extended Lagrangian

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#### Physics
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#### Variational calculus
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#### Geometric quantization
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#### Quantum field theory
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[[!include functorial quantum field theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The notion of _extended Lagrangian_ is the notion of [[Lagrangian]] refined to [[extended quantum field theory]] (or "localized" or "multi-tiered" quantum field theory). At the same time is is the [[prequantum n-bundle]] of the theory in its [[higher geometric quantization]] in top codimension, hence over the point.

## Motivation and Survey
 {#MotivationAndSurvey}

1. [Ingredients of fundamental physics](#IngredientsOfFundamentalPhysics)

1. [Topological local Lagrangian gauge field theory](#TopologicalLocalLagrangianGaugeFieldTheory)

1. [Chern-Simons field theory as a toy example](#ChernSimonsTheoryAsAToyExample)

1. [Topological local field theory in higher category theory](#TopologicalLocalFieldTheoryInHigherGeometry)

1. [Intermezzo: Elements of higher cohesive geometry](#IntermezzoElementsOfHigherCohesiveGeometry)

1. [Chern-Simons theory as the archetypical example](#CSAsArchetypicalExample)

1. [Topological local Lagrangian field theory in higher geometry](#TopologcalLocalLagrangianFieldTheoryInHigherGeometry)

1. [7-dimensional Chern-Simons theory as the next example](#7dCSTheoryAsExample)

1. [∞-Chern-Simons theory as the general example](#InfinityChern-SimonsTheoryAsTheGeneralExample)

### Ingredients of fundamental physics
 {#IngredientsOfFundamentalPhysics}

Ever since [[Issac Newton]], [[theories of physics]] are formulated in the language of [[mathematics]]. Modern physics is formulated in terms of modern mathematics in the most intimate way.
We aim to give here an account at least of central parts of modern physics naturally formulated in the mathematics of _[[higher geometry]]_ that we discussed in _[geometry of physics - Geometry](geometry+of+physics#GEOMETRY)_.

The physics that we are concerned with is _fundamental_ physics,
that part of the large subject of physics which is concerned with elementary  processes and phenomena of physics from which all others are thought to _emerge_ as approximations to collective effects of complex configurations of the elementary
entities.

The best [[theory (physics)|theory]] of fundamental physics as presently understood is known as _[[Einstein-Yang-Mills-Dirac-Higgs theory]]_.
While fundamental, this theory has _free parameters_. These index different flavors of the same general mechanism. Notably the species and the masses of the fundamental [[fermion]] [[particles]] are such parameters, as
is their coupling to the [[force fields]].

Specifying these parameters is called constructing a
_[[model (in theoretical physics)|model]]_ in physics  and specifically a [[phenomenology|phenomenological model]] of fundamental physics,
if aimed at making the observations that are predicted by the theory to closely match those that are being made in experiments. The global choice of these parameters that make the predictions  match all available data to the currently best possible degree is called the _standard model_, specifically the  _[[standard model of particle physics]]_ for the sector in which the large-scale description of the
force of [[gravity]] is approximately negligible, and the
_[[standard model of cosmology]]_
which focuses on that large scale structure.

The following table list some key ingredients of the standard theory of fundamental physics, all of which we discuss in detail further below.

[[!include standard model of fundamental physics - table]]

### Topological local Lagrangian gauge field theory
 {#TopologicalLocalLagrangianGaugeFieldTheory}

While the [[standard model of fundamental physics - table|standard model of fundamental physics]] is a specification of the general  framework of [[Einstein-Yang-Mills-Dirac-Higgs theory]], that theory itself is a particular realization of several deep general principles  of theoretical physics: it is

1. a **[[field (physics)|field]]** theory;
1. a **[[gauge theory|gauge]]** theory;
1. a **[[Lagrangian]]** theory;
1. a **[[local quantum field theory|local]]** theory;
1. a **[[topological field theory|topological]]** theory.

We briefly indicate what this means, a formal discussion of these notions will be given in the  main part below.

* A **[[field (physics)|field]]** theory identifies the basic configurations of a physical system with generalized functions on spacetime, such as [[tensor fields]], and more generally with [[sections]] $\phi$ of some [[fiber bundle]] over spacetime

  $$
    \array{
      && E
      \\
      & {}^{\mathllap{\phi}}{\nearrow} & \downarrow^{\mathrlap{field\;bundle}}
      \\
      X &\stackrel{id}{\to}& X
     }
   $$

  (This is the traditional formulation. At _[[field (physics)]]_ is discussed that the concept of _field [[fiber bundle]]_ needs to be refined to that of field _[[fiber ∞-bundle]]_ in order to accuratly capture all aspects of field theory:   _[The traditional idea of field bundles and its problems](physical+field#IdeaOfFieldBundlesAndItsProblems)_.)

* A **[[gauge theory|gauge]]** theory in the broad sense has not just a [[set]] or [[smooth space]] of field configurations but has a [[groupoid]] or [[stack]]/[[smooth groupoid]] of field configurations: there are equivalences between nominally different field configurations

  $$
    \phi \stackrel{\simeq}{\to} \phi'
  $$

  called _[[gauge transformations]]_.

   A gauge theory in the original sense specifically has a groupoid of [[principal connections]] as its configurations. (Notice that gauge transformations are not in general just a redundancy of the description of the theory, but are genuine data: there are in general non-trivial gauge [[automorphisms]] of a field and the stack of gauge field configurations is not in general equivalent to a plain [[sheaf]].)

* A **[[Lagrangian]]** theory is one which is specified by a function on the space of all its configurations, called the (exponentiated) _[[action functional]]_ $\exp(i S)$. This function is assumed to be the [[integration of differential forms|integral]] of a [[differential form]] $L$ on [[spacetime]] that itself is a function of the field configurations and called the [[Lagrangian]] of the theory.

  $$
    \exp(i S(\phi)) = \exp\left(i \int_X L\left(\phi\right)\left(x\right) \right)
  $$

  The [[critical locus]]

  $$
    \left\{
	 \phi \in \mathbf{Fields}(\Sigma)		|
		d_{\mathrm{var}} S_\phi \simeq 0
	  \right\}
  $$

  of the action functional is called the _[[phase space]]_ of the theory, a point of this is said to be a solution to the _[[equations of motion]]_ of the theory.

* A **[[local quantum field theory|local]]** theory has a Lagrangian form which at each point of spacetime only depends on the values of the fields at that point, and of finitely many derivatives of the fields at that point.

  $$
    \exp(i S(\phi)) = \exp\left(i \int_X L\left(\phi\left(x\right), \nabla \phi\left(x\right)\right)\right)
  $$

  This means that a _[[local Lagrangian]]_ is a horizontal form on the [[jet bundle]] of the field bundle of which the fields are sections.

* A **[[topological field theory|topological]]** theory finally is one where the field bundle is naturally associated to every smooth manifold, typically of some fixed dimension and possibly equipped with  topological [[G-structure]] such as an orientation, but not equipped with any further geometric structure: all geometric structure such as metrics on spacetime are themselves regarded as parts of the field content in a topological theory.

### Chern-Simons field theory as a toy example
 {#ChernSimonsTheoryAsAToyExample}

Most of this was fairly well understood decades ago, by the 1960s. But even with all this conceptual understanding of the structure of fundamental physics available, there were and are a lot of implications of
_[[topological field theory|topological]] [[local  Lagrangian]] [[gauge field theory]]_ which remained elusive.

In the 1980s attention turned to _other_ topological local Lagrangian gauge field theories than that governing the standard model: it turns out that there are many such  [[theories of physics]] that share crucial properties with the standard model but which  otherwise predict, if considered as phenomenological models, physics  radically different from that
which we actually observe experimentally. For instance there are respectable theories of fundamental
physics which describe physics in dimensions other than the three spatial and one time dimension which we clearly observe, theories that contain no force of gravity, others that contain only the force of gravity, theories that contain mirror partners of every species of matter or contain no matter at all, etc.

In short, it was gradually realized, and realized to be important, that, in the platonic world of mathematics and of theoretical physics, there is a whole space of field theories, in which the one underlying the standard model of observed physics is only one single point, even if regarded with its free parameters.

A crucial step forward happened when around 1989 [[Edward Witten]], in this vein,  introduced the study of what is now called _[[Chern-Simons theory]]_ a topological local Lagrangian gauge field theory which, as such, shares
some crucial principles with Einstein-Maxwell-Yang-Mills theory, while at the  same time being  radically different and most importantly:
much simpler to analyze.

The dimension of Chern-Simons theory is 3 instead of 4, but
Chern-Simons theory has a [[gauge field]] just as [[Yang-Mills theory]] does. For $G$ a [[connected topological space|connected]] [[simply connected topological space|simply connected]] [[gauge group]], this is mathematically represented by a
[[infinity-Lie algebroid-valued differential form|Lie algebra-valued differential 1-form]] $A$.

The [[Lagrangian]] of [[Chern-Simons theory]] is the [[Chern-Simons 3-form]]

$$
  L(A)
  \coloneqq
  \mathrm{CS}_3(A)
  \coloneqq
  \langle A \wedge F_A\rangle
  +
  c
  \langle A \wedge [A \wedge A]\rangle
  \,,
$$

whence the name of the theory, and hence the Chern-Simons [[action functional]] is

$$
  \exp(i S(A))
  =
   \exp\left(
     2 \pi i \int_{\Sigma_X} \mathrm{CS}_3(A)
	\right)
$$

At the same time Chern-Simons theory is still being rich in phenomena,
in fact so rich in phenomena that for instance it helped illuminate deep mathematical problems that had not otherwise been understood as much before (properties called _[[knot invariants]]_).

### Topological local field theory in higher category theory
 {#TopologicalLocalFieldTheoryInHigherGeometry}

Another pleasant effect of the introduction of the Chern-Simons "toy model"  for topological field theory was that,
due to its conceptual simplicity, its basic structure could be appreciated also by mathematicians not trained in theoretical physics, who would otherwise fail to see through all the  physics jargon involved in the available discussion of realistic models. Accordingly, immediately after the realization of Chern-Simons field theory,  the first mathematically precise axiomatizations of an $n$-dimensional _[[topological field theory]]_ was proposed, by [[Michael Atiyah]]: "[[functorial quantum field theory]]"

This axiomatization demands in particular that to each [[closed manifold]] of [[dimension]] $(n-1)$ is assigned a [[vector space]]

$$
  Z
   \;\colon\;
  \Sigma_{n-1} \mapsto Z(\Sigma_{n-1}) \in \mathrm{Vect}
$$

thought of as the _[[space of physical states]]_ of the theory on the
spatial slice $\Sigma_{n-1}$ of spacetime.

However, This original axiomatization did not capture the full _[[local quantum field theory|locality]]_ property: the spaces of states are assigned globally to a space $\Sigma_{n-1}$ and are not required to be obtainable by "integrating up local data" over $\Sigma_{n-1}$.

In order to refine this, one needs an axiomatization of field theory that assigns data also to manifolds $\Sigma_{k}$ of [[dimension]] $0 \leq k \leq n$. This should be such that the data assigned to a torus $\Sigma_{k} \times S^1$ is the [[trace]] in a suitable sense, of the data assigned to $\Sigma_{k}$ itself. It turns out that the way to capture this is to think of $Z(\Sigma_k)$ as being a vector space in [[higher category theory]]/[[directed homotopy type theory]]: an [[n-vector space|(n-k)-vector space]].

$$
  \Sigma_k \mapsto Z(\Sigma_k) \in (n-k)\mathrm{Vect}
$$

That something like this should work was hypothetized by [[Dan Freed]] (in _[[Higher Algebraic Structures and Quantization]]_), the precise formulation was later given by [[Jacob Lurie]] (in _[[On the Classification of Topological Field Theories]]_). In the literature the resulting axiomatization is called _[[extended topological field theory]]_ (with "extended" referring to extending the axioms from [[codimension]] 1 to higher codimension) or _[[multi-tiered field theory]]_ (thinking of the data in higher [[codimension]] as being higher "tiers" of the theory).

But in the [[physics]]-community, at least in the [[algebraic quantum field theory]]-community, there is also the well-established term of _[[local quantum field theory]]_ which refers broadly to what the axioms of "extended TQFT" capture. Hence we should think of this as formalizing _topological local field theory_.

### Topological local *Lagrangian* field theory in higher geometry
 {#TopologcalLocalLagrangianFieldTheoryInHigherGeometry}

The formalization of [[extended topological field theory]] thus captures the _[[local quantum field theory|local]]_ aspect of [[field (physics)|field]] [[theory (physics)|theory]]. But we saw [above](#TopologicalLocalLagrangianGaugeFieldTheory) that the theories of fundamental physics have one more crucial property: they are _[[Lagrangian]]_. Here we indicate how to formulate Lagrangian theories that are also local in the sense of [[extended field theory]].

