higher Atiyah groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
=--
=--

\tableofcontents

## Idea

We discuss a refinement of the traditional notion of _[[Atiyah Lie groupoids]]_ (the [[Lie groupoids]] which are the [[Lie integration]] of [[Atiyah Lie algebroids]] of $G$-[[principal bundles]]) from [[differential geometry]] to [[higher differential geometry]] and generally to [[higher geometry]].

* _[General idea](#GeneralIdea)_

* _[In higher prequantum geometry: motivation and survey](#InHigherPrequantumGeometryMotivationAndSurvey)_

### General idea
 {#GeneralIdea}

Briefly, for $G$ an [[∞-group]] in an [[(∞,1)-topos]] and $P \to X$ a $G$-[[principal ∞-bundle]], its _higher Atiyah groupoid_ is the [[groupoid object in an (∞,1)-category|groupoid object]] $At(P)$ such that the

* object of [[objects]] is $X$;

* object of [[morphisms]] is the collection of all $G$-equivariant maps between all pairs of [[fibers]] of $P$.

In these vague words this is precisely the same description as for the traditional Atiyah groupoid. Definition \ref{HigherAtiyahGroupoid} below makes precise what this means in [[higher geometry]].

Besides generalizing the traditional definition to [[homotopy theory]], the notion of higher Atiyah groupoids also generalizes from [[concrete objects]] such as [[Lie groups]] to general objects in an [[(∞,1)-topos]] (general [[∞-stacks]], not necessarily "supported on points"), notably to _[[moduli ∞-stacks]]_ for [[cocycles]] in [[differential cohomology]]. For instance if we assume that the ambient [[(∞,1)-topos]] $\mathbf{H}$ is [[cohesive (∞,1)-topos|cohesive]] and consider $\mathbb{G} \in \mathrm{Grp}(\mathbf{H})$  a _[[sylleptic ∞-group]]_, then there is the [[moduli ∞-stack]] $\mathbf{B}\mathbb{G}_{\mathrm{conn}}$ of _$\mathbb{G}$-[[principal ∞-connections]]_ and this is itself again a [[group object in an (∞,1)-category|group object]].
A $(\mathbf{B}\mathbb{G}_{\mathrm{conn}})$-[[principal ∞-bundle]] is equivalently a $(\mathbf{B}^2\mathbb{G})$-[[principal ∞-connection]] "without [[curving]]". For instance if $\mathbb{G} = U(1)$ is the [[circle group]] in [[smooth ∞-groupoids]], then
$\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})$ classifies [[circle 2-bundle with connection]] without 2-form part: in parts of the literature this is known as "[[bundle gerbes]] with connective structure but without [[curving]]".

So the general definition considered here assigns a higher Atiyah groupoid to a "bundle gerbe with connective structure but no [[curving]]". It turns out that this is the _[[Courant 2-groupoid]]_ which [[Lie integration|Lie integrates]] the [[standard Courant Lie 2-algebroid]] traditionally induced by this data.

The  notion of higher Atiyah groupoids is more general still: the definition does not really require that the object fed into the construction is a plain [[principal ∞-bundle]]. It may notably also be a genuine [[principal ∞-connection]] (hence _with_ "[[curving]]"). We show below that the corresponding higher Atiyah groupoid is that groupoid object whose [[∞-group of bisections]] is the [[quantomorphism n-group]] of the principal $\infty$-connection regarded as a [[prequantum n-bundle]].

In summary, the [[higher geometry|higher geometric]] generalization of the notion of Atiyah groupoids unifies all three of the traditional notions of [[Atiyah groupoid]], of [[Courant 2-groupoid]] and of [[quantomorphism group]] and refines each of these to [[higher geometry]]:

[[!include higher Atiyah groupoid - table]]

At the same time the definition of higher Atiyah groupoids in [[(∞,1)-topos theory]], def. \ref{HigherAtiyahGroupoid} below, is very simple, almost tautological, identifying it as a very fundamental notion in [[(∞,1)-topos theory]]/[[homotopy type theory]].

### In higher prequantum geometry: motivation and survey
 {#InHigherPrequantumGeometryMotivationAndSurvey}

Higher Atiyah groupoids play a central role in [[higher prequantum geometry]].

> under construction

#### 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|charted]] [[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/clsasifying 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]]...)

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

## Definition
 {#Definition}

We now turn to the formal definition of higher Atiyah groupoids and the basic constructions on them.

Let $\mathbf{H}$ be an [[(∞,1)-topos]]. Let $G \in Grp(\mathbf{H})$ be a [[group object in an (∞,1)-category|group object]] in $\mathbf{H}$, an [[∞-group]].

We define now for every $G$-[[principal ∞-bundle]] $P \to X$ in $\mathbf{H}$ a _[[groupoid object in an (∞,1)-category|groupoid object]]_ $At(P) \in Grpd(\mathbf{H})$ in $\mathbf{H}$. In order to do so we invoke two basic facts.

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

The construction of forming the [[Cech nerve]] of a [[morphism]] consitutes an [[equivalence of (∞,1)-categories]] from that of [[1-epimorphisms]] to that of [[groupoid objects in an (∞,1)-category|groupoid objects]] in $\mathbf{H}$:

$$
  (\mathbf{H}^{\Delta^1})_{1epi}
  \stackrel{\simeq}{\to}
  Grpd(\mathbf{H})
 \,.
$$

=--

This is a refined version of one of the [[Giraud-Rezk-Lurie axioms]] characterizing [[(∞,1)-topos]], discussed at _[[groupoid object in an (∞,1)-category]]_.

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

In terms of traditional terminology in the literature on [[topological stacks]]/[[differentiable stacks]] etc, this says that a groupoid object in $\mathbf{H}$ is equivalently an object $X \in \mathbf{H}$ which is equipped with an [[atlas]] $X_0 \to X$.

=--

Write $\mathbf{B}G \in \mathbf{H}$ for the [[delooping]] of $G$ in $\mathbf{H}$ (the [[moduli ∞-stack]] of $G$-[[principal ∞-bundles]], as the following proposition asserts:

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

The operation of forming [[(∞,1)-fibers]] ([[homotopy fibers]]) constitutes an [[equivalence of ∞-groupoids]]

$$
  fib
  \;\colon\;
  \mathbf{H}(X, \mathbf{B}G)
  \to
  G Bund(X)
  \,.
$$

=--

This is discussed at _[[principal ∞-bundle]]_.

Using these two facts we now set:

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

For $P \to X$ a $G$-[[principal ∞-bundle]] in $\mathbf{H}$, its **Atiyah groupoid** is the [[groupoid object in an (∞,1)-category|groupoid object]] $At \in \mathrm{Grpd}(\mathbf{H}) \simeq (\mathbf{H}^{\Delta^1})_{1epi}$ which is the [[1-image]] projection of the classifying map $g \colon X \to \mathbf{B}G$:

$$
  g
  \;\colon\;
  X
   \stackrel{}{\to}
  At(P) \coloneqq im_1(g)
  \hookrightarrow
  \mathbf{B}G
  \,.
$$

=--

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

By the discussion at _[[1-image]]_, the 1-image projection of any morphism $f \colon X \to Y$ in an [[(∞,1)-topos]] is equivalently given as the canonical map given by the [[(∞,1)-colimit]] over the [[Cech nerve]]

$$
  X \to im_1(f) \simeq \underset{\rightarrow}{\lim} (X^{\times^{\bullet+1}_Y})
  \,.
$$

This means that regarded as an object of $Grpd(\mathbf{H})$, the Atiyah groupoid $At(P)$ _is_ simply the [[Cech nerve]] of the classifying map. This means that the definition of Atiyah groupoids in [[higher geometry]] is much more fundamental than in traditional [[geometry]].

=--

## Properties

### Equivalence of Atiyah-groupoid bisections to slice automorphisms

We discuss how the [[∞-group of bisections]] of a higher Atiyah groupoid is canonically equivalent to the $\mathbf{H}$-valued [[automorphism ∞-group]] of the modulating map that gave rise to it, regarded as an object in the [[slice (∞,1)-topos]] over its [[codomain]].

To this end we need the following two definitions

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

For $f \colon X \to Y$ a [[morphism]] in an [[(∞,1)-topos]] $\mathbf{H}$, its **$\mathbf{H}$-valued [[automorphism ∞-group]]** $\mathbf{Aut}_{\mathbf{H}}(f)$ is the [[dependent product]] over $Y$ over the [[automorphism ∞-group]] of $f$ regarded as an object in the [[slice (∞,1)-topos]] $\mathbf{H}_{/Y}$:

$$
  \mathbf{Aut}_{\mathbf{H}}(f)
  \coloneqq
  \underset{Y}{\prod} \mathbf{Aut}_{/Y}(f)
  \,.
$$

=--

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

For [[concrete object|non-concrete]] codomains $Y$ one is usually interested in the [[concretification]] of this group. To be discussed... For an example see at _[The quantomorphism $n$-group](#TheQuantomorphismNGroups)_ below.

=--

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

For $f \colon X \to Y$ a morphism in $\mathbf{H}$, its $\mathbf{H}$-valued slice automorphism $\infty$-group according to prop. \ref{HValuedAutomorphismGroup} sits in an [[(∞,1)-pullback]] [[diagram]]

$$
  \array{
    \mathbf{Aut}_{\mathbf{H}}(f)
    &\to&
    \mathbf{Aut}(X)
    \\
      \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{f \circ (-)}}
    \\
    {*}
    &\stackrel{\vdash f}{\to}&
    [X,Y]
  }
  \,,
$$

where $\mathbf{Aut}(X) \hookrightarrow [X,X]$ is the ordinary [[automorphism ∞-group]] of $X$ in $\mathbf{H}$.

