms of each of the three 2d hypercomplex number systems is 2. Maybe with a bit of handwaving towards _[[groupoid cardinality]]_ one might summarize this as saying that there are "3/2" 2d hypercomplex number systems.

*  Famous hypercomplex number systems of dimension $4$ include the [[quaternion]]s and the [[bicomplex number]]s.

*  In dimension $8$, try the [[octonion]]s and the [[biquaternion]]s.

*  In dimension $16$, try the [[sedenion]]s and the [[bioctonion]]s.

Of course, these are not the only possibilities by any means.  One can always form the [[direct product]] of two hypercomplex number systems to get a hypercomplex number systems with the sum of the dimensions.  Another way to double the dimension is to form the [[tensor product]] with any of the hypercomplex number systems of dimension $2$; in particular, [[complexification]] (the tensor product with the complex numbers) is often denoted by the prefix 'bi‑'.  The [[Cayley–Dickson construction]] will double the dimension of any hypercomplex number system equipped with a (possibly trivial) involution.  Another way to generate associative hypercomplex number systems is through [[Clifford algebra]]s.

There is a thorough list of examples on [the English Wikipedia](https://secure.wikimedia.org/wikipedia/en/wiki/Hypercomplex_number).

[[!redirects hypercomplex number]]
[[!redirects hypercomplex numbers]]
[[!redirects hypercomplex number system]]
[[!redirects hypercomplex number systems]]


hypercomputation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A concept of [[computation]] assuming computable operations more powerful than in traditional [[computability]] (more powerful than [[partial recursive functions]] and [[computable function (analysis)|computable functions in computable analysis]]).

For instance a Turing machine that could not just run forever, but actually process an infinite number of steps "in finite time" would be a hypercomputer. [Bauer](http://math.andrej.com/wp-content/uploads/2011/06/injection.pdf) provides a [[realizability topos]] for infinite time computation; see also [Hamkins](http://arxiv.org/pdf/math/9808093.pdf).

It has been argued that there are [[spacetimes]] -- see at _[[Malament–Hogarth spacetime]]_ -- which are such that they allow [[trajectories]] on which a computer could travel indefinitely together with spacetime points at which an observer could observe the whole infinite history of the computer in finite time. If this indeed were physically realizable it would to some extent contradict the [[physical Church-Turing thesis]], which asserts that no physical process can realize a computer more powerful than a Turing machine.

## Related concepts

* [[Church-Turing thesis]]

* [[computable physics]]

## References

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

* Stanford Encyclopedia of Philosophy, section 4.3 of _[Computation in Physical Systems](http://plato.stanford.edu/entries/computation-physicalsystems/#EvePhySysCom)_

* discussion [Can somebody explain this to me?: The computability of the laws of physics and hypercomputation](http://lesswrong.com/lw/h9c/can_somebody_explain_this_to_me_the_computability/)

[[!redirects hypercomputations]]

[[!redirects hypercomputability]]


hyperconnected geometric morphism

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

# Hyperconnected geometric morphisms
* table of contents
{: toc}

## Idea

A [[geometric morphism]] is _hyperconnected_ if it is (left) [[orthogonal factorization system|orthogonal]] to a [[localic geometric morphism]].

In particular, a _hyperconnected topos_ is a [[topos]] that is "as far from being a [[localic topos]] as possible". In view of the fact that a [[topos]] is a generalized [[space]], while a [[localic topos]] is an ordinary [[topological space]]/[[locale]], regarded as a topos, this means that hyperconnected toposes are the "purely-generalized generalized spaces".

## Definition

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

A [[geometric morphism]] $f\colon E\to F$ between [[toposes]] is called **hyperconnected** if the [[inverse image]] [[functor]] $f^*\colon F\to E$

1. is a [[full and faithful functor]]

1. its [[image]] is closed under [[subquotients]] in $E$.

=--

This appears ([Johnstone, p. 225](#Johnstone)).

## Examples

* If $g\colon C\to D$ is a [[functor]] between [[small categories]] which is both [[essentially surjective functor|essentially surjective]] and [[full functor|full]], then the induced geometric morphism $Set^C\to Set^D$ is hyperconnected.  In fact, instead of essentially surjective it suffices for $g$ to be [[Cauchy surjective functor|Cauchy surjective]], i.e. $D$ is the closure of $g(C)$ under retracts.

* In particular, the [[global sections]] geometric morphism $Set^C\to Set$ on the [[presheaf topos]] is hyperconnected  iff the [[category]] $C$ is strongly connected ([Johnstone, A4.6.9](#Johnstone)), i.e., [[inhabited]] and for any two [[objects]] $A,B$ there exist morphisms $f:A\to B$ and $g:B\to A$.

  This includes for instance the case when $C=M$ is a [[monoid]], and the topos of [[simplicial set|simplicial sets]].

* A [[locally connected topos|locally connected]] and [[local topos]] is hyperconnected precisely if as a [[cohesive topos]] it satisfies _pieces have points_ or equivalently _discrete objects are concrete_. See [[cohesive topos]] for details.

## Properties

+-- {: .un_prop}
###### Observation

Any hyperconnected geometric morphism is [[connected geometric morphism|connected]],

=--

So the name is not unreasonable.

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

Hyperconnected geometric morphisms are the left class of a 2-categorical [[orthogonal factorization system]] on the 2-category of toposes; the right class is the class of [[localic geometric morphisms]].

([[(hyperconnected,localic) factorization system]]).

=--

See ([Johnstone](#Johnstone)).

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

In particular, a [[geometric morphism]] can only be both hyperconnected and [[localic geometric morphism|localic]] if it is an [[equivalence of categories|equivalence]].  Therefore, if we view topoi as generalized [[topological spaces]] (or [[locales]]), the world of hyperconnected topoi and geometric morphisms lives entirely in the "generalized" part.

=--

This is further amplified by the following proposition. Recall that the inclusion $Sh(-) : Locale \simeq LocTopos \hookrightarrow$ [[Topos]] is [[reflective subcategory|reflective]]: it has a [[left adjoint]]: the [[localic reflection]]

$$
  (L \dashv Sh(-)) : Locale \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}
  Topos
  \,.
$$

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

Hyperconnected toposes $\mathcal{E}$ are precisely those whose [[localic reflection]] is the point: $L\mathcal{E} \simeq Sh(*) \simeq Set$.

=--

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

Suppose $\Gamma : \mathcal{E} \to Set$ is hyperconnected. Let $X$ be a [[locale]] and $\mathcal{E} \to Sh(X)$ a [[geometric morphism]]. Notice that this sits in an essentially unique diagram

$$
  \array{
    \mathcal{E} &\to& Sh(X)
    \\
    \downarrow && \downarrow
    \\
    Set &\stackrel{=}{\to}& Set
  }
$$

in [[Topos]], where the vertical morphisms are the essentially unique [[global section]] geometric morphisms (we notationally suppress [[2-morphism|2-isomorophisms]]).

