higher prequantum geometry

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###Context###
#### Higher geometry
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[[!include higher geometry - contents]]
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#### Geometric quantization
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[[!include geometric quantization - contents]]
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> under construction

#Contents#
* table of contents
{:toc}

## Idea

Traditional [[prequantum geometry]] is the [[differential geometry]] of [[smooth manifolds]] which are equipped with a _[[twisted cohomology|twist]]_   in the form of a [[circle group]]-[[principal bundle]] and a circle-[[principal connection]].
In the context of _[[geometric quantization]]_ of [[symplectic manifolds]] these arise as _[[prequantum bundles]]_. Equivalently, prequantum geometry is the _[[contact geometry]]_ of the  total spaces of these bundles, equipped with their [[Ehresmann connection]] [[differential 1-form]] and  thought of as _[[regular contact manifolds]]_.  Prequantum geometry notably studies the [[automorphisms]] of [[prequantum bundles]] covering [[diffeomorphisms]] of the base
-- the _[[prequantum operators]]_ or _[[contactomorphisms]]_ -- and the [[action]] of these on the space of [[sections]] of the [[associated bundle|associated]] [[line bundle]] -- the _[[prequantum states]]_. This is an intermediate step in the genuine _[[geometric quantization]]_ of the [[curvature]] [[differential 2-form]] of these bundles,
which is obtained by "dividing the above data in half" ([[polarization]]), important for instance in the the _[[orbit method]]_.

But prequantum geometry is of interest in its own right.
For instance the above automorphism group naturally provides the
[[Lie integration]] of the [[Poisson bracket]] [[Lie algebra]] of the underlying [[symplectic manifold]], together with the canonical
injection into the [[group of bisections]] of the [[Lie integration]] of the  _[[Atiyah Lie algebroid]]_ which is associated with the given circle bundle, all of which are fundamental objects of interest in the study of [[line bundles]] over [[manifolds]].

For a plethora of applications in [[differential geometry]], one wants to generalize this to _[[higher differential geometry]]_ (see at _[[motivation for higher differential geometry]]_) and accordingly study _higher prequantum geometry_.

## Motivation and survey of results

### Ordinary prequantum geometry in terms of automorphisms in slices
 {#OrdinaryPrequantumGeometryInTermsofAutomorphismsInSlices}

A sequence of time-honored  traditional concepts in [[geometric quantization]]/[[prequantum geometry]] is

| [[Lie groups]]: | [[Heisenberg group]] | $\hookrightarrow$ | [[quantomorphism group]] | $\hookrightarrow$ |   [[gauge group]]  |
|--|--|--|--|--|--|
| [[Lie algebras]]: | [[Heisenberg Lie algebra]] | $\hookrightarrow$ | [[Poisson Lie algebra]] | $\hookrightarrow$ | twisted [[vector fields]] |

For instance in the [[geometric quantization]] of the [[electromagnetic field|electrically]] [[charged particle|charged]] [[particle]] [[sigma-model]] we have a [[prequantum circle bundle]] $P$ with [[connection on a bundle]] $\nabla$ on a [[cotangent bundle]] $X = T^* Y$ which is essentially the [[pullback]] of the [[electromagnetic field]]-bundle on [[target space|target]] [[spacetime]] $Y$. Its _[[quantomorphism group]]_ is the group of [[diffeomorphisms]] $P \stackrel{\simeq}{\to} P$ of the total space of the prequantum bundle which preserve the connection (also called the _[[contactomorphism]]_ of $(P,\nabla)$ regarded as a [[regular contact manifold]]). For the following it is convenient to say this using the language of _[[moduli stacks]]_: we may regard $X$ as a [[representable functor|representable]] [[sheaf]] on the [[site]] of [[smooth manifolds]] (a "[[smooth space]]") and then moreover as a [[representable functor|representable]] [[stack]] on this site (a "[[smooth groupoid]]") and make use of the tautological existence of the [[moduli stack]] of $U(1)$-[[principal connections]], which we write $\mathbf{B}U(1)_{conn}$ (we don't need further details right now, but they can be found for instance at _[[circle n-bundle with connection]]_ for details). By definition this is such that for any $X$ a map $\nabla \colon X \to \mathbf{B}U(1)_{conn}$ is equivalently a $U(1)$-[[principal connection]] and such that a [[homotopy]] $\eta \colon \nabla_1 \to \nabla_2$ between two such maps is equivalently a [[gauge transformation]] between two such connections. With this formulation a [[quantomorphism]] of the [[prequantum bundle]] $\nabla$ is equivalently a diagram of the form as on the right of

$$
  \mathbf{QuantMorph}(\nabla)
  =
  \left\{
  \array{
    X &&\underoverset{\simeq}{\phi}{\to}&& X
    \\
    & \searrow &\swArrow_{\eta}& \swarrow
    \\
    && \mathbf{B}U(1)_{conn}
  }
  \right\}
$$

in the [[(2,1)-category]] of [[stacks]], namely a [[diffeomorphism]] $\phi \colon X \stackrel{\simeq}{\to} X$ of the base space of the bundle together with a [[gauge transformation]] of $U(1)$-[[principal connections]] $\eta \colon \phi^* \nabla \stackrel{\simeq}{\to} \nabla$.

The [[quantomorphism group]] is naturally an ([[infinite-dimensional Lie group|infinite dimensional]]) [[Lie group]]. Its [[Lie algebra]] is the [[Poisson bracket]] [[Lie algebra]]. If $X$ is equipped with the structure of a [[Lie group]] itself (notably if it is a [[vector space]]), then the sub-Lie algebra of that on the [[invariant differential form|invariant vectors]] is the [[Heisenberg Lie algebra]] and the Lie group corresponding to that is the [[Heisenberg group]].

One also says that a triangular diagram as above is an autoequivalence of the "modulating" map $\nabla \colon X \to \mathbf{B}U(1)_{conn}$ in the _[[slice (infinity,1)-category|slice (2,1)-category]]_ of [[stacks]]/[[smooth groupoids]] over $\mathbf{B}U(1)_{conn}$.

Such autoequivalences in slices are familiar from basic concepts of [[Lie groupoid]] theory. For $\mathcal{G} = (\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0)$ a [[Lie groupoid]], we may regard the inclusion of its manifold of objects as an [[atlas]] being a map $p_\mathcal{G} \colon\mathcal{G}_0 \to \mathcal{G}$. Regarding this atlas as an object in the [[slice (infinity,1)-category|slice (2,1)-category]] of [[stacks]]/[[smooth groupoids]] over $\mathcal{G}$, its autoequivalences are diagrams as on the right of

$$
  \mathbf{BiSect}(p_{\mathcal{G}})
  =
  \left\{
    \array{
      \mathcal{G}_0 &&\stackrel{\phi}{\to}&& \mathcal{G}_0
      \\
      & \searrow &\swArrow_\eta & \swarrow
      \\
      && \mathcal{G}
    }
  \right\}
  \,.
$$

This is a [[diffeomorphism]] $\phi \colon \mathcal{G}_0 \stackrel{\simeq}{\to} \mathcal{G}_0$ of the [[smooth manifold]] of [[objects]] equipped with a [[natural transformation]] $\eta$ whose component map is a [[smooth function]] that assigns to each point $q\in\mathcal{G}_0$ a [[morphism]] in $\mathcal{G}$ of the form $\eta_q \colon q \to \phi(q)$. This collection of data is known as a _[[bisection]]_ of a [[Lie groupoid]]. Bisections naturally form a group $\mathbf{BiSect}(p_{\mathcal{G}})$ , which is all the more manifest if we understand them as autoequivalences of the atlas in the slice, called the [[group of bisections]].

This perspective of regarding maps of [[smooth groupoids]] as objects in the slice over their codomain (an elementary step in [[higher category theory]]/[[(infinity,1)-topos theory|higher topos theory]], but not common in traditional differential geometry) turns out to be useful and drives all of the refinements, generalizations and theorems that we discuss in the following: we will see that higher [[prequantum geometry]] is essentially the geometry insice [[slice (infinity,1)-topos|higher slice categories]] of [[infinity-stack|higher stacks]] over [[moduli infinity-stack|higher moduli stacks]] of [[principal infinity-connection|higher principal connections]].

Before we get there, notice the following...

### The need for higher prequantum bundles
 {#TheNeedForHigherPrequantumBundles}

The tools of [[geometric quantization]] mainly apply to [[quantum mechanics]] and only partially to [[quantum field theory]]. In particular in the context of _[[extended prequantum field theory]]_ in [[dimension]] $n$ a [[prequantum bundle]] over the ([[phase space|phase]]-)space of [[field (physics)|fields]] is to be refined (de-[[transgression|transgressed]]) to a _[[prequantum n-bundle]]_ over the [[moduli ∞-stack]] of [[field (physics)|fields]]. Therefore in order to apply [[geometric quantization]] to [[extended prequantum field theory]] to obtain [[extended quantum field theory]] we first need extended/higher prequantum geometry.

For instance the [[prequantum n-bundle|prequantum 3-bundle]] for standard [[3d Chern-Simons theory|3d]] [[Spin group]] [[Chern-Simons theory]] is modulated by the differential [[smooth first fractional Pontryagin class]]

$$
  \array{
    \mathbf{B}Spin_{conn}
     &\stackrel{\tfrac{1}{2}\hat \mathbf{p}_1}{\to}&
    \mathbf{B}^3 U(1)_{conn}
    \\
    \downarrow && \downarrow & forget \; connections
    \\
    \mathbf{B}Spin
     &\stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to}&
    \mathbf{B}^3 U(1)
    \\
    \downarrow && \downarrow & geometric\;realization
    \\
    B Spin &\stackrel{\tfrac{1}{2}p_1}{\to}& K(\mathbb{Z},4)
  }
  \,,
$$

modulating/classifying the universal _[[Chern-Simons circle 3-bundle with connection]]_ (also known as a _[[bundle 2-gerbe]]_) over the [[moduli stack]] of [[field (physics)|fields]] of $G$-Chern-Simons theory, which is the moduli stack $\mathbf{B}G_{conn}$ of $G$-[[principal connection]].

Similarly the [[prequantum n-bundle|prequantum 7-bundle]] for [[7d Chern-Simons theory]] on [[string 2-group]] [[principal infinity-connections|principal 2-connections]] is given by the differential [[smooth second fractional Pontryagin class]]

$$
  \array{
    \mathbf{B}String_{conn}
     &\stackrel{\frac{1}{6}\hat \mathbf{p}_2}{\to}&
    \mathbf{B}^7 U(1)_{conn}
    \\
    \downarrow && \downarrow & forget\; connections
    \\
    \mathbf{B}String
     &\stackrel{\frac{1}{6}\mathbf{p}_2}{\to}&
    \mathbf{B}^7 U(1)
    \\
    \downarrow && \downarrow & geometric\; realization
    \\
    B String &\stackrel{\frac{1}{6}p_2}{\to}& K(\mathbb{Z},8)
  }
  \,,
$$

modulating/classifying the universal _[[Chern-Simons circle 7-bundle with connection]]_ over the moduli 2-stack $\mathbf{B}String_{conn}$ of [[string 2-group]] [[principal infinity-connection|principal 2-connections]].

Therefore we want to lift the [above](#OrdinaryPrequantumGeometryInTermsofAutomorphismsInSlices) table of traditional notions to [[higher geometry]]...

### Brief recollection: Higher geometry

In order to say this, clearly we need some basics of [[higher geometry]]...

$$
  \array{
     && Groupoids
     \\
     & \swarrow && \searrow^{\mathrlap{nerve}}
     \\
     Categories && &&  Kan complexes
     \\
     & \searrow && \swarrow
     \\
     && (\infty,1)-Categories
  }
  \,.
$$

Important construction principle for [[(∞,1)-categories]]: [[simplicial localization]]. For $\mathcal{C}$ a [[category]] with some subset of morphisms $W \hookrightarrow Mor(\mathcal{C})$ declared to be "[[weak equivalences]]", the [[simplicial localization]]

$$
  L_W \mathcal{C} \in (\infty,1)Cat
$$

is the [[universal construction|universal]] $(\infty,1)$-category obtained from $\mathcal{C}$ by universally turning each weak equivalence into an actual [[homotopy equivalence]] in the sense of [[homotopy theory]].

In particular let $C$ be a [[site]], assumed for simplicity to have [[point of a topos|enough points]]. Declare then that in the [[functor category]] $Func(C^{op}, KanCplx)$, hence in [[Kan complex]]-valued presheaves, the weak equivalences are the [[stalk|stalkwise]] [[homotopy equivalences]] of Kan complexes. Then

$$
  \mathbf{H} \coloneqq Sh_{\infty}(C) \coloneqq L_{W} Func(C^{op}, KanCplx)
$$

is called the _[[(∞,1)-topos]]_ of [[(∞,1)-sheaves]]/[[∞-stacks]] on $C$.