The step of passing from a [[Lagrangian]] to the corresponding
[[quantum field theory]] is called _[[quantization]]_. There are two broad approaches to formalization of what this means, called _[[deformation quantization|algebraic deformation quantization]]_ and _[[geometric quantization]]_.  The first of these deals explicitly with producing [[algebras of quantum observables]], while the second describes explicitly how to construct the [[spaces of states]] that we considered above. Both are at least roughly dual to each other, in the spirit of [[Isbell duality|the general duality]] between [[algebra]] and [[geometry]].
For the following discussion we follow the ideas of [[geometric quantization]], which naturally fit a discussion of _[[geometry of physics]]_.
The following table surveys some keywords in this context which we will discuss in detail further below.

[[!include Isbell duality - table]]

In its traditional formulation, one considers the [[symplectic form]] $\omega$ which is naturally induced by a [[local Lagrangian]] on its (reduced) [[covariant phase space]] and says that a _[[geometric prequantization]]_ of this form is the choice of a $U(1)$-[[principal connection]] $\nabla$ on the phase space whose [[curvature]] $F_\nabla$ is $\omega$, hence a lift in

$$
  \array{
    &&& \mathbf{B}U(1)_{conn}
    \\
    {{prequantum} \atop {circle\;bundle}} && {}^{\mathllap{\nabla}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}} & curvature
    \\
    &PhaseSpace
    &\stackrel{\omega}{\to}&
    \Omega^2_{cl}
    \\
    &&
    sympl.\;form
  }
  \,.
$$

A [[section]] of the [[associated bundle|associated]] [[complex line bundle]]

$$
  \array{
    && P \times_{U(1)}\mathbb{C}
    \\
    & {}^{\mathllap{\Psi}}\nearrow & \downarrow
    \\
    X &\stackrel{id}{\to}& X
  }
$$

is called a _[[wave function]]_ on phase space. One is to choose a [[polarization]] of phase space, equipping it locally with "[[canonical coordinates]]" and "[[canonical momenta]]". Then a _[[quantum state]]_ in geometric quantization is a [[wave function]] $\Psi$ which only depends on the [[canonical coordinates]].

$$
  Z(\Sigma_{n-1})
  \coloneqq
  \{wave\; functions\;of\;canonical\;coordinates\}
  \coloneqq
  \left\{
    polarized\;sections\;of\;prequantum\;line\;bundle
  \right\}
  \,.
$$

From this it is clear that in order to refine this to local (extended, multi-tiered)  [[prequantum field theory]] we are to assign _[[circle n-bundle with connection|higher line bundles]]_ in higher codimension, namely a [[prequantum n-bundle|prequantum (n-k)-bundle]] in [[codimension]] $k$.

As a first indication that this makes good sense, notice that the natural assignment to a field in codimension 0, hence to a field configuration on [[spacetime]], is its [[action functional]]. Being a $U(1)$-valued function, this is naturally thought of as a 0-bundle, the _[[prequantum 0-bundle]]_.

So we are after a picture as indicated in the following table.

| | | [[extended prequantum field theory|local/extended Lagrangian field theory]] |  | [[extended quantum field theory|local/extended quantum field theory]]  |
|--|--|--|--|--|
| closed piece of [[spacetime]] of [[dimension]] $0 \leq k \leq n$ |  [[prequantum field theory]] | [[prequantum n-bundle|prequantum (n-k)-bundle]] | [[geometric quantization]] | [[space of states|space of]] $(n-k)$-[[wave functions]] |
| [[closed manifold]] of [[dimension]] $0 \leq k \leq n$ | $\stackrel{local\;prequantum\;field\;theory}{\mapsto}$ | [[circle n-bundle with connection|circle (n-k)-bundle with connection]] on [[moduli stack]] of [[field (physics)|field configurations]] | $\stackrel{polarized\; sections}{\mapsto}$ | [[n-vector space|(n-k)-vector space of states]] |
| $\Sigma_k$ | $\mapsto$ | $\exp(2 \pi i \int_{\Sigma_k}[\Sigma_k,\mathbf{L}]) : [\Sigma_k, \mathbf{Fields}] \to \mathbf{B}^{n-k} U(1)_{conn}$ | $\mapsto$ | $Z(\Sigma_{n-k})$ |

### Intermezzo: Recalling elements of higher cohesive geometry
 {#IntermezzoElementsOfHigherCohesiveGeometry}

The idea of _[[extended prequantum field theory]]_ indicated [above](#TopologcalLocalLagrangianFieldTheoryInHigherGeometry)  is best illustrated by the example of [[Chern-Simons field theory]]. We talk about this in a moment [below](#CSAsArchetypicalExample), but since in order to do so we need now at least some basic notions from [[cohesion|cohesive]] [[higher geometry]], we very briefly recall these first, just for reference. For more discussion see at _[geometry of physics - smooth homotopy types](geometry+of+physics#SmoothnGroupoids)_  and _[geometry of physics - Maurer-Cartan forms](geometry+of+physics#MaurerCartanForms)_ and _[geometry of physics  - principal connections](geometry+of+physics#PrincipalConnections)_

Fix some [[site]] $\mathcal{C}$ of test spaces which encode some notion of  [[geometry]]. The running example to keep in mind is the site [[SmthMfd]] of [[smooth manifolds]], or equivalently its full [[dense subsite]] of [[CartSp]] on manifolds of the form $\mathbb{R}^n$.  Assume for simplicity that the site has _[[point of a topos|enough points]]_ (which [[SmthMfd]] does).

Then the [[(2,1)-category]] of [[(2,1)-sheaves]] ([[stacks]]) on $\mathcal{C}$ is the [[simplicial localization]]

$$
  Sh_{(2,1)}(\mathcal{C})
  \simeq
  L_{le} Func(\mathcal{C}^{op}, Grpd)
  \,.
$$

of that of [[groupoid]]-valued [[presheaves]] on $\mathbf{C}$ at the _local equivalences_. This means that $Sh_{(2,1)}(\mathcal{C})$ is the result of turning [[stalk]]-wise ("local") [[equivalence of groupoids|equivalence of groupoids]] into actual [[equivalences]].

For the running example of $\mathcal{C} \simeq $ [[SmthMfd]] we may think of an object in $Sh_{(2,1)}(\mathcal{C})$ as a _[[smooth groupoid]]_. Examples include [[smooth manifolds]], [[diffeological space]],  [[orbifolds]], [[Lie groupoids]]/[[differentiable stacks]], sheaves of smooth differential forms, etc.

The higher refinement of groupoids which we need are _[[∞-groupoids]]_. The collection of these is equivalently that of [[simplicial sets]] or (compactly generated weakly Hausdorff) [[topological spaces]] with the [[weak homotopy equivalences]] universally turned into actual [[homotopy equivalences]]:

$$
  \infty Grpd \simeq L_{whe} sSet \simeq L_{whe} Top
$$

The combination of these two aspects, the [[geometry]] and the [[homotopy theory]], is that of [[(∞,1)-sheaves]]/[[∞-stacks]], here we write $\mathbf{H}$ for the collection of [[presheaves]] of [[simplicial sets]] ([[simplicial presheaves]]) with the local ([[stalk]]-wise) [[weak homotopy equivalences]] universally turned into actual homotopy equivalences:

$$
  \array{
    \mathbf{H} \coloneqq
    & Sh_\infty(\mathcal{C})
    &
    \stackrel{
      \overset{\infty-stackif.}{\leftarrow}
    }{
      \hookrightarrow
    }
  &
  PSh_\infty(\mathcal{C})
  \\
  & \uparrow^{\mathrlap{\simeq}} && \uparrow^{\mathrlap{\simeq}}
  \\
  & L_{lwhe} Func(\mathcal{C}^{op}, sSet)
  &
  \stackrel{
    \overset{\mathbb{L} id}{\leftarrow}
   }{
     \underset{\mathbb{R} id}{\rightarrow}
  }&
  L_{whe} Func(\mathcal{C}^{op}, sSet)
  }
  \,.
$$

This $\mathbf{H}$ is the _[[(∞,1)-topos]]_ over $\mathcal{C}$. This is the kind of context for the [[higher geometry]] encoded by $\mathcal{C}$ in which all of our constructions take place.

In order to formulate not just [[kinematics]] but also [[dynamics]] in [[physics]], we need to require that $\mathbf{H}$ has an intrinsic notion of what it means to have a _path_ or _[[trajectory]]_ inside one of its objects. This is axiomatized by the notion of  _[[cohesion]]_:

if $\mathcal{C}$ satisfies some properties that make it an an [[infinity-cohesive site]], the the [[global section geometric morphism]] of $\mathbf{H}$ extended to an [[adjoint quadruple]] of [[(∞,1)-functors]]:

$$
  \array{
   \mathbf{H}
    &
   \stackrel{\overset{\Pi}{\to}}{\stackrel{}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}}}
   &
  \infty Grpd
  \\
  \uparrow^{\mathrlap{\simeq}}
  &&
  \uparrow^{\mathrlap{\simeq}}
  \\
  L_{lwhe} Func(\mathcal{C}^{op}, sSet)
  &\stackrel{\overset{hocolim_{\mathcal{C}}}{\to}}{\stackrel{\overset{const}{\leftarrow}}{\stackrel{X \mapsto X(*)}{\stackrel{\overset{}{\to}}{\underset{}{\leftarrow}}}}}&
  L_{whe} sSet
  }
$$

with some mild extra condition on $\Pi$ and $coDisc$. This is the case in particular for $\mathcal{C} = $ [[SmthMfd]] in which case we say that $\mathbf{H}$ is the [[cohesive (∞,1)-topos]] of _[[smooth ∞-groupoids]]_.

The defining [[adjoint quadruple]] of [[(∞,1)-functors]] induces an [[adjoint triple]] of [[comonad|co]][[(∞,1)-monads]]:

| [[shape modality]] | $\dashv$ | [[flat modality]] | $\dashv$ |[[sharp modality]] |
|--|--|--|--|--|
| $\mathbf{\Pi} \coloneqq Disc \circ \Pi$ | | $flat \coloneqq Disc \circ \Gamma$ |   | $\sharp \coloneqq coDisc \circ \Gamma$  |

For $G \in Grp(\mathbf{H})$ a [[group object in an (∞,1)-category|group object]] in $\mathbf{H}$, hence a geometric [[∞-group]], we have by these axioms of [[cohesion]] a [[pasting diagram]] of [[(∞,1)-pullbacks]]

$$
  \array{
    G &\to& *
    \\
    \downarrow^{\mathrlap{\theta_G}} && \downarrow
    \\
    \flat_{dR} \mathbf{B}G
    &\to&
    \flat \mathbf{B}G
    \\
    \downarrow && \downarrow
    \\
    * &\to& \mathbf{B}G
  }
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
  \array{
    \infty-group &\to& *
    \\
    {}^{\mathllap{MC\;form}}\downarrow && \downarrow
    \\
    deRham\, coefficients &\to& flat\;coefficients
    \\
    \downarrow && \downarrow
    \\
    * &\to& principal\,coefficients
  }
  \,.
$$

Here the [[delooping]] $\mathbf{B}G$ is the moduli of geometric $G$-[[principal ∞-bundles]],  $\flat_{dR} \mathbf{B}G$ is that of flat [[infinity-Lie algebroid-valued differential form
|L-∞ algebra valued differential forms]] and $\theta_G$ is the _[[Maurer-Cartan form]]_ on the [[smooth ∞-group]] $G$.

If $\mathbb{G}$ is "slightly abelian" in that it is equipped with the structure of a [[braided ∞-group]], then we write

$$
  curv_{\mathbb{G}}
  \;\colon\;
  \mathbf{B}\mathbb{G}
  \stackrel{\theta_{\mathbf{B}\mathbb{G}}}{\to}
  \flat_{dR}\mathbf{B}^2 \mathbb{G}
$$

for the [[Maurer-Cartan form]] of its [[delooping]] [[∞-group]] and call this the _universal [[curvature]] characteristic_.

Then _$\mathbb{G}$-[[differential cohomology]]_ is
_$curv_{\mathbf{B}\mathbb{G}}$-[[twisted cohomology]]_.
In particular, if we consider globally defined curvature forms

$$
  \Omega_{cl}(-,\mathbb{G}) \to \flat_{dR}\mathbf{B}^2\mathbb{G}
$$

then the [[(∞,1)-pullback]]

$$
  \array{
    \mathbf{B}\mathbb{G}_{conn}
    &\to&
    \Omega(-,\mathbb{G})
    \\
    \downarrow && \downarrow^{\mathrlap{F_{(-)}}}
    \\
    \mathbf{B}\mathbb{G}
    &\stackrel{curv_{\mathbf{B}\mathbb{G}}}{\to}&
    \flat_{dR}\mathbf{B}^2 \mathbb{G}
  }
$$

is the moduli for [[principal ∞-connections]].

