duality between M/F-theory and heterotic string theory

> This entry is about [[M-theory]]/[[F-theory]] [[KK-compactification|compactified]] on [[K3-surfaces]]. For [[M-theory]] on [[MO9]]-planes see instead at _[[Hořava-Witten theory]]_.

***

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#Contents#
* table of contents
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## Idea

A [[duality in string theory]]. The [[non-perturbative effect|non-perturbative]] enhancement of [[duality between heterotic and type II string theory]]:

[[F-theory]]$\,$ "[[KK-compactification|KK-compactified]]" on an [[elliptic fibration|elliptically fibered]] [[K3]] with a [[section]] is supposed to be equivalent to [[heterotic string theory]] [[KK-compactification|KK-compactified]] on a [[2-torus]].

More generally, [[F-theory]] on a [[complex manifold|complex]] $n$-dimensional $X$ fibered $X\to B$ with elliptic [[K3]]-[[fibers]] is supposed to be equivalent to [[heterotic string theory]] on an [[elliptic fibration|elliptically fibered]] [[Calabi-Yau manifold]] $Z \to B$ of [[complex manifold|complex]] [[dimension]] $(n-1)$.

A detailed discussion of the [[equivalence]] of the respective [[moduli spaces]] is originally due to ([Friedman-Morgan-Witten 97](#FriedmanMorganWitten97)). A review of this is in ([Donagi 98](#Donagi98)).

From the abstract of ([Donagi 98](#Donagi98)).

> The [[heterotic string theory|heterotic string]] [[KK-compactification|compactified]] on an $(n-1)$-[[dimension|dimensional]] [[elliptic fibration|elliptically fibered]] [[Calabi-Yau variety|Calabi-Yau]] $Z \to B$ is conjectured to be [[duality in string theory|dual]] to [[F-theory]] [[KK-compactification|compactified]] on an $n$-dimensional [[Calabi-Yau variety|Calabi-Yau]] $X \to B$, fibered over the same base with [[elliptic fibration|elliptic]] [[K3]] fibers. In particular, the [[moduli]] of the two theories should be [[isomorphism|isomorphic]]. The cases most relevant to the physics are $n=2$, $3$, $4$, i.e. the [[KK-compactification|compactification]] is to [[dimensions]] $d=8$, $6$ or $4$ respectively. Mathematically, the richest picture seems to emerge for $n=3$, where the [[moduli space]] involves an analytically [[integrable system]] whose fibers admit rather different descriptions in the two theories.

## Aspects

### Singular locus of the elliptic fibration and 24 D7-branes
 {#SingularLocusAndD7Branes}

In passing from [[M-theory]] to [[type IIA string theory]], the locus of any [[Kaluza-Klein monopole]] in 11d becomes the locus of [[D6-branes]] in 10d. The locus of the [[Kaluza-Klein monopole]] in turn (as discussed there) is the locus where the $S^1_A$-circle fibration degenerates. Hence in F-theory this is the locus where the fiber of the $S^1_A \times S^1_B$-[[elliptic fibration]] degenerates to the [[nodal curve]]. Since the [[T-duality|T-dual]] of [[D6-branes]] are [[D7-branes]], it follows that [[D7-branes]] in F-theory "are" the singular locus of the elliptic fibration.

Now an [[elliptically fibered complex K3-surface]]

$$
  \array{
    T &\longrightarrow& K3
    \\
    && \downarrow
    \\
    && \mathbb{C}\mathbb{P}^1
  }
$$

may be parameterized via the [[Weierstrass elliptic function]] as the solution locus of the equation

$$
  y^2 = x^3 + f(z) x + g(z)
$$

for $x,y,z \in \mathbb{C}\mathbb{P}^1$, with $f$ a [[polynomial]] of degree 8 and $g$ of degree twelve. The [[j-invariant]] of the complex [[elliptic curve]] which this parameterizes for given $z$ is

$$
  j(\tau(z)) = \frac{4 (24 f)^3}{27 g^2 + 4 f^3}
  \,.
$$

The [[poles]] $j\to \infty$ of the [[j-invariant]] correspond to the [[nodal curve]], and hence it is at these poles that the [[D7-branes]] are located.

\begin{imagefromfile}
        "file_name": "K3CobordismBetween24ThreeSpheres.jpg",
        "web": "nlab",
        "width": 260,
        "unit": "px",
        "float": "right",
        "margin": {
            "top": -40,
            "right": 0,
            "bottom": 20,
            "left": 20,
            "unit": "px"
        },
        "alt": "homotopy pasting diagram exhibiting the homotopy Whitehead integral",
        "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

Since the order of the poles is 24 (the polynomial degree of the [[discriminant]] $\Delta = 27 g^2 + 4 f^3$, see at _[[elliptically fibered K3-surface]]  -- [singular points](elliptic+fibration+of+a+K3-surface#SingularPoints)_) there are necessarily _24 D7-branes_ ([Sen 96, page 5](#Sen96), [Sen 97b](#Sen97b), see also [Morrison 04, sections 8 and 17](#Morrison04), [Denef 08, around (3.41)](#Denef08), [Douglas-Park-Schnell 14](duality+between+M%2FF-theory+and+heterotic+string+theory#DouglasParkSchnell14)).

Under [[T-duality]] this translates to 24 [[D6-branes]] in [[type IIA string theory]] on [[K3]] ([Vafa 96, Footnote 2 on p. 6](#Vafa96)).

Notice that the _net charge_ of these 24 D7-branes is supposed to vanish, due to [[S-duality]] effects (e.g. [Denef 08, below (3.41)](#Denef08)).

(This reminds one of the situation for the [[third stable homotopy group of spheres]]...)

For analogous discussion of 24 NS5-branes in [[heterotic string theory]] on [[K3]] see [Schwarz 97, around p. 50](duality+in+string+theory#Schwarz97).

For more see at _[[24 branes transverse to K3]]_.

### From M-branes to F-branes to heterotic strings and NS5-branes

[[!include F-branes -- table]]

### Non-reducible heterotic $E_8$-gauge backgrounds
 {#NonReducibleHeteroticGaugeBackgrounds}

There are some [[F-theory]] backgrounds whose supposed dual in [[heterotic string theory]] involves an [[E8]]-[[principal connection]] which is not [[reduction of the structure group|reducible]] to [[SemiSpin(16)]] ([Distler-Sharpe 10, section 5](#DistlerSharpe10)), while in fact the traditional construction of the heterotic [[worldsheet]] theory only covers this case. In ([Distler-Sharpe 10, section 7-8](#DistlerSharpe10)) it is argued that therefore a more general formulation of [[heterotic string theory]] needs to involve [[parameterized WZW models]]. See also at _[heterotic string -- Properties -- General gauge backgrounds and parameterized WZW models](http://ncatlab.org/nlab/show/heterotic+string+theory#GeneralGaugeBackgroundsAndParameterizedWZWModels)_.

## Related concepts

* [[24 branes transverse to K3]]

\linebreak

[[!include F-theory compactifications -- table]]

\linebreak

[[!include KK-compactifications of M-theory -- table]]

## References

### For type IIA and M-theory

The conjectured duality between [[type IIA string theory]] [[KK-compactification|KK-compactified]] on [[K3]] times an [[n-torus]] and [[heterotic string theory]] on the $(n+2)$-torus is originally due to

* [[Chris Hull]], [[Paul Townsend]], section 6 of: _Unity of Superstring Dualities_, Nucl. Phys. B438:109-137, 1995 ([arXiv:hep-th/9410167](http://arxiv.org/abs/hep-th/9410167))

* {#Witten95} [[Edward Witten]], section 4 of: _[[String Theory Dynamics In Various Dimensions]]_, Nucl. Phys. B443:85-126, 1995 ([arXiv:hep-th/9503124](http://arxiv.org/abs/hep-th/9503124))

* {#Lerche99} [[Wolfgang Lerche]], _On the Heterotic/F-Theory Duality in Eight Dimensions_, In: Baulieu L., Green M., Picco M., Windey P. (eds.) _Progress in String Theory and M-Theory_, NATO Science Series (Series C: Mathematical and Physical Sciences), vol 564. Springer 2001 ([arXiv:hep-th/9910207](https://arxiv.org/abs/hep-th/9910207), [doi:10.1007/978-94-010-0852-5_2](https://doi.org/10.1007/978-94-010-0852-5_2))

Review:

* [[Paul Aspinwall]], [[David Morrison]], _String Theory on K3 Surfaces_, in [[Brian Greene]], [[Shing-Tung Yau]] (eds.), _Mirror Symmetry II_, International Press, Cambridge, 1997, pp. 703-716 ([arXiv:hep-th/9404151](https://arxiv.org/abs/hep-th/9404151))

* {#Aspinwall96} [[Paul Aspinwall]], _K3 Surfaces and String Duality_, in  [[Shing-Tung Yau]] (ed.): _Differential geometry inspired by string theory_ 1-95 ([arXiv:9611137](https://arxiv.org/abs/hep-th/9611137) [spire:426102](http://inspirehep.net/record/426102))

Further discussion:

* [[Paul Aspinwall]], _Enhanced Gauge Symmetries and K3 Surfaces_, Phys.Lett. B357 (1995) 329-334 ([arXiv:hep-th/9507012](http://arxiv.org/abs/hep-th/9507012))

* [[Eric Bergshoeff]], C. Condeescu, G. Pradisi, F. Riccioni, _Heterotic-Type II duality and wrapping rules_, JHEP12(2013)057 ([arXiv:1311.3578](https://arxiv.org/abs/1311.3578))

Specifically in relation to the putative [[K-theory]]-classification of [[D-brane charge]]:

* {#GarciaUranga05} Inaki Garcia-Etxebarria, [[Angel Uranga]], _From F/M-theory to K-theory and back_, JHEP 0602:008,2006 ([arXiv:hep-th/0510073](https://arxiv.org/abs/hep-th/0510073))

Specifically in [[M-theory on G2-manifolds]]:

* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]] section 6.4 of _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))

* [[Bobby Acharya]], Alex Kinsella, [[David Morrison]], *Non-Perturbative Heterotic Duals of M-Theory on $G_2$ Orbifolds* ([arXiv:2106.03886](https://arxiv.org/abs/2106.03886))

Specifically in relation to [[Moonshine]]:

* [[Miranda Cheng]], Sarah M. Harrison, Roberto Volpato, Max Zimet, _K3 String Theory, Lattices and Moonshine_ ([arXiv:1612.04404](https://arxiv.org/abs/1612.04404))

### For F-theory

Discussion for [[F-theory]] includes

* {#MorrisonVafa96a} [[David Morrison]], [[Cumrun Vafa]], _Compactifications of F-Theory on Calabi--Yau Threefolds -- I_, Nucl. Phys. B473 (1996) 74-92 ([arXiv:hep-th/9602114](http://arxiv.org/abs/hep-th/9602114))

* {#MorrisonVafa96b} [[David Morrison]], [[Cumrun Vafa]], _Compactifications of F-Theory on Calabi--Yau Threefolds -- II_, Nucl. Phys. B476:437-469, 1996 ([arXiv:hep-th/9603161](http://arxiv.org/abs/hep-th/9603161))

* {#Sen96} [[Ashoke Sen]], _F-theory and Orientifolds_ ([arXiv:hep-th/9605150](http://arxiv.org/abs/hep-th/9605150))

* {#FriedmanMorganWitten97} Robert Friedman, [[John Morgan]], [[Edward Witten]], _Vector Bundles And F Theory_, Commun.Math.Phys.187:679-743, 1997 ([arXiv:hep-th/9701162](http://arxiv.org/abs/hep-th/9701162))

* {#Aspinwall97} [[Paul Aspinwall]], _M-Theory Versus F-Theory Pictures of the Heterotic String_, Adv.Theor.Math.Phys.1:127-147, 1998 ([arXiv:hep-th/9707014](http://arxiv.org/abs/hep-th/9707014))