By the [above proposition](#HyperconnLocalicFactorizationSystem) there is an essentially unique geometric morphism $p : Set \to Sh(X)$ fitting into this diagram

$$
  \array{
    \mathcal{E} &\to& Sh(X)
    \\
    \downarrow &\nearrow_{\mathrlap{p}}& \downarrow
    \\
    Set &\stackrel{=}{\to}& Set
  }
  \,.
$$

This establishes the [[natural equivalence]]

$$
  Topos(\mathcal{E}, Sh(X)) \simeq LocTopos(Set, Sh(X)) \simeq
  Locale(*, X)
$$

and hence identifies the point as the localic reflection of $\mathcal{E}$.

Conversely, suppose that $\mathcal{E}$ has as localic reflection the point. The [[unit of an adjunction|unit]] of the $(L \dashv Sh(-))$-[[adjunction]] -- the [[reflective subcategory|reflector]] -- $\mathcal{E} \to Set$ is by essential uniqueness the [[global section]] [[geometric morphism]].

Let then

$$
  \array{
    \mathcal{E} &\to& \mathcal{S}
    \\
    \downarrow && \downarrow
    \\
    Set &\stackrel{f}{\to}& \mathcal{T}
  }
$$

be a lifting problem, with the right morphism a [[localic geometric morphism]]. Since these are preserved by [[pullback]] in [[Topos]], this is equivalently a diagram

$$
  \array{
    \mathcal{E} &\to& Sh(X) \simeq f^* \mathcal{S} &\to& \mathcal{S}
    \\
    \downarrow && \downarrow && \downarrow
    \\
    Set &\stackrel{=}{\to}& Set &\stackrel{f}{\to}& \mathcal{T}
  }
  \,.
$$

Since this exhibits $f^* \mathcal{S}$ as a [[localic topos]] $Sh(X)$ for some [[locale]] $X$, we have by the universal property of the adjunction unit an essentially unique lift $p$ in the left square

$$
  \array{
    \mathcal{E} &\to& Sh(X) \simeq f^* \mathcal{S} &\to& \mathcal{S}
    \\
    \downarrow &\nearrow_{\mathrlap{p}}& \downarrow && \downarrow
    \\
    Set = Sh(L(\mathcal{E})) &\stackrel{=}{\to}& Set &\stackrel{f}{\to}& \mathcal{T}
  }
  \,.
$$

By the universal property of the [[pullback]], this is then also an essentially unique solution to the original lifting problem.

=--

## References

* [[Peter Johnstone]], _Factorization theorems for geometric morphisms_ Cahiers, 22, no1 (1981) ([numdam](http://www.numdam.org/item?id=CTGDC_1981__22_1_3_0))
{#Johnstone}

[[!redirects hyperconnected geometric morphism]]
[[!redirects hyperconnected geometric morphisms]]

[[!redirects hyperconnected topos]]
[[!redirects hyperconnected toposes]]


hypercover

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _hypercover_ is the generalization of a [[Čech nerve]] of a [[cover]]: it is a simplicial [[resolution]] of an object obtained by iteratively applying covering families.

## Definition

Let

$$
  (L \dashv i) : Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}
  PSh(C)
$$

be the [[geometric embedding]] defining a [[sheaf topos]] $Sh(C)$ into a [[presheaf topos]] $PSh(C)$.

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

A morphism

$$
  (Y \stackrel{f}{\to} X) \in PSh(C)^{\Delta^{op}}
$$

in the [[category of simplicial objects]] in $PSh(C)$, hence the category of [[simplicial presheaves]], is called a **hypercover** if for all $n \in \mathbb{N}$ the canonical morphism

$$
  Y_{n} \to (\mathbf{cosk}_{n-1} Y)_n \times_{(\mathbf{cosk}_{n-1} X)_n} X_n
$$

in $PSh(C)$ are [[local epimorphisms]] (in other words, $f$ is a "[[Reedy model structure|Reedy]] local-epimorphism").

Here this morphism into the [[fiber product]] is that
induced from the [[naturality square]]

$$
  \array{
    Y &\longrightarrow& X
    \\
    \downarrow && \downarrow
    \\
    \mathbf{cosk}_{n-1} Y &\longrightarrow& \mathbf{cosk}_{n-1} X
  }
$$

of the [[unit of a monad|unit]] of the [[coskeleton]] [[functor]] $\mathbf{cosk}_n : PSh(C)^{\Delta^{op}} \to PSh(C)^{\Delta^{op}}$.

A hypercover is called **bounded** by $n \in \mathbb{N}$ if for all $k \geq n$ the morphisms $Y_{k} \to (\mathbf{cosk}_{k-1} Y)_k \times_{(\mathbf{cosk}_{k-1} X)_k} X_k$ are [[isomorphism]]s.

The smallest $n$ for which this holds is called the **height** of the hypercover.

=--

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

A hypercover that also satisfies the [cofibrancy condition](model+structure+on+simplicial+presheaves#CofibrantObjects) in the projective [[local model structure on simplicial presheaves]] in that

1. it is simplicial-degree wise a [[coproduct]] of [[representable functor|representables]];

1.  degenerate cells split off as a [[coproduct|direct summand]])

is called a **[[split hypercover]]**.

=--

([Dugger-Hollander-Isaksen 02, def. 4.13](#DuggerHollanderIsaksen02)) see also ([Low 14-05-26, 8.2.15](#Low))

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

Definition \ref{CoskeletonDefinition} is equivalent to saying that $f : Y \to X$ is a _local_ acyclic fibration: for all $U \in C$ and $n \in \mathbb{N}$ every lifting problem

$$
  \left(
  \array{
    \partial \Delta[n] \cdot U &\to& Y
    \\
    \downarrow && \downarrow^{\mathrlap{f}}
    \\
    \Delta[n]\cdot U &\to& X
  }
  \right)
  \;\;\simeq
  \;\;
  \left(
  \array{
    \partial \Delta[n] &\to& Y(U)
    \\
    \downarrow && \downarrow^{\mathrlap{f(U)}}
    \\
    \Delta[n] &\to& X(U)
  }
  \right)
$$

has a solution $(\sigma_i)$ after refining to some [[covering]] family $\{U_i \to U\}$ of $U$

$$
  \forall i :
  \left(
  \array{
    \partial \Delta[n] &\to& Y(U_i)
    \\
    \downarrow &{}^{\mathllap{\exists \sigma_i}}\nearrow& \downarrow^{\mathrlap{f(U_i)}}
    \\
    \Delta[n] &\to& X(U_i)
  }
  \right)
  \,,
$$

=--

([Dugger-Hollander-Isaksen 02, prop. 7.2](#DuggerHollanderIsaksen02)).