An [[A-∞ algebra]]-object $G$ in such an $(\infty,1)$-topos such that $\pi_0(G)$ is a [[group]] is called an [[∞-group]] "with geometric structure as encoded by the test spaces $C$". The canonical source of $\infty$-groups are the [[homotopy fiber products]] of point inclusions $* \to X$ of any object X, the [[loop space object]]

$$
  \Omega X \coloneqq {*} \underset{X}{\times} {*}
  \,.
$$

In fact this are _all_ the [[∞-groups]] that there are, up to equivalence: forimg [[loop space objects]] is an [[equivalence of (∞,1)-categories]]

$$
  Grp(\mathbf{H})
   \stackrel{\overset{\Omega}{\leftarrow}}{\underoverset{\mathbf{B}}{\simeq}{\to}}
  \mathbf{H}^{*/}_{\geq 1}
$$

between [[∞-groups]] and [[pointed object|pointed]] [[connected object in an (∞,1)-topos|connected]] objects. The inverse equivalence $\mathbf{B}$ is the _[[delooping]]_ operation.

We say that such an $(\infty,1)$-topos $\mathbf{H}$ is _[[cohesive]]_ if it is equipped with an [[adjoint triple]] of [[idempotent monad|idempotent]] (co)/[[(∞,1)-monads]]

| [[shape modality]] | | [[flat modality]] | | [[sharp modality]]  |
|--|--|--|--|--
| idemp. monad | | idemp. comonad | | idemp. monad |
| $\Pi$ | $\dashv$ | $\flat$ | $\dashv$ | $\sharp$ |

This implies (strictly speaking we need [[differential cohesion]] for that, coming from another adjoint triple of (co)monads) that for every [[braided ∞-group]] $\mathbb{G} \in Grp(\mathbf{H})$ there is a canonical object $\mathbf{B}\mathbb{G}_{conn}$ which modulats $\mathbb{G}$-[[principal ∞-connections]].

### Higher Atiyah groupoids

Looking at the [above](#OrdinaryPrequantumGeometryInTermsofAutomorphismsInSlices) table and noticing the [above](#TheNeedForHigherPrequantumBundles) need for higher prequantum bundles, we should try to find an analogous table of concepts in [[higher geometry]], something like this:

[[!include slice automorphism groups in higher prequantum geometry - table]]

(...)

The way all these notions and theorems work is by considering [[automorphism ∞-groups]] of the classifying (or rather: modulating) maps $\nabla \colon X \to \mathbf{B}\mathbb{G}_{conn}$ of a [[prequantum ∞-bundle]] in the [[slice (∞,1)-topos]] over the domain. For instance

$$
  \mathbf{QuantMorph}(\nabla)
  =
  \left\{
    \array{
      X && \underoverset{\simeq}{\phi}{\to} && X
      \\
      & {}_{\mathllap{\nabla}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla}}
      \\
      && \mathbf{B}\mathbb{G}_{conn}
    }
  \right\}
  \,.
$$

The others are obtained by succesively forgetting connection data. For instance

$$
  \BiSect(Cou(\nabla))
  =
  \left\{
    \array{
      X && \underoverset{\simeq}{\phi}{\to} && X
      \\
      & {}_{\mathllap{\nabla_1}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla_1}}
      \\
      && \mathbf{B}(\mathbf{B}\mathbb{G}_{conn})
     }
   \right\}
   \,.
$$

and

$$
  \BiSect(At(\nabla))
  =
  \left\{
    \array{
      X && \underoverset{\simeq}{\phi}{\to} && X
      \\
      & {}_{\mathllap{\nabla_0}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla_0}}
      \\
      && \mathbf{B}\mathbb{G}
    }
  \right\}
  \,.
$$

The extension sequence is then schematically simply the following

$$
  \left\{
    \array{
      && X
      \\
      & \swarrow &  & \searrow
      \\
      & \searrow &\swArrow& \swarrow
      \\
      && \mathbf{B}\mathbb{G}_{conn}
    }
  \right\}
  \; \to \;
  \left\{
    \array{
      X &&\stackrel{\simeq}{\to}&& X
      \\
      & {}_{\mathllap{\nabla}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla}}
      \\
      && \mathbf{B}\mathbb{G}_{conn}
    }
  \right\}
  \;
  \to
  \;
  \left\{
    \array{
      X && \stackrel{\simeq}{\to} && X
    }
  \right\}
$$

in this generality this now includes various other notions, too:

[[!include higher Atiyah groupoid - table]]

### The central theorem: Quantomorphism $\infty$-group extensions

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

For $\mathbb{G}$ a [[braided ∞-group]] and $\nabla \colon X \to \mathbf{B}\mathbb{G}_{conn}$ a higher prequantum geometry with respect to $\mathbb{G}$ there is a long [[homotopy fiber sequence]]

$$
  \left(\Omega \mathbb{G}\right)\mathbf{FlatConn}\left(\nabla\right)
  \to
  \mathbf{QuantMorph}(\nabla)
  \to
  \mathbf{HamSympl}(\nabla)
  \stackrel{\nabla \circ (-)}{\to}
  \mathbf{B}\left(\left(\Omega \mathbb{G}\right)\mathbf{FlatConn}\left(\nabla\right) \right)
  \,.
$$

Similarly there is the [[Heisenberg infinity-group]] extension

$$
  (\Omega \mathbb{G})\mathbf{FlatConn}(X)
   \to
  \mathbf{Heis}(\nabla)
   \to
  G
$$

=--

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

The [[Lie differentiation]] of the [[∞-group extension]] sequence of prop. \ref{QuantomorphismExtension} is a [[homotopy fiber sequence]] of [[L-∞ algebras]]

$$
  \mathbf{H}(X, \flat \mathbf{B}^{n-1}\mathbb{R})
  \to
  \mathfrak{Poisson}(X,\omega) \to \mathcal{X}_{Ham}(X,\omega)
  \stackrel{\iota_{(-)\omega}}{\to}
  \mathbf{B}\mathbf{H}(X, \flat \mathbf{B}^{n-1}\mathbb{R})
  \,,
$$

where

* $\mathfrak{Poisson}(X,\omega)$ is the [[Poisson Lie n-algebra]] as defined in ([Rogers 11](#Rogers11)).

* $\mathcal{X}_{Ham}$ is the Lie algebra of [[vector fields]] restricted to the [[Hamiltonian vector fields]];

* $\mathbf{H}(X, \flat (\mathbf{B}^{n-1})\mathbb{R})$ is the [[chain complex]] for flat [[de Rham cohomology]] in the given degree, regarded as an abelian [[L-∞ algebra]].

=--

The following table shows what this sequence reduces to when one chooses $\mathbb{G} = \mathbf{B}^{n-1}U(1)$.

[[!include geometric quantization extensions - table]]

### Examples: $String$ and $Fivebrane$ as Heisenberg $\infty$-groups

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

For $G$ a simply connected semisimple compact Lie group such as the [[spin group]], let

$$
  \nabla \coloneqq
   \exp\left(2 \pi i \int_{S^1} [S^1, \tfrac{1}{2}\hat \mathbf{p}_1]\right)
  \;\colon\; G \to \mathbf{B}^2 U(1)_{conn}
$$

be the canonical [[circle 2-bundle with connection]] over it. Then the [[Heisenberg infinity-group|Heisenberg 2-group]] [[infinity-group extension|extension]]

$$
  U(1)\mathbf{FlatConn}(G)
   \to
  \mathbf{Heis}(\nabla)
   \to
  G
$$

is the [[string 2-group]] extension

$$
  \mathbf{B}U(1) \to String(G) \to G
  \,.
$$

=--

(by classification of extensions by cohomology... by Lie 2-algebra computation...)

(and analogously for [[fivebrane 6-group]]...)

$$
 \mathbf{B}^6 U\left(1\right)
   \to
  \mathbf{Heis}\left(\exp\left(2 \pi i \int_{S^1} \left[S^1, \tfrac{1}{2}\hat \mathbf{p}_2\right] \right)\right) \to String
$$

## Constructions in higher prequantum geometry

[[!include slice automorphism groups in higher prequantum geometry - table]]

[[!include higher Atiyah groupoid - table]]

[[!include geometric quantization extensions - table]]

## Related entries

* [[Poisson bracket Lie n-algebra]]

* [[definite parameterization of WZW terms]]

* [[definite globalization of WZW terms]]

* [[higher structures]]

* [[basic bundle gerbe]]

## References

> See also the references at _[[n-plectic geometry]]_ and at _[[higher geometric quantization]]_.

* {#FRS13a} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher geometric prequantum theory|Higher $U(1)$-gerbe connections in geometric prequantization]]_, Rev. Math. Phys. **28** 06 1650012 (2016) [[arXiv:1304.0236](http://arxiv.org/abs/1304.0236), [doi:10.1142/S0129055X16500124](https://doi.org/10.1142/S0129055X16500124)]

* {#FRS13b} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]_, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 – 142 ([arXiv:1304.6292](http://arxiv.org/abs/1304.6292))

* {#Schreiber13} [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ ([arXiv:1310.7930](http://arxiv.org/abs/1310.7930))

[[!redirects higher prequantization]]
[[!redirects higher prequantizations]]

[[!redirects higher pre-quantization]]
[[!redirects higher pre-quantizations]]


higher regulator

[[!redirects Beilinson regulator]]

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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[arithmetic geometry]] a _higher regulator_, or just _regulator_ for short, is a [[homomorphism]] from [[algebraic K-theory]] to a suitable [[ordinary cohomology]] theory. (It makes sense to think that it "regulates" [[cocycles]] in [[algebraic K-theory]], which tend to be hard to analyze, to become cocycles in ordinary cohomology, about which typically more may be said.) This generalizes the original concept of a [[regulator of a number field]] which is a measure for the [[group of units]] of the [[ring of integers]] of the number field, in view of the fact that the [[determinant]] function provides a canonical homomorphism $K_1 \to GL_1$ from the first [[algebraic K-theory]] to the [[group of units]].

Regulators are used in the study of [[L-functions]] for instance in the context of the [[Beilinson conjectures]] and the [[Bloch-Kato conjecture]].

The simplest example of such regulators are

* the [[analytic class number formula]];

* the [[Dedekind zeta-function]] of $K$;

* the Dirichlet regulator map, discussed below.

These regulators may be understood as essentially being the [[Chern characters]] in [[algebraic K-theory]] ([Deninger Scholl (2.6)](#DeningerScholl), [Tamme 12](#Tamme12)).  Based on this observation they serve to define [[differential cohomology]]-refinements of algebraic K-theory, namely _[[differential algebraic K-theory]]_ ([Bunke-Tamme 12](#BunkeTamme12)).

> we usually call transformations from K-theory to [[cyclic homology]] Chern characters, and transformations from K-theory to differential forms regulators. There is one exception, namely the usual Chern character from topological K-theory to cohomology with complex coefficients calculated by the de Rham cohomology. ([Bunke 14, remark 2.20](#Bunke14))

## Definition

### Dirichlet and Borel regulator
 {#BorelRegulator}

For $R$ a [[ring]], $K R$ denoting its [[algebraic K-theory]] [[spectrum]] and $\Sigma^n H \mathbb{R}$ a [[suspension]] of the [[Eilenberg-MacLane spectrum]] of the [[real numbers]], then a regulator of $K R$-[[cohomology theory]] is a [[homomorphism]] of [[spectra]]

$$
  r_{\sigma,p} \;\colon\; K R \longrightarrow \Sigma^p H \mathbb{R}
$$

or equivalently is the induced [[cohomology operations]]. e.g. ([Bunke-Tamme 12, 1.2](#BunkeTamme12))

If here $R$ is the [[ring of integers]] of a [[number field]] and $\sigma \colon R \hookrightarrow \mathbb{C}$ is a choice of embedding into the [[complex numbers]], then the _Borel regulator_ ([Borel 74](#Borel74)) is of this form, for odd $p$, such that its induced [[cohomology operation]]

$$
  r_{\sigma,p} \;\colon\; K_1(R) \longrightarrow \mathbb{R}
$$

is the _Dirichlet regulator_ given by

$$
  u \mapsto log {\vert \sigma(u) \vert}
  \,.
$$

A description of this in [[differential algebraic K-theory]] is in ([Bunke-Tamme  12, 1.2](#BunkeTamme12)):

for $X$ a [[smooth manifold]], then a class in $K R^0(X)$ is represented by  a [[finitely generated module|finitely generated]] [[projective module|projective]] $R$-[[module bundle]] $V \to X$. Write

$$
  cycl(V) \in K R^0(X)
$$

for this class.