=--

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

For $\mathcal{G} \in Grpd(\mathbf{H})$ a [[groupoid object in an (∞,1)-category|groupoid object]] in an [[(∞,1)-topos]] $\mathbf{H}$, its **[[∞-group of bisections]]** $\mathbf{BiSect}(\mathcal{G}) \in Grpd(\mathbf{H})$ is the $\mathbf{H}$-valued automorphism $\infty$-group, def. \ref{HValuedAutomorphismGroup}, of the [[atlas]] $\phi_{\mathcal{G}} \colon \mathcal{G}_0 \to \mathcal{G}$ of $\mathcal{G}$ under prop. \ref{EffectiveEpisAreEquivalentlyGroupoids}:

$$
  \mathbf{BiSect}(\mathcal{G})
  \coloneqq
  \mathbf{Aut}_{\mathbf{H}}(\phi_{\mathcal{G}})
  \,.
$$

=--

With this the following proposition is immediate, but important for the interpretation of higher Atiyah groupoids:

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

For $c \;\colon\; X \to \mathbf{F}$ a [[morphism]] in an [[(∞,1)-topos]] $\mathbf{H}$, modulating an [[fiber ∞-bundle]] $P \to X$, there is an canonical [[equivalence in an (infinity,1)-category|equivalence]] of [[∞-groups]]

$$
  \mathbf{BiSect}(At(P))
  \simeq
  \mathbf{Aut}_{\mathbf{H}}(c)
  \,.
$$

=--

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

We may read this as saying that the higher Atiyah groupoid of an [[fiber ∞-bundle]] is the [[universal property|universal]] solution to the problem of finding a [[groupoid object in an (∞,1)-category|groupoid object]] whose [[∞-group of bisections]] reproduces a given slice [[automorphism ∞-group]]. In many applications, this is indeed the crucial property that drives the interest in higher Atiyah groupoids, see the _[Examples](#Examples)_ below.

=--

### Sequences of inclusions of Atiyah-bisection $\infty$-groups
 {#SequencesOfInclusionsOfGroupsOfBisections}

Let $\mathbf{H}$ be an [[(∞,1)-topos]] which is [[cohesive (∞,1)-topos|cohesive]]. As discussed there, this implies that there is an internal notion of [[differential cohomology]] and in particular of [[principal ∞-connections]] in $\mathbf{H}$. We note here how the canonical forgetful maps between [[moduli ∞-stacks]] of [[principal ∞-bundles]] equipped with differing degree of differential refinement induce canonical inclusions of the corresponding higher Atiyah groupoids.

Let $\mathbb{G} \in Grp(\mathbf{H}) be a $[[braided ∞-group]]. Then there exists, by [[cohesion]], a canonical notion of $\mathbb{G}$-[[principal ∞-connections]], whose [[moduli ∞-stack]] we denote $\mathbf{B}\mathbb{G}_{\mathrm{conn}}$. This is equipped with a canonical map

$$
  \mathbf{B}\mathbb{G}_{conn} \to \mathbf{B}\mathbb{G}
$$

which "forgets the connection". Then for $\nabla \colon X \to \mathbf{B}\mathbb{G}_{conn}$ a $\mathbb{G}$-[[principal ∞-connection]] we write

$$
  \nabla_0 \;\colon\; X \stackrel{\nabla}{\to}
   \mathbf{B}\mathbb{G}_{conn} \to \mathbf{B}\mathbb{G}
$$

for the corresponding underlying map.

+-- {: .num_prop #InclusionOfNableBisectionsIntoNable0Bisections}
###### Proposition

The [[dependent sum]] along this map induces a canonical map of [[∞-groups]]

$$
  \mathbf{BiSect}(At(\nabla))
  \to
  \mathbf{BiSect}(At(\nabla_0))
  \,.
$$

=--

If we regard $\nabla$ as a [[prequantum n-bundle]] then this is a canonical inclusion of the [[quantomorphism n-group]] into the $\infty$-group of "$\nabla_0$-twisted diffeomorphisms".

If $\mathbb{G}$ is even a [[sylleptic ∞-group]], then the above moduli $\infty$-stacks have a further [[delooping]] and we obtain a 2-step sequence of forgetful maps

$$
  \mathbf{B}^2 \mathbb{G}_{conn}
  \to
  \mathbf{B}(\mathbf{B}\mathbb{G}_{conn})
  \to
   \mathbf{B}^2 \mathbb{G}
  \,.
$$

Accordingly:

+-- {: .num_prop #InclusionOfNableBisectionsIntoNable1AndNabla0Bisections}
###### Proposition

The [[dependent sum]] along these maps induces inclusions of $\infty$-groups

$$
  \mathbf{BiSect}(At(\nabla))
  \to
  \mathbf{BiSect}(At(\nabla_1))
  \to
  \mathbf{BiSect}(At(\nabla_0))
  \,.
$$

=--

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

This now interprets as the inclusion

1. of the [[quantomorphism n-group]] into the $\infty$-group of bisections of the [[higher Courant groupoid]];

1. $\infty$-group of bisections of the [[higher Courant groupoid]] into that of "$\nabla_0$-twisted diffeomorphisms".

=--

In summary we have the following table of inclusions

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

See below at _[Examples-- The traditional Courant Lie 2-algebroid](#TheTraditionalCourantLie2Algebroid)_ for more on this.

## Examples
 {#Examples}

We first show how the general notion of _higher Atiyah groupoid_ reproduces various traditonal structures.

* _[The traditonal Atiyah Lie groupoid](#TheTraditionalAtiyagLieGroupoid)_

* _[The traditional Courant Lie 2-algebroid](#TheTraditionalCourantLie2Algebroid)_

* _[The traditional quantomorphism group](#TheTraditionalQuantomorphismGroup)_

* _[The quantomorphism n-groups](#TheQuantomorphismNGroups)_

* _[The traditional Heisenberg group](#TheTraditionalHeisenbergGroup)

* _[The Heisenberg n-group](#TheHeisenbergnGroup)

### The traditional Atiyah Lie groupoid
 {#TheTraditionalAtiyagLieGroupoid}

We discuss how the traditional notion of [[Atiyah Lie groupoids]] in traditional [[differential geometry]] is a special case of higher Atiyah groupoids of def. \ref{HigherAtiyahGroupoid}.

To set this up we take the ambient [[(∞,1)-topos]] to be

$\mathbf{H} \coloneqq$ [[Smooth∞Grpd]] and make use of the canonical embeddings

[[SmthMfd]] $\hookrightarrow$ [[diffeological space|DiffeologicalSpace]] $\hookrightarrow$ [[smooth spaces|SmoothSpace]] $\hookrightarrow$ [[Smooth∞Grpd]],

and

[[Lie groupoid|LieGrpd]] $\simeq$ [[differentiable stack]] $\hookrightarrow$ [[Smooth∞Grpd]]

which are understood in the following.

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

For $G$ a [[Lie group]], $X$ a [[smooth manifold]] and $P \to X$ a $G$-[[principal bundle]], the traditional [[Atiyah Lie groupoid]] of $P$ is equivalent to that of def. \ref{HigherAtiyahGroupoid}.

=--

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

Write $g \colon X \to \mathbf{B}G$ for the classifying map of $P \to X$, by prop. \ref{ClassificationOfGPrincipalBundles}.

By remark \ref{AtiyahGroupoidIsCechNerve} the higher Atiyah groupoid $At(P)$ is simply the [[Cech nerve]] of this map. Since $G$ and $X$ and hence $P$ are all [[truncated object in an (infinity,1)-category|0-truncated objects]], hence $\mathbf{B}G$ a [[1-truncated]] object, this Cech nerve is [[coskeleton|2-coskeletal]] and hence is sufficient to consider the first three degrees. By definition these are

$$
  At(P)
  \simeq
  \left(
    X \underset{\mathbf{B}G}{\times}X \underset{\mathbf{B}G}{\times}X
      \stackrel{\to}{\stackrel{\to}{\to}}
    X \underset{\mathbf{B}G}{\times}X
      \stackrel{\to}{\to}
    X
  \right)
  \,,
$$

where $X \underset{\mathbf{B}G}{\times}X $ denotes the [[homotopy fiber product]] of $g$ with itself, and so forth. To see what this object is, pick any $U \in $ [[CartSp]], and observe that

$$
  \mathbf{H}(U, X\underset{\mathbf{B}G}{\times}X )
  \simeq
  \mathbf{H}(U,X) \underset{\mathbf{H}(U,\mathbf{B}G)}{\times} \mathbf{H}(U,X)
$$

(using that the [[(∞,1)-categorical hom]]-[[(∞,1)-functor]] $\mathbf{H}(U,-)$ preserves [[(∞,1)-limits]]) is equivalently the [[set]] of [[triples]] consisting of two [[smooth functions]] $\phi_1, \phi_2 \colon X$ and a [[gauge transformation]] between the [[pullback|pulled-back bundles]] $\eta \colon \phi_1^* P \to \phi_2^* P$ on $U$.

Since $U$ is topologically [[contractible topological space|contractible]], and hence every $G$-[[principal bundle]] over $U$ admits a [[section]], every such triple induces a [[function]], in fact a [[bijection]], from the set of lifts $\hat \phi_1 \colon U \to P$ of $\phi_1$ to the set of lifts $\hat \phi_2 \colon U \to P$ which are $C^\infty(U,G)$-equivariant. By $G$-equivariant every pair consisting of a single lift $\hat \phi_1$ and its image $\eta(\hat \phi_1)$ already uniquely determes $\eta$. Therefore the above set of triples is [[natural isomorphism|naturally isomorphic]] to the set of [[smooth functions]] $U \to P \times_G P \coloneqq (P \times P)/_{diag} G$. This is precisely the [[smooth manifold]] of [[morphisms]] of the traditional [[Atiyah Lie groupoid]]. Since this is true for all $U \in $ [[CartSp]] and [[natural transformation|naturally]] so, and since [[CartSp]] is a [[site]] of definition of [[Smooth∞Grpd]] it follows by the [[(∞,1)-Yoneda lemma]] (which in the present cases reduces to the ordinary [[Yoneda lemma]]), we have a [[natural equivalence]]

$$
  X \underset{\mathbf{B}G}{\times} X
  \simeq
  P \times_G P
  \,.
$$

In this manner it is immediate to check that this identification respects all the structure maps, and hence the above Cech nerve is indeed identified as the [[simplicial manifold]] which is the [[nerve]] of the traditional [[Atiyah Lie groupoid]] $(P \times_G P \stackrel{\to}{\to} X)$.

