infinity-Lie algebra cohomology

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#### Cohomology
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#### $\infty$-Lie theory
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[[!include infinity-Lie theory - contents]]
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#### $\infty$-Chern-Weil theory
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[[!include infinity-Chern-Weil theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The notion of _$\infty$-Lie algebra cohomology_ generalizes the notion of [[Lie algebra cohomology]] from [[Lie algebra]]s to [[∞-Lie algebra]]s.

## Definition

For $\mathbf{H}$ an [[(∞,1)-topos]] over duals of algebras over an abelian [[Lawvere theory]] $T$, we have by the theory of [[function algebras on ∞-stacks]] a [[reflective (∞,1)-subcategory]]

$$
  \mathbf{L} \stackrel{\leftarrow}{\hookrightarrow}
  \mathbf{H}
$$

obtained as the [[localization of an (∞,1)-category|localization]] of $\mathbf{H}$ at morphisms that induces [[isomorphism]]s in [[cohomology]] with coefficients in the canonical [[line object]] $\mathbb{A}$, where the small objects in $\mathbf{L}$ are modeled by duals of [[cosimplicial algebra]]s.

We may think of $\mathbf{L}$ as the [[(∞,1)-category]] of all [[∞-Lie algebroid]]s inside the [[∞-Lie groupoid]]s which are the objects of $\mathbf{H}$. For instance for $T$ the theory of commutative [[associative algebra]]s over a field, the [[monoidal Dold-Kan correspondence]] identified cosimplicial algebras with [[dg-algebra]]s, which we may think of as the [[Chevalley-Eilenberg algebra]]s of the [[∞-Lie algebroid]]s.

An [[∞-Lie algebra]] $\mathfrak{g}$ is a [[connected]] object in $\mathbf{L}$ and **$\infty$-Lie algebra cohomology** is the [[cohomology|intrinsic cohomology]] of $\mathbf{H}$ restricted to $\mathbf{L}$.

Typically $\mathbf{L}$ is [[presentable (∞,1)-category|presented]] by the [[opposite category|opposite]] of a [[function algebras on ∞-stacks|model structure on cosimplicial/cochain algebras]]: the [[Chevalley-Eilenberg algebra]]s of the given [[∞-Lie algebroid]]s. In terms of that model cocycle in $\infty$-Lie algebra cohomology have explicit and familiar algebraic expressions. These we discuss in

* [Explicit definition](#ExplicitDefinition) .

A discussion of details of how exactly this models the general abstract definition is in

* [(∞,1)-Topos theoretic interpretation](#Topos).

### Explicit definition {#ExplicitDefinition}

For $\mathfrak{g}$ an [[∞-Lie algebra]] and $n \in \mathbb{N}$, a [[cocycle]] on $\mathfrak{g}$ in degree $n$ with coefficients in the trivial module is a morphism

$$
  \mu : \mathfrak{g} \to b^{n-1}\mathbb{R}
$$

to the [[line Lie n-algebra]].

Dually this is a [[dg-algebra]] morphism

$$
  CE(\mathfrak{g}) \leftarrow CE(b^{n-1}\mathbb{R}) : \mu
$$

of [[Chevalley-Eilenberg algebra]]s. here $CE(b^{n-1} \mathbb{R})$ is the [[semifree dga]] on a single generator in degree $n$ with vanishing differential. So this is equivalently an element

$$
  \mu \in \wedge^n \mathfrak{g}^*
$$

which is closed in $CE(\mathfrak{g})$. For $\mathfrak{g}$ an ordinary Lie algebra, this latter description reproduces the traditional definition of cocycles in [[Lie algebra cohomology]].

For the moment see

* [[Chevalley-Eilenberg algebra]]

* [[Weil algebra]]

* [[invariant polynomial]]

for more.

### $(\infty,1)$-topos theoretic interpretation {#Topos}

We may understand the above definitions of $\infty$-Lie algebra cocycles as a special case of the general notion of the [[cohomology|intrinsic cohomology]] of an [[(∞,1)-topos]] by embedding $\infty$-Lie algebras as [[infinitesimal space|infinitesimal]] [[∞-Lie group]]s into the [[(∞,1)-topos]] $\mathbf{H} = $ [[?LieGrpd]] of [[∞-Lie groupoids]].

For a general recognition principle of [[homotopy fibers]] in the [[model structure for L-infinity algebras]] see also ([Fiorenza-Rogers-Schreiber 13, theorem 3.1.13](#FiorenzaRogersSchreiber13)).

Recall from [[function algebras on ∞-stacks]] that [[∞-Lie algebroids]] form the [[reflective sub-(∞,1)-category]]

$$
  \mathbf{L}
  \stackrel{\leftarrow}{\hookrightarrow}
  \mathbf{H}
$$

of a corresponding [[(∞,1)-topos]] $\mathbf{H}$ of structure $\infty$-groupoids.

As described at [[?LieGrpd]], one realization of this general situation for genuine $\infty$-Lie groupoids is as follows:

Let [[ThCartSp]] be the [[site]] of [[infinitesimal object|infinitesimally]] thickened [[Cartesian space]]s. This is the site for the [[Cahiers topos]]. Then the [[(∞,1)-category of (∞,1)-sheaves]] $\mathbf{H} = Sh(ThCartSp)$ we may take to be the $(\infty,1)$-topos of
[[synthetic differential ∞-groupoid]]s. We have then a [[simplicial Quillen adjunction]]

$$
  (C^\infty Alg^{\Delta})^{op} \stackrel{\leftarrow}{\hookrightarrow}
  [ThCartSp^{op}, sSet]_{proj,loc}
$$

between the [[opposite category|opposite]] of the [[function algebras on ∞-stacks|model structure on cosimplicial smooth algebras]]. This models the reflective inclusion of [[∞-Lie algebroid]]s into all synthetic differential $\infty$-groupoids

$$
  \infty LieAlg stackrel{\leftarrow}{\hookrightarrow}
  \infty SDGrpd
  \,.
$$

Details on this are at [[function algebras on ∞-stacks]]. But the [[function algebras on ∞-stacks|model structure on cosimplicial smooth algebras]]s is the [[transferred model structure]] of the [[model structure on cosimplicial rings]], and for the following discussion we can essentially just as well use the analogous Quillen adjunction without the smooth structure originally considered by [[Bertrand Toen]]

$$
  (CAlg^\Delta)^{op} \stackrel{\leftarrow}{\hookrightarrow}
  [CAlg,sSet]_{proj,cov}
$$

that is referenced and reviewed in some detail at [[rational homotopy theory in an (∞,1)-topos]].

Notice that the embedding map is just degreewise the [[Yoneda embedding]].

Notice moreover that by the [[monoidal Dold-Kan correspondence]] (see there for details) we have that the dual Dold-Kan functor $\Xi : Ch^\bullet_+ \to Ab^\Delta$ extends to the right adjoint part in a [[Quillen equivalence]] between the opposite of the [[model structure on dg-algebras]] and the opposite [[model structure on cosimplicial rings|model structure on cosimplicial algebras]]

$$
  \Xi^{op} : dgAlg^{op} \stackrel{\simeq_{Quillen}}{\to}
   (CAlg^\Delta)^{op}
  \,.
$$

In total this gives a right Quillen functor

$$
  R :
  dgAlg^{op} \stackrel{\Xi^{op}}{\to} (CAlg^\Delta)^{op}
  \to
  [CAlg, sSet]_{proj,cov}
$$

that models the embedding of $\infty$-Lie algebroids into a [[(∞,1)-topos]] of $\infty$-Lie groupoids. When restricted to [[∞-Lie algebra]]s ($\infty$-Lie algebroids over the point) the difference between the sites $CAlg^{op}$ and [[ThCartSp]] plays no role. In fact for that case we could just as well restrict to a site of only [[infinitesimal space]]s, because all homs from a finite non-thickened space into an infinitesimal space are trivial anyway.

Therefor for $\mathfrak{g}$ and $\mathfrak{h}$ $\infty$-Lie algebras, a [[cocycle]] on $\mathfrak{g}$ with values in $\mathfrak{h}$ is just a morphism

$$
  (c : \mathfrak{g} \to \mathfrak{h}) \in \infty LieAlg \subset \infty LieGrpd
$$

and the [[∞-groupoid]] of cocycles is

$$
  \infty LieGrpd(\mathfrak{g}, \mathfrak{h})
  \simeq
  \infty LieAlg(\mathfrak{g}, \mathfrak{h})
  \,.
$$

Such cocycles are modeled by morphisms in $dgAlg^{op}$ from a cofibrant representative of $\mathfrak{g}$ to a fibrant representative of $\mathfrak{h}$. Since in $dgAlg$ all objects are fibrant, in $dgAlg^{op}$ all objects are cofibrant. The cofibrant objects in the [[model structure on dg-algebras]] are the [[Sullivan algebra]]s $CE(\mathfrak{h})$. In particular for $\mathfrak{h} = b^{n-1}\mathbb{R}$ we have that $CE(b^{n-1}\mathbb{R})$ is a Sullivan algebra, so $b^{n-1} \mathbb{R}$ is fibrant in $dgAlg^{op}$.

In summary, this says that morphisms

$$
  CE(\mathfrak{g}) \leftarrow CE(b^{n-1}\mathbb{R})
$$

indeed model the abstract intrinsic $(\infty,1)$-topos theoretic notion of cocycles in $\infty Lie Algd \subset \infty Lie Grpd$.

### Examples

Special cases of $\infty$-Lie algebra cohomology are of course

* [[Lie algebra cohomology]]

* [[nonabelian Lie algebra cohomology]].

Specific examples include:

* cohomology of [[Poisson Lie algebroids]] is _[[Poisson cohomology]]_.

## Transgression between invariant polynomials and cocycles via Chern-Simons elements

We recall the procedure by which to an [[∞-Lie algebroid]] [[invariant polynomial]] $\omega$ we associate an [[∞-Lie algebroid]] cocycle $\nu$ that is _in transgression with_ $\omega$.

The [[dg-algebra]] of [[invariant polynomial]]s is a sub-dg-alghebra of the [[kernel]] of the canonical morphism $W(\mathfrak{a}) \to CE(\mathfrak{a})$ from the [[Weil algebra]] to the [[Chevalley-Eilenberg algebra]] of $\mathfrak{a}$

$$
  inv(\mathfrak{a}) \subset CE(\Sigma \mathfrak{a}) = ker(W(\mathfrak{a}) \to CE(\mathfrak{a}))
  \,.
$$

From the [[short exact sequence]]

$$
  CE(\Sigma \mathfrak{a}) \to W(\mathfrak{a}) \to
  CE(\mathfrak{a})
$$

we obtain the long exact sequence in [[chain homology and cohomology|cohomology]]

$$
  \cdots
  \to
  H^{n+1}(CE(\mathfrak{a}))
  \stackrel{\delta}{\to} H^{n+2}(CE(\Sigma \mathfrak{a}))
  \to \cdots
  \,.
$$

We say that $\mu \in CE(\mathfrak{a})$ is in transgression with $\omega \in inv(\mathfrak{a}) \subset CE(\Sigma \mathfrak{a})$ if their classes map to each other under the connecting homomorphism $\delta$:

$$
  \delta : [\mu] \mapsto [\omega]
  \,.
$$

The following spells out in detail how one finds to a given invariant polynomial $\omega$ the cocycle that it is in transgression with.

1. We first regard the [[invariant polynomial]] $\omega$ as an element of the [[Weil algebra]] $W(\mathfrak{a})$ under the inclusion $inv(\mathfrak{a}) \hookrightarrow W(\mathfrak{a})$, where, by the very definiton of invariant polynomials, it is closed: $d_{W(\mathfrak{a})} \omega = 0$.

1. then we find an element $cs_\omega \in W(\mathfrak{a})$ with the property that $d_{W(\mathfrak{a})} cs_\omega = \omega$. This is guranteed to exist because $W(\mathfrak{a})$ has trivial cohomology.

1. then we send this element $cs_\omega\in W(\mathfrak{a})$ along the restriction map $W(\mathfrak{a}) \to CS(\mathfrak{a})$ to an elemeent we call $\nu$.

The procedure is illustarted by the following diagram

$$
  \array{
    0 && \omega &\leftarrow & \omega
    \\
    \;\;\uparrow^{\mathrlap{d_{CE(\mathfrak{a})}}}
    &&
    \;\;\uparrow^{\mathrlap{d_{W(\mathfrak{a})}}}
    \\
    \nu &\leftarrow& cs(\omega)
    \\
    \\
    \\
    \\
    CE(\mathfrak{a})
    &\leftarrow&
    W(\mathfrak{a})
    &\leftarrow&
    inv(\mathfrak{a})
  }
$$

From the fact that all morphisms involved respect the differential and from the fact that the image of $\omega$ in $CE(\mathfrak{a})$ vanishes it follows that

* this element $\nu$ satisfies $d_{CE(\mathfrak{a})} \nu = 0$, hence that it is an $\infty$-Lie algebroid cocycle.