Now for

$$
  \mathbf{c}
  \;\colon\;
  \mathbf{B}G
  \to
  \mathbf{B}\mathbb{G}
$$

a [[universal characteristic map]] we say that a _differential refinement_ is a choice of $\mathbf{B}G_{conn}$ and $\hat \mathbf{c}$ in

$$
  \array{
    \flat \mathbf{B}G
    &\stackrel{\flat \mathbf{c}}{\to}&
    \mathbf{B}\mathbb{G}
    \\
    \downarrow && \downarrow^{}
    \\
    \mathbf{B}G_{conn} &\stackrel{\hat \mathbf{c}}{\to}&
    \mathbf{B}\mathbb{G}_{conn}
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}G
    &\stackrel{\mathbf{c}}{\to}&
    \mathbf{B}\mathbb{G}
  }
  \,.
$$

These are the [[extended Lagrangians]] in our discussion of [[extended prequantum field theory]].

In particular if, in the context of [[smooth ∞-groupoids]], we consider $\mathbb{G} = \mathbf{B}^{n-1}U(1)$ the [[circle n-group]] then

$$
  \mathbf{B}\mathbb{G}_{conn}
  \simeq
  \mathbf{B}^n U(1)_{conn}
  \simeq
  DK(  U(1) \to \Omega^1 \to \cdots \to \Omega^n)
$$

is given as the [[∞-stackification]] of the image under the [[Dold-Kan correspondence]] of the _[[Deligne complex]]_ for [[ordinary differential cohomology]].

In this case there is [[fiber integration in ordinary differential cohomology]] refined to moduli $\infty$-stacks: for $\Sigma_k$ an oriented [[closed manifold]] of [[dimension]] $k$ we have a map of [[moduli ∞-stacks]]

$$
  \exp\left(
   2 \pi i \int_{\Sigma} (-)
  \right)
  \;\colon\;
  [\Sigma_k, \mathbf{B}^n U(1)_{conn}]
  \to
  \mathbf{B}^{n-k} U(1)_{conn}
  \,.
$$

Composed with the [[internal hom]] this is _[[transgression]]_ of differential cocycles. The transgressions of an [[extended Lagrangian]] as above are (the moduli for) the higher [[prequantum n-bundles]] that we consider here.

### Chern-Simons theory as the archetypical example
 {#CSAsArchetypicalExample}

For illustration purposes, we survey how the [[Chern-Simons field theory]] considered as a toy model for topological local Lagrangian gauge field theory considered [above](#ChernSimonsTheoryAsAToyExample) has in [[higher geometry]] a [[prequantum n-bundle|prequantum (n-k)-bundle]] associated in [[codimension]] $k$.

First consider again $G$ to be a [[connected topological space|connected]] and [[simply connected topological space|simply connected]] [[simple Lie group]]. Then the [[integral cohomology]] of the [[classifying space]] of $G$ is

$$
  H^4(BG , \mathbb{Z}) \simeq \mathbb{Z}
  \,.
$$

An element in here is a [[universal characteristic class]] of $G$-[[principal bundles]]. In Chern-Simons theory this is going to be what traditionally is called the [[level (Chern-Simons theory)|level]] of the theory.

We write

$$
  c : B G \to  B^3 U(1) \simeq K(\mathbb{Z},4)
$$

for a representative [[universal characteristic map]] of the corresponding degree-4 [[universal characteristic class]].

Under [[geometric realization of cohesive ∞-groupoids|geometric realization]] as just [recalled](#IntermezzoElementsOfHigherCohesiveGeometry) this has an essentially unique lift to _smooth cohomology_, hence to a morphism of [[moduli ∞-stacks]]

$$
  \mathbf{c}
  \;\colon\;
  \mathbf{B}G
  \to
  \mathbf{B}^3 U(1)
$$

which, in a precise sense, is the [[Lie integration]] of the canonical 3-[[cocycle]]

$$
  \langle -, [-,-]\rangle \;\colon\; \mathfrak{g} \to \mathbf{B}^2 \mathbb{R}
$$

in the [[Lie algebra cohomology]] of the [[Lie algebra]] $\mathfrak{g}$ of $G$ in that

$$
  \mathbf{c} \simeq \tau_1 \exp(\mu)
  \,.
$$

Moreover, this smooth universal characteristic map  has a further lift to [[differential cohomology]]

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

This modulates a [[circle n-bundle with connection|circle 3-bundle with connection]] -- the "[[Chern-Simons circle 3-bundle with connection]]" -- on the [[smooth infinity-groupoid|smooth]] [[moduli stack]] of smooth $G$-[[principal connections]].

$$
  \array{
    \mathbf{B}G_{conn}
    &\stackrel{\hat \mathbf{c}}{\to}&
    \mathbf{B}^3 U(1)_{conn}
    \\
    \downarrow && \downarrow &&&&& \in  \mathbf{H}
    \\
    \mathbf{B}G
    &\stackrel{\mathbf{c}}{\to}&
    \mathbf{B}^3 U(1)
    \\
    && &&& \downarrow^{\mathrlap{{\vert \Pi(-)\vert}}}
    \\
    B G &\stackrel{c}{\to}& B^3 U(1) &&&&& \in L_{whe} Top
  }
$$

One finds that the [[transgression]] of this to codimension 0 is the Chern-Simons action functional discussed above. The other transgressions are

| [[dimension]] |  | [[prequantum n-bundle|prequantum (3-k)-bundle]] |  |
|---------------|--|-------------------------------------------------|--|
| $k = 0$ | [[twisted differential string structure|differential fractional first Pontryagin class]] | $\hat \mathbf{c} \;\colon\; \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$ |  ([FSS](#FSS))  |
| $k = 1$ | [[WZW model]] [[background gauge field|background]] [[B-field]] | $G \to [S^1, \mathbf{B}G_{conn}] \stackrel{[S^1, \hat \mathbf{c}]}{\to} [S^1, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{S^1} (-))}{\to} \mathbf{B}^2 U(1)_{conn}$ |  |
| $k = 2$ | off-sheff [[prequantum bundle]] | $[\Sigma_2, \mathbf{B}G_{conn}] \stackrel{[\Sigma_2, \hat \mathbf{c}]}{\to} [\Sigma_2, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_2} (-))}{\to} \mathbf{B} U(1)_{conn}$ | |
| $k = 3$ | [[action functional]] | $[\Sigma_3, \mathbf{B}G_{conn}] \stackrel{[\Sigma_3, \hat \mathbf{c}]}{\to} [\Sigma_3, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_3} (-))}{\to} U(1)$

Here the [[WZW model|WZW]] [[B-field]] is the topological part of a 2-dimensional local Lagrangian field theory called the _[[Wess-Zumino-Witten model]]_. One says that the WZW model is related by the [[holographic principle]] with [[3d Chern-Simons theory]].

If the [[gauge group]] $G$ is not simple and simply connected, then there are other canonical [[universal characteristic classes]] in $H^4(B G, \mathbb{Z})$. If these have a differential refinement, they give rise to the corresponding Chern-Simons theory.

Notably in the case that $G = U(1)$ is the [[circle group]] (hence not [[simply connected topological space|simply connected]]) there is the [[cup product]] of the universal [[first Chern class]] with itself

$$
  c_1 \cup c_1 \;\colon\; B U(1) \to B^3 U(1)
  \,.
$$

The smooth and differential refinement of the [[first Chern class]] itself $c_1 \;\colon\; B U(1) \to B U(1) \simeq K(\mathbb{Z},2) $ is tautological: this is simply the identity

$$
  \widehat \mathbf{c}_1 \coloneqq id
  \;\colon\;
  \mathbf{B}U(1)_{conn}
  \to
  \mathbf{B}U(1)_{conn}
  \,.
$$

The nontrivial part is that, as one can see, the  [[cup product]] on [[differential cohomology]] refines to [[moduli ∞-stacks]] of [[circle n-bundle with connection]] as the [[∞-stackification]] of what is called the [[Beilinson-Deligne cup product]], and so we have

$$
  \widehat \mathbf{c}_1 \cup \widehat \mathbf{c}_1
  \;\colon\;
  \mathbf{B}U(1)_{conn}
  \to
  \mathbf{B}^3 U(1)_{conn}
  \,.
$$

This is the [[extended Lagrangian]] for abelian Chern-Simons theory. On field configuration which happens to be given by a globally defined 1-form $A$ its value is

$$
  \exp\left(
    2\pi i \int_{\Sigma_3} A \wedge d A
  \right)
  \in
  U(1)
  \,.
$$

But now not every field configuration is necessarily given by a globally defined 1-form anymore, and more generally the action functional has a more sophisticated expression.

### 7-dimensional Chern-Simons theory as the next example
 {#7dCSTheoryAsExample}

So far we have considered general motivation for [[extended prequantum field theory]], and indication how its application to the well-known case of [[3d Chern-Simons theory]] naturally captures the characteristics of the theory. Now we consider further examples whose higher geometry has not necessarily been considered traditionally.

An immediate class of further examples is obtained by abelian [[higher Chern-Simons theory]] induced by the [[cup product]] of the [[first Chern class]] with itself

$$
  [DD_{2k} \cup DD_{2k}] \in H^{4(k+1)}(B^{2k+1} U(1), \mathbb{Z})
  \,.
$$

This has a refinement ([FSS cup](#FSSCup)) to a [[differential characteristic class]] by the
_[[Beilinson-Deligne cup product]]_ of the [[differential characteristic class|differential refinement]] of the higher [[Dixmier-Douady class]] with itself:

$$
  \widehat \mathbf{DD}_{2k}
  \cup
  \widehat \mathbf{DD}_{2k}
  \;\colon\;
  \mathbf{B}^{2k+1} U(1)_{conn}
  \to
  \mathbf{B}^{4k+3} U(1)_{conn}
  \,.
$$

In particular there is the abelian [[7d Chern-Simons theory]]

$$
  \widehat \mathbf{DD}_2
  \cup
  \widehat \mathbf{DD}_2
  \;\colon\;
  \mathbf{B}^3 U(1)_{conn}
  \to
  \mathbf{B}^7 U(1)_{conn}
  \,.
$$

An argument in ([Witten 98](#AdS-CFT#Witten98)) says that this abelian
[[7d Chern-Simons theory]] (or more precisely a [[quadratic refinement]] of it) is related to the _[[self-dual higher gauge theory]]_ sector of the [[6d (2,0)-supersymmetric QFT]] on the [[worldvolume]] of a single [[M5-brane]]_  in analogy to how [[3d Chern-Simons theory]] is related to the [[WZW model]]:

| [[schreiber:∞-Chern-Simons theory]] | $\leftarrow$[[holographic principle]] $\rightarrow$| [[schreiber:∞-Wess-Zumino-Witten theory ]] | [[Kaluza-Klein reduction]] $\to$ |   |
|--|--|--|--|--|
| [[3d Chern-Simons theory]]  | [3dCS-2dCFT](holographic+principle#3dCS-2dCFT) | [[WZW model]] | | |
| [[7d Chern-Simons theory]]  | [AdS7/CFT6 duality](AdS-CFT#AdS7CFT6) | [[6d (2,0)-supersymmetric QFT]] | | [[N=4 D=4 super Yang-Mills theory]] |

The argument in ([Witten 98](#AdS-CFT#Witten98)) also says that in view of [AdS7/CFT6 duality](AdS-CFT#AdS7CFT6) the [[field (physics)|field]] of the 7d theory here is to be identified withe the _[[supergravity C-field]]_ in [[11-dimensional supergravity]]/[[M-theory]].

But the [[supergravity C-field]] is known to have moduli that are in general more complicated than just $\mathbf{B}^3 U(1)_{conn}$ (due to a [[Green-Schwarz anomaly cancellation]] mechanism in [[11-dimensional supergravity]] and a "flux quantization" condition). In ([FSS 7dCD](#FSS7dCS)) the corrected moduli are dicussed and found to involve a coupling of the abelian 7d theory with a nonabelian 7d Chern-Simons theory whose [[field (physics)|fields]] are twisted higher spin connections: "[[twisted differential string structures]]".

To see what this is, recall that [above](#CSAsArchetypicalExample) we interpreted the [[extended Lagrangian]] of [[3d Chern-Simons theory]] for a simply connected compact simple gauge group $G$ as the smooth and moreover differential refinement of the canonical [[universal characteristic class]] in $H^4(B G, \mathbb{Z})$. Specifically for $G$ the [[spin group]] this is the [[first fractional Pontryagin class]], represented by a [[characteristic map]]

$$
  \tfrac{1}{2}p_1 \;\colon\; B Spin \to B^3 U(1)
  \,.
$$

This can be understood as a higher analog of the [[second Stiefel-Whitney class]] $w_2 \;\colon\; B SO \to B^2 \mathbb{Z}$ which is the universal [[obstruction]] to [[spin structure]]. We say that $\tfrac{1}{2}p_1$ is the universal [[obstruction]] to a _[[higher spin structure]]_ called _[[string structure]]_ (this and the motivation for this terminology is discussed in detail in _[geometry of physics - Fields - Examples - Fields of gravity and G-structure](geometry+pf+physics##FieldsOfGravityAndGeneralizedGeometry)_). These higher obstructions are the (dual) [[k-invariants]] of the [[Whitehead tower]] of $B O$:

[[!include higher spin structure - table]]

From the point of view of [[extended prequantum field theory]] we are to regard each horizontal map in this tower as classifying the potential underlying [[instanton sector]] of an [[extended Lagrangian]]/[[prequantum n-bundle]], and we may ask if these lift to actual extended Lagrangian.