=--

### The traditional Courant Lie 2-algebroid
 {#TheTraditionalCourantLie2Algebroid}

There is a traditional construction which assigns to a [[bundle gerbe]] "with connective structure but without [[curving]]" a [[Courant Lie 2-algebroid]]. We discuss here how this is the [[Lie differentiation]] of the corresponding higher Atiyah groupoid.

In order to do so, we pick again, as [above](#TheTraditionalAtiyagLieGroupoid), as ambient context $\mathbf{H} = $ [[Smooth∞Grpd]].

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

For $\mathbb{G} \coloneqq U(1) \in LieGrp \hookrightarrow Grp(\mathbf{H})$ the [[circle Lie group]] (which is in particular a [[sylleptic ∞-group]]), the sequence of maps of [[moduli ∞-stacks]]

$$
  \mathbf{B}^2 U(1)_{conn}
  \to
  \mathbf{B}(\mathbf{B}U(1)_{conn})
  \to
  \mathbf{B}^2 U(1)
$$

of prop. \ref{InclusionOfNableBisectionsIntoNable1AndNabla0Bisections} is presented under the canonical equivalence [[Smooth∞Grpd]] $\simeq _{L_{lwhe}} Func(CartSp^{op}_{smooth}, sSet)$ by the image under the [[Dold-Kan correspondence]] of the evident sequence of [[chain maps]]

$$
  \array{
    U(1) &\stackrel{id}{\to}& U(1) &\stackrel{id}{\to}& U(1)
    \\
    \downarrow^{\mathrlap{d log}} && \downarrow^{\mathrlap{d log}} && \downarrow^{\mathrlap{0}}
    \\
    \Omega^1 &\stackrel{id}{\to}& \Omega^1 &\stackrel{\mathrlap{0}}{\to}& 0
    \\
    \downarrow^{\mathrlap{d}} && \downarrow^{\mathrlap{0}} && \downarrow^{\mathrlap{0}}
    \\
    \Omega^2 &\to& 0 &\to& 0
  }
  \,,
$$

where on the left we have the [[Deligne complex]] for degree-3-[[ordinary differential cohomology]].

=--

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

This is a direct consequence of the discussion at _[[circle n-bundle with connection]]_.

=--

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

This makes precise how

* $\mathbf{B}^2 U(1)_{conn}$ is the [[moduli infinity-stack|moduli 2-stack]] of [[circle 2-bundles with connection]];

* $\mathbf{B}(\mathbf{B}U(1)_{conn})$ is the moduli 2-stack of circle 2-bundle "with connection but without [[curving]]";

* $\mathbf{B}^2 U(1)$ is the moduli 2-stack of [[circle 2-group]]-[[principal 2-bundles]].

=--

Let

$$
  \nabla_1 \colon X \to \mathbf{B}(\mathbf{B}U(1)_{conn})
$$

be the map modulating [[circle 2-bundle with connection]] but "without [[curving]]". Then then higher Atiyah groupoid of th $(\mathbf{B}U(1)_{conn})$-[[principal 2-bundle]] classified by this map has as higher Atiyah groupoid the corresponding _[[Courant Lie 2-groupoid]]_: the object which is the [[Lie integration]] of the traditional [[Courant Lie 2-algebroid]] associated with $\nabla_1$.

To see we observe that the corresponding [[∞-group of bisections|2-group of bisections]] is

$$
  \mathbf{Aut}_{\mathbf{H}}(\nabla_1)
  \coloneqq
  \underset{\mathbf{B}(\mathbf{B}U(1)_{conn})}{\prod}
  \mathbf{Aut}_{/\mathbf{B}(\mathbf{B}U(1)_{conn})}(\nabla_1)
  \,.
$$

This has as objects [[diagrams]] in $\mathbf{H}$ of the form

$$
  \array{
     X &&\underoverset{\simeq}{\phi}{\to}&& X
     \\
     & {}_{\mathllap{\nabla_1}}\searrow &\swArrow_\eta& \swarrow_{\mathrlap{\nabla_1}}
     \\
     && \mathbf{B}(\mathbf{B}U(1)_{conn})
  }
  \,,
$$

hece equivalently pairs consisting of a [[diffeomorphism]] $\phi \colon X \to X$ and a [[gauge transformation]] (of 2-connections without [[curving]])

$$
  \eta
    \;\colon \;
  \phi^* \nabla_1 \to \nabla_1
  \,.
$$

The morphisms are accordingly the suitable [[natural transformations]] of these diagrams.

This is precisely the 2-group of "bundle gerbe symmetries" of $\nabla_1$ which is studient in ([Collier](#Collier)). With this identification the main result there is the above claim.

Moreover, the canonical inclusions of smooth 2-groups of prop. \ref{InclusionOfNableBisectionsIntoNable1AndNabla0Bisections} reproduces, under [[Lie differentiation]], the inclusion of the [[Poisson Lie 2-algebra]] into that [[Lie 2-algebra]] of sections of the corresponding [[Courant Lie 2-algebroid]] observed in ([Rogers 10](#Rogers10)).

### The traditional quantomorphism group
 {#TheTraditionalQuantomorphismGroup}

For $\nabla \colon X \to \mathbf{B}U(1)_{conn}$
the [[group of bisections]] of the corresponding Atiyah groupoid is the [[quantomorphism group]] of $\nabla$ regarded as a [[prequantum bundle]].

(...)

### The quantomorphism $n$-groups
 {#TheQuantomorphismNGroups}

In ([Rogers 11](#Rogers)) is a proposal for the generalization of the notion of [[Poisson bracket]] [[Lie algebra]] of a [[symplectic manifold]] to a notion of [[Poisson Lie n-algebra]] induced by an _[[n-plectic manifold]]_. Since the [[Lie integration]] of the [[Poisson bracket]] is traditionally known as the _[[quantomorphism group]]_, the Lie integration of these [[Poisson Lie n-algebras]] should be called an _[[quantomorphism n-group]]_.

We here discuss a general abstract theory of [[quantomorphism n-groups]] as [[∞-groups of bisections]] of a higher Atiyah groupoid associated with a [[principal ∞-connections]]. Then we show that under [[Lie differentiation]] this reproduces the construction in ([Rogers 11](#Rogers11)).

[[!include geometric quantization extensions - table]]

For all of the following, let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]] equipped with [[differential cohesion]]. Let $\mathbb{G} \on Grpd(\mathbf{H})$ be equipped with the structure of a [[braided ∞-group]]. Then there is a canonical object $\mathbf{B}\mathbb{G}_{conn} \in \mathbf{H}$ which is the [[moduli ∞-stack]] of $\mathbb{G}$-[[principal ∞-connections]].

Fox such a [[principal ∞-connection]] given by a map

$$
  \nabla \;\colon\; X \to \mathbf{B}\mathbb{G}_{conn}
  \,.
$$

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

By prop. \ref{HValuedAutomorphismAsFiber} the $\mathbf{H}$-valued automorphism $\infty$-group $\mathbf{Aut}_{\mathbf{H}}(\nabla)$ according to def. \ref{HValuedAutomorphismGroup} sits in an [[(∞,1)-pullback]] diagram of the form

$$
  \array{
    \mathbf{Aut}_{\mathbf{H}}(\nabla)
    &\to&
    \mathbf{Aut}(X)
    \\
    \downarrow && \downarrow^{\mathrlap{\nabla \circ (-)}}
    \\
    {*}
    &\stackrel{\vdash \nabla}{\to}& [X, \mathbf{B}\mathbb{G}_{conn}]
  }
  \,.
$$

By remark \ref{Concretification} we want to pass to its [[concretification]]. Indeed, in the above diagram the [[mapping ∞-stack]] $[X, \mathbf{B}\mathbb{G}_{conn}]$ is not quite yet the correct [[moduli ∞-stack]] for $\mathbb{G}$-[[principal ∞-connections]] on $X$, but instead its [[differential concretification]] $\mathbb{G}\mathbf{Conn}(X)$ is, as defined at _[concretification - Examples - Of differential moduli](concretification#ConcretificationOfDifferentialModuli)_. Therefore the following definition states the above pullback diagram with that replacement.

=--

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

Let $\mathbb{G}$ be a [[braided ∞-group]] as above and let $\nabla \;\colon\; X \to \mathbf{B}\mathbb{G}_{conn}$ be a $\mathbb{G}$-[[principal ∞-connection]].

The **[[quantomorphism ∞-group]]** $QuantMorph(\nabla) \in \mathrm{Grp}(\mathbf{H})$ of a $\nabla$ is the object fitting into the [[(∞,1)-pullback]]

$$
  \array{
    \mathbf{QuantMorph}(\nabla)
    &\to&
    \mathbf{Aut}(X)
    \\
    \downarrow && \downarrow^{\mathrlap{\nabla \circ (-)}}
    \\
    {*}
    &\stackrel{\vdash \nabla}{\to}& \mathbb{G}\mathbf{Conn}(X)
  }
  \,.
$$

=--

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

The [[Hamiltonian symplectomorphism ∞-group]] $\mathbf{HamSympl}(\nabla)$ is the [[1-image]] of the canonical map $\mathbf{QuantMorph}(\nabla) \to \mathbf{Aut}(X)$.