* any two different choices of $cs_\omega$ lead to cocylces $\mu$ that are cohomologous.

We say $\nu$ is a cocycle _in transgression with_ $\omega$.
We may call $cs_{\omega}$ here a _Chern-Simons element_ of $\omega$. Because for $A : T X \to \mathfrak{a}$ any collection of [[∞-Lie algebroid valued differential forms]] coming dually from a dg-morphism $\Omega^\bullet(X) \leftarrow W(\mathfrak{a}) : A$ the image $\omega(A)$ of $\omega$ will be a curvature characteristic form and the image $cs_\omega(A)$ its corresponding Chern-Simons form.

In the case where $\mathfrak{g}$ is an ordinary semisimple [[nLab:Lie algebra|Lie algebra]], this reduces to the ordinary study of ordinary Chern-Simons 3-forms associated with $\mathfrak{g}$-valued 1-forms. This is described in the section _Semisimple Lie algebras_ .

### Examples

For $\mathfrak{g}$ an [[semisimple Lie algebra]], the transgression between the [[Killing form]]-[[invariant polynomial]] and the 3-cocycle $\langle -, [-,-] \rangle$ is exhibited by the "ordinary" Chern-Simons element, which gives these [[action functional]] of ordinary [[Chern-Simons theory]].

A [[symplectic Lie n-algebroid]] is an [[∞-Lie algebroid]] equipped with a nondegenerate binary invariant polynomial in degree $n+2$.

Examples are

* [[symplectic manifold]]s

* [[Poisson Lie algebroid]]s

* [[Courant algebroid]]s.

The coresponding Chern-Simons elements exhibiting the transgression of these invariant polynomials give [[action functional]]s for generalized [[Chern-Simons theory]] (see the above entries for more details).

## Extensions {#Extensions}

In any  [[(∞,1)-topos]] with its intrinsic notion of [[cohomology]], a [[cocycle]] $c : X \to \mathbf{B}^{n+1} A$ classifies an _extension_ $\mathbf{B}^n A \to \hat X \to X$. This $\hat X$ is nothing but the [[homotopy fiber]] of $c$, or equivalently the $\mathbf{B}^n A$-[[principal ∞-bundle]] classified by $c$.

After embedding [[∞-Lie algebra]]s into the [[(∞,1)-topos]] of [[∞-Lie groupoid]]s as described [above](#Topos), the same abstract reasoning applies to $\infty$-Lie algebra cocycles and the extensions of $\infty$-Lie algebras that these classify: for $c : \mathfrak{g} \to b^n \mathbb{R} $ a cocycle of $\infty$-Lie algebras, the extension $b^{n-1} \mathbb{R} \to \hat \mathfrak{g} \to \mathfrak{g}$ is the [[homotopy fiber]] of this morphism in [[?LieGrpd]].

> a more systematic discussion is now in the section Cohomology of ∞-Lie algebroids at [[synthetic differential ∞-groupoid]].

For $\mathfrak{g}$ an ordinary [[Lie algebra]], this reproduces the ordinary notions of extensions from [[Lie algebra cohomology]] and [[nonabelian Lie algebra cohomology]].

**Observation**

For $c : \mathfrak{g} \to b^n \mathbb{R}$ an $(n+1)$-cocycle of an $\infty$-Lie algebra $\mathfrak{g}$, the ordinary [[pullback]] in $dgAlg^{op}$

$$
  \array{
    \hat g &\to& inn(b^{n-1}\mathbb{R})
    \\
    \downarrow && \downarrow
    \\
    \mathfrak{g} &\stackrel{c}{\to}& b^n \mathfrak{R}
  }
$$

maps under $R$ to a pullback diagram of simplicial presheaves which exhibits $R(\hat \mathfrak{g})$ as isomorphic to the [[homotopy pullback]] in the [[homotopy category]].

Here the right morphism denotes the dual of the generating cofibration in $dgAlg$, which models the $b^n \mathfrak{R}$-[[universal principal ∞-bundle]].

**Proof**

Being a right Quillen functor, $R$ preserves fibrations and pullbacks, hence

$$
  \array{
    R \hat g &\to& R inn(b^{n-1}\mathbb{R})
    \\
    \downarrow && \downarrow
    \\
    R \mathfrak{g} &\stackrel{R c}{\to}& R b^n \mathfrak{R}
  }
$$

is a pullback of a fibration. Since $[ThCartSp^{op},sSet]_{proj}$ is a [[proper model category|right proper model category]] this is a [[homotopy pullback]], even if $R \mathfrak{g}$ is possibly not fibrant. (The detailed argument for that is reproduced at [[proper model category]].)

Since [[∞-stackification]] preserves finite [[(∞,1)-limit]]s, this is sufficient to deduce that $R \hat \mathfrak{g}$ represents in the [[homotopy category]] $Ho([ThCartSp, sSet]_{proj,cov})$ the [[homotopy fiber]] of $R c : R \mathfrak{g} \to R b^n \mathbb{R}$.

### Examples

* For $\mathfrak{g}$ and $\mathfrak{h}$ ordinary [[Lie algebra]]s, and $der(\mathfrak{h})$ the Lie algebra of [[derivation]]s of $\mathfrak{h}$, a morphism $\mathfrak{g}\to der(\mathfrak{h})$ is a cocycle in [[nonabelian Lie algebra cohomology]] and the extension it classifies is an ordinary [[Lie algebra extension]].

* The [[string Lie 2-algebra]] is the $b \mathbb{R}$-extension of a semisimple [[Lie algebra]] $\mathfrak{g}$ with bilinear [[invariant polynomial]] $\langle -,-\rangle$ corresponding to the 3-cocycle $\langle -,[-,-]\rangle \in CE(\mathfrak{g})$.

* Similarly for the [[supergravity Lie 3-algebra]] and the [[supergravity Lie 6-algebra]] and all the other extensions in [[schreiber:The brane bouquet]].

## Related concepts

A comprehensive discusson of an ambient $\infty$-topos in which $\infty$-Lie algebroid cohomology lives is at

* [[synthetic differential ∞-groupoid]].

Other notions related to $\infty$-Lie algebroid cohomology include

* **$\infty$-Lie algebra cocycle**

* [[Chern-Simons element]]

* [[invariant polynomial]]

* [[signs in supergeometry]]

## References

Discussion of cohomology of $L_\infty$-algebras is in

* Michael Penkava, _$L_\infty$-algebras and their cohomology_ ([arXiv:q-alg/9512014](http://arxiv.org/abs/q-alg/9512014))

The relation between $L_\infty$-cohomology and extension of $L_\infty$-algebras is discussed around theorem 3.8 of

* [[Andrey Lazarev]], _Models for classifying spaces and derived deformation theory_ ([arXiv:1209.3866](http://arxiv.org/abs/1209.3866))

The general structure of the threory of $\infty$-Lie algebroid cohomology and [[transgression]] between $\infty$-Lie algebroid [[invariant polynomial]]s and -cocycles via [[Chern-Simons element]] was given in

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _$L_\infty$-connections_ (web)

A recognition principle for [[homotopy fibers]] of $L_\infty$-homomorphisms appears as theorem 3.1.13 in

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]_ ([arXiv:1304.6292](http://arxiv.org/abs/1304.6292))
 {#FiorenzaRogersSchreiber13}

Discussion of extensions of [[super L-∞ algebras]] based on the [[super Poincare Lie algebra]] is in

* [[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]]_

The hypercohomology of the Chevalley-Eilenberg-de Rham complex of a Lie algebroid L over a scheme with coefficients in an L-module can be expressed as a [[derived functor]] as shown in

* [[Ugo Bruzzo]], _Lie algebroid cohomology as a derived functor_ ([arxiv/1606.02487](http://arxiv.org/abs/1606.02487))

[[!redirects ∞-Lie algebra cohomology]]
[[!redirects L-∞-algebra cohomology]]
[[!redirects L-∞ algebra cohomology]]
[[!redirects Lie ∞-algebra cohomology]]

[[!redirects L-infinity algebra cohomology]]

[[!redirects ∞-Lie algebra cocycle]]
[[!redirects L-∞-algebra cocycle]]
[[!redirects L-∞ algebra cocycle]]
[[!redirects Lie ∞-algebra cocoycle]]

[[!redirects ∞-Lie algebra cocycles]]
[[!redirects infinity-Lie algebra cocycles]]
[[!redirects L-∞-algebra cocycles]]
[[!redirects L-∞ algebra cocycles]]
[[!redirects Lie ∞-algebra cocoycles]]

[[!redirects ∞-Lie algebroid cohomology]]
[[!redirects L-∞-algebroid cohomology]]
[[!redirects L-∞ algebroid cohomology]]
[[!redirects Lie ∞-algebroid cohomology]]

[[!redirects ∞-Lie algebroid cocycle]]
[[!redirects L-∞-algebroid cocycle]]
[[!redirects L-∞ algebroid cocycle]]
[[!redirects Lie ∞-algebroid cocoycle]]

[[!redirects L-∞ cocycle]]
[[!redirects L-∞ cocycles]]
[[!redirects L-infinity cocycle]]
[[!redirects L-infinity cocycles]]

[[!redirects L-∞ extension]]
[[!redirects L-∞ extensions]]
[[!redirects L-infinity extension]]
[[!redirects L-infinity extensions]]

[[!redirects Lie algebroid cohomology]]
[[!redirects Lie algebroid cocycle]]
[[!redirects Lie algebroid cocycles]]

[[!redirects ∞-Lie algebroid cocycles]]
[[!redirects L-∞-algebroid cocycles]]
[[!redirects L-∞ algebroid cocycles]]
[[!redirects Lie ∞-algebroid cocoycles]]

[[!redirects ∞-Lie algebroid cohomology]]
[[!redirects infinity-Lie algebroid cohomology]]

[[!redirects L-∞ algebra extension]]
[[!redirects L-∞ algebra extensions]]

[[!redirects L-infinity algebra extension]]
[[!redirects L-infinity algebra extensions]]


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

#Contents#
* table of contents
{:toc}

## Idea

For $\mathfrak{g}$ an [[∞-Lie algebra]] (or more generally [[∞-Lie algebroid]]), the **$\infty$-groupoid of $\mathfrak{g}$-valued forms** is the [[∞-groupoid]] whose

* [[object]]s are [[differential forms]] with values in $\mathfrak{g}$;

* [[morphism]]s are [[gauge transformation]]s between these;

* [[k-morphisms]] are order $k$ higher gauge transformations.

This naturally refines to a non-[[concrete sheaf|concrete]] [[∞-Lie groupoid]] whose $U$-parameterized smooth families of objects are [[∞-Lie algebroid]]-valued [[differential form]]s on $Z$.

A [[cocycle]] with coefficients in this is a [[connection on an ∞-bundle]].

For an introduction see the section ∞-Lie algebra valued forms at [[∞-Chern-Weil theory introduction]].

## Definition

For $X$ a [[smooth manifold]] and $\mathfrak{g}$ an [[∞-Lie algebra]] or more generally an [[∞-Lie algebroid]], a  **$\infty$-Lie algebroid valued differential form** on $X$ is a morphism of [[dg-algebra]]s

$$
  \Omega^\bullet(X)
  \leftarrow
  W(\mathfrak{g})
  :
  A
$$

from the [[Weil algebra]] of $\mathfrak{g}$ to the [[de Rham complex]] of $X$. Dually this is a morphism of [[∞-Lie algebroid]]s

$$
  A : T X \to inn(\mathfrak{g})
$$

from the [[tangent Lie algebroid]] to the [[Weil algebra|inner automorphism ∞-Lie algebra]].

Its [[curvature]] is the composite of morphisms of [[graded vector space]]s

$$
  \Omega^\bullet(X) \stackrel{A}{\leftarrow}
  W(\mathfrak{g})
  \stackrel{F_{(-)}}{\leftarrow}
  \wedge^1 \mathfrak{g}^*
  :
  F_{A}
  \,.
$$

Precisely if the curvatures vanish does the morphism factor through the [[Chevalley-Eilenberg algebra]] $W(\mathfrak{g}) \to CE(\mathfrak{g})$.

$$
  (F_A = 0)
  \;\;\Leftrightarrow
  \;\;
  \left(
  \array{
     && CE(\mathfrak{g})
     \\
     & {}^{\mathllap{\exists A_{flat}}}\swarrow & \uparrow
     \\
     \Omega^\bullet(X) &\stackrel{A}{\leftarrow}& W(\mathfrak{g})
  }
  \right)
$$

in which case we call $A$ **flat**.