Under the chosen embedding $\sigma$ we have the [[complexification]] $V \otimes_\sigma \mathbb{C}$ of this [[module bundle]], which is a [[complex vector bundle]] with (because $R$ is geometrically [[discrete object|discrete]]) a [[flat connection]] $\nabla_{V_\sigma}$.

The choice of hermitean structure on $V_\sigma$, hence a [[reduction of the structure group]] to the [[unitary group]] induces an adjoint connection $\nabla_{V_\sigma}^\ast$. Write then

$$
  CS_p(\nabla_{V_\sigma}^\ast, \nabla_{V_\sigma})
$$

for the relative [[Chern-Simons form]] between these two connections, hence for the [[transgression]] of the relative [[Chern character]] in degree $p+1$.

This is a closed differential form, (the Kamber-Tondeur form, see [Bismut-Lott 95](#BismutLott95)).

This differential form represents, via the [[de Rham theorem]] [[isomorphism]], the Dirichlet regulator above

$$
  r_{\sigma,p}(cycl(V))
  \simeq
  [CS_p(\nabla_{V_\sigma}^\ast, \nabla_{V_\sigma})]
  \in
  H_{dR}^p(X) \simeq H^p(X,\mathbb{R})
  \,.
$$

In [[differential algebraic K-theory]] this construction can be refined from landing in [[de Rham cohomology]] to landing in genuine [[ordinary differential cohomology]] ([[higher prequantization]]), hence with $CS_p(\nabla_{V_\sigma}^\ast, \nabla_{V_\sigma})$ itself realized as the [[curvature]] of a [[circle n-bundle with connection|circle (p-1)-bundle with connection]].

### Regulators of generalized algebraic K-theories
 {#ForGeneralizedAlgebraicKTheories}

Based on the [above](#BorelRegulator) abstract formulation of
the classical Beilinson- and Borel-regulators, the following
general definition suggests itsef:

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

Consider the following data:

1. For $\mathcal{C}^\otimes$ a [[symmetric monoidal (∞,1)-category]] write $\mathcal{K}(\mathcal{C})$ for its [[algebraic K-theory of a symmetric monoidal (∞,1)-category]].

1. For $A \in Ch_\bullet(Ab)$ any [[chain complex]] write $H A$ for its [[Eilenberg-MacLane spectrum]] given by the [[stable Dold-Kan correspondence]].

Then a _regulator_ with [[coefficients]] in $A$ of the [[algebraic K-theory]] [[Brown representability theorem|represented]] by $\mathcal{K}(\mathcal{C})$ is a [[homomorphism]] of [[spectra]] (hence a [[cohomology operation]])

$$
  r \;\colon\; \mathcal{K}(\mathcal{C}) \longrightarrow H A
  \,.
$$

Accordingly the [[spectrum]] of $A$-regulators of $\mathcal{K}(\mathcal{C})$ is the [[mapping spectrum]] $[\mathcal{K}(\mathcal{C}), H A]$.

=--

([Bunke-Tamme 12, def. 23](#BunkeTamme12)).

### Beilinson regulators

(...)

## Examples

### Constructions in terms of line $n$-bundles ($n-1$-bundle gerbes)
 {#GeometricConstruction}

Some special cases of Beilinson regulators have known "geometric" constructions in terms of maps relating [[holomorphic line n-bundles]] for various $n$.

The regulator

$$
  c_{2,2} \colon K_2(X) \longrightarrow H^2(X, \mathbb{Z}(2)_{\mathcal{D}})
$$

is given by sending pairs of non-vanishing [[holomorphic functions]] to the [[holomorphic line bundle]] which is their [[Beilinson-Deligne cup product]] (the "[[Deligne line bundle]]") ([Bloch 78](#Bloch78)).

Moreover, the regulator

$$
  c_{1,2} \colon K_1(X) \longrightarrow H^3(X, \mathbb{Z}(2)_{\mathcal{D}})
$$

or rather its component

$$
  c_{1,2} \colon H^1(X, \mathbf{K}_2) \longrightarrow H^3(X, \mathbb{Z}(2)_{\mathcal{D}})
$$

is given by sending functions constituing a [[cocycle]] in the relevant [[Gersten complex]] to a [[bundle gerbe]] whose transition line bundles are [[Deligne line bundles]] built from these functions ([Brylinski 94, theorem 3.3](#Brylinski94)).

Notice that $c_{2,1}$ is the regulator that interpolates the [[string 2-group]]/[[universal Chern-Simons line 3-bundle]] for a [[reductive algebraic group]] from the algebraic to the complex-analytic realm, see at _[universal Chern-Simons line 3-bundles -- For reductive algebraic groups](universal+Chern-Simons+line+3-bundle#ForReductiveAlgebraicGroups)_.

## Properties

### Becker-Gottlieb transfer and GRR for algebraic K-theory

 {#BeckerGottliebTransfer}

For $\pi \colon X \to B$ a [[proper map|proper]] [[submersion]] of [[smooth manifolds]], there is a variant of [[fiber integration in generalized cohomology]] given by the [[Becker-Gottlieb transfer]] in some $E$-[[cohomology theory]]

$$
  tr_\pi \;\colon\; E^\ast(X) \longrightarrow E^\ast(B)
  \,.
$$

Moreover, for the above [[sheaves]] of $R$-[[modules]] $cycl(V)$ we have the [[direct image]] sheaves $\pi_\ast V$ and there is an identity

$$
  \underset{i \geq 0}{\sum} (-1)^i cycl(R^i \pi_\ast(V))
  \simeq
  tr_\pi(cycl(V))
$$

in $K R(B)$. The [above](#BorelRegulator) Borel-Dirichlet regulator $r_{\sigma,p}$ is such that it preseves this as an identity in $H(X,\mathbb{R})$. Hence it plays a role here analogous that of the [[Chern character]] in the [[Grothendieck-Riemann-Roch theorem]].

The [[Becker-Gottlieb transfer]] refines in turn to [[differential cohomology]], hence [[differential algebraic K-theory]] mapping to [[ordinary differential cohomology]], according to ([Bunke-Gepner 13](BunkeGepner13)).

However, the above relation between direct image of sheaves and push-forward in cohomology receives a correction when refined to [[differential algebraic K-theory]], a correction by a term in the image of the inclusion $a(-)$ of differential forms into [[differential cohomology]], by the [[transfer index conjecture]] one has

$$
  \underset{i \geq 0}{\sum} (-1)^i \widehat{cycl}(R^i \pi_\ast(V))
   + a(something)
  \simeq
  \widehat{tr}_\pi(\widehat{cycl}(V))
$$

where the hats denote the [[differential cohomology]] refinement.

See at _[[Becker-Gottlieb transfer]]_.

### Relation to complex volumes and Bloch group
 {#RelationToComplexVolumes}

There is some relation between the Borel regulators and [[complex volumes]] of [[hyperbolic manifolds]] via maps out of the [[Bloch group]] ([Suslin 90](#Suslin90), [Neumann-Yang 97, p. 17](#NeumannYang97), [Zickert 07, p. 3](#Zickert07), [Zickert 09](#Zickert09)).

For $k$ an [[algebraic number field]] and $\sigma_1, \cdots, \sigma_{r_2}\colon k \to \mathbb{C}$ its complex embeddings up to [[conjugation]], then write

$$
  vol_j \coloneqq vol \circ (\sigma_j)\colon H_3(PSL(2,k), \mathbb{Z})
  \to \mathbb{R}
$$

Then then map

$$
  (vol_1, \cdots, vol_{r_2}) \colon H_3(PSL(2,k),\mathbb{Z}) \longrightarrow \mathbb{R}^{r_2}
$$

is the Borel regulator  ([Neumann 11, p. 6](#Neumann11)).

For the moment, see at _[[Bloch group]]_ for more details.

## Related concepts

* [[Beilinson conjecture]]

## References

The Borel regulator is due to

* {#Borel71} [[Armand Borel]], _Cohomologie de $SL_n$ et valeurs de fonctions de zeta_, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 417 (1971), 613–636.

* {#Borel74} [[Armand Borel]], _Stable real cohomology of arithmetic groups_, Ann. Sci. Ecole Norm. Sup. (4) 7 (1974), 235-272 (1975). MR 0387496

The Beilinson regulator with values in [[Deligne cohomology]] is due to

* {#Bloch78} [[Spencer Bloch]], _Applications of the dilogarithm function in algebraic K-theory and algebraic geometry_, in: Proc. of the International Symp. on Alg. Geometry, Kinokuniya, Tokyo, 1978

* {#Bloch81} [[Spencer Bloch]], _The dilogarithm and extensions of Lie algebras_, Algebraic K-Theory Evanston 1980, Lecture Notes in Mathematics Volume 854, 1981, pp 1-23

* {#Beilinson85} [[Alexander Beilinson]] _Higher regulators and values of L-functions_, Journal of Soviet Mathematics 30 (1985), 2036-2070, ([mathnet (Russian)](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intd&paperid=73&option_lang=eng), [DOI](http://dx.doi.org/10.1007%2FBF02105861))

* {#Beilinson80} [[Alexander Beilinson]], _Higher regulators of curves_, Funct. Anal. Appl. 14 (1980), 116-118, [mathnet (Russian)](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=faa&paperid=1800&option_lang=eng).

* {#Beilinson87} [[Alexander Beilinson]], _Height pairing between algebraic cycles_, in _K-Theory, Arithmetic and Geometry_, Lecture Notes in Mathematics Volume 1289, 1987, pp 1-26, [DOI](http://dx.doi.org/10.1007/BFb0078364).

reviewed in

* {#Soule84} [[Christophe Soulé]], _Régulateurs_ Séminaire Bourbaki, 27 (1984-1985), Exp. No. 644, 17 p. ([Numdam](http://www.numdam.org/item?id=SB_1984-1985__27__237_0))

* {#EsnaultViehweg88} [[Hélène Esnault]], [[Eckart Viehweg]], _Deligne-Beilinson cohomology_ in Rapoport, Schappacher, Schneider (eds.) _Beilinson's Conjectures on Special Values of L-Functions_ . Perspectives in Math. 4, Academic Press (1988) 43 - 91 ([pdf](http://www.uni-due.de/~mat903/preprints/ec/deligne_beilinson.pdf))

* {#Goncharov04} [[Alexander Goncharov]], _Regulators_, Algebraic K-theory Handbook ([arXiv:0407308](http://arxiv.org/abs/math/0407308))

A discussion of the Beilinson regulator on $K_2$ in terms of [[bundle gerbes]] is in

* {#Brylinski94} [[Jean-Luc Brylinski]], _Holomorphic gerbes and the Beilinson regulator_, Astérisque 226 (1994): 145-174 ([[Brylinski94.pdf:file]])

See also

* [[Christophe Soulé]], _On higher p-adic regulators_, Algebraic K-Theory Evanston 1980 Lecture Notes in Mathematics Volume 854, 1981, pp 372-401_ ([publisher page](http://link.springer.com/chapter/10.1007%2FBFb0089530))

The relation to [[Chern characters]] is made very explicit in

* {#Tamme12} [[Georg Tamme]], _Karoubi's relative Chern character and Beilinson's regulator_, Ann. Sci. Ec. Norm. Super. (4) 45 (2012), no. 4, 601-636. ([[TammeRegulator.pdf:file]])

see also

* {#DeningerScholl} [[Christopher Deninger]], [[Anthony Scholl]], section 2.6 of _The Beilinson conjectures_ ([pdf](https://www.dpmms.cam.ac.uk/~ajs1005/preprints/d-s.pdf))

The interpretation of these regulator Chern characters in [[differential algebraic K-theory]] is due to

* {#BunkeGepner13} [[Ulrich Bunke]], [[David Gepner]], _Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory_ ([arXiv:1306.0247](http://arxiv.org/abs/1306.0247))

* {#BunkeTamme12} [[Ulrich Bunke]], [[Georg Tamme]], _Regulators and cycle maps in higher-dimensional differential algebraic K-theory_ ([arXiv:1209.6451](http://arxiv.org/abs/1209.6451))

* {#BunkeNikolausTamme15} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Georg Tamme]], _The Beilinson regulator is a map of ring spectra_ ([arXivL1509.05667](http://arxiv.org/abs/1509.05667))

* {#Bunke14} [[Ulrich Bunke]], _A regulator for smooth manifolds and an index theorem_ ([arXiv:1407.1379](http://arxiv.org/abs/1407.1379))

based on

* {#BismutLott95} [[Jean-Michel Bismut]], [[John Lott]], _Flat vector bundles, direct images and
higher real analytic torsion_, J. Amer. Math. Soc. 8 (1995), no. 2, 291-363.

