geometry of physics -- G-structure and Cartan geometry

> This entry contains one chapter of _[[geometry of physics]]_. See there for background and context

> previous chapter _[[geometry of physics -- manifolds and orbifolds]]_

#Contents#
* table of contents
{:toc}

## **$G$-Structure and Cartan geometry**

### Model Layer

#### $G$-Structure

$$
  \mathbf{B}G \to \mathbf{B}K
$$

given a $K$-[[principal bundle]]

$$
  \array{
    \tilde X &\to &\mathbf{B}K
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    X
  }
$$

a [[reduction of the structure group]] along $G \to K$ is

$$
  \array{
    \tilde X &&\to&& \mathbf{B}G
    \\
    & \searrow &\swArrow_{e}& \swarrow
    \\
    && \mathbf{B}K
  }
$$

#### Examples

##### Vielbein, orthogonal structure, Riemannian geometry
 {#RiemannianGeometry}

* [[vielbein]], [[orthogonal structure]]

[[reduction of the structure group]] along

$\mathbf{B}O(n) \to \mathbf{B}GL(n)$

$$
  \array{
    \tilde X &&\to&& \mathbf{B}O(n)
    \\
    & {}_{\mathllap{\vdash T \Sigma}}\searrow &\swArrow_{e}& \swarrow
    \\
    && \mathbf{B}GL(n)
  }
$$

$e$ is [[vielbein]]: definition of an [[orthonormal frame]]
at each point

###### Electromagnetism in gravitational background

example: the other 2 [[Maxwell equations]]: $\mathbf{d} \star F = j_{el}$.

[[Einstein-Maxwell theory]]

##### Almost complex structure

* [[almost complex structure]]

##### Almost Hermitean structure

* [[almost Hermitean structure]]

##### Almost symplectic structure

* [[almost symplectic structure]]

##### Metaplectic structure

* [[metaplectic structure]]

##### Metalinear structure

* [[metalinear structure]]

##### Generalized complex geometry

* [[generalized complex geometry]]

##### Type II geometry

* [[type II geometry]]

##### Generalized Calabi-Yau structure

* [[generalized Calabi-Yau manifold]]

##### Exceptional generalized geometry

* [[exceptional generalized geometry]]

##### Spin structure, String structure, Fivebrane structure

* [[spin structure]]

* [[string structure]]

* [[fivebrane structure]]

### Semantics Layer

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

Given a homomorphism of groups $G \longrightarrow GL(V)$, a _[[G-structure]]_ on a $V$-manifold $X$ is a lift $\mathbf{c}$ of the [[frame bundle]] $\tau_X$ of prop. \ref{FrameBundle} through this map

$$
  \array{
    X && \stackrel{}{\longrightarrow} && G
    \\
    & {}_{\mathllap{\tau_X}}\searrow &\swArrow& \swarrow
    \\
    && \mathbf{B}GL(V)
  }
  \,.
$$

=--

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

As in remark \ref{ModuliForFramings}, it is useful to express def. \ref{GStructure} in terms of the [[slice topos]] $\mathbf{H}_{/\mathbf{B}GL(V)}$. Write $G\mathbf{Struc}\in \mathbf{H}_{/\mathbf{B}GL(V)}$ for the given map $\mathbf{B}G\to \mathbf{B}GL(V)$ regarded as an object in the slice. Then a $G$-structure according to def. \ref{GStructure} is equivalently a choice of morphism in $\mathbf{H}_{/\mathbf{B}GL(V)}$ of the form

$$
  \mathbf{c} \;\colon\; \tau_X \longrightarrow G\mathbf{Struc}
  \,.
$$

In other words, $G\mathbf{Struc} \in \mathbf{H}_{/\mathbf{B}GL(v)}$ is the _[[moduli stack]]_ for $G$-structures.

=--

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

A choice of [[framing]] $\phi$, def. \ref{Framing}, on a $V$-manifold $X$ induces a [[G-structure]] for any $G$, given by the [[pasting diagram]] in $\mathbf{H}$

$$
  \array{
     X &\longrightarrow& \ast &\longrightarrow&
     \\
     & \searrow & \downarrow & \swarrow
     \\
     && \mathbf{B}GL(V)
  }
$$

or equivalently, via remark \ref{ModuliForFramings} and remark \ref{ModuliForGStructures}, given as the [[composition]]

$$
  \mathbf{c}_{li}
  \;\colon\;
  \tau_X \stackrel{\phi}{\longrightarrow} V\mathbf{Frame} \longrightarrow G\mathbf{Struc}\,.
$$

We call this the _left invariant $G$-structure_.

=--

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

For $X$ a $V$-manifold, then
a [[G-structure]] on $X$, def. \ref{GStructure}, is
_[[integrable G-structure|integrable]]_ if for any $V$-[[atlas]] $V \leftarrow U \rightarrow X$  the pullback of the $G$-structure on $X$ to $V$ is equivalent there to  the left-inavariant $G$-structure on $V$ of example \ref{GStructureFromLeftTranslationFraming}, i.e. if we have an [[correspondence]] in the double [[slice topos]] $(\mathbf{H}_{/\mathbf{B}GL(V)})_{/G\mathbf{Struc}}$ of the form

$$
  \array{
     && \tau_U
     \\
     & \swarrow && \searrow
     \\
     \tau_V && \swArrow && \tau_X
     \\
     & {}_{\mathllap{\mathbf{c}_{li}}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}}
     \\
     && G \mathbf{Struc}
  }
  \,.
$$