The [[curvature characteristic forms]] of $A$ are the composite

$$
  \Omega^\bullet(X)
  \stackrel{A}{\leftarrow}
  W(\mathfrak{g})
  \stackrel{\langle F_{(-)} \rangle}{\leftarrow}
  inv(\mathfrak{g})
  :
  \langle F_A\rangle
  \,,
$$

where $inv(\mathfrak{g}) \to W(\mathfrak{g})$ is the inclusion of the [[invariant polynomial]]s.

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

For $U$ a [[smooth manifold]], the **$\infty$-groupoid of $\mathfrak{g}$-valued forms is the [[Kan complex]]

$$
  \exp(\mathfrak{g})_{conn}(U)
  :
  [k]
  \mapsto
  \left\{
     \Omega^\bullet(U \times \Delta^k)
      \stackrel{A}{\leftarrow}
     W(\mathfrak{g})
     \;\;
     |
     \;\;
       \forall v \in \Gamma(T \Delta^k) : \iota_v F_A = 0
  \right\}
$$

whose [[k-morphism]]s are $\mathfrak{g}$-valued forms $A$ on $U \times \Delta^k$ with sitting instants, and with the property that their [[curvature]] vanishes on vertical vectors.

The canonical morphism

$$
  \exp(\mathfrak{g})_{conn} \to \exp(\mathfrak{g})
$$

to the untruncated [[Lie integration]] of $\mathfrak{g}$ is given by restriction of $A$ to [[vertical differential form]]s (see below).

=--

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

Here we are thinking of $U \times \Delta^k \to U$ as a trivial [[bundle]].

The _first_ [[Ehresmann connection|Ehresmann condition]] will be identified with the conditions on lifts $\nabla$ in [[∞-anafunctor]]s

$$
  \array{
    && \exp(\mathfrak{g})_{conn}
    \\
    & {}^{\mathllap{\nabla}}\nearrow & \downarrow
    \\
    C(U) &\stackrel{g}{\to}& \exp(\mathfrak{g})
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    X
  }
$$

that define [[connections on ∞-bundles]]. More on this in the [Properties](#Properties)-section below.

=--

## Properties {#Properties}

### Curvature characteristics

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

For $A \in \exp(\mathfrak{g})_{conn}(U,[k])$ a $\mathfrak{g}$-valued form on $U \times \Delta^k$ and for $\langle - \rangle \in W(\mathfrak{g})$ any [[invariant polynomial]], the corresponding [[curvature characteristic form]] $\langle F_A \rangle \in \Omega^\bullet(U \times \Delta^k)$ descends down to $U$.

=--

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

It is sufficient to show that for all $v \in \Gamma(T \Delta^k)$ we have

1. $\iota_v \langle F_A \rangle = 0$;

1. $\mathcal{L}_v \langle F_A \rangle = 0$.

The first condition is evidently satisfied if already $\iota_v F_A = 0$. The second condition follows with [[Cartan calculus]] and using that $d_{dR} \langle F_A\rangle = 0$:

$$
  \mathcal{L}_v \langle F_A \rangle =
  d \iota_v \langle F_A \rangle
  +
  \iota_v d \langle F_A \rangle
  = 0
  \,.
$$

=--

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

For a general $\infty$-Lie algebra $\mathfrak{g}$ the curvature forms $F_A$ themselves are not closed, hence requiring them to have no component along the simplex does not imply that they descend. This is different for abelian $\infty$-Lie algebras: for them the curvature forms themselves are already closed, and hence are themselves already curvature characteristics that do descent.

=--

It is useful to organize the $\mathfrak{g}$-valued form $A$, together with its restriction $A_{vert}$ to [[vertical differential form]]s and with its [[curvature characteristic form]]s in the [[commuting diagram]] (following _[Weil algebra -- Characterization in the smooth infinity-topos](Weil+algebra#CharacterizationInSmoothTopos)_)

$$
    \array{
      \Omega^\bullet(U \times \Delta^k)_{vert}
      &\stackrel{A_{vert}}{\leftarrow}&
      CE(\mathfrak{g})
      &&&
      gauge\;transformation
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U \times \Delta^k)
      &\stackrel{A}{\leftarrow}&
      W(\mathfrak{g})
      &&&
      \mathfrak{g}-valued\;form
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U)
      &\stackrel{\langle F_A\rangle}{\leftarrow}&
      inv(\mathfrak{g})
      &&&
      curvature\;characteristic\;forms
    }
$$

in [[dgAlg]].

The commutativity of this diagram is implied by $\iota_v F_A = 0$.

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

Write $\exp(\mathfrak{g})_{CW}(U)$ for the $\infty$-groupoid of $\mathfrak{g}$-valued forms fitting into such diagrams.

$$
  \exp(\mathfrak{g})_{CW}(U)
  :
  [k]
  \mapsto
  \left\{
    \array{
      \Omega^\bullet(U \times \Delta^k)_{vert}
      &\stackrel{A_{vert}}{\leftarrow}&
      CE(\mathfrak{g})
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U \times \Delta^k)
      &\stackrel{A}{\leftarrow}&
      W(\mathfrak{g})
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U)
      &\stackrel{\langle F_A\rangle}{\leftarrow}&
      inv(\mathfrak{g})
    }
  \right\}
  \,.
$$

=--

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

If we just consider the top horizontal morphism in this diagram we obtain the object

$$
  \exp(\mathfrak{g})(U)
  :
  [k]
  \mapsto
  \left\{
    \array{
      \Omega^\bullet(U \times \Delta^k)_{vert}
      &\stackrel{A_{vert}}{\leftarrow}&
      CE(\mathfrak{g})
     }
  \right\}
$$

discussed in detail at [[Lie integration]]. If we consider the top square of the diagram we obtain the object

$$
  \exp(\mathfrak{g})_{diff}(U)
  :
  [k]
  \mapsto
  \left\{
    \array{
      \Omega^\bullet(U \times \Delta^k)_{vert}
      &\stackrel{A_{vert}}{\leftarrow}&
      CE(\mathfrak{g})
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U \times \Delta^k)
      &\stackrel{A}{\leftarrow}&
      W(\mathfrak{g})
   }
  \right\}
  \,.
$$

This forms a [[resolution]] of $\exp(\mathfrak{g})$ and may be thought of as the $\infty$-groupoid of [[pseudo-connection]]s.

We have an evident sequence of morphisms

$$
  \array{
    \exp(\mathfrak{g})_{conn} &&& genuine\;connections
    \\
    \downarrow
    \\
    \exp(\mathfrak{g})_{CW} &&& pseudo-connection\;with\;global\;curvature\;characteristics
    \\
    \downarrow
    \\
    \exp(\mathfrak{g})_{diff} &&& pseudo-connections
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \exp(\mathfrak{g}) &&& bare bundles
  }
  \,,
$$

where we label the objects by the structures they classify, as discussed at [[∞-Chern-Weil theory]].

Here the botton morphism is a weak equivalence and the others are [[monomorphism]]s.

Notice that in full [[∞-Chern-Weil theory]] the fundamental object of interest is really $\exp(\mathfrak{g})_{diff}$ -- the object of [[pseudo-connection]]s. The other objects only serve the purpose of picking particularly nice representatives:

the object $\exp(\mathfrak{g})_{CW}$ may contain pseudo-connections, those for which at least the associated [[circle n-bundles with connection]] given by the $\infty$-Chern Weil homomorphism are genuine connections.

This distinction is important: over objects $X \in $ [[?LieGrpd]] that are not [[smooth manifold]]s but for instance [[orbifold]]s, the genuine connections for higher Lie algebras do _not_ exhaust all nonabelian differential cocycles. This just means that not every differential class in this case does have a nice representative in the usual sense.

=--

### 1-Morphisms: integration of infinitesimal gauge transformations {#InfGaugeTrafo}

The 1-[[morphism]]s in $\exp(\mathfrak{g})(U)$ may be thought of as [[gauge transformation]]s between $\mathfrak{g}$-valued forms. We unwind what these look like concretely.

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

Given a 1-morphism in $\exp(\mathfrak{g})(X)$, represented by $\mathfrak{g}$-valued forms

$$
  \Omega^\bullet(U \times \Delta^1)
  \leftarrow
  W(\mathfrak{g})
  :
  A
$$

consider the unique decomposition

$$
  A = A_U + ( A_{vert} := \lambda \wedge d t)  \; \;
  \,,
$$

with $A_U$ the horizonal differential form component and $t : \Delta^1 = [0,1] \to \mathbb{R}$ the canonical coordinate.

We call $\lambda$ the **gauge parameter** . This is a function on $\Delta^1$ with values in 0-forms on $U$ for $\mathfrak{g}$ an ordinary [[Lie algebra]], plus 1-forms on $U$ for $\mathfrak{g}$ a [[Lie 2-algebra]], plus 2-forms for a Lie 3-algebra, and so forth.

=--

We describe now how this enccodes a gauge transformation

$$
  A_0(s=1) \stackrel{\lambda}{\to} A_U(s = 1)
  \,.
$$

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

By the nature of the [[Weil algebra]] we have

$$
  \frac{d}{d s} A_U
  =
  d_U \lambda + [\lambda \wedge A] + [\lambda \wedge A \wedge A] + \cdots
  +
  \iota_s F_A
  \,,
$$

where the sum is over all higher brackets of the [[∞-Lie algebra]] $\mathfrak{g}$.

=--

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

In [[Cartan calculus]] for $\mathfrak{g}$ an ordinary Lie algebra may write (see [here](Ehresmann+connection#FormulationInTermsOfCurvatureForm)) the corresponding  [[Ehresmann connection|second Ehresmann condition]] $\iota_{\partial_s} F_A = 0$ equivalently

$$
  \mathcal{L}_{\partial_s} A = ad_\lambda A
  \,.
$$

=--

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

Define the **[[covariant derivative]] of the gauge parameter** to be

$$
  \nabla \lambda := d \lambda + [A \wedge \lambda] + [A \wedge A \wedge \lambda] + \cdots
  \,.
$$

=--

In this notation we have

* the general identity

  \[
    \frac{d}{d s} A_U = \nabla \lambda + (F_A)_s
    \label{ShiftedGaugeTrafo}
  \]

* the **horizontality** or **[[Ehresmann connection|second Ehresmann condition]]** (or "strict [rheonomy](D%27Auria-Fre+formulation+of+supergravity#Rheonomy)") $\iota_{\partial_s} F_A = 0$, the [[differential equation]]

  \[
    \frac{d}{d s} A_U = \nabla \lambda
    \label{GaugeTrafo}
    \,.
  \]

This is known as the equation for **infinitesimal [[gauge transformation]]s** of an $\infty$-Lie algebra valued form.

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

By [[Lie integration]] we have that $A_{vert}$ -- and hence $\lambda$ -- defines an element $\exp(\lambda)$ in the [[∞-Lie group]] that integrates $\mathfrak{g}$.

The unique solution $A_U(s = 1)$ of the above [[differential equation]] at $s = 1$ for the initial values $A_U(s = 0)$ we may think of as the result of acting on $A_U(0)$ with the [[gauge transformation]] $\exp(\lambda)$.

=--

## Examples

### Lie algebra valued 1-forms

+-- {: .num_prop}
###### Proposition
**(connections on ordinary bundles)**

For $\mathfrak{g}$ an ordinary [[Lie algebra]] with simply connected [[Lie group]] $G$ and for $\mathbf{B}G_{conn}$ the [[groupoid of Lie algebra-valued forms]] we have an [[isomorphism]]

$$
  \tau_1 \exp(\mathfrak{g})_{conn} = \mathbf{B}G_{conn}
$$

=--

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

To see this, first note that the sheaves of objects on both sides are manifestly isomorphic, both are the sheaf of $\Omega^1(-,\mathfrak{g})$. For morphisms, observe that for a form $\Omega^\bullet(U \times \Delta^1) \leftarow W(\mathfrak{g}) : A$ which we may decompose into a horizontal and a verical pice as $A = A_U + \lambda \wedge d t$ the condition $\iota_{\partial_t} F_A = 0$ is equivalent to the [[differential equation]]

$$
  \frac{\partial}{\partial t} A
  =
  d_U \lambda + [\lambda, A]
  \,.
$$

For any initial value $A(0)$ this has the unique solution

$$
  A(t) = g(t)^{-1} (A + d_{U}) g(t)
  \,,
$$

where $g : [0,1] \to G$ is the [[parallel transport]] of $\lambda$:

$$
  \begin{aligned}
    &
    \frac{\partial}{\partial t}
    \left(
       g_(t)^{-1} (A + d_{U}) g(t)
     \right)
     \\
     = &
     g(t)^{-1} (A + d_{U}) \lambda g(t)
     -
     g(t)^{-1} \lambda (A + d_{U}) g(t)
   \end{aligned}
$$

(where for ease of notaton we write actions as if $G$ were a [[matrix Lie group]]).