The next step up the ladder is the universal [[second fractional Pontryagin class]]. Its [[twisted differential string structure|differential refinement]] has been constructed in ([FSS diff coc](#FiorenzaSchreiberStasheff)).

$$
  \array{
    \mathbf{B}String_{conn}
    &\stackrel{\widehat {\tfrac{1}{6}\mathbf{p}_2}}{\to}&
    \mathbf{B}^7 U(1)_{conn}
    \\
    \downarrow && \downarrow &&&&& \in \mathbf{H}
    \\
    \mathbf{B}G
    &\stackrel{ \tfrac{1}{6}\mathbf{p}_2 }{\to}&
    \mathbf{B}^7 U(1)
    \\
    && &&& \downarrow^{\mathrlap{{\vert \Pi(-)\vert}}}
    \\
    B String &\stackrel{\tfrac{1}{6}p_2}{\to}& B^7 U(1) &&&&& \in L_{whe} Top
  }
  \,.
$$

This now defines an [[extended Lagrangian]] for a  [[7d Chern-Simons theory]] on [[String 2-group]]-[[2-connections]].

Coming back to the above arguments about the relation of the 7d theory to a 6d theory, these suggest that the 6d theory involves  [[prequantum n-bundle|prequantum 6-bundle]]-analog of the [[WZW model|WZW 2-bundle]].

Indeed, the assignment of a higher Wess-Zumino-Witten-type [[prequantum n-bundle|prequantum (n-1)-bundle]] to an extended prequantum Lagrangian works very generally: the _prequantum $(n-1)$-bundle_ is the WZW-type $(n-1)$-bundle of the theory.
Its underlying $(n-1)$-bundle is simply the [[looping]] of the $n$-bundle underlying the [[extended Lagrangian]]. Specifically, the [[string 2-group]]-[[7d Chern-Simons theory]] has a WZW- prequantum 6-bundle on the [[string 2-group]] which is modulated by the [[looping]] $\Omega(\tfrac{1}{6}\mathbf{p}_2)$ of the differential characteristic map $\tfrac{1}{2}\mathbf{p}_2$

$$
  \array{
    String
    && &\stackrel{\Omega \left( \tfrac{1}{6}\mathbf{p_2}\right)}{\to}& &&
    \mathbf{B}^6 U(1)
    \\
    \downarrow
    && && && \uparrow
    \\
    String //_{Ad} String
    &
    \stackrel{\simeq}{\to}
    &
    [\mathbf{\Pi}(S^1), \mathbf{B}String]
    &
    \stackrel{[\mathbf{\Pi}(S^1), \tfrac{1}{2}\mathbf{p}_2]}{\to}
    &
    [\mathbf{\Pi}(S^1), \mathbf{B}^7 U(1)]
    &\simeq&
    \mathbf{B}^6 U(1) // \mathbf{B}^6 U(1)
  }
  \,.
$$

Here the lower factorization of this map shows canonically that this 6-bundle is [[equivariant]] with respect to the [[adjoint action|adjoint]] [[∞-action]] of the [[string 2-group]] on itself, the analog of the ordinary adjoint equivariance of the ordinary WZW 2-bundle.

Therefore in summary we find the following identifications of [[extended prequantum field theory]] with (semi-)traditional notions:

[[!include extended prequantum field theory - table]]

### $\infty$-Chern-Simons theory as the general example
 {#InfinityCSAsGeneralExample}

In the examples of [[2d Chern-Simons theory]] and [[7d Chern-Simons theory]] that we have discussed above, the fully extended Lagrangian is equivalently the [[prequantum n-bundle]] on the universal [[moduli ∞-stack]] of [[field (physics)|fields]], for $n = 3$ and $n = 7$, respectively.

But in fact this is the general situation for fully extended Lagrangians:

if $\mathbf{Fields} \in \mathbf{H}$ is the object that serves as the [[moduli ∞-stack]] of [[field (physics)|fields]] for some $n$-dimensional [[theory (physics)|theory]] in that for any [[spacetime]]/[[worldvolume]] $\Sigma_n$ of [[dimension]] $n$ the space of fields is the [[mapping stack]] $[\Sigma_n, \mathbf{Fields}]$, then the corresponding [[prequantum n-bundle]] has to be over this object $\mathbf{Fields}$ and hence to be modulated by a map of moduli $\infty$-stacks of the form

$$
  \mathbf{L} \;\colon\;
  \mathbf{Fields}
  \to
  \mathbf{B}^n U(1)_{conn}
$$

hence by a fully extended Lagrangian.

Above we considered the case that $\mathbf{Fields} \simeq \mathbf{B}G_{conn}$ for a [[Lie group]] $G$ or $\mathbf{Fields} \simeq \mathbf{B}String_{conn}$ for the [[string 2-group]]. In general for $G \in Grpd(\mathbf{H})$ a [[smooth ∞-group]] an extended Lagrangian as above
encodes an [[extended prequantum field theory|extended prequantum]] _[[higher gauge theory]] of Chern-Simons type_:

the [[curvature]] $(n+1)$-form of $\mathbf{L}$ (i.e. the [[n-plectic geometry|pre n-plectic form]] of the [[n-plectic geometry|n-plectic]] [[∞-stack]] $\mathbf{Fields}$ )

$$
  F_{\mathbf{L}}
  \;\colon\;
  \mathbf{B}G_{conn}
  \stackrel{\mathbf{L}}{\to}
  \mathbf{B}^n U(1)_{conn}
  \stackrel{F_{(-)}}{\to}
  \Omega^{n+1}_{cl}
$$

is -- by [[∞-Chern-Weil theory]] -- an [[invariant polynomial]] of the [[∞-Lie algebra]] $\mathfrak{g}$ of $G$. Accordingly, the [[action functional]]

$$
  \exp\left(
    2 \pi i \int_{\Sigma_n} [\Sigma_n, \mathbf{L}]
  \right)
  \;\colon\;
  [\Sigma_n, \mathbf{B}G_{conn}]
  \to
  U(1)
$$

locally takes an [[infinity-Lie algebroid-valued differential form
|L-∞ algebra valued differential form]] $A$ (the [[higher gauge fields]]) to a [[Lagrangian]] $CS(A)$ which locally is a differential form with the property that $ d CS(A) = F_A$.
This means that
$\exp\left( 2 \pi i \int_{\Sigma_n} [\Sigma_n, \mathbf{L}] \right)$ is a [[higher gauge theory]] _of Chern-Simons type_. We say that it is an _[[schreiber:∞-Chern-Simons theory]]_.

This remains true if $\mathbf{Fields}$ is of more general form, so that fields are [[infinity-Lie algebroid-valued differential form
|L-∞ algebroid valued differential form]].

This means that fully [[extended prequantum field theory]] is considerably more restrictive than the traditional notion of [[local Lagrangian]] field theory, just as [[extended topological quantum field theory]] is much more restrictive than non-extended [[TFT]].

1. **fewer choices in prequantization** Traditional [[geometric quantization]] involves making considerable choices. Notably the choice of [[prequantum bundle]] lifting the [[symplectic form]]. In [[extended prequantum field theory]] the prequantum bundles in codimension 1 are constrained to all arise by [[transgression]] from a [[prequantum n-bundle]] in codimension $n$. This is a strong coherence condition that drastically reduces the available choices.

   For instance for $G$ a compact Lie group, there is an [[isomorphism]]

   $$
     H^n(B G, \mathbb{Z})
     \simeq
     \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n U(1))
     \,.
   $$

   This means that if $H^n(B G, \mathbb{Z})$ has no [[torsion subgroup|torsion]], then there is _at most one_ extended prequantization of a given invariant polynomial.

1. **selecting physically natural theories** While the traditional formalization of [[local Lagrangian]] field theory does capture crucial aspects of those [[theories (physics)|theories]] that one is interested in in [[physics]], it still admits many "unnatural theories". The _generic_ [[local Lagrangian]] (in the sense of a horizontal form on the [[jet bundle]] of the [[field bundle]]) contains many higher order derivatives of fields, non-polynomial interactions etc., properties that the typical [[theory (physics)|theories]] of relevance in theoretical physics do no share.

The remaining question then is how accurately [[extended prequantum field theory]] constrains the space of possible [[local Lagrangians]]. In other words: which theories of interest can not be thought of as [[schreiber:∞-Chern-Simons theories]] or [[schreiber:∞-Wess-Zumino-Witten theory ]]?

As a hint to the answer of this question, here is a list of names examples of [[schreiber:∞-Chern-Simons theories]]. We discuss the items of the list in detail in the following.

* [[higher dimensional Chern-Simons theory]]

  * [[1d Chern-Simons theory]]

    * [[Wilson loop]]

  * [[2d Chern-Simons theory]]

    * [[Poisson sigma-model]],

    * [[A-model]], [[B-model]] [[topological string]]

  * [[3d Chern-Simons theory]]

    * [[Chern-Simons theory]]

    * [[Dijkgraaf-Witten theory]]

    * [[Courant sigma-model]]

    * [[3d quantum gravity]]

  * [[4d Chern-Simons theory]]

    * [[Yetter model]]

    * [[BF-theory]]

  * [[5d Chern-Simons theory]]

  * [[6d Chern-Simons theory]]

  * [[7d Chern-Simons theory]]

    * topological [AdS7/CFT6](http://ncatlab.org/nlab/show/AdS-CFT#AdS7CFT6)-sector

    * [[topological M-theory]]

* [[schreiber:∞-Chern-Simons theory]]

  * [[∞-Dijkgraaf-Witten theory]]

  * [[higher dimensional Chern-Simons gravity]]

  * [[AKSZ sigma-model]]

  * [[string field theory]]

The last example here brings us back full circle to where we started this motivation with [above](#IngredientsOfFundamentalPhysics): by the central property of [[string theory]], the action functional for [[string field theory]] (or some [[KK-reduction]] thereof) contains [[Einstein-Yang-Mills theory]].

(...)

## Definition

Let $\mathbf{H}$ be an ambient [[(∞,1)-topos]], typically a [[cohesive (∞,1)-topos]] such as [[Smooth∞Grpd]] or one of its [[slice (∞,1)-topos|slices]]. For $n \in \mathbb{N}$ let $\mathbf{B}^n U(1)_{conn} \in \mathbf{H}$ be the [[moduli ∞-stack]] of [[circle n-bundles with connection]]. Let moreover $\mathbf{Fields} \in \mathbf{H}$ be the given [[moduli ∞-stack]] of [[field (physics)|field configurations]], so that for $\Sigma$ some [[worldvolume]] [[manifold]] a [[field (physics)|field]] configuration on $\Sigma$ is a map in $\mathbf{H}$ of the form

$$
  \phi \colon \Sigma \to \mathbf{Fields}
$$

a [[gauge transformation]] $\phi \Rightarrow \phi'$ is a [[homotopy]] between two such maps, and a [[higher gauge transformation]] is a higher homotopy.

Then an **extended Lagrangian** or [[prequantum n-bundle]] for an $n$-dimensional [[quantum field theory]] with fields classified by $\mathbf{Fields}$ is a map in $\mathbf{H}$ of the form

$$
  \mathbf{L}
  \;\colon\;
  \mathbf{Fields}
  \to
  \mathbf{B}^n U(1)_{conn}
  \,.
$$

By post-[[composition]] this sends a field $\pho \colon \Sigma_n \to \mathbf{Fields}$ to the [[circle n-bundle with connection]] on $\Sigma_n$ modulated by

$$
  \mathbf{L}(\phi)
  \;\colon\;
  \Sigma_n
  \stackrel{\phi}{\to}
  \mathbf{Fields}
  \stackrel{\mathbf{L}}{\to}
  \mathbf{B}^n U(1)_{conn}
  \,.
$$

This is locally  -- restricted to some [[good cover]] $U \to \Sigma_n$ of $\Sigma_b$) given by a [[differential n-form]]

$$
  L(\phi) \omega \in \Omega^n(U)
  \,,
$$

where $L(\phi) \in C^\infty(U)$ is a [[smooth function]] and where $\omega \in \Omega^n(U)$ is some [[volume form]] or some other prespecified [[measure]]. This function $L(\phi)$ is the ordinary [[Lagrangian]].

