exponential object

> This entry is about the concept in [[category theory]]. For (co)exponential functions see at _[[exponential map]]_ and _[[coexponential map]]_.

***

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#### Category theory
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# Contents
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## Idea

An __exponential object__ $X^Y$ is an [[internal hom]] $[Y,X]$ in a [[cartesian closed category]].  It generalises the notion of [[function set]], which is an exponential object in [[Set]].

More generally, in a category with [[finite products]], an exponential object $X^Y$ is a [[representing object]] for the functor $\hom(- \times Y, X)$.

{#YetMoreGenerally} Yet more generally, an exponential object $X^Y$ in a [[category]] $\mathcal{C}$ is a [[representing object]] for the functor $y(X)^{y(Y)}$, where $y \,\colon\, \mathcal{C} \longrightarrow PSh(\mathcal{C})$ denotes the [[Yoneda embedding]] and the latter exponentiation takes place in the [[category of presheaves]] $PSh(\mathcal{C})$, where it always exists (see at *[[closed monoidal structure on presheaves]]).

## Definition

The above is actually a complete definition, but here we spell it out.

Let $X$ and $Y$ be objects of a [[category]] $C$ such that all binary [[products]] with $Y$ exist.  (Usually, $C$ actually has all binary products.)  Then an __exponential object__ is an object $X^Y$ equipped with an __[[evaluation map]]__ $\mathrm{ev}\colon X^Y \times Y \to X$ which is universal in the sense that, given any object $Z$ and map $e\colon Z \times Y \to X$, there exists a unique map $u\colon Z \to X^Y$ such that
$$ Z \times Y \stackrel{u \times \mathrm{id}_Y}\to X^Y \times Y \stackrel{\mathrm{ev}}\to X $$
equals $e$.

Equivalently, this data can be repackaged as a natural isomorphism $\hom_C(-, X^Y) \cong \hom_C(- \times Y, X)$ (where $\mathrm{ev}$ is the image of the identity arrow on $X^Y$), so that exponential objects are representations of the latter as a [[representable functor]].

The map $u$ is known by various names, such as the _exponential transpose_ or *[[currying]]* of $e$.  It is sometimes denoted $\lambda(e)$ in a hat tip to the [[lambda-calculus]], since in the [[internal logic]] of a cartesian closed category this is the operation corresponding to $\lambda$-abstraction.  It is also sometimes denoted $e^\flat$ (as in music notation), being an instance of the more general notion of [[adjunct]] or [[mate]].

As with other [[universal construction]]s, an exponential object, if any exists, is [[generalized the|unique up to unique isomorphism]].  It can also be characterized as a [[distributivity pullback]].

### Without finite products

In a category without finite products, an exponential object is given by an object $X^Y$ and a universal natural transformation $C(-, X^Y) \times C(-, X) \to C(-, Y)$. See, for instance, [this](https://github.com/punkdit/categories/blob/master/www.mta.ca/cat-dist/archive/2008/08-1#L751-L782) [[categories mailing list]] post by [[Richard Garner]].

## Related notions

As before, let $C$ be a category and $X,Y\in C$.

* If $X^Y$ exists, then we say that $X$ __exponentiates__ $Y$.

* If $Y$ is such that $X^Y$ exists for all $X$, we say that $Y$ is __exponentiable__ (or *powerful*, cf. Street-Verity [pdf](http://www.emis.de/journals/TAC/volumes/23/3/23-03.pdf)).  Then $C$ is __[[cartesian closed category|cartesian closed]]__ if it has a [[terminal object]] and every object is exponentiable (which requires that all binary products exist too).

* More generally, a morphism $f\colon Y \to A$ is __exponentiable__ (or *powerful*) when it is exponentiable in the [[over category]] $C/A$.  This is equivalent to saying that all pullbacks along $f$ exist and that the resulting [[base change]] functor $f^* : C/A \to C/Y$ has a [[right adjoint]], usually denoted $\Pi_f$ and called a [[dependent product]].  In particular, $C$ is __[[locally cartesian closed category|locally cartesian closed]]__ iff every morphism is exponentiable, iff all pullback functors have right adjoints.  (Sometimes locally cartesian closed categories are also required to have a terminal object, and hence to also be cartesian closed.)

* Conversely, if $X$ is such that $X^Y$ exists for all $Y$, we say that $X$ is __exponentiating__.  (This requires that $C$ have all binary products.)  Again, $C$ is __cartesian closed__ if it has a terminal object and every object is exponentiating.  (The reader should beware that some authors say "exponentiable" for what is here called "exponentiating.")

Dually, a __[[coexponential object]]__ in $C$ is an exponential object in the [[opposite category]] $C^{op}$.  A __[[cocartesian coclosed category]]__ has all of these (and an [[initial object]]).  Some coexponential objects occur naturally in algebraic categories (such as [[rings]] or [[frames]]) whose opposites are viewed as categories of spaces (such as [[schemes]] or [[locales]]).  Cf. also [[cocartesian closed category]].

{#WhenCIsNotCartesian} When $C$ is not cartesian but merely monoidal, then the analogous notion is that of a [[internal hom|left/right internal hom]].

## Properties

* If $C$ has equalizers of coreflexive pairs, then any pullback of an exponentiable morphism is exponentiable.  This follows from the [[adjoint triangle theorem]], since the left adjoint $\Sigma_f$ of pullback is [[comonadic functor|comonadic]].

## Examples

Of course, in any cartesian closed category every object is exponentiable and exponentiating.  In general, exponentiable objects are more common and important than exponentiating ones, since the existence of $X^Y$ is usually more related to properties of $Y$ than properties of $X$.

### Exponentiation of sets and of numbers

In the cartesian closed category [[Set]] of [[set]]s, for $X,S \in Set$ to sets, their exponentiation $X^S$ is the set of [[function]]s $S\to X$.

Restricted to [[finite set]]s and under the [[cardinality]] operation $|-| : FinSet \to \mathbb{N}$ this induces an exponentiation operation on [[natural number]]s

$$
  \left|X^S\right| = |X|^{|S|}
  \,.
$$

This exponentiation operation on numbers $(-)^{(-)} : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ is therefore the [[decategorification]] of the canonically defined [[internal hom]] of sets. It sends numbers $a,b \in \mathbb{N}$ to the product

$$
  a^b = a \times a \times \cdots \times a \;\; (b \; \text{factors})
  \,.
$$

If $b = 0$ is [[zero]], the expression on the right is 1, reflecting the fact that $0$ is the [[cardinality]] of the [[empty set]], which is the [[initial object]] in [[Set]].

When the natural numbers are embedded into larger [[rig]]s or [[ring]]s, the operation of exponentiation may extend to these larger context. It yields for instance an exponentiation operation on the [[positive real numbers]].

### More examples

* In [[Top]] (the category of _all_ topological spaces), the exponentiable spaces are precisely the [[core-compact spaces]] (see [Day and Kelly](#DayKelly)).  In particular, this includes [[locally compact Hausdorff space]]s.  However, most [[nice category of spaces|nice categories of spaces]] are cartesian closed, so that all objects are exponentiable; note that usually the cartesian product in such categories has a slightly different topology than it does in $Top$.

The condition that a topological space be core-compact (i.e. exponentiable) is in fact a condition on its underlying locale. More precisely, a topological space is core-compact if and only if its underlying locale is a [[continuous poset]]. In fact, a topological space is exponentiable if and only if its underlying locale is exponentiable:

* A [[locale]] is exponentiable if and only if it is a [[continuous poset]] (see [Hyland](#Hyland)) -- this is sometimes taken as the _definition_ of a [[locally compact locale]]). The notion of continuous poset generalizes straightforward to that of [[continuous category]] and [[continuous ∞-category]]. Using these notions, one has analogous characterization of those [[toposes]] and [[(∞,1)-toposes]] which are exponentiable (see [[metastably locally compact locale]] and [[continuous category]] as well as [[exponentiable topos]]).

* In [[algebraic set theory]] (see [[category with class structure]] for one example) one often assumes that only small objects (and morphisms) are exponentiable. analogous to how in material [[set theory]] one can talk about the class of functions $Y\to X$ when $Y$ is a set and $X$ a class, but not the other way round.

* In a [[type theory]] with [[dependent product]]s, every display morphism is exponentiable in the category of [[context]]s ---even in a type theory without identity types, so that not every morphism is display and the relevant slice category need not have all products.

* In a [[functor category]] $D^C$, a [[natural transformation]] $\alpha:F\to G$ is exponentiable if it is [[cartesian natural transformation|cartesian]] and each component $\alpha_c:F c \to G c$ is exponentiable in $D$.  Given $H\to F$, we define $\Pi_\alpha(H)(c) = \Pi_{\alpha_c}(H c)$; then for $u:c\to c'$ to obtain a map $\Pi_{\alpha_c}(H c) \to \Pi_{\alpha_{c'}}(H c')$ we need a map $\alpha_{c'}^*(\Pi_{\alpha_c}(H c)) \to H c'$.  But since $\alpha$ is cartesian, $\alpha_{c'}^*(\Pi_{\alpha_c}(H c)) \cong \alpha_c^* (\Pi_{\alpha_c}(H c))$, so we have the counit $\alpha_c^* (\Pi_{\alpha_c}(H c)) \to H c$ that we can compose with $H u$.  (This is certainly not an if-and-only-if, however: for instance, if $C$ is small, then all morphisms of $Set^C$ are exponentiable, whether or not they are cartesian.)

However, exponentiating objects do matter sometimes.

* In [Abstract Stone Duality](http://www.paultaylor.eu/ASD/), [[Sierpinski space]] is exponentiating.

* [[Toby Bartels]] has [argued](http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html#c023187) that [[predicative mathematics]] can have a set of [[truth value]]s as long as this set is not exponentiating (or even exponentiates only [[finite set]]s).

* A [[dialogue category]] is a [[symmetric monoidal category]] equipped with the non-cartesian monoidal analogue of an exponentiating object.

### Relative Examples

Let $\mathcal{C}$ be a category with finite limits and $f: C \to D$ a morphism in $\mathcal{C}$. Then $f$ is exponentiable as an object of the slice category $\mathcal{C}\downarrow D$ if and only if the [[base change]] functor $f^\ast: \mathcal{C} \downarrow D \to \mathcal{C} \downarrow C$ has a right adjoint. In this case, we say that $f$ is an [[exponentiable morphism]] in $\mathcal{C}$.

* The exponentiable morphisms in $Top$ were characterized by [Niefield](#Niefield82). In particular, a subspace inclusion $C \to D$ is exponentiable if and only if it is [[locally closed]].

* The exponentiable morphisms in $Locale$ and $Topos$ which are embeddings were also characterized by [Niefield](#Niefield81). It seems(?) that no complete characterization of exponentiable morphisms in $Locale$ or $Topos$ appears in the literature.

* The exponentiable morphisms in $Cat$ are the [[Conduché functors]].

## Properties

As with other internal homs, the __[[currying]]__ isomorphism
$$ hom_C(Z,X^Y) \cong hom_C(Z \times Y,X) $$
is a [[natural isomorphism]] of sets. By the usual [[Yoneda lemma|Yoneda]] arguments, this isomorphism can be internalized to an isomorphism in $C$:
$$ (X^Y)^Z \cong X^{Y\times Z}. $$
Similarly, $X \cong X^1$, where $1$ is a [[terminal object]].  Thus, a product of exponentiable objects is exponentiable.

Other natural isomorphisms that match equations from ordinary algebra include:

*  $(X \times Y)^Z = X^Z \times Y^Z$;
*  $1^Z \cong 1$.

These show that, in a cartesian monoidal category, a product of exponentiating objects is also exponentiating.

Now suppose that $C$ is a [[distributive category]].  Then we have these isomorphisms:

*  $X^{Y + Z} \cong X^Y \times X^Z$;
*  $X^0 \cong 1$.

Here $Y + Z$ is a [[coproduct]] of $Y$ and $Z$, while $0$ is an [[initial object]].  Thus in a distributive category, the exponentiable objects are closed under coproducts.

Note that any cartesian closed category with finite coproducts must be distributive, so all of the isomorphisms above hold in any closed [[2-rig]] (such as [[Set]], of course).