=--

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

The [[quantomorphism ∞-group]] $\mathbf{QuantMorph}(\nabla)$ in an [[∞-group extension]] of the Hamiltonian symplectomorphism $\infty$-group of def. \ref{HamiltonianSymplectomorphismInfinityGroup} by the [[∞-group]] $(\Omega\mathbb{G})\mathbf{FlatConn}(X)$ of [[concretification|concretified]] [[flat ∞-connections]] on $X$: we have a [[homotopy fiber sequence]]

$$
  (\Omega\mathbb{G})\mathbf{FlatConn}(X)
  \to
  \mathbf{QuantMorph}(\nabla)
  \to
  \mathbf{HamSympl}(\nabla)
  \,.
$$

Moreover, at least at the level of the underlying objects, this extension is classified by the [[cocycle]] $\mathbf{HamSympl}(\nabla) \stackrel{\nabla \circ (-)}{\to} \mathbf{B}((\Omega\mathbb{G})\mathbf{FlatConn}(X))$ in that we have a long [[homotopy fiber sequence]]

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

=--

We now restrict this to a special case and describe it more in detail:

Let $\mathbf{H} = $ [[Smooth∞Grpd]], let $X \in $ [[SmthMfd]] $\hookrightarrow$ [[Smooth∞Grpd]] and let $\mathbb{G} \coloneqq \mathbf{B}^{n-1}U(1) \in \mathrm{Grp}(\mathbf{H})$ be the [[circle n-group]]. Finally let $\omega \colon X \to \Omega^{n+1}$ be an [[n-plectic form]] and $\nabla \;\colon\; X \to \mathbf{B}^n U(1)_{conn}$ a prequantization by a [[prequantum circle n-bundle]].

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

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 traditional Heisenberg group
 {#TheTraditionalHeisenbergGroup}

(...) [[Heisenberg group]] (...)

### The Heisenberg $n$-group
 {#TheHeisenbergnGroup}

If $X = G \in \mathbf{H}$ has itself [[∞-group]] structure, then it is natural to restrict the [[quantomorphism ∞-group]] to that subgroup of the [[Hamiltonian symplectomorphism ∞-group]] whose elements come from the $G$-[[∞-action]] on itself. This is the corresponding _[[Heisenberg ∞-group]]_.

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

With all assumtions as [above](#TheQuantomorphismNGroups), let $G \in Grp(\mathbf{H})$ be an [[∞-group]] and let

$$
  G \hookrightarrow \mathbf{Aut}(G)
$$

(where on the right we have the [[automorphism ∞-group]] of the underlying object $G \in \mathbf{H}$) the inclusion that exhibits the left $G$-[[∞-action]] on itself.

The the **[[Heisenberg ∞-group]]** $\mathbf{Heis}(\nabla)$ is the [[(∞,1)-pullback]] in the [[diagram]]

$$
  \array{
    \mathbf{Heis}(\nabla)
    &\to&
    \mathbf{QuantMorph}(\nabla)
    \\
    \downarrow && \downarrow
    \\
    G &\to& \mathbf{Aut}(G)
  }
  \,.
$$

=--

The following is an immediate consequence of the definition

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

The [[Heisenberg ∞-group]] $\mathbf{Heis}(\nabla)$ is an [[∞-group extension]] of $G$ by $(\Omega \mathbb{G})\mathbf{FlatConn}(G)$: we have a [[homotopy fiber sequence]] of [[∞-groups]]

$$
  (\Omega \mathbb{G})\mathbf{FlatConn}(G)
  \to
  \mathbf{QuantMorph}(\nabla)
  \to
  G
  \,.
$$

=--

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

In $\mathbf{H} = $ [[Smooth∞Grpd]] consider $G \in LieGrp \hookrightarrow Grp(\mathbf{H})$ a [[connected topological space|connected]], [[simply connected topological space|simply connected]] [[compact Lie group|compact]] [[semisimple Lie group]], say the [[Spin group]] $G = Spin$. Then the [[Killing form]] [[invariant polynomial]] is a pre-3-plectic form on the [[moduli stack]] of $G$-[[principal connections]]:

$$
  \langle -,-\rangle
  \;\colon\;
  \mathbf{B}G_{conn} \to \mathbf{\Omega}_{cl}^4
  \,.
$$

This has a [[higher geometric prequantization]] by the [[smooth first fractional Pontryagin class]], a map

$$
  \tfrac{1}{2}\hat\mathbf{p}_1
  \;\colon\;
  \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}
  \,.
$$

The [[transgression]] of this to maps oout of the [[circle]] yields a [[circle 2-bundle with connection]]

$$
  \nabla
   \;\colon\;
  G \to [S^1, G]
   \stackrel{[S^1, \frac{1}{2}\hat\mathbf{p}_1]}{\to}
  [S^1, \mathbf{B}^3 U(1)_{conn}]
   \stackrel{\exp(2 \pi i \int_{S^1}(-))}{\to}
  \mathbf{B}^2 U(1)
  \,.
$$

This is a [[prequantum circle 2-bundle]] which prequantizes the canonical [[differential 3-form]] on $G$, the one which is [[invariant differential form|left invariant]] and at the neutral element is $\langle -,[-,-]\rangle$.

Consider now the [[higher prequantum geometry]] of this 2-connection. So now $\mathbb{G} = \mathbf{B}U(1)$.

Observe that

$$
  \begin{aligned}
    (\Omega \mathbb{G})\mathbf{FlatConn}(G)
    & \simeq
    U(1) \mathbf{FlatConn}(G)
    \\
    & =
    \mathbf{B}U(1)
  \end{aligned}
$$

because $G$ is assumed to be [[simply connected topological space|simply connected]]. (Notice that $\mathbf{B}U(1)$ does appear here with its canonical smooth structure: while a [[gauge transformation]] from the trivial $U(1)$-[[principal connection]] to itself is a constant function along $X$, the smooth structure in $U(1)\mathbf{FlatConn}(G)$ comes from how this may vary in parameterized collections ).

Therefore by prop. \ref{HeisenbergInfinityGroupExtension} we have an [[∞-group extension]]

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

This exhibits the Heisenberg 2-group here as the [[string 2-group]] $String(G)$:

$$
  \mathbf{Heis}(\nabla) \simeq String(G)
  \,.
$$

=--

## Related concepts

[[!include higher Atiyah groupoid - table]]

## References

The [above](#TheTraditionalCourantLie2Algebroid) identification of higher Atiyah groupoids of "bundle gerbes with connective structure but without [[curving]]" with those [[Lie integration|Lie integrating]] the corresponding [[standard Courant Lie 2-algebroid]] is directly implied (under the above translations) by the main result in

* [[Braxton Collier]], _Infinitesimal Symmetries of Dixmier-Douady Gerbes_ ([arXiv:1108.1525](http://arxiv.org/abs/1108.1525))
 {#Collier}

The corresponding inclusion of the [[Poisson Lie 2-algebra]] into the [[Lie 2-algebra]] of bisections of the [[Courant Lie 2-algebroid]] was first observed in

* [[Chris Rogers]], _2-plectic geometry, Courant algebroids, and categorified prequantization_ ,  [arXiv:1009.2975](http://arxiv.org/abs/1009.2975).
 {#Rogers10}

in the context of _[[2-plectic geometry]]_ over [[smooth manifolds]].

The [[Poisson Lie n-algebra]] over an [[n-plectic manifold]], which by prop. \ref{spring} is the [[Lie differentiation]] of the [[quantomorphism n-group]] of any [[prequantum circle n-bundle]] prequantizing the $n$-plectic form, has been proposed in

* [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis (2011) ([arXiv:1106.4068](http://arxiv.org/abs/1106.4068))
 {#Rogers11}

Most further statements here will appear in

* [[Domenico Fiorenza]], [[Urs Schreiber]], _[[schreiber:Higher geometric prequantum theory]]_
 {#FiorenzaRogersSchreiber}

[[!redirects higher Atiyah groupoids]]

[[!redirects higher gauge groupoid]]
[[!redirects higher gauge groupoids]]


higher Atiyah groupoid - table

**[[higher Atiyah groupoid]]**

| [[higher Atiyah groupoid]]: | standard [[higher Atiyah groupoid]] | [[higher Courant groupoid]] | groupoid version of [[quantomorphism n-group]] |
|--|--|--|--|
| [[coefficient]] for [[cohomology]]: | $\mathbf{B}\mathbb{G}$  | $\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})$  | $\mathbf{B} \mathbb{G}_{conn}$ |
| type of [[fiber ∞-bundle]]: | [[principal ∞-bundle]] | [[principal ∞-connection]] without top-degree connection form | [[principal ∞-connection]] |


higher Cartan geometry

{:goal: .un_remark style="border:solid #0000cc;background: #add8e6;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Higher Cartan geometry_ is supposed to be the generalization of [[Cartan geometry]] to [[higher geometry]]; hence the theory of [[geometric homotopy types]] ([[manifolds]], [[orbifolds]], [[Lie groupoids]], [[geometric stacks]], [[smooth groupoids]], [[smooth infinity-groupoids]], ...) which are locally modeled on [[homotopy quotients]] $G/H$ of geometric [[infinity-groups]] -- the globalized version of [[higher Klein geometry]] (see also the [survey table](#Survey) below).

More in detail, this means that given a morphism $H \to G$ of suitably geometric [[∞-groups]] then a _higher Cartan geometry_ modeled on the [[homotopy quotient]] $G/H$ is a higher geometric space $X$ such as an [[orbifold]], [[geometric ∞-stack]], [[derived scheme]] etc. which, in some suitable sense, has its [[tangent spaces]] identified with an [[infinitesimal neighbourhood]] in $G/H$.

Just like traditional [[Cartan geometry]] (in particular in the guise of [[G-structures]]) captures a plethora of relevant kinds of geometries (([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian geometry]] ([Cartan 23](#Cartan23)), [[conformal geometry]], ... [[complex geometry]], [[symplectic geometry]], ..., [[parabolic geometry]]) so higher Cartan geometry is supposed to similarly govern types of [[higher differential geometry]].

A class of examples where aspects of higher Cartan geometry may be seen to secretly underlie traditional discussion is the theory of [[super p-brane sigma-models]] on [[supergravity]] [[target space|target]]-[[super-spacetimes]]. This we consider in the examples [below](#SuperBraneGeometry). See also at _[[super-Cartan geometry]]_.