See also

* {#Weibel84} [[Charles Weibel]], _Algebraic K-theory and the Adams e-invariant_, in _Algebraic K-theory, number theory, geometry and analysis _(Bielefeld, 1982), volume 1046 of Lecture Notes in Math., pages 442-450. Springer, Berlin, 1984. 61 [pdf](http://www.math.illinois.edu/K-theory/handbook/1-139-190.pdf)

* {#Suslin84} A.A. Suslin, _On the K-theory of local fields_, In Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983), volume 34, pages 301-318, 1984.

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

For more references see also at _[[Beilinson conjecture]]_.

Relation of the Borel regulator to the [[Bloch group]], the [[Cheeger-Simons class]]/[[complex volumes]] of [[hyperbolic manifolds]] is discussed in

* {#Suslin90} [[Andrei Suslin]]. _$K_3$ of a field, and the Bloch group_. Trudy Mat. Inst. Steklov., 183:180–199, 229, 1990. Translated in Proc. Steklov Inst. Math. 1991, no. 4, 217–239, Galois theory, rings, algebraic groups and their applications (Russian).

* {#NeumannYang97} [[Walter Neumann]], Jun Yang, _Bloch invariants of hyperbolic 3-manifolds_, Duke Math. J. Volume 96, Number 1 (1999), 29-59. ([arXiv:math/9712224](http://arxiv.org/abs/math/9712224), [Euclid](http://projecteuclid.org/euclid.dmj/1077228942))

* {#Zickert07} [[Christian Zickert]], _The volume and Chern-Simons invariant of a representation_, Duke Math. J., 150 (3):489-532, 2009 ([arXiv:0710.2049](http://arxiv.org/abs/0710.2049), [Euclid](http://projecteuclid.org/euclid.dmj/1259332507))

* {#Zickert09} [[Christian Zickert]], _The extended Bloch group and algebraic K-theory_ ([arXiv:0910.4005](http://arxiv.org/abs/0910.4005))

* {#Neumann11} [[Walter Neumann]], _Realizing arithmetic invariants of hyperbolic 3-manifolds_, Contemporary Math 541 (Amer. Math. Soc. 2011), 233--246 ([arXiv:1108.0062](http://arxiv.org/abs/1108.0062))

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#Contents#
* table of contents
{:toc}

This page is supposed to be a review of the seminal article

* [[Alexander Beilinson]],

  _Higher regulators and values of L-functions_,

  Journal of Soviet Mathematics 30 (1985), 2036-2070,

  ([mathnet (Russian)](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intd&paperid=73&option_lang=eng), [DOI](http://dx.doi.org/10.1007%2FBF02105861))

introducing [[Beilinson-Deligne cohomology]], [[Beilinson regulators]], [[higher regulators]], and the [[Beilinson conjectures]].

Please note that it is currently in a very preliminary state, having been prepared quickly as notes for a seminar talk on the first section of the paper.
It follows the paper very closely, and an interested reader might like to rewrite from the [[nPOV]].

## Overview

### Notation

By [[analytic space]] we will mean [[real analytic space]].
Let $An$ denote the real [[analytic site]].
Consider the category $Ab(Sh(An))$ of [[abelian sheaves]] on $An$.
Beilinson denotes by $D^+(An)$ its bounded above [[derived category]], i.e. the category of [[connective]] [[cochain complexes]] up to [[quasi-isomorphism]].
Given an analytic space $X \in An$, we will also consider its [[petit topos]] $X^\sim$ of sheaves on the site $Ouv(X)$ of open subsets.

In $D^+(An)$ we have the complex $\Omega^\bullet$, of [[de Rham complex]]es of [[holomorphic forms]].  Let $\Omega^{\ge i}$ denote the "stupid" [[filtration]].

Fix a subring $A \subset \mathbf{R}$ and let $A(p) = A \cdot (2\pi i)^p \subset \mathbf{C}$ for $p \in \mathbf{Z}$.  We will abuse notation and write $\mathbf{C} \in D^+(An)$ also for the constant sheaf valued in the complex concentrated in degree zero.  Similarly we will view $A(p)$ as an object of $D^+(An)$.

We will write $H^*_B(X, \mathbf{C})$ for the [[Betti cohomology]] of $X \in An$.

### Main constructions and conjectures

#### Deligne cohomology of analytic spaces

For each $p \in \mathbf{Z}$, the inclusions $\Omega^{\ge p} \hookrightarrow \Omega^\bullet$ and $A(p) \hookrightarrow \mathbf{C} \hookrightarrow \Omega^\bullet$ induce a canonical morphism
  $$ \Omega^{\ge p} \oplus A(p) \longrightarrow \Omega^\bullet $$
(given by the difference of the two inclusions).

+-- {: .num_defn}
###### Definition
For $p \in \mathbf{Z}$, the **Deligne complex of weight $p$**, denoted $A(p)_\D$, is defined as the [[mapping cone]] of the above morphism shifted by -1, hence fitting in the [[distinguished triangle]]
  $$ (\Omega^{\ge p} \oplus A(p))[-1] \longrightarrow \Omega^\bullet[-1] \longrightarrow A(p)_\D \longrightarrow \Omega^{\ge p} \oplus A(p). $$
=--

The Deligne cohomology is just the [[hypercohomology]] of this complex.  That is, consider the [[right derived functor]] $R\Gamma(-, A(p)_D)$ of the functor of [[global sections]] on $An$ with values in $D^+(A-mod)$.

+-- {: .num_defn}
###### Definition
The **Deligne cohomology** of $X \in An$ of weight $p$ and degree $q$ with coefficients in $A$ is
  $$ H_D^q(X, A(p)) = H^q R\Gamma(X, A(p)_D). $$
=--

+-- {: .num_prop}
###### Proposition
There is a canonical [[long exact sequence]]
  $$ \cdots \to H^{q-1}_B(X, \mathbf{C}) \to H^q_D(X, A(p)) \to \Omega^{\ge p} H^q_B(X, \mathbf{C}) \oplus H^q(X, A(p)) \to \cdots $$
=--

This follows by applying the [[cohomological functor]] $R\Gamma(X, -)$ to the above [[distinguished triangle]].

+-- {: .num_lemma}
###### Lemma
For $p \le 0$ there is a canonical [[quasi-isomorphism]]
  $$ A(p)_D \stackrel{\sim}{\longrightarrow} \Omega^{\ge p}. $$
For $p \gt 0$ there are canonical [[quasi-isomorphisms]]
  $$ \begin{aligned}
    A(p)_D &\stackrel{\sim}{\longrightarrow} (A(p) \to \mathcal{O} \to \Omega^1 \to \cdots \to \Omega^{p-1}) & (\ast) \\
    &\stackrel{\sim}{\longrightarrow} (0 \to \mathcal{O}/A(p) \to \Omega^1 \to \cdots \to \Omega^{p-1}) &
  \end{aligned} $$
=--

#### Multiplicative structure

+-- {: .num_prop}
###### Proposition
There exists a canonical morphism in $D^+(An)$
  $$ - \cup - : A(i)_D \otimes^L A(j)_D \longrightarrow A(i+j)_D $$
inducing the structure of a [[graded commutative ring]] on
  $$ H^*_D(X, A(*)) = \bigoplus_{p,q} H^p_D(X, A(p)) $$
for all $X \in An$.
=--

+-- {: .proof}
###### Proof
Beilinson gives an explicit formula using the usual explicit model for the [[mapping cone]].
He also remarks shortly that the product can be defined by observing that the obvious multiplicative structures on $\Omega^\bullet$, $\Omega^{\ge *}$, $A(*)$, turn each into a [[monoid object]] in the [[symmetric monoidal category]] of [[cochain complexes]] (of [[abelian sheaves]]), that is, into [[dg-algebras]] (of complexes of sheaves).
Consider then the [[homotopy pullback]] of the diagram $A(*) \to \Omega \leftarrow \Omega^{\ge *}$; Beilinson claims that the underlying complex of this [[dg-algebra]] has in each degree $p$ the Deligne complex $A(p)_D$ of weight $p$.
This point is expanded on in [Hopkins-Quick](#HopkinsQuick).

Here we will simply give a formula for the quasi-isomorphic complex $(*)$ of Lemma 1 (assuming $i,j \gt 0$), which we will denote for the moment by $A(i)_E$.
For $X \in An$, take $x \in \Gamma(X, A(i)_E)$, $y \in \Gamma(X, A(j)_E)$, and define

  $$ x \cup y = \begin{cases}
    x \cdot y & \deg(x) = 0\quad\text{or}\quad\deg(y) = 0 \\
    x \wedge dy & \deg(x) \gt 0\quad\text{and}\quad\deg(y) = j \gt 0 \\
    0 & \text{otherwise}
  \end{cases} $$

We omit the various verifications, that this defines a morphism of complexes, is associative, commutative, etc.
One gets a [[monoid object]] in the category of cochain complexes of [[abelian sheaves]].  It only remains to see that $R\Gamma(X, -)$ preserves monoids, so that one gets the structure of a graded commutative ring on the [[hypercohomology]] groups $H^*_D(X, A(*))$.
=--

#### The Bloch regulator

Beilinson uses this cup product for $i=j=1$ to recover the [[Bloch regulator]].

+-- {: .num_theorem}
###### Theorem
**(Bloch)**.  For each [[algebraic curve]] $X$ over $\mathbf{R}$, there is a canonical functorial homomorphism
  $$ r_X : K_2(X) \to H^1_B(X, \mathbf{C}^*) $$
from the second [[algebraic K-theory]] group to the first [[Betti cohomology]] group with coefficients in $\mathbf{C}^*$.
=--

+-- {: .proof}
###### Sketch of proof
By Lemma 1, there are quasi-isomorphisms
  $$ \mathbf{Z}(1)_D \stackrel{\sim}{\to} \mathcal{O}^*[-1] $$
induced by the [[exponential map]], and
  $$ \mathbf{Z}(2)_D \stackrel{\sim}{\to} (\mathcal{O}^* \stackrel{d \log}{\to} \Omega^1)[-1] $$
induced by $x \mapsto \exp(x/2\pi i)$.
It follows that the cup product
  $$ \cup : H^1_D(X, \mathbf{Z}(1)) \otimes H^1_D(X, \mathbf{Z}(1)) \longrightarrow H^2_D(X, \mathbf{Z}(2)) $$
corresponds to a canonical homomorphism
  $$ \mathcal{O}^*(X) \otimes \mathcal{O}^*(X) \to H^1(X, \mathcal{O}^* \to \Omega^1). $$
According to Deligne, the RHS classifies isomorphism classes of [[line bundles]] with [[holomorphic connection]].
Since $\dim(X) = 1$, all [[connections]] are [[integrable connection|integrable]] and this group is identified with $H^1_B(X, \mathbf{C}^*)$.

Now by Matsumoto's theorem, giving a presentation of the $K_2$ group of a [[field]] by two generators and certain relations, one has
  $$ K_2(\mathcal{O}(X)) = (\mathcal{O}(X)^* \otimes \mathcal{O}(X)^*)/\lt t \otimes (1-t) \gt_{t \ne 0,1}. $$
On the other hand, one has the Steinberg identity $t \cup (1-t) = 0$ for $t \in \mathcal{O}^*(X)$.
It follows that the homomorphism above factors through $K_2(\mathcal{O}(X)) = K_2(\eta)$ for $\eta$ a generic point.
To extend it to all of $X$, one uses the commutative diagram of localization [[exact sequences]]
  $$ \begin{array}{ccccc}
    K_2(X) & \longrightarrow & K_2(\eta) & \longrightarrow & \oplus_{x \in X(\mathbf{C})} \mathbf{C}^* \\
    \downarrow & & \downarrow & & \downarrow \\
    H^1_B(X, \mathbf{C}) & \longrightarrow & H^1(\eta, \mathbf{C}^*) & \longrightarrow & \oplus_{x \in X(\mathbf{C})} \mathbf{C}^*
  \end{array} $$
The first row comes from the [[Gersten-Quillen resolution]] for K-theory.
=--

#### Relative cohomology

If $S$ and $T$ are [[toposes]] and $u^* : S \rightleftarrows T : u_*$ is a [[geometric morphism]], consider the [[Artin gluing]], i.e. the [[topos]] $(id_S/u_*) = (u^*/id_T)$ whose objects are morphisms $u^*(F) \to G$ for $F \in S$, $G \in T$, and morphisms are commutative diagrams.
Write $(S, T)$ for this topos.