## Related concepts

* [[exponential ideal]]

* [[exponentiable topos]]

* [[reflexive object]]

## References

Discussion with focus on application to [[topological spaces]] and [[compactly generated topological spaces]]:

* {#DayKelly} [[Brian Day]], [[G. Max Kelly]], *On topological quotients preserved by pullback or products*, Mathematical Proceedings of the Cambridge Philosophical Society **67** 3 (1970) 553 - 558  ([doi:10.1017/S0305004100045850](https://doi.org/10.1017/S0305004100045850))

* [[Susan Niefield]], *Cartesianness*, PhD thesis, Rutgers 1978 ([proquest:302920643](https://www.proquest.com/docview/302920643))

* {#Niefield82} [[Susan Niefield]], _Cartesianness: topological spaces, uniform spaces, and affine schemes._, Journal of Pure and Applied Algebra, 23.2, 1982, pp. 147-167.(doi:10.1016/0022-4049(82)90004-4, [pdf](https://core.ac.uk/download/pdf/81936509.pdf))

* {#Niefield81} [[Susan Niefield]], _Cartesian inclusion: locales and toposes._, Communications in Algebra, 9.16, 1981, pp. 1639-1671  ([doi:10.1080/00927878108822672](https://doi.org/10.1080/00927878108822672))

* [[Francis Borceux]], Section 7.1 of: *Categories and Structures*, Vol. 2 of: *[[Handbook of Categorical Algebra]]*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865))

Discussion for [[locales]]:

* {#Hyland} [[Martin Hyland]], _Function Spaces in the Category of Locales._ ([doi:10.1080/00927878108822672](https://doi.org/10.1080/00927878108822672))

Discussion for [[internal groupoids]] in [[Top]] ([[topological groupoids]]):

* [[Susan B. Niefield]], [[Dorette A. Pronk]], *Internal groupoids and exponentiability*, Cahiers de topologie et géométrie différentielle catégoriques, [**LX** 4 (2019)](http://cahierstgdc.com/index.php/volume-lx/) ([pdf](http://cahierstgdc.com/wp-content/uploads/2019/10/Niefield-Pronk-LX-4.pdf))

Exponentiable [[relational structures]] are considered in

* [[Jason Parker]], *Exponentiability in categories of relational structures*  (2022) [[arXiv:2208:07350](https://arxiv.org/abs/2208.07350)]

[[!redirects exponential object]]
[[!redirects exponential objects]]
[[!redirects exponentiable object]]
[[!redirects exponentiable objects]]
[[!redirects exponentiable morphism]]
[[!redirects exponentiable morphisms]]
[[!redirects powerful morphism]]
[[!redirects powerful morphisms]]
[[!redirects powerful object]]
[[!redirects powerful objects]]

[[!redirects exponentiating object]]
[[!redirects exponentiating objects]]


exponential ring

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# Contenta
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## Idea

Exponential rings are [[rings]] that behave like the [[real numbers]] with its [[exponential map]].

## Definitions

A [[ring]] $R$ is a __exponential ring__ if it is equipped with a [[monoid]] [[homomorphism]] $e^{(-)}:R \to R$ from the additive monoid of $R$ to the multiplicative monoid of $R$:

$$e^{0} = 1$$

$$e^{a + b} = e^a \cdot e^b$$

## Properties

The monoid homomorphism is an [[group]] homomorphism into the multiplicative subgroup of two-sided [[units]] of $R$, due to the fact that the additive monoid of $R$ is a [[group]].

### Trigonometric functions

If an exponential ring $R$ has elements $i$ and $\frac{1}{2}$ in $R$ such that $i \cdot i = -1$ and $\frac{1}{2} + \frac{1}{2} = 1$, then the __[[sine]]__ and __[[cosine]]__ could be defined as

$$
\sin{x} \coloneqq \frac{1}{2}\cdot i\cdot (e^{-i\cdot x} - e^{i\cdot x}) = \frac{1}{2}\cdot (e^{-x \cdot i} - e^{x \cdot i}) \cdot i
$$

$$
\cos{x} \coloneqq \frac{1}{2}\cdot (e^{i\cdot x} + e^{-i\cdot x})
$$

It could be proven from these definitions that the sine and cosine satisfy various [[trigonometric identities]].

## Examples

* Every ring can be made into a exponential ring by defining $e^a \coloneqq 1$ for all $a$ in $R$.

* Given a natural number $n$, the [[integers modulo n|integers modulo $2n$]] $\mathbb{Z}/2n\mathbb{Z}$ with $e^{(-)}$ defined such that $e^{a} = 1$ for $a \equiv 0 \mod 2$ and $e^{a} = -1$ for $a \equiv 1 \mod 2$ is an exponential ring.

* The real numbers and complex numbers with the usual [[exponential map]] $\exp$ are a exponential ring.

* An exponential ring that is a [[field]] is a __exponential field__.

## Related concepts

* [[exponential map]]

* [[Pythagorean ring]]

## External links

* Wikipedia, *[Exponential ring](https://en.wikipedia.org/wiki/Exponential_ring)*


exponentiation > history

< [[exponentiation]]


Ext

> under construction

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#### Homological algebra
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#Contents#
* table of contents
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## Idea

In the context of [[homological algebra]] the [[right derived functor]] of the [[hom-functor]] is called the _$Ext$-functor_ . It derives its name from the fact that the [[derived hom-functor]] between [[abelian groups]] classifies abelian [[group extensions]] of $A$ by $K$. (This is a special case of the general classification of [[principal ∞-bundles]]/[[∞-group extensions]] by general [[cohomology]]/[[group cohomology]].)

Together with the [[Tor]]-functor it is one of the central objects of interest in homological algebra.

Given an [[abelian category]] $\mathcal{A}$ we may consider the [[hom-functor]] $Hom_{\mathcal{A}} : \mathcal{A}^{op}\times \mathcal{A}\to $[[Ab]] either as a functor in first or in second argument, and compute the corresponding [[right derived functors]].

If they exist, the classical right derived functors of either functor agree and also agree with the [[homology]] of the mixed [[double complex]] obtained by taking simultaneously a  [[projective resolution]] of the first contravariant argument and an [[injective resolution]] of the second covariant argument. The last construction is called the _balanced $Ext$._

Alternatively, one can consider the [[derived category]] $D(\mathcal{A})$ and define

$$
  Ext^p(X,A) \coloneqq Hom_{D(A)}(X,A[p])
$$

or define $Ext^i$-groups as groups of abelian extensions of length $i$, discussed below at _[Relation to extensions](#RelationToGroupExtensions)_.

## Definition

We give the definition following the discussion at _[[derived functors in homological algebra]]_.

### Contravariant $Ext$ on an ordinary object
 {#ContravariantExtOnObject}

Let $\mathcal{A}$ be an [[abelian category]] with
[[projective object|enough projectives]]. And let $A \in \mathcal{A}$ be any object. Consider the [[contravariant functor|contravariant]] [[hom-functor]]

$$
  Hom_{\mathcal{A}}(-, A) : \mathcal{A}^{op} \to Ab
  \,.
$$

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

This is a [[left exact functor]].

Therefore to derive it by [[resolutions]] we need to consider
[[injective resolutions]] in the [[opposite category]] $\mathcal{A}^{op}$.
But these are [[projective resolutions]] in $\mathcal{A}$ itself.

=--

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

For $X \in \mathcal{A}$ any object and $((Q X) \to X) \in Ch_{\bullet \geq 0}(\mathcal{A})$ a [[projective resolution]], and for $n \in \mathbb{N}$, the **$n$th $Ext$-group** of $X$ with [[coefficients]] in $A$ is the degree-$n$ [[cochain cohomology]]

$$
  Ext^n(X,A)
   \coloneqq
  H^n ( Hom_{\mathcal{A}}((Q X)_\bullet, A))
$$

of the [[cochain complex]] $Hom((Q X)_\bullet, A)$.

=--

The following proposition expands a bit on the meaning of this definition. Write

$$
  [-,-] : Ch_{\bullet}(\mathcal{A})^{op} \times Ch_\bullet(\mathcal{A})
  \to
  Ch_\bullet(Ab)
$$

for the [[internal hom of chain complexes|enriched hom of chain complexes]].

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

The $n$th Ext-group is canonically identified with the 0-th [[homology]] of this enriched hom from the resolution $Q X$ of $X$ to the $n$-fold [[delooping]]/[[suspension]] chain complex of $A$
$\mathbf{B}^n A = A[n]$ (concentrated on $A$ in degree $n$):

$$
  Ext^n(X,A)
  \simeq
  H_0 [(Q X), A[n] ]
  \,;
$$

or equivalently, if we think of degree [[chain homology]] as the 0th [[homotopy group]] (under [[Dold-Kan correspondence]]) and write the $n$-fold [[suspension]]/[[delooping]] of $A$ as $\mathbf{B}^n A$:

$$
  Ext^n(X,A)
  \simeq
  \pi_0 [(Q X), \mathbf{B}^n A ]
  \,.
$$

=--

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

This is a special case of the general discussion at [[cochain cohomology]].

By the discussion at _[[internal hom of chain complexes]]_, the 0-[[cycles]] of $[(Q X), \mathbf{B}^n A ]$ are [[chain maps]] of the form

$$
  \array{
    \vdots && \vdots
    \\
    \downarrow && \downarrow
    \\
    (Q X)_{n+1} &\stackrel{f_{n+1}}{\to}& 0
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}} && \downarrow
    \\
    (Q X)_{n} &\stackrel{f_{n}}{\to}&  A
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}} && \downarrow
    \\
    (Q X)_{n-1} &\stackrel{f_{n-1}}{\to}& 0
    \\
    \downarrow && \downarrow
    \\
    \vdots && \vdots
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_1}} && \downarrow
    \\
    (Q X)_1 &\stackrel{f_1}{\to}& 0
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_0}} && \downarrow
    \\
    (Q X)_0 &\stackrel{f_0}{\to}& 0
  }
  \,.
$$

By the definition of chain maps this are precisely those morphisms
$f_n : (Q X)_n \to A$ such that

$$
  d^n f_n \coloneqq f_n \circ \partial^{Q X}_n = 0
$$

which exhibits $f_n$ as a degree-$n$ [[cochain]] in the [[cochain complex]] $Hom((Q X)_\bullet, A)$.

Similarly, the $0$-[[boundaries]] in $[(Q X), \mathbf{B}^n A]$ come from [[chain homotopies]] $\lambda : 0 \Rightarrow f$:

$$
  \array{
    \vdots && \vdots
    \\
    \downarrow && \downarrow
    \\
    (Q X)_{n+1} &\stackrel{f_{n+1}}{\to}& 0
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}}
     &\nearrow_{\mathrlap{\lambda_{n+1}}}& \downarrow
    \\
    (Q X)_{n} &\stackrel{f_{n}}{\to}&  A
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_{n-1}}}
    &\nearrow_{\mathrlap{\lambda_{n}}}& \downarrow
    \\
    (Q X)_{n-1} &\stackrel{f_{n-1}}{\to}& 0
    \\
    \downarrow && \downarrow
    \\
    \vdots && \vdots
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_1}}
    &\nearrow_{\mathrlap{\lambda_{2}}}& \downarrow
    \\
    (Q X)_1 &\stackrel{f_1}{\to}& 0
    \\
    \downarrow^{\mathrlap{\partial^{Q X}_0}}
    &\nearrow_{\mathrlap{\lambda_{1}}}& \downarrow
    \\
    (Q X)_0 &\stackrel{f_0}{\to}& 0
  }
  \,.
$$

in that

$$
  f_n = \lambda_n \circ \partial^{Q X}_{n-1}
  \,.
$$

This are precisely the degree-$n$ [[coboundaries]] in
$Hom((Q X)_\bullet, A)$.

=--

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

This perspective on the $Ext^n$-group as being the [[homotopy classes]] of maps out of (a resolution of) $X$ to $\mathbf{B}^n A$ is made more manifest in the discussion [in terms of derived categories](#InTermsOfDerivedCategories) below. It connects $Ext$-groups and their [relation to extensions](#RelationToGroupExtensions) to the general context of _[[cohomology]]_ and _[[∞-group extensions]]_. See at _[[abelian sheaf cohomology]]_ for more on this.

=--

### In terms of derived categories
 {#InTermsOfDerivedCategories}

(...)