The $G$-structure is _infintesimally integrable_ if this holds true at at after restriction along the [[relative shape modality]] $\flat^{rel} U \to U$, def. \ref{RelativeFlat}, to all the infinitesimal disks in $U$:

$$
  \array{
     && \tau_{\flat^{rel}U}
     \\
     & \swarrow && \searrow
     \\
     \tau_V && \swArrow && \tau_X
     \\
     & {}_{\mathllap{\mathbf{c}_{li}}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}}
     \\
     && G \mathbf{Struc}
  }
  \,.
$$

=--

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

Consider an [[infinity-action]] of $GL(V)$ on $V$ which linearizes to the canonical $GL(V)$-action on $\mathbb{D}^V_e$ by def. \ref{GeneralLinearGroup}. Form the [[semidirect product]] $GL(V) \rtimes V$. Consider any group homomorphism $G\to GL(V)$.

A _$(G\to G\rtimes V)$-[[Cartan geometry]]_ is a $V$-manifold $X$ equipped with a $G$-structure, def. \ref{GStructure}. The Cartan geometry is called _(infinitesimally) integrable_ if the $G$-structure is so, according to def. \ref{IntegrabilityOfGStructure}.

=--

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

For $V$ an [[abelian group]], then in traditional contexts the infinitesimal integrability of def. \ref{IntegrabilityOfGStructure} comes down to the [[torsion of a G-structure]] vanishing. But for $V$ a [[nonabelian group]], this definition instead enforces that the torsion is on each [[infinitesimal disk]] the intrinsic left-invariant torsion of $V$ itself.

Traditionally this is rarely considered, matching the fact that ordinary [[vector spaces]], regarded as [[translation groups]] $V$, are [[abelian groups]]. But [[super vector spaces]] regarded (in suitable dimension) as [[super translation groups]] are _[[nonabelian groups]]_ (we discuss this in detail below in _[The super-Klein geometry: super-Minkowski spacetime](#SuperMinkowskiSpacetime)_). Therefore super-vector spaces $V$ may carry intrinsic torsion, and therefore first-order integrable $G$-structures on $V$-manifolds are torsion-ful.

Indeed, this is a phenomenon known as the [[torsion constraints in supergravity]]. Curiously, as discussed there, for the case of [[11-dimensional supergravity]] the [[equations of motion]] of the gravity theory are _equivalent_ to the super-Cartan geometry satisfying this torsion constraint. This way super-Cartan geometry gives a direct general abstract route right into the heart of [[M-theory]].

=--

### Syntax Layer

(...)


geometry of physics -- geometric quantization with KU-coefficients

[[!redirects geometric quantization with KU-coefficients]]

> This entry is a chapter of _[[geometry of physics]]_. See there for context.

> previous chapters: _[[geometry of physics -- modules|modules]]_

***

#Contents#
* table of contents
{:toc}

## Introduction

In the section _[[prequantized Lagrangian correspondences]]_ we had seen how [[classical mechanics]] is captured by [[smooth groupoids]] slices over $\mathbf{B}U(1)$.

We consider this now as the [[boundary field theory]] of the [[2d Poisson-Chern-Simons theory]] in [[smooth infinity-groupoids]] over $\mathbf{B}^2 U(1)$ and discuss [[geometric quantization]] from the point of view of the holographic/[[motivic quantization]] of this system.

That is we consider the [[symplectic groupoid]] $SG(X,\pi)$ of a [[Poisson manifold]] $(X,\pi)$ as equipped with its [[prequantum line n-bundle|prequantum line 2-bundle]]. $SH(X,\pi) \to \mathbf{B}^2 U(1)$. We linearize this to an [[(infinity,1)-module bundle]] by composing with $\mathbf{B}^2 U(1)\to B GL_1(KU)$. Then we perform the boundary [[path integral quantization]] by [[fiber integration in K-theory]] and show how this reproduces, on the boundary, traditional [[geometric quantization]] such as in particular the [[orbit method]].

$$
  \array{
    && X
    \\
    & \swarrow && \searrow
    \\
    \ast && && SG(X,\pi)  && && pre-quantum
    \\
    & \searrow & \swArrow & \swarrow
    \\
    && \mathbf{B}^2 U(1)
    \\
    && \downarrow && && && \downarrow^{\mathrlap{quantization}}
    \\
    && B GL_1(KU)
    \\
    \\
    KU(\ast) &&\longrightarrow && KU(SG(X,\pi)) && && quantum
  }
$$

The abstract picture behind this cohomological [[motivic quantization]] is discussed at _[[dependent linear type theory]]_. Here we focus on explicit details for the case of [[KU]]-quantization.

## KU-theory

### Twisted KU-cohomology of manifolds

([Rosenberg 04](twisted+K-theory#Rosenberg89), [Atiyah-Segal 04](twisted+K-theory#AtiyahSegal04)), review is in ([Nuiten section 3.2.1](#Nuiten))

### Twisted KU-cohomology of smooth local quotient stacks

([FHT I](#FHT)), reviewed in ([Nuiten section 3.3.2](#Nuiten))

Write $H_0$ for the $\mathbb{Z}/2\mathbb{Z}$-graded [[separable Hilbert space]].