+-- {: .num_defn}
###### Definition
The functor of [[global sections]] on $(S, T)$ is the [[left exact functor]]
  $$ \Gamma(S, T, -) : Ch^+(Ab(S, T)) \longrightarrow Ch^+(Ab) $$
defined by
  $$ \Gamma(S, T, F) = Cone(\Gamma(T, F_T) \to \Gamma(S, F_S))[-1] $$
for each sheaf $F \in (S, T)$ given by $u^*(F_S) \to F_T$ with $F_S \in S$ and $F_T \in T$.
Let $R\Gamma(S, T, -) : D^+(Ab(S, T)) \to D^+(Ab)$ denote its [[right derived functor]].
=--

The rest of this section is just defining a [[monoidal product]] on $Ch^+(Ab((S, T)))$, and explaining that a [[monoid]] in $D^+(Ab((S,T)))$ will induce a [[monoid]] in $D^+(Ab)$, i.e. a [[dg-algebra]], after taking [[cohomology]].

#### Complexes with logarithmic singularities

Let $\Pi$ denote the category of pairs $(X, \overline{X})$ with $\overline{X} \in An$ [[smooth manifold|smooth]] and $j : X \hookrightarrow \overline{X}$ an [[open subspace]] such that the complement $\overline{X} - X$ is a [[normal crossing divisor]].
The [[open immersion]] $j$ induces a functor $\Ouv(X) \to \Ouv(\overline{X})$ on the [[petit sites]] by mapping an open subspace $U \subset X$ to $U \subset X \subset \overline{X}$.
This induces a [[geometric morphism]] $j^* : \overline{X}^{\sim} \rightleftarrows X^{\sim} : j_*$ on the [[petit toposes]].
Following the discussion of the previous section, we make the

+-- {: .num_defn}
###### Definition
The **topos of the pair** $(X, \overline{X})$ is defined to be the [[Artin gluing]] $(j^*/id)$, and will be denoted $(X, \overline{X})^{\sim}$.
Hence a **sheaf on the pair** $(X, \overline{X})$ is a sheaf $F$ on $X$, a sheaf $G$ on $\overline{X}$, and a connecting morphism $j^*(G) \to F$.
=--

Let $\Omega^\bullet_{X,\overline{X}} \in \Ch^+(Ab((X, \overline{X})^{\sim}))$ denote the [[de Rham complex]] of [[holomorphic forms]] on $C$ with [[logarithmic singularities]] along $\overline{X} - X$.
That is, $\Omega^\bullet_{X,\overline{X}}$ is an object of $Ch^+(Ab(\overline{X}^\sim))$ and we view it as a complex on $(X, \overline{X})^\sim$ by taking the part on $X^\sim$ to be trivial.
Let $\Omega^{\ge p}_{X, \overline{X}}$ denote the stupid [[filtration]].

Now we define a complex of abelian sheaves $A(p)_D$ in $Ch^+(Ab((X, \overline{X})^{\sim}))$ as follows.

+-- {: .num_defn}
###### Definition
The **Beilinson-Deligne complex with logarithmic singularities** of weight $p$ of the pair $(X, \overline{X})$ is a complex of abelian sheaves on $(X, \overline{X})^\sim$, denoted $A(p)_D \in Ch^+(Ab((X, \overline{X})^\sim))$ and defined as follows.
In degree $n \in \mathbf{Z}$, take the sheaf
  $$ A(p)_{D,X} = Cone^n(A(p) \to \Omega^\bullet_X) $$
in $X^\sim$ (where the [[mapping cone]] is taken in $Ch^+(Ab(X^\sim))$), and the sheaf
  $$ A(p)_{D,\overline{X}} = \Omega^{\ge p}_{X, \overline{X}} $$
in $\overline{X}^\sim$, together with the connecting morphism induced by the inclusion $j^*(\Omega^{\ge p}) \hookrightarrow \Omega^\bullet_X$.
=--

+-- {: .num_defn}
###### Definition
The **Beilinson-Deligne cohomology with logarithmic singularities** of the pair $(X, \overline{X})$ in weight $p$ and degree $q$ and with coefficients in $A$, is the [[hypercohomology]]
  $$ H^q_D(X, \overline{X}, A(p)) = H^q R\Gamma((X, \overline{X})^\sim, A(p)_D). $$
=--

One defines a cup product on these complexes in the same way as above, and gets a graded commutative ring structure on Beilinson-Deligne cohomology with logarithmic singularities.

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

=--

#### Deligne cohomology for algebraic varieties

Let $\Pi' \subset \Pi$ denote the full subcategory spanned by pairs $(X, \overline{X})$ for which $\overline{X}$ is a [[smooth variety|smooth]] [[projective variety|projective]] [[algebraic variety]].
Let $V = V_\mathbf{R}$ denote the category of [[smooth variety|smooth]] [[quasi-projective variety|quasi-projective]] [[schemes]] over $\mathbf{R}$.
By the [[GAGA]] principle, we have a functor $\sigma : \Pi' \to V$ which sends a pair $(X, \overline{X})$ to $X$.
Conversely given $X \in V$, by [[Hironaka]] there exists a pair $(X, \overline{X}) \in \Pi'$ (a compactification).

+-- {: .num_defn}
###### Definition
Let $X \in V$ be a smooth quasi-projective algebraic variety over $\mathbf{R}$.
Let $(X, \overline{X}) \in \Pi'$ be a compactification and define the **Beilinson-Deligne cohomology** of $X$ as
  $$ H_D^q(X, A(p)) = H^q R\Gamma(\overline{X}, X, A(p)_D) $$
and
  $$ H_B^q(X, A(p)) = H^q R\Gamma(X, A(p)). $$
=--

One shows that these definitions are independent of the chosen compactification.
By the above, one gets a [[cup product]] also on these cohomology groups.

Next Beilinson shows that $H_D$ can be defined as cohomologies of certain complexes of [[Zariski topology|Zariski sheaves]].
He notes that this is not necessary for the remainder of the paper, so we omit this here.

#### Chern classes of vector bundles

There is a canonical morphism
  $$ c_1 : R\Gamma(X, \mathcal{O}^*)[-1] \longrightarrow H_D(X, A(1)) $$
for each $X \in V$.
This induces a canonical homomorphism
  $$ Pic(X) = H^1(X, \mathcal{O}^*) \longrightarrow H^2(X, A(1)) $$
from the [[Picard group]] of [[invertible sheaves]].

+-- {: .num_defn}
###### Definition
For an [[invertible sheaf]] $\mathcal{L}$ on $X$, its **first Chern class** is defined to be the image of the class of $\mathcal{L}$ under the above homomorphism.
=--

One can show that for $A = \mathbf{Z}$, this homomorphism is injective, and further surjective if $X$ is [[compact]].

Next Beilinson shows the [[projective bundle formula]] for Beilinson-Deligne cohomology.

+-- {: .num_prop}
###### Proposition
**(Projective bundle formula)**.
Let $E$ be an [[vector bundle]] of rank $r$ on $X$, let $\pi : \mathbf{P}(E) \to X$ be the associated [[projective bundle]], and $\mathcal{O}(1)$ the [[tautological sheaf]] on $\mathbf{P}(E)$.
The homomorphism
  $$ \oplus c_1(\mathcal{O}(1))^j \cup \pi^*
       : \bigoplus_{j=0}^{r-1} H_D(X, A(i-j))[2j]
       \longrightarrow H_D(\mathbf{P}(E), A(i)) $$
is invertible.
=--

+-- {: .proof}
###### Proof
By definition we have the distinguished triangle
  $$ H_B(X)[-1] \to R\Gamma(X, A(i))_D) \to R\Gamma(X, F^i) \oplus R\Gamma(X, A(i)) \to. $$
One checks that it is compatible with the cup product.
Since the morphism $A(i)_D \to A(i)$ sends first [[Chern classes]] in Deligne cohomology to first Chern classes in [[Betti cohomology]], the [[projective bundle formula]]s for [[Betti cohomology]] and [[de Rham cohomology]] show that the map in question induces an isomorphism on the leftmost and rightmost members of the triangle.
Hence the result follows.
=--

After the projective bundle theorem, one can define [[Chern classes]] of [[vector bundles]] following [[Grothendieck]].
In particular one gets the [[Chern character]]
  $$ ch : K_0(X) \longrightarrow \bigoplus_i H^{2i}(X, A \otimes \mathbf{Q}(i)). $$

+-- {: .num_lemma}
###### Lemma
For each $i$, there exists a unique assignment to a [[vector bundle]] $E$ over $X \in \mathcal{V}$ a class
  $$ c_i(E) \in H^{2i}_D(X, A(i)) $$
that is [[functorial]] with respect to [[inverse images]] and for which $A(i)_D \to A(i)$ sends $c_i(E)$ to the usual Chern class in [[Betti cohomology]].
=--

+-- {: .proof}
###### Proof
We omit the proof and just recall the construction due to Grothendieck of the Chern classes.
Let $r = rk(E)$ and write $P = \mathbf{P}(E)$.
By the projective bundle formula one has
  $$ H_D^{2r}(P, \mathbf{Z}(r)) = \bigoplus_{i=0}^{r-1} H^{2j+2r}_D(X, \mathbf{Z}(r-j)) $$
There exist $\gamma_i$ such that
  $$ \Sigma_{i=0}^r \pi^* \gamma_i \cup c_1(\mathcal{O}_P(1))^{r-i} = 0 $$
with $\gamma_i \in H_D^{2i}(X, \mathbf{Z}(i))$ and $\gamma_0 = 1$.
We define $c_i(E) = \gamma_i$.
=--

#### Homologies

In this paragraph, our goal is to define the [[homology theory]] dual to Deligne cohomology, for schemes over $\mathbf{R}$.
To do this, we first define functorial complexes on $\Pi_*$.
Then they extend, more or less formally, to the category of finite type schemes over $\mathbf{R}$ and [[proper maps]].
Then we establish [[Poincare duality]].

##### for smooth analytic spaces

+-- {: .num_defn}
###### Definition
Let $X \in An$ be smooth.
Let
  $$ \mathcal{A}^\bullet_X \qquad \text{(resp.} \quad \mathcal{D}^\bullet_X \text{)} $$
denote the complex of $C^\infty$ [[differential forms|forms]] (resp. with [[distribution]] coefficients on $\mathcal{A}^{-p,-q}_{X}$).
This is the [[totalization]] of the [[double complex]] $\mathcal{A}^{*,*}_X$ (resp. $\mathcal{D}^{*,*}_X$) formed by sheaves of $(p,q)$-forms (resp. with [[distribution]] coefficients).
Let $\mathcal{A}^{\ge *}_{X}$ and $\mathcal{D}^{\ge *}_{X}$ denote the respective induced [[filtrations]].

Let
  $$ C'^\bullet(X, A(p)) $$
denote the complex of $C^\infty$ [[singular chains]] with coefficents in the [[constant sheaf]] $A(p)$.

Let
  $$ \mathcal{D}^\bullet_X = \Gamma_c(X, \mathcal{D}_X) $$
denote the complex of [[global sections]] with [[compact support]].
(Here we view $\mathcal{D}_X$ as a sheaf on $X$.)
=--

##### for pairs (logarithmic singularity)

Let $\Pi_* \subset \Pi$ denote the subcategory with the same objects and only morphisms $f : (X, \overline{X}) \to (Y, \overline{Y})$ which satisfy $f(\overline{X} - X) \subset \overline{Y} - Y$.

+-- {: .num_defn}
###### Definition
Let $(X, \overline{X}) \in \Pi_*$.
Define the complexes
  $$ \mathcal{A}^\bullet_{(X, \overline{X})} = \mathcal{A}^\bullet_{\overline{X}} \otimes_{\Omega^\bullet_{\overline{X}}} \Omega^\bullet_{(X, \overline{X})} $$
and
  $$ \mathcal{D}^\bullet_{(X, \overline{X})} = \mathcal{D}^\bullet_{\overline{X}} \otimes_{\Omega^\bullet_{\overline{X}}} \Omega^\bullet_{(X, \overline{X})} $$
with the induced filtrations $\mathcal{A}^{\ge *}_{(X, \overline{X})}$ and $\mathcal{D}^{\ge *}_{(X, \overline{X})}$.