It is therefore maybe curious to note that while [[Cartan geometry]] as originating in ([Cartan 23](#Cartan23)) drew its motivation from the mathematical formulation of the theory of [[Einstein gravity]], higher Cartan geometry is well motivated by higher dimensional [[supergravity]] such as 10d [[type II supergravity]] and [[heterotic supergravity]] as well as [[11-dimensional supergravity]].

## Motivation
 {#Motivation}

Here we informally survey motivation for higher Cartan geometry from phenomena and open problems visible in traditional geometry.

1. [Pre-quantization of symplectic geometry](#MotivationPrequantizationOfSymplectic)

1. [Higher pre-quantization and Parameterized WZW terms](#MotivationDefiniteParameterizationOfWZWTerms)

1. [Higher pre-quantization and Globalized WZW terms](#MotivationDefiniteGlobalizationOfWZWTerms)

   $\,$

1. [Interlude: Super Cartan geometry](#MotivationSuperCartanGeometry)

1. [Higher Cartan connections and stacky Cartan geometry](#HigherCartanConnectionsAndStackyCartanGeometries)

1. [Definite higher WZW terms on stacky Cartan geometries](#DefiniteWZWTermsOnStackyCartanGeometries)

$\,$

Alternatively, higher Cartan geometry may be motivated intrinsically simply as the result of [[synthetic geometry|synthetically]] formulating [[Cartan geometry]] in [[homotopy type theory]]. This is the way in which the _[definition below](#Definition)_ proceeds. In the _[Examples](#Examples)_ we discuss how this abstract theory indeed serves to inform the motivating phenomena listed here.

### Pre-quantization of symplectic geometry
 {#MotivationPrequantizationOfSymplectic}

While a [[symplectic manifold]] structure $(X,\omega)$ is an example of an ([[integrable G-structure|integrable]]) [[G-structure]], hence of a [[Cartan geometry]], in many applications [[symplectic forms]] $\omega$ are to be refined to [[complex line bundles]] with [[connection on a bundle|connection]], equivalently [[circle-bundles with connection]] $\mathbf{L}$ (with [[curvature]] $F_{\mathbf{L}} = \omega$), a refinement known as _[[geometric prequantization]]_.

$$
  (X,\omega) \stackrel{pre-quantization}{\mapsto} (X,\mathbf{L})
$$

While two [[differential forms]] on $X$ are either [[equal]] or not, two [[principal connections]] on $X$ may be different and still [[equivalence|equivalent]]. The connection $\nabla$ may have non-trivial _[[automorphisms]]_, while a differential form $\omega$ does not. (Readers may be more familiar with this kind of phenomenon from the discussion of the _[[moduli stack of elliptic curves]]_.)

Hence while there is just a [[set]] and hence a [[homotopy 0-type]] of [[symplectic forms]] on $X$, there is a [[groupoid]] and hence a [[homotopy 1-type]] of [[principal connections]] on $X$. It is in this sense that the pair $(X,\nabla)$ involves [[higher geometry]], namely [[homotopy n-types]] for $n \gt 0$.

$$
  \array{
     \left\{ \omega \right\} && \left\{ \mathbf{L} \right\}
     \\
     0-type && 1-type
  }
$$

This implies notably that where $\omega$ has a [[stabilizer group]] under the [[diffeomorphism]] [[action]] on $X$ -- the [[symplectomorphism group]] --, $\mathbf{L}$ instead has a "[[homotopy stabilizer group]]",

$$
  Stab^h_{Diff(X)}(\mathbf{L})
  =
  \left\{
    \left(
       \array{
         \phi \colon X \stackrel{\simeq}{\to} X\,,
         \\
         \eta \colon \phi^\ast \mathbf{L} \stackrel{\simeq}{\to} \mathbf{L}
       }
    \right)
  \right\}
$$

consisting of pairs of a [[diffeomorphism]] $\phi$ and an [[isomorphism]] $\phi^\ast \mathbf{L} \stackrel{\simeq}{\to} \mathbf{L}$. This is called the _[[quantomorphism group]]_.

$$
  \array{
    Stab^h_{Diff(X)}(\mathbf{L}) = QuantMorph(\mathbf{L})
    \\
    Stab_{Diff(X)}(\omega) = SymplMorph(\omega)
  }
$$

Hence the [[prequantum geometry]] $(X,\mathbf{L})$ is still clearly a geometry of sorts, but not a Cartan geometry. On the other hand, it is still similar enough to be usefully regarded form this perspective:

Just like, by the [[Darboux theorem]], every symplectic manifold $(X,\omega)$ has an [[atlas]] by [[charts]] [[isomorphism|isomorphic]] to a [[symplectic vector space]]

$$
  (V \simeq \mathbb{R}^{2n}, \omega_V = \mathbf{d}p_i \wedge \mathbf{d}q^i)
  \,,
$$

so every [[prequantum line bundle]] $\mathbf{L}$ on $X$ refining $\omega$ is [[equivalence|equivalent]] over this atlas to the $U(1)$-[[principal connection]] given by the globally defined connection for

$$
  \mathbf{L}_{V} \coloneqq p_i \wedge \mathbf{d}q^i
  \,.
$$

Moreover, just like the [[symplectic group]] $Sp(V,\omega_V)$ is the [[stabilizer group]] of $\omega_V$ under the canonical [[general linear group]]-[[action]] on $V$, so the [[homotopy stabilizer group]] of $\mathbf{L}_V$ (the part of the [[quantomorphism group]] $QuantMorph(\mathbf{L}_V)$ covering this) is the [[Mp^c]]-group, $Mp^c(V,\omega_V) = Mp(V,\omega_V)\underset{\mathbb{Z}/2\mathbb{Z}}{\times}U(1)$, the $U(1)$-version of the [[metaplectic group]] $Mp(V,\omega_V)$  ,

$$
  \array{
     && Stab^h_{GL(V)}(\mathbf{L}_V)
     \\
     && \simeq
     \\
     &&
     Mp^c(V,\omega_V)
     &\hookrightarrow&
     QuantMorph(V,\mathbf{L}_V)
     \\
     \downarrow && \downarrow && \downarrow
     \\
     &&
     Sp(V,\omega_V)
     &\hookrightarrow&
     SympMorph(V,\omega_V)
     \\
     && \simeq
     \\
     && Stab_{GL(V)}(\omega_V)
  }
$$

In this sense [[metaplectic quantization]] is a higher analog of symplectic geometry.

While one may well reason, evidently, about pre-quantization of symplectic manifolds without a general theory of higher Cartan geometry in hand, this class of examples serves as a first blueprint for what higher Cartan geometry should be like, and points the way to its higher-degree generalizations considered [below](#MotivationDefiniteParameterizationOfWZWTerms).

In particular, recurring themes are

1. [[circle n-bundles with connection]] $\mathbf{L}$ [[higher prequantization|higher prequantizing]] [[definite forms]] $\omega$;

1. their [[homotopy stabilizer groups]]/[[higher quantomorphism groups]] and the [[infinity-group extension]] they form and the higher [[G-structures]] associated with them.

### Higher pre-quantization and Parameterized WZW terms
 {#MotivationDefiniteParameterizationOfWZWTerms}

A particularly interesting example of a pre-quantization as above is the Kac-Moody [[central extension]] of [[loop groups]] of [[compact Lie group|compact]] [[semisimple Lie group]] $G$ (see [here](loop+group#ByGeometricQuantization)).

Loop groups are naturally symplectic geometries, whose symplectic form is the [[transgression]] of the canonical [[left invariant form|left invariant]] [[differential 3-form]] $\omega_3 = \langle-,[-,-]\rangle$ on $G$:

$$
  (G, \omega_3)
  \stackrel{transgression}{\mapsto}
  (L G, \omega_2)
$$

Similarly their central extension is the [[transgression]] to [[loop space]] of a  higher-degree analog of traditional [[pre-quantization]] down on $G$: the canonical [[left invariant form|left invariant]] [[differential 3-form]] $\omega_3 = \langle-,[-,-]\rangle$ lifts to a [[circle 2-bundle with connection]] $\mathbf{L}_3$, whose [[curvature]] 3-form is $F_{\mathbf{L}_3} = \omega_3$:

$$
  (G, \mathbf{L}_3)
  \stackrel{transgression}{\mapsto}
  (L G, \mathbf{L}_2)
$$

This $\mathbf{L}_3$ is also called the _[[WZW gerbe]]_ or _WZW term_, as its [[volume holonomy]] serves as the [[gauge interaction]] [[action functional]] for the [[Wess-Zumino-Witten sigma model]] with [[target space]] $G$.

Now the 2-connections on $G$ form a [[2-groupoid]] hence a [[homotopy 2-type]], the pair $(G,\nabla^G)$ may be regarded as being an object in yet a bit higher differential geometry.

$$
  \array{
     \left\{ \omega_3 \right\} && \left\{ \mathbf{L}_3 \right\}
     \\
     0-type && 2-type
  }
$$

Now given a $G$-[[principal bundle]]

$$
  \array{
    \mathbf{L}_3 && \mathbf{L}_3^P
    \\
    G &\stackrel{}{\longrightarrow}& P
    \\
    && \downarrow
    \\
    && X
  }
$$

then a natural question is whether there is a _[[parameterized WZW  term|definite parameterization]]_ $\mathbf{L}_3^P$ of $\mathbf{L}_3$ to a 2-form connection on $P$ which restricts fiberwise to $\nabla^G$ in a suitable sense up to [[gauge transformation]]. Such _[[parameterized WZW terms]]_ play a key role in [[heterotic string theory]] and [[equivariant elliptic cohomology]].