Over an [[orientation|oriented]] [[closed manifold]] $\Sigma_n$ of [[dimension]] $n$ the corresponding [[action functional]] is the [[transgression]] of $\mathbf{L}$ to the mapping stack/[[internal hom]] out of $\Sigma_n$, hence the composite

$$
  \exp\left(
    2\pi i \int_{\Sigma_n} [\Sigma_n, \mathbf{L}]
  \right)
  \;\colon\;
  [\Sigma_n, \mathbf{Fields}]
  \stackrel{[\Sigma_n, \mathbf{L}]}{\to}
  [\Sigma_n, \mathbf{B}^n U(1)_{conn}]
  \stackrel{\exp(2 \pi in \int_{\Sigma_n})(-) }{\to}
  U(1)
  \,,
$$

where the first map is the [[internal hom]] on morphisms and the second is the [[higher holonomy]] map of a [[circle n-bundle with connection]] over $\Sigma_n$, hence the [[fiber integration in ordinary differential cohomology]].

For $\phi : \Sigma_n \to \mathbf{Fields}$ a field such that the [[circle n-group|circle (n-1)-group]] [[principal ∞-bundle]] underlying $\mathbf{L}(\phi)$ is trivializable and trivialized, the differential form $L(\phi)\omega$ discussed above descends to $X$ itself and the [[action functional]] is simply the [[integration of differential forms]]

$$
  \phi \mapsto \exp\left(2 \pi i \int_{\Sigma_n} L(\phi) \omega \right)
  \,.
$$

This special case is the relation between [[Lagrangians]] and [[action functionals]] as traditionally considered.

Generally, for $0 \leq k \leq n$ and $\Sigma_{k}$ an oriented [[closed manifold]] of [[dimension]] $k$, transgression as above yields a map in $\mathbf{H}$ of the form

$$
  \exp\left(
    2\pi i \int_{\Sigma_k} [\Sigma_n, \mathbf{L}]
  \right)
  \;\colon\;
  [\Sigma_k, \mathbf{Fields}]
  \stackrel{[\Sigma_n, \mathbf{L}]}{\to}
  [\Sigma_l, \mathbf{B}^n U(1)_{conn}]
  \stackrel{\exp(2 \pi in \int_{\Sigma_k})(-) }{\to}
  \mathbf{B}^{n-k}U(1)_{conn}
$$

which modulates a [[circle n-bundle with connection|circle (n-k)-bundle with connection]] on the [[moduli infinity-stack]] of fields on $\Sigma_k$. At least for large classes of theories, this is the (off-shell) [[prequantum n-bundle|prequantum (n-k)-bundle]] of the theory.

## Properties

### Relation to universal characteristic classes

By the discussion at _[[field (physics)]]_ in the section _[Properties - Relation to twisted cohomology](field%20%28physics%29#RelationToTwistedCohomology)_ a field configuration in any [[theory of physics]] may equivalently be thought of as a [[cocycle]] in [[twisted cohomology|twisted]] general [[cohomology]]. Here we discuss that evaluating an extended Lagrangian on such a field configuration is equivalently producing a [[characteristic class]]. In other words, under this identification with [[cohomology]] an extended Lagrangian is equivalently a [[universal characteristic map]] and its [[equivalence class]] is the corresponding [[universal characteristic map]].

[[!include gauge field - table]]

## Examples

### For discrete higher gauge groups: $\infty$-Dijkgraaf-Witten theory

$G$ a [[discrete ∞-group]]

extended Lagrangians are cocycles in [[group cohomology]]:

$$
  \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n U(1))
  \simeq
  \pi_0 Top(B G, B^n U(1)_{disc})
  \simeq
  H_{Grp}^n(G,U(1))
$$

these are the extendd Lagrangians of [[infinity-Dijkgraaf-Witten theory]].

### For ordinary gauge theory with

We consider $G$ a [[Lie group]], to be regarded as the [[gauge group]] of a [[gauge theory]]. We write $\mathbf{B}G_{conn} \in \mathbf{H}$ for the [[smooth infinity-groupoid|smooth]] [[moduli stack]] of $G$-[[principal connections]].

Setting $\mathbf{Fields} \coloneqq \mathbf{B}G_{conn}$ this is the [[moduli stack]] for [[field (physics)|fields]] in a pure $G$-gauge theory. An extended Lagrangian is hence to be a map

$$
  \mathbf{L}
    \;\colon\;
  \mathbf{B}G_{conn}
   \stackrel{}{\to}
  \mathbf{B}^n U(1)_{conn}
  \,.
$$

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

For $G \in \mathbf{H} \coloneqq $ [[Smooth∞Grpd]] a [[compact Lie group]] and $n \in \mathbb{N}$, then [[geometric realization of cohesive ∞-groupoids]] ${\vert - \vert} \colon \mathbf{H} \to \infty Grpd \simeq L_{whe} Top$ induces a [[natural isomorphism]]

$$
  {\vert- \vert}
  \;\colon\;
  \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n U(1))
  \stackrel{\simeq}{\to}
  \pi_0 L_{whe} Top(B G, B^n U(1))
  \simeq
  H^{n+1}(B G, \mathbb{Z})
  \,.
$$

=--

This is discussed at _[[Lie group cohomology]]_. The proof is in ([dcct](#dcct)).

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

This means that the [[instanton sectors]] of extended Lagrangians for pure ordinary gauge theory are labeled precisely by the [[integral cohomology]] of the [[classifying space]] $B G$ of the [[gauge group]]. In ordinary [[Chern-Simons theory]] this is called the **level** of the theory.

=--

### Higher gauge theory

If $G \in Grp(\mathbf{H})$ is an [[∞-group]] (a [[smooth ∞-group]] if $\mathbf{H}$ is [[Smooth∞Grpd]]) and

$$
  \mathbf{Fields}
  \simeq
  \mathbf{B}G_{conn}
$$

is a [[moduli ∞-stack]] of $G$-[[principal ∞-bundle|principal]] [[connection on an ∞-bundle|∞-connections]], then the theory in question is a ([[higher gauge theory|higher]]) [[gauge theory]]. In this case the map

$$
  \mathbf{c} \;\colon\; \mathbf{B}G \to \mathbf{B}^n U(1)
$$

underlying the extended Lagrangian is a [[cocycle]] that induces an [[extension of ∞-groups]], fitting a [[homotopy fiber sequence]]

$$
  \mathbf{B}^{n-1}U(1) \to \mathbf{B}\hat G
\stackrel{}{\to} \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n U(1)
  \,.
$$

In this case the extended Lagrangian also plays the role as the universal differential refinement of the [[obstruction]] class to [[lift of structure groups]] through $\hat G \to G$. More generally, its [[homotopy fiber products]] are [[moduli ∞-stacks]] of [[twisted differential c-structures]]. See there for more examples.

### Construction of extended Lagrangians by Lie integration
 {#ByLieIntegration}

[[Lie integration]]:

for $\mathbb{g}$ an [[L-∞ algebra]] and

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

a [[homomorphism]] of $L_\infty$-algebras into the [[line Lie n-algebra]], hence a [[cocycle]] in [[L-∞ algebra cohomology]], the operation of differentially refined [[Lie integration]] ([FSS diff coc](#FiorenzaSchreiberStasheff)) produces a morphism of [[moduli ∞-stacks]]

$$
  \widehat {\exp(\mu)}
  \;\colon\;
  \exp(\mathfrak{g})_{conn}
  \to
  \mathbf{B}^n \mathbb{R}_{conn}
$$

Under [[truncated object in an (∞,1)-category|truncation]] this yields

$$
  \mathbf{B}G_{conn}
  \to
  \mathbf{B}^n (\mathbb{R}/\Gamma)
  \,,
$$

where $G \coloneqq \Omega \tau_n \exp(\mathfrak{g})$ and where $\mathbf{B}G_{conn}$ is the moduli of $G$-[[connections on an ∞-bundle|∞-connections]]. ([FSS](#FiorenzaSchreiberStasheff))

### Higher Chern-Simons theory

Various examples are discussed at

* [[schreiber:∞-Chern-Simons theory]]

* [[higher dimensional Chern-Simons theory]]

  * [[1d Chern-Simons theory]]

  * [[Chern-Simons theory]]

  * [[5d Chern-Simons theory]]

  * [[7d Chern-Simons theory]]

  * [[AKSZ sigma-models]]

  * [[string field theory]]

  * [[infinite-dimensional Chern-Simons theory]]

## Related concepts

[[!include extended prequantum field theory - table]]

## References

Expositions, examples and further pointers are in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Extended higher cup-product Chern-Simons theories]]_

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:A higher stacky perspective on Chern-Simons theory]]_

Lecture notes are in _[[geometry of physics]]_, and section 1.2 of

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

A general abstract discussion is in section 3 there, further discussion of examples in section 5.

The construction of extended Lagrangian by [[Lie integration]] of [[L-∞ algebra cocycles]] is due to

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech Cocycles for Differential characteristic Classes]]_
 {#FiorenzaSchreiberStasheff}

Construction of differential cup-product theories is in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Extended higher cup-product Chern-Simons theories]]_
 {#FSSCup}

The discussion of the abelian [[7d Chern-Simons theory]] involved in [AdS7/CFT6 duality](AdS-CFT#AdS7CFT6) is due to ([Witten 98](#AdS-CFT#Witten98)). The discussion of the non-abelian quantum-corrected version is due to

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:7d Chern-Simons theory and the 5-brane]]_
 {#FSS7dCS}

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The moduli 3-stack of the C-field]]_

[[!redirects extended Lagrangians]]


extended natural number

# Extended natural numbers
* table of contents
{: toc}

## Idea

The __extended natural number system__, denoted $\bar{\mathbb{N}}$, consists of all of the [[natural numbers]] together with an extra number representing [[infinity]].

## Definition

Classically, $\bar{\mathbb{N}}$ is the [[disjoint union]] of the set $\mathbb{N}$ of natural numbers and a [[point]] $\{\infty\}$.  That is,
$$ \bar{\mathbb{N}} = \{0,1,2,\ldots,\infty\} .$$

In [[constructive mathematics]], a more careful definition is required: an __extended natural number__ is an [[infinite sequence]] $x$ of [[binary digits]] (each $0$ or $1$) with the property that $x_i = 1$ if $x_j = 1$ for any $j \leq i$; that is, the sequence is [[monotone function|monotone]].  Then the natural number $n$ is identified with a sequence of $n$ copies of $0$ followed by $1$s, while infinity is identified with a sequence of all $0$s.  It is *not* constructively valid that every natural number is either finite or infinite, but it is valid that any that is not finite is infinite, while [[Markov's principle]] is the converse.

## Universal property

$\bar{\mathbb{N}}$ comes naturally equipped with a map $pred\colon \bar{\mathbb{N}} \to 1 + \bar{\mathbb{N}}$ as defined below:
$$
pred(x) = \begin{cases}
*      & if\; x = 0 ,\\
n      & if\; x = n + 1 ,\\
\infty & if\; x = \infty .\end{cases}
$$

Thus, it is a [[coalgebra]] for the endofunctor $H(X) = 1 + X$ on [[Set]], and indeed is the [[terminal coalgebra]] for $H$.  That is, given any set $S$ and map $p\colon S \to 1 + S$, there is a unique map $corec_S p\colon S \to \bar{\mathbb{N}}$ such that
$$ \array {
S                      & \stackrel{p}\to    & 1 + S \\
\downarrow_{corec_S p} &                    & \downarrow_{\id_1 + corec_S p} \\
\bar{\mathbb{N}}       & \stackrel{pred}\to & 1 + \bar{\mathbb{N}}
} $$
commutes.  Indeed, $corec_S p$ is defined corecursively by $corec_S p(a) = 0$ if $p(a) = *$ and $\pred(\corec_S p(a)) = \corec_S p(a^\prime)$ if $p(a) = a^\prime \in S$.  In this way, $\bar{\mathbb{N}}$ is [[duality|dual]] to the [[natural number | natural number system]] $\mathbb{N}$ in its guise as a [[natural numbers object]].

You can think of $corec_S p$ as mapping an element $a$ of $S$ to the maximum number of times that $p$ can be applied in succession, starting from $a$, before being taken out of $S$.  Since this may never occur, we need $\infty$ as a possible value.  At the other extreme, if $p(a) = *$ then $p$ cannot be applied at all to $a$ before leaving $S$, so $corec_S p(a) = 0$.

Note that this universal property also holds constructively (which is why we can be sure that the constructive definition above is correct).  We define $pred$ constructively as follows:
$$
pred(x) = \begin{cases}
*                & if\; x_0 = 1 ,\\
(x_1,x_2,\ldots) & if\; x_0 = 0 .\end{cases}
$$

## Formalization in proof assistants

With the [[Coq]] [[proof assistant]]:

```coq
CoInductive conat : Set :=
| cozero : conat
| cosucc : conat -> conat.

CoFixpoint omega : conat := cosucc omega.
```

## Topology

We may naturally give $\bar{\mathbb{N}}$ a [[topological structure|topology]] giving it the structure of a [[compact Hausdorff space]]; unusually, this works even in weak constructive foundations (without having to use the [[fan theorem]] or pass to a [[locale]]).