([Kashiwara-Shapira](#KashiwaraShapira))

(...)

## Properties

### Basic properties
 {#BasicProperties}

\begin{proposition}
  \label{ExtCompatibleWithDirectSumsAndProducts}

For all $n \in \mathbb{N}$ and $R \in Rings$, the $Ext^n_R(-,-)$-functor sends [[direct sums]] in the first variable and [[direct products]] in the second variable to direct products:

$$
  Ext^n_R
  \big(
    \oplus_i A_i
    \,,\,
    B
  \big)
  \;\simeq\;
  \prod_i
  Ext^n_R
  \big(
    A_i
    \,,\,
    B
  \big)
$$

and

$$
  Ext^n_R
  \big(
    A
    \,,\,
    \prod_i B_i
  \big)
  \;\simeq\;
  \prod_i
  Ext^n_R
  \big(
    A
    \,,\,
    B_i
  \big)
  \,.
$$

\end{proposition}

(e.g. [Weibel, Prop. 3.3.4](#Weibel))

In particular, $Ext^n_R$ preserves the [[finite colimit|finite]] [[coproduct]] (a [[biproduct]]) in both variables:

$$
  Ext^n_R
  \big(
    A
    \,,\,
    B_1 \oplus B_2
  \big)
  \;\simeq\;
  Ext^n_R
  \big(
    A
    \,,\,
    B_1
  \big)
  \oplus
  Ext^n_R
  \big(
    A
    \,,\,
    B_2
  \big)
  \,.
$$

### Relation to extensions
 {#RelationToGroupExtensions}

We discuss how the group $Ext^n(X,A)$ is identified with the group of [[extensions]] of $X$ by $\mathbf{B}^{n-1} A = A[n-1]$. In particular for $n = 1$ and $\mathcal{A} = $ [[Ab]] this means that $Ext^1(X,A)$ classified ordinary [[group extensions]] of $X$ by $A$.

This is the relation that the name "$Ext$" derives from. At _[[infinity-group extension]]_ is discussed how this relation is a special case of the more general relation that identifies [[derived hom-spaces]] $\mathbf{H}(X,\mathbf{B}^{n+1} A)$ with $\mathbf{B}^n A$-[[principal ∞-bundles]] over $X$.

#### 1-Extensions over single objects

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

For $X,A \in \mathcal{A}$ two objects, an **[[extension]]** of $X$ by $A$ is a [[short exact sequence]]

$$
  0 \to A \to P \to X \to 0
  \,.
$$

An [[homomorphism]] of two such extensions $P_1$ and $P_2$ is a [[morphism]] $P_1 \to P_2$ in $\mathcal{A}$ fitting into a [[commuting diagram]] of the form

$$
  \array{
     && P_1
     \\
     & \nearrow & & \searrow
     \\
     A && \downarrow && X
     \\
     & \searrow & & \nearrow
     \\
     && P_2
  }
  \,.
$$

=--

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

All these homomophisms are necessarily [[isomorphisms]], by the [[short five lemma]].

=--

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

Write $Ext(X,A)$ for the set of [[isomorphism classes]] of such extensions.

=--

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

Under [[Baer sum]] $Ext(X,A)$ becomes an [[abelian group]].

=--

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

Define a morphism

$$
  ExtractCocycle
  \;\colon\;
  Extensions(X,A)
  \longrightarrow
  Ext^1(X,A)
$$

by the following construction:

1. Choose a [[projective presentation]] $N \hookrightarrow Q \to X$ of $X$. Then  for $A \to P \to X$ an [[algebra extension|extension]] consider the [[commuting diagram]]

   $$
     \array{
       N
         &\longrightarrow&
       Q
         &\longrightarrow&
       X
       \\
       \big\downarrow {}^{\mathrlap{\sigma|_N}}
         &&
       \big\downarrow {}^{\mathrlap{\sigma}}
         &&
       \big\downarrow {}^{\mathrlap{=}}
       \\
       A
         &\longrightarrow&
       P
         &\longrightarrow&
       X
     }
     \,,
   $$

   where

   * $\sigma$ is any choice of [[lift]] of $Q \to X$ through $P \to X$, which exists by definition since $Q$ is a [[projective object]];

   * $\sigma|_N$ is the induced morphism on the [[fibers]], which exists by the [[exact sequence|exactness]] of the two sequences.

1. By prop. \ref{Ext1FromProjectivePresentation}, $\sigma|_N$ represents an element $ \big[ \sigma|_N \big] \in Ext^1(X,A)$. Let this be the [[image]] of the map to be defined:

$$
  ExtractCocycle
    \;\colon\;
  (A \to P \to X)
    \;\mapsto\;
  \big[ \sigma|_N \big]
  \,.
$$

This construction is independent of the choice of $P$ and $\sigma$ involved and hence the map is well-defined.

=--

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

The map from def. \ref{ExtractCocycleFromExtension}
is a [[natural isomorphism]] of abelian groups

$$
  Ext(X,A) \stackrel{\simeq}{\longrightarrow} Ext^1(X,A)
  \,.
$$

=--

#### Higher extensions over general chain complexes

(...)

Given $[g] \in \mathbb{R}Hom(X, A[n])$.

Let $Q \to X$ be a [[projective resolution]].
Let $g : Q \to A[n]$ be a representative of $[g]$.
Consider the [[pullback]]

$$
  \array{
    P &\to& cone(0 \to A[n])
    \\
    \downarrow && \downarrow
    \\
    Q &\stackrel{g}{\to}& A[n]
    \\
    \downarrow
    \\
    X
  }
$$

(...)

### Relation to group cohomology
 {#RelationToGroupCohomology}

For $G$ a [[discrete group]] with $\mathbb{Z}[G]$ its [[group ring]], over the [[integers]], and for $N$ a linear $G$-[[representation]], hence a $\mathbb{Z}[G]$-[[module]], the [[group cohomology]] of $G$ with [[coefficients]] in $N$ is

$$
  Ext^\bullet_{\mathbb{Z}[G]Mod}(\mathbb{Z}, N)
  \,.
$$

(...)

### Localization
 {#Locatization}

(...)

### Techniques for constructing $Ext^n$

We discuss some facts helpful for the construction of $Ext^n$-groups in certain situations.

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

If $X \in \mathcal{A}$ is a [[projective object]], then

$$
  Ext^n(X, -)  = 0
$$

is the [[zero object|zero]]-[[functor]] for all $n \geq 1$.

=--

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

The covariant [[hom-functor]] $Hom(X,-)$ is generally a [[left exact functor]]. By the construction of $Ext^n$ via [[projective resolutions]], def. \ref{OfSingleObjByProjResolution}, it is sufficient to show that it is also a [[right exact functor]] if $P$ is projective. In fact, this is one of the equivalent characterizations of _[[projective objects]]_ (ee the section [projective object -- in abelian categories -- equivalent characterizations](projective+object#EquivalentCharacterizationInAbelianCats) for details).

=--

+-- {: .num_prop #Ext1FromProjectivePresentation}
###### Proposition

For $X, A \in \mathcal{A}$ two objects, and

$$
  0 \to N \stackrel{i}{\hookrightarrow} P \stackrel{p}{\to} X \to 0
$$

a [[short exact sequence]] with $P$ a [[projective object]], hence exhibiting a [[projective presentation]] $X \simeq coker(N \hookrightarrow P)$ of $X$, there is an [[exact sequence]]

$$
  0
   \to
  Hom(X,A)
    \stackrel{Hom(p,A)}{\to}
  Hom(P, A)
    \stackrel{Hom(i,A)}{\to}
  Hom(N,A)
    \to
  Ext^1(X,A)
    \to
  0
$$

exhibiting $Ext^1(X,A)$ as the [[cokernel]] of $Hom(i,A)$.

=--

### Yoneda product

The [[Yoneda product]] is a pairing

$$
  Ext^n(A,M) \otimes Ext^m(A,N) \to Ext^{n+m}(A,M\otimes_A N).
$$

(...)

## Applications in cohomology

A [[derived hom-functor]] such as the $Ext$ on chain complexes compute general notions of _[[cohomology]]_ (see the discussion there). Here we list some specific incarnations of the $Ext$-construction in the context of cohomology.

### Universal coefficient theorem

The _[[universal coefficient theorem]]_ identifies, under suitable conditions, [[cohomology]] to the [[duality|dual]] of [[homology]] up to $Ext^1$-groups.

### Various notions of cohomology expressed by $Ext$

Various notions of [[cohomology groups]] in the context of [[algebra]] can be expressed as $Ext$-groups, for instance:

* For $G$ a [[discrete group]], with $K[G]$ its [[group ring]] over a commutative ring K, and for $N$ a $K$-linear $G$-[[representation]]   (hence a $K[G]$-[[module]]), the [[group cohomology]] of $G$ with [[coefficients]] in $N$ is

  $$
    Ext^\bullet_{K[G]Mod}(K, N)
    \,.
  $$

* For $A$ an [[associative algebra]] over some [[field]] $k$ and $N$ an $A$-[[bimodule]], hence an $A \otimes A^{op}$-[[module]],

  $$
    Ext^\bullet_{(A \otimes A^{op})Mod}(A, N)
  $$

  is the [[Hochschild cohomology]] of $A$ with [[coefficients]] in $N$.

* For $\mathfrak{g}$ a [[Lie algebra]] over a commutative ring $K$, with [[universal enveloping algebra]] $\mathcal{U}(\mathfrak{g})$, and $N$ a Lie algebra module (hence an $\mathcal{U}(\mathfrak{g})$-module), the [[Lie algebra cohomology]] of $\mathfrak{g}$ with [[coefficients]] in $N$ is

  $$
    Ext^\bullet_{\mathcal{U}(\mathfrak{g}) Mod}(K, N)
    \,.
  $$

## Examples
 {#Examples}

+-- {: .num_prop #ExtensionsOfTheIntegersAreTrivial}
###### Proposition
**([[group extensions]] of the [[integers]] are [[trivial group|trivial]])**

The $Ext^1$-group of the [[integers]] with coefficients in any $A \in $ [[AbelianGroups]] is [[trivial group|trivial]]:

$$
  Ext^1\big(
    \mathbb{Z},
    \,
    A
  \big)
  \;\simeq\; 0
  \,.
$$

=--

(since the integers are already projective,
e.g. [Boardman, Prop. 19](#Boardman))

+-- {: .num_prop #GroupExtensionsOfFiniteCyclicGroups}
###### Proposition
**([[group extensions]] of [[finite group|finite]] [[cyclic groups]])**

The $Ext^1$-group of the [[cyclic group]] of [[order of a group|order]] $n \in \mathbb{N}$ with coefficients in any $A \in $ [[AbelianGroups]] is the [[quotient group]] $A/n A \coloneqq coker\Big( A \overset{n \cdot(-)}{\longrightarrow} A \Big)$:

$$
  Ext^1
  \big(
    \mathbb{Z} / n\mathbb{Z},
    \,
    A
  \big)
    \;\simeq\;
  A / n A
  \,.
$$

=--

(e.g. [Boardman, Prop. 20](#Boardman))

But:

\begin{prop}
$$
  Ext^{\geq 2}
  \big(
    \mathbb{Z} / n\mathbb{Z},
    \,
    A
  \big)
    \;\simeq\;
  0
  \,.
$$
\end{prop}

(e.g. [here](https://www.math.purdue.edu/~arapura/algebra/homological3.pdf#page=8))

+-- {: .num_example #ExamplesOfExtGroupsOfFiniteCyclicGroups}
###### Examples

As special cases of Prop. \ref{GroupExtensionsOfFiniteCyclicGroups} we have:

* $
    Ext^1
    \big(
      \mathbb{Z} / n\mathbb{Z},
      \,
      \mathbb{Z}
    \big)
    \;\simeq\;
    \mathbb{Z} / n\mathbb{Z}
   \,.
  $

* $
    Ext^1
    \big(
      \mathbb{Z} / n\mathbb{Z},
      \,
      \mathbb{Z} / m\mathbb{Z}
    \big)
    \;\simeq\;
    \mathbb{Z} / d\mathbb{Z}
    \,,
  $

  where $d \coloneqq$ [[gcd]]$(n,m)$ .