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

If the [[topos]] $Sh(C)$ has [[point of a topos|enough points]] a morphism $f : Y \to X$ in $Sh(C)^{\Delta^{op}}$ is a hypercover if all its [[stalk]]s are acyclic [[Kan fibration]]s.

=--

In this form the notion of hypercover appears for instance in ([Brown 73](#Brown73)).

In some situations, we may be interested primarily in hypercovers that are built out of data entirely in the site $C$.  We obtain such hypercovers by restricting $X$ to be a discrete simplicial object which is representable, and each $Y_n$ to be a coproduct of representables.  This notion can equivalently be formulated in terms of diagrams $(\Delta/A) \to C$, where $A$ is some simplicial set and $(\Delta/A)$ is its [[category of simplices]].

## Examples

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

Consider the case that $X = const X_0$ is simplicially constant. Then the conditions on a morphism $Y \to X$ to be a hypercover is as follows.

* in degree 0: $Y_0 \to X_0$ is a local epimorphism.

* in degree 1: The commuting diagram in question is

  $$
    \array{
      Y_1 &\to& X_0
      \\
      \downarrow && \downarrow^{\mathrlap{diag}}
      \\
      Y_0 \times Y_0 &\to& X_0 \times X_0
    }
    \,.
  $$

  Its [[pullback]] is $(Y_0 \times Y_0)\times_{X_0 \times X_0} X_0 \simeq Y_0 \times_{X_0} Y_0$, hence the condition is that

  $Y_1 \to Y_0 \times_{X_0} Y_0$ is a local epimorphism.

* in degree 2: The commuting diagram in question is

  $$
    \array{
      Y_2 &\to& X_0
      \\
      \downarrow && \downarrow^{Id}
      \\
      (Y_1 \times_{Y_0} Y_1 \times_{Y_0}Y_1)_{\times_{Y_0 \times Y_0}}  Y_0
      &\to& X_0
    }
    \,.
  $$

  So the condition is that the vertical morphism is a local epi.

* Similarly, in any degree $n \geq 2$ the condition is that

  $$
    Y_n \to (\mathbf{cosk}_{n-1} Y)_n
  $$

  is a local epimorphism.

=--

## Properties

### Existence and refinements

#### Čech nerves

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

For $U = \{U_i \to X\}$ a [[cover]], the [[Čech nerve]] projection $C(U) \to X$ is a hypercover of height 0.

=--

#### Over general sites
 {#OverGeneralSites}

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

Given any [[site]] $(\mathcal{C},J)$ and given a diagram of simplicial presheaves

$$
  \array{
    && Y
    \\
    && \downarrow^{\mathrlap{hcov}}
    \\
    X' &\longrightarrow& X
  }
$$

where the vertical morphism is a hypercover, then there exists a completion to a [[commuting diagram]]

$$
  \array{
    Y' &\longrightarrow& Y
    \\
    \downarrow^{\mathrlap{hcov}} && \downarrow^{\mathrlap{hcov}}
    \\
    X' &\longrightarrow& X
  }
$$

where the left vertical morphism is a split hypercover, def. \ref{SplitHypercover}.

Moreover, if $(\mathcal{D}, K)\to (\mathcal{C},J)$ is a [[dense subsite]] then $Y'$ as above exists such that it is simplicial-degree wise a [[coproduct]] of ([[representable functor|representables]] by) objects of $\mathcal{D}$.

=--

(e.g. [Low 14-05-26, lemma 8.2.20](#Low))

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

In particular taking $X'\to X$ in prop. \ref{ExistenceOfSplitRefinements} to be an identity, the proposition says that every hypercover may be refined by a split hypercover.

=--

(see also [Low 14-05-26, lemma 8.2.23](#Low))

#### Over Verdier sites
 {#OverVerdierSites}

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

A **[[Verdier site]]** is a [[small category]] with finite [[pullbacks]] equipped with a [[basis for a Grothendieck topology]] such that the generating [[covering]] maps $U_i \to X$ all have the property that their [[diagonal]]

$$
  U_i \to U_i \times_X U_i
$$

is also a generating covering. We say that $U_i \to X$ is **basal**.

=--

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

It is sufficient that all the $U_i \to X$ are [[monomorphism]]s.

Examples include the standard [[open cover]]-topology on [[Top]].

=--

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

A **basal hypercover** over a [[Verdier site]] is a hypercover $U \to X$ such that for all $n \in \mathbb{N}$ the components of the maps into the matching object $U_n \to M U_n$ are basal maps, as above.

=--

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

Over a Verdier site, every hypercover may be refined by a split (def. \ref{SplitHypercover}) and basal hypercover (def. \ref{BasalHypercover}).

=--

This is ([Dugger-Hollander-Isaksen 02, theorem 8.6](#DuggerHollanderIsaksen02)).

### Hypercover homology {#HypercoverHomology}

Let $f : Y \to X$ be a hypercover. We may regard this as an object in the [[overcategory]] $Sh(C)/X$. By the discussion here this is equivalently $Sh(C/X)$. Write $Ab(Sh(C/X))$ for the category of abelian [[group object]]s in the [[sheaf topos]] $Sh(C/X)$. This is an [[abelian category]].

Forming in the sheaf topos the [[free functor|free]] abelian group on $f_n$ for each $n \in \mathbb{N}$, we obtain a simplicial abelian group object $\bar f \in Ab(Sh(C/X))^{\Delta}$. As such this has a [[Moore complex|normalized chain complex]] $N_\bullet(\bar f)$.

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

For $f : Y \to X$ a hypercover, the [[chain homology]] of $N(\bar f)$ vanishes in positive degree and is the group of [[integer]]s in degree 0, as an object in $Ab(Sh(C)(X)$:

$$
  H_p(N(f)) \simeq
  \left\{
    \array{
       0 & for \; p \geq 1
       \\
       \mathbb{Z} & for \; p = 0
    }
  \right.
  \,.
$$

=--

### Descent and cohomology
 {#DescentAndCohomology}

The following theorem characterizes the [[∞-stack]]/[[(∞,1)-sheaf]]-condition for the [[presentable (∞,1)-category|presentation]] of an [[(∞,1)-topos]] by a [[local model structure on simplicial presheaves]] in terms of descent along hypercovers.

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

In the [[local model structure on simplicial presheaves]] $PSh(C)^{\Delta^{op}}$ an object is fibrant precisely if it is fibrant in the global [[model structure on simplicial presheaves]] and in addition satisfies [[descent]] along all hypercovers over representables that are degreewise [[coproduct]]s of representables.

=--

This is the central theorem in ([Dugger-Hollander-Isaksen 02](#DuggerHollanderIsaksen02)).

The following theorem is a corollary of this theorem, using the discussion at [[abelian sheaf cohomology]]. But historically it predates the above- theorem.