Review of ([Friedman-Morgan-Witten 97](#FriedmanMorganWitten97)) is in

* {#Donagi98} [[Ron Donagi]], _ICMP lecture on heterotic/F-theory duality_ ([arXiv:hep-th/9802093](http://arxiv.org/abs/hep-th/9802093))

* Björn Andreas, _$N=1$ Heterotic/F-theory duality_, PhD thesis ([pdf](http://edoc.hu-berlin.de/dissertationen/physik/andreas-bjoern/PDF/Andreas.pdf))

with more details in

* {#DonagiMarkman95} [[Ron Donagi]], [[Eyal Markman]], _Spectral curves, algebraically completely integrable Hamiltonian systems, and moduli of bundles_ ([arXiv:alg-geom/9507017](http://arxiv.org/abs/alg-geom/9507017))

The issue with non-reducible $E_8$-gauge connections is highligted in

* {#DistlerSharpe10} [[Jacques Distler]], [[Eric Sharpe]], section 5 of _Heterotic compactifications with principal bundles for general groups and general levels_, Adv. Theor. Math. Phys. 14:335-398, 2010 ([arXiv:hep-th/0701244](http://arxiv.org/abs/hep-th/0701244))

On a subtlety in the application of the [[Narasimhan-Seshadri theorem]] in the duality:

* Herbert Clemens, Stuart Raby, _Heterotic/F-theory Duality and Narasimhan-Seshadri Equivalence_ ([arxiv:1906.07238](https://arxiv.org/abs/1906.07238))

See also:

* {#DouglasParkSchnell14} [[Michael Douglas]], Daniel S. Park, Christian Schnell, _The Cremmer-Scherk Mechanism in F-theory Compactifications on K3 Manifolds_, JHEP05 (2014) 135 ([arXiv:1403.1595](https://arxiv.org/abs/1403.1595))

* Herbert Clemens, Stuart Raby, *Heterotic/F-theory Duality and Narasimhan-Seshadri Equivalence* $[$[arXiv:1906.07238](https://arxiv.org/abs/1906.07238)$]$

### For both

* Yusuke Kimura, _New perspectives in the duality of M-theory, heterotic strings, and F-theory_ ([arXiv:2103.03088](https://arxiv.org/abs/2103.03088))

[[!redirects duality between F-theory and heterotic string theory]]

[[!redirects F/M-theory on elliptically fibered Calabi-Yau 2-folds]]

[[!redirects duality between heterotic string theory and M/F-theory]]

[[!redirects duality between heterotic string theory and F-theory]]

[[!redirects F-theory on K3]]


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#### String theory
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#Contents#
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## Idea

one of the [[dualities in string theory]]: between [[type I string theory]] and [[heterotic string theory]]

(...)

## Properties

### Gauge group

For [[type I string theory]] on [[flat orbifold|flat]] [[target spacetime]] [[orientifolds]] $\mathbb{R}^{9,1}$
[[RR-field tadpole cancellation]] requires 32 [[D-branes]] to cancel the [[O-plane charge]] of -32 ([here](orientifold+plane#OPlaneChargeForFlatOrientifolds)).

Under the [[duality between type I and heterotic string theory]] this translates to the [[semi-spin group|semi-spin]] [[gauge group]] [[SemiSpin(32)]] of [[heterotic string theory]].

## References

* [[Arkady Tseytlin]], _Heterotic - type I superstring duality and low-energy effective actions_, Nucl.Phys.B467:383-398, 1996 ([arXiv:hep-th/9512081](https://arxiv.org/abs/hep-th/9512081))

* [[Oren Bergman]], [[Matthias Gaberdiel]], G. Lifschytz, _String Creation and Heterotic-Type I' Duality_, Nucl. Phys. B524 (1998) 524-544 ([arXiv:hep-th/9711098](https://arxiv.org/abs/hep-th/9711098))

* {#APT98} I. Antoniadis, H. Partouche, T.R. Taylor, _Lectures on Heterotic-Type I Duality_, Nucl.Phys.Proc.Suppl. 61A (1998) 58-71 ([arXiv:hep-th/9706211](https://arxiv.org/abs/hep-th/9706211))

See also

* K. Foerger, _On Heterotic/Type I Duality in d=8_ ([arXiv:hep-th/9812154](https://arxiv.org/abs/hep-th/9812154))

[[!redirects duality between heterotic and type I string theory]]

[[!redirects I - HET duality]]
[[!redirects I - HET]]
[[!redirects HET - I duality]]
[[!redirects HET - I]]


duality in physics

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#### Physics
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#Contents#
* table of contents
{:toc}

## Idea

In fundamental physics, notably in [[quantum field theory]] and [[string theory]] one often says that a non-trivial [[equivalence]] of [[quantum field theories]] between two [[models (in theoretical physics)]] is a "[[duality]]".

## Examples

* [[quark-hadron duality]]

[[!include duality in string theory -- contents]]

## Formalization and Relation to mathematical duality
 {#Formalization}

One should beware that the use of the word "duality" in physics is in general different from concepts called "[[duality]]" in [[mathematics]].

For instance in [[T-duality]] only simple cases exhibit such obviously "dual" behaviour and in general cases such as [[U-duality]] really only the notion of _[[equivalence]]_ remains. This more closely resembles the mathematical concept of [[Morita equivalence]], see [Relation to Morita equivalence](#Morita). However, in some cases such as _[[Montonen-Olive duality]]/[[S-duality]]_ the equivalence involves some actual [[duality]] in the mathematical sense, as in replacing the [[gauge group]] by its [[Langlands dual group]].

### General formulation
 {#FormalizationGeneral}

One way to accurately formalize most of what is usually meant by "duality" in physics might instead be the following.

\linebreak

Write $LagrangianData$ for a [[moduli stack]] of [[prequantum field theory]] data consisting of species of [[field (physics)|fields]] and of [[Lagrangians]]/[[action functionals]] defined  on these.

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

For the well-understood case of [[mirror symmetry]] this would be the usual [[moduli space]] of [[Calabi-Yau manifolds]] regarded as the Lagrangian data for the [[2d (2,0)-superconformal QFT]].

=--

One imagines that [[quantization]] gives a map from such [[prequantum field theory|prequantum data]] to a moduli stack $QFTs$ of actual [[quantum field theories]]

$$
  quantization \;\colon\; LagrangianData \longrightarrow QFTs
  \,.
$$

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

Continuing example \ref{LagrangianDataInCaseOfMirrorSymmetry}
in the case of [[mirror symmetry]] this would be the [[TCFT]]-construction that takes a [[Calabi-Yau manifold]] to its [[Calabi-Yau A-∞ category]] ("of branes") which defines the corresponding [[2d TQFT]] via the [noncompact version](cobordism%20hypothesis#ForNoncompactCobordisms) of the [[cobordism hypothesis]].

=--

The [[1-image]] of this map would be the moduli space of [[Lagrangian field theory|Lagrangian quantum field theories]]

$$
  quantization
    \;\colon\;
  LagrangianData \longrightarrow LagrangianQFTs
  \hookrightarrow
  QFTs
  \,.
$$

By assumption this is now a [[1-epimorphism]] and hence an [[atlas]] of [[moduli stacks]].

The physical concept of duality, such as in [[mirror symmetry]], says that two points $L_1, L_2 \colon \ast \to LagrangianData$ in the space of Lagrangian data are "dual" to each other, if they become equivalent as quantum field theories after [[quantization]].

Mathematically this means that the space of such "dualities" is the [[homotopy fiber product]]

$$
  \array{
     && Dualities
     \\
     & \swarrow && \searrow
     \\
     LagrangianData && \swArrow_{\simeq} && LagrangianData
     \\
     & {}_{\mathllap{quantization}}\searrow && \swarrow_{\mathrlap{quantization}}
     \\
     && LagrangianQFTs
  }
$$

By definition, an element of $Dualities$ is two Lagrangians and a choice of equivalence of their associated quantum field theories:

$$
  quantization(L_1) \simeq quantization(L_2)
  \,.
$$

This construction is the first step in associating the [[groupoid object in an (∞,1)-category]] which is induced by the [[atlas]] "quantization" via [[Giraud's theorem]] of [[Higher Topos Theory]].

It continues in the way that [[Cech covers]] do (whence one speaks of the _[[Cech nerve]]_ construction of the quantization map $LagrangianData \to LagrangianQFTs$): above "$Dualities$" there is the space of triples of Lagrangian data that all have the same quantization, equipped with dualities between any two of them, and equipped with an equivalence of dualities (hence a "duality of dualities") between the composite of two of these and the third:

$$
  \array{
    && \vdots
   \\
  DualitiesOfdualities &\simeq& LagrangianData \underset{LagrangianQFT}{\times} LagrangianData \underset{LagrangianQFT}{\times} LagrangianDat
   \\
    && \downarrow \downarrow \downarrow
    \\
  Dualities &\simeq& LagrangianData \underset{LagrangianQFT}{\times} LagrangianData
   \\
   && \downarrow \downarrow
   \\
   && LagrangianData
  }
$$

It continues this way through all $n$-fold dualities of dualities.
The resulting $\infty$-groupoid object has as moduli stack of objects $LagrangianData$ and as moduli stack of 1-morphisms $Dualities$. Its corresponding [[stack]] realization is $LagrangianQFTs$ and so the corresponding [[augmented simplicial set|augmented]] [[simplicial object]] looks as

$$
  \array{
     \vdots
     \\
     \downarrow\downarrow\downarrow\downarrow
     \\
     DualitiesOfDualities
     \\
     \downarrow\downarrow\downarrow
     \\
     Dualities
     \\
     \downarrow \downarrow
     \\
     LagrangianData
     \\
     \downarrow^{\mathrlap{quantization}}
     \\
     LagrangianQFTs
  }
  \,.
$$

Such towers are to be thought of as the incarnation of [[equivalence relations]] as we pass to [[(∞,1)-category theory]]: A plain [[equivalence relation]] is just the first stage of such a tower

$$
  \array{
     Dualities
     \\
     \downarrow \downarrow
     \\
     Lagrangians
  }
$$

The conditions on an equivalence relation -- reflexivity, transitivity, symmetry -- may be read as those on a [[groupoid object]] -- identity, composition, inverses. So now in homotopy logic this is boosted to an [[groupoid object in an (∞,1)-category]] by relaxing all three to hold only up to higher coherent homotopies.

The bottom-most arrow

$$
  \array{
     LagrangianData
     \\
     \downarrow^{\mathrlap{quantization}}
     \\
     LagrangianQFTs
  }
$$

is the quotient projection of the equivalence relation. In 1-logic this would be its [[cokernel]], here in homotopy logic it is the [[homotopy colimit]] over the full [[simplicial object|simplicial]] diagram.

So the perspective of the full diagram gives the usual way of speaking in QFT also a reverse:

instead of saying

a) that two Lagrangians are dual if there is an equivalence between the QFTs which they induce under quantization,

we may turn this around and say that therefore

b) quantization is the result of forming the [[homotopy quotient]] of the space of Lagrangian data by these duality relations.

It is one of the clauses of the [[Giraud theorem]] in [[(∞,1)-topos theory]] that these two perspectives are equivalent.

### Examples
 {#FormalizationExamples}

For a concrete example of the [above](#FormalizationGeneral), consider the class of [[Chern-Simons theories]] and [[Rozansky-Witten theories]]. As [[Lagrangian field theories]] they are very different, but after [[perturbative quantum field theory|perturbative quantization]], their [[Lagrangian density|Lagrangian data]] maps to the same  abstract vector space of [[quantum observables]]: the [[Vassiliev knot invariants]].

