Milnor-Quillen theorem on MU

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#### Cohomology
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[[!include cohomology - contents]]
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#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#### Higher algebra
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[[!include higher algebra - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _Milnor-Quillen theorem on $MU$_ determines the structure of the [[graded ring]] $\pi_{\bullet}(MU)$ of [[stable homotopy groups]] of the universal complex [[Thom spectrum]] [[MU]].

In ([Milnor 60](#Milnor60)) it was shown this is the [[polynomial ring]] $\mathbb{Z}[y_1, y_2, \cdots]$ on generators $y_{n+1}$ in degree $2(n+1)$ for all $n \in \mathbb{N}$.

Notice that by [[Thom's theorem]], this is also isomorphic to the [[cobordism ring]] $\Omega_\bullet^U \simeq \pi_\bullet(M U)$ of [[smooth manifolds]] equipped with stable [[almost complex structure]].

Moreover, by [[Lazard's theorem]], this graded ring is also abstractly [[isomorphism|isomorphic]] to the [[Lazard ring]] $L$.

But the [[universal complex orientation on MU]] induces a preferred ring [[homomorphism]]

$$
  L \longrightarrow \pi_\bullet(M U) \simeq \Omega_\bullet^U \simeq \mathbb{Z}[y_1, y_2, \cdots]
  \,.
$$

A priori it is not clear whether this particular canonical homomorphism exhibits the [[isomorphism]]. But it does, this is the result of ([Quillen 69](#Quillen69)).

## Statement

Write $MU$ for the [[E-∞ ring]] [[spectrum]] of [[complex cobordism cohomology theory]]. Since this is a [[complex oriented cohomology theory]], by [[Lazard's theorem]] there is associated a commutative 1-dimensional [[formal group law]] classified by a [[ring]] homomorphism of the form

$$
  L \longrightarrow \pi_\bullet(MU)
$$

from the [[Lazard ring]] $L$.

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

This canonical homomorphism is an [[isomorphism]]

$$
  L \stackrel{\simeq}{\longrightarrow} \pi_\bullet(MU)
  \,.
$$

=--

This is due to ([Quillen 69](#Quillen69)), based on ([Milnor 60](#Milnor60)), reproduced e.g. as ([Kochman 96, theorem 3.7.7, theorem 4.4.13](#Kochman96)).

**Proof strategy:** (for Milnor's part)

Apply the [[Boardman homomorphism]] to get the statement over the [[rational numbers]]. Deduce that $\pi_\bullet(MU)$ is finitely generated so that it is now sufficient to prove it over the [[p-adic integers]] for all $p$.

Now use the $H\mathbb{F}_p$-[[Adams spectral sequence]], which, on its second page, expresses these homotopy groups by the $H\mathbb{F}_p$-[[homology of MU]].

The [[homology of MU]] may be computed by reducing, via the [[Thom isomorphism]], to computation of the homology of the [[classifying space]] $B U$, which in turn is given by [[Kronecker pairing]] from the [[Conner-Floyd Chern classes]].

Using this and applying the [[change of rings theorem]], the [[Adams spectral sequence]] is seen to collapse right away, and so the result may now be obtained by explicitly computing the relevant comodule [[Ext]]-groups of the [[homology of MU]]

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

The dual generalized [[Steenrod algebra]] $MU_\bullet(MU)$ has a structure of
[[commutative Hopf algebroid]] over the [[Lazard ring]]. This is the content of the [[Landweber-Novikov theorem]].

=--

## Related theorems

* [[Lazard's theorem]]

* [[Landweber exact functor theorem]]

* [[Landweber-Novikov theorem]]

## References

The computation of $\pi_\bullet(M U)$ was first due to

* {#Milnor60} [[John Milnor]], _On the Cobordism Ring ft* and a Complex Analogue_, Amer. J.  Math., 82 (1960), 505-521.

the proof that the canonical morphism $L \to \pi_\bullet(M U)$ is an isomorphism is due to

* {#Quillen69} [[Daniel Quillen]], _On the formal group laws of unoriented and complex cobordism theory_, Bull. Amer. Math. Soc. 75 (1969), 1293–1298. ([Euclid](http://projecteuclid.org/euclid.bams/1183530915))