+-- {: .num_theorem}
###### Theorem
**(Verdier's hypercovering theorem)

For $X$ a [[topological space]] and $F$ a [[sheaf]] of [[abelian group]]s on $X$, we have that the [[abelian sheaf cohomology]] of $X$ with coefficients in $F$ is given

$$
  H^q(X, F) \simeq {\lim_{\to}}_{Y \to X} H^q(Hom_{Sh}(Y^\bullet,F))
$$

by computing for each hypercover $Y \to X$ the [[cochain cohomology]] of the [[Moore complex]] of the cosimplicial abelian group obtained by evaluating $F$ degreewise on the hypercover, and then taking  the [[colimit]] of the result over the [[poset]] of all hypercovers over $X$.

=--

A proof of this result in terms of the structure of a [[category of fibrant objects]] on the category of simplicial presheaves appears in ([Brown 73, section 3](#Brown73)).

## Related entries

* [[hypersheaf]]

## Reference

The concept of hypercovers was introduced for [[abelian sheaf cohomology]] in

*  [[Jean-Louis Verdier]], Exposé V, sect. 7 of  [[SGA4]],

An early standard reference founding [[étale homotopy theory]] is

* [[Michael Artin]], [[Barry Mazur]], _Étale Homotopy_ , Lecture Notes in Mathematics 100, Springer- Verlag, Berline-Heidelberg-New York (1972).

The modern reformulation of their notion of hypercover in terms of simplicial presheaves is mentioned for instance at the end of section 2, on [p. 6](http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf#page=6) of

* [[John Frederick Jardine]], _Fields Lectures: Simplicial presheaves_ ([pdf](http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf))

A discussion of hypercovers of [[topological spaces]] and relation to [[étale homotopy type]] of [[smooth schemes]] and [[A1-homotopy theory]] is in

* {#Isaksen01}[[Daniel Isaksen]], _Étale realization of the $\mathbb{A}^1$-homotopy theory of schemes_, 2001 ([K-theory archive](http://www.math.uiuc.edu/K-theory/0495/))

* {#DuggerIsaksen05} [[Daniel Dugger]], [[Daniel Isaksen]], _Hypercovers in topology_, 2005 ([pdf](http://www.math.uiuc.edu/K-theory/0528/hypercover.pdf), [K-Theory archive](http://www.math.uiuc.edu/K-theory/0528/))

A discussion in a topos with enough points in in

* {#Brown73} [[Kenneth Brown]], _[[BrownAHT|Abstract homotopy theory and generalized sheaf cohomology]]_, 1973

A thorough discussion of hypercovers over representables and their role in [[descent]] for simplicial presheaves is in

* {#DuggerHollanderIsaksen02} [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], _Hypercovers and simplicial presheaves_, Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 136. No. 1., 2004. ([arXiv:0205027](http://arxiv.org/abs/math/0205027), [K-theory archive](http://www.math.uiuc.edu/K-theory/0563/))

On the Verdier hypercovering theorem see

* [[Kenneth Brown]], _[[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf Cohomology]]_

* [[John Frederick Jardine]], _The Verdier hypercovering theorem_ ([doi](https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/verdier-hypercovering-theorem/13FA7ACF61C486C2FD5C06FFF6483562))

* [[Zhen Lin Low]], _Cocycles in categories of fibrant objects_, ([pdf](http://arxiv.org/abs/1502.03925v3))

Split hypercover refinement over general sites is discussed in

* {#Low} [[Zhen Lin Low]], _[[Notes on homotopical algebra]]_

[[!redirects hypercovers]]

[[!redirects Verdier hypercover theorem]]
[[!redirects Verdier hypercovering theorem]]


hypercrossed complex

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

#Contents#
* table of contents
{:toc}

## Idea

 The [[Moore complex]] of a [[simplicial group]] (or, more generally, any [[simplicially enriched groupoid]]), carries a lot more structure that just a [[chain complex]] of group(oid)s. This structure is its structure as a **hypercrossed complex**.  The notion was introduced by [[Pilar Carrasco]] in her thesis.

## Related concepts

* [[group]]

* [[2-group]], [[crossed module]], [[differential crossed module]]

* [[3-group]], [[2-crossed module]] / [[crossed square]], [[differential 2-crossed module]]

* [[n-group]]

* [[∞-group]], [[simplicial group]], [[crossed complex]], **hypercrossed complex**

## References

* [[Pilar Carrasco]], 1987, _Complejos Hipercruzados, Cohomologia y Extensiones_, Ph.D. thesis, Universidad de Granada.

* [[Pilar Carrasco|P. Carrasco]] and [[Antonio Cegarra|A. M. Cegarra]], _Group-theoretic Algebraic Models for Homotopy Types_, J. Pure Appl. Alg., 75, (1991), 195 – 235. <https://doi.org/10.1016/0022-4049(91)90133-M>

[[!redirects hypercrossed complexes]]


hypercube

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

#Contents#
* table of contents
{:toc}

## Idea

A _hypercube_ is a [[cube]] in higher [[dimension]].

Regarded as a [[topological space]], the $n$-dimensional hypercube for $m \in \mathbb{N}$ is $[0,1]^n$, the $n$-fold [[Cartesian product]] of the standard [[interval]] with itself.

## Related concepts

* [[hyperplane]]

## References

See also:

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

[[!redirects hypercubes]]


hyperdescent

## Idea

The **hyperdescent** condition is used to refer to a specific [[∞-descent]] condition for [[∞-presheaves]].

Concretely, hyperdescent can be implemented by performing the [[left Bousfield localization]] of the [[model category]] of [[simplicial presheaves]] equipped with the [[projective model structure]] (say) at [[hypercovers]].
Equivalently, we can localize at those morphisms of [[simplicial presheaves]] that induce isomorphisms on all sheaves of homotopy groups.
Then [[fibrant objects]] in the resulting [[model structure]] will be precisely [[presheaves]] of [[Kan complexes]] that satisfy the hyperdescent condition.

Model structures on [[simplicial sheaves]] implementing hyperdescent were first constructed by [[André Joyal]] in mid-1980s.  [[John F. Jardine]] then constructed analogous model structures for [[simplicial presheaves]].

Hyperdescent should be contrasted with [[Čech descent]].
The [[Čech descent]] condition is weaker than the hyperdescent condition, since it uses [[Čech nerves]] of [[covering families]] instead of arbitrary [[hypercovers]].
However, for [[hypercomplete sites]] such as [[Zariski site]], [[Nisnevich site]], [[CartSp]], [[Stein site]], and most of the other sites used in differential and complex geometry, the [[hyperdescent]] condition coincides with the Čech descent condition.

## Related concepts

* [[Čech descent]]

* [[∞-presheaf]], [[∞-sheaf]], [[simplicial presheaf]]

* [[hypercover]]


hyperdeterminant

#Contents#
* table of contents
{:toc}

## Idea

The concept of __Hyperdeterminant__ (or Cayley determinant) is a generalization of the concent of [[determinant]] to multidimensional matrices ([[hypermatrices]], i.e. representations of [[tensors]] in a choice of [[linear basis]]).