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

For $X$ a [[smooth manifold]] and $G$ a [[compact Lie group]] with [[action]] on $X$, then the [[Hilbert bundle]]

$$
  E \coloneqq X \times H_0 \otimes L^2(G) \to X
$$

equipped with the $G$-action on the fibers given on $L^2(G)$ by pullback along the right $G$-action on itself is a universal equivariant Hilbert bundle, meaning that the space of equivariant sections of the associated Fredholm bundle is the $G$-[[equivariant K-theory]] of $X$

$$
  K^\bullet(X//G)
  \simeq
  \Gamma(X, Fred^\bullet(E))
  \,.
$$

=--

### Twisted integration with KU-coefficients

[[fiber integration in K-theory]]

along maps of [[manifolds]] ([CareyWang08](#CareyWang08))

along [[representable morphisms]] of [[local quotient stacks]] ([FHT I](#FHT))

review is in [Nuiten section 4.2](#Nuiten)

## Geometric quantization

[Bongers, section 2](#Bongers)

### Symplectic manifold with Hamiltonian $G$-action

([Nuiten, section 5.2.1](#Nuiten))

### Universal orbit quantization

([FHT II, section 1](#FHT)) ([Nuiten, section 5.2.2](#Nuiten))

## References

* {#Bongers} Stephan Bongers, _[[schreiber:master thesis Bongers|Geometric quantization of symplectic and Poisson manifolds]]_, 2014

* {#Nuiten} [[Joost Nuiten]], _[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]_, 2013

* {#CareyWang08} [[Alan Carey]], [[Bai-Ling Wang]], _Thom isomorphism and push-forward map in twisted K-theory_, J. K-Theory 1(2), 357-393 (2008) ([arXiv:0507414](http://arxiv.org/abs/math/0507414))

* {#FHT} [[Daniel Freed]], [[Mike Hopkins]], [[Constantin Teleman]], _[[Loop Groups and Twisted K-Theory]]_ I (2011) ([arXiv:0711.1906](http://arxiv.org/abs/0711.1906)), II (2013) ([arXiv:0511232](http://arxiv.org/abs/math/0511232))


geometry of physics -- groups

> This entry contains one chapter of _[[geometry of physics]]_.

> previous chapters: _[[geometry of physics -- homotopy types|homotopy types]]_, _[[geometry of physics -- smooth homotopy types|smooth homotopy types]]_, _[[geometry of physics -- stable homotopy types]]_

> next chapters, _[[geometry of physics -- principal bundles|principal bundles]]_, _[[geometry of physics -- representations and associated bundles|representations and associated bundles]]_

***

#Contents#
* table of contents
{:toc}

## Groups

The modern mathematical terminology _group_ is short for _group of [[symmetries]]_ (e.g. [Klein 1872](Klein+geometry#Klein1872)). Mathematicians and physicist tend to marvel at the ubiquity and profoundness that the concept of symmetry groups has turned out to exhibit since its conception in the 19th century. Indeed it is a fundamental concept in a sense whose full depth becomes clear (only) in [[homotopy theory]]: _groups_ are equivalently the [[pointed object|pointed]] [[connected object in an (infinity,1)-topos|connected]] [[homotopy types]].

In traditional literature this fact is fully appreciated typically only in rather advanced corners of [[algebraic topology]], and even that mostly just somewhat secretly. But while it is true that this fact has very sophisticated consequences, at its heart it is a simple fundamental fact that is visible and useful already in elementary [[group theory]] and [[representation theory]].

We being in

* _[1-Groups](#1Groups)_

with a discussion of the simplest case of ordinary discrete groups from this natural perspective of regarding them as pointed connected [[homotopy 1-types]], their [[delooping]] [[groupoids]]. Then we gradually generalize to the study of [[infinity-group]] objects in general [[(infinity,1)-toposes]].

### Model Layer

#### 1-Groups
 {#1Groups}

##### Discrete groups as pointed connected groupoids

We discuss how, via [[looping and delooping]], [[discrete groups]] are equivalent to pointed connected [[groupoids]].

Write

* [[Grpd]] for the [[(2,1)-category]] of [[groupoids]] ([[objects]] are [[groupoids]], [[1-morphisms]] are [[functors]] between these and [[2-morphisms]] are [[natural transformations]] between those, which are nessecarily [[natural isomorphisms]]),

* [[Grp]] for the [[1-category]] of [[groups]] ([[discrete groups]]), also regarded as a [[(2,1)-category]];

* $Grpd^{\ast/}$ for the $(2,1)$-category of [[pointed objects]] in [[Grpd]],

* $Grpd_{\geq 1} \hookrightarrow Grpd$ for the [[full sub-(infinity,1)-category|full sub-(2,1)-category]] on [[connected object in an (infinity,1)-topos|connected]] groupoids, those for which $\pi_0 \simeq \ast$;

* $Grp^{\ast/}_{\geq 1}$ for the [[pointed objects]] in connected groupoids.

* $\pi_1(X,x) \in Grp$ for the [[fundamental group]] of a pointed groupoid $(\ast \stackrel{x}{\to} X) \in Grpd^{\ast/}$ at the given basepoint.

* $\mathbf{B}G \in Grpd$, given a group $G$, for the groupoid $(G\stackrel{\longrightarrow}{\longrightarrow} \ast)$, with composition given by the product in the group. There are two possible choices of conventions, we agree that

  $$
    \array{
      && \ast
      \\
      & {}^{\mathllap{g_1}}\nearrow && \searrow^{\mathrlap{g_2}}
      \\
      \ast && \underset{g_1 \cdot g_2}{\longrightarrow} && \ast
    }
    \,.
  $$

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

The [[(2,1)-category]] $Grp_{\geq 1}$ of connected groupoids is equivalent to its [[full sub-(infinity,1)-category|full sub-(2,1)-category]] on those objects of the form $\mathbf{B}G$, for $G$ a group.