Define the complex of relative $C^\infty$ singular chains on $(X, \overline{X})$ as the [[quotient chain complex]]
  $$ C'^\bullet(X, \overline{X}, A(i)) := C'^\bullet(\overline{X}, A(i))/C'^\bullet(\overline{X}-X, A(i)). $$

Let
  $$ \mathcal{D}^\bullet(X, \overline{X}) = \Gamma_c(\overline{X}, \mathcal{D}^\bullet_{(X, \overline{X})}) $$
denote the complex of sections with [[compact support]], with the induced filtration $\mathcal{D}^{\ge *}(X, \overline{X})$.
=--

Finally, define the complex $C'_D(X, \overline{X}, A(p))$ (this is the complex that will give us the Deligne homology groups $H'_D^*(X, \overline{X}, A(p))$).

+-- {: .num_defn}
###### Definition
For $(X, \overline{X}) \in \Pi_*$, define the complex
  $$ C'_D(X, \overline{X}, A(p)) = Cone(\mathcal{D}^{\ge p}(X, \overline{X}) \oplus C'^\bullet(X, \overline{X}, A(i)) \longrightarrow \mathcal{D}^\bullet(X, \overline{X})). $$
=--

This is functorial on $\Pi_*$.

##### for schemes

Let $Sch_*$ denote the category of [[finite type]] schemes over $\mathbf{R}$ and [[proper maps]].
Let $V_* \subset Sch_*$ denote the subcategory of smooth quasi-projective schemes.
Let $\Pi'_* = \Pi' \cap \Pi_*$ denote the category of pairs $(X, \overline{X})$ with $\overline{X}$ smooth projective, $X \subset \overline{X}$ open with $\overline{X} - X$ a normal crossing divisor, and with morphisms $f : (X, \overline{X}) \to (Y, \overline{Y})$ such that $f(\overline{X} - X) \subset \overline{Y} - Y$.

+-- {: .num_lemma}
###### Lemma
The functor $C'^\bullet_D(-, A(p))$ on the category $\Pi'_*$ extends uniquely to a functor on $Sch_*$.
In particular one gets has a [[distinguished triangle]] in $D^+(A-mod)$
  $$ \longrightarrow H'_{dR}(X) \longrightarrow H'_D(X, A(p)) \longrightarrow F^i H'_{dR}(X) \oplus H'_B(X, A(p)) \longrightarrow, $$
where $H'_{dR}$ is [[de Rham homology]], $F^i$ is the [[Hodge filtration]], $H'_B$ is the [[Borel-Moore homology]] with coefficients in the [[constant sheaf]] $A(p)$, and $H'_D(X, A(p))$ is the [[Deligne homology]], defined by the complex $C'_D(X, A(p))$.
=--

+-- {: .num_lemma}
###### Lemma
**(Poincare duality).**
Let $X$ be a smooth scheme of dimension $n$.
There is a canonical isomorphism
  $$ H'_D(X, A(p)) = H_D(X, A(p+n))[2n]. $$
=--

+-- {: .proof}
###### Proof
Let $(X, \overline{X}) \in \Pi'_*$ be a compactification of $X$.
Consider the presheaf on $\overline{X}$ of complexes of abelian groups, defined by
  $$ U \mapsto C'(\overline{X}, A(p))/C'(\overline{X} - (X \cap U), A(p)). $$
Take its associated sheaf, and consider it as a complex of abelian sheaves, $\overline{C}'_{X, \overline{X}}(A(p))$.
First of all note that
  $$ \overline{C}'_{X, \overline{X}}(A(p)) = j_*j^* \overline{C}'_{X, \overline{X}}(A(p)) $$
where $j : X \hookrightarrow \overline{X}$ is the open immersion.
Note that $j^* \overline{C}'_{X, \overline{X}}(A(p))$ is a [[flasque resolution]] of the sheaf $A(p+n)[2n]$ on $X$.
Since the embedding
  $$ \Omega_{X, \overline{X}} \hookrightarrow \mathcal{D}_{X, \overline{X}}[-2n] $$
is a [[filtered quasi-isomorphism]], and the [[associated graded objects]] $gr^p \mathcal{D}_{X, \overline{X}}$ are [[soft sheaves]], one gets
  $$ R\Gamma(\Omega_{X, \overline{X}}, \Omega^{\ge p}_{X, \overline{X}}) = \Gamma(\overline{X}, (\mathcal{D}_{X, \overline{X}}, \mathcal{D}^{\ge p+n}_{X, \overline{X}}))[-2n] $$

=--

#### Cycles

Let $X$ be a scheme and $Y \in Z_n(X)$ an [[irreducible topological space|irreducible]] subscheme of dimension $n$.
Note that the canonical homomorphism
  $$ H'_D^{-2n}(Y, A(-n)) \longrightarrow H'_B^{-2n}(Y, A(-n)) = A $$
is invertible.
Let $cl_D(Y)$ denote the element of $H'_D^{-2n}(Y, A(-n))$ corresponding to the unit $1 \in A$.
Hence one gets a homomorphism
  $$ cl_D : Z_n(X) \longrightarrow H'_D^{-2n}(X, A(-n)) $$
given by $cl_D[Y] = i_*(\cl_D(Y))$.
If $X$ is smooth, by Poincare duality this corresponds to a homomorphism
  $$ cl_D : Z^n(X) \longrightarrow H_D^{2n}(X, A(n)) $$
on the group of [[algebraic cyles]] of [[codimension]] $n$.

+-- {: .num_lemma}
###### Lemma
If $X$ is smooth and compact, then for each $Y \in Z_n(X)$, if $cl_B(Y) \in H'_B^{-2n}(X, \mathbf{Z}(-n))$ is equal to 0, then $cl_D(Y)$ coincides with the Abel-Jacobi-Griffiths periods of the cycle $Y$.
=--

+-- {: .proof}
###### Proof
The distinguished triangle defining $\mathbf{Z}(n)_D$ induces, after passing to the associated [[long exact sequence]], a [[short exact sequence]]
  $$ 0 \to \mathcal{I}^n(X) \to H_D^{2n}(X, \mathbf{Z}(n)) \to Hdg^n(X) \to 0 $$
where $\mathcal{I}^n$ is the $n$th [[intermediate Jacobian]] of [[Griffiths]], defined as
  $$ \mathcal{I}^n = H^{2n-1}_B(X, \mathbf{C})/(H^{2n-1}_B(X, \mathbf{Z}(n)) \oplus F^n H^{2n-1}_B(X, \mathbf{C})) $$
and $\Hdn^n(X)$ is the group of integral [[Hodge cycles]] of type $(n, n)$.

Using the usual explicit model for the [[mapping cone]], $cl_D(Y)$ is the homology class of the cycle
  $$ (cl_F(Y), i_* cl_B(Y), 0) \in C'_D^{-2n}(X, \mathbf{Z}(-n)) $$
where $i : Y \hookrightarrow X$ denotes the closed immersion, and $cl_F(Y) \in F^{-n}\mathcal{D}^{-2n}(X)$ is a distribution defined by integration over $Y$.
Since $cl_B(Y) = 0$, we can choose $s \in C'^{-2n-1}(X, \mathbf{Z}(-n))$ such that $d(s) = i_*(cl_B(Y))$.
By subtracting from $cl_D(Y)$ the boundary $(0, s, 0)$, we see that $cl_D(Y) = (cl_F(Y), 0, s)$.
But the latter is precisely the definition of the periods of the cycle $Y$.
=--

#### Hodge conjecture for Deligne cohomology

### Regulators

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

=--
+-- {: .num_lemma}
###### Lemma

=--
...

## References

* {#Beilinson85} [[Alexander Beilinson]] _Higher regulators and values of L-functions_, Journal of Soviet Mathematics 30 (1985), 2036-2070, ([mathnet (Russian)](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intd&paperid=73&option_lang=eng), [DOI](http://dx.doi.org/10.1007%2FBF02105861)) (reviewed in [Esnault-Viehweg 88](#EsnaultViehweg88))

* {#EsnaultViehweg88} [[Hélène Esnault]], [[Eckart Viehweg]], _Deligne-Beilinson cohomology_ in Rapoport, Schappacher, Schneider (eds.) _Beilinson's Conjectures on Special Values of L-Functions_ . Perspectives in Math. 4, Academic Press (1988) 43 - 91 ([pdf](http://www.uni-due.de/~mat903/preprints/ec/deligne_beilinson.pdf))

* {#HopkinsQuick} [[Michael Hopkins]], [[Gereon Quick]], _Hodge filtered complex bordism_, [arXiv](http://arxiv.org/abs/1212.2173v3).


higher Segal space

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#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Internal $(\infty,1)$-Categories
+--{: .hide}
[[!include internal infinity-categories contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Warning
There are several unrelated generalizations of the concept of a [[Segal space]] which might be thought of as "higher Segal spaces". For example, one might discuss

 - [[$n$-fold Segal spaces]], a model for $(\infty,n)$-categories.

 - [[$n$-uple Segal spaces]], a model for cubical $(\infty,n)$-categories.

 - [[$d$-Segal spaces]] in the sense of Dyckerhoff and Kapranov, a model for something like an $(\infty,1)$-category, but without uniqueness of composites (for $d \geq 2$) and with higher associativity only in dimension $d$ and above.

This article discusses $d$-Segal spaces in the sense of Dyckerhoff and Kapranov.

## Idea

There are several ways to think about $d$-Segal spaces:

### Higher associativity parameterized by polyhedra

A $1$-Segal space $C$ is a [[Segal space]], i.e. a simplicial space satisfying the [[Segal condition]]. We think of the Segal condition in the following way. For every subdivision of an interval $I$ into subintervals $I_1,\dots,I_n$, and for any choice of labelings of the endpoints of these intervals by objects $c_0,\dots,c_n$, and any choice of labelings $\gamma_1 \in C(c_0,c_1),\dots,\gamma_n \in C(c_{n-1},c_n)$ of the intervals $I_1, \dots, I_n$, the Segal condition provides us with a "composite" labeling $\gamma_n \circ \dots \circ \gamma_1$ of the whole interval $I$, in a coherent way. "Coherence" here means that the composition is continuous in the $\gamma_i$'s, but moreover that it is associative: if we compose our labelings in two steps, for example, we get the same result as if we compose our labelings in one step: $\gamma_3 \circ (\gamma_2 \circ \gamma_1) = \gamma_3 \circ \gamma_2 \circ \gamma_1$.

A $2$-Segal space is, like a $1$-Segal space, a simplicial space, but it satisfies only a weakened version of the Segal condition. Instead of stipulating that labelings of triangualtions of 1-dimensional intervals may be coherently composed, we stipulate that labelings of triangulations of 2-dimensional polygons may be coherently composed.

Similarly $d$-Segal spaces are simplicial spaces with higher associativity data parameterized by triangulations of $d$-dimensional polyhedra.

### Categories with multivalued composition

A 2-Segal space is a "category with multivalued composition", or a category enriched in [[Span]]. A composite of two morphisms $a \to b \to c$ need not exist, and if it does it may not be unique. But whatever composites there are satisfy all "higher associativity conditions" one could want.

## Definition

### Dyckerhoff-Kapranov
 {#DyckerhoffKapranov}

In ([DyckerhoffKapranov 12](#DyckerhoffKapranov12)) a 2-Segal space is defined to be a simplicial space with a higher analog of the weak composition operation known from [[Segal spaces]].

Let $X$ be a [[simplicial topological space]] or [[bisimplicial set]] or generally a [[simplicial object]] in a suitable [[simplicial model category]].

For $n \in \mathbb{N}$ let $P_n$ be the $n$-[[polygon]]. For any [[triangulation]] $T$ of $P_n$ let $\Delta^T$ be the corresponding [[simplicial set]]. Regarding $\Delta^n$ as the cellular [[boundary]] of that polygon provides a morphism of simplicial sets $\Delta^T \to \Delta^n$.

Say that $X$ is a **2-Segal object** if for all $n$ and all $T$ as above, the induced [[morphisms]]

$$
  X_n := [\Delta^n, X]
  \to
  X_T := [\Delta^T,X]
$$

are [[weak equivalences]].

**Warning.** A Dyckerhoff-Kapranov "2-Segal spaces" is not itself a model for an [[(∞,2)-category]]. Instead, it is a model for an [[(∞,1)-operad]] ([Dyckerhoff-Kapranov 12, section 3.6](#DyckerhoffKapranov12)).

Under some conditions DW 2-Segal spaces $X_\bullet$ induce [[Hall algebra]] structures on $X_1$ ([Dyckerhoff-Kapranov 12, section 8](#DyckerhoffKapranov12)).