## References

The concept originates with [[Cayley]].

* [[Israel Gelfand|Israel M. Gelfand]], [[Mikhail Kapranov]], [[Andrei Zelevinsky]], _Discriminants, Resultants, and Multidimensional Determinants [gBooks](http://books.google.fr/books?id=n6ZxAKevSbsC&lpg=PP1&hl=fr&pg=PP1#v=onepage&q&f=false)

* wikipedia [hyperdeterminant](http://en.wikipedia.org/wiki/Hyperdeterminant) (lists also the original references)

* Giorgio Ottaviani, _Introduction to the hyperdeterminant and to the rank of multidimensional matrices_, [arxiv/1301.0472](http://arxiv.org/abs/1301.0472)

MathOverflow discussion on a possible volume interpretation of hyperdeterminants:

* [Joseph O'Rourke](http://mathoverflow.net/users/6094/joseph-orourke), _Does the hyperdeterminant calculate a quantity akin to the volume of a parallelepiped?_, MathOverflow. [URL](http://mathoverflow.net/q/220331) (version: 2015-10-07).

[[!redirects hyperdeterminants]]


hyperdoctrine

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of a _hyperdoctrine_ is essentially an axiomatization of the collection of [[slice category|slices]] of a [[locally cartesian closed category]] (or something similar): a [[category]] $T$ together with a functorial assignment of "slice-like"-categories to each of its objects, satisfying some conditions.

In its use in [[mathematical logic]] ("categorical logic" ([Lawvere 69](#Lawvere69))) a hyperdoctrine is thought of (under [[categorical semantics]] of [[logic]]/[[type theory]]) as a collection of [[contexts]] together with the operations of [[context extension]]/[[substitution]] and [[quantifier|quantification]] on the categories of [[propositions]] or [[types]] in each context.  Therefore specifying the structure of a hyperdoctrine over a given [[category]] is a way of equipping that with a given kind of [[logic]].

Specifically, a hyperdoctrine on a category $T$ for a given notion of logic $L$ is a [[functor]]

$$
  P \colon T^{op} \to \mathbf{C}
$$

to some [[2-category]] (or even [[higher category]]) $\mathbf{C}$, whose objects are categories whose [[internal logic]] corresponds to $L$.  Thus, each object $A$ of $T$ is assigned an "$L$-logic" (the internal logic of $P(A)$).

In the most classical case, $L$ is [[propositional logic]], and $\mathbf{C}$ is a 2-category of [[posets]] (e.g. [[lattices]], [[Heyting algebras]], or [[frames]]).  A hyperdoctrine is then an incarnation of [[first-order logic|first-order]] [[predicate logic]].  A canonical class of examples of this case is where $P$ sends $A \in T$ to the [[poset of subobjects]] $Sub_T(A)$ of $A$.  The predicate logic we obtain in this way is the usual sort of [[internal logic]] of $T$.

We generally require also that for every morphism $f \colon A \to B$ the morphism $P(f)$ has both a [[left adjoint]] as well as a [[right adjoint]], typically required to satisfy [[Frobenius reciprocity]] and the [[Beck-Chevalley condition]].  These adjoints are regarded as the action of [[quantifiers]] along $f$.  Thus, a hyperdoctrine can also be regarded as a way of "adding quantifiers" to a given kind of logic.

More precisely, one thinks of

* $T$ as a category of [[types]] or rather [[contexts]];

* $P$ as assigning

  * to each context $X \in T$ the [[lattice]] of [[propositions]] in this context;

  * to each morphism $f \colon X \to Y$ in $T$ the operation of  "substitution of variables" / "extension of contexts" for propositions $P(Y) \to P(X)$;

* the left adjoint to $P(f)$ gives the application of the [[existential quantifier]];

* the right adjoint to $P(f)$ gives the application of the [[universal quantifier]] (see there for the interpretation of quantifiers in terms of adjoints).

* The Beck-Chevalley condition ensures that quantification interacts with substitution of variables as expected

* Frobenius reciprocity expresses the derivation rules.

So, in particular, a hyperdoctrine is a kind of [[indexed category]] or [[Grothendieck fibration|fibered category]].

The general concept of hyperdoctrines does for [[predicate logic]] precisely what [[Lindenbaum-Tarski algebras]] do for [[propositional logic]], positioning the [[categorical semantics|categorical formulation]] of [[logic]] as a natural extension of the algebraization of logic.

## Properties
 {#Properties}

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

The [[functors]]

* $Cont$, that form a [[category of contexts]] of a [[first-order logic|first-order theory]];

* $Lang$ that forms the [[internal language]] of a [[hyperdoctrine]]

constitute an [[equivalence of categories]]

$$
  FirstOrderTheories
    \stackrel{\overset{Lang}{\leftarrow}}{\underset{Cont}{\to}}
  Hyperdoctrines
  \,.
$$

=--

This is due to ([Seely, 1984a](#SeelyA)). For more details see _[[relation between type theory and category theory]]_.

## Examples

### Classes of hyperdoctrines

* [[first-order hyperdoctrine]]

* [[modal hyperdoctrine]]

* [[regular hyperdoctrine]]

* [[coherent hyperdoctrine]]

* [[Boolean hyperdoctrine]]

* [[geometric hyperdoctrine]]

* [[linear hyperdoctrine]]

### Special cases

* $T$ = the category of contexts, $P(X)$ is the category of formulas. "Given any theory (several sorted, intuitionistic or [[first-order hyperdoctrine|classical]]) ..."

* $T$ = the category [[Set]] of [[small sets]], $P(X) = 2^X =$ the [[power set]] functor, assigning the [[poset]] of all propositional functions

  ("or one may take suitable 'homotopy classes' of deductions").