=--

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

Given a connected groupoid $X$, pick any basepoint $x\in X$ and consider the canonical inclusion $\mathbf{B}\pi_1(X,x) \longrightarrow X$. By construction this is [[fully faithful functor|fully faithful]] and by assumption of connectedness it is [[essentially surjective functor|essentially surjective]], hence it is an [[equivalence of groupoids]].

=--

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

The [[hom-groupoids]] between connected groupoids with fundamental groups $G$ and $H$, respectively, are equivalent to the [[action groupoids]] of the set of group [[homomorphisms]] $G \to H$ [[action|acted]] on by [[conjugation]] with elements of $H$:

$$
  Grpd(\mathbf{B}G, \mathbf{B}H)
  \simeq
   Grp(G,H)//_{ad}H
$$

Given two group homomorphisms $\phi_1, \phi_2 \colon G \longrightarrow H$ then an [[isomorphism]] between them in this hom-groupoid is an element $h \in H$ such that

$$
  \phi_2 = Ad_h \circ \phi_1 \coloneqq h^{-1}\cdot \phi_1(-) \cdot h
  \,.
$$

=--

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

By direct inspection of the naturality square for the [[natural transformations]] which are the morphisms in $Grpd(\mathbf{B}G, \mathbf{B}H)$:

$$
  \array{
    \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast
    \\
    \downarrow^{\mathrlap{g_1}} && && \downarrow^{\mathrlap{\phi_1(g_1)}} &&
    \downarrow^{\mathrlap{\phi_2(g_1)}}
    \\
    \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast
    \\
    \downarrow^{\mathrlap{g_2}}
    && &&
    \downarrow^{\mathrlap{\phi_1(g_2)}}
    &&
    \downarrow^{\mathrlap{\phi_2(g_2)}}
    \\
    \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast
  }
  \,.
$$

=--

+-- {: .num_remark #piAs2Functor}
###### Remark

The operation of forming $\pi_1$ is equivalently the operation of forming the [[homotopy fiber product]] of the point inclusion with itself, and hence extends to a [[(2,1)-functor]]

$$
  \pi_1 \colon Grpd^{\ast/} \longrightarrow Grp
  \,.
$$

=--

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

Restricted to [[connected object in an (infinity,1)-topos|connected groupoids]] among the pointed groupoids, the functor $\pi_1 \colon Grpd^{\ast/}_{\geq 1} \longrightarrow Grp$ of remark \ref{piAs2Functor} is an [[equivalence of (2,1)-categories]].

=--

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

It is clear that the functor is essentially surjective: for $G$ any [[group]] then  $\pi_1(\mathbf{B}G,\ast) \simeq G$.

The more interesting point to notice is that $\pi_1$ is indeed a fully faithful [[(2,1)-functor]], in that for any $(X,x), (Y,y) \in Grpd^{\ast/}_{\geq 1}$ then the functor

$$
  (\pi_1)_{X,Y}
  \colon
  Grpd^{\ast/}((X,y),(Y,y))
  \longrightarrow
  Grp(\pi_1(X,x), \pi_1(Y,y))
$$

is an [[equivalence of groupoids|equivalence]] of [[hom-groupoids]]. By prop. \ref{SkeletalRepresentativesForConnectedGroupoids} it is sufficient to check this for $X = \mathbf{B}G$ and $Y = \mathbf{B}H$ with their canonical basepoints, hence to check that for any two groups  $G,H$ the functor

$$
  (\pi_1)_{X,Y}
  \;\colon\;
  Grpd^{\ast/}((\mathbf{B}G,\ast),(\mathbf{B}H,\ast))
  \longrightarrow
  Grp(G,H)
$$

is an equivalence.

To see this, observe that, by definition of [[pointed objects]] via the [[undercategory]] under the point, a morphism in $Grpd^{\ast/}$ between groupoids of this form $\mathbf{B}(-)$ is a diagram in $Grp$ (unpointed) of the form

$$
  \array{
    && \ast
    \\
    & \swarrow &\swArrow_{h}& \searrow
    \\
    \mathbf{B}G
    &&
    \underset{\mathbf{B}\phi}{\longrightarrow}
    &&
    \mathbf{B}H
  }
$$

where the [[natural isomorphism]] is equivalently just the choice of an element $h \in H$. Hence these morphisms are pairs $(\phi,h)$ of a group homomorphism and an element of the domain.

We claim that the [[(2,1)-functor]] $\pi_1$ takes such $(\phi,h)$ to the homomorphism $Ad_{h^{-1}} \circ \phi \;\colon\; G \longrightarrow H$. To see this, consider via remark \ref{piAs2Functor} this functor as forming loops:

$$
  \pi_1(\mathbf{B}G,\ast)
  =
  \left\{
    \array{
      && \ast
      \\
      & \swarrow && \searrow
      \\
      \ast && \swArrow_{\mathrlap{g}} && \ast
      \\
      & \searrow && \swarrow
      \\
      && \mathbf{B}G
    }
  \right\}_{g\in G}
  \,.
$$

This shows that on a morphism as above this acts by forming the [[pasting]]

$$
  \array{
    && \ast
    \\
    & \swarrow && \searrow
    \\
    \ast && \swArrow_{\mathrlap{g}} && \ast
    \\
    & \searrow && \swarrow &\swArrow_{\mathrlap{h}}& \searrow
    \\
    && \mathbf{B}G && \underset{\phi}{\longrightarrow} && \mathbf{B}H
  }
  \;\;\;\;
  =
  \:\;\;\;
  \array{
    && && \ast
    \\
    && & \swarrow && \searrow
    \\
    && \ast && \swArrow_{\mathrlap{h\phi(g)h^{-1}}} && \ast
    \\
    & \swarrow & \swArrow_{\mathrlap{h}} & \searrow && \swarrow
    \\
    \mathbf{B}G && \underset{\phi}{\longrightarrow} && \mathbf{B}H
  }
  \,.
$$

Unwinding the [[whiskering]] of [[natural transformations]] here, the claim follows, as indicated by the label of the upper 2-morphisms on the right.