According to

* {#Hopkins99} [[Mike Hopkins]], section 4 of _Complex oriented cohomology theories and the language of stacks_, course notes 1999 ([pdf](http://www.math.rochester.edu/u/faculty/doug/otherpapers/coctalos.pdf))

the best reference as of turn of the millennium was still

* {#Adams74} [[Frank Adams]], part II, section 6-8 of _[[Stable homotopy and generalised homology]]_, Chicago lectures in mathematics, 1974

But see

* {#Kochman96} [[Stanley Kochman]], section 4.4 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Lurie10} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010,

  Lecture 7 _The homology of MU_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture7.pdf))

  Lecture 9 _The Adams spectral sequence for MU_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture9.pdf))

  Lecture 10 _The proof of Quillen's theorem_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture10.pdf))

Other review includes

* {#Kochman96} [[Stanley Kochman]], theorem 3.7.7 (Milnor's theorem) of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_, chapter 4 _$B P$-Theory and the Adams-Novikov spectral sequence_, [pdf](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenel4.pdf)

also

* [[Charmaine Sia]], section 2 of _Calculating the $E_2$-term of the Adams spectral sequence_ ([pdf](http://www.math.harvard.edu/~sia/notes/classical_topology_adams.pdf))

* [[Sam Nolen]], sections 1 and 2 of _The Adams-Novikov spectral sequence_ ([pdf](http://samnolen.com/Adams%20Novikov.pdf))

* [[Akhil Mathew]], _[Quillen's theorem on the formal group law of MU](https://amathew.wordpress.com/2012/05/28/quillens-theorem-on-the-formal-group-law-of-mu/#more-3345)_

[[!redirects Milnor's theorem on MU]]

[[!redirects Quillen's theorem on MU]]
[[!redirects Milnor-Quillen's theorem on MU]]

[[!redirects Quillen's theorem about MU]]


Milo Moses

I am a [[mathematics|mathematician]] specializing in the algebraic theory of [[topological quantum computation]], and the related underlying fields of pure mathematics. In particular, I am enthusiastic about the use of category theory to illuminate concepts in condensed matter physics and error correcting codes.

\begin{imagefromfile}
"file_name": "milo-picture.jpg",
"float": "right",
"width": 250,
"unit": "px"
\end{imagefromfile}

category: people


min-entropy

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### Context
#### Measure and probability theory
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[[!include measure theory - contents]]
=--
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=--

# Contents
* table of contents
{: toc}

## Idea

A notion of [[entropy]].

In the context of [[probability theory]], the **min-entropy** of a discrete probability distribution is the negative [[logarithm]] of the probability of the most likely outcome. It is never greater than the ordinary [entropy](entropy#entropy_of_a_partition_of_a_discrete_probability_space) of that distribution.

For a [[quantum system]] with [[density matrix]] $\rho$, the min-entropy is minus the [[logarithm]] of the [[maximum]] of its [[operator spectrum]], hence (since density matrices are [[self-adjoint operators]]) of its [[operator norm]] (which in the case of a [[finite-dimensional vector space|finite-dimensional]] [[space of states]] this is the maximum [[eigenvalue]]):

$$
  S_{min}(\rho)
  \;\coloneqq\;
  -
  ln
  \Big(
    max
    \big(
       Spectrum(\rho)
    \big)
  \Big)
  \;=\;
  -
  ln
  \big(
    \left\Vert
      A
    \right\Vert
  \big)
  \,.
$$

(e.g. [Chen 19, Def. 5.2.2](#Chen19))

## Properties

### Relation to Reny entropy

Min-entropy is the [[limit of a sequence|limit]] of [[Rényi entropy]] at order $\alpha$ as $\alpha \to \infty$.