* $T$ = the category of small sets, $P(X) = Set^X$ ... "This hyperdoctrine
may be viewed as a kind of set-theoretical surrogate of proof
theory"
* "honest proof theory would presumably yield a hyperdoctrine with
nontrivial $P(X)$, but a syntactically presented one".
* $T$ = [[Cat]], the category of small categories, $P(B) = 2^B$
* $T$ = [[Cat]] the category of small categories, $P(B) = Set^B$
* $T$ = [[Grpd]] the category of small groupoids, $P(B) = Set^B$

## Related notions
 {#RelatedNotions}

* [[comprehension]]

* [[tripos]]

* [[modal hyperdoctrine]]

[[!include categories and logic - table]]

## References

The notion was introduced in

* {#Lawvere69} [[Bill Lawvere]], _Adjointness in Foundations_, ([TAC](http://www.tac.mta.ca/tac/reprints/articles/16/tr16abs.html)), Dialectica 23 (1969), 281-296

and further developed in

* [[Bill Lawvere]], _[[Equality in hyperdoctrines and comprehension schema as an adjoint functor]]_, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14. ([pdf](https://ncatlab.org/nlab/files/LawvereComprehension.pdf))

and with further emphasis on the role of the [[Beck-Chevalley condition]]:

* {#SeelyA} [[Robert A. G. Seely]], *Hyperdoctrines, Natural Deduction and the Beck Condition*, Zeitschr. f. math Logik und Grundlagen d. Math. **29** (1983) 505-542 [[pdf](https://www.math.mcgill.ca/seely/ZML/ZML.PDF)]

Surveys and reviews include

* [[Anders Kock]], [[Gonzalo Reyes]], _Doctrines in categorical logic_, in J. Barwise (ed.) _Handbook of Mathematical Logic_ (North Holland, Amsterdam, 1977) 283-313

* [[Peter Dybjer]], _(What I know about) the history of the identity type_ (2006) ([pdf slides](http://www.cse.chalmers.se/~peterd/papers/historyidentitytype.pdf))

A [[string diagram]] calculus for monoidal hyperdoctrines is discussed in

* Geraldine Brady, [[Todd Trimble]], _[[A string diagram calculus for predicate logic]]_

[[!redirects hyperdoctrines]]


hyperfunction

#Contents#
* table of contents
{:toc}

##Idea
The theory of hyperfunctions, created by the Japanese school of [[Mikio Sato]], [[Masaki Kashiwara]] et al. is one of the many variants of the theory of [[generalized functions]]. Unlike the earlier theory of [[distributions]] of Schwarz et alt. it is not based on duals of some spaces of smooth functions but rather on boundary values of [[holomorphic functions]]. This allows usage of sheaf theory and includes Schwartz distributions as a special case. For example in Schwarz's theory apart from measures one has functions with discrete support which are linear combinations of finite derivatives of delta functions. Here one can allow things like the exponential of a differential operator applied to a delta function. So in a sense one has distributions of infinite order.

Hyperfunctions are a very useful tool in the study of [[D-modules]], holonomic systems of differential equations, and especially some aspects of [[symplectic geometry]] and [[harmonic analysis]] that are part of [[microlocal analysis]], especially [[algebraic microlocalization]].

This page is about hyperfunctions of one variable only, for the multiple variable case see [[hyperfunction of multiple variables]].

## Abstract
We define hyperfunctions and explain some basic properties.

The theory of hyperfunctions of one variable is considerably easier than the one for multiple variables, for this reason we will split the exposition to handle the one dimensional case first, and then generalize to multiple dimensions: [[hyperfunction of multiple variables]].

The exposition of the one dimensional theory will try to illuminate the following points:

1. Hyperfunctions are more general than [[distributions]]: In a certain sense, the space of test functions (for compactly supported hyperfunctions) is restricted to real analytic functions instead of smooth functions. Distributions are "functions that are meromorphic on the real line", while hyperfunctions are allowed to have essential singularities. The meaning of this will be explained by an [example] (/nlab/show/hyperfunction#Hyerfunction_Not_Distribution).

A basic strategy of the exposition will therefore be to stress both similarities and differences of hyperfunctions and distributions (real analytic and smooth setting).

2. Hyperfunctions require less technical machinery, at least in one dimension, since they can be defined and studied with basic complex analysis only. That will nevertheless make it possible to give a rigorous definitions of otherwise only formal expressions that are often used by engineers and physicists, like $\int_a^b \delta (x) dx$.

3. Hyperfunctions and their [[microfunction]]s both form [[flabby sheaves]], which is the starting point of a study of singularities of hyperfunctions and the [[algebraic analysis]] of systems of [[differential equations]].

##Definition

Let $I = (a, b) \subseteq \mathbb{R}$ be an open interval, a complex neighbourhood of $I$ is an open set $U \subset \mathbb{C}$ such that $U \bigcap \mathbb{R} = I$. For $V \subset \mathbb{C}$ open let $\mathcal{O}(V)$ be the [[sheaf]] of holomorphic functions on $V$.

+-- {: .un_def}
###### Definition
The set of hyperfunctions on $I$ for a complex neighbourhood $U$ of $I$ is defined to be the quotient
$$
\mathcal{B}(I) \coloneqq \frac{ \mathcal{O}(U \setminus I) }{\mathcal{O}(U)}
$$
=--

The set $\mathcal{B}(I)$ does not depend on the chosen complex neighbourhood $U$, which is a consequence of [[Mittag-Leffler's theorem]] in the one-dimensional case. The definition easily generalizes to open subsets of $\mathbb{R}$.

Let $U^+ \coloneqq \{u \in U : \operatorname{Im}(u) \gt 0 \}$ and $U^- \coloneqq \{u \in U : \operatorname{Im}(u) \lt 0 \}$, then a nontrivial hyperfunction can be described by two functions $F^+$ holomorphic on $U^+$ and $F^-$ holomorphic on $U^-$ such that there is no function $F$ holomorphic on $U$ that restricts to $F^+$ and $F^-$. In this sense hyperfunctions are "differences" of holomorphic functions on a "boundary" (we did not specify an algebraic object that contains both $F^+$ and $F^-$, so strictly speaking we cannot subtract them).

In the following we will often write $F = (F^+, F^-)$ for a hyperfunction $F$, with $(F^+, F^-)$ being a representative of the equivalence class that is $F$ in $\mathcal{B}$ on some complex neighbourhood. If the algebraic expression of $F^+$ and $F^-$ coincides we can further simplify the notation and simply write $[F]$.

###Examples
The $\delta$ distribution can be represented by $\delta = [\frac{i}{2 \pi z}]$. We can prove that this equality is simply a restatement of Cauchy's integral formula after we define the concept of integration for a compactly supported hyperfunction below.

The Heaviside function $H(x)$ has as a representation:
$$
     H(x) = [- \frac{1}{2 \pi i} log(-z)]
$$
Here we use the main branch of the [[logarithm]] that is defined on the complex plane minus the negative real axis, then $log(-z)$ takes the value $log(|z|) - \pi \, i$ on the upper side of the positive real axis and $log(|z|) + \pi \, i$ on the lower side of the positive real axis, while being holomorphic on the negative real axis. This results in
$$
        H(x) = \lim_{\epsilon \to 0+} (F^+ (x + i \epsilon) - F^- (x - i \epsilon)) =

\begin{cases}

  1, & \text{if}\; x \gt 0  \\
  0, & \text{if}\; x \lt 0

\end{cases}

$$
and a singularity in $x=0$.

##Basic Properties and Definitions {#Basic_Properties_One_Variable}

Let $U \subseteq \mathbb{R}$ be open.

###Derivative

+-- {: .un_def}
###### Definition
The **derivative** of a hyperfunction $F = [F^+, F^-]$ is
$$
F' = [\frac{dF^+}{dz}, \frac{dF^-}{dz}]
$$
=--
We immediatly see that the derivative of a hyperfunction is again a hyperfunction and that all hyperfunctions are indefinitly differentiable.