Concretely, in both cases the map from [[Lagrangian density|Lagrangian data]] to [[quantum observables]] is given by the assignment of [[weight systems]] on [[Jacobi diagrams]], namely:

1) [[Lie algebra weight systems]] for [[Chern-Simons theory]]

2) [[Rozansky-Witten weight systems]] for [[Rozansky-Witten theory]]

The corresponding [[Wilson loop]] [[quantum observables]] are the evaluation of these [[weight systems]] on the [[universal Vassiliev invariant]]

1) [[Chern-Simons theory]] [[Wilson loop observables]]:

2) [[Rozansky-Witten theory]] [[Wilson loop observables]]:

Hence in total we have the following quantization map from Lagrangian data to observables:

To the extent that all observables of these theories, on the [[3-sphere]], are [[Wilson loop observables]], we must regard as physically equivalent any two elements on the left of this map (Lagrangian field theories) whose image on the right coincides. The space of such pairs is the [[fiber product]] of this quantization map with itself:

### Relation to Morita equivalence
 {#Morita}

According to [Schwarz 04](#Schwarz04):

> I am convinced that the mathematical notion of [[Morita equivalence]] of [[associative algebras]] and its generalization for [[differential algebra|differential associative algebras]] should be regarded as the mathematical foundation of [[duality in string theory|dualities in string/M-theory]].

[[Albert Schwarz|Schwarz]] showed that [[KK-compactifications]] on [[Morita equivalence|Morita equivalent]] [[noncommutative tori]] are physically equivalent ([Schwarz98](#Schwarz98)). This work is followed up in relation to [[T-duality]] in [Pioline99](#Pioline99) and [CNS11](#CNS11).

[BMRS08](#BMRS08) discusses an axiomatic definition of [[topological T-duality]] generalizing and refining T-duality between noncommutative spaces in terms of Morita equivalence to a special type of [[KK-theory|KK-equivalence]], which defines a T-duality action that is of order two up to Morita equivalence.

In [Okada09](#Okada09) a variant of [[mirror symmetry]] is shown to be a form of [[derived Morita equivalence]].

Morita theoretic ideas are also involved in [[factorization homology]], the [[blob complex]] and premodular TQFTS, see [MW10](#MW10), [BZBJ15](#BZBJ15), and [Scheimbauer](#Scheimbauer) for the Morita $(\infty, n)$-category of $E_n$-algebras.

## Duality after compactification

Informal comments on a possible general picture of duality in physics arising from [[duality in string theory]], especially after [[KK-compactification]], from [Brennan, Carta & Vafa 17, p. 7](duality+in+string+theory#BrennanCartaVafa17):

> Since we have seen that the full [[string theories]] are all interrelated by a sequence of [[duality in string theory|dualities]], one would expect that their [[KK-compactification|compactifications]] are also related by dualities. As it turns out, these relations are so abundant that we can make the following observation:

> **"Conjecture"**: Whenever the [[dimension]], [[number of supersymmetries|number of preserved supercharges]], and [[chiral fermion|chiralities]] of two different [[string theory vacuum|compactifications of string theory]] match, there are choices of [[KK-compactification|compactification geometries]] such that they are dual descriptions of the same physical theory.

> Surprisingly, we are aware of no known counter examples. In this sense, dualities in lower dimensional theories are not hard to find, but rather are  hard to prevent! One rationale for the existence of dualities is as [[Sergio Cecotti]] puts it, “the scarcity of rich structures”. In particular the very existence of quantum systems of gravity is hard to arrange and if we succeed to get more than one theory with a given symmetry, there is a good chance we have landed on the same theory.

## Related concepts

* [[parent action functional]]

There is also a duality in the _description_ of physics:

[[!include Isbell duality - table]]

## References

### General

* {#AlvarezGaume98} [[Luis Alvarez-Gaumé]], Frederic Zamora, _Duality in Quantum Field Theory (and String Theory)_, AIP Conference Proceedings 423, 46 (1998) ([arXiv:hep-th/9709180](https://arxiv.org/abs/hep-th/9709180))

* {#Corfield17} [[David Corfield]], _Duality as a category-theoretic concept_, Studies in History and Philosophy of Modern Physics Volume 59, August 2017, Pages 55-61 ([doi:10.1016/j.shpsb.2015.07.004](https://doi.org/10.1016/j.shpsb.2015.07.004))

### As Morita equivalence

On the idea that duality in physics may be captured by the notion of [[Morita equivalence]]:

* {#Schwarz98} [[Albert Schwarz]], _Morita equivalence and duality_ ([arXiv:hep-th/9805034](http://arxiv.org/abs/hep-th/9805034))

* {#Pioline99} B. Pioline, [[Albert Schwarz]], _Morita equivalence and T-duality (or $B$ versus $\Theta$)_ ([arXiv:hep-th/9908019](http://arxiv.org/abs/hep-th/9908019))

See also

* {#Schwarz04} [[Albert Schwarz]], _My Life In Science_, 2004 ([pdf](https://www.math.ucdavis.edu/~schwarz/bion.pdf), [[AlbertSchwarzLifeInScience.pdf:file]])

Specifically concerning [[T-duality]]:

* {#CNS11} Ee Chang-Young, Hiroaki Nakajima, Hyeonjoon Shin, _Fermionic T-duality and Morita Equivalence_ ([arXiv:1101.0473](http://arxiv.org/abs/1101.0473))

* {#BMRS08} [[Jacek Brodzki]], [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], _D-branes, KK-theory and duality on noncommutative spaces_, ([pdf](http://eprints.soton.ac.uk/46524/1/D-branes,_KK-theory_and__duality_on_noncommutative_spaces_BRMS_Published_JoP.pdf))

Specifically concerning [[mirror symmetry]]:

* {#Okada09} So Okada, _Homological mirror symmetry of Fermat polynomials_ ([arXiv:0910.2014](http://arxiv.org/abs/0910.2014))

See also more general discussion of [[Morita equivalence]]:

* {#BZBJ15} [[David Ben-Zvi]], Adrien Brochier, David Jordan, _Integrating quantum groups over surfaces: quantum character varieties and topological field theory_, ([arXiv:1501.04652](http://arxiv.org/abs/1501.04652))

* {#MW10} [[Scott Morrison]], [[Kevin Walker]], _The blob complex_, ([arXiv:1009.5025](http://arxiv.org/abs/1009.5025))

* {#Scheimbauer} Claudia Scheimbauer, _Factorization Homology as a Fully Extended Topological Field Theory_, ([pdf](http://math.unice.fr/~cazanave/Gdt/FH/ScheimbauerThesisJuly4FINAL.pdf))

### Duality in string theory

On [[duality in string theory]] (but see there fore more):

* {#Polchinski14} [[Joseph Polchinski]], _Dualities_ ([arXiv:1412.5704](http://arxiv.org/abs/1412.5704))

* [[Cumrun Vafa]], around 3:30, 12:00 of _On Mathematical Aspects of String Theory_ ([video](https://www.youtube.com/watch?v=yreUdrIbt2Q))

[[!redirects dualities in physics]]


duality in string theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Whatever it is otherwise, [[string theory]] turns out to be an organizational principle that subsumes a wealth of [[effective quantum field theories]] together with hints for their [[UV-completion]]. As such, string theory reveals a multitude of equivalences between superficially very different-looking (classes of) quantum field theories, or between various limits (notably strong/weak coupling limits) of various different-looking quantum field theories. Large classes of these equivalences go by the name of "dualities":

## Examples

[[!include duality in string theory -- contents]]

## Overview
 {#Overview}

> under construction

[[!include heterotic S-duality -- table]]

[[!include gauge theory from AdS-CFT -- table]]

## Related concepts

* [[duality in physics]]

* [[Gopakumar-Vafa duality]]

* [[mysterious duality]]

* [[magic pyramid]]

## References

### Original articles

The original article suggesting the relation of 11d sugra to string theory is

* [[Edward Witten]], _[[String Theory Dynamics In Various Dimensions]]_, Nucl.Phys.B443:85-126,1995 ([arXiv:hep-th/9503124](http://arxiv.org/abs/hep-th/9503124))

An original article collecting all the weak/strong electric/magnetic dualities is

* [[Mike Duff]], _M-theory (the theory formerly known as string)_, Int.J.Mod.Phys. A11 (1996) 5623-5642 ([arXiv:hep-th/9608117](http://arxiv.org/abs/hep-th/9608117))

### Surveys and lecture notes

* {#Schwarz97} [[John Schwarz]], _Lectures on Superstring and M Theory Dualities_, Nucl. Phys. Proc. Suppl. 55B:1-32, 1997 ([arXiv:hep-th/9607201](https://arxiv.org/abs/hep-th/9607201))

* [[Michael Dine]], _String Theory Dualities_ [[arXiv:hep-th/9609051](https://arxiv.org/abs/hep-th/9609051)]

* {#AlvarezGaume98} [[Luis Alvarez-Gaumé]], Frederic Zamora, _Duality in Quantum Field Theory (and String Theory)_, AIP Conference Proceedings 423, 46 (1998) ([arXiv:hep-th/9709180](https://arxiv.org/abs/hep-th/9709180))

* Stefan Forste, [[Jan Louis]], _Duality in String Theory_, Nucl. Phys. Proc. Suppl. 61A:3-22, 1998 ([arXiv:hep-th/9612192](https://arxiv.org/abs/hep-th/9612192))

* [[Ashoke Sen]], _Duality symmetries in string theory_, Current Science Vol. 77, No. 12, 25 December 1999 ([jstor:24104472](https://www.jstor.org/stable/24104472))

* [[Mike Duff]], chapter 6 of _[[The World in Eleven Dimensions]]: Supgergravity, Supermembranes and M-theory_, IoP 1999

* {#Polchinski14} [[Joseph Polchinski]], _Dualities_ ([arXiv:1412.5704](http://arxiv.org/abs/1412.5704))

More on dualities induced on [[K3]]-compactifications:

* [[Paul Aspinwall]], _K3-surfaces and string duality_ ([arXiv:hep-th/9611137](http://arxiv.org/abs/hep-th/9611137))

More in view of the [[landscape of string theory vacua]]:

* {#BrennanCartaVafa17} T. Daniel Brennan, Federico Carta, [[Cumrun Vafa]], _The String Landscape, the Swampland, and the Missing Corner_ ([arXiv:1711.00864](https://arxiv.org/abs/1711.00864))

Also

* [[Cumrun Vafa]], around 3:30, 12:00 of _On Mathematical Aspects of String Theory_ ([video](https://www.youtube.com/watch?v=yreUdrIbt2Q))

Discussion amplifying the role of [[category theory]], and [[higher geometry]] is in

* {#Sharpe19} [[Eric Sharpe]], _Categorical Equivalence and the Renormalization Group_, Proceedings of LMS/EPSRC Symposium _[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html)_, Fortschritte der Physik 2019 ([arXiv:1903.02880](https://arxiv.org/abs/1903.02880))

[[!redirects dualities in string theory]]

[[!redirects string duality]]
[[!redirects string dualities]]


duality in string theory -- contents

**[[duality in string theory]]**

**general mechanisms**

* **[[T-duality]]** ([[topological T-duality]], [[non-abelian T-duality]])

  [[mirror symmetry]]

* **[[S-duality]]**

  [[electric-magnetic duality]], [[Montonen-Olive duality]], [[geometric Langlands duality]]

* **[[U-duality]]**

  [[exceptional generalized geometry]]

**string-fivebrane duality**

* [[dual heterotic string theory]]

**[[string-string dualities]]**

* [[HET - I]]

* [[HET - II]]

* [[HET - M]]

**[[M-theory]]**

* [[duality between M-theory and type IIA superstring theory|M - IIA]].

* [[duality between M-theory on Z2-orbifolds and type IIB string theory on K3-fibrations|M- IIB]]

**[[F-theory]]**

* [[duality between M-theory and F-theory|M - F]]

* [[duality between F-theory and heterotic string theory|F - HET]]

**string-QFT duality**

* [[AdS-CFT duality]]

  (an [[open/closed string duality]])

  * [[AdS2/CFT1 duality]]

  * [[AdS3/CFT2 duality]]

    * [[p-adic AdS/CFT duality]]

  * [[AdS-QCD correspondence]]

* [[Chern-Simons theory as topological string theory]]

**QFT-QFT duality**:

* [[effective QFT]] incarnations of [[open/closed string duality]],

  relating ([[supergravity|super]]-)[[gravity]] to ([[super Yang-Mills theory|super]]-)[[Yang-Mills theory]]:

  * [[KLT relations]]

  * [[BCJ relations]]

  * [[classical double copy]]

* [[Kramers-Wannier duality]]

* [[Seiberg duality]] (swapping [[NS5-branes]])

* [[AGT correspondence]] ([[wrapped brane|wrapped]] [[M5-branes]])

* [[3d-3d correspondence]] ([[wrapped brane|wrapped]] [[M5-branes]])


duality involution

# Duality involutions

* table of contents
{: toc}

## Idea

A **duality involution** is an abstract operation that generalizes the notion of [[opposite category]].  What this means precisely depends on the underlying abstract structure.