In particular this implies that the endpoints of the path of $\mathfrak{g}$-valued 1-forms are related by the usual cocycle condition in $\mathbf{B}G_{conn}$

$$
  A(1) = g(1)^{-1} (A + d_U) g(1)
  \,.
$$

In the same fashion one sees that given 2-cell in $\exp(\mathfrak{g})(U)$ and any 1-form on $U$ at one vertex, there is a unique lift to a 2-cell in $\exp(\mathfrak{g})_{conn}$, obtained by parallel transporting the form around. The claim then follows from the previous statement of [[Lie integration]] that $\tau_1 \exp(\mathfrak{g}) = \mathbf{B}G$.

=--

### Lie 2-algebra valued forms

* For $\mathfrak{g}$ [[Lie 2-algebra]], a $\mathfrak{g}$-valued differential form in the sense described here is precisely an [[Lie 2-algebra valued form]].

### Ordinary $n$-forms and the de Rham complex

For $n \in \mathbb{N}$, $n \geq 1$ we have that $b^{n-1}\mathbb{R}$-valued differential forms are in natural bijection to ordinary closed [[differential form]]s in degree $n$

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

Notice that under addition of differential forms, $\exp(b^{n-1}\mathbb{R})_{conn}$ is over each $U \in CartSp$ an abelian [[simplicial group]].

Under the [[Dold-Kan correspondence]] $Ch_\bullet^+ \stackrel{\overset{N}{\leftarrow}}{\underset{\Xi}{\to}} sAb$ we may therefore identify $\exp(b^{n-1}\mathbb{R})_{conn}$ with a presheaf $N \exp(b^{n-1}\mathbb{R})_{conn}$ of chain complexes.

=--

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

The degreewise [[fiber integration]] of differential forms over simplices
constitutes a morphism

$$
  \int_{\Delta^\bullet} :
  N\exp(b^{n-1}\mathbb{R})_{conn}
  \to
  \left(
    C^\infty(-, \mathbb{R})
    \stackrel{d_{dR}}{\to}
    \Omega^1(-)
    \stackrel{d_{dR}}{\to}
    \cdots
    \stackrel{d_{dR}}{\to}
    \Omega^n(-)
  \right)
  \,.
$$

that is a weak equivalence.

=--

This is shown at circle n-bundle with connection -- from Lie intgeration based on the discussion at ∞-Lie groupoid -- Lie-integrated ∞-groups -- differential coefficients.

### Supergravity fields

What is called an  "extended soft group manifold" in the literature on the [[D'Auria-Fre formulation of supergravity]] is really precisely a collection of $\infty$-Lie algebroid valued forms with values in a super $\infty$-Lie algebra such as the [[supergravity Lie 3-algebra]] (for 11-dimensional [[supergravity]]). The way [[curvature]] and [[Bianchi identity]] are read off from "extded soft group manifolds" in this literature is -- apart from this difference in terminology -- precisely what is described above.

## Related concepts

* [[groupoid of Lie-algebra valued forms]]

* [[2-groupoid of Lie 2-algebra valued forms]]

* [[3-groupoid of Lie 3-algebra valued forms]]

* **$\infty$-groupoid of ∞-Lie-algebra valued forms**

## References

The (obvious but conceptually important) observation that [[Lie algebra-valued 1-forms]] regarded as morphisms of [[graded vector space]]s $\Omega^\bullet(X) \leftarrow \wedge^1 \mathfrak{g}^* : A$ are equivalently morphisms of dg-algebras out of the [[Weil algebra]] $\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A$ and that one may think of as the identity $W(\mathfrak{g}) \leftarrow W(\mathfrak{g}) : Id$ as the _universal $\mathfrak{g}$-connection_ appears in early articles for instance highlighted on p. 15 of

* Franz W. Kamber; Philippe Tondeur, _Semisimplicial Weil algebras and characteristic classes for foliated bundles in Čech cohomology_ , Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, pp. 283--294. Amer. Math. Soc., Providence, R.I., (1975).

following [[Eli Cartan]]'s influential work (see [[Weil algebra]] for more references).

The (evident) generalization to Weil algebras of [[∞-Lie algebra]]s and [[∞-Lie algebroid]]s is considered explicitly in

* Hisham Sati, Urs Schreiber, Jim Staasheff, _$L_\infty$-algebra valued connections_ (web)

but -- somewhat implicitly -- this construction appears earlier, notably in the [[D'Auria-Fre formulation of supergravity]]. A collection of such precursors to these notions is collected at

* [[schreiber:differential cohomology in an (∞,1)-topos -- references]]: ∞-Lie algebra valued forms

The structure of the formula \eqref{GaugeTrafo} for infinitesimal gauge transformations of higher forms is widely known in the literature on [[supergravity]] and [[string theory]], if maybe not formalized in terms of $\infty$-Lie algebra theory as we do here. One exception is the remarkable book

*  [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fre]], _[[Supergravity and Superstrings - A Geometric Perspective]]_ .

In this old book no $\infty$-Lie algebras are mentioned explicitly, but the [[dg-algebra]] computations that are considered are easily seen to be precisely their [[Chevalley-Eilenberg algebra]]-incarnations.

The authors use the term _extended [[soft group manifold]]_ for what here we identify as an $\infty$-Lie algebra valued form $T X \to inn(\mathfrak{g})$.

With this terminological translation understood, and observing that all their constructions straightforwardly generalize to more general dg-algebras than these authors conisder explicitly, we find that

* our equation \eqref{ShiftedGaugeTrafo} for the possibly shifted gauge transformation is their equation I.3.136;

* our equation \eqref{GaugeTrafo} for the genuine gauge transfomation is their equation for _horizontal_ or [[rheonomy|rheonomic]] gauge transformations III.3.23 .

In fact their full rheonomy constraint III.3.32 is essentialy the same horizontality constraint, but applied not just to the 1-simplex $\Delta^1$, but to the [[supermanifold|super simplex]] $\Delta^{1|p}$.

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infinity-Lie group

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

#Contents#
* table of contents
{:toc}

## Definition

An _smooth $\infty$-group_ is a [[group object in an (∞,1)-category|group object]] in the [[cohesive (∞,1)-topos]] [[Smooth∞Grpd]] of [[smooth ∞-groupoid]]s.

## Examples

### Lie groups

An ordinary [[Lie group]] is a 0-[[truncated]] $\infty$-Lie group.

### Lie 2-groups

A [[Lie 2-group]] is a 1-[[truncated]] $\infty$-Lie group.

* [[string 2-group]]

### Lie 6-groups

A Lie 6-group is a 5-[[truncated]] $\infty$-Lie group.

* [[fivebrane 6-group]].

## Related concepts

* [[∞-group]]

* [[higher gauge theory]]

  * [[higher gauge field]]

  * [[higher gauge transformation]]

* [[cohesive (∞,1)-topos]]

  * [[discrete ∞-groupoid]]

  * [[Euclidean-topological ∞-groupoid]]

  * [[smooth ∞-groupoid]]

  * [[synthetic differential ∞-groupoid]]

## References

See [[smooth ∞-groupoid]].

[[!redirects ∞-Lie group]]

[[!redirects ∞-Lie groups]]
[[!redirects infinity-Lie groups]]

[[!redirects Lie n-group]]
[[!redirects Lie n-groups]]

[[!redirects Lie ∞-group]]
[[!redirects Lie ∞-groups]]


infinity-Lie theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Higher Lie theory_ is the refinement of [[Lie theory]] to [[higher geometry]] ([[higher differential geometry]]).  It studies [[smooth ∞-groupoids]] and [[L-∞ algebroids]] and their relation via [[Lie integration]] and [[Lie differentiation]].

## Related concepts

* [[higher structures]]

* [[higher differential geometry]]

  * [[motivation for higher differential geometry]]

[[!redirects ∞-Lie theory]]

[[!redirects higher Lie theory]]


infinity-Lie theory - contents

**[[∞-Lie theory]]** ([[higher geometry]])

**Background**

*Smooth structure*

* [[generalized smooth space]]

  * [[smooth manifold]]

  * [[diffeological space]]

  * [[Frölicher space]]

* [[smooth topos]]

  * [[Cahiers topos]]

* [[smooth ∞-groupoid]], [[concrete smooth ∞-groupoid]]

* [[synthetic differential ∞-groupoid]]

*Higher groupoids*

* [[∞-groupoid]]

  * [[groupoid]]

  * [[2-groupoid]]

  * [[strict ∞-groupoid]]

    * [[crossed complex]]

* [[∞-group]]

  * [[simplicial group]]

*Lie theory*

* [[Lie theory]]

  * [[Lie integration]], [[Lie differentiation]]

  * [[Lie's three theorems]]

  * [[Lie theory for stacky Lie groupoids]]

**∞-Lie groupoids**

* [[∞-Lie groupoid]]

  * [[∞-Lie groupoid|strict ∞-Lie groupoid]]

  * [[Lie groupoid]]

    * [[differentiable stack]]

    * [[orbifold]]

* [[∞-Lie group]]

  * [[Lie group]]

    * [[simple Lie group]], [[semisimple Lie group]]

  * [[Lie 2-group]]

**∞-Lie algebroids**

* [[∞-Lie algebroid]]

  * [[Lie algebroid]]

* [[Lie ∞-algebroid representation]]

* [[L-∞-algebra]]

  * [[model structure for L-∞ algebras]]: [[model structure on dg-Lie algebras|on dg-Lie algebras]],  [[model structure on dg-coalgebras|on dg-coalgebras]], [[model structure on simplicial Lie algebras|on simplicial Lie algebras]]

  * [[Lie algebra]]

    * [[semisimple Lie algebra]], [[compact Lie algebra]]

  * [[Lie 2-algebra]]

    * [[strict Lie 2-algebra]]

    * [[differential crossed module]]

* [[Lie 3-algebra]]

  * [[differential 2-crossed module]]

* [[dg-Lie algebra]], [[simplicial Lie algebra]]

* [[super L-∞ algebra]]

  * [[super Lie algebra]]

**Formal Lie groupoids**

* [[formal group]], [[formal groupoid]]

**Cohomology**

* [[Lie algebra cohomology]]

  * [[Chevalley-Eilenberg algebra]]

  * [[Weil algebra]]

  * [[invariant polynomial]]

    * [[Killing form]]

* [[nonabelian Lie algebra cohomology]]

**Homotopy**

* [[homotopy groups of a Lie groupoid]]

**Related topics**

* [[∞-Chern-Weil theory]]

**Examples**

*$\infty$-Lie groupoids*

* [[Atiyah Lie groupoid]]

* [[fundamental ∞-groupoid]]

  * [[path groupoid]]

  * [[path n-groupoid]]

* [[principal ∞-bundle|smooth principal ∞-bundle]]

*$\infty$-Lie groups*

* [[orthogonal group]]

  * [[special orthogonal group]]

  * [[spin group]]

  * [[string 2-group]]

  * [[fivebrane 6-group]]

* [[unitary group]]

  * [[special unitary group]]

* [[∞-Lie groupoid|circle Lie n-group]]

  * [[circle group]]

*$\infty$-Lie algebroids*

* [[tangent Lie algebroid]]

* [[action Lie algebroid]]

* [[Atiyah Lie algebroid]]

* [[symplectic Lie n-algebroid]]

  * [[symplectic manifold]]

  * [[Poisson Lie algebroid]]

  * [[Courant Lie algebroid]]

    * [[generalized complex geometry]]

*$\infty$-Lie algebras*

* [[general linear Lie algebra]]

* [[orthogonal Lie algebra]], [[special orthogonal Lie algebra]]

* [[endomorphism L-∞ algebra]]

* [[automorphism ∞-Lie algebra]]

* [[string Lie 2-algebra]]

* [[fivebrane Lie 6-algebra]]

* [[supergravity Lie 3-algebra]]

* [[supergravity Lie 6-algebra]]

* [[line Lie n-algebra]]


infinity-Lie-groupoid > history

< [[infinity-Lie-groupoid]]
[[!redirects infinity-Lie-groupoid -- history]]


infinity-limits - contents

**[[limit|limits and colimits]]**

## 1-Categorical

* [[limit|limit and colimit]]

  * [[limits and colimits by example]]

  * [[commutativity of limits and colimits]]

  * [[small limit]]

  * [[filtered colimit]]

    * [[directed colimit]]

      * [[sequential colimit]]

  * [[sifted colimit]]

  * [[connected limit]], [[wide pullback]]

  * [[preserved limit]], [[reflected limit]], [[created limit]]

  * [[product]], [[fiber product]], [[base change]], [[coproduct]], [[pullback]], [[pushout]], [[cobase change]], [[equalizer]], [[coequalizer]], [[join]], [[meet]], [[terminal object]], [[initial object]], [[direct product]], [[direct sum]]

  * [[finite limit]]

    * [[exact functor]]

* [[Kan extension]]

  * [[Yoneda extension]]

* [[weighted limit]]

* [[end|end and coend]]

* [[fibered limit]]

## 2-Categorical

* [[2-limit]]

  * [[inserter]]

  * [[isoinserter]]

  * [[equifier]]

  * [[inverter]]

  * [[PIE-limit]]

* [[2-pullback]], [[comma object]]

## (∞,1)-Categorical

* [[(∞,1)-limit]]

  * [[(∞,1)-pullback]]

    * [[fiber sequence]]

### Model-categorical

* [[homotopy Kan extension]]

* [[homotopy limit]]

  * [[homotopy product]]

  * [[homotopy equalizer]]

  * [[homotopy fiber]]

  * [[mapping cone]]

  * [[homotopy pullback]]

  * [[homotopy totalization]]

  * [[homotopy end]]

* [[homotopy colimit]]

  * [[homotopy coproduct]]

  * [[homotopy coequalizer]]

  * [[homotopy cofiber]]

  * [[mapping cocone]]

  * [[homotopy pushout]]

  * [[homotopy realization]]

  * [[homotopy coend]]

[Edit this sidebar](/nlab/edit/infinity-limits+-+contents)


infinity-local site

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

#Contents#
* table of contents
{:toc}

## Idea

A [[site]] is **$\infty$-local** if it satisfies sufficient conditions for the [[(∞,1)-sheaf (∞,1)-topos]] over it to be a [[local (∞,1)-topos]].