One finds that such [[parameterized WZW term|definite parameterizations]] are equivalent to [[lift of structure group|lifts of structure group]] of the bundle from $G$ to the [[homotopy stabilizer group]] of $\mathbf{L}_3$ under the right $G$-action on itself, and this turns out to be the [[string 2-group]] $String(G)$, which is itself the [[homotopy quotient]] of the group of based paths of $G$ by the Kac-Moody loop group of $G$. By the above we may also think of this as a [[Heisenberg 2-group]]:

$$
  Heis(\mathbf{L}_3)
  =
  Stab^h_{G}(\mathbf{L}_3)
  \simeq
  String(G)
  \simeq
  (P_\ast G) // \widehat{L G}
$$

Hence a definite parameterization of $\mathbf{L}_3$ over $P$ is a [[string structure]] on $P$. The [[obstruction]] to that is

* for $G = SU(N)$: the [[second Chern class]] $\c_2(P)$

* for $G = Spin$: the [[first fractional Pontryagin class]] $\tfrac{1}{2}p_1(P)$.

These are the obstructions famous from [[Green-Schwarz anomaly cancellation]] in [[heterotic supergravity]].

While this class of examples is not yet Cartan geometry proper (higher or not) since the bundle $P$ here is not a [[tangent bundle]], it contains in it the key aspect of [[parameterized WZW model|definite parameterizations]] of higher pre-quantized forms related to higher [[G-structures]]. Such definite parameterizations turn out to be part of genuine examples of higher Cartan geometry, to which we turn [below](#MotivationDefiniteGlobalizationOfWZWTerms) and key ingredients of higher Cartan geometry apply to both cases.

But more generally, one considers this situation for WZW terms on [[coset spaces]] $G/H$, relevant in  _[[gauged WZW model]]_.

+-- {: goal}
#### Goal 1 of higher Cartan geometry

Provide [[obstruction]] classes for [[definite parameterizations of higher WZW terms]].

=--

This we consider [below](#ObstructionClassesForDefiniteParameterizations)

### Higher pre-quantization and Globalized WZW terms
 {#MotivationDefiniteGlobalizationOfWZWTerms}

Often one wants to consider definite parameterizations as above along the tangent bundle of a $V$-manifold $X$, such that the parameterization comes from a _global_ $\mathbf{L}$ on $X$, a [[definite globalization of a WZW term]].

Given a [[vector space]] $V$ equipped with a (constant, i.e. translationally [[left invariant form|left invariant]]) [[differential n-form|differential (p+2)-form]]

$$
  \omega_V \in \Omega^{p+2}(V)
$$

a natural question to ask is for a $V$-manifold $X$ (i.e. an $n$-dimensional manifold if $V \simeq \mathbb{R}^n$) to carry a differential form

$$
  \omega \in \Omega^2(X)
$$

which is a [[definite form]], definite on $\omega_V$, in that its restriction to each [[tangent space]] is equal, up to a $GL(V)$-transformation, to $\omega_V$.

Standard theory of [[G-structures]] easily shows that such definite forms correspond to $Stab_{GL(V)}(\omega_V)$-structures on $X$, for $Stab_{GL(V)}(\omega_V)$ the [[stabilizer group]] of $\omega_V$ under the canonical $GL(V)$-[[action]] (by [[pullback of differential forms]]).

For instance if $V = \mathbb{R}^7$ and $\omega_V \in \Omega^3(V)$ is the [[associative 3-form]], then $Stab_{GL(V)}(\omega_V) = G_2$ is the [[exceptional Lie group]] [[G2]] and this yields [[G2-structures]].

But in view of the [above](#MotivationDefiniteParameterizationOfWZWTerms) discussion one is led to re-state this question for the case that $(V,\omega_V)$ is refined to a [[prequantum n-bundle|prequantum (p+1)-bundle]] $(V,\mathbf{L}_{p+2})$.

Just as a 1-connection is precisely the data needed to define [[holonomy|line holonomy]], so an $(p+1)$-connection is precisely the data needed to define $(p+1)$-[[volume holonomy]]

$$
  \array{
     \left\{ \omega_{p+2} \right\} && \left\{ \mathbf{L}_{p+2} \right\}
     \\
     0-type && (p+1)-type
  }
$$

A _[[definite globalization]]_ of such $\mathbf{L}_{p+2}$ over a $V$-manifold $X$ should be a [[circle n-bundle with connection|circle (p+1)-connection]] $\mathbf{L}_{p+1}^X$ on $X$ which suitably, up to the relevant [[higher gauge transformations]], restricts locally to $\mathbf{L}_V$.

For instance for first-order integrable such globalizations one would require that (in particular) for each [[infinitesimal disk]] $\mathbb{D}$ in a $V$-cover $U$ we have an [[equivalence]]

$$
  \array{
    & \mathbf{L}_{p+2}|_{|\mathbb{D}} & \simeq & \mathbf{L}_{p+2}^X|_{|\mathbb{D}}
    \\
    && \mathbb{D}
    \\
    && \downarrow
    \\
    && U && &&
    \\
    & \swarrow && \searrow
    \\
    V && && X
    \\
    \mathbf{L}_{p+2} && && \mathbf{L}_{p+2}^X
  }
$$

This problem indeed appears in the formulation of [[super p-brane sigma models]] on [[target space|target]] [[super-spacetimes]]. Here $V$ is a [[super Minkowski spacetime]], $\omega_V$ is an exceptional [[super Lie algebra]] [[Lie algebra cohomology|cocycle]] of degree $(p+2)$ and the formulation of the [[Green-Schwarz sigma model]] requires that it is refined (higher pre-quantized) to a higher WZW term, a $p$-form connection. The [[supergravity]] [[equations of motion]] imply a [[definite globalization]] $\omega$ of $\omega_V$ of a [[super-spacetime]], but to globally define the GS-WZW model one hence needs to lift this globalization to a $(p+1)$-connection, too (thereby "canceling the [[classical anomaly]]" of the model).

These [[definite globalizations]] are in particular [[parameterized WZW term|definite parameterizations]], as above, of the restriction of the higher WZW term to the [[infinitesimal disk]]-bundle of spacetime,

$$
  \array{
     \mathbf{L}_{p+2}|_{|\mathbb{D}} && \mathbf{L}_{p+2}^X|_{T_{inf}X}
     \\
     \mathbb{D} & \longrightarrow& T_{inf}X
     \\
     && \downarrow
     \\
     && X
  }
$$

Notice that for the [[infinitesimal disk]] every [[diffeomorphism]] is a [[linear transformation]], hence

$$
  Aut(\mathbb{D}) \simeq GL(V)
$$

and therefore by the above a definite globalization determines a [[G-structure]] for $G = Stab^h_{GL(V)}(\mathbf{L}_{p+2})$. Conversely, the [[obstruction]] to such a structure is an obstruction to a definite globalization.

This construction extends to [[forgetful functor]] (an [[(infinity,1)-functor]])

$$
  \left\{
    \array{
       definite \; globalizations \; \mathbf{L}_{p+2}^X
       \\
       of\; \mathbf{L}_{p+2} \; over \; X
    }
  \right\}
  \longrightarrow
  \left\{
     \array{
        Stab^h_{GL(V)}(\mathbf{L}_{p+2}|_{\mathbb{D}})-structures
        \\
        on \; X
     }
  \right\}
  \,.
$$

(For instance in the case of applications to [[supergravity]] that we turn to [below](#MotivationSuperCartanGeometry), these structures are extensions of strutures given by solutions to the super-[[Einstein equations]].

It is here that developing a theory of higher Cartan geometry has much potential, since, while the globalizations of the forms $\omega_V$ have been extensively studied in the literature, the globalization of their pre-quantized refinement to higher WZW-terms $\mathbf{L}_{p+2}$ has traditionally received almost no attention yet. A brief mentioning of the necessity of considering appears for instance in ([Witten 86, p. 17](#Witten86)), but traditional tools do get one very far in this question.

More precisely, this is the situation for all those [[branes]] in the old [[brane scan]] which have no tensor-multiplets on the [[worldvolume]], equivalently those on which no other branes may end (such as the [[string]] or the [[M2-brane]], but not the [[D-branes]] and not the [[M5-brane]]). For more general branes, it turns out that the target space itself is a higher geometric space. This leads us to higher Cartan geometry proper. This we turn to [below](#MotivationSuperCartanGeometry).)

Accordingly, now the symmetries of $\mathbf{L}_{p+2}^X$ form an extension of the _[[isometries]]_ of the induced $Stab^h_{GL(V)}(\mathbf{L}_{p+2}|_{\mathbb{D}})$-structure.

$$
  (\mathbf{B}^p U(1))FlatConn(X)
  \longrightarrow
  Stab^h_{Diff(X)}(\mathbf{L}_{p+2}^X)
  \longrightarrow
  Isom(X)
$$

One finds that after [[Lie differentiation]] these extensions are of the kind konwn in the phyiscs literature as [[BPS charge]] extensions.

+-- {: goal}
### Goal 2 of higher Cartan geometry

Classify these refined and generalized [[BPS states]].

=--

This we turn to [below](#HigherExtended IsometryGroups).

So far these examples point to higher Cartan geometry modeled on homomorphisms

$$
  Stab^h_{GL(V)}(\mathbf{L}_{p+2})
  \longrightarrow
  V \rtimes Stab^h_{GL(V)}(\mathbf{L}_{p+2})
$$

where the [[homotopy stabilizer group]] $Stab^h_{GL(V)}(\mathbf{L}_{p+2})$ is a [[infinity-group]], but where $V$ is still an ordinary manifold. We now turn ([below](#HigherCartanConnectionsAndStackyCartanGeometries)) to examples that also turn the local model space $V$ into an higher [[geometric homotopy type]]. But first we need a little [interlude](#MotivationSuperCartanGeometry).

### Interlude: Super-Cartan geometry
 {#MotivationSuperCartanGeometry}

Before further motivating ever higher Cartan geometry, it serves to pause and realize that while passing from manifolds to [[stacks]], we are in particular first of all generalizing to [[sheaves]]. So even before going higher in homotopy degree, one may ask how much of Cartan geometry may be formulated in [[sheaf toposes]], first over the [[site]] of [[smooth manifolds]] itself, which leads to Cartan geometry in the generality of [[smooth spaces]], and next over [[sites]] other than that of [[smooth manifolds]] -- _[[super-Cartan geometry]]_.