We may define the topology simply (and constructively) as follows: a [[subset]] $G$ of $\bar{\mathbb{N}}$ is _[[open subset|open]]_ if, whenever $\infty \in G$, there is a finite $n$ such that $m \in G$ whenever $m \geq n$.  In other words, $G$ is a [[neighbourhood]] of $\infty$ just when almost every finite number also belongs to $G$.

In this way, $\bar{\mathbb{N}}$ is the [[Alexandroff compactification]] of the [[discrete space]] $\mathbb{N}$.  The space is obviously compact because, given an [[open cover]] $\mathcal{U}$, we have $\infty \in G \in \mathcal{U}$ for some $G$, so $G$ alone contains almost every point, and only finitely many more open sets are needed.

It is sometimes convenient to represent $\bar{\mathbb{N}}$ as a [[subspace]] of the [[real line]] $\mathbb{R}$, which we can do by interpreting the natural number $n$ as $2^{-n}$ and $\infty$ as $0$.  Constructively, the monotone bit sequence $x$ becomes the real number
$$
\frac{1}{2} {\sum_{i=0}^\infty x_i 2^{-i}}
,$$
which always converges.  Another common representation uses $1/(n+1)$ instead of $1/2^n$.

Given any topological space $X$, an [[infinite sequence]] in $X$ may be thought of as a [[continuous map]] to $X$ from the discrete space $\mathbb{N}$.  Then this sequence [[convergent sequence|converges]] iff this map can be extended to a continuous map on $\bar{\mathbb{N}}$.  For this reason, $\bar{\mathbb{N}}$ is sometimes called the __universal convergent sequence__.  (Strictly speaking, unless $X$ is at least [[sequentially Hausdorff space|sequentially Hausdorff]], a map to $X$ from $\bar{\mathbb{N}}$ contains more information than a sequence in $X$ with the property of convergence.)

This may all be generalised from sequences to other [[nets]]; given a [[directed set]] $D$, we form $\bar{D}$ by adjoining $\infty$ and taking $G \subseteq D$ as a neighbourhood of $\infty$ iff $G$ owns almost all of $D$.  (Constructively, this may require using locales for the general case.)

## In higher category theory

Concepts in [[higher category theory]] often come in $n$-versions where $n$ is an extended natural number.  (Sometimes it's also possible to give $n$ a few negative values as well; see [[negative thinking]].)  Typically, the $\infty$-version is all-encompassing, with the $n$-versions as special cases.  On the other hand, the $1$-version is usually more familiar outside of category theory.

See also [[categorification]].

## In constructive mathematics

The claim that every extended natural number is either finite or infinite is equivalent to the [[limited principle of omniscience]] ($LPO$) for natural numbers.  On the other hand, the $LPO$ for extended natural numbers is simply true; given any function from $\bar{\mathbb{N}}$ to $\{0,1\}$, it is either all $0$s or has a $1$.  See [Escardó (2011)](#Escardo).

## Related concepts

* [[extended natural numbers object]]

## References

*  [[Martín Escardó]] (2011) via constructivenews, [Infinite sets that satisfy the principle of omniscience in all varieties of constructive mathematics](http://www.cs.bham.ac.uk/~mhe/papers/omniscient-2011-07-06.pdf) (pdf)
{#Escardo}

[[!redirects extended natural number]]
[[!redirects extended natural numbers]]
[[!redirects extended natural number system]]

[[!redirects conatural number]]
[[!redirects conatural numbers]]
[[!redirects conat]]
[[!redirects conumber]]
[[!redirects conumbers]]


extended natural numbers > history

< [[extended natural numbers]]
[[!redirects extended natural numbers -- history]]


extended natural numbers object

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

\tableofcontents

## Idea

Recall that a [[topos]] is a [[category]] that behaves likes the category [[Set]] of [[set|sets]].

An **extended natural numbers object** (extended NNO) in a topos is an [[object]] that behaves [[internalization|internal to]] that topos like the set $\overline{\mathbb{N}}$ of [[extended natural number|extended natural numbers]] does in [[Set]].

## Definition

### In a topos or cartesian closed distributive category

An **extended natural numbers object** in any [[topos]] (or any [[cartesian closed category|cartesian closed]] [[distributive category]]) $E$ with [[terminal object]] $1$ and [[coproduct]] $A + B$ is

* an object $\overline{\mathbb{N}}$ in $E$

* equipped with a morphism $p:\overline{\mathbb{N}} \to (1 + \overline{\mathbb{N}})$

* such that for every other object $A$ with morphism $q:A \to (1 + A)$, there is a unique morphism $\mu_A:A \to \overline{\mathbb{N}}$ such that this diagram commutes:

$$\array{& A & \overset{q}\rightarrow & 1 + A & \\
          \mu_A & \downarrow &&\downarrow & \mu_A\\
          &\overline{\mathbb{N}} & \underset{p}\rightarrow& 1 + \overline{\mathbb{N}} & \\
}$$

Thus, the extended NNO is the [[terminal coalgebra]] for the [[endofunctor]] $A \mapsto 1 + A$ on $E$.

### In a symmetric closed distributive monoidal category

The above definition makes sense in any [[symmetric monoidal category|symmetric]] [[closed monoidal category|closed]] [[distributive monoidal category]] $C$, provided that we use the [[tensor unit]] and [[tensor product]] instead of the [[terminal object]] and [[cartesian product]].

An **extended natural numbers object** in a symmetric closed distributive monoidal category $C$ with [[tensor unit]] $I$ and [[coproduct]] $A \oplus B$ is

* an object $\overline{N}$ in $C$

* equipped with a morphism $p:\overline{N} \to (I \oplus \overline{N})$

* such that for every other object $A$ with morphism $q:A \to (I \oplus A)$, there is a unique morphism $\mu_A:A \to \overline{N}$ such that this diagram commutes:

$$\array{& A & \overset{q}\rightarrow & I \oplus A & \\
          \mu_A & \downarrow &&\downarrow & \mu_A\\
          &\overline{N} & \underset{p}\rightarrow& I \oplus \overline{N} & \\
}$$

## Examples

* The set of [[extended natural numbers]] is the extended natural numbers object in [[Set]]

* Given a [[field]] $K$, the underlying [[vector space]] of the [[sequence algebra]] $K^\mathbb{N}$ is the extended natural numbers object in [[Vect|$\mathrm{Vect}_K$]].

## Properties

If the topos is a [[Boolean topos]] with a [[natural numbers object]], then the extended natural numbers object  $\overline{\mathbb{N}}$ is isomorphic to the object $\mathbb{N} + 1$.

## Related concepts

* [[extended natural numbers]]
* [[natural numbers object]]

[[!redirects conatural numbers object]]
[[!redirects conatural numbers objects]]
[[!redirects conatural number object]]
[[!redirects conatural number objects]]
[[!redirects conatural-numbers object]]
[[!redirects conatural-numbers objects]]
[[!redirects conatural-number object]]
[[!redirects conatural-number objects]]

[[!redirects co-natural numbers object]]
[[!redirects co-natural numbers objects]]
[[!redirects co-natural number object]]
[[!redirects co-natural number objects]]
[[!redirects co-natural-numbers object]]
[[!redirects co-natural-numbers objects]]
[[!redirects co-natural-number object]]
[[!redirects co-natural-number objects]]

[[!redirects extended natural numbers object]]
[[!redirects extended natural numbers objects]]
[[!redirects extended natural number object]]
[[!redirects extended natural number objects]]
[[!redirects extended natural-numbers object]]
[[!redirects extended natural-numbers objects]]
[[!redirects extended natural-number object]]
[[!redirects extended natural-number objects]]
[[!redirects extended NNO]]
[[!redirects extended NNOs]]


extended power

#Contents#
* table of contents
{:toc}

## Definition

Write $\mathbb{F}_2 \coloneqq \mathbb{Z}/2\mathbb{Z}$ for the [[field]] with two elements.

For $V$ an $\mathbb{F}_2$-[[module]], hence an $\mathbb{F}_2$-[[vector space]], and for $n \in \mathbb{N}$, write

$$
  V^{\otimes n}_{h \Sigma_n} \in \mathbb{F}_2 Mod
$$

for the [[homotopy quotient]] of the $n$-fold [[tensor product]] of $V$ with itself by the [[action]] of the [[symmetric group]]. Explicitly this is presented, up to [[quasi-isomorphism]] by the ordinary [[coinvariants]] $D_n(V)$ of the tensor product of $V^{\otimes n}$ with a [[free resolution]] $E \Sigma_n^\bullet$ of $\mathbb{F}_2$:

$$
  V^{\otimes n}_{h \Sigma_n}
  \simeq
  D_n(V)
  \coloneqq
  (V^{\otimes n} \otimes E\Sigma_n)_{\Sigma_n}
  \,.
$$

This is called the $n$th _[[extended power]]_ of $V$.

## Application

* [[Steenrod square]]

## Related concepts

* [[tensor power]]

* [[H-infinity ring spectrum]]

## References

* [[Jacob Lurie]], 18.917 _[Topics in Algebraic Topology: The Sullivan Conjecture, Fall 2007](http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007)_. (MIT OpenCourseWare: Massachusetts Institute of Technology), _[Lecture notes](http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/)_

  Lecture 2 _Steenrod operations_ ([pdf](http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture2.pdf))

[[!redirects extended powers]]


extended prequantum field theory - table

**[[extended prequantum field theory]]**

|  $0 \leq k \leq n$ | (off-shell) [[prequantum n-bundle|prequantum (n-k)-bundle]] | traditional terminology |
|--|--|--|
| $0$ |  [[differential cohomology|differential]] [[universal characteristic map]]  | [[level (Chern-Simons theory)|level]]  |
| $1$ | [[prequantum n-bundle|prequantum (n-1)-bundle]] | [[infinity-Wess-Zumino-Witten model|WZW bundle (n-2)-gerbe]] |
| $k$ | [[prequantum n-bundle|prequantum (n-k)-bundle]] |  |
| $n-1$ | [[prequantum n-bundle|prequantum 1-bundle]] | (off-shell) [[prequantum bundle]]  |
| $n$ | [[prequantum 0-bundle]] | [[action functional]] |


extended probabilistic powerdomain

[[!redirects Extended probabilistic powerdomain]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The extended probabilistic powerdomain is a [[monad of valuations]] on [[topological spaces]]. Its functor part assigns to a given [[topological space]] the space of [[continuous valuations]] over it.

The idea of this monad was first given by Kirch for the case of [[domains]] (see [Kirch '93](#kirch), in German, or [AJK '06](#Jung), in English). It was extended to all of [[Top]] and given its current form by Heckmann (see [Heckmann '96](#Heckmann96)).

## Definitions

### Spaces of valuations

Given a [[topological space]] $X$, denote by $V X$ the space whose points are [[continuous valuations]] on $X$ with values in $[0,\infty]$. Equip $V X$ with the [[topology]] [[topological basis|generated]] by the sets in the form
$$
\theta(U,r) \coloneqq \{ \nu \in V X \,|\, \nu(U) \gt r \},
$$
for $r\ge 0$ and $U\subseteq X$ [[open]], or equivalently by the sets in the form
$$
\Theta(g,r) \coloneqq \left\{ \nu \in V X \,\Big|\, \int g \,d\nu \gt r \right\}
$$
for $r\ge 0$ and $g:X\to[0,\infty]$ [[lower semicontinuous]].

This topology can be seen as the [[pointwise topology]] if we view valuations either as functions on the [[open sets]] or as [[functionals]] on lower semicontinuous functions (via [[continuous valuation#integration|integration]]).
It is also the [[initial topology]] of either evaluation of open sets or integration of functions, meaning that it is the coarsest topology for which either the assignments
$$
\nu \mapsto \nu(U)
$$
for every open $U\subseteq X$, or
$$
\nu \mapsto \int g \, d\nu
$$
for every lower semicontinuous $g:X\to[0,\infty]$, are lower semicontinuous.
Lower semicontinuity, in some sense, plays the role that [[measurable function|measurability]] plays for the [[Giry monad]] (see also [[correspondence between measure and valuation theory]]).

The [[specialization preorder]] of this topology is known as the [[stochastic order]], and can be seen as the [[pointwise order]] of valuations as functions on the open sets.

It is known (see [Jung '04](#jung2)) that if $X$ is [[stably compact]], then $V X$ is stably compact too, so that the monad $V$ restrict to the subcategory of [[stably compact spaces]].

### Functoriality

Given a [[continuous map]] $f:X\to Y$, define the map $ f:V X\to V Y$ as the one assigning to a continuous valuation $\nu\in V X$ its [[continuous valuation#pushforward|pushforward]] along $f$.