## On 2-categories

A **duality involution** on a [[2-category]] $K$ is a 2-functor $(-)^\circ : K^{co} \to K$ that is coherently (or strictly) self-inverse.  A precise definition, and [[coherence theorem]], can be found in [Shulman 2016](#Shulman2016).  This sort of duality involution is [[stuff, structure, property|structure rather than a property]], but it can be given a universal property relative to a [[2-category with contravariance]].

## On bicategories

In the [[bicategory]] [[Prof]] of categories and profunctors, $A^{op}$ is the [[dual object]] of $A$ relative to a monoidal structure, making $Prof$ a [[compact closed bicategory]].  Thus, in any compact closed bicategory, the operation taking each object to its dual may be called a **duality involution**.  Note that being compact closed is a property and not a structure on a bicategory.

## On equipments

It should be possible to combine a duality involution on a 2-category and on a bicategory to obtain a notion of duality involution on a [[proarrow equipment]], but it is not clear exactly what the coherence conditions should be.  [Weber 2007](#Weber2007) extends a duality involution on a 2-category to its [[virtual equipment]] of discrete [[two-sided fibrations]] by asking for equivalences $DFib(A\times B,C) \simeq DFib(A,B^{\circ}\times C)$ that are pseudonatural in $A,B,C$.  But perhaps in general some compatibility with the composition of profunctors should also be assumed?

## References

* [[Mike Shulman]], *Contravariance through enrichment*, [arXiv](https://arxiv.org/abs/1606.05058), 2016
 {#Shulman2016}

* [[Mark Weber]], *Yoneda structures from 2-toposes*,  Applied Categorical Structures, Vol. 15, p259-323, 2007, [web](https://sites.google.com/site/markwebersmaths/home/yoneda-structures-from-2-toposes)

[[!redirects duality involutions]]


dualizable module

## Idea

## Definition

A [[module]] $M$ over a [[commutative ring]] $R$
is __dualizable__ if it is a [[dualizable object]]
in the [[symmetric monoidal category]] of $R$-[[modules]]
equipped with the [[tensor product]] over $R$.

Since this [[symmetric monoidal category]]
is a [[closed monoidal category]],
the [[dual object]] to $M$
is necessarily $Hom_R(M,R)$.

Furthermore, the abstract [[evaluation map]]
$$Hom_R(M,R)\otimes_R M\to R$$
must coincide with the map induced by the [[bilinear map]]
$$Hom_R(M,R)\times_R M\to R$$
that sends $(f,m)$ to $f(m)$.

## Characterization

\begin{theorem}
An $R$-module is dualizable if and only if
it is a [[finitely generated]] [[projective module]].
\end{theorem}

\begin{proof}
First, [[dualizable objects]] are closed under [[retracts]]
and finite [[direct sums]].
Any [[finitely generated]] [[projective module]]
is a [[retract]] of $R^n$ for some $n\ge0$,
so to show that
[[finitely generated]] [[projective modules]] are [[dualizable]],
it suffices to observe that $R$ is dualizable as an $R$-[[module]].

Conversely, we show that [[dualizable objects]] are [[finitely generated]]
[[projective modules]].
Unfolding the definition of a [[dualizable object]],
an $R$-[[module]] $M$ is dualizable
if the coevaluation map
$$coev: R \to M\otimes Hom_R(M,R)$$
and the evaluation map
$$ev: Hom_R(M,R)\otimes M\to R$$
satisfy the [[triangle identities]]:
$$(id_M \otimes ev)\circ (coev\otimes id_M) = id_M,$$
$$(ev \otimes id_{Hom(M,R)})\circ (id_{Hom(M,R)}\otimes coev) = id_{Hom(M,R)}.$$

The coevaluation map sends $1\in R$ to a _finite_ sum
$$\sum_{i\in I} m_i\otimes f_i.$$
The [[triangle identities]] now read
$$\sum_{i\in I} m_i f_i(p) = p,\qquad p\in M$$
$$\sum_{i\in I} r(m_i) f_i = r, \qquad r\in Hom_R(M,R).$$
The first identity implies that $m_i$ ($i\in I$) generate $M$ as an $R$-module,
i.e., $M$ is [[finitely generated]].

Consider the map $a: R^I\to M$ that sends $(r_i)_{i\in I}$
to $\sum_{i\in I} m_i r_i$.
Consider also the map $b: M\to R^I$ that sends $p\in M$
to $(f_i(p))_{i\in I}\in R^I$.
The first triangle identity now reads $b a = id_M$.
Thus, $M$ is a retract of $R^I$, i.e., $M$ is a [[projective module]].
\end{proof}

## Geometric interpretation

See also [[Serre–Swan theorem]] and [[smooth Serre–Swan theorem]].

\begin{theorem}
(Serre, 1955.)
The [[category]] of dualizable modules over a [[commutative ring]] $R$
is equivalent to the [[category]] of [[algebraic vector bundles]]
(defined as [[locally free sheaves]] over the [[structure sheaf of rings]])
over the [[Zariski spectrum]] of $R$.
\end{theorem}

\begin{theorem}
(Swan, 1962.)
Given a [[compact Hausdorff space]] $X$,
the [[category]] of dualizable modules over the real [[algebra]]
of [[continuous maps]] $X\to\mathbf{R}$
is equivalent to the category of finite-dimensional continuous [[vector bundles]] over $X$.
\end{theorem}

\begin{theorem}
(See, e.g., Nestruev 2003, 11.33.)
Given a [[smooth manifold]] $X$,
the [[category]] of dualizable modules over the real [[algebra]]
of [[smooth maps]] $X\to\mathbf{R}$
is equivalent to the category of finite-dimensional smooth [[vector bundles]] over $X$.
\end{theorem}

## Related concepts

* [[dualizable object]]

* [[perfect chain complex]]

* [[perfect module]]

* [[Serre–Swan theorem]]

* [[smooth Serre–Swan theorem]]

* [[duality between geometry and algebra]]

[[!redirects dualizable modules]]


dualizable object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Monoidal categories
+-- {: .hide}
[[!include monoidal categories - contents]]
=--
#### Duality
+--{: .hide}
[[!include duality - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A (left/right) _dual_ to an [[object]] in a [[monoidal category]] $\mathcal{C}$ is a [[left adjoint|left]]/[[right adjoint|right]] [[adjoint]] to the object regarded as a [[morphism]] in the [[delooping]] [[2-category]] $\mathbf{B}\mathcal{C}$. If a dual exists, the object is called _dualizable_.

Being *dualizable* may often be thought of as a [[category theory|category-theoretic]] notion of *finiteness* [[Pareigis 1976 p. 113](#Pareigis76)] for [[objects]] in a [[monoidal category]].  For instance:

* a [[vector space]] is dualizable in [[Vect]] with its standard [[tensor product of vector spaces]] precisely if it is a [[finite-dimensional vector space]] (cf. Exp. \ref{FiniteDimensionalVectorSpaces}),

* a [[spectrum]] is dualizable in the [[stable homotopy category]] with its [[smash product]] precisely if it is a [[finite spectrum]].

A more precise intuition is that an object is dualizable if its "size" is no larger than the "additivity" of the monoidal category.  For instance, since [[Vect]] and the [[stable homotopy category]] are finitely [[additive category|additive]], but not infinitely so, dualizability there is indeed a notion of plain finiteness, as is the case for many monoidal categories in which one considers dualizability.

However, in a monoidal category which is not additive at all, such as [[Set]] (or any cartesian monoidal category), only the [[terminal object]] is dualizable --- whereas in an "infinitely additive" monoidal category such as [[Rel]] or [[SupLat]], many "infinite" objects are dualizable.  (In $Rel$, *all* objects are dualizable.)

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

Beware that there are other notions of "dual object", distinct from this one. See for example _[[dual object in a closed category]]_ (eq:DualityInClosedCategory), and also the discussion at _[[category with duals]]_.

=--

## In a monoidal category

### Definition

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

An object $A$ in a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$ is **dualizable** if it has an [[adjunction|adjoint]] when regarded as a [[morphism]] in the one-object [[delooping]] [[bicategory]] $\mathbf{B}\mathcal{C}$ corresponding to $\mathcal{C}$.  Its adjoint in $\mathbf{B}\mathcal{C}$ is called its **dual** in $C$ and often written as $A^*$.

(This notion, though not the terminology, is due to [Lindner 1978](#Lindner78).)

If $C$ is [[braided monoidal category|braided]] then left and right adjoints in $\mathbf{B}C$ are equivalent; otherwise one speaks of $A$ being **left dualizable** or **right dualizable**.

Explicitly this means the following:

A right **[[duality]]** between objects $A, A^\ast \in (\mathcal{C}, \otimes, \mathbb{1})$

consists of

1. a [[morphism]] of the form

   \[
     \label{EvaluationMap}
     ev_A
       \;\colon\;
     A^\ast \otimes A \longrightarrow \mathbb{1}
   \]

   called the _counit_ of the duality, or the _[[evaluation map]]_;

1. a [[morphism]] of the form

   \[
     \label{CoEvaluationMap}
     i_A
       \;\colon\;
     \mathbb{1}
       \longrightarrow
     A \otimes A^\ast
   \]

   called the _unit_ or _coevaluation map_

such that

* ([[triangle identity]]) the following [[commuting diagram|diagrams commute]]

  $$
    \array{
       A^\ast \otimes (A \otimes A^\ast)
         &\overset{id_{A^\ast} \otimes i_A}{\longleftarrow}&
       A^\ast \otimes \mathbb{1}
       \\
       {}^{\mathllap{\alpha^{-1}_{A^\ast,A, A^\ast}}}_{\mathllap{\simeq}}\downarrow
       &&
       \downarrow^{\mathrlap{\ell^{-1}_{A^\ast} \circ r_{A^\ast}}}_{\mathrlap{\simeq}}
       \\
       (A^\ast \otimes A) \otimes A^\ast
         &\underset{ev_A \otimes id_{A^\ast}}{\longrightarrow}&
       \mathbb{1} \otimes A^\ast
    }
  $$

  and

  $$
    \array{
       (A \otimes A^\ast)  \otimes A
         &\overset{i_A \otimes id_A}{\longleftarrow}&
       \mathbb{1} \otimes A
       \\
       {}^{\mathllap{\alpha_{A,A^\ast, A}}}_{\mathllap{\simeq}}\downarrow
       &&
       \downarrow^{\mathrlap{r_A^{-1}\circ \ell_A}}_{\mathrlap{\simeq}}
       \\
       A \otimes (A^\ast \otimes A)
         &\underset{id_A \otimes ev_A}{\longrightarrow}&
       A \otimes \mathbb{1}
       \mathrlap{\,,}
    }
  $$

  where $\alpha$ denotes the [[associator]] of the [[monoidal category]] $\mathcal{C}$, and $\ell$ and $r$ denote the left and right [[unitors]], respectively.

=--

\begin{remark}
**(Terminology)**
\linebreak
Unfortunately, conventions on left and right vary and sometimes contradict their use for adjoints.  A common convention is that a *right dual* of $A$ is an object $A^*$ equipped with a **[[unit of an adjunction|unit]]** (or *[[coevaluation]]*)
$$
  i \,\colon\, \mathbb{1} \to A \otimes A^*
$$
and **counit** (or *[[evaluation]]*)
$$
  e \,\colon\, A^* \otimes A \to \mathbb{1}
$$
satisfying the '[[triangle identities]]' familiar from [[adjunctions]].