* $
    Ext^1
    \big(
      \mathbb{Z} / n\mathbb{Z},
      \,
      \mathbb{Q}
    \big)
    \;\simeq\;
    0
    \,.
  $

=--

(e.g. [Boardman, Cor. 21](#Boardman))

In fact the last case of Example \ref{ExamplesOfExtGroupsOfFiniteCyclicGroups} generalizes beyond cyclic groups:

+-- {: .num_prop #ExtensionsByRationalNumebrsAreTrivial}
###### Proposition
**([[group extension]] by the [[rational numbers]] are trivial)**

The $Ext^1$-group of any $A \in $ [[AbelianGroups]] with coefficients in the [[rational numbers]] $\mathbb{Q}$ is trivial:

$$
  Ext^1\big(
    A,
    \,
    \mathbb{Q}
  \big)
  \;\simeq\; 0
  \,.
$$

=--

(e.g. [Boardman, Prop. 22](#Boardman))

Less immediate is this example:

+-- {: .num_prop #ExtensionsByRationalNumebrsAreTrivial}
###### Proposition
**([[group extension]] of [[rational numbers]] by the [[integers]])**

The $Ext^1$-group of the [[rational numbers]] by the [[integers]] is the [[quotient group]] $\mathbb{A}/\mathbb{Q}$

* of the [[ring of adeles|group of adeles]] (without the [[real numbers]]-factor), i.e. the [[restricted product]] $\mathbb{A} \,\coloneqq\, \underoverset{p\;prime}{\prime}{\prod} \mathbb{Q}_p$ for $\mathbb{Q}_p$ the [[p-adic numbers]] restricted along the inclusion $\mathbb{Z}_p \to \mathbb{Q}_p$ of the [[p-adic integers]];

* by the [[rational numbers]] $\mathbb{Q}$ (...):

$$
  Ext^1\big(
    \mathbb{Q},
    \,
    \mathbb{Z}
  \big)
  \;\simeq\;
  \mathbb{A}/\mathbb{Q}
  \,.
$$

=--

([Boardman, Theorem 25](#Boardman))

## Related concepts

[[!include homotopy-homology-cohomology]]

## References

Original discussion:

* {#EilenbergMacLane42} [[Samuel Eilenberg]], [[Saunders MacLane]], *Group Extensions and Homology*, Annals of Mathematics **43** 4 (1942) 757-831 [[doi:10.2307/1968966](https://doi.org/10.2307/1968966), [jstor:1968966](https://www.jstor.org/stable/1968966)]

* [[Nobuo Yoneda]], *On ext and exact sequences*, PhD thesis, Journal of the Faculty of Science, Imperial University of Tokyo, 1960 ([pdf](http://alpha.math.uga.edu/~lorenz/YonedaExtExactSequences.pdf), [CiNii:naid/500000325773](https://ci.nii.ac.jp/naid/500000325773))

* [[Saunders MacLane]], _Group Extensions by primary abelian groups_, Transactions of the American Mathematical Society, **95** 1 (1960) 1-16 [[jstor:1993327](http://www.jstor.org/stable/1993327)]

Texbook accounts (see also most references at _[[homological algebra]]_):

* {#Weibel} [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_,  Cambridge Studies in Adv. Math. 38, CUP 1994

* [[Henri Cartan]], [[Samuel Eilenberg]], _Homological algebra_, Princeton Univ. Press 1956.

* S. I . Gelfand, [[Yuri Manin]], _Methods of homological algebra_

A systematic discussion from the point of view of [[derived categories]] is in

* {#KashiwaraShapira} [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_, Springer (2000)

Lecture notes:

* [[Kiyoshi Igusa]], _25 The Ext Functor_ ([pdf](http://people.brandeis.edu/~igusa/Math101b/Ext.pdf))

* {#Boardman} [[Michael Boardman]], _Some Common $Tor$ and $Ext$ Groups_ ([pdf](http://math.jhu.edu/~jmb/note/torext.pdf), [[BoardmanTorAndExtGroups.pdf:file]])

* {#May} [[Peter May]], Section 4 of: _Notes on Tor and Ext_ ([pdf](http://www.math.uchicago.edu/~may/MISC/TorExt.pdf))

* [[Patrick Morandi]], _Ext Groups and Ext Functors_, ([pdf](http://sierra.nmsu.edu/morandi/oldwebpages/math683fall2002/Ext.pdf))

  (warning: the last section on resolutions for cocycles for group (abelian) extensions is not correct)

See also:

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

[[!redirects Ext group]]
[[!redirects Ext-group]]
[[!redirects Ext-groups]]
[[!redirects Ext groups]]
[[!redirects Ext functor]]
[[!redirects Ext functors]]
[[!redirects Ext-functor]]
[[!redirects Ext-functors]]


extended affine symplectic group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
#### Geometric quantization
+--{: .hide}
[[!include geometric quantization - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Given a [[symplectic vector space]] $(V,\omega)$ one may consider the restriction of its [[quantomorphism group]] to the [[affine symplectic group]] $ASp(V,\omega)$ ([Robbin-Salamon 93, corollary 9.3](#RobbinSalamon93))

$$
  \array{
     ESp(V,\omega) &\hookrightarrow& QuantMorph(V,\omega)
     \\
     \downarrow && \downarrow
     \\
     ASp(V,\omega)
     &\hookrightarrow&
     HamSympl(V,\omega)
  }
$$

Sometimes (e.g. [Robbin-Salamon 93, p. 30](#RobbinSalamon93)) this $ESp(V,\omega)$ is called the _extended symplectic group_, but maybe to be more specific one should at the very least say "[[extended affine symplectic group]]" or "extended inhomogeneous symplectic group" ([ARZ 06, prop. V.1](#ARZ06)).

Notice that the further restriction to $V$ regarded as the [[translation group]] over itself is the [[Heisenberg group]] $Heis(V,\omega)$

$$
  \array{
     Heis(V,\omega)
     &\hookrightarrow&
     ESp(V,\omega) &\hookrightarrow& QuantMorph(V,\omega)
     \\
     \downarrow && \downarrow && \downarrow
     \\
     V
     &\hookrightarrow&
     ASp(V,\omega)
     &\hookrightarrow&
     HamSympl(V,\omega)
  }
$$

The group $ESp(V,\omega)$ is that of those [[quantomorphisms]] which come from [[Hamiltonians]] that are [[quadratic Hamiltonians]]. Those elements covering elements in the [[symplectic group]] instead of the [[affine symplectic group]] come from [[homogeneously quadratic Hamiltonians]] (e.g. [Robbin-Salamon 93, prop. 10.1](#RobbinSalamon93)). In fact $ESp$ is the [[semidirect product]] of the [[metaplectic group]] $Mp(V,\omega)$ with the [[Heisenberg group]] ([ARZ 06, prop. V.1](#ARZ06), see also [Low 12](#Low12))

$$
  ESp(V,\omega) \simeq Heis(V,\omega) \rtimes Mp(V,\omega)
  \,.
$$

## Example

Let $(V,\omega) = (\mathbb{R}^2, \mathbf{d}p\wedge \mathbf{d}q)$
be the 2-dimensional [[symplectic vector space]].

Write

$$
  q,p \colon \mathbb{R}^2 \longrightarrow \mathbb{R}
$$

for its two canonical coordinate functions (the "[[canonical coordinates]] and [[canonical momenta]]").

Write

$$
  i \colon \mathbb{R}^2 \longrightarrow \mathbb{R}
$$

for the [[constant function]] with value 1.

The [[Poisson bracket]] is

$$
  \{p.q\} = i
  \,.
$$

Any [[smooth function]] $H \colon \mathbb{R}^2 \to \mathbb{R}$ we may call a [[Hamiltonian]]. Given a Hamiltonian $H$, its [[Hamiltonian flow]] is the [[flow]] given by the [[vector field]] (the [[Hamiltonian vector field]]) corresponding to the [[derivation]] $\{H,-\}$ on $C^\infty(\mathbb{R}^2)$.

Those Hamiltonians whose Hamiltonian flows are [[linear functions]] on $\mathbb{R}^2$ are precisely the [[homogeneously quadratic Hamiltonians]]:

$$
  \exp(t\{\tfrac{1}{2}p^2,-\})
  \colon
  \left[
    \array{
       q
       \\
       p
    }
  \right]
  \mapsto
  \left[
    \array{
       q +t p
       \\
       p
    }
  \right]
$$

$$
  \exp(\{\tfrac{1}{2}q^2,-\})
  \colon
  \left[
    \array{
       q
       \\
       p
    }
  \right]
  \mapsto
  \left[
    \array{
       q
       \\
       p + t q
    }
  \right]
$$

$$
  \exp(t \{q p,-\})
  \colon
  \left[
    \array{
       q
       \\
       p
    }
  \right]
  \mapsto
  \left[
    \array{
       e^t q
       \\
       e^{-t} p
    }
  \right]
$$

The general element of the [[metaplectic group]] $Mp(\mathbb{R}^2,\mathbf{d}q \wedge \mathbf{d}p)$ is hence

$$
  \exp(t_1 \tfrac{1}{2}p^2 + t_2 \tfrac{1}{2}q^2 + t_3 q p)
$$

By [[differentiation|differentiating]] this by $t$ at $t = 0$ we obtain a [[basis]] for the [[Lie algebra]]  $\mathfrak{sp}(V,\omega)$ of, both, the [[symplectic group]] $Sp(V,\omega)$ as well as its [[metaplectic group]] $Mp(V,\omega)$

$$
  \left[
    \array{
      0 & 0
      \\
      1 & 0
    }
  \right]
  \,,
  \;\;\;
  \left[
    \array{
       0 & 1
       \\
       0 & 0
     }
  \right]
  \,,
  \;\;\;
  \left[
     \array{
       1 & 0
       \\
       0 & -1
     }
  \right]
$$

The Hamiltonians that generate translations are precisely the homogeneously linear Hamiltonians:

$$
  \exp(t\{p,-\})
  \colon
  \left[
    \array{
       q
       \\
       p
    }
  \right]
  \mapsto
  \left[
    \array{
       q + t
       \\
       p
    }
  \right]
$$

$$
  \exp(t\{q,-\})
  \colon
  \left[
    \array{
       q
       \\
       p
    }
  \right]
  \mapsto
  \left[
    \array{
       q
       \\
       p - t
    }
  \right]
$$

This is the key point where the extension appears: While these two linear translation operations themselves (i.e. the underlying [[symplectomorphisms]]) of course commute with each other, their generating Hamiltonians do not Poisson commute but instead form the [[Heisenberg algebra]] extension of the [[translation group]].

The general element of the extended affine symplectic group $ESp(\mathbb{R}^2, \mathbf{d}q \wedge \mathbf{d}p)$ is

$$
  \exp(t_1 \tfrac{1}{2}p^2 + t_2 \tfrac{1}{2}q^2 + t_3 q p + t_4 p + t_5 q + t_6 i)
  \,.
$$

## References

* {#RobbinSalamon93} [[Joel Robbin]],  [[Dietmar Salamon]], _Feynman path integrals on phase space and the metaplectic representation_, Math. Z. __221__ (1996), no. 2, 307–-335, ([MR98f:58051](http://www.ams.org/mathscinet-getitem?mr=98f:58051), [doi](http://dx.doi.org/10.1007/BF02622118), [[RobbinSalamonMetaplectic.pdf:file]]), also  in [[Dietmar Salamon]] (ed.), _Symplectic Geometry_, LMS Lecture Note series 192 (1993)

* {#ARZ06} [[Sergio Albeverio]], J. Rezende and J.-C. Zambrini, _Probability and Quantum Symmetries. II. The Theorem of Noether in quantum mechanics_, Journal of Mathematical Physics 47, 062107 (2006) ([pdf](http://gfm.cii.fc.ul.pt/people/jczambrini/JMathPhys-47-062107.pdf))

* {#Low12} Stephen G. Low, _Maximal quantum mechanical symmetry: Projective representations of the inhomogenous symplectic group_, J. Math. Phys. 55, 022105 (2014) ([arXiv:1207.6787](http://arxiv.org/abs/1207.6787))

[[!redirects extended inhomogeneous symplectic group]]


extended cobordism

[[!include manifolds and cobordisms - contents]]
***
[[!include functorial quantum field theory - contents]]

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

#Idea#

In so far as a [[cobordism]] connects its boundary components with each other, the idea of an _extended cobordism_ is that its (pieces of) boundary components are themselves extended cobordisms between their (pieces of) boundary components.