One observes now that these extra labels $h$ are precisely the information that "trivializes" the conjugation action which in prop. \ref{HomsBetweenBGs} prevents the bare set of group homomorphism: a [[2-morphism]] $(\phi_1, h_1) \Rightarrow (\phi_2,h_2)$ in $Grp^{\ast/}$  is a natural isomorphism of groupoids

$$
  \array{
    \mathbf{B}G
    &\stackrel{\phi_1}{\longrightarrow}&
    \mathbf{B}H
    \\
    {}^{\mathllap{id}}\downarrow &\Downarrow^{\mathrlap{h}}& \downarrow^{\mathrlap{id}}
    \\
    \mathbf{B}G
    &\underset{\phi_2}{\longrightarrow}&
    \mathbf{B}H
  }
$$

(encoding a conjugation relation $\phi_2 = Ad_{h} \circ \phi_1$ as above) such that we have the [[pasting]] relation

$$
  \array{
    && \ast
    \\
    & \swarrow &\swArrow_{h_1}& \searrow
    \\
    \mathbf{B}G
    &&
    \stackrel{\phi_1}{\longrightarrow}
    &&
    \mathbf{B}H
    \\
    {}^{\mathllap{id}}\downarrow && \Downarrow^{\mathrlap{h}} && \downarrow^{\mathrlap{id}}
    \\
    \mathbf{B}G
    &&\underset{\phi_2}{\longrightarrow} &&
    \mathbf{B}H
  }
  \;\;\;\;\;
  =
  \;\;\;\;\;
  \array{
    && \ast
    \\
    & \swarrow &\swArrow_{h_2}& \searrow
    \\
    \mathbf{B}G
    &&
    \underset{\phi_2}{\longrightarrow}
    &&
    \mathbf{B}H
  }
  \,.
$$

But this says in components that $h_2 = h_1\cdot h$. Hence there is a _at most one_ morphism in $Grpd^{\ast/}((\mathbf{B}G,\ast),(\mathbf{B}H,\ast))$ from $(\phi_1,h_1)$ to $(\phi_2,h_2)$: it exists if $\phi_2 = Ad_h \circ \phi_1$ and $h_2 = h_1\cdot h$.

But since, by the previous argument, the functor $\pi_1$ takes $(\phi_1,h_1)$ to $Ad_{h_1^{-1}} \circ \phi_1$, this means that such a morphism exists precisely if both $(\phi_1,h_1)$ and $(\phi_2,h_2)$ are taken to the same group homomorphism by $\pi_1$

$$
  Ad_{h_2^{-1}} \circ \phi_2
  =
  Ad_{h^{-1}\cdot h_1^{-1}}\circ \phi_2
  =
  Ad_{h_1^{-1}} \circ \phi_1
  \,.
$$

This establishes that $\pi_1$ is alspo an equivalence on all [[hom-groupoids]].

=--

This proof also shows that $\mathbf{B}(-)$ is in fact the inverse equivalence:

+-- {: .num_cor #1GroupsEquivalentToPointedConnected1Types}
###### Corollary

There is an [[equivalence of (2,1)-categories]] between pointed connected groupoids and plain groups

$$
  Grp
  \stackrel{\underoverset{\simeq}{\pi_1 = \Omega_\ast}{\longleftarrow}}{\underset{\mathbf{B}}{\longrightarrow}}
  Grpd^{\ast/}_{\geq 1}
$$

given by forming [[loop space objects]] and by forming deloopings.

=--

##### Topological and Lie groups

* [[topological group]]

* [[Lie group]]

* [[smooth group]]

#### 2-Groups

With the above perspective on ordinary groups, it is now essentially clear what [[infinity-groups|higher groups]] in [[homotopy theory]] are to be. We discuss this in full generality below, but of course it serves to highlight the first higher case, that of [[2-groups]].

The idea is clear: where a group is equivalently a pointed connected [[homotopy 1-type]]/[[groupoid]] in that it is the [[loop space object]] of such a pointed connected type, a [[2-group]] is equivalently a pointed connected [[homotopy 2-type]]/[[2-groupoid]] in that it is its [[loop space object]].

In general this means that a [[2-group]] is a [[groupoid]] that is equipped with the structure of a group for which the usual axioms ([[associativity]], [[inverses]]) hold (only) up to [[coherence|coherent]] [[homotopy]]. One hence speaks of _[[weak 2-groups]]_.

But it turns out that in this low degree there is not too much space for such weakening to happen. Indeed, every [[2-groupoid]]/[[homotopy 2-type]] has a model by a [[strict 2-groupoid]], see also at _[homotopy hypothesis -- for homotopy 2-types](homotopy+hypothesis#For2Groupoids)_. Accordingly, every (discrete) [[2-group]] is equivalent to one which is a [[groupoid]] equipped with strict group structure, where the axioms hold "on the nose". The algebraic data encoded by such is known as a _[[crossed module]]_ of groups.