## Definition

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

A [[site]] $C$ is **$\infty$-local** if

* it has a [[terminal object]] $*$;

* the [[limit]]-functor $\lim_\leftarrow : [C^{op}, sSet] \to$ [[sSet]]
  sends [[Cech nerve]] projections $C(U) \to X$ over [[covering]] families
  $\{U_i \to X\}$ to [[weak homotopy equivalence]]s:

  $$
    Hom_C(*, C(U)) \stackrel{\simeq}{\to} Hom_C(*,X)
    \,.
  $$

=--

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

If $C$ is also a [[strongly ∞-connected site]] then it is an [[∞-cohesive site]].

=--

## Proposition

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

For $C$ an $\infty$-local site, the [[(∞,1)-sheaf (∞,1)-topos]] $Sh_{(\infty,1)}(C)$ over it is a [[local (∞,1)-topos]], in that the [[global section]] [[(∞,1)-geometric morphism]] has a further right [[adjoint (∞,1)-functor]]

$$
  Sh_{(\infty,1)}
  \stackrel{\overset{\nabla}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{\nabla}{\leftarrow}}}
  \infty Grpd
  \,.
$$

=--

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

We may [[presentable (∞,1)-category|present]] the [[(∞,1)-sheaf (∞,1)-topos]] by the local [[model structure on simplicial presheaves]]

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

For the notation see the details of the analagous proof at [[∞-connected site]]. As discussed there, the functor $\Gamma$ is given by evaliation on the terminal object. At the level of [[simplicial presheaves]] the [[sSet]]-[[enriched functor|enriched]] [[right adjoint]] to $\Gamma$ is given by

$$
  \nabla S : U \mapsto sSet(\Gamma(U), S)
$$

as confirmed by the following [[end]]/[[coend]] calculus computation:

$$
  \begin{aligned}
    [C^{op}, sSet](X, \nabla(S))
    & =
    \int_{U \in C} sSet(X(U), sSet(\Gamma(U), S)
    \\
    & = \int_{U \in C} sSet(X(U) \times \Gamma(U), S)
    \\
    & = sSet( \int^{U \in C} X(U) \times \Gamma(U), \;\; S )
    \\
    & = sSet( \int^{U \in C } X(U) \times Hom_C(*, U), \;\; S)
    \\
    & = sSet(X(*), S)
    \\
    & = sSet(\Gamma(X), S)
  \end{aligned}
  \,,
$$

where in the second but last step we used the [[co-Yoneda lemma]].

It is clear that

$$
  (\Gamma \dashv \nabla)
    :
  [C^{op}, sSet]_{proj} \stackrel{\overset{\Gamma}{\to}}{\underset{\nabla}{\leftarrow}}
  sSet_{Quillen}
$$

is a [[Quillen adjunction]], since $\nabla$ manifestly preserves fibrations and acyclic fibrations. Since $[C^{op}, sSet]_{proj,loc}$ is a [[left proper model category]] to see that this descends to a Quillen adjunction on the
[[local model structure on simplicial presheaves]] it is sufficient to check that $\nabla : sSet_{Quillen} \to  [C^{op}, sSet]_{proj,loc}$ preserves fibrant objects, in that for $S$ a [[Kan complex]] we have that $\nabla S$ satisfies [[descent]] along [[Cech nerve]]s of covering families.

This follows from the second defining condition on the $\infty$-local site $C$, that $Hom_C(*,C(U)) \simeq Hom_C(*,U)$. Using this we have for fibrant $S \in sSet_{Quillen}$ the descent weak equivalence

$$
  [C^{op}, sSet](U, \nabla S)
   =
  sSet(Hom_C(*,U), S)
   \simeq
  sSet(Hom_C(*,C(U)), S)
   =
  [C^{op}, sSet](C(U), \nabla S)
  \,,
$$

where we use in the middle step that $sSet_{Quillen}$ is a [[simplicial model category]] so that homming the weak equivalence between cofibrant objects into the fibrant object $S$ indeed yields a weak equivalence (using the [[factorization lemma]]).

=--

## Examples

* [[CartSp]], [[ThCartSp]]

## Related concepts

* [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

  * [[connected topos]] / [[∞-connected (∞,1)-topos]]

  * [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* [[locally connected site]] / [[locally ∞-connected site]]

  * [[connected site]] / [[∞-connected site]]

  * [[strongly connected site]] / [[strongly ∞-connected site]]

  * [[totally connected site]] / [[totally ∞-connected site]]

* [[local site]] / **∞-local site**

* [[cohesive site]], [[∞-cohesive site]]

[[!redirects ∞-local site]]

[[!redirects ∞-local sites]]
[[!redirects infinity-local sites]]


infinity-permutation representation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[∞-representation]] on objects of [[∞Grpd]].

The analog on [[(∞,1)-category theory]] of a [[permutation representation]].

## Properties

### Tannaka duality

See at [[Tannaka duality]] the section [For ∞-permutation representations](http://ncatlab.org/nlab/show
/Tannaka%20duality#ForInfinityPermutations).

### Grothendieck construction

By the [[(∞,1)-Grothendieck construction]] $\infty$-permutation representations

$$
  \mathbf{B}G \to \infty \mathrm{Grpd}
$$

correspond to [[∞-functors]]

$$
  V//G \to \mathbf{B}G
$$

sitting in [[fiber sequences]]

$$
  V \to V//G \to \mathbf{B}G
  \,,
$$

where $V//G$ is the corresponding [[action ∞-groupoid]].

[[!redirects ∞-permutation representation]]
[[!redirects ∞-permutation representations]]
[[!redirects infinity-permutation representations]]


infinity-quantity

For now, see [[schreiber:∞-quantity]].

[[!redirects ∞-quantity]]


infinity-quotient

#Contents#
* table of contents
{:toc}

## Idea

The generalization of the notion of [[quotient]] from [[category theory]] to [[(∞,1)-category theory]].

## Related concepts

* [[groupoid object in an (∞,1)-category]], [[(∞,1)-colimit]]

* [[∞-action]], [[principal ∞-bundle]]

[[!redirects ∞-quotient]]
[[!redirects infinity-quotient]]
[[!redirects infinity-quotients]]


infinity-representation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In full generality, an **$\infty$-representation** is an [[∞-action]] of some [[higher algebra|higher algebraic]] structure on some object in a [[higher category theory|higher category]] up to [[coherent]] [[homotopy]]. One also speaks of _representation up to homotopy_ or maybe _sh-representations_ .

We motivate the general notion of $\infty$-representation by recalling first some [[category theory|category theoretic]] aspects of the ordinary notion of representation, and then leading over to the analogous [[(∞,1)-category theory|(∞,1)-category theoretic]] notions.

### Representations as functors

Recall that for $G$ a [[discrete group]] with [[delooping]] [[groupoid]] $\mathbf{B}G = (G \stackrel{\to}{\to})$, and $k$[[Vect]] the category of [[vector space]]s (over some base [[field]] $k$), ordinary linear [[representation]]s of $G$ are equivalently [[functor]]s

$$
  \rho : \mathbf{B}G \to k Vect
  \,.
$$

Such a functor takes the single object of $\mathbf{B}G$ to some vector space $V$ and takes every morphism $(* \stackrel{g}{\to} * )$ in $\mathbf{B}G$ labeled by an element $g \in G$ to a linear [[automorphism]] $\rho(g) : V \to V$ such that composition and the identity is respected. We have an [[equivalence of categories]]

$$
  Func(\mathbf{B}G, k Vect) \simeq Rep_k(G)
  \,.
$$

Here the category $Vect$ could be replaced by other categories and they need not be [[abelian categories]] or otherwise linear for the above to make sense. For instance if we take instead the category [[Set]] itself, then a functor

$$
  \rho : \mathbf{B}G \to Set
$$

is what is called a [[permutation representation]]. In [[topology]] one is interested in representations in [[Top]]

$$
  \rho : \mathbf{B}G \to Top
  \,.
$$

(However, rarely is it sufficient to regard these just as functors to the 1-category $Top$. Instead, in order to speak about topological [[fiber bundle]]s and fibrations, one needs to regard [[Top]] here as an [[(∞,1)-category]] and regard $\rho : \mathbf{B}G \to Top$ as an [[(∞,1)-functor]]. This we come to below in [∞-Representations](#IdeaInfRepresentations))

Moreover, we may replace $\mathbf{B}G$ by a more general [[groupoid]]. For $K$ any groupoid, a functor

$$
  \rho : K \to k Vect
$$

is called a linear representation of $K$. This now picks not just a single vector space $V \in Vect$, but one vector space $V_x$ for each [[object]] $x \in K$. And to each morphism $(x \stackrel{g}{\to} y)$ in $K$ is assigns a [[linear map]] $\rho(g) : V_x \to V_y$.

For instance if $K = \Pi_1(X)$ is the [[fundamental groupoid]] of a [[manifold]] $X$, then a representation

$$
  \rho : \Pi(X) \to vect
$$

is a [[vector bundle]] over $X$ with [[curvature|flat]] [[connection on a bundle]].

And we do not even need to assume that $K$ here is a groupoid. For instance if $D$ is a [[directed graph]] (or [[quiver]]) and $F(D)$ its [[path category]], then a functor

$$
  F(D) \to Vect
$$

is called a [[quiver]]-representation of $D$.

One could in principle therefore speak of a functor

$$
  F(D) \to Set
$$

as a "quiver permutation representation", but there does not seem to be much use of this terminology in practice. The examples do show, however, that there is considerable overlap between the notion of _[[representation]]_ and of _[[functor]]_ .

### Structured representations as morphisms in 2-toposes

Still a bit more generally, we can speak of representations that preserve extra structure, such as [[smooth structure]]. For instance for $G$ a [[Lie group]] we have that $\mathbf{B}G$ is a [[Lie groupoid]]: an object in the [[(2,1)-topos]] of [[(2,1)-sheaves]] over the [[site]] $C = $ [[CartSp]] or $C =$ [[Diff]].

We may also promote the category [[Vect]] to this $(2,1)$-topos, by replacing it by the [[stack]] $VectBund$, which assigns to each test manifold $U \in C$ the groupoid of smooth [[vector bundle]]s over $U$. Then a morphism

$$
  \rho : \mathbf{B}G \to VectBund
$$

in the [[(2,1)-topos]] $Sh_{(2,1)}(C)$ is a _smooth representation_ of $G$, in that the linear automorphisms $\rho(g) : V \to V$ depend smoothly on the point $g \in G$.

We can recover the underlying ordinary representation by applying the [[global section]] functor $\Gamma : Sh_{(2,1)}(C) \to Grpd$. This is given by evaluating every thing on the [[terminal object]] ${*} \in C$, which here is just the ordinary [[point]], regarded as a smmoth manifold.

This yields the underlying bare representation

$$
  \Gamma(\rho) : \mathbf{B}G \to Vect
  \,.
$$

Conversely, one finds that extending such a bare representation from the point to all test spaces in $C$ amounts to equipping it with smooth structure.