[[!include Renyi entropies -- table ]]

## Related entries

* [[max-entropy]]

## References

Review:

* {#Chen19} Yi-Hsiu Chen, *Computational Notions of Entropy: Classical, Quantum, and Applications*, 2019 ([pdf](https://dash.harvard.edu/bitstream/handle/1/42029772/CHEN-DISSERTATION-2019.pdf?sequence=1&isAllowed=y))

* Robert Koenig, [[Renato Renner]], Christian Schaffner, *The operational meaning of min- and max-entropy*, IEEE Trans. Inf. Th., vol. 55, no. 9 (2009) [[arXiv:0807.1338](https://arxiv.org/abs/0807.1338), [doi:10.1109/TIT.2009.2025545](https://doi.org/10.1109/TIT.2009.2025545)]

See also:

* Wikipedia, *[Min-entropy](https://en.wikipedia.org/wiki/Min-entropy)*

* Scholarpedia, *[Quantum entropies -- Min entropy](http://www.scholarpedia.org/article/Quantum_entropies#Min_entropy)*

Generalization to condition entropies:

* Fabian Furrer, Johan Åberg, Renato Renner, *Min- and Max-Entropy in Infinite Dimensions*, Communications in Mathematical Physics volume 306, pages 165–186 (2011) ([doi:10.1007/s00220-011-1282-1](https://doi.org/10.1007/s00220-011-1282-1))

In the context of [[holographic entanglement entropy]]:

* [[Ning Bao]], [[Geoffrey Penington]], [[Jonathan Sorce]], [[Aron C. Wall]], Section 2.3 of: _Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT_,  JHEP 2019:69 ([arXiv:1812.01171](https://arxiv.org/abs/1812.01171))

[[!redirects min-entropy]]
[[!redirects min-entropies]]

[[!redirects min entropy]]
[[!redirects min entropies]]


min-plus algebra

The min-plus algebra is the same as the [[tropical semiring|tropical algebra]]. This name is used when it is considered in applications to systems theory.

There is also a [[max-plus algebra]].  They are both examples of [[idempotent semiring]]s or [[dioid]]s.


Mina Aganagic

* [webpage](https://math.berkeley.edu/people/faculty/mina-aganagic)

## Selected writings

On the [[Green-Schwarz sigma-model]] of the [[M5-brane]] including the [[self-dual higher gauge field]] and [[double dimensional reduction]] to the [[D4-brane]]:

* [[Mina Aganagic]], [[Jaemo Park]], Costin Popescu, [[John Schwarz]], _World-Volume Action of the M Theory Five-Brane_ ([arXiv:hep-th/9701166](http://arxiv.org/abs/hep-th/9701166))

On [[Green-Schwarz sigma models]] for [[D-branes]]:

* {#APPS97b} [[Mina Aganagic]], [[Jaemo Park]], Costin Popescu, [[John Schwarz]], _Dual D-Brane Actions_, Nucl. Phys. B496 (1997) 215-230 ([arXiv:hep-th/9702133](https://arxiv.org/abs/hep-th/9702133))

On [[topological string theory]] and [[integrable systems]]:

* [[Mina Aganagic]], [[Robbert Dijkgraaf]], [[Albrecht Klemm]], [[Marcos Marino]], [[Cumrun Vafa]], *Topological Strings and Integrable Hierarchies*, Commun. Math. Phys. **261** (2006) 451-516 [[arXiv:hep-th/0312085](https://arxiv.org/abs/hep-th/0312085), [doi:10.1007/s00220-005-1448-9](https://doi.org/10.1007/s00220-005-1448-9)]

* [[Mina Aganagic]], [[Miranda Cheng]], [[Robbert Dijkgraaf]], Daniel Krefl, [[Cumrun Vafa]], *Quantum Geometry of Refined Topological Strings*, J. High Energ. Phys. **2012** 19 (2012) [[arXiv:1105.0630](https://arxiv.org/abs/1105.0630), doi:10.1007/JHEP11(2012)019]

On a refinement of $SU(N)$ [[Chern-Simons theory]] on a [[3-manifold]] via the refined topological string and the [[D=6 N=(2,0) SCFT|(2,0)]] theory on $N$ [[M5-branes]]:

* [[Mina Aganagic]], Shamil Shakirov, _Knot Homology from Refined Chern-Simons Theory_, ([arXiv:1105.5117](https://arxiv.org/abs/1105.5117))

On [[string theory]] and [[mathematics]]

* _String Theory and Math: Why This Marriage May Last_ ([arxiv:1508.06642](http://arxiv.org/abs/1508.06642))

On [[equivariant elliptic cohomology]] of [[quiver varieties]] in relation to the [[AGT correspondence]]:

* [[Mina Aganagic]], [[Andrei Okounkov]], _Elliptic stable envelopes_ ([arXiv:1604.00423](https://arxiv.org/abs/1604.00423))

category: people


mind

## References

* {#Penrose94} [[Roger Penrose]], _Shadows of the mind_, Oxford University Press, New York, 1994 ([Wikipedia entry](https://en.wikipedia.org/wiki/Shadows_of_the_Mind))

See also

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

[[!redirects consciousness]]


Ming Ng

* [website](https://puzzledoyster.github.io/)

## Selected publications

On [[point-free topology]]:

* [[Ming Ng]], [[Steve Vickers]], *Point-free Construction of Real Exponentiation*, Logical Methods in Computer Science, Volume 18, Issue 3 (August 2, 2022), ([doi:10.46298/lmcs-18(3:15)2022](https://doi.org/10.46298/lmcs-18%283%3A15%292022), [arXiv:2104.00162](https://arxiv.org/abs/2104.00162))

* [[Ming Ng]], [[Steven Vickers]], *A Point-Free Look at Ostrowski's Theorem and Absolute Values* ([arXiv:2308.14758](https://arxiv.org/abs/2308.14758))

* [[Ming Ng]], *Logical Berkovich Geometry: A Point-free Perspective* ([arXiv:2308.16472](https://arxiv.org/abs/2308.16472))

* [[Ming Ng]], *Adelic Geometry via Topos Theory*, PhD thesis ([pdf](https://puzzledoyster.github.io/publications/FINALSUBMISSION.pdf))

category: people


Mingsheng Ying

* [personal page](http://cs.tsinghua.edu.cn/~yingmsh)

* [Institute page](https://www.cs.tsinghua.edu.cn/csen/info/1186/4020.htm)

## Selected writings

On [[software verification]] for [[quantum programming languages]]:

* [[Mingsheng Ying]], [[Yuan Feng]], *Model Checking Quantum Systems --- A Survey* [[arXiv:1807.09466](https://arxiv.org/abs/1807.09466)]

* [[Mingsheng Ying]], *Model Checking for Verification of Quantum Circuits*, in: *Formal Methods. FM 2021*, Lecture Notes in Computer Science **13047**, Springer (2021) $[$[arXiv:2104.11359](https://arxiv.org/abs/2104.11359), [doi:10.1007/978-3-030-90870-6_2](https://doi.org/10.1007/978-3-030-90870-6_2)$]$

* [[Mingsheng Ying]], Zhengfeng Ji, *Symbolic Verification of Quantum Circuits*,  $[$[arXiv:2010.03032](https://arxiv.org/abs/2010.03032)$]$

* {#YingFeng21} [[Mingsheng Ying]], [[Yuan Feng]], *Model Checking Quantum Systems -- Principles and Algorithms*, Cambridge University Press (2021) [[ISBN:9781108484305](https://www.cambridge.org/ae/academic/subjects/computer-science/programming-languages-and-applied-logic/model-checking-quantum-systems-principles-and-algorithms?format=HB)]

and introducing the [[quantum programming language]] *[[CoqQ]]*:

* [[Li Zhou]], [[Gilles Barthe]], Pierre-Yves Strub, Junyi Liu, [[Mingsheng Ying]], *CoqQ: Foundational Verification of Quantum Programs* [[arXiv:2207.11350](https://arxiv.org/abs/2207.11350)]

  > (in [[Coq]])

Understanding the [[linear logic]]-sector of [[bunched logic]] as a kind of [[quantum logic]]:

* [[Li Zhou]], [[Gilles Barthe]], [[Justin Hsu]], [[Mingsheng Ying]], [[Nengkun Yu]], *A Quantum Interpretation of Bunched Logic for Quantum Separation Logic*, LICS '21: Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science  75 (2021) 1–14 [[arXiv:2102.00329](https://arxiv.org/abs/2102.00329), [doi:10.1109/LICS52264.2021.9470673](https://doi.org/10.1109/LICS52264.2021.9470673)]

## Related entries

* [[software verification]]

* [[quantum programming language]]

* [[QPMC]]

category: people


Minhyong Kim

Minhyong Kim is Professor of Number Theory at the University of Oxford.