One key example for this is [[supergeometry]]. Where a major application of traditional Cartan geometry is its restriction to [[orthogonal structures]] encoding ([[pseudo-Riemannian geometry|pseudo]]-)[[Riemannian geometry]] of particular relevance in the theory of [[gravity]], the analogous orthogonal structures in [[supergeometry]] serve to set up the theory of [[supergravity]].

More in detail, after picking a [[dimension]] $d\in \mathbb{N}$ and writing $\mathfrak{Iso}(\mathbb{R}^{d-1,1})$ for the [[Poincaré Lie algebra]], then a choice of "number of supersymmetries" is a choice of [real spin representation](spin+representation#RealIrreducibleSpinRepresentationInLorentzSignature) $N$. Then the [[direct sum]]

$$
  \mathfrak{Iso}(\mathbb{R}^{d-1,1|N})
  \coloneqq
  \underbrace{\mathfrak{Iso}(\mathbb{R}^{d-1,1})}_{even} \oplus \underbrace{N}_{odd}
$$

regarded as a [[super vector space]] with $N$ in odd degree becomes a [[super Lie algebra]] by letting the $[even,odd]$ bracket to be given by the defining [[action]] and by letting the $[odd,odd]$ bracket be given by a canonically induced bilinear and $\mathfrak{o}$-equivariant pairing -- the [[super Poincaré Lie algebra]]. This still canonical contains the [[Lorentz Lie algebra]] $\mathfrak{o}(\mathbb{R}^{d-1,1})$ and the [[quotient]]

$$
  \mathbb{R}^{d-1,1|N}
  \coloneqq
  \mathfrak{Iso}(\mathbb{R}^{d-1,1|N})/\mathfrak{o}(\mathbb{R}^{d-1,1})
$$

is called [[super Minkowski spacetime]] (equipped with its [[super translation Lie algebra]] structure).

From this, a [[super-Cartan geometry]] is defined in direct analogy to the Cartan formulation of Riemannian geometry

| [[Cartan geometry]] | $\mathfrak{g}$ | $\mathfrak{h}$ | $\mathfrak{g}/\mathfrak{h}$ |
|---------------------|----------------|----------------|-----------------------------|
| [[pseudo-Riemannian geometry]]/[[Einstein gravity]]  | $\mathfrak{Iso}(\mathbb{R}^{d-1,1})$ | $\mathfrak{o}(d-1,1)$ | $\mathbb{R}^{d-1,1}$ |
| [[supergravity]] | $\mathfrak{Iso}(\mathbb{R}^{d-1,1\vert N})$ | $\mathfrak{o}(d-1,1)$ | $\mathbb{R}^{d-1,1\vert N}$ |

Indeed, all the traditional literature on supergravity (e.g. ([Castellani-D'Auria-Fré 91](#CastellaniDAuriaFre91))) is phrased, more or less explicitly, in terms of [[Cartan connections]] for the inclusion of the [[Lorentz group]] into the [[super Poincaré group]], this being the formalization of what physicists mean when saying that they pass to "local supersymmetry".

It so happens that from within such super-Cartan geometry there appear some of the most interesting examples of what should be higher Cartan geometry, hence _higher super-Cartan geometry_. This we turn to [below](#MotivationDefiniteGlobalizationOfWZWTerms).

### Higher Cartan connections and Stacky Cartan geometries
 {#HigherCartanConnectionsAndStackyCartanGeometries}

A traditional [[Cartan connection]], being a [[principal connection]] satisfying some extra conditions, is locally (on some [[chart]] $U \to X$) in particular a [[Lie algebra valued differential form]] $A \in \Omega^1(U,\mathfrak{g})$. Following Cartan, this is equivalently a [[homomorphism]] of [[dg-algebras]] of the form

$$
  \Omega^\bullet(U) \longleftarrow W(\mathfrak{g}) \colon A
$$

from the [[Weil algebra]] of the [[Lie algebra]] $\mathfrak{g}$ to the [[de Rham complex]] of $U$, equivalently a homomorphism of just [[graded algebras]]

$$
  \Omega^\bullet(U) \longleftarrow CE(\mathfrak{g}) \colon A
$$

from the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$. (Requiring this second morphism to also respect the dg-algebra structure, hence the differential, is equivalent to requiring the [[curvature]] form $F_A$ to vanish, hence to the connection being a [[flat connection]]).

In particular for the description of [[supergravity]] [[superspacetimes]] one considers this for $\mathfrak{g} = \mathfrak{Iso}(\mathbb{R}^{d-1,1|N})$ the [[super Poincaré Lie algebra]] of some [[super Minkowski spacetime]] $\mathbb{R}^{d-1|N}$. This serves to encode a [[Levi-Civita connection]] as for ordinary [[gravity]] modeled by ordinary [[orthogonal structure]] [[Cartan geometry]], together with the [[gravitino]] field.

In detail, the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{Iso}(\mathbb{R}^{10,1|N=1}))$ for 11-dimensional [[Minkowski spacetime]] turned super via the unique irreducible 32-dimensional [[spin representation]] (see [here](spin+representation#RealIrreducibleSpinRepresentationInLorentzSignature)) is freely generated as a [[graded commutative algebra|graded commutative]] [[superalgebra]] on

* elements $\{e^a\}_{a = 1}^{11}$ and $\{\omega^{ a b}\}$ of degree $(1,even)$;

* and elements $\{\psi^\alpha\}_{\alpha = 1}^{32}$ of degree $(1,odd)$

and as a [[differential graded algebra]] its [[differential]] $d_{CE}$ is determined by the equations

$$
  d_{CE} \, \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}
$$

$$
  d_{CE} \, e^{a } = \omega^a{}_b \wedge e^b + \frac{i}{2}\bar \psi \Gamma^a \psi
  \,.
$$

An algebra homomorphism as above sends these generators to differential forms of the corresponding degree, the [[vielbein]]

$$
  E^a \coloneqq A(e^a) \in \Omega^{(1,even)}(U)
  \,,
$$

whe [[spin connection]]

$$
  \Omega^{a}{}_{b} \coloneqq A(\omega^a{}_b) \in \Omega^{(1,even)}(U)
$$

and the [[gravitino]]

$$
  \Psi^\alpha \coloneqq A(\psi^\alpha) \in \Omega^{(1,odd)}(U)
  \,.
$$

But a key aspect of higher dimensional [[supergravity]] theories is that their [[field (physics)|field]] content necessarily includes, in addition to the [[graviton]] and the [[gravitino]], higher [[differential n-form]] fields, notably the 2-fom [[B-field]] of 10-dimensional [[type II supergravity]] and [[heterotic supergravity]] as well as the 3-form [[C-field]] of [[11-dimensional supergravity]].

This means that these higher dimensional supergravity theories are not in fact entirely described  by super-[[Cartan geometry]]. This is to be contrasted with the fact that the very motivation for Cartan geometry, in the original article ([Cartan 23](#Cartan23)), was the mathematical formulation of the [[theory (physics)|theory]] of [[gravity]] ([[general relativity]]).

Now a key insight due to ([D'Auria-Fré-Regge 80](#DAuriaFreRegge80), [D'Auria-Fré 82](#DAuriaFre82)) was that the "tensor multiplet" fields of higher dimensional supergravity theories as above are naturally brought into the previous perspective if only one allows more general [[Chevalley-Eilenberg algebras]].

Namely, we may add to the above CE-algabra

* a single generator $c_3$ of degree $(3,even)$

and extend the differential to that by the formula

$$
  d_{CE} \, c_3 = \frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b
  \,.
$$

This still squares to zero due to the remarkable property of 11d [[super Minkowski spacetime]] by which $\frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b \in CE^4(\mathfrak{Iso}(10,1|N=1))$ is a representative of an exception [[super Lie algebra|super]]-[[Lie algebra cohomology]] class. (The collection of all these exceptional classes constitutes what is known as the _[[brane scan]]_).

In the textbook ([Castellani-D'Auria-Fré 91](#CastellaniDAuriaFre91)) a beautiful algorithm for constructing and handling higher supergravity theories based on such generalized CE-algebras is presented, but it seems fair to say that the authors struggle a bit with the right mathematical perspective to describe what is really happening here.

But from a modern perspective this becomes crystal clear: these generalized CE algebras are CE-algebras not of [[Lie algebras]] but of [[strong homotopy Lie algebra]], hence of [[L-infinity algebras]], in fact of [[Lie n-algebras|Lie (p+1)-algebras]] for $(p+1)$ the degree of the relevant differential form field.

Specifically, we may write the above generalized CE-algebra with the extra degree-3 generator $c_3$ as the CE-algebra $CE(\mathfrak{m}2\mathfrak{brane}) $

of the _[[supergravity Lie 3-algebra]]_ $\mathfrak{m}2\mathfrak{brane}$.

Now a morphism

$$
  \Omega^\bullet(U) \stackrel{}{\longleftarrow} CE(\mathfrak{m}2\mathfrak{brane}) \;\colon\; A
$$

encodes [[graviton]] and [[gravitino]] fields as above, but in addition it encodes a 3-form

$$
  C_3 \coloneqq A(c_3) \in \Omega^{(3,even)}(U)
$$

whose [[curvature]]

$$
  G_4 = \mathbf{d}C_3 + \frac{1}{2}\bar \Psi \Gamma^{a b} \wedge \Psi \wedge E_a \wedge E_b
$$

satisfies a modified [[Bianchi identity]], crucial for the theory of [[11-dimensional supergravity]] ([D'Auria-Fré 82](#DAuriaFre82)).

So this collection of differential form data is no longer a [[Lie algebra valued differential form]], it is an [[L-infinity algebra valued differential form]], with values in the [[supergravity Lie 3-algebra]].

The quotient

$$
  \widehat{\mathbb{R}}^{10,1|N=1}
  \coloneqq
  \mathfrak{g}/\mathfrak{h}
  =
  \mathfrak{m}2\mathfrak{brane} / \mathfrak{o}(\mathbb{R}^{10,1|N=1})
$$

is known as _[[extended super Minkowski spacetime]]_.