The idea arose in the context of [[FQFT|extended quantum field theories]] -- which were originally thought of as representations of ordinary cobordism categories -- when it was realized that a decent description of these QFTs requires assigning data to pieces of arbitrary codimension to a manifold. In the context of QFT this is often thought of as an incarnation of the notion of _locality_ in physics, which says that every data assigned to a chunk of space must already be fixed by what is assigned to all pieces of any of its decomopositions.

Indeed, the idea of extended cobordisms led to the [[generalized tangle hypothesis]] which takes this localization concept to its extreme by asserting, roughly,  that the representation of fully extended cobordisms is already entirely determined by what is assigned to the _point_.

#Formalizations#

For a long time only partial progress was made with formalizing the idea of extended cobordisms, which are expected to form an [[infinity-category]] or [[omega-category]] of sorts. One approach using the notion of [[Trimble n-category]] as

* Eugenia Cheng and Nick Gurski, _Towards an $n$-category of cobordisms_ ([tac](http://www.tac.mta.ca/tac/volumes/18/10/18-10abs.html))

Another formalization is by Marco Grandis, using [[multi-cospans]], see

* [[Cospans in Algebraic Topology]]

This is in principle very general, indicating that a general extended cobordisms should be a  [[multispan|multi-cospan]] in [[Top]]. But details are worked out only in dimension up to 2.

In this dimension the resulting structure is closely related to the bicategory of cobordisms described in

* [[Jeffrey Morton]], _A Double Bicategory of Cobordisms With Corners_ ([arXiv](http://arxiv.org/abs/math.CT/0611930v1))

More recently

* [[Jacob Lurie]] has presented a complete formalization in terms of an [[(infinity,n)-category of cobordisms]].

This generalizes an idea which for $n=2$ was for instance also applied by Chris Schommer-Pries, see

* [Schommer Pries on Classification of 2-Dimensional Extended TFTs](http://golem.ph.utexas.edu/category/2008/06/schommerpries.html)

[[!redirects extended cobordisms]]


extended functorial field theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functorial Quantum Field Theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Extended quantum field theory_ (or _multi-tiered quantum field theory_) is the fully [[local quantum field theory|local]] formulation of [[functorial quantum field theory]], stated using [[higher category theory]]

Whereas a

* [[category theory|1-categorical]] [[functorial field theory]] may be regarded as a rule that allows one to compute invariants $Z(\Sigma)$ assigned to $d$-dimensional [[manifold]]s by cutting these manifolds into a sequence $\{\Sigma_i\}$ of $d$-dimensional composable [[cobordism]]s with $(d-1)$-dimensional boundaries $\partial \Sigma_i$, e.g.  $\Sigma = \Sigma_2 \coprod_{\partial \Sigma_1 = \partial \Sigma_2} \Sigma_1$, then assigning quantities $Z(\Sigma_i)$ to each of these and then composing these quantities in some way, e.g. as $Z(\Sigma) = Z(\Sigma_2)\circ Z(\Sigma_1)$;

we have that

* in extended [[functorial field theory]] $Z(\Sigma)$ may be computed by decomposing $\Sigma$ into $d$-dimensional pieces with piecewise smooth boundaries, whose boundary strata are of arbitrary codimension $k$.

For that reason extended functorial field theory is also sometimes called **local** or **localized** functorial field theory. In fact, the notion of locality in [[quantum field theory]] is precisely this notion of locality. And, as also discussed at [[FQFT]], this higher dimensional version of locality is naturally encoded in terms of [[higher category theory|n-functoriality]] of $Z$ regarded as a functor on a [[higher category theory|higher category]] of [[cobordism]]s.

##  Definition

### The category of extended cobordisms

The definition of a $j$-cobordism is recursive. A $(j+1)$-cobordism between $j$-cobordisms is a [[compact space|compact]] [[orientation|oriented]] $(j+1)$-dimensional [[smooth manifold]] with corners whose the boundary is the [[disjoint union]] of the target $j$-cobordism and the orientation reversal of the source $j$-cobordism.  (The base case of the recursion is the [[empty set]], thought of as a $(-1)$-dimensional manifold.)

$n Cob_d$ is an $n$-category with smooth compact oriented $(d-n)$-manifolds as objects and cobordisms of cobordisms up to $n$-cobordisms, up to diffeomorphism, as morphisms.

There are various suggestions with more or less detail for a precise definition of a higher category $n Cob_n$ of fully extended $n$-dimensional cobordisms.

A very general (and very natural) one consists in taking a further step in categorification: one takes $n$-cobordisms as $n$-morphisms and smooth homotopy classes of diffeomorphisms beween them as $(n+1)$-morphisms. Next one iterates this; see details at [[(∞,n)-category of cobordisms]].

See

* [[extended cobordism]].

### Extended functorial field theory

Fix a [[base ring]] $R$, and let $C$ be the [[symmetric monoidal category|symmetric monoidal]] $n$-category of $n$-$R$-modules.

An $n$-extended $C$-valued functorial field theory of dimension $d$ is a symmetric $n$-tensor functor $Z: n Cob_d \rightarrow C$ that maps

* smooth compact oriented $d$-manifolds to elements of $R$

* smooth compact oriented $(d-1)$-manifolds to $R$-modules

* cobordisms of smooth compact oriented $(d-1)$-manifolds to $R$-linear maps between $R$-modules

* smooth compact oriented $(d-2)$-manifolds to $R$-linear [[additive categories]]

* cobordisms of smooth compact oriented $(d-2)$-manifolds to functors between $R$-linear categories

* etc ...

* smooth compact oriented $(d-n)$-manifolds to $R$-linear $(n-1)$-categories

* cobordisms of smooth compact oriented $(d-n)$-manifolds to $(n-1)$-functors between $R$-linear $(n-1)$-categories

with compatibility conditions and gluing formulas that must be satisfied... For instance, since the functor $Z$ is required to be monoidal, it sends monoidal units to monoidal units. Therefore, the $d$-dimensional [[vacuum]] is mapped to the unit element of $R$, the $(d-1)$-dimensional vacuum to the $R$-module $R$, the $(d-2)$-dimensional vacuum to the category of $R$-modules, etc.

Here $n$ can range between $0$ and $d$. This generalizes to an arbitrary symmetric monoidal category $C$ as codomain category.

## Examples

### Classes of examples by dimension

$n=1$ gives ordinary [[TQFT]].

The most common case is when $R = \mathbb{C}$ (the [[complex numbers]]), giving unitary ETQFT.

The most common cases for $C$ are

*  $C = n Hilb(R)$, the category of $n$-[[n-Hilbert space|Hilbert spaces]] over a topological field $R$. As far as we know this is only defined up to $n=2$.

*  $C = n Vect(R)$, the category of $n$-[[n-vector space|vector spaces]] over a field $R$.

*  $C = n Mod(R)$, the (conjectured?) category of $n$-[[n-module|modules]] over a commutative ring $R$.

3d: [[Turaev-Viro model]]

### Generic examples

By the [[cobordism hypothesis]]-theorem every [[fully dualizable object]] in a symmetric monoidal $(\infty,n)$-category with duals provides an example.

### Specific examples

* [[Levin-Wen model]]

* [Chern-Simons theory as a fully extended TQFT](Topological+Quantum+Field+Theories+from+Compact+Lie+Groups#3dCSFullyExtended)

* ...many more...

See also at _[[TCFT]]_.

## Properties

### Construction of ETQFT's

* By generators and relations

* By path integrals (this is Daniel Freed's approach)

* By modular tensor n-categories?

### Classification of ETQFT's

Assume $Z: n Cob_d \rightarrow n Vect(R)$ is an extended TQFT. Since $Z$ maps the $(d-1)$-dimensional vacuum to $R$ as an $R$-vector space, by functoriality $Z$ is forced to map a $d$-dimensional closed manifold to an element of $R$.
Iterating this argument, one is naturally led to conjecture that, under the correct categorical hypothesis, the behaviour of $Z$ is enterely determined by its behaviour on $(d-n)$-dimensional manifolds. See details at [[cobordism hypothesis]].

### Relation of ETQFT to AQFT

See

* [[topological chiral homology]].

also

* _[AQFT from n-functorial QFT](http://arxiv.org/abs/0806.1079)_.

## Related concepts

* [[local quantum field theory]]

* [[extended Lagrangian]]

More on extended QFTs is also at

* [[FQFT]]

[[!include Isbell duality - table]]

## References

### General

* Ruth J. Lawrence, _Triangulations, categories and extended topological field theories_. Quantum topology, 191–208, Ser. Knots Everything, 3, World Sci. Publ., River Edge, NJ, 1993.  [doi](https://doi.org/10.1142/9789812796387_0011).
Presented at the AMS Meeting 876, held in Dayton, Ohio, on October 31, 1992.

* [[Daniel S. Freed]], _Higher Algebraic Structures and Quantization_ ([arXiv:hep-th/9212115](http://arxiv.org/abs/hep-th/9212115))

* [[Daniel S. Freed]], _Quantum Groups from Path Integrals_ ([arXiv:q-alg/9501025](http://arxiv.org/abs/q-alg/9501025))

* [[John Baez]] and [[James Dolan]], _Higher-dimensional Algebra and Topological Quantum Field Theory_  ([arXiv:q-alg/9503002](http://arxiv.org/abs/q-alg/9503002))

* [[Daniel S. Freed]], _Remarks on Chern-Simons theory_, [arXiv:0808.2507](https://arxiv.org/abs/0808.2507).

* [[Jacob Lurie]],  _[[On the Classification of Topological Field Theories]]_. [arXiv](http://arxiv.org/abs/0905.0465)

With an eye towards the full extension of [[Chern-Simons theory]]:

* [[Daniel S. Freed]], _Remarks on Fully Extended 3-Dimensional Topological Field Theories_ (2011) ([pdf](http://www.ma.utexas.edu/users/dafr/stringsmath_np.pdf))

* [[Daniel S. Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]],  _[[Topological Quantum Field Theories from Compact Lie Groups]]_ , in P. R. Kotiuga (ed.) _A celebration of the mathematical legacy of Raoul Bott_ AMS (2010) ([arXiv](http://arxiv.org/abs/0905.0731))

Review:

* [[Luigi Alfonsi]], §6 in: *Higher geometry in physics*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.07308](https://arxiv.org/abs/2312.07308)]

For TQFTs appearing in [[solid state physics]] in the context of [[topological order]] (see also at *[[K-theory classification of topological phases of matter]]*):

* {#FreedMoore12} [[Daniel S. Freed]], [[Gregory Moore]], _Twisted equivariant matter_ ([arxiv/1208.5055](http://arxiv.org/abs/1208.5055))

  > (uses [[equivariant K-theory]] to classify free fermion gapped phases with symmetry)

* {#Freed14} [[Daniel S. Freed]], _Short-range entanglement and invertible field theories_ ([arXiv:1406.7278](http://arxiv.org/abs/1406.7278))

* [[Daniel Freed]], [[Michael Hopkins]], *Reflection positivity and invertible topological phases*, Geometry & Topology ([arXiv:1604.06527](https://arxiv.org/abs/1604.06527))

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], *Condensations in higher categories* ([arXiv:1905.09566](https://arxiv.org/abs/1905.09566))

For its relation to the notion of [[generalized symmetries]]:

* [[Daniel S. Freed]], [[Gregory W. Moore]], [[Constantin Teleman]], *Topological symmetry in quantum field theory* [[arXiv:2209.07471](https://arxiv.org/abs/2209.07471)]

[[!include D=6 N=(2,0) SCFT as an extended functorial field theory -- references]]

[[!redirects extended topological quantum field theory]]

[[!redirects EQFT]]
[[!redirects extended TQFT]]

[[!redirects extended quantum field theory]]
[[!redirects extended QFT]]

[[!redirects extended quantum field theories]]
[[!redirects extended topological quantum field theories]]
[[!redirects extended QFTs]]

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[[!redirects extended topological field theories]]
[[!redirects extended field theory]]
[[!redirects extended field theories]]

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[[!redirects multi-tiered field theories]]

[[!redirects multi-tiered quantum field theory]]
[[!redirects multi-tiered quantum field theories]]

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[[!redirects extended TFT]]
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[[!redirects fully extended TQFTs]]


extended geometric quantization of 2d Chern-Simons theory

> under construction, notes related to material that appears in detail elsewhere, see the [References](#References).