####Examples
As the derivative of the Heaviside function we obtain:
$$
    H'(x) = [\frac{d}{dz} (- \frac{1}{2 \pi i} log(-z))] = [- \frac{1}{2 \pi \, i\, z}] = \delta (x)
$$

Note that we can derive this relationship without any resort to dualities of [[TVS]].

###Sheaf Structure

+-- {: .un_theorem}
###### Theorem
**(hyperfunctions are a module over analytic functions)**
Let $\mathcal{A}(U)$ be the algebra of real analytic functions on $U$, then $\mathcal{B}(U)$ is a module over $\mathcal{A}(U)$.
=--

In fact every real analytic function $f$ is naturally an element of $\mathcal{B}(U)$ represented by $[f, 0]$ or $[0, -f]$ on a complex neighbourhood. Given any hyperfunction $F = (F^+, F^-)$ we can define the product by $f F := (f F^+, F^-) = (F^+, -f F^-)$ which can be shown to be independent of the various choices involved (representation of $F$, complex neighbourhood of $U$, domain of the extension of the domain of $f$).

+-- {: .un_theorem}
###### Theorem
**(hyperfunctions form a [[flabby sheaf]])**
The sets of hyperfunctions of open subsets of $U$ form a [[flabby sheaf]] of [[vector spaces]] over $U$.
=--

+-- {: .un_theorem}
###### Theorem
**(distributions are a subsheaf)**
There is a linear injection from the space of distributions $\mathcal{D}'(U)$ to the space of hyperfunctions $\mathcal{B}(U)$. The sheaf of germs of distributions is therefore a subsheaf of the sheaf of germs of hyperfunctions.
=--

###Multiplication
There is no product of hyperfunctions that reduces to the ordinary product for real analytic functions and respects the chain rule of differential calculus, just as in the case of distributions.
Nevertheless certain hyperfunctions may be multiplied, more on that later.

First let us note that the naive definition of the product of two hyperfunctions $F = [F^+, F^-], G = [G^+, G^-]$ via
$$
FG := [F^+ G^+, F^- G^-]
$$
is obviously not well defined on equivalence classes.
...

###Support
In general we cannot speak of the value of a hyperfunction at a certain point, much like we cannot speak of the value of a distribution at a certain point. We say that a hyperfunction $F$ vanishes on an open subset $U'$ of $U$ if it coincides with the zero hyperfunction, which is of course equivalent to stating that there is a representation of $F = (F^+, F^-)$ such that the boundary values of $F^+$ and $F^-$ coincide on $U'$ and therefore define a real analytic function on $U'$.

+-- {: .un_def}
###### Definition
The **support** of a hyperfunction $F$ is the complement of the largest open subset of $U$ on which $F$ vanishes in the sense described above.
=--

For any compact $K$ we will denote by $\mathcal{B}_K$ all hyperfunctions whose support is contained in $K$.

####Integration for Compactly Supported Hyperfunctions
Since we defined hyperfunctions for open subsets only, we do not yet know anything about hyperfunctions $\mathcal{B}_K$ with support in a compact subset $K$, but the following theorem tells us that there is no difference:

+-- {: .un_theorem}
###### Theorem
**(characterization of compactly supported hyperfunctions)**
Let $\mathcal{B}_K$ be all hyperfunctions with support contained in $K \subset \mathbb{R}$ compact and $V$ be any complex neighbourhood of $K$. Then we have the following isomorphism:
$$
\mathcal{B}_K \cong \frac{ \mathcal{O}(V\setminus K) }{\mathcal{O}(V)}
$$
=--

Let $a, b \in \mathbb{R}$ with $a \leq b$ and suppose that $F$ is a hyperfunction that is analytical in $a$ and $b$. Then we can choose a complex neighbourhood $V$ of $[a, b]$, paths $\tau^+$ in $V^+$  and $\tau^-$ in $V^-$ from a to b and a representation of $F = (F^+, F^-)$ such that we can define the integral

$$
\int_a^b F(x) dx := \int_{\tau^+} F^+ (z) dz - \int_{\tau^-} F^- (z) dz
$$

independently of all the arbitrary choices we made, thanks to the holomorphy of the functions on the right side (and Cauchy's integral formula for holomorphic functions, of course).

####Example of Integration for Hyperfunctions
Using the definition of an integral of a hyperfunction above, we can easily prove
$$
      \int_a^b F'(x) dx = F(b) - F(b)
$$
for all hyperfunctions $F$ that are analytic in $a$ and $b$. Therefore we just made rigorous sense of the formula
$$
\int_a^b \delta (x) dx = H(b) - H(a)
$$
for $a, b \neq 0$ and $H$ the Heaviside function.

###Example of a Hyperfunction that is not a Distribution {#Hyerfunction_Not_Distribution}
Warning: As of this moment this paragraph consists of some handwaving.

A well known structure theorem in the theory of distributions says that every distribution whose support consists of one single point $P$ is in fact a finite linear combination of distributional derivatives of the $\delta$ function at $P$.

As an example of a hyperfunction that is not a distribution we therefore have $F = [\exp{\frac{1}{z}}]$ with $\supp{F} = \{ 0 \}$.

Using the representation of the $\delta$ distribution, we see that
$$
\delta^{(n)}(x) = [\frac{(-1)^{n+1} n!}{2 \pi i} \frac{1}{z^{n+1}}]
$$

Using this representation, for any real analytic test function $a$ we see that the following identity holds, which motivates the use of the $\delta$ as in distribution theory:
$$
\int a(x) \delta^{n}(x) dx = a^{(n)} (0)
$$

Comparing the Laurent series of $F$ with the representation of $\delta^{(n)}$ we get
$$
[\exp{\frac{1}{z}}] = [1 + 2 \pi i  \sum_{n=0}^{\infty}  \frac{1}{n! (n-1)!} \delta^{(n)}(z)]
$$
with $(-1)! := 1$.
Given the structure theorem about distributions cited above, we see that the hyperfunction $[\exp{\frac{1}{z}}]$ cannot be a distribution. There is an analog structure theorem for hyperfunctions, however:

+-- {: .un_theorem}
###### Theorem
**(structure of hyperfunctions supported at a single point)**
Let $F$ be a hyperfunction supported at the origin, then $F$ has a representation as
$$
       F(x) = [ \sum_{n=0}^{\infty} b_n \delta^{(n)} (x)]
$$
where the coefficients satisfy
$$
\lim_{n \to \infty} \sqrt{|b_n n!|} = 0
$$
=--

We include a sketch of the proof of the first statement to illustrate the ease of the use of hyperfunctions.