The [[Lie integration]] of this is a [[smooth infinity-group|smooth 3-group]] $G$ which receives a map from the [[Lorentz group]].

This means that a global description of the geometry which ([Castellani-D'Auria-Fré 91](#CastellaniDAuriaFre91)) discuss locally on [[charts]] has to be a higher kind of Cartan geometry which is locally modeled not just on [[cosets]], but on the [[homotopy quotients]] of ([[smooth infinity-group|smooth]], [[smooth super infinity-groupoid|supergeometric]], ...) [[infinity-groups]].

### Definite higher WZW terms on stacky Cartan geometries
 {#DefiniteWZWTermsOnStackyCartanGeometries}

Once such a higher Cartan super-spacetime $X$ as [above](#HigherCartanConnectionsAndStackyCartanGeometries) has been obtained, then we are back to the [above](#MotivationDefiniteGlobalizationOfWZWTerms) question of constructing [[definite globalizations of WZW terms]] over it.

Indeed, the [[super p-brane sigma-models]] of the [[D-branes]] and the [[M5-brane]] have [[WZW terms]] defined not on plain [[super Minkowski spacetimes]], but on the above [[extended super Minkowski spacetimes]].  For instance the WZW term of the [[M5-brane]] sigma model is a [[higher prequantization]] of the following 7-form  ([D'Auria-Fré 82](#DAuriaFre82))

$$
  \omega_7
  \;\coloneqq\;
  \frac{1}{2} \bar \psi \wedge \Gamma^{a_1 \cdots a_5} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_5}
   +
  \frac{13}{2} \bar \psi \Gamma^{a_1 a_2} \psi \wedge e_{a_1} \wedge e_{a_2} \wedge c_3
 \;\;\;
 \in
  CE^7(\widehat{\mathbb{R}}^{10,1|N=1})
$$

on the above [[extended super Minkowski spacetime]], where $c_3$ is the extra degree-3 generator discussed above.

Under [[Lie integration]] this becomes ([FSS 13](#FSS13)) a degree-7 WZW term defined on a [[smooth super infinity-groupoid|supergeometric 3-group]] $G/H$ and defining the M5-brane sigma model on a curved supergravity target space means to construct [[definite globalizations]] of this over higher Cartan geometries $X$ modeled on this [[homotopy quotient]] $G/H$.

The result $(X,\mathbf{L}^X_7)$ is a pair which is still analogous to the [[symplectic geometry|symplectic geometries]] that we started with, but is now in higher geometric homotopy theory in every possible sense.

Computing for this case the higher extensions of isometries as [above](#MotivationDefiniteGlobalizationOfWZWTerms), one finds ([dcct, sections 1.2.11.3 and 1.2.15.3.3](#dcct)) the [[quantomorphism n-group]]
for the [[smooth super infinity-groupoid|supergeometric 7-group]] with is the [[Lie integration]] of the [[M-theory Lie algebra]] of $X$, witnessing the degree of $X$ being a "[[BPS state]]" of 11d supergravity. These BPS states are known to be an immensely rich mathematical topic (e.g. via their "[[wall crossing]] phenomena"), but one sees here that it is but the local and infinitesimal shadow of a much richer structure: higher isometries in higher super-Cartan geometry.

In terms of the physics this refinement corresponds to [[classical anomaly]]-cancellation of [[super p-brane sigma models]], a problem that is by and large open.

+-- {: goal}
#### Goal 3 of higher Cartan geometry

Provide [[classical anomaly]] cancellation for [[super p-brane sigma-models]] such as the [[M5-brane]].

=--

## Definition
 {#Definition}

(...)

see at _[[differential cohesion]]_ the section _[structures](differential+cohesive+%28infinity%2C1%29-topos#structures)_.

(...)

## Properties

### Obstruction theorems
 {#ObstructionClassesForDefiniteParameterizations}

(...)

### Higher extended isometry groups (BPS)
 {#HigherExtended IsometryGroups}

(...)

## Examples
 {#Examples}

### Survey
 {#Survey}

[[!include local and global geometry - table]]

### Super-Poincaré-geometry, Super $p$-brane geometry
 {#SuperBraneGeometry}

Notice that ordinary [[gravity]] can be understood as the theory of $(O(d,1) \hookrightarrow Iso(d,1))$-[[Cartan geometry]], where $Iso(d,1)$ is the [[Poincare group]] and $O(d,1)$ the [[orthogonal group]] of [[Minkowski space]]. This is called the [[first order formulation of gravity]].

One can read the [[D'Auria-Fre formulation of supergravity]] as saying that higher dimensional [[supergravity]] is analogously given by higher Cartan supergeometry. See there and see the examples at [[higher Klein geometry]] for more on this.

## Related concepts

* [[higher differential geometry]]

* [[higher Klein geometry]]

## References

Traditional Cartan geometry goes back to

* {#Cartan23} [[Élie Cartan]] _Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie)_. Annales scientifiques de l'École Normale Supérieure, Sér. 3, 40 (1923), p. 325-412  ([NUMDAM](http://www.numdam.org/item?id=ASENS_1923_3_40__325_0))

There is secretly a good bit of higher super-Cartan geometry in the [[supergravity]] textbook

* {#CastellaniDAuriaFre91} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

based on results and observations due to

* {#DAuriaFreRegge80} [[Riccardo D'Auria]], [[Pietro Fré]] [[Tullio Regge]], _Graded Lie algebra, cohomology and supergravity_, Riv. Nuov. Cim. 3, fasc. 12 (1980) ([spire](http://inspirehep.net/record/156191))

* {#DAuriaFre82}  [[Riccardo D'Auria]], [[Pietro Fré]] _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982) 101-140

Mentioning of the need for [[definite globalizations of WZW terms]] is (ever so briefly) in

* {#Witten86} [[Edward Witten]], p. 17 of _Twistor - Like Transform in Ten-Dimensions_, Nucl. Phys. B266 (1986) 245 ([spire](http://inspirehep.net/record/214192/?ln=en))

That there ought to be a systematic study of [[higher Klein geometry]] and higher Cartan geometry has been amplified by [[David Corfield]] since 2006.

Construction of the higher WZW terms on homotopy quotients $G/H$ of higher super-groups is due to

* {#FSS13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_, International Journal of Geometric Methods in Modern Physics, Volume 12, Issue 02 (2015) 1550018 ([arXiv:1308.5264](http://arxiv.org/abs/1308.5264))

with more details in ([dcct](#dcct)).

A formalization of higher Cartan geometry via [[differential cohesion]] is in

* {#dcct} [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_

* [[Urs Schreiber]], _[[schreiber:Higher Cartan Geometry]]_, Harmonic analysis seminar, Charles University Prague, 2015

Formalization of this in [[homotopy type theory]] is in

* {#Wellen17} [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_, 2017

[[!redirects Cartan 2-geometry]]

[[!redirects ∞-Cartan connection]]
[[!redirects ∞-Cartan connections]]

[[!redirects higher Cartan connection]]
[[!redirects higher Cartan connections]]


Higher categories and homotopical algebra

This page is about the book:

* [[Denis-Charles Cisinski]]:

  \linebreak

  **Higher Categories and Homotopical Algebra**

  \linebreak

  Cambridge Studies in Advanced Mathematics **180**

  Cambridge University Press (2019)

  \linebreak

  ISBN: 9781108588737

  [doi:10.1017/9781108588737](https://doi.org/10.1017/9781108588737)

  [pdf](https://cisinski.app.uni-regensburg.de/CatLR.pdf)

  [MAA review](https://www.maa.org/press/maa-reviews/higher-categories-and-homotopical-algebra)

on [[(infinity,1)-category theory|$(\infty,1)$-category theory]] and [[homotopical algebra]].

\linebreak

**Contents**

1\. Prelude

1.1. Presheaves

1.2. The category of simplicial sets

1.3. Cellular filtrations

1.4. Nerves

1.5. Definition of ∞-categories

1.6. The Boardmann-Vogt construction

2\. Basic homotopical algebra

2.1. Factorisation systems

2.2. Model Categories

2.3. Derived functors

2.4. Model structures ex nihilo

2.5. Absolute weak equivalences

3\. The homotopy theory of ∞-categories

3.1. Kan fibrations and the Kan-Quillen model structure

3.2. Inner anodyne extensions

3.3. The Joyal model category structure

3.4. Left or right fibrations, joins and slices

3.5. Invertible natural transformations

3.6. ∞-categories as fibrant objects

3.7. The Boardman-Vogt construction, revisited

3.8. Serre’s long exact sequence

3.9. Fully faithful and essentially surjective functors

4\. Presheaves: externally

4.1. Catégories fibrées en ∞-groupoïdes

4.2. Mapping spaces as fibres of slices

4.3. Final objects

4.4. Grothendieck base change formulas and Quillen’s Theorem A

4.5. Fully faithful and essentially surjective functors, revisited

4.6. Locally constant functors and Quillen’s Theorem B

5\. Presheaves: internally

5.1. Minimal fibrations

5.2. The universal left fibration

5.3. Homotopy classification of left fibrations

5.4. Rectification of morphisms

5.5. Bivariant model category structures

5.6. The twisted diagonal

5.7. Locally small ∞-categories

5.8. The Yoneda Lemma

6\. Adjoints, limits and Kan extensions

6.1. Adjoints

6.2. Limits and colimits

6.3. Extensions of functors by colimits

6.4. Kan extensions

6.5. The Cartesian product

6.6. Fibre products

6.7. Duality

7\. Homotopical algebra

7.1. Localisation

7.2. Calculus of fractions

7.3. Constructions of limits

7.4. Finite direct diagrams

7.5. Derived functors

7.6. Equivalences of ∞-categories with finite limits

7.7. Homotopy completeness

7.8. The homotopy hypothesis

7.9. Homotopy limits as limits

7.10. Mapping spaces in locally small localisations

7.11. Presentable ∞-categories

Bibliography

Notations

Index

\linebreak

category: reference

[[!redirects Higher Categories and Homotopical Algebra]]