## Examples

A central motivating example comes from $K$-theory. If $C$ is a [[Quillen-exact category]], then $S_\bullet C$ is a 2-Segal space. Here $S_\bullet$ is the [[Waldhausen S-construction]]. There is one object of $S_\bullet C$, denoted $0$. There is a morphism $0 \to 0$ for each object of $C$. A composite of in $S_\bullet C$ of two objects $c,c' \in C$ is an object $c'' \in C$ equipped with a short exact sequence $0 \to c \to c'' \to c' \to 0$. Thus the composite is generally not unique, but it does satisfy all the higher associativity conditions required of a 2-Segal space.

## References

For more references along these lines do *not* see at _[[n-fold complete Segal space]]_ -- that is a different concept.

The Dyckerhoff-Kapranov "higher Segal spaces" [above](#DyckerhoffKapranov) are discussed in

* Tobias Dyckerhoff, [[Mikhail Kapranov]], _Higher Segal spaces I_, [arxiv:1212.3563](https://arxiv.org/abs/1212.3563); now part of the book T. Dyckerhoff, M. Kapranov, _Higher Segal spaces_, Springer LNM 2244 (2019) [doi](https://doi.org/10.1007/978-3-030-27124-4)
 {#DyckerhoffKapranov12}

* Tobias Dyckerhoff, _Higher Segal spaces_, talk at Steklov Mathematical Institute (2011) ([video](http://www.mathnet.ru/php/presentation.phtml?presentid=3718&option_lang=eng))
 {#Dyckerhoff}

* [[Mikhail Kapranov]], _Higher Segal spaces _, talk at IHES (2012) ([video](http://www.dailymotion.com/video/xpm3at_mikhail-kapranov-a-meeting-on-the-occasion-of-the-75th-birthday-of-yuri-ivanovich-manin_tech))
 {#Kapranov}

* Tashi Walde, _On the theory of higher Segal spaces_, thesis, Brexen 2020 [pdf](https://bonndoc.ulb.uni-bonn.de/xmlui/bitstream/handle/20.500.11811/8705/5938.pdf?sequence=1)

* [[Matthew B. Young]], *Relative 2-Segal spaces*, Algebraic & Geometric Topology **18** (2018) 975-1039 [[doi:10.2140/agt.2018.18.975](https://doi.org/10.2140/agt.2018.18.975)]

  > We introduce a relative version of the 2–Segal simplicial spaces defined by Dyckerhoff and Kapranov, and Gálvez-Carrillo, Kock and Tonks. Examples of relative 2–Segal spaces include the categorified unoriented cyclic nerve, real pseudoholomorphic polygons in almost complex manifolds and the $\mathcal{R}_\bullet$-construction from Grothendieck–Witt theory. We show that a relative 2–Segal space defines a categorical representation of the Hall algebra associated to the base 2–Segal space. In this way, after decategorification we recover a number of known constructions of Hall algebra representations. We also describe some higher categorical interpretations of relative 2–Segal spaces.

The notion of unital 2-Segal space is also discovered independently under the name of a __decomposition space__ in

* [[Imma Gálvez-Carrillo]], [[Joachim Kock]], [[Andrew Tonks]], _Decomposition spaces, incidence algebras and Möbius inversion_, [arXiv:1404.3202](https://arxiv.org/abs/1404.3202)

There are many sequels including

* Joachim Kock, David I. Spivak, _Decomposition-space slices are toposes_, [arXiv:1807.06000](https://arxiv.org/abs/1807.06000)
* Louis Carlier, [[Joachim Kock]], _Antipodes of monoidal decomposition spaces_, [arXiv:1807.11858](https://arxiv.org/abs/1807.11858)
* Matthew Feller, [[Richard Garner]], [[Joachim Kock]], May U. Proulx, [[Mark Weber]], _Every 2-Segal space is unital_, [arXiv:1905.09580](https://arxiv.org/abs/1905.09580)
* Nicolas Behr, [[Joachim Kock]], _Tracelet Hopf algebras and decomposition spaces_, [arXiv:2105.06186](https://arxiv.org/abs/2105.06186)
* Philip Hackney, [[Joachim Kock]], _Culf maps and edgewise subdivision_, [arXiv:2210.11191](https://arxiv.org/abs/2210.11191); _Free decomposition spaces_, [arXiv:2210.11192](https://arxiv.org/abs/2210.11192)

[[!redirects higher Segal space]]
[[!redirects higher Segal spaces]]

[[!redirects 2-Segal space]]
[[!redirects 2-Segal spaces]]

[[!redirects higher complete Segal space]]

[[!redirects decomposition space]]
[[!redirects decomposition spaces]]


higher spin gauge theory

+-- {: .rightHandSide}
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#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Generally by a _higher spin field theory_ is meant a [[quantum field theory]] that involves [[field (physics)|fields]] of spin $\gt 2$ (recalling that a [[spinor field]] has spin 1/2, a [[gauge field]] has spin 1, a [[gravitino]] field has spin 3/2 and the field of [[gravity]] has spin 2).

Folklore had it that all higher spin field theories with _[[mass]]-less_ higher spin fields are inconsistent due to negative norm states ("ghost") appearing in their [[quantization]]. (This argument underlies the dominance of $\mathcal{N} = 1$ [[11-dimensional supergravity]], see the introduction to the entry _[[12-dimensional supergravity]]_ for more on this).

But then it was discovered that there is a higher spin gauge theory ([Vasiliev 96](#Vasiliev96)) which is a kind of [[higher gauge theory]] whose [[field (physics)|field content]] is an _infinite tower_ of massless fields of ever higher [[spin]].

Ever since the refine folklore says that higher spin theories with a _finite_ number of higher spin field species is inconsistent, but that an infinite tower can fix the problem (...add reference...).

One way that higher spin gauge theories are thought to naturally arise is as the limiting case of [[string field theory]] when the [[string tension]] is sent to zero ([Henneaux-Teitelboim 87, section 2](#HenneauxTeitelboim87), [Gross 88](#Gross88), [Sagnotti-Tsulaia 03](#SagnottiTsulaia03), [Bonelli 03](#Bonelli03)): The excitation spectrum of the [[string]] [[sigma-model]] contains beyond the massless particles of the [[effective QFT|effective]] [[supergravity]] theory an infinite tower of massive excitations, of ever higher [[spin]]. There are, however, certain limits in which all these masses become negligible to a reference energy scale -- the _tensionless limit_ -- this is notably so for compactifications on [[anti de Sitter spaces]] of small radius. In this limit the string spectrum looks like an infinite collection of massless [[spinning particles]] for ever higher spin. Due to their common origin in the string, these share intricate relations among each other, which are argued to be described by higher spin gauge theory. (Notice that at least closed bosonic [[string field theory]] is itself already a [[higher gauge theory]], even without sending the [[string tension]] to zero, see at _[closed string field theory -- As an ∞-Chern-Simons theory](string+field+theory#AsAnInfinityCSTheory)_.)

At present there are several classes of higher spin theories that evade the no-go theorems: (1) massless and conformal higher spin theories in three dimensions can be formulated as Chern-Simons theories; (2) conformal higher spin gravity in four dimensions is a higher spin extension of the Weyl gravity; (3) Chiral higher spin gravity is a higher spin extension of self-dual Yang-Mills theory and self-dual Gravity. At the same time the holographic higher spin theories face certain difficulties due non-locality implied by the AdS/CFT duality to vector models

## References

### General

Original articles include

* {#Vasiliev96} [[Mikhail Vasiliev]], _Higher-Spin Gauge Theories in Four, Three and Two Dimensions_ ([arXiv:hep-th/9611024](http://arxiv.org/abs/hep-th/9611024))

Reviews and lecture notes include the following:

* {#Vasiliev03} [[Mikhail Vasiliev]], _Higher Spin Gauge Theories in Various Dimensions_, 27th Johns Hopkins Workshop on Current Problems in Particle Theory: Symmetries and Mysteries of M Theory ([arXiv:hep-th/0401177](https://arxiv.org/abs/hep-th/0401177))

* [[Mikhail Vasiliev]], _Higher spin gauge theories in any dimension_ talk at String2004 in Moscow ([arXiv:hep-th/0409260](https://arxiv.org/abs/hep-th/0409260))

* R. Argurio, [[Glenn Barnich]], G. Bonelli, M. Grigoriev (eds.) _Higher spin gauge theories_ Solvay Workshops and Symposia  (2004) ([pdf](http://www.ulb.ac.be/sciences/ptm/pmif/Solvay1proc.pdf))

* [[Xavier Bekaert]], S. Cnockaert, Carlo Iazeolla, [[Mikhail Vasiliev]], _Nonlinear higher spin theories in various dimensions_ ([arXiv:0503128](https://arxiv.org/abs/hep-th/0503128))

* V.E. Didenko, E.D. Skvortsov, _Elements of Vasiliev theory_ ([arXiv:1401.2975](https://arxiv.org/abs/1401.2975))

* Rakibur Rahman, Massimo Taronna, _From Higher Spins to Strings: A Primer_ ([arXiv:1512.07932](https://arxiv.org/abs/1512.07932) )

* Pan Kessel, _The Very Basics of Higher-Spin Theory_ ([arXiv:1702.03694](https://arxiv.org/abs/1702.03694))

* Ivan Vuković, _Higher spin theory_ ([arXiv:1809.02179](https://arxiv.org/abs/1809.02179))

* Dmitry Ponomarev, *Basic introduction to higher-spin theories* [[arXiv:2206.15385](https://arxiv.org/abs/2206.15385)]

Further developments include for instance

* Johan Engquist, Olaf Hohm, _Geometry and dynamics of higher-spin frame fields_ ([arXiv:0708.1391](http://arxiv.org/abs/0708.1391))

### Concrete higher spin gravities

* [[Miles Blencowe]], _A consistent interacting massless higher-spin field theory in $D=2+1$_ Classical and quantum gravity, volume 6 no 4 (1998)

* C.N. Pope, P.K. Townsend, Conformal Higher Spin in (2+1)-dimensions, Phys.Lett. B225 (1989) 245-250.

* E. S. Fradkin,  V. Ya. Linetsky, _a Superconformal Theory of Massless Higher Spin Fields in D=2+1_ ([web](http://adsabs.harvard.edu/abs/1989MPLA....4..731F))

* Maxim Grigoriev, Karapet Mkrtchyan, Evgeny Skvortsov, Matter-free higher spin gravities in 3D: Partially-massless fields and general structure  ([arXiv:2005.05931](https://arxiv.org/abs/2005.05931))

* Maxim Grigoriev, Iva Lovrekovic, Evgeny Skvortsov, New Conformal Higher Spin Gravities in $3d$ ([arXiv:1909.13305](https://arxiv.org/abs/1909.13305))

* Arkady A. Tseytlin, On limits of superstring in AdS(5) x S**5 ([arXiv:0201112](https://arxiv.org/abs/hep-th/0201112))

* Arkady Y. Segal, Conformal higher spin theory ([arXiv:0207212](https://arxiv.org/abs/hep-th/0207212))

* Dmitry Ponomarev, E.D. Skvortsov, Light-Front Higher-Spin Theories in Flat Space ([arXiv:1609.04655](https://arxiv.org/abs/1609.04655))

### Relation to string theory
 {#ReferencesRelationToStringTheory}

The idea that higher spin gauge theory appears as the limiting case of [[string field theory]] where the [[string tension]] vanishes goes back to

* {#HenneauxTeitelboim87} [[Marc Henneaux]], [[Claudio Teitelboim]], section 2 of _First And Second Quantized Point Particles Of Any Spin_, in [[Claudio Teitelboim]], [[Jorge Zanelli]] (eds.) _Santiago 1987, Proceedings, Quantum mechanics of fundamental systems 2_, pp. 113-152. Plenum Press.