As before, this is not restricted to [[connected]] objects: we may replace $\mathbf{B}G$ here with any [[Lie groupoid]]. For instance for $X$ a [[smooth manifold]] and $\mathbf{P}_1(X)$ is smooth [[path groupoid]] a representation

$$
  \rho : \mathbf{P}_1(X) \to VectBund
$$

as a morphism in the 2-topos is a [[vector bundle]] on $X$ with [[connection on a bundle]]. Or if we consider non-linear representations, a representation

$$
  \rho : \mathbf{P}_1(X) \to \mathbf{B}G
$$

is a $G$-[[principal bundle]] with [[connection on a bundle]] on $X$. See [[parallel transport]] for more details and references.

By just changing the [[site]] here, we can implement other geometric structures. For instance for $G$ an [[algebraic group]] we may think of $\mathbf{B}G$ as an [[algebraic stack]] over something like the [[fppf-site]]  structure $C = CAlg_k^{op}$ on formal duals of commutative $k$-algebras or similar.

In this case there is a well-known good generalization of $VectBund$: instead of just vector bundles we can consider their completion to [[quasicoherent sheaves]]. The stack of these is the object in the [[(2,1)-topos]] given by

$$
  QC : Spec A \mapsto A Mod
  \,,
$$

where on the right we have the groupoid of [[module]]s over $A$. Over the point this is again just a $k$-module hence a [[vector space]] and hence a representation

$$
  \rho : \mathbf{B}G \to QC
$$

of an algebraic group is a representation of $G$ on a vector space.

But also here we may allow the represented structure to have more than one object. For instance for $X$ any [[scheme]] regarded as an object in $Sh_{(2,1)}(C)$ a representation of $X$ in the context of $QC$ is a morphism

$$
  \rho : X \to QC
  \,,
$$

which is equivalently a [[quasicoherent sheaf]] of modules on $X$. As before, we may think of this as assigning to each point of $X$ one representation space, only that in a scheme $X$ there are no morphisms that would act on these.

But more generally for $K$ an [[algebraic stack]], a representation

$$
  \rho : K \to QC
$$

assigns to each point of $K_0$ a representation space, such that these glue together to a quasicoherent sheaf of modules, and to each [[morphism]] in $K$ a morphism between the corresponding representation spaces, as before.

Analogous constructions are available for more general sites, effectively we can take $C$ to be the [[opposite category]] of $T$-[[algebras over a Lawvere theory]] for $T$ any [[algebraic theory]] that contains the theory of [[abelian group]]s. If for instance we take $T =$ [[CartSp]] we are back to the smooth case discussed before.

Also notice that if we take the site to be the point, $C = *$, then sheaves over it are just sets and stacks over it are just bare groupoids, so that we recover the discussion at the very beginning.

### $\infty$-Representations  {#IdeaInfRepresentations}

We have found above the term "representation" is, to a large extent, congruent with the term "morphism in a (2,1)-topos with codomain a stack of modules".

This way of thinking of representations has an immediate generalization to [[higher category theory]] and in particular to [[(∞,1)-category theory]]/[[homotopy theory]].

To start with the simple discussion over the point again, a model for a notion of a category of $\infty$-modules that is useful is an [[(∞,1)-category]] $Ch_\bullet(k)$ that is [[presentable (∞,1)-category|presented]] by a [[model structure on chain complexes]].

If again $G$ is a [[discrete group]], then an  [[(∞,1)-functor]] (equivalently: a "strong homotopy-functor" or "homotopy coherent functor", see there for details)

$$
  \rho : \mathbf{B}G \to Ch_\bullet(k)
$$

assigns

* to the single obect of $\mathbf{B}G$ a chain complex $V_\bullet$;

* to a group element $g \in G$ a chain map $\rho(g) : V_\bullet \to V_\bullet$;

* to a pair of group elements $g, g'$ a chain homotopy

  $\rho(g,g') : \rho(g')\circ\rho(g) \Rightarrow \rho(g' g)$;

* to a triple of group elements a homotopy of homotopies between composites of $\rho(g,g'), \rho(g,g'')$ and $\rho(g',g'')$ and so on

  (see the diagrams at [[group cohomology]] for more details in low degree).

In other words, this is much like representation of $G$ as before on an ordinary vector space, only that now the action property of $\rho$ holds only _up to [[coherent]] [[homotopy]]_ . Therefore people also speak of _representations up to homotopy_ ([AbadCrainic](#AbadCrainic)) as well as
_strong homotopy representations_ and many other variants.

As before, there is in principle no reason to restrict oneself to representations of groupoids here. For $K$ any [[∞-groupoid]] or even [[(∞,1)-category]] (recall the [[quiver]] representations) and for $Mod$ any [[(∞,1)-category]] of $\infty$-modules (for instance as [[presentable (∞,1)-category|presented]] by a [[model structure on modules over an algebra over an operad]]) we may call an [[(∞,1)-functor]]

$$
  \rho : K \to Mod
$$

an $\infty$-representation of $K$.

If we wish to consider $\infty$-generalizations of [[permutation representation]]s we can also consider more general codomain $(\infty,1)$-categories here. For instance if we take [[∞Grpd]] itself, then an  $\infty$-permutation representation

$$
  K \to \infty Grpd
$$

is known as an [[(∞,1)-presheaf]]. For $K$ the [[delooping]] of an ordinary group or the [[orbit category]] of a [[topological group]], such appear genuinely from the point of view of representations for instance in [[equivariant cohomology]] and [[orbit category|equivariant homotopy]]  theory. Notice that by the [[homotopy hypothesis]]-theorem we have an [[equivalence of (∞,1)-categories]]

$$
  \infty Grpd \simeq Top
$$

so that the above is equivalently an [[(∞,1)-functor]]

$$
  K \to Top
$$

hence literally a _representation up to homotopy_ in the classical sense of [[homotopy theory]].

As before, all this may be lifted from the point into large classes of [[(∞,1)-topos]]es  to equip the notion of $\infty$-representation with geometric structure (algebraic structure, smooth structure, etc.)

There are then analogs of the above relation between representations of path groupoids and connections on bundles. For more on this see [[higher parallel transport]]. Quite generally, in every [[locally ∞-connected (∞,1)-topos]] $\mathbf{H}$ there is a notion of [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos|fundamental ∞-groupoid]] $\mathbf{P}(X)$ of any object $X$. Representation of $\mathbf{\Pi}(X)$ define general abstract flat differential cohomology and [[local system]]s on $X$, generally also in [[nonabelian cohomology]] (see there for some more properties and examples).

For instance [[dg-geometry]] is the study of the [[(∞,1)-topos]] over an [[(∞,1)-site]] of formal duals of [[dg-algebra]]s. Again there is the canonical [[∞-stack]]

$$
  QC : Spec A \mapsto A Mod
$$

on this site, where now however $A Mod$ denotes the [[∞-groupoid]] (or [[(∞,1)-category]] if we do a more comprehensive discussion) of [[chain complex]]es equipped with the structure of a [[module]] ove the dg-algebra $A$.

For $X$ any [[∞-stack]] then a morphism

$$
  \rho : X \to QC
$$

is a equivalently a [[quasicoherent ∞-stack]] of modules on $X$, or an $\infty$-representation with "dg-algebraic structure".

If one replaces $X$ here by its [[de Rham space|de Rham stack]] $X_{dR}$ then dg-algebraic $\infty$-representations

$$
  \rho : X_{dR} \to QC
$$

are [[D-module]]s on $X$.

A discussion of these higher categorical structure in [[representation theory]] is in ([Ben-ZviNadler](#Ben-ZviNadler)).

### $n$-Representations

If the codomain $Mod$ is  an [[(∞,1)-category]] that is just a [[(n,1)-category]] (all [[k-morphisms]]s for $k \gt n$ are effectively identities) then an $\infty$-representation is called an **$n$-representation**. These are _representations up to homotopy_ where from degree $n$ on all homotopies are actually identities: $n$-[[truncated]] representations up to homotopy.

As always in [[higher category theory]], the cases for low $n$ are more restrictive but typically admit a more tractable detailed analysis and construction.

2-representations of [[2-groups]] and [[Lie 2-group]]s on various variants of [[2-vector space]]s have been considered for instance in ([Schreiber](#Schreiber),  [BaezBaratinFreidelWise](#BaezBaratinFreidelWise), and other places).

In analogy to the case for $n=1$, 2-Representations $\mathbf{P}_2(X) \to 2 Vect$ of the smooth [[path 2-groupoid]] of a [[smooth manifold]] describe [[connections on a 2-bundle]]. See there for more details.

## Definition
 {#Definition}

### In homotopy type theory
 {#InHomotopyTypeTheory}

In a general abstract context of [[homotopy type theory]] we may define $\infty$-representations as follows.

For $\mathbf{H}$ an [[(∞,1)-topos]], let $G \in Grp(\mathbf{H})$ be a [[group object in an (∞,1)-category|group object]] in $\mathbf{H}$, hence an [[∞-group]]. Then the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}G}$ over its [[delooping]] is the [[(∞,1)-category]] of [[∞-actions]] of $G$

$$
  Act(G) \simeq \mathbf{H}_{/\mathbf{B}G}
  \,,
$$

hence of possibly "non-linear" $\infty$-representations. (See at _[[∞-action]]_ for details). A genuine (linear) $\infty$-representation is then an  [[abelian ∞-group]] object in $Act(G)$.

## Related concepts

* [[(∞,1)-module bundle]]

* [[action]], [[∞-action]]

* [[module]], [[∞-module]]

* [[representation]], **$\infty$-representation**

  * [[2-representation]],

* [[associated bundle]], [[associated ∞-bundle]]

[[!include homotopy type representation theory -- table]]

## References

2-representations of [[2-groups]] and [[Lie 2-group]]s such as the [[string 2-group]] on [[2-vector space]]s are disscussed in

* {#Schreiber}[[Urs Schreiber]], _AQFT from $n$-functorial QFT_
  ([arXiv:0806.1079](http://arxiv.org/abs/0806.1079)) (appendix)

* {#BaezBaratinFreidelWise} [[John Baez]], Aristide Baratin, Laurent Freidel, Derek Wise, _Infinite-Dimensional Representations of 2-Groups_ ([arXiv:0812.4969](http://arxiv.org/abs/0812.4969))

References for 2- and 3-representations of [[path n-groupoid]]s are at [[higher parallel transport]].

$\infty$-Representations of [[groupoids]] and [[Lie algebroid]]s on $(\infty,1)$-categories of chain complexes are discussed under the term  _representations up to homotopy_ in

* {#AbadCrainic} [[Camilo Arias Abad]], [[Marius Crainic]] , _Representations up to homotopy and Bott's spectral sequence for Lie groupoids_ ([arXiv](http://arxiv.org/abs/0911.2859))

A discussion of quasicoherent $\infty$-stacks and D-modules in the context of [[representation theory]] is for instance in

* {#Ben-ZviNadler} [[David Ben-Zvi]], [[David Nadler]],

  _The character theory of a complex group_ ([arXiv:0904.1247](http://arxiv.org/abs/0904.1247))

  _Loop Spaces and Representations_ ([arXiv:1004.5120](http://arxiv.org/abs/1004.5120))

[[!redirects infinity-representations]]
[[!redirects ∞-representation]]
[[!redirects ∞-representations]]
[[!redirects representation up to homotopy]]
[[!redirects representations up to homotopy]]


infinity-space

<div class="rightHandSide toc">
[[!include higher geometry - contents]]
</div>

#Idea#

In the sense of [[space and quantity]] a _space_ is, quite generally, a [[presheaf]] on some category of test objects.

An _$\infty$-space_ is some [[vertical categorification|∞-categorification]] of this.

With some [[category]] $Sp \subset PSh(C)$ of spaces (1-spaces, that is) fixed, an $\infty$-space may for instance be modeled as an [[∞-groupoid]] [[internalization|internal to]] $Sp$.

If $Sp$ is for instance a category of _smooth_ test objects, such as [[Diff]] or [[CartSp]], then we have

* a smooth 1-space is a [[smooth space]];

* a smooth 2-space is a [[Lie groupoid]];

* a smooth $\infty$-space is a [[Lie ∞-groupoid]] or [[smooth ∞-stack]].

Generally, the notion of [[∞-groupoid]] [[internalization|internal to]] $PSh(C)$ is naturally interpreted as a [[simplicial presheaf]]. Indeed, when equipped with the [[model structure on simplicial presheaves]], simplicial presheaves are good [[models for ∞-stack (∞,1)-toposes]].

The $\infty$-stack terminology is possibly more familiar than that of $\infty$-spaces. To some extent the usefulness of both terminologies depends on whether $Sp$ is a [[petit topos]] of presheaves of open subsets in some fixed [[topological space]] or a [[gros topos]] of presheaves on general test spaces, such as on (an [[essentially small category|essentially small]]) version of [[Top]] itself.