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Local 2d prequantum boundary field theory and KK-Quantization

### **1)** Local 2d prequantum field theory

ambient [[(∞,1)-topos]] $\mathbf{H} = $ [[Smooth∞Grpd]].

We use in there mostly just the [[full sub-(∞,1)-category]]

$$
  DiffStack \hookrightarrow SmoothGrpd \hookrightarrow Smooth \infty Grpd
$$

of [[differentiable stacks]], i.e. the full subcategory on object that are presented by [[Lie groupoids]] but as [[coefficients]] at least we also need [[smooth infinity-groupoid|smooth 2-groupoids]].

In particular, write $\mathbf{B}^2 U(1)$ for the [[smooth 2-groupoid]] which is the [[circle 3-group]].

For $\mathbf{Fields} \in \mathbf{H}$ a [[moduli stack]] of [[field (physics)|fields]], we say that a map

$$
  \array{
    \mathbf{Fields}
    \\
    \downarrow^{\mathrlap{exp(i S)}}
    \\
    \mathbf{B}^2 U(1)
  }
$$

is a [[local action functional]].

(More generally we consider [[smooth super ∞-groupoids]] and replace $\mathbf{B}^2 U(1)$ by the moduli stack of [[super 2-line bundles]].)

Since this is necessarily a [[dualizable objects]] in the [[(∞,2)-category]] of [[(∞,n)-category of correspondences|correspondences]] $Corr_2(\mathbf{H}, \mathbf{B}^2 U(1))$ it induces by the [[cobordism theorem]] a **[[local prequantum field theory]]** given by a [[monoidal (∞,n)-functor|monoidal (∞,2)-functor]]

$$
  \array{
    && Corr_2(\mathbf{H},\mathbf{B}^2 U(1)) &\to& Corr_2(\mathbf{H}, KU Mod)
    \\
    & {}^{\mathllap{\exp(i S)}}\nearrow&  \downarrow
    \\
    Bord_n &\stackrel{\mathbf{Fields}}{\to}& Corr_2(\mathbf{H})
  }
$$

Here we postcomposed with [[geometric realization]]

$$
  \mathbf{B}^2 U(1)
  \to
  {\vert \mathbf{B}^2 U(1)\vert}
  \simeq
  K(\mathbb{Z},3)
  \to
  B gl_1(KU)
  \to
  KU Mod
  \,.
$$

### **2)** Local boundary 2d prequantum field theory

Considering one marked boundary condition we have [[boundary field theory]] defined in addition to the data of $\exp(i S) \colon \mathbf{Fields} \to KU Mod$ by a morphisms in $Corr_2(\mathbf{H}, KU Mod)$ which as a [[diagram]] in $\mathbf{H}$ is of the form

$$
  \array{
    && Q
    \\
    & \swarrow && \searrow^{\mathrlap{i}}
    \\
    \ast && \swArrow_{\mathrlap{\xi}} && \mathbf{Fields}
    \\
    & {}_{\mathllap{\mathbb{I}}}\searrow
    &&
    \swarrow_{\mathrlap{\exp(i S)}}
  }
  \,.
$$

This is the _[[boundary condition]]_.

### **3)** Quantization in KK-Theory
 {#QuantizationInKKTheory}

We [[quantization|quantize]] by interpreting the [[path integral]] as [[push-forward in generalized cohomology]] in [[complex topological K-theory]] $KU$.

Specifically we quantize a correspondence as above by applying the following procedure

1. Decompose the correspondence explicitly as

   $$
     \array{
       && Q
       \\
       & \swarrow && \searrow^{\mathrlap{i}}
       \\
       \ast &\swArrow_{\mathrlap{\xi}}& \downarrow^{\mathrlap{i^\ast \chi}}  && \mathbf{Fields}
       \\
       & {}_{\mathllap{\mathbb{I}}}\searrow
       &&
       \swarrow_{\mathrlap{\chi}}
       \\
       && KU Mod
     }
     \,.
   $$

1. Form [[twisted groupoid convolution algebras]] to produce a [[co-span]] of [[Hilbert bimodules]]

   $$
     \mathbb{C} \stackrel{\xi}{\to}
     C_{i^\ast \chi}(Q)
     \stackrel{i^\ast}{\leftarrow}
     C_\chi(X)
     \,.
   $$

   Regard this as a cospan in [[KK-theory]].

1. Assume that the middle and right objects are [[dualizable objects]]. Choose a map in [[KK-theory]]

   $$
     Th(Q)
     \colon
     \left(C_{i^\ast \chi}\left(Q\right)\right)
     \to
    \left(C_{i^\ast \chi}\left(Q\right)\right)^{\vee}
   $$

   to the dual [[C*-algebra]]. (If this is an KK-[[equivalence]] then this is the [[Thom isomorphism]] in this context.)

1. Consider the [[dual morphism]] to $i^\ast$, to be denoted

   $$
     i_! \;\colon\;
     \left(C_{i^\ast}\left(Q\right)\right)^{\vee}
     \to
     \left(C_\chi\left(X\right)\right)^\vee
   $$

   and then turn the above cospan into the following consecutive composite in [[KK-theory]]

   $$
     i_! Th\left(Q_{i^\ast \chi}\right) \xi
     \colon
     \mathbb{C}
     \stackrel{\xi}{\to}
     C_{i^\ast \chi}(Q)
     \stackrel{Th}{\to}
     (C_{i^\ast \chi}(Q))^\vee
     \stackrel{i_!}{\to}
     (C_{\chi}(X))^\vee
     \,.
   $$

   This we regard as the quantization of the boundary correspondence.

(...)

## 2d Poisson Chern-Simons theory -- Idea

### General

We discuss aspects of the [[extended geometric quantization]] of one of the simplest and yet interesting examples of [[schreiber:∞-Chern-Simons theory]], namely [[2d Chern-Simons theory]] and here specifically the case induced by a binary and non-degenerate [[invariant polynomial]], namely the [[Lie integration|Lie integrated]] version of the [[Poisson sigma-model]].

It turns out that several aspects of the [[extended geometric quantization]] of this 2d [[theory (physics)|theory]] are known in the literature already, albeit in disguise: the [[2-plectic geometry|2-plectic]] [[moduli stack]] of [[field (physics)|fields]] is equivalently what in the literature is known as a _[[symplectic groupoid]]_ and its [[higher geometric quantization]] is essentially what is known as the _[[geometric quantization of symplectic groupoids]]_.

The suggestion that there should be such a relation is contained already in ([Cattaneo-Felder 01](symplectic%20groupoid#CattaneoFelder01)), but at the time of that writing the [[geometric quantization of symplectic groupoids]] had not been performed yet.

To some extent in the following we review the traditional [[geometric quantization of symplectic groupoids]] while showing at the same time how its ingredients are (more) naturally interpreted in [[higher symplectic geometry]]. This makes quantization of symplectic groupoids a good test case against which to check notions of [[higher geometric quantization]].

Specifically, we discuss what should be the first nontrivial case of the Chern-Simons-type _[[holographic principle]]_ realized in [[higher geometric quantization]]: The [[moduli stack]] of the [[2d Chern-Simons theory]] is the [[Lie integration]] of the [[Poisson Lie algebroid]] associated to a [[Poisson manifold]]. The latter we may think of as defining a [[quantum mechanical system]], hence a $(2-1) = 1$-dimensional [[quantum field theory]]. The higher geometric quantization of the 2-d theory yields a [[2-vector space]] [[space of states|of quantum 2-states]] (assigned to the point n codimension 2). Under the identification of [[2-vector spaces]] with [[categories of modules]] over an [[associative algebra]], this space of quantum 2-states identifies (the [[Morita equivalence]]-class of) an algebra. Suitably re-interpreting traditional results about the quantization of symplectic groupods shows that this algebra serves as a kind of [[strict deformation quantization]] of that [[Poisson manifold]] [[Hawkins (2008)](#Hawkins08)].

Notice that at the level of just infinitesimal ("formal") [[deformation quantization]] a similar [[holographic principle|holographic]] relation between quantization of a Poisson manifold and of its associated 2d [[sigma-model]] QFT has famously been shown by [[Alberto Cattaneo]] and [[Giovanni Felder]] to underly [[Kontsevich]]'s construction of deformation quantization (see at _[[Poisson sigma-model]]_). We suggest that the discussion here provides the refinement of this relation to [[strict deformation quantization]].

All of this should be a blueprint for an analogous situaton in one dimension higher, where the analogous procedure should reproduce the famous holographic quantization of the [[2d Wess-Zumino-Witten theory]] in terms of that of a [[3d Chern-Simons theory]].

### Identifying "symplectic groupoids" as 2-plectic groupoids

We briefly indicate the basis for re-interpreting traditional [[symplectic groupoid]]-theory in terms of [[higher symplectic geometry]].

The identification of the traditional notion of "[[symplectic groupoid]]" as really a [[2-plectic geometry|2-plectic]] structure is evident as soon as one translates the traditional definition of a [[symplectic groupoid]] to more intrinsic language of [[smooth infinity-groupoid|higher differential geometry]]. We discuss this in detail below, but in brief it works as follows:

A [[symplectic groupoid]] $(\mathbf{X},\omega)$ is traditionally defined to be a [[Lie groupoid]] $\mathbf{X}$ which is equipped with a (non-degenerate) [[differential 2-form]] $\omega \in \Omega^2(\mathbf{X}_1)$ on its [[smooth manifold]] of [[morphisms]], such that it is annihilated by the [[de Rham differential]] as well as by
the operator $\delta$ that sums the [[pullback of differential forms|pullback]] of $\omega$ to the space $\mathbf{X}_2$ of composable morphisms along the [[source]] and [[target]] maps and minus that along the [[composition]] map.
But together this just means that regarded as a triple $(0, \omega, 0) \in \underset{k = 0,1,2}{\oplus} \Omega^{3-k}(\mathbf{X}_k)$ the symplectic form is a  [[cocycle]] in the [[simplicial de Rham complex]] over the [[simplicial manifold]] which is the [[nerve]] of the Lie groupoid. And this finally means fully intrinsically that $\omega$ is a degree-3 [[cocycle]] in the [intrinsic de Rham cohomology](cohesive+%28infinity,1%29-topos+--+structures#deRhamCohomology) [of smooth ∞-groupoids](smooth+infinity-groupoid+--+structures#deRhamCoefficientsInBnU1) over $\mathbf{X}$, which we denote by

$$
  \omega \;\colon\; \mathbf{X} \to \flat_{dR} \mathbf{B}^3 U(1)
  \,.
$$

This simple observation shows that [[symplectic groupoids]], which in traditional literature are treated as a topic in [[symplectic geometry]], are really objects in [[higher symplectic geometry]], namely in [[2-plectic geometry]].

More specifically, if $(X,\pi)$ is a [[Poisson manifold]], there is canonically associated with it a [[Lie algebroid]] $\mathfrak{P}$, the _[[Poisson Lie algebroid]]_. This is an example of a [[symplectic Lie n-algebroid]] and as such it carries a canonical binary [[invariant polynomial]]. Together this serves as the [[target space]] and [[background gauge field]] of what is called the [[Poisson sigma-model]]. But moreover, the [[Lie integration]] of this data is the [[extended Lagrangian]] of an [[schreiber:∞-Chern-Simons theory]], namely a map

$$
  \mathbf{L}
  \;\colon\;
  \tau_1\exp(\mathfrak{P})_{conn}
  \to
  \mathbf{B}^2 (\mathbb{R}/\Gamma)_{conn}
$$

from the [[moduli stack]] of [[field (physics)|fields]] of the [[2d Chern-Simons theory]] to that of [[circle 2-bundles with connection]].