It can be proven that the map $V f$ is continuous too, so that $V$ is an [[endofunctor]] of [[Top]].

### Unit and multiplication

We can define the unit of the [[monad]] as follows. Given a space $X$, define the map $\delta:X\to V X$ as the one assigning to the point $x\in X$ the [[valuation (measure theory)#dirac_valuation|Dirac valuation]] at $x$.
This map $\delta$ is [[continuous map|continuous]], and [[natural transformation|natural]] in $X$.

The multiplication map makes use of the concept of [[valuation (measure theory)#integration|integration over a valuation]]. Given a valuation $\xi\in V V X$, we can define the valuation $E \xi\in V X$ as the one mapping an [[open]] set $U\subseteq X$ to
$$
E \xi (U) \coloneqq \int_X \nu(U) \, d\xi(\nu).
$$
This integral is well defined, since the assignment $U\mapsto \nu(U)$ is [[lower semicontinuous]]. The assignment $U\mapsto E \xi (U)$ gives a [[continuous valuation]] on $X$, and the resulting map $E: V V X \to V X$ is [[continuous map|continuous]] and [[natural transformation|natural]] in $X$.

The maps $\delta$ and $E$ satisfy the usual [[axioms]] of a [[monad]]. The monad $(V,\delta,E)$ is usually called the **extended probabilistic powerdomain**.

This construction, especially the way the unit and multiplications are defined, can be thought of as a topological analogue of the [[Giry monad]].

## Monoidal structure

(...)

## Algebras

(...)

## Notable submonads

There are a number of monads that can be constructed as submonads of $V$. The [[monoidal functor|monoidal structure]] of $V$ is inherited by these submonads too, allowing the formation of joints and marginals.

See also [[monads of probability, measures, and valuations#detailed_list|monads of probability, measures, and valuations - detailed list]].

### Normalized valuations

If one restricts to _normalized_ valuations, i.e.~those $\nu\in V X$ with $\nu(X)=1$, one obtains a submonad of $V$ which can be thought of as the one of [[probability]] valuations.

### The measure monad on Top

One can restrict $V$ only to those valuations which are [[valuation (measure theory)#extending_valuations_to_measures|extendable to measures]]. The resulting subspace $M X\subseteq V X$ (for every topological space $X$) is the space of [[tau-additive measures]] on $X$, with the subspace topology inherited by $V X$. For probability measures, this topology is sometimes known as the **A-topology**, after Alexandrov (not to be confused with the [[Alexandrov topology]], which is a different concept), for example in [Bogachev, section 8.10.iv](#bogachev2). The [[specialization preorder]] is again the [[stochastic order]].
Since extendable valuations are stable with respect to pushforwards and integrations, $M$ forms a submonad of $V$, the **measure monad on [[Top]]**.

More details can be found in [Fritz-Perrone-Rezagholi '19](#support), worked out explicitly for the normalized case (see below).

See also [[correspondence between measure and valuation theory]].

### The probability monad on Top

If one restricts the measure monad above to the $\tau$-smooth _probability_ measures (i.e. normalized), one obtains again a submonad, which seems to be the most general [[probability monad]] on [[Top]].

If a topological space is [[Tychonoff space|Tychonoff]] (for example a [[metric space]] or a [[compactum|compact Hausdorff space]]), the A-topology for probability measures coincides with the usual [[weak topology]] of measures with respect to continuous functions.
In particular, on the [[subcategory]] of [[compactum|compact Hausdorff spaces]], this monad restricts to the [[Radon monad]].

### The monad of topological cones

If one restrict to [[valuation (measure theory)#simple_valuations|simple valuations]], i.e. those that are linear combinations of deltas, one obtains again a submonad of $V$, which can be thought of as the free [[topological cone]] monad (or free internal $[0,\infty]$-[[module object]] monad).

### The monad of topological convex spaces

If one further restricts to _normalized_ simple valuations, one obtains as submonad the free [[topological convex space]] monad.

## Related concepts

* [[monads of probability, measures, and valuations]]
* [[monads in computer science]]

* [[valuation (measure theory)]]
* [[correspondence between measure and valuation theory]]

* [[Giry monad]], [[Radon monad]], [[probabilistic powerdomain]], [[valuation monad on locales]], [[distribution monad]]

* [[convex space]], [[conical space]]

## References

* {#kirch} Olaf Kirch, _Bereiche und Bewertungen_ (in German), Master Thesis, Technische Hochschule Darmstadt, 1993 ([ps.gz](http://fb04286.mathematik.tu-darmstadt.de/fbereiche/logik/research/Domains/papers/kirch/diplom.ps.gz))

* {#Heckmann96} Reinhold Heckmann, _Spaces of valuations_, Papers on General Topology and Ap-plications, 1996 ([doi:10.1111/j.1749-6632.1996.tb49168.x](https://doi.org/10.1111/j.1749-6632.1996.tb49168.x),[pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.45.5845&rep=rep1&type=pdf))

* {#jung2} Achim Jung, _Stably compact spaces and the probabilistic powerspace construction_, ENTCS 87, 2004 ([doi:10.1016/j.entcs.2004.10.001](https://doi.org/10.1016/j.entcs.2004.10.001)).

* {#Jung}Mauricio Alvarez-Manilla, Achim Jung, [[Klaus Keimel]], _The probabilistic powerdomain for stably compact spaces_, Theoretical Computer Science 328, 2004 ([doi:10.1016/j.tcs.2004.06.021](https://doi.org/10.1016/j.tcs.2004.06.021))

* {#algebras} [[Jean Goubault-Larrecq]] and Xiaodong Jia, _Algebras of the extended probabilistic powerdomain monad_, ENTCS 345, 2019
([doi:10.1016/j.entcs.2019.07.015](https://doi.org/10.1016/j.entcs.2019.07.015))

* {#support} [[Tobias Fritz]], [[Paolo Perrone]] and Sharwin Rezagholi, _Probability, valuations, hyperspace: Three monads on Top and the support as a morphism_, 2019 ([arXiv:1910.03752](https://arxiv.org/abs/1910.03752))

* {#bogachev2} V. Bogachev, _Measure Theory_, vol. 2 (2007).

[[!redirects extended probabilistic powerdomain]]
[[!redirects valuation monad on top]]
[[!redirects valuation monad on topological spaces]]


extended real number

# Extended real numbers
* table of contents
{: toc}

## Idea

An extended real number is usually a [[real number]], but it might be $\pm\infty$.  We can also think of an extended real number as a number of the form $\log(1/t - 1)$ for $t \in [0,1]$ (including $t = 0,1$) or a number of the form $\tan t$ for $t$ a real number (including an odd multiple of $\pi/2$).

There are actually two slightly different notions that both go by the name 'extended real number': one in which $\infty$ and $-\infty$ are distinct (the [[logarithm]]), and one in which they are identified (the [[tangent]]).  The latter kind forms a [[quotient set|quotient space]] of the former.

## Definitions

Classically, one simply defines an extended real number to be either a real number, $\infty$, or (for the version in which these are distinguished) $-\infty$.  However, one has to take more care for a definition that works in [[constructive mathematics]].  Such a definition may also be interesting to a classical mathematician for its own sake, of course.

### As extended Dedekind cuts

One way to define extended real numbers is to adjust the notion of [[Dedekind cut]].  This naturally defines the sort of extended real number in which $\infty \ne -\infty$.

So let an __extended Dedekind cut__ be a pair $(L,U)$ of sets of [[rational numbers]] such that:

*  For each $a \in L$, there is some $b \in L$ such that $a \lt b$.
*  For each $b \in U$, there is some $a \in U$ such that $a \lt b$.
*  Whenever $a \lt b$ are rational numbers, either $a \in L$ or $b \in U$ (non-exclusively).
*  If $a \in L$ and $b \in U$, then $a \lt b$.

(The usual slight variations available in the notion of Dedekind cut apply here as well, in the same ways.)

If $L$ is empty, then $U$ must be the set of all rational numbers; this extended cut represents $-\infty$.  If $U$ is empty, then $L$ must be the set of all rational numbers; this extended cut represents $\infty$.  An extended Dedekind cut is __bounded__ if instead $L$ and $U$ are both [[inhabited set|inhabited]].  The bounded cuts are the usual Dedekind cuts that represent real numbers.  So the only change is that we no longer require this boundedness condition.

The space of extended real numbers in this sense is often denoted $\overline{\mathbb{R}}$.  As a [[topological space]] (or [[locale]], constructively), it is the [[end compactification]] of the [[locally compact space]] $\mathbb{R}$.  Geometrically, this is a [[line segment]], the __extended real number line segment__.

### As ratios of real numbers

Another way to define extended real numbers is to represent them as ratios of real numbers.  This naturally defines the sort of extended real number in which $\infty = -\infty$.

So let a __nontrivial ratio__ of real numbers be a pair $(a,b)$ of real numbers such that either $a \ne 0$ or $b \ne 0$.  (By $\ne$ we mean the usual [[apartness relation]] on real numbers.)  We write the ratio as $a ∶ b$, and we consider $a ∶ b$ and $c ∶ d$ to be __equivalent__, written $a ∶ b ∷ c ∶ d$, if $a d = b c$.  This is an [[equivalence relation]] on nontrivial ratios; modulo this equivalence relation, the nontrivial ratios represent extended real numbers.

If $b \ne 0$, then the nontrivial ratio $a ∶ b$ is equivalent to $a/b ∶ 1$ and represents the real number $a/b$.  The nontrivial ratios $a ∶ 0$ for $a \ne 0$ are all equivalent and represent $\infty$.  (The ratio $0 ∶ 0$ is considered trivial and does not represent any extended real number.  If we want to include it, then we get a [[wheel]].)

This is a special case of forming the [[projective line]] of a [[field]], in this case the field of real numbers.  Accordingly, the space of extended real numbers in this sense is often denoted $\mathbb{P}\mathbb{R}^1$.  As a [[topological space]] (or [[locale]], constructively), it is the [[one-point compactification]] of the [[locally compact space]] $\mathbb{R}$.  Geometrically, this is a [[circle]], the __extended real number circle__.

## Orders

We can put a [[linear order]] on $\overline{\mathbb{R}}$ by setting $-\infty \lt x \lt \infty$ for any real number $x$.  In terms of Dedekind cuts, this is the straightforward extension of the usual definition of $\lt$ on such cuts.

We can put a [[cyclic order]] $R$ on $\mathbb{P}\mathbb{R}^1$ by setting $R(x,y,z)$ true if any of the following hold:

*  $x \lt y \lt z$ are all real numbers,
*  $y \lt z \lt x$ are all real numbers,
*  $z \lt x \lt y$ are all real numbers,
*  $y \lt z$ are real numbers and $x = \infty$,
*  $z \lt x$ are real numbers and $y = \infty$,
*  $x \lt y$ are real numbers and $z = \infty$.

(These orders can be defined constructively too, with a little more work.)

## Arithmetic

In $\overline{\mathbb{R}}$, we take $x + \infty$, $\infty + x$, and $\infty + \infty$ to be $\infty$ (for $x$ a real number) and similarly $x - \infty$, $-\infty + x$, and $-\infty - \infty$ to be $-\infty$.  Of course, $\infty$ and $-\infty$ are opposites (additive inverses).  These are all straightforward extensions of the usual definition of addition for Dedekind cuts.  However, when we try $\infty - \infty$ or $-\infty + \infty$, the Dedekind cut calculation gives us $(\empty,\empty)$, which is not a cut, even extended.  (Although some sources define these sums, it is most common to leave them undefined, I think.)

In $\mathbb{P}\mathbb{R}^1$, we similarly take $x + \infty$ and $\infty + x$ to be $\infty$ (for $x$ a real number).  Now $\infty$ is its own opposite.  These all agree with the additive rule $(a ∶ b) + (c ∶ d) = (a d + b c ∶ b d)$.  If we apply this to $\infty + \infty$, however, the result is $0 ∶ 0$, so again we take that to be undefined.  (However, in this case, the definition $\infty + \infty = \infty$ is also widely seen, I think.)

Note that although we have blithely referred to 'additive inverses', addition does not form a [[group]] in either case.  The reason is that the opposites of $\pm\infty$ are not really additive inverses, since the result of the relevant addition is undefined rather than zero.  (If you define the sum to be zero so that they become inverses, then you lose associativity; ultimately, there is no way to make a group, since $0 + \infty = 1 + \infty$ shows that the operation is not cancellative.)

In $\overline{\mathbb{R}}$, we take $x \cdot {\pm \infty}$ to be $\pm \infty$ when $x \gt 0$ but $x \cdot {\pm \infty} = \mp \infty$ for $x \lt 0$.  (This continues to work even if $x$ is infinite too.)  But now $0 \cdot {\pm \infty}$ is undefined.  (Many sources define this to be zero, although I think this is really because they are using $0 \cdot \infty = 0$ in the nonnegative [[lower reals]], at least in applications to [[measure theory]].)