##### Discrete

* [[2-group]]

* [[crossed module]]

##### Braided and abelian

* [[braided 2-group]]

* [[symmetric 2-group]]

#### Simplicial groups
 {#SimplicialGroups}

##### Discrete

* [[simplicial group]]

##### Abelian

* [[abelian group]]

##### Topological and Lie simplicial groups

* [[simplicial topological space]]

* [[geometric realization of simplicial topological spaces]]

#### Chain complexes and Spectra

##### Chain complexes and the Dold-Kan correspondence

At _[[geometry of physics -- homotopy types]]_
is discussed how the [[Dold-Kan correspondence]] allows to think of [[chain complexes]] in non-negative degree as [[homotopy types]], namely as [[Kan complexes]] underlying [[simplicial abelian groups]] which are [[equivalence of categories|equivalent]] to the chain complexes, via the [[normalized chain complex]] operation, see the section _[geometry of physics -- homotopy types -- Dold-Kan correspondence](geometry%20of%20physics%20--%20homotopy%20types#DoldKanCorrespondence)_.

But by the discussion [above](#SimplicialGroups) of [[simplicial groups]],
this means that actually the Dold-Kan correspondence identifies chain complexes with a certain class of [[abelian ∞-groups]].

(...)

##### Spectra and the stable Dold-Kan correspondence

While, under the [[Dold-Kan correspondences]],
chain complexes are [[abelian ∞-groups]], they do not exhaust the space of all objects that deserve to be called such. The general concept of [[abelian ∞-groups]]
is the concept called _[[spectrum]]_ in [[algebraic topology]], [[stable homotopy types]].

* [[stable Dold-Kan correspondence]]

#### Group cohomology

##### Discrete group cohomology

One of the remarkable conceptual simplifications brought about by general [[homotopy theory]] pertains to the general concept of [[cohomology]]: effectively every flavor of cohomology that has been considered turns out to be nothing but the theory of [[derived hom space|(infinity,1)-categorical hom spaces]] in a suitable [[(infinity,1)-topos]].

Specifically for the case of [[group cohomology]], this is the following simple statement.

Let $A$ be an [[abelian group]]. For $n \in \mathbb{N}$ write

$$
  (\mathbf{B}^n A)_\bullet \in KanCplx
$$

for the [[Kan complex]] underlying the [[simplicial group]] which is the image of the [[chain complex]] $A[-1]$ concentrated on $A$ in degree $n$ under the [[Dold-Kan correspondence]] $DK \colon Ch_{\bullet \geq 0}(Ab)\stackrel{\simeq}{\longrightarrow} sAb \stackrel{forget}{\longrightarrow} KanCplx$

$$
  (\mathbf{B}^n A)_\bullet
  \coloneqq
  DK(A[-n])
  \,.
$$

For $G$ any [[discrete group]], not necessarily [[abelian group|abelian]], write

$$
  (\mathbf{B}G)_\bullet \in KanCplx
$$

for the [[nerve]] of the [[groupoid]] ($G\stackrel{\longrightarrow}{\longrightarrow} \ast$).

See at _[[geometry of physics -- homotopy types]]_ for detailed discussion of these two constructions.

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

Given a [[discrete group]] $G$, then a degree-$n$ [[cocycle]] in [[group cohomology]] of $G$ with [[coefficients]] in $A$ is equivalently a morphism

$$
  (\mathbf{B}G)_\bullet \longrightarrow (\mathbf{B}^n A)_\bullet
  \,.
$$

A [[homotopy]] between two such morphisms is equivalently a [[coboundary]] between two such cocycles.

The [[group cohomology]] [[cohomology group|group]] of $G$ with coefficients in $A$ is the [[equivalence classes]] of cocycles modulo coboundaries, hence is the connected components of the [[hom-groupoid]]:

$$
  H^n_{Grp}(G,A)
  \simeq
  \pi_0 Hom(\mathbf{B}G, \mathbf{B}^n A)
  \,.
$$

=--

It is instructive to spell this out in low degree.

+-- {: .num_prop #2CocyclesAndCoboundaries}
###### Proposition

Let $G$ be a [[discrete group]] and $A$ an [[abelian group|abelian]] [[discrete group]], regarded as being equipped with the trivial $G$-[[action]].

Then a **group 2-[[cocycle]]** on $G$ with coefficients in $A$ is a [[function]]

$$
  c \colon G \times G \to A
$$

such that for all $(g_1, g_2, g_3) \in G \times G \times G$ it satisfies the [[equation]]

\[
  \label{2CocycleConditionForCentralExtension}
  c(g_1, g_2)
  -
  c(g_1, g_2 \cdot g_3)
  +
  c(g_1 \cdot g_2, g_3)
  -
  c(g_2, g_3)
  =
  0
  \;\;\;\; \in A
\]

(called the **group 2-cocycle condition**).

For $c, \tilde c$ two such cocycles, a **[[coboundary]]** $h \colon c \to \tilde c$ between them is a [[function]]

$$
  h \colon G \to A
$$

such that for all $(g_1,g_2) \in G \times G$ the [[equation]]

\[
  \label{Group2CoboundaryEquation}
  \tilde c(g_1,g_2)
  =
  c(g_1,g_2) + (d h)(g_1,g_2)
\]

holds in $A$, where

$$
  (d h)(g_1, g_2) \coloneqq h(g_1 g_2) - h(g_1) - h(g_2)
$$

is the **group 2-coboundary** encoded by $h$.