If we [[concretify]] the moduli stack by forgetting the connection data here and just consider the underlying [[instanton sectors]], then this is precisely the symplectic groupoid data above

$$
  \array{
    \tau_1\exp(\mathfrak{P})_{conn}
    &\stackrel{\mathbf{L}}{\to}&
    \mathbf{B}^2 (\mathbb{R}/\Gamma)_{conn}
    \\
    {}^{\mathllap{concretify}}\downarrow && \downarrow
    \\
    \mathbf{X} &\stackrel{\omega}{\to}& \flat_{dR} \mathbf{B}^3 U(1)
  }
  \,.
$$

It is in this way that we may identify the [[symplectic groupoid]] of a Poisson manifold

$$
  \mathbf{X} \simeq \tau_1 \exp(\mathfrak{P})
$$

as the [[moduli stack]] of the [[extended prequantum field theory|extended prequantum]] [[2d Chern-Simons theory]] which refines the [[Poisson sigma-model]].

Hence we identify the [[geometric quantization]] of $(\mathbf{X}, \omega)$ as the extended geometric quantization of the [[2d Chern-Simons theory]]. But traditional literature shows (and this was motivation for introducing [[symplectic groupoids]] in the first place) that this encodes the [[quantization]] of the [[Poisson manifold]] $(X, \pi)$ itself, regarded as a [[quantum mechanical system]]. This is the traditional topic of _[[geometric quantization of symplectic groupoids]]_.

Taken together this says: the [[extended geometric quantization]] of the [[extended prequantum field theory|extended prequantum]] [[Poisson sigma-model]] computes the ordinary [[geometric quantization]] of the underlying Poisson manifold.

This statement we recognize as a geometric quantization analog of a famous relation in [[algebraic deformation quantization]]. As discussed there, the construction by [[Kontsevich]] of the algebraic deformation quantization of any Poisson manifold was identified by [[Cattaneo]] and [[Felder]] as a limiting case of the  [[n-point function|3-point function]] in the [[perturbation theory|perturbative quantization]] of the corresponding 2d Poisson $\sigma$-model.

These relations to traditional theory we use in the following to explore aspects of the [[extended geometric quantization]] of [[2d Chern-Simons theory]].

### Summary and survey

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## General theory

We use the general tools for formulating notions in [[physics]] in terms of [[higher differential geometry]] as discussed at _[[geometry of physics]]_. Here we just collect some of the main ingredients needed below.

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

We write

$$
  DL
  \;\colon\;
  [CartSp^{op}, Ch_\bullet(Ab)]
  \stackrel{\simeq}{\to}
  [CartSp^{op}, Ab^{\Delta^{op}}]
  \stackrel{forget}{\to}
  [CartSp^{op}, KanCplx]
  \stackrel{}{\to}
  L_{lhe}
  [CartSp^{op}, KanCplx]
  \simeq
  Smooth\infty Grpd
$$

where the first equivalence is the [[Dold-Kan correspondence]] and the last map is the map to the [[simplicial localization]].

=--

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

For $k,n \in \mathbb{N}$, $k \leq n$ we write

$$
  \mathbf{B}^n U(1)_{conn^k}
  \coloneqq
  DK\left[
    \underline{U}(1)
    \stackrel{\mathbf{d}}{\to}
    \Omega^1
    \stackrel{\mathbf{d}}{\to}
    \cdots
    \stackrel{\mathbf{d}}{\to}
    \Omega^k
    \stackrel{}{\to}
    0
    \stackrel{}{\to}
    \cdots
    \stackrel{}{\to}
    0
  \right]
$$

with $\Omega^k$ in degree $(n-k)$ (hence with $(n-k)$ vanishing entries on the right).

In particular

* $\mathbf{B}^n U(1) \simeq \mathbf{B}^n U(1)_{conn^0}$

* $\mathbf{B}^n U(1)_{conn} \simeq \mathbf{B}^n U(1)_{conn^n}$.

=--

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

For $n,k \in \mathbb{N}$, $k \lt n$ we have

1. $\Omega (\mathbf{B} U(1)_{conn}) \simeq \flat \mathbf{B}^{n-1}$

1. $\Omega (\mathbf{B}^n U(1)_{conn}^k) \simeq \mathbf{B}^{n-1} U(1)_{conn}^k$.

=--

(...)

$$
  \array{
    field space
    &&
    \tau_n \exp(\mathfrak{a})_{conn}
    &\to&
    \mathbf{B}^n U(1)_{conn}
    \\
    && \downarrow && \downarrow
    \\
    delooped phase space && \tau_n \exp(\mathfrak{a})
    &\to&
    \mathbf{B}^n U(1)_{conn^{n-1}}
  }
$$

$$
  \array{
    C_0 &\to& *
    \\
    \downarrow && \downarrow
    \\
    \tau_n \exp(\mathfrak{a}) &\to& \mathbf{B}^n U(1)_{conn^{n-1}}
    \\
    \uparrow && \uparrow
    \\
    C_1 &\to& *
  }
$$

under [[homotopy fiber product]] this yields the phase space prequantum bundle

$$
  Phase(C_0, C_1) \to \mathbf{B}^{n-1}U(1)_{conn}
  \,.
$$

(...)

## The setup

Let $(X, \pi)$ be a [[Poisson manifold]], which we tend to think of as defining a _[[quantum mechanical system]]_. This canonically induces the structure of [[Lie algebroid]] over $X$, the _[[Poisson Lie algebroid]]_ $(\mathfrak{P}, \mathbf{\omega})$ over $X$. This is a [[symplectic Lie n-algebroid|symplectic Lie algebroid]], with graded symplectic form (binary [[invariant polynomial]]) $\mathbf{\omega}$ which is in [[transgression]] with the Poisson tensor $\pi$, regarded as a [[Lie algebroid cohomology|Lie algebroid cocycle]]. The transgression is witnessed by a [[Chern-Simons element]] $cs_\pi$.

### Moduli stack of fields

By the general construction of [[schreiber:infinity-Chern-Simons theory]] this means that there is a universal differential characteristic map

$$
  \mathbf{L}
  \;\colon\;
  \exp(\mathfrak{P})_{conn}
  \to
  \mathbf{B}^2 \mathbb{R}_{conn}
$$

and its truncation ([[Lie integration]])

$$
  \mathbf{L}
  \;\colon\;
  \tau_1 \exp(\mathfrak{P})_{conn}
  \to
  \mathbf{B}^2 (\mathbb{R}/\Gamma)_{conn}
  \,.
$$

Here $\tau_1 \exp(\mathfrak{P})_{conn}$ is the [[smooth groupoid|smooth]] [[moduli stack]] of fields of the 2d Poisson-Chern-Simons theory and
$\mathbf{H}$ is the [[extended Lagrangian]] of a [[2d Chern-Simons theory]] ([[schreiber:Higher Chern-Weil Derivation of AKSZ Sigma-Models|FRS]]). Infinitesimally this yields the [[Poisson sigma-model]].

We discuss here the [[higher geometric quantization]] of this [[theory (physics)|theory]] defined by $\mathbf{L}$.

### The Phase space stack
 {#ThePhaseSpaceStack}

There is the canonical universal forgetful map

$$
  \tau_1 \exp(\mathfrak{P})_{conn}
  \to
  \tau_1 \exp(\mathfrak{P})
$$

which forgets the conection data of fields and only retains their "[[instanton sector]]". The [[smooth groupoid]] $\tau_1 \exp(\mathfrak{P})$ is the [[symplectic groupoid]] which is the [[Lie integration]] of the [[Poisson Lie algebroid]] $\mathfrak{P}$. (In the traditional literature this is called the symplectic groupoid only if it happens to be representable by a [[Lie groupoid]], hence if $\mathfrak{P}$ is integrable in the traditional sense of Lie groupoid theory.)

Notice that this is equivalently [[differential concretification]] over the point: $\tau_1 \exp(\mathfrak{P})$ may be understood as the [[smooth groupoid|smooth]] [[moduli stack]] of $\mathfrak{P}$-[[Lie algebroid valued differential forms|valued]] forms on the point.

But moreover, we may think of $\tau_1 \exp(\mathfrak{P})$ as being the [[phase space]] of the [[open string]] 2d Poisson-Chern-Simons [[sigma-model]] for the space-filling [[D-brane]] boundary condition: in this interpretation the [[objects]] of $\tau_1 \exp(\mathfrak{P})$ are the endpoints of open strings, the [[morphisms]] are solutions to the [[Euler-Lagrange equations]] of motion along an interval (an initial spatial slice of string) and [[composition]] is string concatenation.

This is the point of view suggested in ([Cattaneo-Felder 01](symplectic%20groupoid#CattaneoFelder01)), there argued for by the observation that the space of [[morphisms]] of $\tau_1 \exp(\mathfrak{P})$ is naturally the [[symplectic reduction]] of (in our language) of that of $\tau_1 \exp(\mathfrak{P})$ for the [[moment map]] that characterizes the [[equations of motion]] and [[gauge symmetries]] of the [[Poisson sigma-model]].

### D-branes
 {#DBranes}

Let $\mathfrak{C}_0, \mathfrak{C}_1 \hookrightarrow \mathfrak{P}$ be two [[Lagrangian dg-submanifold|Lagrangian]] sub-[[Lie algebroids]] of the given [[Poisson Lie algebroid]]. These correspond to [[coisotropic submanifolds]] of the underlying [[Poisson manifold]].

Accoring to ([Cattaneo-Felder 03](coisotropic+submanifold#CattaneoFelder03)) we are two think of these as being two [[D-brane]] inclusions for the [[open string]] Poisson-Chern-Simons model.

Their [[Lie integration]] induces two Lagrangian sub-groupoids of the [[symplectic groupoid]]

$$
  tau_1 \exp(\mathfrak{C}_0)
  \to
  \tau_1 \exp(\mathfrak{P})
  \leftarrow
  \tau_1 \exp(\mathfrak{C}_1)
  \,.
$$

This means that the [[moduli stack]] of [[open strings]] with endpoints restricted to these two [[D-branes]] is the [[homotopy fiber product]] of [[smooth groupoids]]

$$
  \array{
    & & \mathbf{Fields}_{\mathfrak{C}_0, \mathfrak{C}_1}
    \\
    & \swarrow && \searrow
    \\
    \tau_1 \exp(\mathfrak{C}_0) && && \tau_1 \exp(\mathfrak{C}_1)
    \\
    & \searrow && \swarrow
    \\
    && \tau_1 \exp(\mathfrak{P})
  }
  \,.
$$

One finds that this [[homotopy pullback]] is equivalently the [[symplectic reduction]] considered in ([Cattaneo-Felder 03, page 6, 7](coisotropic+submanifold#CattaneoFelder03)).

That for space-filling branes $\mathfrak{C}_0, \mathfrak{C}_1 \coloneqq \mathfrak{P}$ one recovers the original moduli stack

$$
  \mathbf{Fields}_{\mathfrak{P}, \mathfrak{P}}
  \simeq
  \tau_1 \exp(\mathfrak{P})
$$

is just the statement that the homotopy pullback of an equivalence is an equivalence.

## Constructions

### Moduli of 2d CS fields and symplectic groupoids

We discuss how the [[moduli stack]] of the [[2d Chern-Simons theory]] obtained by [[Lie integration]] of the [[Poisson sigma-model]] is a [[symplectic groupoid]].

Let $\mathfrak{P}$ be the [[Poisson Lie algebroid]] corresponding to a [[Poisson manifold]] that comes from a [[symplectic manifold]] $(X,\omega)$.

The [[symplectic groupoid]] associated to this is (by the discussion there) supposed to be the [[fundamental groupoid]]
$\Pi_1(X)$ of $X$ equipped on its space of morphisms with the differential form $p_1^* \omega - p_2^* \omega$, where $p_1,p_2$ are the two endpoint projections from paths in $X$ to $X$.

We demonstrate in the following how this is indeed the result of applying the [[∞-Chern-Weil homomorphism]] to this situation.

For simplicity we shall start with the simple situation where $(X,\omega)$ has a global [[Darboux coordinate chart]] $\{x^i\}$. Write $\{\omega_{i j}\}$ for the components of the [[symplectic form]] in these coordinates, and $\{\omega^{i j}\}$ for the components of the [[inverse]].