In the former case of a [[petit topos]], the $\infty$-stack terminology may be more suggestive. In the latter case of a [[gros topos]] the $\infty$-space perspective typically conveys the right intuition for the objects under consideration much better.

# Warning on terminology #

The idea, that "space" is fundamentally to be interpreted in the sense of [[space and quantity]] is only partially compatible with wide-spread use of the word "space" to mean concretely [[topological space]].

In fact, in the context of "$\infty$-space" there is a subtle confusion of terms (possible) here, which deserves to be carefully sorted out:

In [[higher category theory]] [[topological space]]s $X$ are often, via the [[homotopy hypothesis]], effectively _identified_ with their [[fundamental ∞-groupoid]]s $\Pi(X)$. The [[∞-groupoid]] $\Pi(X)$ -- as an $\infty$-groupoid [[internalization|internal to]] [[Set]] -- is really a _discrete_ $\infty$-space in the above sense, namely an $\infty$-groupoid in the category of presheaves on the point.

# $\infty$-Quantities #

Following the [[duality]] of [[space and quantity]], the concept dual to $\infty$-space is [[∞-quantity]]. See there for more details.

# References #

One place where the conceptual usefulness of interpreting categories internal to [[smooth space]]s as smooth **2-spaces** (and de-emphasizing conceptually their _realization_ as simplicial sheaves) has been particularly amplified is the work

* [[Toby Bartels]], _2-Bundles_ ([web](http://tobybartels.name/2bundles/))

[[!redirects ∞-space]]


infinity-stack

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of, equivalently

* _$\infty$-stack_,

and specifically of

* _[[(∞,1)-sheaf]]_,

* _[[geometric homotopy type]]_

is the $\infty$-[[vertical categorification|categorification]] of the notion of, equivalently

* [[sheaf]] and [[stack]]

* [[geometric homotopy type]].

Where a sheaf is a [[presheaf]] with values in [[Set]] that satisfies the sheaf condition, an [[higher category theory|∞-category]]-valued ([[pseudofunctor|pseudo]])[[presheaf]] is an _$\infty$-stack_ if it "satisfies descent" in that its assignment to a space $X$ is equivalent to its [[descent]] data for any [[cover]] or [[hypercover]] $Y^\bullet \to X$: if the canonical morphism

$$
  \mathbf{A}(X) \to
  Desc(Y^\bullet, \mathbf{A})
$$

is an equivalence. This is the _descent condition_.

One important motivation for $\infty$-stacks is that they generalize the notion of [[Grothendieck topos]] from [[category theory|1-categorical]] to [[higher category theory|higher categorical context]].

This is a central [[motivation for sheaves, cohomology and higher stacks|motivation for considering higher stacks]]. They may also be thought of as [[internal ∞-groupoid]]s in a [[Grothendieck topos|sheaf topos]].

## Definition

A well developed theory exists for $\infty$-stacks that are sheaves with values in [[∞-groupoids]]. Given that ordinary sheaves may be thought of as sheaves of [[0-category|0-categories]] and that $\infty$-groupoid-values sheaves may be thought of as sheaves of [[(infinity,0)-category|(∞,0)-categories]], these may be called [[(infinity,1)-sheaf|(∞,1)-sheaves]]. In the case that these $\infty$-groupoids have vanishing [[homotopy groups]] above some degree $n$, these are sometimes also called [[sheaves of n-types|sheaf of n-types]].

The currently most complete picture of [[(infinity,1)-sheaf|(∞,1)-sheaves]] appears in

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

but is based on a long development by other authors, some of which is indicated in the list of references below.

With the general machinery of [[(∞,1)-category]] theory in place, the definition of the [[(infinity,1)-category of (infinity,1)-sheaves|(∞,1)-category of ∞-stacks]] is literally the same as that of a [[category of sheaves]]: it is a [[reflective (∞,1)-subcategory]]

$$
  \infty Stacks(C) \simeq Sh_\infty(C)
  \stackrel{\stackrel{\bar{(\cdot)}}{\leftarrow}}{\to}
  PSh_\infty(C)
$$

of the [[(infinity,1)-category of (infinity,1)-functors|(∞,1)-category of (∞,1)-presheaves]] with values in [[∞Grpd]], such that the left adjoint [[(∞,1)-functor]] $\bar {(\cdot)}$ -- the [[∞-stackification]] operation -- is left exact.

One of the main theorems of [[Higher Topos Theory]] says that the old [[model structure on simplicial presheaves|model structures on simplicial presheaves]] are the canonical

* [[models for ∞-stack (∞,1)-toposes]].

This allows to regard various old technical results in a new conceptual light and provides powerful tools for actually handling $\infty$-stacks.

In particular this implies that the old definition of [[abelian sheaf cohomology]] is secretly the computation of [[∞-stackification]] for $\infty$-stacks that are in the image of the [[Dold-Kan correspondence|Dold-Kan embedding]] of [[chain complex]]es of sheaves into [[simplicial presheaves|simplicial sheaves]].

### Derived $\infty$-stacks

Notice that an $\infty$-stack is a [[(∞,1)-presheaf]] for which not only the codomain is an [[(∞,1)-category]], but where also the domain, the [[site]], may be an [[(∞,1)-category]].

To emphasize that one considers $\infty$-stacks on higher categorical sites one speaks of [[derived stacks]].

### Higher $\infty$-stacks

The above concerns $\infty$-stacks with values in [[∞-groupoids]], i.e, [[(∞,0)-category|(∞,0)-categories]]. More generally there should be notions of $\infty$-stacks with values in  [[(n,r)-category|(n,r)-categories]]. These are expected to be modeled by the [[model structure on homotopical presheaves]] with values in the category of [[Theta spaces]].

### Quasicoherent $\infty$-stacks

An archetypical class of examples of $\infty$-stacks are [[quasicoherent ∞-stack]]s of [[module]]s, being the [[vertical categorification|categorification]] of the notion of [[quasicoherent sheaf]]. By their nature these are really $(\infty,1)$-stacks in that they take values not in [[∞-groupoid]]s but in [[(∞,1)-categories]], but often only their [[∞-groupoid]]al [[core]] is considered.

### Affine $\infty$-stacks

In

* [[Bertrand Toen]], _Affine stacks (Champs affines)_ ([arXiv:math/0012219](http://arxiv.org/abs/math/0012219))

for the [[site]] $C = Alg_k^{op}$ with a suitable topology a [[Quillen adjunction]]

$$
  \mathcal{O} : sPSh(C)_{loc} \stackrel{\leftarrow}{\to}
  [\Delta^{op},Alg_k] \simeq dgAlg_k^{+}
  :
  Spec
$$

is presented, where $\mathcal{O}$ sends and $\infty$-stack to its global [[dg-algebra]] of functions and $Spec$ constructs the simplicial presheaf "represented" degreewise by a cosimplicial algebra (under the [[monoidal Dold-Kan correspondence]] these are equivalent to dg-algebras).

An $\infty$-stack in the image of $Spec : dgAlg_k^+ \to sPSh(C)$ is an **affine $\infty$-stack**. The image of an arbitrary $\infty$-stack under the composite

$$
  Aff : sPSh(C) \stackrel{\mathcal{O}}{\to} dgAlg_k^+ \stackrel{Spec}{\to} sPSh(C)
$$

is its **affinization**.

This notion was considered in the full [[(∞,1)-category]] picture in

* [[David Ben-Zvi]], [[David Nadler]], _Loop Spaces and Connections_ ([arXiv:math/1002.3636](http://arxiv.org/abs/1002.3636))

where it is also generalized to [[derived stack]]s, i.e. to the [[(∞,1)-site]] $dgAlg_k^-$ of cochain [[dg-algebra]]s in non-positive degree, where the pair of [[adjoint (∞,1)-functor]]s is

$$
  \mathcal{O} :
  Sh_{(\infty,1)}((dgAlg_k^-)^{op})
  \stackrel{\leftarrow}{\to}
  [\Delta^{op},dgAlg_k^-]
  \simeq
  dgAlg_k
  :
  Spec
$$

with $\mathcal{O}$ taking values in _unbounded_ dg-algebras.

In detail, $\mathcal{O}$ acts as follows: every [[∞-stack]] $X$ may be written as a ([[homotopy colimit|colimit]]) over [[representable functor|representable]] $Spec A_i \in dgAlg_i$

$$
  X \simeq \lim_{\to^i} Y(Spec A_i)
  \,,
$$

where $Y : (dgAlg^-)^{op} \to \mathbf{H}$ is the [[(∞,1)-Yoneda embedding]].

The functor $\mathcal{O}$ takes any such colimit-description, and simply reinterprets the colimit in $dgAlg^{op}$, i.e. the limit in $dgAlg$:

$$
  \mathcal{O}(X) = \lim_{\leftarrow^i} A_i
  \,.
$$

## Related concepts

* [[sheaf]]

* [[2-sheaf]] / [[stack]]

* [[(∞,1)-sheaf]] / **$\infty$-stack**,

  * [[sheaf of spectra]]

  * [[sheaf of L-∞ algebras]]

* [[(∞,2)-sheaf]]

* [[(∞,n)-sheaf]]

[[!include homotopy n-types - table]]

## References

The study of $\infty$-stacks is known in parts as the study of [[nonabelian cohomology]]. See there for further references.

The search for $\infty$-stacks probably began with [[Alexander Grothendieck]] in _[[Pursuing Stacks]]_.

The notion of $\infty$-stacks can be set up in various notions of $\infty$-categories. [[Andre Joyal]], Jardine, [[Bertrand Toen]] and others have developed the theory of $\infty$-stacks in the context of [[simplicial presheaf|simplicial presheaves]] and also in [[Segal category|Segal categories]].

* [[Bertrand Toën]], [[Gabriele Vezzosi]]; _Homotopical algebraic geometry. I. Topos theory_, Adv. Math. 193 (2005), no. 2, 257--372, [doi](http://dx.doi.org/10.1016/j.aim.2004.05.004), _Homotopical Algebraic Geometry II: geometric stacks and applications_, [math.AG/0404373](http://front.math.ucdavis.edu/0404.5373)

* [[Bertrand Toën]], [[Gabriele Vezzosi]]; _Segal topoi and stacks over Segal categories_, [math.AG/0212330](http://arxiv.org/abs/math.AG/0212330).

* [[Bertrand Toën]], _Higher and derived stacks: a global overview_, In: _Algebraic Geometry Seattle 2005_, Proceedings of Symposia in Pure Mathematics, Vol. 80.1, AMS 2009 ([arXiv:math/0604504](http://arxiv.org/abs/math/0604504), [doi:10.1090/pspum/080.1](https://doi.org/10.1090/pspum/080.1))

This concerns $\infty$-stacks with values in [[∞-groupoids]], i.e. $(\infty,0)$-categories. More generally [[descent]] conditions for $n$-stacks and $(\infty,n)$-stacks with values in [[(infinity,n)-category|(∞,n)-categories]] have been earlier discussed in

* [[Andre Hirschowitz]], [[Carlos Simpson]]; _Descente pour les $n$-champs_ ([arXiv](http://arxiv.org/abs/math/9807049))

All this has been embedded into a coherent global theory in the setting of [[quasicategory|quasicategories]] in

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

Textbook account on presentation by [[model structures on simplicial presheaves]]:

* {#Jardine15} [[John F. Jardine]], *[[Local homotopy theory]]*, Springer Monographs in Mathematics (2015) [[doi:10.1007/978-1-4939-2300-7](https://doi.org/10.1007/978-1-4939-2300-7)]

[[!redirects infinity-stacks]]
[[!redirects ∞-stack]]
[[!redirects ∞-stacks]]
[[!redirects infinity stack]]


infinity-stack homotopically

see at _[[infinity-stack]]_


infinity-stackification

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##  Idea

$\infinity$-Stackification is another term for [[(infinity,1)-category of (infinity,1)-sheaves|(∞,1)-sheafification]]. It is the direct [[(infinity,1)-category|(∞,1)-categorical]] analog of the following 1-categorical situation.

Recall that for $S$ a [[site]], [[sheafification]] is the [[functor]]
$$
  \bar{(-)} : PSh(S) \to Sh(S) \hookrightarrow PSh(S)
$$
which sends every [[presheaf]] $F$ on $S$ to another [[presheaf]] $\bar F$ which is weakly equivalent to $F$ with respect to the [[homotopical category]] structure on $PSh(S)$ induced from the [[Grothendieck topology]] on $X$. The presheaf $\bar F$ respects weak equivalences and satisfies [[descent]] in that the [[hom-functor]] $Hom_{PSh(S)}(-,\bar F)$ sends weak equivalences (the [[local isomorphism]]s) to weak equivalences.