With this convention, if $\otimes$ in $C$ is interpreted as composition in $\mathbf{B} C$ in [[diagrammatic order]], then right duals in $C$ are the same as right adjoints in $\mathbf{B}C$ --- whereas if $\otimes$ in $C$ is interpreted as composition in $\mathbf{B} C$ in classical 'Leibnizian' order, then right duals in $C$ are the same as *left* adjoints in $\mathbf{B} C$.

Of course, in a [[symmetric monoidal category]], there is no difference between left and right duals.
\end{remark}

\begin{remark}
There are various equivalent definitions of dualizability, some of which are apparently weaker than the explicit definition in terms of both unit and counit, or which assume only one of them together with a [[universal property]] for it.

Regarding the equivalent characterization via adjointable tensor products, beware Rem. \ref{TensorAdjunctabilityDoesNotImplyDualizability} below.

\end{remark}

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

A dualizable object $A$, def. \ref{DualizableObject}, for which the structure unit/counit maps between $A \otimes A^\ast$ and the [[unit object]] are [[isomorphisms]] is called an **[[invertible object]]**.

=--

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

If every object of $C$ has a left and right dual, then $C$ is called a [[rigid monoidal category]] or an [[autonomous monoidal category]].  If moreover it is [[symmetric monoidal category|symmetric]], it is called a [[compact closed category]].

=--

See [[category with duals]] for more discussion.

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

Given a [[morphism]] $f \colon X \to Y$ between two dualizable objects in a [[symmetric monoidal category]], the corresponding [[dual morphism]]

$$
  f^\ast \colon Y^\ast \to X^\ast
$$

is the one obtained by $f$ by composing the duality unit, the counit and the [[braiding]]...

=--

### Examples
 {#Examples}

\begin{definition}\label{FiniteDimensionalVectorSpaces}
**([[finite-dimensional vector spaces]])**
\linebreak
Let $V$ be a [[finite-dimensional vector space]] over a [[field]] $k$, and let $V^* = Hom(V,k)$ be its usual [[dual vector space]].  We can define $\varepsilon\colon V^* \otimes V \to k$ to be the obvious pairing.  If we also choose a finite basis $\{v_i\}$ of $V$, and let $\{v_i^*\}$ be the [[dual basis]] of $V^*$, then we can define $\eta\colon k \to V\otimes V^*$ by sending $1$ to $\sum_i v_i \otimes v_i^*$.  It is easy to check the [[triangle identities]], so $V^*$ is a dual of $V$ in $Vect_k$.
\end{definition}

\begin{example}\label{DualizableModules}
**([[dualizable modules]])**
\linebreak
More generally, in the [[symmetric monoidal category]] of [[modules]] over a [[commutative ring]], dualizable objects are precisely [[finitely generated module|finitely generated]] [[projective modules]]. [[Dold & Puppe 1984, Ex. 1.4](#DoldPuppe84)].
See at *[[dualizable module]]* for more.
\end{example}

\begin{example}\label{DualizableChainComplexes}
**(dualizable [[chain complexes]])**
Yet more generally, in the [[category of chain complexes]] of [[modules]] over a [[commutative ring]], with respect to the [[tensor product of chain complexes]], an object is dualizable iff it is a [[bounded chain complex]] of [[dualizable modules]], hence (by Ex. \ref{DualizableModules}) a bounded chain complex of [[finitely generated module|finitely generated]] [[projective modules]] [[Dold & Puppe 1984, Prop. 1.6](#DoldPuppe84)].
\end{example}

\begin{example}
**([[Spanier-Whitehead duality]])**
\linebreak
Let $M$ be a finite-dimensional [[smooth manifold]], [[Whitney embedding theorem|choose]] an [[embedding of smooth manifolds|embedding]] $M\hookrightarrow \mathbb{R}^n$ for some $n$, and let $Th(N X)$ be the [[Thom spectrum]] of the [[normal bundle]] of this embedding.  Then the [[Thom collapse]] map defines an $\eta$ which exhibits $Th(N X)$ as a dual of $\Sigma_+^\infty M$ in the [[stable homotopy category]].  This is a version of *[[Spanier-Whitehead duality]].*
\end{example}

\begin{definition}
\label{AdjointEndofunctors}
**([[adjoint functor|adjoint]] [[endofunctors]])**
\linebreak
Given any [[category]] $\mathscr{C}$, the [[endofunctor|endo]]-[[functor category]] $End(\mathcal{C}) \,\coloneqq\, \text{Fun}(\mathscr{C},\mathscr{C})$  canonically becomes a [[monoidal category]], with [[tensor product]] given by [[composition]] of [[functors]] and [[horizontal composition]] of [[natural transformations]] (*[[whiskering]]*).

Under this identification, an object in $End(\mathcal{C})$ is dualizable iff it is an [[adjoint functor]], with [[evaluation]] and [[co-evaluation]] given by the [[counit of the adjunction]] and the [[unit of the adjunction]], respectively.
\end{definition}

\begin{example}
**([[Poincaré duality algebras]])**
\linebreak
A [[C*-algebra]] is a *[[Poincaré duality algebra]]*
if it is a dualizable object in the  [[symmetric monoidal category]] [[KK-theory|KK]] with dual its [[opposite algebra]].

See at _[KK-theory -- Poincare duality](KK-theory#PoincareDualityAndThomIsomorphism)_.
\end{example}

\begin{example}
For $E$ an [[E-∞ ring]], then in the [[(∞,1)-category of (∞,1)-modules]] $E Mod$ the dualizable objects coincide with the [[compact objects]] and the [[perfect objects]].

See at _[(∞,1)-category of (∞,1)-modules -- Compact generation](%28?%2C1%29-category+of+%28?%2C1%29-modules#CompactGeneration)_ for more.
\end{example}

### Properties

#### Relation to adjunctable tensor products

\begin{proposition}
\label{RelationToAdjunctableTensorProducts}
In a [[symmetric monoidal category]], the following are equivalent:

* an object $A$ is dual to $A^\ast$ in the sense of Def. \ref{DualizableObject}, with [[evaluation map]] $ev_A \,\colon\, A^\ast \otimes A \to \mathbb{1}$ (eq:EvaluationMap),

* the map

  \[
    \label{HomIsoInducedByDualizability}
    Hom\big( X,\, Y \otimes A^\ast \big)
    \xrightarrow{\;
      (\text{-}) \otimes A
    \;}
    Hom\big( X \otimes A ,\, Y \otimes A^\ast \otimes A \big)
    \xrightarrow{\;
      id_Y \otimes ev_A \circ (\text{-})
    \;}
    Hom\big( X \otimes A ,\, Y \big)
  \]

  is an [[isomorphism]] (a [[bijection]] of [[hom sets]]) for all objects $X$, $Y$.

\end{proposition}
This is [Dold & Puppe 1984, Thm. 1.3 (b) and (c)](#DoldPuppe84).

\begin{remark}
  Prop. \ref{RelationToAdjunctableTensorProducts} says *in particular* that for dual objects $A$, $A^\ast$ the map (eq:HomIsoInducedByDualizability) exhibits the tensor product [[functor]] $(\text{-}) \otimes A^\ast$ as a [[right adjoint|right]] [[adjoint functor|adjoint]] to $(\text{-}) \otimes A$ (hence: an [[internal hom]]-functor):

$$
  (\text{-}) \otimes A
  \;\;
    \dashv
  \;\;
  (\text{-}) \otimes A^\ast
  \,.
$$

If the ambient category is indeed [[closed monoidal category|closed monoidal]] with [[internal hom]] denoted $[-,-]$ and unit denoted by $\mathbb{1}$, this means that $(\text{-}) \otimes A^\ast \,\simeq\, [A,\text{-}]$ (by [essential uniqueness of adjoints](adjoint+functor#UniquenessOfAdjoints)) and hence in particular that:
$$
  A^\ast
    \;\simeq\;
  \mathbb{1} \otimes A^\ast
    \,\simeq\,
  [A,\mathbb{1}]
  \,.
$$

But since (by [[symmetric monoidal category|symmetry]]) the object $A \,\simeq\, (A^{\ast})^\ast$ is also the dual object to $A^\ast$, there is

1. a *further* right adjoint to the [[internal hom]] (an "[[amazing right adjoint]]")

1. which coincides with the left adjoint to make an [[ambidextrous adjunction]]:

$$
  (\text{-}) \otimes A
  \;\;
    \dashv
  \;\;
  (\text{-}) \otimes A^\ast
  \;\;
    \dashv
  \;\;
  (\text{-}) \otimes A
  \,.
$$
\end{remark}

\begin{remark}\label{TensorAdjunctabilityDoesNotImplyDualizability}
**(Tensor-adjuncatibility does not imply dualizability)**
\linebreak
Prop. \ref{RelationToAdjunctableTensorProducts} does *not* claim that for $A$ to be dualizable it is sufficient that $(\text{-}) \otimes A$ has a [[right adjoint]].

A [[counterexample]] is indicated by [[Noah Snyder]] in [math.SE:a/692318](https://math.stackexchange.com/a/692318/58526), referring to Exp. 2.20 in [arXiv:1406.4204](https://arxiv.org/abs/1406.4204).
\end{remark}

#### Trace

Dualizable objects support a good abstract notion of [[trace]]. (...)

#### Relation to cobordism hypothesis

Dualizable objects in an [[symmetric monoidal (∞,1)-category]] are already [[fully dualizable objects]]. The [[cobordism hypothesis]] implies that there is a canonical $O(1) \simeq \mathbb{Z}/2\mathbb{Z}$-action on the [[∞-groupoid]] of dualizable objects, and this is just the dualizing operation. See at _[cobordism hypothesis -- Framed version -- Implications: Canonical O(n)-action](cobordism+hypothesis#TheCanonicalOnAction)_.

## In a closed category

In a [[closed category]] $(\mathcal{C}, [-,-], 1)$ the (weak) dual to an object $X \in \mathcal{C}$ is defined to be the [[internal hom]] into the [[unit object]]

\[
  \label{DualityInClosedCategory}
  \mathbb{D}X \coloneqq [X,1]
  \,.
\]

## In a closed monoidal category

In a [[closed monoidal category]] $\mathbb{D}X$ (eq:DualityInClosedCategory) is also called the _weak dual_ of $X$ [[Becker & Gottlieb, p. 5](#BeckerGottlieb)] (in contrast with the monoidal dual of Def. \ref{DualizableObject}, which would then be called the *strong dual* [[Dold & Puppe 1984 Def. 1.2](#DoldPuppe84)]).

\begin{definition}\label{ReflexiveDualizableObject}
**(reflexive dualizable object)**
\linebreak
If the [[adjunct]]
$$
  X \longrightarrow \mathbb{D}\mathbb{D}X
$$
of
$$
  X \otimes \mathbb{D}X
  \xrightarrow{\;\; \sigma \;\;}
  \mathbb{D}X \otimes X
  \xrightarrow{\;\; ev \;\;}
  \mathbb{1}
$$
is an [[isomorphism]] (identifying $X$ with its double dual)
then $X$ is called *reflexive* as a weak dualizable object.
[[Deligne & Milne 1982 p 111](#DeligneMilne82), [Dold & Puppe 1984 Def. 1.2](#DoldPuppe84)]
\end{definition}

\begin{remark}
If $\mathcal{C}$ is a [[compact closed category]], def. \ref{RigidAndCompactClosed}, then the weak dual $\mathbb{D}X$ is also [[generalized the|the]] strong dual object $X^\ast$ to $X$ in the above monoidal sense. Here dualization exhibits $\mathcal{C}$ as a [[star-autonomous category]] ($\mathbb{D}(-) = (-)^\ast$ is the star-operation).
\end{remark}

\begin{remark}
The property of $X$ being dualizable can be expressed as a property of the weak dual, namely that the induced map $\mathbb{D}X \otimes X \to [X,X]$ is an isomorphsim.
\end{remark}

## In a symmetric monoidal $(\infty,n)$-category

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

An [[object]] in a [[symmetric monoidal (∞,n)-category]] $C$ is called **dualizable** if it is so as an object in the ordinary [[symmetric monoidal category|symmetric monoidal]] [[homotopy category]] $Ho(C)$.