In $\mathbb{P}\mathbb{R}^1$, we take $x \cdot \infty$ to be $\infty$ as long as $x \ne 0$; but $0 \cdot \infty$ is undefined (although it is defined in the [[wheel]] over $\mathbb{R}$).  We may write $1/\infty = 0$ and $1/0 = \infty$ to define division, but again these are not really multiplicative inverses.  (With $0 \cdot 1 = 0 \cdot 2$ and $\infty \cdot 1 = \infty \cdot 2$, multiplication is not cancellative either.)

## Constructive aspects

The definitions as phrased above give us *[[located real number|located]]* extended real numbers.  We can of course consider unlocated extended real numbers, either [[one-sided real number|one-sided]] or [[MacNeille real number|MacNeille]].  The natural definitions of these are at least somewhat extended; the lower one-sided reals include $\infty$, the upper one-sided reals include $-\infty$, and the MacNeille reals include both (with $\infty \ne -\infty$).  However, one can always restrict to bounded numbers or alternatively fully extend the one-sided numbers, so being extended and being located are technically independent.  Still, whenever $\pm\infty$ shows up constructively (especially if only one of them appears), then this is a sign that one may want to generalise to one-sided (or even MacNeille) reals.

This has implications for arithmetic; for example, while $0 \cdot \infty$ is undefined in the extended located real numbers (and in the MacNeille real numbers), it is $0$ in the nonnegative lower real numbers (and $\infty$ in the extended nonnegative upper real numbers).

[[!redirects extended real]]
[[!redirects extended reals]]
[[!redirects extended real number]]
[[!redirects extended real numbers]]

[[!redirects extended real line]]
[[!redirects extended real number line]]
[[!redirects extended real line segment]]
[[!redirects extended real number line segment]]
[[!redirects extended real circle]]
[[!redirects extended real number circle]]


extended super Minkowski spacetime

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###Context###
#### Physics
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[[!include physicscontents]]
=--
#### $\infty$-Chern-Weil theory
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[[!include infinity-Chern-Weil theory - contents]]
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#### Super-Geometry
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[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

As opposed to ordinary [[Minkowski spacetime]], [[super Minkowski spacetime]] has, when regarded as the [[super translation Lie algebra]], exceptional [[cocycles]]. The [[super L-infinity algebra]] extensions classified by these serve as _extended super-spacetimes_ which are [[target spaces]] for super $p$-[[branes]].

Notably the [[supergravity Lie 3-algebra]] is the extended Minkowski spacetime on which [[superspacetimes]] of [[11-dimensional supergravity]] are modeled.

## Related concepts

* [[extended supersymmetry]]

* [[M-theory supersymmetry algebra]]

* [[type II supersymmetry algebra]]

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

## References

Discussion in the language of the  [[D'Auria-Fre formulation of supergravity]] ("FDA"s) and the [[brane scan]] is in

* C. Chryssomalakos, [[José de Azcárraga]], [[José M. Izquierdo]], C. Pérez Bueno, _The geometry of branes and extended superspaces_, Nucl. Phys. B **567** (2000) 293-330  [[arXiv:hep-th/9904137](http://arxiv.org/abs/hep-th/9904137), doi:10.1016/S0550-3213(99)00512-X]

* Makoto Sakaguchi, _IIB-Branes and New Spacetime Superalgebras_, JHEP 0004 (2000) 019 ([arXiv:hep-th/9909143](https://arxiv.org/abs/hep-th/9909143))

* [[José de Azcárraga]], J. M. Izquierdo, _Superalgebra cohomology, the geometry of extended superspaces and superbranes_, Nucl.Phys.B567:293-330,2000 ([arXiv:hep-th/0105125](http://arxiv.org/abs/hep-th/0105125))

and particularly with emphasis of the extended super Minkowski spacetimes as targets for the [[Green-Schwarz super-p brane sigma models]] with gauge fields on the [[worldvolume]] (tensor multiplets) in

* {#Azcarraga05} [[José de Azcárraga]], _Superbranes, $D=11$ CJS supergravity and enlarged superspace coordinates/fields correspondence_, AIP Conf.Proc.767:243-267,2005 ([arXiv:hep-th/0501198](http://arxiv.org/abs/hep-th/0501198))

Discussion that makes the [[super L-infinity algebra]] [[homotopy theory]] underlying this manifest 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]]_,  International Journal of Geometric Methods in Modern Physics. Volume 12, Issue 02 (2015) 1550018 (35 pages) 2013 ([arXiv:1308.5264](http://arxiv.org/abs/1308.5264))

[[!redirects extended super Minkowski spacetimes]]
[[!redirects extended super-spacetime]]

[[!redirects extended super spacetime]]
[[!redirects extended super spacetimes]]

[[!redirects extended Minkowski super spacetime]]
[[!redirects extended Minkowski super spacetimes]]

[[!redirects extended super Minkowski  spacetime]]
[[!redirects extended superspacetime]]
[[!redirects extended superspacetime]]

[[!redirects extended super-Minkowski spacetime]]
[[!redirects extended super-Minkowski spacetimes]]

[[!redirects extended Minkowski spacetime]]


extended supersymmetry

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###Context###
#### $\infty$-Lie theory
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[[!include infinity-Lie theory - contents]]
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#### Super-Geometry
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[[!include supergeometry - contents]]
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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The super-Poincaré Lie algebra has certain exceptional super [[Lie algebra extensions]] by "[[central charges]]" which commute with the generators of the [[super translation Lie algebra]] and form a [[tensor]] [[representation]] of the remaining [[Lorentz group]] generators.

Here "central charges" is in quotation marks (and typically is so in the literature) since these charges will be in the [[center]] of the [[super translation Lie algebra]] but not in general in the center of the full [[super Poincare Lie algebra]].

## Examples

The "[[M-theory Lie algebra]]" is a polyvector extension of the super Poincaré Lie algebra $\mathfrak{siso}_{N=1}(10,1)$ by polyvectors of rank $p = 2$ and $p=5$ (the [[M2-brane]] and the [[M5-brane]] in the [[brane scan]]), see below [Polyvector extensions as automorphism Lie algebras](#PolyvectorExtensionsAsAutomorphismLieAlgebras).

Similarly the _[[type II supersymmetry algebra]]_ in [[type II supergravity]].

## Properties

### Short multiplets

Extended supersymmetry algebras in general have
"short" [[supermultiplets]]. These are called _[[BPS states]]_.

### As automorphism Lie algebras of Lie $n$-superalgebras
 {#PolyvectorExtensionsAsAutomorphismLieAlgebras}

At least some of the [polyvector extensions](#PolyvectorExtensions) of the super Poincaré Lie algebra arise as the [[automorphism]] super Lie algebras of the [[Lie n-algebra]]
[[infinity-Lie algebra cohomology|extensions]] classified by the cocycles discussed above.

For instance the automorphisms of the [[supergravity Lie 3-algebra]] gives the "[[M-theory Lie algebra]]"-extension of super-Poincaré in 11-dimensions ([FSS 13](#FSS13)). This is also discussed at _[supergravity Lie 3-algebra -- Polyvector extensions](supergravity+Lie+3-algebra#Polyvector)_.

## Related concepts

* [[extended super Minkowski spacetime]]

## References

### General

Introductions and lecture notes include

* José D. Edelstein, _Lecture 6: Extended supersymmetry_, talk at _Supersymmetry_, Santiago de Compostela, November 20, 2012  ([pdf](http://www-fp.usc.es/~edels/SUSY/L6_SUSY.pdf))

* Sven Krippendorf, Fernando Quevedo, Oliver Schlotterer, section 2.3 of _Cambridge Lectures on Supersymmetry and Extra Dimensions_ ([arXiv:1011.1491](http://arxiv.org/abs/1011.1491))

* _Theories with extended supersymmetry_ ([pdf](http://people.sissa.it/~bertmat/lect6.pdf))

A comprehensive account and classification of the polyvector extensions of the super Poincaré Lie algebras is in

* {#ACDP} [[Dmitry Vladimirovich Alekseevsky]], [[Vicente Cortés]], [[Chandrashekar Devchand]], [[Antoine Van Proeyen]], _Polyvector Super-Poincaré Algebras_ Commun.Math.Phys. 253 (2004) 385-422 ([arXiv:hep-th/0311107](http://arxiv.org/abs/hep-th/0311107))

### From super Lie $n$-algebras

The derivation of the extended supersymmetry [[M-theory Lie algebra]] as the [[derivations]] of the [[supergravity Lie 3-algebra]] is, somewhat implicitly, in

* [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], _[[Supergravity and Superstrings - A Geometric Perspective]]_
{#CastellaniDAuriaFre}

A more explicit discussion of this in terms of [[super L-infinity algebra]] [[homotopy theory]] 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]]_
 {#FSS13}


extendible cardinal

# Extendible cardinals
* table of contents
{:toc}

## Idea

An *extendible cardinal* is a type of [[large cardinal]].

## Definition

+-- {: .un_defn}
###### Definition
If $\kappa\lt\lambda$, we say $\kappa$ is **$\lambda$-extendible** if there is an [[elementary embedding]] $j:V_\lambda \to V_\mu$ with critical point $\kappa$ such that $j(\kappa)\gt \lambda$.

We say $\kappa$ is **extendible** if it is $\lambda$-extendible for all $\lambda\gt\kappa$.
=--

[[!redirects extendible cardinals]]


extension

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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

> This entry is about two senses of _extension_: extension of morphisms, [[formal dual|dual]] to [[lift]], and extension of objects ([[algebra extension]]).  In [[foundations]] and [[formal logic]]  there is also [[extension (semantics)]] and [[context extension]].

#Contents#
* table of contents
{:toc}

## Idea

### Extension of morphisms

An **extension** of a morphism $f: A\to Y$ along a [[monomorphism]] $i: A\hookrightarrow X$ is a morphism $\tilde{f}:X\to Y$ such that $\tilde{f}\circ i = f$. One sometimes, extends along more general morphisms than monomorphisms.

The dual problem is the problem of [[lifting]] a morphism $f: Y\to B$ through an epimorphism (or more general map) $p:X\to B$, giving a morphism $\tilde{f}: Y\to X$ such that $f = p\circ\tilde{f}$.

### Extension of an object by another object

In a [[category]] $C$ with a notion of [[short exact sequence]] (e.g. any [[semiabelian category]], [[Quillen exact category]] etc.) an **extension** of an object $Q$ by an object $K$ is any object $X$ fitting in a [[short exact sequence]] of the form

$$
  K
  \stackrel{i}
  \to
  X
  \stackrel{p}
  \to
  Q
  \,.
$$

For further cases, such as _[[group extension]]_, _[[Lie algebra extension]]_, _[[infinitesimal extension]]_ etc., see at _[[algebra extension]]_.

Classification of extensions in many categories is obtained using a [[forgetful functor]] $C\to D$ to a simpler category $D$, which preserves short exact sequences. For example, if all extensions in $D$ are isomorphic to $K\coprod Q$, then one looks for an additional structure in $C$ needed to equip the coproduct $K \coprod Q$ with a structure of an object in $C$ such that the $i$ and $p$ are morphisms in
$C$ making above a short exact sequence in $C$.

### Other notions of extension

* [[extension (semantics)]]

* [[context extension]]

* [[absolute extensor]]

* [[extension of distributions]]

## Examples

### Extension of functions

The [[Tietze extension theorem]] is about *extensions of continuous maps* from a subspace to a normal toplogical space.

[[!include extension theorems -- table]]

### Group extensions

For example, in the category [[Grp]] of (possibly nonabelian) groups one has a short exact sequence usually denoted $1\to X\to Z\to Y\to 1$ corresponding to a [[group extension]].

## Related concepts

* [[homotopy extension property]]

* [[Kan extension]]

* [[lifts and extensions]]

## References

General discussion with an eye towards [[algebraic topology]] and the [[Tietze extension theorem]]:

* [[Norman Steenrod]], _Cohomology operations, and obstructions to extending continuous functions_, Advances in Mathematics Volume 8, Issue 3, June 1972, Pages 371-416 (doi:10.1016/0001-8708(72)90004-7, [pdf](http://www.maths.ed.ac.uk/%7Eaar/papers/steen6.pdf))

[[!redirects extensions]]


extension (semantics)

> This entry is about the notion of extension in [[semantics]]. For the notion of [[extension]] in [[algebra]], see there.

#Contents#
* table of contents
{:toc}

## Idea

Given a [[proposition]] $P$ on [[terms]] of [[type]] $X$, its _extension_ is the type $\{x \in X | P(x)\}$ of all elements $x$ for which $P(x)$ is [[true]].

## Related entries

* [[extensional type theory]]

* [[comprehension]]

## References

* wikipedia, [extension (semantics)](http://en.wikipedia.org/wiki/Extension_%28semantics%29)