Essentially by definition (according to [[Higher Topos Theory]]) the situation for $\infty$-stacks is entirely analogous, as described at [[(∞,1)-category of (∞,1)-sheaves]].

(Noticing that "$\infty$-stack" is synonymous to "[[(∞,1)-sheaf]]", "$\infty$-stackification" to "$(\infty,1)$-sheafification", and so on.

## Examples

* A concrete model for such $\infty$-stacks is the Brown-Joyal-Jardine-Toën model using a [[model structure on simplicial presheaves]]. With respect to this [[model category]] structure on [[simplicial presheaf|simplicial presheaves]] the _fibrant objects_
are the globally [[Kan complex]]-valued (i.e. [[∞-groupoid]]-valued) presheaves that satisfy [[descent for simplicial presheaves]]. Therefore here $\infty$-stackification is given by _[[fibrant resolution|fibrant replacement]]_ in the model category.

## Related concepts

* [[sheafification]], [[plus-construction on presheaves]]

* [[(∞,1)-sheafification]] / **∞-stackification**

## References

For instance section 6.5.3 of

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

[[!redirects ∞-stackification]]


infinity-topos

A notion of _$\infty$-topos_ is a generalization of that of [[topos]] to [[higher category theory]].

If by $\infty$-category one means an $(\infty,\infty)$-category, then such a notion is still undefined.  However, nowadays many people write "$\infty$-category" to mean what here on the nLab we usually still call an [[(∞,1)-category]], and thus an "$\infty$-topos" means an [[(∞,1)-topos]].  See also [[higher topos theory]] for more.

[[!redirects infinity topos]]
[[!redirects infinity-topoi]]
[[!redirects ∞-topos]]
[[!redirects ∞-toposes]]
[[!redirects ∞-topoi]]


infinity-Wess-Zumino-Witten model

* [[Wess-Zumino-Witten model]]


infinity-Wess-Zumino-Witten theory - contents

* [[∞-Chern-Weil theory]]

* [[infinity-Chern-Simons theory - contents|∞-Chern-Simons theory]]

* **[[schreiber:∞-Wess-Zumino-Witten theory]]**

## Models

* [[1d WZW model]]

* 2d [[Wess-Zumino-Witten model]]

  * [[current algebra]]

  * [[Green-Schwarz superstring]]

* [[higher dimensional WZW theory]]

  * [[4d WZW theory]]

  * [[Green-Schwarz action functional]]

* [[WZW-type superstring field theory]]

[Edit this sidebar](/nlab/edit/infinity-Wess-Zumino-Witten+theory+-+contents)


information

#Contents#
* table of contents
{:toc}

## Idea

(...)

## References

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

On [[modal logic]] in information science and [[game theory]]:

* [[Wiebe van der Hoek]], [[Marc Pauly]], *Modal logic for games and information*, Ch. 20 in: *The Handbook of Modal Logic*, Studies in Logic and Practical Reasoning **3** (2007) 85-183 $[$[doi:10.1016/S1570-2464(07)80023-1](https://doi.org/10.1016/S1570-2464(07)80023-1), [book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/)$]$

[[!redirects information science]]


information geometry

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

## Idea

Information geometry aims to apply the techniques of [[differential geometry]] to [[statistics]]. Often it is useful to think of a family of probability distributions as a _statistical_ [[manifold]]. For example, normal [[Gaussian distribution]]s form a 2-[[dimension]]al manifold, parameterised by $(\mu, \sigma)$, mean and standard deviation. On such manifolds there are notions of [[Riemannian metric]], [[connection]], [[curvature]], and so on, of statistical relevance.

More precisely,

> [[Kullback-Leibler information]], or [[relative entropy]], features as a measure of [[divergence]] (not quite a [[metric]], because it's asymmetric), and [[Fisher information]] takes the role of [[curvature]]. One useful aspect of information geometry is that it gives a means to prove results about statistical models, simply by considering them as well-behaved geometrical objects. For instance, it's basically a tautology to say that a [[manifold]] is not changing much in the vicinity of points of low curvature, and changing greatly near points of high curvature. Stated more precisely, and then translated back into probabilistic language, this becomes the [[Cramer-Rao inequality]], that the variance of a parameter estimator is at least the reciprocal of the [[Fisher information]]. ([Shalizi](#Shalizi))

Founders of the systematical theory are N. N. Chentsov and [[Shun-ichi Amari]].

## Definitions

For $X$ a [[measurable space]] let $S$ be (a subspace of) the space of probability [[measure]]s on $X$, equipped with the structure of a [[smooth manifold]].

The **Fisher metric** on $S$ is the [[Riemannian metric]] given on two [[vector field]]s $v,w \in T S$ by

$$
  g(v,w)_s := E_s( v(log s) w(log s))
  \,,
$$

where $E_s(\cdots)$ denotes the [[expectation value]] under the measure $s \in S$ of the function $x \mapsto  v(log s)_x w(log s)_x$ on $X$.

For instance ([Amari, Section 2.1](#AmariTextbook)).

## Related concepts

* [[entropy]]
* [[Fisher metric]]
* [[quantum information]]
* [[relative entropy]], also known as the Kullback-Leibler divergence

## References

See also [[Fisher metric]], where Fisher metric in other contexts and quantum generalizations are treated. See also [[quantum information]].

Textbooks providing the big picture

* Nihat Ay, Jürgen Jost, [[Hông Vân Lê]], Lorenz Schwachhöfer, _Information geometry_, Ergeb. der Mathematik and ihrer Grenzgebiete 3. Folge, 64, Springer 2017
* [[Shun-ichi Amari]], O. E. Barndorff-Nielsen, R. E. Kass, S. L. Lauritzen, C. R. Rao, _Differential geometry in statistical inference_ ([project Euclid](http://projecteuclid.org/euclid.lnms/1215467056))
 {#AmariTextbook}
* [[Shun-ichi Amari]], _Information geometry and its applications_, Applied Mathematical Sciences 194, Springer 2016
* [[Shun-ichi Amari]], Hiroshi Nagaoka, _Methods of information geometry_, Transactions of mathematical monographs; v. 191, American Mathematical Society, 2000.

For a series of articles, see

* [[John Baez]], [Information Geometry](http://math.ucr.edu/home/baez/information/)

Lecture notes include

*  [[Shun-ichi Amari]], MaxEnt 2007 [lecture](http://video.google.com/videoplay?docid=2175767568645553282#)
*  [[Shun-ichi Amari]], ETVC08 [lecture](http://videolectures.net/etvc08_amari_igaia/).
*  Sanjoy Dasgupta, [lecture](http://videolectures.net/mlss05us_dasgupta_ig/).

See also

*  [[Hông Vân Lê|Hông Vân Lê]], Statistical manifolds are statistical models, Journal of Geometry 84(1-2) (2006) 83--93.

*  Blog [post](http://golem.ph.utexas.edu/category/2006/11/infinitedimensional_exponentia.html).

A brief introduction with more references is

* Cosma Shalizi, _Information geometry_ ([webpage](http://www.cscs.umich.edu/~crshalizi/notabene/info-geo.html))
 {#Shalizi}

Several people have noted an equivalence between statistic inference as parametric model selection and statistical mechanics on a statistical manifold, e.g.,

* [[Vijay Balasubramanian]], _Statistical Inference, Occam’s Razor and Statistical Mechanics on the Space of Probability Distributions_, Neural Comp.9(2)(1997) 349–368, ([arXiv:cond-mat/9601030](https://arxiv.org/abs/cond-mat/9601030)).

The interpretation of a [[quantum field theory]] as a probability distribution on the space of field configurations so as to allow the conversion of techniques from information geometry to analogous measures of proximity between QFTs is in

* [[Vijay Balasubramanian]], [[Jonathan Heckman]], Alexander Maloney, _Relative Entropy and Proximity of Quantum Field Theories_, ([arXiv:1410.6809](https://arxiv.org/abs/1410.6809))

A treatment of collective statistical inference as resulting in the partition function of a non-linear [[sigma model]] is in

* [[Jonathan Heckman]], _Statistical Inference and String Theory_, ([arXiv:1305.3621](https://arxiv.org/abs/1305.3621))

More in the context of [[quantum field theory]]:

* Johanna Erdmenger, Kevin T. Grosvenor, Ro Jefferson, _Information geometry in quantum field theory: lessons from simple examples_ ([arXiv:2001.02683](https://arxiv.org/abs/2001.02683))


information metric

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## Idea

In [[information geometry]], a _(Fisher-)information metric_ is a [[Riemannian metric]] on a [[manifold]] of [[probability distribution]]s over some probability space $X$ (the latter often assumed to be finite).

## Definition

On a finite probability space $X \in $ [[Set]] a positive [[measure]] is a [[function]] $\rho : X \to \mathbb{R}_+$ and a [[probability distribution]] is one such that $\sum_{x \in X} \rho(x) = 1$.

This space is actually a [[submanifold]] of $\mathbb{R}_{\geq 0}^{\vert X \vert}$. For $\{\frac{\partial}{\partial x^i}\}$ the canonical [[basis]] of [[tangent vector]]s on this wedge of [[Cartesian space]], the information metric $g$ is given by

$$
  g(\frac{\partial}{\partial x^i}, \frac{\partial }{\partial x^j})(\rho) = \frac{1}{\rho(x^i)} \delta_{i j}
  \,.
$$

## Related concepts

* [[Wasserstein metric]]

## References

* L. L. Campbell, _An extended Čencov characterization of the information metric_
Journal: Proc. Amer. Math. Soc. 98 (1986), 135-141.  ([AMS](http://www.ams.org/journals/proc/1986-098-01/S0002-9939-1986-0848890-5/home.html))

[[!redirects information metrics]]

[[!redirects Fisher information metric]]
[[!redirects Fisher information metrics]]


information theory

# Information theory
* table of contents
{: toc}

## Overview

There are many theories explicating specific features of naive or intuitive notions of information.

The academic sites of these theories range over philosophy, mathematics (type theory, measure theory / probability theory / statistics), computer science, physics, communication sciences, psychology / sociology, economics, semiotics, cybernetics, etc.

Information theories in methodological proximity to mathematics are e.g.

* statistical information theory

*  semantic information theory

* algorithmic information theory

* [[constructive-type-theoretical information theory]]

## Related topics

* [[analytic principle]]

* [[coding theory]]

* [[computational complexity theory]]

* [[infon]]

* meaning

* knowledge

* [[structure]]

## Contributors to information theory

*  Statistical information theory: Claude Shannon, Warren Weaver, Ronald Fisher

* Semantic information theory: Yehoshua Bar-Hillel, Rudolf Carnap

*  Algorithmic information theory: Ray Solomonoff, Andrey Kolmogorov, Gregory Chaitin

* Constructive-type-theoretical information theory: [[Giuseppe Primiero]], Tijn Borghuis, Fairouz Kamareddine, [[Rod Nedepelt]]

* Miscellaneous: Fred Dretske, Keith Devlin, Jon Barwise, Jeremy Seligman, Jaakko Hintikka

## References

* [[Giuseppe Primiero]], [[spahn:information and knowledge - a constructive type-theoretical approach]], Springer, 2008{#Primiero}

* {#Burgin} [[Mark Burgin]], *Theory of information*, World Scientific (2009) [[doi:10.1142/7048](https://doi.org/10.1142/7048)]

[[!redirects information theory]]


infra-Bayesianism

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## Idea

InfraBayesianisn a theory of reinforcement learning in the nonrealizable setting based on imprecise probability by Vanessa Kosoy and Alex Appel.
It is part of Vanessa Kosoy's Learning-Theoretic Agenda in AI alignment.

Non-realizable in this context refers to the situation where the 'true' distribution does not necessarily lie in the model class of the (reinforcement) agent.

Infradistributions generalize the credal sets of imprecise probability which in turn generalize probability distributions.

## Related concepts

* [[imprecise probability]]

## References

* ["Basic inframeasure Theory"](https://www.lesswrong.com/posts/YAa4qcMyoucRS2Ykr/basic-inframeasure-theory), Alex Appel


infraparticle

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## Related entries

* [[infrared divergence]]

* [[quantum electrodynamics]]

## References

* {#Schroer63} [[Bert Schroer]], _Infrateilchen in der Quantenfeldtheorie_, Fortschr. Phys. 173 (1963), 1527

In relation to the [[Gauss law]]:

* {#Buchholz86} [[Detlev Buchholz]], _Gauss’ law and the infraparticle problem_, Phys.  Lett.  B **174** (1986) 331-334 \[doi:10.1016/0370-2693(86)91110-X\]

See also:

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

[[!redirects infraparticles]]

[[!redirects infra-particle]]
[[!redirects infra-particles]]