=--

This appears as ([Lurie, def. 2.3.5](#Lurie)).

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

This means that an object in $C$ is dualizable if there exists unit and counit 1-morphism that satisfy the [[triangle identity]] up to [[homotopy]]. The definition does not demand that this homotopy is [[coherent]] (that it satisfies itself higher order relations up to higher order [[k-morphism]]s).

If the structure morphisms of the adjunction of a dualizable object _have_ themselves all adjoints, then the object is called a [[fully dualizable object]].

=--

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

As before, we may equivalently state this after [[delooping]] the monoidal structure and passing to the $(\infty,n+1)$-category $\mathbf{B}C$. Then $C$ has duals for objects precisely if $\mathbf{B}C$ has all adjoints.

=--

## In a linearly distributive category

In a [[linearly distributive category]], duality is naturally defined by mixing the two tensors $(\otimes,\top)$ and $(\parr,\bot)$: the unit is $i : \top \to A \parr A^* $ and the counit is $ev:A^* \otimes A \to \bot$.  The triangle identities make sense by inserting the linear distributivities; they assert that the following composites are identities:

$$ A \cong \top \otimes A \xrightarrow{i} (A \parr A^*) \otimes A \xrightarrow{\delta} A \parr (A^* \otimes A) \xrightarrow{ev} A \parr \bot \cong A $$

$$ A^* \cong A^* \otimes \top \xrightarrow{i} A^* \otimes (A \parr A^*) \xrightarrow{\delta} (A^* \otimes A) \parr A^* \xrightarrow{ev} \bot \parr A^* \cong A^*. $$

A symmetric linearly distributive category is (symmetric) [[star-autonomous category|star-autonomous]] if and only if all objects have duals in this sense.  The same is true in the non-symmetric case if we require both left and right duals.

This notion of duality generalizes to that of [[linear adjoints]] in a [[linear bicategory]], and also to dual objects in a [[polycategory]].

## Related concepts

* [[dualizable module]]

* [[dualizing object]]

* [[self-dual object]]

* [[fully dualizable object]]

* [[rigid monoidal category]],

* [[star-autonomous category]]

[[!include finite objects -- table]]

## References

The strong-duality of [[finite-dimensional vector spaces]] is the lead-in example used in

* {#EilenbergMacLane45} [[Samuel Eilenberg]], [[Saunders MacLane]], first page of: *[[General Theory of Natural Equivalences]]*,  Transactions of the American Mathematical Society **58** 2 (1945) 231-294 [[doi:10.1090/S0002-9947-1945-0013131-6](https://doi.org/10.1090/S0002-9947-1945-0013131-6), [jstor:1990284](http://www.jstor.org/stable/1990284)]

to motivate [[category theory]] in the first place.

Further original articles with dedicated discussion of the notion of dualizable objects in [[monoidal categories]]:

* {#Pareigis76} [[Bodo Pareigis]], p. 113 in: *Non-additive ring and module theory IV: The Brauer group of a symmetric monoidal category*, Lecture Notes in Mathematics **549** (1976) [[doi:10.1007/BFb0077339]( https://doi.org/10.1007/BFb0077339)]

  > (calls dualizable objects "finite objects")

* Thomas S. Ligon, *Galois-Theorie in monoidalen Kategorien*, Munich (1978) [[pdf](https://edoc.ub.uni-muenchen.de/14958/1/Ligon_Thomas.pdf), [doi:10.5282/edoc.14958](https://doi.org/10.5282/edoc.14958)]

  transl: *Galois theory in monoidal categories* (2019) [[pdf](https://edoc.ub.uni-muenchen.de/24952/7/Ligon_Thomas.pdf), [doi:10.5282/edoc.24952](https://doi.org/10.5282/edoc.24952)]

  > (follows [Pareigis 1976](#Pareigis76))

* {#Lindner78} [[Harald Lindner]], *Adjunctions in monoidal categories*, Manuscripta Mathematica **26** 1-2 (1978)  123-139 [[doi:10.1007/BF01167969](https://doi.org/10.1007/BF01167969)]

  > (discusses dual objects in a monoidal category as [[adjunctions]] in its [[delooping]] [[2-category]])

* {#DeligneMilne82} [[Pierre Deligne]], [[James Milne]]: §1  pp. 110 of: *Tannakian categories*,  in: *Hodge Cycles, Motives, and Shimura Varieties*, Lecture Notes in Mathematics **900**, Springer (1982) [[doi:10.1007/978-3-540-38955-2_4](https://doi.org/10.1007/978-3-540-38955-2_4), [webpage](http://www.jmilne.org/math/xnotes/tc.html)]

* {#DoldPuppe84} [[Albrecht Dold]], [[Dieter Puppe]], §1 of: *Duality, Trace and Transfer*, Proceedings of the Steklov Institute of Mathematics, **154** (1984) 85–103 [[mathnet:tm2435](http://mi.mathnet.ru/tm2435), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/doldpup2.pdf)]

Early history with an eye towards formulating [[Becker-Gottlieb transfer]]:

* {#BeckerGottlieb} [[James Becker]], [[Daniel Gottlieb]], from p. 5. in _A History of Duality in Algebraic Topology_ [[pdf](http://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf), [[BeckerGottlieb-DualityHistory.pdf:file]]]

Further developments:

* [[Mark Hovey]], [[John Palmieri]], [[Neil Strickland]]: Def. 1.1.2 and Thm. 2.1.3 in: *Axiomatic stable homotopy theory*, Memoirs Amer. Math. Soc. **610** (1997) [[ISBN:978-1-4704-0195-5](https://bookstore.ams.org/memo-128-610), [pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/axiomatic.pdf)]

* {#LewisMaySteinberger86} [[L. Gaunce Lewis]], [[Peter May]], [[Mark Steinberger]] (with contributions by [[James McClure|J.E. McClure]]): §III.1 in: *Equivariant stable homotopy theory*, Springer Lecture Notes in Mathematics  **1213** (1986) [[pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf), [doi:10.1007/BFb0075778](https://link.springer.com/book/10.1007/BFb0075778)]

Review:

* [[Peter May]], §2 in: *Picard Groups, Grothendieck Rings, and Burnside Rings of Categories*, Advances in Mathematics **163** 1 (2001) 1-16 [[pdf](https://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=86E4370463A9D802F6489B99282943AF?doi=10.1.1.205.7625&rep=rep1&type=pdf), [doi:10.1006/aima.2001.1996](https://doi.org/10.1006/aima.2001.1996)]

* [[Dominic Culver]], [[Mitchell Faulk]], *Duality Notes*, lecture notes for *West Coast Algebraic Topology Summer School* (2014) [[pdf](https://mathtube.org/sites/default/files/lecture-notes/Duality%20Notes.pdf), [[CulverFaulk-DualityNotes.pdf:file]] [mathtube](https://www.mathtube.org/lecture/notes/duality-notes)]

  > (towards [[fully dualizable objects]])

* {#HeunenVicary19} §3.1 in: [[Chris Heunen]], [[Jamie Vicary]], *Categories for Quantum Theory*, Oxford University Press 2019 [[ISBN:9780198739616](https://global.oup.com/academic/product/categories-for-quantum-theory-9780198739616)]

  based on:

  {#HeunenVicary12} [[Chris Heunen]], [[Jamie Vicary]], *Lectures on categorical quantum mechanics* (2012) [[pdf](https://www.cs.ox.ac.uk/files/4551/cqm-notes.pdf), [[HeunenVicary-QuantumLectures.pdf:file]]]

  > (in the context of [[quantum information theory via dagger-compact categories]])

See also:

* Wikipedia, *[Dual object](https://en.wikipedia.org/wiki/Dual_object)*

Further examples:

* [[Kate Ponto]] and [[Mike Shulman]], _Traces in symmetric monoidal categories_, ([arXiv:1107.6032](http://arxiv.org/abs/1107.6032))

On [[self-dual objects]] and the corresponding [[inner products]] and [[dagger-category|dagger-structure]]:

* {#Selinger12} [[Peter Selinger]], _Autonomous categories in which $A \simeq A^\ast$_, talk at QPL 2012 ([[SelingerSelfDual.pdf:file]])

Monoidal categories with freely adjoint duals:

*  Kevin Coulembier, [[Ross Street]], Michel van den Bergh, _Freely adjoining monoidal duals_  [[arXiv:2004.09697](https://arxiv.org/abs/2004.09697)]

The notion of [[fully dualizable objects]] in a [[symmetric monoidal (infinity,n)-category|symmetric monoidal $(\infty,n)$-category]]:

* {#Lurie} [[Jacob Lurie]], Section 2.3  _[[On the Classification of Topological Field Theories]]_

On the connection between dualisability, finiteness, and enrichment, see:

* [[Chris Heunen]], _Semimodule enrichment_, Electronic Notes in Theoretical Computer Science **218** (2008) 193-208 [[doi:10.1016/j.entcs.2008.10.012](https://doi.org/10.1016/j.entcs.2008.10.012)]

[[!redirects dualizable objects]]

[[!redirects dualisable object]]
[[!redirects dualisable objects]]

[[!redirects dual object]]
[[!redirects dual objects]]

[[!redirects left dual object]]
[[!redirects left dual objects]]
[[!redirects left dual]]
[[!redirects left duals]]

[[!redirects right dual object]]
[[!redirects right dual objects]]
[[!redirects right dual]]
[[!redirects right duals]]

[[!redirects weak dual]]
[[!redirects weak duals]]

[[!redirects strong dual]]
[[!redirects strong duals]]

[[!redirects weak dual object]]
[[!redirects weak dual objects]]

[[!redirects strong dual object]]
[[!redirects strong dual object]]

[[!redirects weakly dual object]]
[[!redirects weakly dual objects]]

[[!redirects strongly dual object]]
[[!redirects strongly dual object]]

[[!redirects dual type]]
[[!redirects dual types]]

[[!redirects dual pair]]
[[!redirects dual pairs]]


dualizing module

#Contents#
* table of contents
{:toc}

## Idea

A [[dualizing object in a closed category]] in a context of [[Verdier duality]]/[[six operations]].

## References

Discussion in [[E-infinity geometry]] is in

* [[Jacob Lurie]], section 4.2 of _[[Representability theorems]]_

[[!redirects dualizing modules]]

[[!redirects dualizing complex]]
[[!redirects dualizing complexes]]


dualizing object

> This entry is about a concept of duality in general [[category theory]]. For the concept of dualizing objects in a [[closed category]] as used in [[homological algebra]] and [[stable homotopy theory]] see at _[[dualizing object in a closed category]]_.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Duality
+-- {: .hide}
[[!include duality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _dualizing object_ is an object $a$ which can be regarded as being an object of two different [[categories]] $A$ and $B$, such that the concrete [[duality]] which is induced by homming into that object induces dual [[adjunctions]] between $A$ and $B$, schematically:

$$
  T : A^{op} \stackrel{Hom_A(-,a)}{\to} B
$$
$$
  S: B^{op} \stackrel{Hom_B(-,a)}{\to} A
  \,.
$$

Many famous dualities are induced this way, for instance [[Stone duality]] and [[Gelfand-Naimark duality]].

## Remark on terminology

There are various different terms for "dualizing objects". As recalled in [Porst-Tholen 91, p. 112](#PorstTholen91)

* Isbell speaks of _objects keeping summer and winter homes_;

* Lawvere speaks of _objects sitting in two categories_;

* Simmons speaks of _schizophrenic objects_.

It has been convincingly argued by [[Tom Leinster]] (blog comment [here](http://golem.ph.utexas.edu/category/2007/01/more_on_duality.html#c007089)) that the term "schizophrenic" should not be used. [[Todd Trimble]] then suggested the term "ambimorphic object."  Another suggestion was "Janusian object."