Then the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{P})$ is generated from $\{x^i\}$ in degree 0 and $\{\partial_i\}$ in degree 1, with differential given by

$$
  d_{CE} x^i = - \omega^{i j} \partial_j
$$

$$
  d_{CE} \partial_i = \frac{\partial \pi^{j k}}{\partial x^i} \partial_j \wedge \partial_k = 0
  \,.
$$

The differential in the corresponding [[Weil algebra]] is hence

$$
  d_{W} x^i = - \omega^{i j} \partial_j + \mathbf{d}x^i
$$

$$
  d_{W} \partial_i = \mathbf{d} \partial_i
  \,.
$$

By the discussion at [[Poisson Lie algebroid]], the symplectic [[invariant polynomial]] is

$$
  \mathbf{\omega} = \mathbf{d} x^i \wedge \mathbf{d} \partial_i \in W(\mathfrak{P})
  \,.
$$

Clearly it is useful to introduce a new basis of generators with

$$
  \partial^i := -\omega^{i j} \partial_j
  \,.
$$

In this new basis we have a manifest isomorphism

$$
  CE(\mathfrak{P}) = CE(\mathfrak{T}X)
$$

with the [[Chevalley-Eilenberg algebra]] of the [[tangent Lie algebroid]] of $X$.

Therefore the [[Lie integration]] of $\mathfrak{P}$ is the [[fundamental groupoid]] of $X$, which, since we have assumed global Darboux oordinates and hence [[contractible]] $X$, is just the [[pair groupoid]]:

$$
  \tau_1 \exp(\mathfrak{P}) = \Pi_1(X) = (X \times X \stackrel{\overset{p_2}{\to}}{\underset{p_1}{\to}} X)
  \,.
$$

It remains to show that the symplectic form on $\mathfrak{P}$ makes this a [[symplectic groupoid]].

Notice that in the new basis the invariant polynomial reads

$$
  \begin{aligned}
    \mathbf{\omega}
      &=
     - \omega_{i j} \mathbf{d}x^i \wedge \mathbf{d} \partial^j
     \\
      & = \mathbf{d}( \omega_{i j} \partial^i \wedge \mathbf{d}x^j)
   \end{aligned}
$$

and that we may regard this as a morphism of $L_\infty$-algebroids

$$
  \mathbf{\omega}
   :
  \mathfrak{T}\mathfrak{P} \to \mathfrak{T}b^3 \mathbb{R}
$$

The corresponding [[infinity-Chern-Weil theory|infinity-Chern-Weil homomorphism]] that we need to compute is given by the [[∞-anafunctor]]

$$
  \array{
    \exp(\mathfrak{P})_{diff}
     &\stackrel{\exp(\mathbf{\omega})}{\to}&
    \exp(b \mathbb{R})_{dR}
    &\stackrel{\int_{\Delta^\bullet}}{\to}&
    \mathbf{\flat}_{dR}\mathbf{B}^3 \mathbb{R}
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \exp(\mathfrak{P})
  }
  \,.
$$

Over a test space $U$ in degree 1 an element in $\exp(\mathfrak{P})_{diff}$ is a pair
$(X^i, \eta^i)$

$$
  X^i \in C^\infty(U \times \Delta^1)
$$

$$
  \eta^i \in \Omega^1_{vert}(U \times \Delta^1)
$$

subject to the verticality constraint, which says that along $\Delta^1$ we have

$$
  d_{\Delta^1} X^i + \eta^i_{\Delta^1} = 0
  \,.
$$

The vertical morphism $\exp(\mathfrak{P})_{diff} \to \exp(\mathfrak{P})$ has in fact a [[section]] whose image is given by those pairs for which $\eta^i$ has no leg along $U$. We therefore find the desired form on $\exp(\mathfrak{P})$ by evaluating the top morphism on pairs of this form.

Such a pair is taken by the top morphism to

$$
  \begin{aligned}
    (X^i, \eta^j)
     & \mapsto
   \int_{\Delta^1}
    \omega_{i j} F_{X^i} \wedge F_{\partial^j}
   \\
    & =
  \int_{\Delta^1}
    \omega_{i j} (d_{dR} X^i + \eta^i) \wedge d_{dR} \eta^j
  \in
  \Omega^3(U)
  \end{aligned}
  \,.
$$

Using the above verticality constraint and the condition that $\eta^i$ has no leg along $U$, this becomes

$$
  \cdots =
  \int_{\Delta^1}
  \omega_{i j}
    d_U X^i
  \wedge
    d_U d_{\Delta^1} X^j
  \,.
$$

By the [[Stokes theorem]] the integration over $\Delta^1$ yields

$$
  \cdots
    =
  \omega_{i j} d_{dR} x^i \wedge \eta^j|_{0}
   -
  \omega_{i j} d_{dR} x^i \wedge \eta^j|_{1}
  \,.
$$

This completes the proof.

$$
  \array{
    X &\to& *
    \\
    \downarrow && \downarrow
    \\
    \tau_1 \exp(\mathfrak{P})
    &\stackrel{\omega}{\to}&
    \flat_{dR} \mathbf{B}^3 U(1)
  }
  \,.
$$

### Prequantum 2-states

Generally, given a [[prequantum line 2-bundle]] the corresponding (local) [[prequantum n-states]] are the (local) [[sections]] of this [[line 2-bundle]], and these in turn are equivalently the [[twisted unitary bundles]] (maybe best known for the [[WZW model]] where these are the [[Chan-Paton gauge fields]]). These 2-sections/twisted bundles form a [[2-vector space]] of prequantum 2-states. This is equivalently a [[category of modules]] over some [[associative algebra]], well defined up to [[Morita equivalence]].

A canonical way of constructing this algebra is as follows: present the [[line 2-bundle]] as a [[bundle 2-gerbe]], hence as a multiplicative [[line bundle]] over the morphisms of a [[Lie groupoid]], then the algebra is the [[groupoid algebra]] (see there for details) of sections of this line bundle. A module over this is manifestly a [[bundle gerbe module]], which in turn is equivalently a unitary bundle twisted by the line 2-bundle.

More in detail:

Let $\mathbf{X}$ be a smooth groupoid and $\mathbf{X} \to \mathbf{B}^2 U(1)$ the map modulating a [[circle 2-group]]-[[principal 2-bundle]] $P \to \mathbf{X}$.

Let $(\coprod_n \mathbf{B}U(n))//\mathbf{B}U(1) \to \mathbf{B}^2 U(1)$ the canonical [[infinity-action|2-representation]], the sections of the [[associated infinity-bundle|associated 2-bundle]] are unitary [[twisted bundles]] equivariant on $\mathbf{X}$.

If we present the 2-bundle by a [[bundle gerbe]] exhibited as a multiplicative [[line bundle]] over the space of morphisms $\mathbf{X}_1$, then there is the [[groupoid algebra|convolution algabra]] $\mathcal{A}$ of sections of this line bundle. Twisted unitary vector bundles are equivalently projective [[modules]] over this algebra.

This means that under the identification

$$
  \phi \colon Alg_k \stackrel{\simeq}{\to} 2 Vect_k
$$

of the [[2-category]] of [[2-vector spaces]] with that of [[algebras]], [[bimodules]] and intertwiners (see at _[[n-vector space]]_), the convolution algebra $\mathcal{A}$ _is_ the 2-vector space of sections

$$
  \phi\left(\mathcal{A}\right)
   \simeq
  \Gamma_{\mathbf{X}}\left(P \times_{\mathbf{B}U\left(1\right)} \coprod_n \mathbf{B}U\left(n\right) \right)
  \,.
$$

### Polarizations and branes

The full generalization of the notion of _[[polarization]]_ from traditional [[geometric quantization]] to [[higher geometric quantization]] may need more thinking, but in the case of 2d Chern-Simons theory we can apply the following plausible shortcut.

A [[Poisson Lie algebroid]] $(\mathfrak{P}, \mathbf{\omega})$ is a [[symplectic Lie n-algebroid]] for $n = 1$. This means that if we regard it as a [[dg-manifold]] (the dg-manifold whose [[dg-algebra]] of functions is the [[Chevalley-Eilenberg algebra]] $(CE(\mathfrak{P})$) then the [[invariant polynomial]] $\mathbf{\omega}$ constitutes a graded [[symplectic form]] on $\mathfrak{P}$. Since a [[real polarization]] of an ordinary [[symplectic manifold]] is equivalently a [[foliation]] by [[Lagrangian submanifolds]], it hence makes sense to take a [[real polarization]] of $(\mathfrak{P}, \mathbf{\omega})$ to consist of [[Lagrangian dg-submanifolds]]. As discussed there, for the [[Poisson Lie algebroid]] these corespond to the [[coisotropic submanifolds]] of the underlying [[Poisson manifold]] $(X, \pi)$. The same definition of polarization has been used/obtained in the study of [[geometric quantization of symplectic groupoids]] (see there).

The [[branes]] of the [[2d Chern-Simons theory]] should be those leaves of the foliation which satisfy an extra [[BS-leaf|intrability condition]].

Indeed, according to [[branes]] of the [[Poisson sigma-model]] are supposed to be coisotropic submanifolds, see at [Poisson sigma-model -- Properties -- Branes](Poisson+sigma-model#Branes).

### Quantum 2-states

Summing up, the convolution subalgebra $\mathcal{A}_q \hookrightarrow \mathcal{A}$ of polarized sections is under $\phi$ the actual 2-vector space of states.

In _[[geometric quantization of symplectic groupoids]]_ it is show that this is the algebra of observables of the [[quantum mechanical system]] of the underlying [[Poisson manifold]] (its [[strict deformation quantization]]).

In fact we have to quantize the whole [[atlas]] of the [[symplectic groupoid]]

$$
  X \to \tau_1 \exp(\mathfrak{P})
  \,.
$$

By the discussion at [[groupoid convolution algebra]] this yields a [[bimodule]]

#### Symplectic case

Consider the simple case where $(X,\omega)$ is a [[symplectic manifold]] and in fact a [[symplectic vector space]].

Here the [[symplectic groupoid]] is the [[pair groupoid]] $Pair(X)_\bullet$ carrying the trivial twist. The [[atlas]] is the object inclusion

$$
  X \to Pair(X)
  ,.
$$

Notice that this is equivalent ([[Morita equivalence|Morita equivalent]]) to just the terminal map

$$
  X \to *
  \,.
$$

Accordingly, the [[groupoid convolution algebra]] of $Pair(X)_\bullet$ is the [[C-star-algebra]] of [[compact operators]] $\mathcal{K}(X)$. A [[Morita equivalence]] [[bimodule]] from there to the ground field is (...)

The [[groupoid-principal bundle|groupoid-principal]]-[[bibundle]] corresponding to this atlas regarded as a [[Morita morphism]] is just the [[projection]] $X \times X \stackrel{p_1}{\to} X$. Hence the $C^\ast$-bimodule here is that generated from [[integral kernels]].

In conclusion, up to equivalence the bimodule is $L^2(X)$ regarded as a $C(X)-\mathbb{C}$-bimodule.

(...)

## References
 {#References}

The above discussion was inspired by

* {#Hawkins08} [[Eli Hawkins]], *A Groupoid Approach to Quantization*, J. Symplectic Geom. **6** 1  (2008) 61-125 [[arXiv:math/0612363](https://arxiv.org/abs/math/0612363), [euclid](https://projecteuclid.org/journals/journal-of-symplectic-geometry/volume-6/issue-1/A-groupoid-approach-to-quantization/jsg/1215032733.full)]

and details have been written out in these two theses:

* {#Bongers} Stefan Bongers, _[[schreiber:master thesis Bongers|Geometric quantization of symplectic and Poisson manifolds]]_, master thesis, Utrecht (2013)

* {#Nuiten13} [[Joost Nuiten]], _[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]_, master thesis, Utrecht (2013)

Here section 4 of [Bongers (2013)](#Bongers) recalls and makes explicit relevant details in the construction of the [[symplectic groupoid]], and section 5.2.2 of [Nuiten (2013)](#Nuiten13) essentially provides a re-casting of the quantization construction in [Hawkins (2008)](#Hawkins08) which gives it an invariant home and brings out the above holographic picture more explicitly.

This reveals a general abstract nature of the construction which was meant to be further brought out in:

* [[Urs Schreiber]], *[[schreiber:Quantization via Linear homotopy types]]* [[arXiv:1402.7041](https://arxiv.org/abs/1402.7041)]

where the above situation corresponds to "running example a)", Example 7.3 ([p. 70](https://arxiv.org/pdf/1402.7041.pdf#page=70)).

[[!redirects 2d Poisson-Chern-Simons theory]]
[[!redirects 2d Poisson-Chern-Simons theories]]