The degree-2 **group cohomology** is the set

$$
  H^2_{Grp}(G,A) = 2Cocycles(G,A) / Coboundaries(G,A)
$$

of [[equivalence classes]] of group 2-cocycles modulo group 2-coboundaries. This is itself naturally an [[abelian group]] under pointwise addition of cocycles in $A$

$$
  [c_1] + [c_2] = [c_1 + c_2]
$$

where

$$
  c_1 + c_2 \colon (g_1, g_2) \mapsto c_1(g_1,g_2) + c_2(g_1, g_2)
  \,.
$$

=--

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

The 2-simplices in $(\mathb{B}G)_\bullet$ are

$$
  (\mathbf{B}G)_2
  =
  \left\{
  \left.
  \array{
    && {*}
    \\
    & {}^{g_1}\nearrow && \searrow^{g_2}
    \\
    {*}
    &&\stackrel{g_1 g_2}{\to}&& {*}
  }
  \right|
  g_1, g_2 \in G
  \right\}
  \,,
$$

and the 3-simplices are

$$
  (\mathbf{B}G)_3
  =
  \left\{
  \left.
  \array{
    {*} &&\stackrel{g_2}{\to}&& {*}
    \\
    \uparrow^{\mathrlap{g_1}} &&{}^{g_1 g_2}\nearrow&& \downarrow^{\mathrlap{g_3}}
    \\
    {*}
    &&\stackrel{g_1 g_2 g_3}{\to}&&
    {*}
  }
  \;\;\;\;
  \Rightarrow
  \;\;\;\;
  \array{
    {*} &&\stackrel{g_2}{\to}&& {*}
    \\
    \uparrow^{\,mathrlap{g_1}} &&\searrow^{g_2 g_3}&& \downarrow^{\mathrlap{g_3}}
    \\
    {*}
    &&\stackrel{g_1 g_2 g_3}{\to}&&
    {*}
  }
  \right|
  g_1, g_2, g_3 \in G
  \right\}
  \,.
$$

The 2-simplices in $(\mathb{B}^2 A)_\bullet$ are

$$
  (\mathbf{B}^2 A)_2
  =
  \left\{
  \left.
  \array{
    && {*}
    \\
    & {}^{}\nearrow &\Downarrow^{\mathrlap{c}}& \searrow^{}
    \\
    {*}
    &&\stackrel{}{\to}&& {*}
  }
  \right|
  c\in A
  \right\}
  \,,
$$

and the 3-simplices are

$$
  (\mathbf{B}^2 A)_3
  =
  \left\{
  \left.
  \array{
    {*} &&\stackrel{}{\to}&& {*}
    \\
    \uparrow^{\mathrlap{}} &\swArrow_{c_1}&{}^{}\nearrow&\swArrow_{c_2}& \downarrow^{\mathrlap{}}
    \\
    {*}
    &&\stackrel{}{\to}&&
    {*}
  }
  \;\;\;\;
  \Rightarrow
  \;\;\;\;
  \array{
    {*} &&\stackrel{}{\to}&& {*}
    \\
    \uparrow^{\mathrlap{}} &\swArrow_{c_3}&\searrow^{}&\swArrow_{c_4}& \downarrow^{\mathrlap{}}
    \\
    {*}
    &&\stackrel{}{\to}&&
    {*}
  }
  \right|
  c_1, c_2, c_3 \in A;
  c_4 = c^3 - c_1 - c_2
  \right\}
  \,.
$$

Therefore a homomorphism of [[Kan complexes]]/[[simplicial sets]] $c \colon (\mathbf{B}G)_\bullet \to (\mathbf{B}^2 A)_\bullet$ is in degree 2 a function

$$
  c_2
  \;\;
  :
  \;\;
  \left(
  \array{
    && {*}
    \\
    & {}^{g_1}\nearrow && \searrow^{g_2}
    \\
    {*}
    &&\stackrel{g_2 g_1}{\to}&& {*}
  }
  \right)
  \;\;\;
  \mapsto
  \;\;\;
  \left(
  \array{
    && {*}
    \\
    & {}^{{*}}\nearrow &\Downarrow^{c(g_1,g_2)}& \searrow^{{*}}
    \\
    {*}
    &&\stackrel{{*}}{\to}&& {*}
  }
  \right)
$$

i.e. a map $c \;\colon\; G \times G \to K$. To be a simplicial homomorphism this has to extend to 3-[[simplices]] as:

$$
  \begin{aligned}
  c_3
  \;\;\;
  &:
  \;\;\;
  \left(
  \array{
    {*} && \stackrel{g_2}{\longrightarrow} && {*}
    \\
    \uparrow^{g_1} &&{}^{g_2 g_1}\nearrow&& \downarrow^{g_3}
    \\
    {*}
    && \stackrel{g_3 g_2 g_1}{\longrightarrow} && {*}
  }
  \;\;\;\;
  \Rightarrow
  \;\;\;\;
  \array{
    {*} &&\stackrel{g_2}{\to}&& {*}
    \\
    \uparrow^{g_1} &&\searrow^{g_3 g_2}&& \downarrow^{g_3}
    \\
    {*}
    &&\stackrel{g_3 g_2 g_1}{\to}&&{*}
  }
  \right)
  \\
  & \mapsto
  \left(
  \array{
    {*} &\stackrel{}{\longrightarrow} & &\stackrel{}{\longrightarrow}& {*}
    \\
    \uparrow &\Downarrow^{c(g_1,g_2)}
    &\nearrow&\Downarrow^{c(g_2,g_3)}&
    \downarrow
    \\
    {*}
    &\longrightarrow&&\longrightarrow&{*}
  }
  \;\;\;\;
  \stackrel{}{\Rightarrow}
  \;\;\;\;
  \array{
    {*} &\longrightarrow&&\longrightarrow& {*}
    \\
    \uparrow &\Downarrow^{c(g_1,g_2 g_3)}
    &\searrow &\Downarrow^{c(g_2, g_3)}& \downarrow
    \\
    {*}
    &\longrightarrow&&\longrightarrow&{*}
  }
  \right)
  \end{aligned}
  \,.
$$

Hence this is the cocycle condition.