## Definition

Let $A$ and $B$ be two categories over $Set$, given by functors

$$
  U : A \to Set
$$
$$
  V : B \to Set
  \,.
$$

(Traditionally in what follows, one assumed $U$ and $V$ to be faithful, i.e., one considered $A$ and $B$ to be categories of sets with [[stuff, structure, property|extra structure]], aka [[concrete categories]], although we won't actually need this. Or, sometimes one sees the hypothesis that $U, V$ are representable, but we won't require this either.)

The situation described in the Idea section asks us to consider contravariantly [[adjoint functors]] $T: A \to B$, $S: B \to A$ such that $U S: B \to Set$ and $V T: A \to Set$ are representable, i.e., are given by isomorphisms $B(-, \mathbf{b}) \cong U S$ (with representing element $\phi \in U S \mathbf{b}$) and $A(-, \mathbf{a}) \cong V T$ (with representing element $\psi \in V T \mathbf{a}$). Let $\eta: 1_A \to S T$ and $\epsilon: 1_B \to T S$ denote the universal maps of the adjunction (so that if we write $T$ and $S$ in covariant form as $T: A \to B^{op}$ and $S: B^{op} \to A$, we would have $T \dashv S$ with $\eta$ as unit and $\epsilon$ as counit).

+-- {: .num_prop #sitting}
###### Proposition
The representing data $\phi \in U S \mathbf{b}, \psi \in V T \mathbf{a}$ induce a canonical isomorphism $\omega: U \mathbf{a} \stackrel{\sim}{\to} V \mathbf{b}$. It is given by the evident composite

$$U \mathbf{a} \stackrel{U\eta \mathbf{a}}{\to} U S T \mathbf{a} \stackrel{\sim}{\to} B(T \mathbf{a}, \mathbf{b}) \to Set(V T \mathbf{a}, V \mathbf{b}) \stackrel{eval_\psi}{\to} V \mathbf{b}.$$

The inverse is given by a similar composite but switching the roles of $\mathbf{a}$ and $\mathbf{b}$, $\phi$ and $\psi$, $\eta$ and $\epsilon$, $S$ and $T$, and $U$ and $V$.
=--

+-- {: .proof}
###### Proof
Letting $x$ be an element of $U \mathbf{a}$, the element $\omega(x) \in V \mathbf{b}$ is by definition $(V g)(\psi)$ where $g: T \mathbf{a} \to \mathbf{b}$ is the unique map (by representability) such that $(U \eta \mathbf{a})(x) = (U S g)(\phi)$. The alleged inverse $\widebar{\omega}: V \mathbf{b} \to U \mathbf{a}$ takes an element $y \in V \mathbf{b}$ to $(U f)(\phi)$, where $f: S \mathbf{b} \to \mathbf{a}$ is the unique map such that $(V T f)(\psi) = (V \epsilon \mathbf{b})(y)$.

To check these maps are inverse, one may check simply $(\omega \widebar{\omega})(y) = y$ for any $y \in V \mathbf{b}$; the other equation $\widebar{\omega} \omega = id$ follows by symmetry. With notation as above, put $x = \widebar{\omega}(y)$, so $x = (U f)(\phi) \in U \mathbf{a}$. Now we embark on a diagram chase, starting with

$$\array{
U S \mathbf{b} & \stackrel{U f}{\to} & U \mathbf{a} & \stackrel{U \eta \mathbf{a}}{\to} & U S T \mathbf{a} \\
 _\mathllap{\phi} \uparrow & \searrow_\mathrlap{U \eta S \mathbf{b}} & & _\mathllap{U S T f} \nearrow & \uparrow_\mathrlap{\cong} \\
1 & & U S T S \mathbf{b} & & B(T \mathbf{a}, \mathbf{b}) \\
 & _\mathllap{\xi} \searrow & \uparrow_\mathrlap{\cong} & \nearrow _\mathrlap{B(T f, \mathbf{b})} \\
 & & B(T S \mathbf{b}, \mathbf{b}) & &
}$$

where the top and right quadrilaterals commute by naturality, and $\xi: T S \mathbf{b} \to \mathbf{b}$ is the unique arrow making the left quadrilateral commute. In fact $\xi$ is the structure map for a $T S$-algebra structure on $\mathbf{b}$, although for our purposes we will only need the unit axiom for such a structure. Following the perimeter of this diagram, we see

$$g = (T \mathbf{a} \stackrel{T f}{\to} T S \mathbf{b} \stackrel{\xi}{\to} \mathbf{b})$$

is the unique arrow $g: T \mathbf{a} \to \mathbf{b}$ such that $(U S g)(\phi) = (U \eta \mathbf{a})(x)$. Continuing on our way, applying $V$ to $g$ and evaluating at $\psi$, we have only to check that the composite

$$1 \stackrel{\psi}{\to} V T \mathbf{a} \stackrel{V T f}{\to} V T S \mathbf{b} \stackrel{V \xi}{\to} V \mathbf{b}$$

equals $y: 1 \to V \mathbf{b}$. But already we said $(V T f)(\psi) = (V \epsilon \mathbf{b})(y)$, so the last composite is

$$1 \stackrel{y}{\to} V \mathbf{b} \stackrel{V \epsilon \mathbf{b}}{\to} V T S \mathbf{b} \stackrel{V \xi}{\to} V \mathbf{b}$$

and all we need to do now is check the unit equation $\xi \circ (\epsilon \mathbf{b}) = 1_\mathbf{b}$. But this follows from representability of $U S$, applied to the diagram

$$\array{
U S \mathbf{b} & \stackrel{U \eta S \mathbf{b}}{\to} & U S T S \mathbf{b} & \stackrel{U S \epsilon \mathbf{b}}{\to} & U S \mathbf{b} \\
 _\mathllap{\phi} \uparrow & & \uparrow_\mathrlap{\cong} & & \uparrow_\mathrlap{\cong} \\
1 & \underset{\xi}{\to} & B(T S \mathbf{b}, \mathbf{b}) & \underset{B(\epsilon \mathbf{b}, \mathbf{b})}{\to} & B(\mathbf{b}, \mathbf{b})
}$$

where the top horizontal composite is an identity, according to a triangular equation.
=--

The canonical identification of sets $\omega: U \mathbf{a} = V \mathbf{b}$ means that we have an object "sitting in two categories" (Lawvere), namely an $A$-structure $\mathbf{a}$ on this set and a $B$-structure $\mathbf{b}$ on the same set, with $T: A^{op} \to B$ providing a lift of $A(-, \mathbf{a}): A^{op} \to Set$ through $V: B \to Set$, and $S: B^{op} \to A$ lifting $B(-, \mathbf{b}): B^{op} \to Set$ through $U: A \to Set$.

Given functors $U: A \to Set$, $V: B \to Set$, we define $Adj_{rep}(U, V)$ to be the category whose objects are contravariantly adjoint pairs $(S: B \to A, T: A \to B)$ such that $U S$ and $V T$ are representable. Morphisms $(S, T) \to (S', T')$ are pairs of natural transformations $\alpha: S \to S'$, $\beta: T \to T'$ such that the diagrams

$$\array{
1_A & \stackrel{\eta'}{\to} & S' T' & & & & & & T S' & \stackrel{T \alpha}{\to} & T S \\
 _\mathllap{\eta} \downarrow & & \downarrow_\mathrlap{S' \beta} & & & & & & _\mathllap{\beta S'} \downarrow & & \downarrow_\mathrlap{\epsilon} \\
S T & \underset{\alpha T}{\to} & S' T & & & & & & T' S' & \underset{\epsilon'}{\to} & 1_B
}$$

commute. We define $2\text{-}Pull(U, V)$ to be the [[2-pullback]] of $U$ and $V$ in $Cat$ (i.e., the bi-iso-comma-object). In conjunction with the [[Yoneda lemma]], the preceding proposition can be read as giving the construction of a functor $\Phi: Adj_{rep}(U, V) \to 2\text{-}Pull(U, V)$.

Here then is one key definition.

+-- {: .num_defn #ambi}
###### Definition
Given $U: A \to Set$, $V: B \to Set$, a $(U, V)$-**dualizing object** (or $(U, V)$-*ambimorphic object*) is a triple $(\mathbf{a}, \mathbf{b}, \omega: U \mathbf{a} \stackrel{\sim}{\to} V \mathbf{b})$ in the essential image of $\Phi$.
=--

### Naturally represented adjunctions

Definition \ref{ambi} is reasonably general and is one of several notions of "schizophrenic object" given in [Dimov-Tholen93](#DT93). A somewhat tighter notion, also considered in [Dimov-Tholen 89](#DT89) and [Porst-Tholen 91](#PorstTholen91) and which covers many cases that arise in practice, involves initial lifts through the functors $U$, $V$.

Again, suppose given contravariant adjoint functors $S, T$ with $U S$ and $V T$ representable as above. We have maps

$$\gamma \coloneqq (U \stackrel{U \eta}{\to} U S T \cong B(T-, \mathbf{b})), \qquad \delta \coloneqq (V \stackrel{V \epsilon}{\to} V T S \cong A(U-, \mathbf{a}))$$

so that for each object $a$ of $A$ and $x \in U a$, we have a corresponding map $(\gamma a)(x): T a \to \mathbf{b}$ (playing the role of "$g$" in the proof of Proposition \ref{sitting}), and similarly for each object $b$ of $B$ and $y \in V b$, we have a map $(\delta b)(y): S b \to \mathbf{a}$ (playing the role of "$f$" in Proposition \ref{sitting}).

+-- {: .num_defn}
###### Definition
The adjunction between $S$ and $T$ is *naturally* represented if the family
$\{(\gamma a)(x): T a \to \mathbf{b}\}_{x \in U a}$ is $V$-[[topological functor|initial]] for each $a \in Ob(A)$, and the family $\{(\delta b)(y): S b \to \mathbf{a}\}_{y \in V b}$ is $U$-initial for each $b \in Ob(B)$.
=--

+-- {: .num_prop}
###### Proposition
The restriction of $\Phi: Adj_{rep}(U, V) \to 2\text{-}Pull(U, V)$ to the full subcategory whose objects are naturally represented adjoint pairs is full and faithful.
=--

See [Dimov-Tholen](#DT93), Proposition 2.3, where the proof is sketched. Thus naturally represented adjoint pairs could be equivalently described as certain types of ambimorphic objects. We now describe these.

Let us denote the action of $A$ on the module $U: A \to Set$, with components $A(a, a') \times U a \to U a'$, by the notation $(f, x) \mapsto f \cdot_U x$. Thus each $x \in U a$ induces a map $- \cdot_U x: A(a, \mathbf{a}) \to U \mathbf{a}$. Similarly, each $y \in V b$ induces a map $- \cdot_V y: B(b, \mathbf{b}) \to V \mathbf{b}$.

+-- {: .num_defn #natural}
###### Definition
A $(U, V)$-ambimorphic object $(\mathbf{a}, \mathbf{b}, \omega: U \mathbf{a} \stackrel{\sim}{\to} V \mathbf{b})$ is *natural* if for every $a \in Ob(A)$, the $V$-structured source diagram

$$\{A(a, \mathbf{a}) \stackrel{- \cdot_U x}{\to} U \mathbf{a} \stackrel{\omega}{\to} V \mathbf{b}\}_{x \in U a}$$

admits an [[topological concrete category|initial lift]] targeted at $\mathbf{b}$ (so a suitable diagram of the form $\gamma_x: T a \to \mathbf{b}$ indexed over $x \in U a$, for some object $T a$ of $B$), and for every $b \in Ob(B)$, the $U$-structured source diagram

$$\{B(b, \mathbf{b}) \stackrel{- \cdot_V y}{\to} V \mathbf{b} \stackrel{\omega^{-1}}{\to} U \mathbf{a}\}_{y \in V b}$$

admits an initial lift (a diagram of type $\delta_y: S b \to \mathbf{a}$), targeted at $\mathbf{a}$.
=--

Thus the category of naturally represented adjoint pairs relative to $(U, V)$ is equivalent to the category of natural $(U, V)$-dualizing objects. This is the notion of "schizophrenic object" given by [Porst-Tholen 91](#PorstTholen91); a reasonably detailed proof of the equivalence with naturally represented adjoint pairs is given in their Theorem 1.7.