* {#Gross88} [[David Gross]], _High-Energy Symmetries Of String Theory_, Phys. Rev. Lett. 60 (1988) 1229.

and is further developed for instance in

* {#SagnottiTsulaia03} [[Augusto Sagnotti]], M. Tsulaia, _On higher spins and the tensionless limit of String Theory_, Nucl. Phys. B682:83-116, 2004 ([arXiv:hep-th/0311257](http://arxiv.org/abs/hep-th/0311257))

* {#Bonelli03} G. Bonelli, _On the Tensionless Limit of Bosonic Strings, Infinite Symmetries and Higher Spins_, Nucl. Phys. B669 (2003) 159-172 ([arXiv:hep-th/0305155](http://arxiv.org/abs/hep-th/0305155))

* [[Augusto Sagnotti]], M. Taronna, _String Lessons for Higher-Spin Interactions_, Nucl. Phys. B842:299-361,2011 ([arXiv:1006.5242](http://arxiv.org/abs/1006.5242))

* {#Sagnotti11} [[Augusto Sagnotti]], _Notes on Strings and Higher Spins_ ([arXiv:1112.4285](http://arxiv.org/abs/1112.4285))

And conversely:

* Rakibur Rahman, Massimo Taronna, _From Higher Spins to Strings: A Primer_ in [[Stefan Fredenhagen]] (ed.) _Introduction to Higher Spin Theory_ ([arXiv:1512.07932](https://arxiv.org/abs/1512.07932))

* [[Matthias Gaberdiel]], [[Rajesh Gopakumar]], _String Theory as a Higher Spin Theory_, J. High Energ. Phys. (2016) 2016: 85 ([arXiv:1512.07237](https://arxiv.org/abs/1512.07237), doi:10.1007/JHEP09(2016)085)

* {#CKSZ16} Simon Caron-Huot, Zohar Komargodski, Amit Sever, Alexander Zhiboedov, _Strings from Massive Higher Spins: The Asymptotic Uniqueness of the Veneziano Amplitude_, JHEP10(2017)026 ([arXiv:1607.04253](https://arxiv.org/abs/1607.04253))

* {#SeverZhiboedov17} Amit Sever, Alexander Zhiboedov, _On Fine Structure of Strings: The Universal Correction to the Veneziano Amplitude_, JHEP06(2018)054 ([arXiv:1707.05270](https://arxiv.org/abs/1707.05270))

### Relation with double field theory

Discussion of the appearance of spin-2 Fierz-Pauli fields in [[double field theory]]:

* Chen-Te Ma, Franco Pezzella, _More Stringy Effects in Target Space from Double Field Theory_, JHEP 08 (2020) 113 ([arXiv:1909.00411](https://arxiv.org/abs/1909.00411v2))

### Relation to other systems

Relation to [[Kac-Moody algebras]] is discussed in

* [[Marc Henneaux]], [[Axel Kleinschmidt]], [[Hermann Nicolai]], _Higher spin gauge fields and extended Kac-Moody symmetries_ ([arXiv:1110.4460](http://arxiv.org/abs/1110.4460))

Expression in terms of [[AKSZ sigma-models]] is discussed in

* {#AlkalevGrigorievSkvortsov14} K.B. Alkalaev, Maxim Grigoriev, E.D. Skvortsov, _Uniformizing higher-spin equations_ ([arXiv:1409.6507](http://arxiv.org/abs/1409.6507))

### Higher spin Chern-Simons theory

[[Chern-Simons theory]] for higher spin fields is discussed in

* [[Miles Blencowe]], _A consistent interacting massless higher-spin field theory in $D=2+1$_ Classical and quantum gravity, volume 6 no 4 (1998)

* E. S. Fradkin,  V. Ya. Linetsky, _a Superconformal Theory of Massless Higher Spin Fields in D=2+1_ ([web](http://adsabs.harvard.edu/abs/1989MPLA....4..731F))

* Johan Engquist, Olaf Hohm, _Higher-spin Chern-Simons theories in odd dimensions_ ([arXiv:0705.3714](http://arxiv.org/abs/0705.3714))

### Relation to holography

We list references that discuss the relation of higher spin gauge theory to the [[AdS/CFT correspondence]].

* Simone Giombi, Xi Yin, _Higher Spin Gauge Theory and Holography: The Three-Point Functions_ ([arXiv:0912.3462](http://arxiv.org/abs/0912.3462))

* Simone Giombi, Xi Yin, _Higher Spins in AdS and Twistorial Holography_ ([arXiv:1004.3736](http://arxiv.org/abs/1004.3736))

* Simone Giombi, _TASI Lectures on the Higher Spin - CFT duality_  ([arXiv:1607.02967](https://arxiv.org/abs/1607.02967))

* Charlotte Sleight, _Lectures on Metric-like Methods in Higher Spin Holography_ ([arXiv:1701.08360](https://arxiv.org/abs/1701.08360))

[[!redirects higher spin gauge theories]]

[[!redirects higher spin field theory]]
[[!redirects higher spin field theories]]

[[!redirects higher spin quantum field theory]]
[[!redirects higher spin quantum field theories]]


higher spin geometry - contents

**[[spin geometry]]**, **[[string geometry]]**, **[[fivebrane geometry]]** ...

## Ingredients

* [[supergeometry]]

* [[higher differential geometry]]

## Spin geometry

[[spin geometry]]

* [[spin group]]

  [[pin group]]

  [[semi-spin group]]

  [[central product spin group]]

  [[spin^c group]]

* [[spin representation]],

  * [[spinor]], [[fermion]]

  * [[charge conjugation matrix]], [[Fierz identity]]

  * [[real spin representation]]

  * [[Dirac conjugate]]

  * [[Dirac spinor]], [[Weyl spinor]], [[Majorana spinor]]

* [[spin structure]], [[spin^c structure]]

* [[spinor bundle]]

* [[Dirac operator]]

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

* [[Dirac equation]]

* [[Dirac field]]

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

## String geometry

[[string geometry]]

* [[string 2-group]], [[string^c 2-group]]

* [[string structure]], [[string^c structure]]

## Fivebrane geometry

* [[fivebrane 6-group]]

* [[fivebrane structure]]

## Ninebrane geometry

* [[ninebrane 10-group]]


higher spin structure

[[!include higher spin structure - table]]

[[!redirects higher spin structures]]
[[!redirects higher Spin structure]]
[[!redirects higher Spin structures]]


higher spin structure - table

| [[smooth ∞-group|smooth]] [[∞-group]] | [[Whitehead tower]] of [[smooth ∞-groupoid|smooth]] [[moduli ∞-stacks]] | [[G-structure]]/[[higher spin structure]] | [[obstruction]] |
|--|--|--|--
|  | $\vdots$ | | |
|  | $\downarrow$ | | |
| [[ninebrane 10-group]] | $\mathbf{B}Ninebrane $ | [[ninebrane structure]] | [[third fractional Pontryagin class]] |
|  | $\downarrow$ | | |
| [[fivebrane 6-group]] | $\mathbf{B}Fivebrane \stackrel{\tfrac{1}{n} p_3}{\to}  \mathbf{B}^{11}U(1)$ | [[fivebrane structure]] | [[second fractional Pontryagin class]] |
|  | $\downarrow$ | | |
| [[string 2-group]] | $\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)$ | [[string structure]] | [[first fractional Pontryagin class]] |
|  | $\downarrow$ | | |
| [[spin group]] | $\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)$ | [[spin structure]] | [[second Stiefel-Whitney class]] |
|  | $\downarrow$ | | |
| [[special orthogonal group]] | $\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2$ | [[orientation structure]] | [[first Stiefel-Whitney class]] |
|  | $\downarrow$ | | |
| [[orthogonal group]] | $\mathbf{B}O \stackrel{\mathbf{w}_1}{\to} \mathbf{B}\mathbb{Z}_2$ | [[orthogonal structure]]/[[vielbein]]/[[Riemannian metric]] |  |
|  | $\downarrow$ | | |
| [[general linear group]] | $\mathbf{B}GL$  | [[smooth manifold]] | |

(all hooks are [[homotopy fiber sequences]])


higher stack

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

#Contents#
* table of contents
{:toc}

## Idea

Classically, the theory of [[stacks]] was motivated by the study of [[moduli problems]] for which objects are classified up to [[isomorphism]].  Higher stacks are a generalization where objects are classified up to some notion of [[equivalence]], like [[complexes]] up to [[quasi-isomorphism]], [[topological spaces]] up to [[weak homotopy equivalence]], or [[abelian categories]] up to [[equivalence of categories]].

## See also

* [[stack]]
* [[n-stack]]
* [[infinity-stack]]
* [[derived stack]]
* [[derived algebraic geometry]]

## References

* [[Carlos Simpson]], _Algebraic (geometric) $n$-stacks_, 1996, [arXiv:alg-geom/9609014](http://arxiv.org/abs/alg-geom/9609014).

* [[André Hirschowitz]], [[Carlos Simpson]], _Descente pour les n-champs (Descent for n-stacks)_, 1998, [arXiv:math/9807049](http://arxiv.org/abs/math/9807049).

[[!redirects higher stacks]]


higher structure

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

#Contents#
* table of contents
{:toc}

## Idea

The term _higher structure_ (as used in numerous conference titles in recent years, e.g. [CGX 11](#CGX11), [BBM 16](#BBM16), [SBBM 18](#SBBM18), [JSSW 18](#JSSW18) and by a journal [Higher Structures](#HSJ)) has come to mean essentially what elsewhere was and is called _[[categorification|categorified]] [[mathematical structure]]_ or _[[mathematical structure]] [[internalization|in]] [[higher category theory]]_, or, more specifically, _[[mathematical structure]] [[internalization|in]] [[(∞,1)-category theory]]_, hence _[[mathematical structure]] [[internalization|in]] [[homotopy theory]]_ ([[higher algebra]]/[[homotopical algebra]], [[higher geometry]]/[[derived algebraic geometry|derived geometry]], etc.).

For example, [[L-∞ algebras]], which are [[Lie algebra objects]] [[internalization|in]] the [[(∞,1)-category|(∞,1)-]][[category of chain complexes]], would be a core topic in _higher structures in [[Lie theory]]_ (elsewhere: _[[∞-Lie theory]]_).

## Related entries

* [[higher homotopy]]

* [[higher category theory]], [[higher topos theory]], [[(∞,1)-categorification]]

* [[homotopy type theory]], [[univalent foundations for mathematics]]

* [[higher geometry]]

* [[higher differential geometry]]

* [[higher Lie theory]]

* [[higher prequantum geometry]]

* [[higher complex analytic geometry]]

* [[higher algebra]], [[homotopical algebra]]

* [[higher linear algebra]]

* [[higher category theory and physics]]

* [[L-infinity algebras in physics|L-∞ algebras in physics]]

* [[homotopical algebraic quantum field theory]]

## References

General exposition:

* [[Urs Schreiber]], _[[schreiber:Higher Structures|Higher Structures in Mathematics and Physics]]_, talk at [Oberwolfach Workshop 1651a](https://www.mfo.de/occasion/1651a/www_view), 2016 Dec. 18-23

Discussion in [[string theory]]/[[M-theory]]:

* [[Branislav Jurčo]], [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]], _Higher Structures in M-Theory_, Introduction to _[[Higher Structures in M-Theory 2018]]_, Fortsch. d. Phys. **67** (2019) 8-9 [[arXiv:1903.02807](https://arxiv.org/abs/1903.02807), [doi:10.1002/prop.201910001](https://doi.org/10.1002/prop.201910001)]

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The rational higher structure of M-theory]]_
in _Proceedings of the [LMS-EPSRC Durham Symposium](http://www.maths.dur.ac.uk/lms/)_: _[[Higher Structures in M-Theory 2018]]_, August 2018, Fortschritte der Physik **67** 8-9 (2019) [[arXiv:1903.02834](https://arxiv.org/abs/1903.02834), [doi:10.1002/prop.201910017](https://doi.org/10.1002/prop.201910017)]

Conferences and proceedings:

* {#CGX11} [[Alberto Cattaneo]], Anthony Giaquinto, [[Ping Xu]], _Higher Structures in Geometry and Physics -- In Honor of [[Murray Gerstenhaber]] and [[Jim Stasheff]]_, Progress in Mathematics 287, Birkhäuser 2001 ([doi:10.1007/978-0-8176-4735-3](https://link.springer.com/book/10.1007/978-0-8176-4735-3))

* {#BBM16} [[Michael Batanin]], [[Christian Blohmann]], [[Martin Markl]], _[Program on Higher Structures in Geometry and Physics](https://www.mpim-bonn.mpg.de/node/5883)_ MPI Bonn (2016)

* Vincent Coll, Anthony Giaquinto, [[Tom Lada]], [[Ping Xu]], _[Higher Structures 2](https://www.higherstructures2.com/)_ (2018)

* {#SBBM18} [[Stefan Schwede]], [[Clark Barwick]], [[Julie Bergner]], [[Ieke Moerdijk]], _[Higher structures in homotopy theory](https://www.newton.ac.uk/event/hhhw01)_, Isaac Newton Institute 2018

* {#JSSW18} [[Branislav Jurčo]], [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]], _[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html)_, LMS EPSRC Durham Symposium 2018

* *[Higher Structures and Field Theory](https://www.esi.ac.at/events/e430/)* ESI, Vienna (2022)

Journals:

* {#HSJ} _[Higher Structures](https://higher-structures.math.cas.cz/)_, Journal ISSN 2209-0606

[[!redirects higher structures]]