A similar argument gives the coboundaries.

=--

We discuss now how in the computation of $H^2_{Grp}(G,A)$ one may concentrate on the _normalized_ cocycles.

+-- {: .num_defn #Normalized2Cocycle}
###### Definition

A group 2-cocycle $c \colon G \times G \to A$, def. \ref{2CocyclesAndCoboundaries} is called **normalized** if

$$
  \forall_{g_0,g_1 \in G}
  \;\;
  \left(g_0 = e \;or\; g_1 = e \right)
  \Rightarrow
  \left( c(g_0,g_1) = e \right)
  \,.
$$

=--

+-- {: .num_lemma #2CocycleOngeisSameasOnee}
###### Lemma

For $c \colon G \times G \to A$ a group 2-cocycle, we have for all $g \in G$
that

$$
  c(e,g) = c(e,e) = c(g,e)
  \,.
$$

=--

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

The cocycle condition (eq:2CocycleConditionForCentralExtension) evaluated on

$$
  (g^{-1},  g,  e) \in G^3
$$

says that

$$
 c(g^{-1}, g) + c(e, e) = c(g, e) + c(g^{-1}, g )
$$

hence that

$$
  c(e,e) = c(g, e)
  \,.
$$

Similarly the 2-cocycle condition applied to

$$
  (e,  g,  g^{-1}) \in G^3
$$

says that

$$
  c(e,g) + c(g,g^{-1}) = c(g,g^{-1}) + c(e,e)
$$

hence that

$$
  c(e,g) = c(e,e)
  \,.
$$

=--

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

Every group 2-cocycle $c \colon G \times G \to A$ is
cohomologous to a normalized one, def. \ref{Normalized2Cocycle}.

=--

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

By lemma \ref{2CocycleOngeisSameasOnee} it is sufficient to
show that $c$ is cohomologous to a cocycle $\tilde c$ satisfying
$\tilde c(e,e) = e$.
Now given $c$, Let $h \colon G \to A$ be given by

$$
  h(g) \coloneqq c(g,g)
  \,.
$$

Then $\tilde c \coloneqq c + d c$ has the desired property, with (eq:Group2CoboundaryEquation):

$$
  \begin{aligned}
    \tilde c(e,e)
    & \coloneqq
    (c + d h)(e,e)
    \\
    & =
    c(e,e)  + c(e \cdot e, e \cdot e) - c(e,e) - c(e,e)
    \\
    & =
    0
  \end{aligned}
  \,.
$$

=--

### Semantic Layer
 {#SemanticLayer}

Above in corollary \ref{1GroupsEquivalentToPointedConnected1Types} we had seen that ordinary groups $G$ are equivalent to pointed connected [[homotopy 1-types]], their [[deloopings]] $\ast \to \mathbf{B}G$.

This statement has an immediate generalization to any [[(∞,1)-topos]] $\mathbf{H}$: while one may define [[∞-group]] algebraically as follows, it is most convenient to define them via looping, as below.

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

Given an [[(∞,1)-topos]] $\mathbf{H}$, then [[group object in an (∞,1)-category|group object]] in $\mathbf{H}$ (an [[∞-group]]) is an object $G \in \mathbf{H}$ equipped with the structure of an [[A-∞ algebra]] (i.e. a product operation which satisfies [[associativity]] up to higher [[coherence|coherent]] [[homotopy]]), such that the [[n-truncated object in an (∞,1)-category|0-truncation]] $\tau_0 G$ is an ordinary [[group object]].

=--

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

Given any object $X$ with a base point $x \colon \ast \to X$, then the [[loop space object]] $\Omega_x X$ canonically has the structure of an [[∞-group]], def. \ref{InfinityGroupAsAInfinityObject}, where the product operation is given by concatenation of loops.

=--

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

The [[looping]] operation of example \ref{LoopSpaceObjectAsGroupObject} constitutes an [[equivalence of (∞,1)-categories]]

$$
  Grp(\mathbf{H})
  \stackrel{\overset{\Omega}{\longleftarrow}}{\underoverset{\mathbf{B}}{\simeq}{\longrightarrow}}
  \mathbf{H}_{\geq 1}^{\ast /}
$$

between [[pointed object|pointed]] [[n-connected object in an (∞,1)-topos|connected objects]] in $\mathbf{H}$ and [[∞-group]] objects in $\mathbf{H}$.

The inverse equivalence $\mathbf{B}$ we call the _[[delooping]]_ operation.

=--

For $\mathbf{H} = $ [[∞Grpd]] this is the [[May recognition theorem]]. For general $\mathbf{H}$  this is [[Higher Algebra|Lurie, "Higher Algebra", theorem 5.1.3.6]].

### Syntactic Layer

#### Pointed connected types

* [[connected type]]

#### Identity types of connected types

(...)