## Examples

One easy general example of the notion given in Definition \ref{ambi} is where $C$ is a [[symmetric monoidal category|symmetric]] [[monoidal closed category]] and $d$ is an object therein. Here we may take $U = V = C(I, -): C \to Set$ where $I$ is the monoidal unit, and the contravariant representable functor $C(-, d): C \to Set$ lifts to a contravariant enriched hom $[-, d]: C \to C$; the symmetry isomorphism can be exploited to show how $[-, d]$ is adjoint to itself. See [here](http://ncatlab.org/nlab/show/duality#concrete_dualities) for more.

Further examples appear at

* [[Stone duality]]

* [[Gelfand duality]] (see at _[[center of an adjunction]]_ the section _[Examples - Gelfand duality](fixed+point+of+an+adjunction#GelfandDuality)_)

* [[star-autonomous category]]

* [[Chu construction]]

* [[rational homotopy theory]]

* [[dualizing complex]], [[dualizing module]]

## References

* {#DT89} G. D. Dimov, [[Walter Tholen]], _A Characterization of Representable Dualities,_ In: _Categorical Topology and its Relation to Analysis, Algebra and Combinatorics,_ Prague, Czechoslovakia 22-27 August 1988,  J. Adamek and S. MacLane (eds.), World Scientific, Singapore, New Jersey, London, Hong Kong, 1989, pp. 336-357.

* {#DT93} G. D. Dimov, [[Walter Tholen]], _Groups of Dualities,_ Trans. Amer. Math. Soc.,
336 (2), 901-913, 1993. ([pdf](http://www.ams.org/journals/tran/1993-336-02/S0002-9947-1993-1100693-0/S0002-9947-1993-1100693-0.pdf))

* {#PorstTholen91} [[Hans-E. Porst]], [[Walter Tholen]], _Concrete Dualities_ in H. Herrlich, [[Hans-E. Porst]] (eds.) _Category Theory at Work_, Heldermann Verlag 1991 ([pdf](http://www.heldermann.de/R&E/RAE18/ctw07.pdf))

* [[Michael Barr]], John F. Kennison, R. Raphael, _Isbell Duality_ ([pdf](http://www.tac.mta.ca/tac/volumes/20/15/20-15.pdf))

[[!redirects dualizing objects]]
[[!redirects ambimorphic object]]
[[!redirects ambimorphic objects]]
[[!redirects schizophrenic object]]
[[!redirects schizophrenic objects]]


dualizing object in a closed category

> This entry is about dualizing objects in closed (monoidal) categories in the sense of [[homological algebra]] and [[stable homotopy theory]] (e.g. [[dualizing modules]]). For a more general concept see at _[[dualizing object]]_.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Duality
+-- {: .hide}
[[!include duality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _dualizing object_ $D$ in a [[closed category]] $\mathcal{C}$ is an [[object]] such that the [[internal hom]] $[-,D] \colon \mathcal{C} \to \mathcal{C}^{op}$ into it serves as an [[involution|involutive]] [[duality]] operation on $\mathcal{C}$, or at least on a suitable [[full subcategory]] $\mathcal{C}_D \hookrightarrow \mathcal{C}$, i.e. it induces an [[equivalence of categories]] $[-,D] \colon \mathcal{C}_D \to \mathcal{C}_D^{op}$.

## Definition

Let $\mathcal{C}$ be a [[closed category]], $D\in \mathcal{C}$ an object, and $\mathcal{C}_D \hookrightarrow \mathcal{C}$ a [[full subcategory]].  We say that $D$ is a **dualizing object on $\mathcal{C}_D$** if the induced functor

$$ [-,D] : \mathcal{C}_D \to \mathcal{C}_D^{op} $$

is an [[equivalence of categories]].  Note that this includes the assumption that $[-,D]$ maps $\mathcal{C}_D$ into itself, which is not automatic.

Note that we do not in general assume $D\in \mathcal{C}_D$.  In [[stable homotopy theory]] the subcategory in question is typically that of [[homotopy types with finite homotopy groups]] (e.g. for [[Anderson duality]]). Specifically in [[homological algebra]] one speaks also of _[[dualizing modules]]_. (See for instance ([Heard-Stojanoska 14, def. 3.1](#HeardStojanoska14) and [[Representability Theorems|Lurie, section 4.2]]).)  Notably in a [[Grothendieck-Verdier context]] $(f^\ast \dashv f_\ast)$, $(f_! \dashv f^!)$ of [[six operations]] the functor $f^!$ typically preserves dualizing objects in this sense, which is a crucial ingredient of [[Verdier duality]].

If $\mathcal{C} = \mathcal{C}_D$, we say that $D$ is a **global dualizing object**.  A biclosed monoidal category $\mathcal{C}$ with a global dualizing object is one definition of a [[star-autonomous category]].

### In closed monoidal categories

If $\mathcal{C}$ is a [[closed symmetric monoidal category]], then $[-,D]$ is adjoint to itself on the right, i.e. we have a natural isomorphism $\mathcal{C}(A,[B,D]) \cong \mathcal{C}(B,[A,D])$.  Thus, if $[-,D]$ maps $\mathcal{C}_D$ into itself, then its restriction to $\mathcal{C}_D$ is an equivalence of categories if and only if the unit and counit of this adjunction are isomorphisms.  But these unit and counit are both the "double-dualization" map

$$A \to [[A,D],D]$$

(the [[adjunct]] of the [[evaluation map]] $[A,D] \otimes A \to D$, which in turn is the adjunct of the identity map $[A,D] \to [A,D]$) for $A\in \mathcal{C}_D$; so $D$ is a dualizing object on $\mathcal{C}_D$ if and only if $[-,D]$ preserves $\mathcal{C}_D$ and these maps are isomorphisms for all $A\in \mathcal{C}_D$.

If $\mathcal{C}$ is a non-symmetric [[biclosed monoidal category]], with two internal-homs $(A\otimes -) \dashv [A,-]$ and $(-\otimes A) \dashv \langle A,-\rangle$, then $[-,D]$ is adjoint on the right not to itself but to $\langle-,D\rangle$.  A similar argument then shows that $D$ is dualizing if and only if $[-,D]$ and $\langle-,D\rangle$ both map $\mathcal{C}_D$ to itself, and both double-dualization maps

$$ A\to \langle [A,D],D\rangle\qquad and \qquad A \to [\langle A,D\rangle,D]$$

are isomorphisms.

## Examples

### In a cartesian closed category
 {#InACartesianClosedCategory}

A [[cartesian closed category]] that with a global dualizing object is necessarily just a [[preorder]].  This statement is often known as _[[Joyal]]'s lemma_, recalled for instance in [Abramsky 09](#Abramsky09).  It can be slightly strengthened as follows:

+-- {: .num_prop}
###### Proposition
A cartesian closed category $C$ that is self-dual (carries an equivalence $N: C^{op} \to C$) is necessarily a preorder (whose posetal reflection is then a [[Heyting algebra]]). If the self-duality comes from a dualizing object, then the Heyting algebra is a Boolean algebra.
=--

+-- {: .proof}
###### Proof
For the first statement, let $1$ be terminal; then $N(1)$ is an initial object $0$, and similarly $N$ takes finite products to finite coproducts (coproducts are necessary for a Heyting algebra). So it remains to show $C$ is a preorder. Let $x, y$ be any two objects. The number of morphisms $x \to y$ is the number of morphisms $1 \to [x, y]$, which is the number of morphisms $N([x, y]) \to N(1) = 0$. But this is at most one since the initial object is strict (if there is any $z \to 0$, then $z$ is a retract of $0 \times z \cong 0$, hence $z \cong 0$; thus there is at most one morphism $z \to 0$).

The second statement is immediate: if $d$ is the dualizing object, then

$$d \cong [1, d] = N(1) \cong 0$$

so that $N(x) = [x, 0]$ is the negation and the hypothesis becomes the condition that double negation on the Heyting algebra is the identity, i.e., the Heyting algebra is a Boolean algebra.
=--

### In the category of spectra -- Anderson duality

In the [[stable (infinity,1)-category of spectra]], the [[sphere spectrum]] (which induces [[Spanier-Whitehead duality]] on spectra which are [[dualizable objects]] with respect to the [[smash product of spectra]]) is not a dualizing object.  However, the [[Anderson spectrum]] $I_{\mathbb{Z}}$ is a dualizing object on a suitable subcategory of finite spectra ([[Representability Theorems|Lurie, Example 4.3.9]]).  The [[duality]] operation $[-,I_{\mathbb{Z}}]$ that it induces is [[Anderson duality]].

For instance, the Anderson dual of [[KU]] is (complex conjugation-equivariantly) the 4-fold [[suspension spectrum]] $\Sigma^4 KU$ ([Heard-Stojanoska 14, theorem 8.2](#HeardStojanoska14)); and [[tmf]]$[1/2]$ is Anderson dual to its 21-fold suspension ([Stojanoska 12](#Stojanoska12))

### In the category of suplattices

In the category [[Sup]] of [[suplattices]], the opposite $\Omega^{op}$ of the poset $\Omega$ of [[truth values]] is a global dualizing object (and hence $Sup$ is [[star-autonomous category|star-autonomous]]).  The internal-hom $[A,B]$ is the suplattice of sup-preserving maps $A\to B$, hence $[A,\Omega^{op}]$ is the suplattice of contravariant maps $A \to \Omega^{op}$ that take suprema in $A$ to infima in $\Omega$.  But by the [[adjoint functor theorem]] for posets, any such map is representable by an element of $A$; thus $[A,\Omega^{op}] \cong A^{op}$.  The double-dualization is therefore isomorphic to the identity $A\cong (A^{op})^{op}$.

### Chu spaces

The [[Chu construction]] is a (co)universal way of "making any object into a dualizing object".

## Related concepts

* [[dual object in a closed category]]
* [[star-autonomous category]]
* [[Chu construction]]
* [[ambimorphic object]]

## References

General discussion is in

* {#BoryachenkoDrinfeld11} [[Mitya Boryachenko]], [[Vladimir Drinfeld]], _A duality formalism  in the spirit of Grothendieck and Verdier_ ([arXiv:1108.6020](http://arxiv.org/abs/1108.6020))

* [[Jacob Lurie]], section 4.2 of _[[Representability Theorems]]_

Reviews of the general concept and then discussion of [[Anderson duality]] is in

* {#Stojanoska12} [[Vesna Stojanoska]], _Duality for Topological Modular Forms_, Doc. Math. 17 (2012) 271-311 ([arXiv:1105.3968](http://arxiv.org/abs/1105.3968))

* {#HeardStojanoska14} [[Drew Heard]], [[Vesna Stojanoska]], _K-theory, reality, and duality_ ([arXiv:1401.2581](http://arxiv.org/abs/1401.2581))

Discussion in the context of the [[linear logic]]/[[quantum logic]] of [[quantum physics]] is in

* {#Abramsky09} [[Samson Abramsky]], _No-Cloning in categorical quantum mechanics_, (2008) in I. Mackie and S. Gay (eds), _Semantic Techniques for Quantum Computation_ , Cambridge University Press ([arXiv:0910.2401](http://arxiv.org/abs/0910.2401))

[[!redirects dualizing objects in a closed category]]
[[!redirects dualizing objects in closed categories]]


Duan-Lu Zhou

* [institute page](http://theory.iphy.ac.cn/english/direction/zhouduanlu.htm)

## Selected writings

On [[quantum information theory]] ([[entanglement entropy]]) applied to [[topological phases of matter]]/[[topological order]]:

* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:

  *[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$

category: people