+-- {: .proof}
###### Proof
Let $F$ be a hyperfunction supported at the origin, that means that $F$ is analytic in a complex neighbourhood of the origin (excluding the origin itself). Therefore $F$ has a representation by a [[Laurent series]] in a neighbourhood of the origin:
$$
    F(x) = [\sum_{- \infty}^{\infty} a_n z^n] = [\sum_{- \infty}^{-1} a_n z^n]
$$
In the representation of $F$ we may remove the holomorphic part of the Laurent series. Now we may insert the representation of the $\delta$ hyperfunctions and get
$$
F(x) = [\sum_{- \infty}^{-1} b_n \delta^{(n)} (z)] = \sum_{- \infty}^{-1} b_n [\delta^{(n)} (z)]
$$
The last sum is understood to converge in the topology of uniform convergence on compact subsets where the summands are holomorph.
=--

###Topology and Duality for Compactly Supported Hyperfunctions
We can introduce a canonical topology on the space $\mathcal{B}_K$ of hyperfunctions with support in a compact $K$ and show that the topological dual is the space $\mathcal{A}_K$ of germs of real analytical functions, that is the inductive limit of the spaces of holomorphic functions on a cofinal set of complex neighbourhoods of $K$, and vice versa.

The algebraic statement is known as **Köthe's (duality) theorem**.

This will also hint at the fact that we will not be able to introduce a topology on $\mathcal{B}(\mathbb{R})$ that resembles the situation in the theory of distributions, since its [[dual vector space|dual space]] would be something like "the space of compactly supported real analytic functions".

...

##Main Theoretical Results
One striking example of the use of hyperfunctions in the theory of differential equations with real analytic coefficients is this result due to Sato:

Let $U \subseteq \mathbb{R}$ be open and $P$ be a linear, finite order differential operator with real analytic coefficients defined on $U$:
$$
       P(x, \frac{d}{dx}) = \sum_{i=0}^n a(x) (\frac{d}{dx})^{i}
$$
with $a_n \neq 0$.

+-- {: .un_theorem}
###### Theorem
**(solvability of differential equations)**
For every $f \in \mathcal{B}(U)$ there is a solution $u \in \mathcal{B}(U)$ of the equation $P u = f$. Every such solution can be extended to an open set $V \supset U$ iff the coefficients $a_i$ and $f$ can be extended to $V$.
=--

Briefly: $P$ is surjective on hyperfunctions.

Two related results are:

+-- {: .un_theorem}
###### Theorem
$P$ is a sheaf endomorphismus, both of the sheaf of real analytic functions and of the sheaf of hyperfunctions.
=--

+-- {: .un_theorem}
###### Theorem
$P$ does not enlarge the support of hyperfunctions.
=--

##Singularities and Microfunctions
Much information about a given hyperfunction is encoded in the kind of singularities that it has. A first step into the theory of singularities is the following definition:

+-- {: .un_def}
###### Definition
The **singular support sing supp($F$)** of a hyperfunction $F$ is the complement of the largest open set on which $F$ is real analytic.
=--

This definition does not seem particularly useful, since it only pinpoints the singular locus of a hyperfunction, without explaining the differences between various singularities. One question of crucial importance to physics is for example "when can two [[distributions]]" be multiplied?

The singular support does not help much: The $\delta$ distribution cannot be squared, while the distribution defined by $[\frac{1}{x + \epsilon i}]$ can. (The latter is a distribution defined as a [[Cauchy principal value]]). The singular support of both consists of the origin. But in a certain sense the singularity of the $\delta$ distribution is worse than that of $[\frac{1}{x + \epsilon i}]$. Going a step further requires the notion of [[wavefront sets]] in the smooth setting. Note that the definition of a [[wavefront set]] needs the concept of a cutoff function, that is a smooth function with compact support, which cannot be used in the real analytic setting.

...

## Related concepts

* [[microlocal analysis]], [[wave front set]]

##References

* Springer Online Enc. of Math.: [hyperfunction](http://eom.springer.de/h/h048420.htm)

A gentle introduction with examples is the booklet

* K. Yosida, _Operational calculus. A theory of hyperfunctions_, Applied Mathematical Sciences, 55. Springer-Verlag, New York, 1984. x+170 pp.

* [[Mikio Sato]], _Theory of hyperfunctions I_, [pdf scan](https://repository.dl.itc.u-tokyo.ac.jp/?action=repository_uri&item_id=39927&file_id=19&file_no=1)

* M. [[Kashiwara]],   T. Kawai,   T. Kimura,  _Foundation of algebraic analysis_, Princeton Univ. Press  (1986)  ((translated from the Japanese))

* Mitsuo Morimoto: _An introduction to Sato's hyperfunctions._ ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0811.46034&format=complete))

* A. Eida, S. Pilipović, _On the microlocal decomposition of same classes of hyperfunctions_, Math. Proc. Camb. Phil. Soc. __125__, 455-461 (1999)

It is possible to use hyperfunctions as an introduction to generalized functions for physicists and engineers with a minimal background in complex analysis, see

* Urs Graf: _Introduction to hyperfunctions and their integral transforms_ ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:pre05662238&format=complete) [pdf](https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B_dBawvWxGGQODUyNDhjNzktNjM3Mi00NzM0LTg4ZDYtMWIzNGIwNWZkMzdi&hl=en_US)
category: analysis
[[!redirects hyperfunction]]
[[!redirects hyperfunctions]]
[[!redirects Sato's hyperfunction]]
[[!redirects Sato's hyperfunctions]]


hypergeometric construction of KZ solutions

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A large class of solutions of the [[Knizhnik-Zamolodchikov equation]], hence a large class of [[braid group representations]], arises as the [[Dolbeault cohomology|holomorphic]] [[twisted de Rham cohomology]]  (twisted by a "[[local system]]") of [[configuration spaces of points]] in the punctured [[plane]] ([Schechtman & Varchenko 89](#SchechtmanVarchenko89), [90](#SchechtmanVarchenko90), [91](#SchechtmanVarchenko91)). This subsumes in particular the construction of [[conformal blocks]] (on the punctured [[Riemann sphere]]) for [[2d CFTs]] of [[affine Lie algebra]]/[[WZW model]]-type ([Feigin, Schechtman & Varchenko 90](#FeiginSchechtmanVarchenko90), [94](#FeiginSchechtmanVarchenko90), [95](#FeiginSchechtmanVarchenko95)).

Since the canonical [[differential form]]-representatives in this construction may be regarded as generalized [[hypergeometric functions]], the original authors referred to this approach as "hypergeometric solutions" to the [[KZ equation]] ([Schechtman & Varchenko 90](#SchechtmanVarchenko90)). In hindsight, it is not the theory of hypergeometric functions that drives the theory, but the peculiarities of (holomorphic) [[local systems]] and their [[twisted de Rham cohomology]] of [[configuration spaces of points]].

## References

[[!include hypergeometric KZ-solutions -- references]]