* [Webpage](http://people.maths.ox.ac.uk/kimm/)

## Selected writings

On a [[Lefschetz trace formula]] in [[equivariant cohomology]]:

* [[Minhyong Kim]], _A Lefschetz trace formula for equivariant cohomology_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 28 no. 6 (1995), p. 669-688 ([numdam:ASENS_1995_4_28_6_669_0](http://www.numdam.org/item?id=ASENS_1995_4_28_6_669_0), [MR97d:55012](http://www.ams.org/mathscinet-getitem?mr=97d:55012))

Kim has notably written about conceptual questions that appear in [[Diophantine geometry]] ([[arithmetic geometry]]), [[anabelian geometry]] and the [[Langlands program]], for instance in

* {#MK09} _Galois Theory and Diophantine geometry_, 2009 ([pdf](http://people.maths.ox.ac.uk/kimm/papers/cambridgews.pdf))

A proposal for [[arithmetic Chern-Simons theory]]:

* {#Kim1} [[Minhyong Kim]], *Arithmetic Chern-Simons Theory I*, in: *Galois Covers, Grothendieck-Teichmüller Theory and Dessins d'Enfants*, Proceedings in Mathematics & Statistics **330**,  Spinger (2020) [[arXiv:1510.05818](http://arxiv.org/abs/1510.05818), [doi:10.1007/978-3-030-51795-3_8](https://doi.org/10.1007/978-3-030-51795-3_8)]

On [[arithmetic gauge theory]]:

* {#Kim18} [[Minhyong Kim]], *Arithmetic Gauge Theory: A Brief Introduction*,  Modern Physics Letters A **33** 29 (2018) 1830012 [[arxiv:1712.07602](https://arxiv.org/abs/1712.07602), [doi:10.1142/S0217732318300124](https://doi.org/10.1142/S0217732318300124)]

On [[BF-theory]] in [[arithmetic geometry]]:

* [[Magnus Carlson]], [[Minhyong Kim]], _A note on abelian arithmetic BF-theory_, ([arXiv:1911.02236](https://arxiv.org/abs/1911.02236))

category:people


mini-workshop on Poisson geometry

## Mini-workshop in Poisson Geometry ##

***
April 3, Wuhan at [the Center for Mathematical Sciences](http://mathcenter.hust.edu.cn/) of [Huazhong University of Science and Technology](http://english.hust.edu.cn/).

***
####[Official Webpage](http://mathcenter.hust.edu.cn/activities/meeting/51.html) (some information in Chinese)

***
####Speakers####

Zhangju Liu (Beijing University)

Chengming Bai (Chern Institute, Nankai University)

Zhujun Zheng (South China University of Technology)

Yunhe Sheng (Jilin University)

####Participants####

Zhangju Liu (Beijing University)

Chengming Bai (Chern Institute, Nankai University)

Zhujun Zheng (South China University of Technology)

Yanhui Bi (Nanchang University)

Yunhe Sheng (Jilin University)

Tao Zhang (Henan Normal University)

Deshou Zhong (China Youth University of Political Studies)

####Organizers####
[Jinqiao Duan](http://mypages.iit.edu/~duan/)  (HUST, UCLA, IIT), [Yunhe Sheng](http://math.jlu.edu.cn/?act=view&id=741&mod=info) (Jilin University) and [[Chenchang Zhu]] (Gottingen, Germany)

###Schedule###

|        |            |        |
|-----|----|-----|
| Time | Speaker|Title|
| 9:00--10:00   |   Zhangju Liu   | Geometric structure in classical mechenics  |
| 10:30--11:30  | Chengming Bai             |TBA |
| 15:00--16:00  | Yunhe Sheng|Homotopy Poisson structures|
| 16:30--17:30  |Zhujun Zheng|Quantum computation|


minimal coupling

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

## Idea

In [[quantum field theory]], the term _minimal coupling_ refers to the kind of interaction betweem [[fermion|fermionic]] [[particles]] and [[force]] [[gauge fields]].

The [[force]] [[gauge fields]] are modeled by [[connection on a bundle|principal connections]] on a $G$-[[principal bundle]] where $G$ is the [[gauge group]] of the given [[gauge theory]]. (For instance $G = U(1) \times SU(2) \times SU(3)$ in the [[standard model of particle physics]]).

The [[matter]] fields are [[sections]] $\psi$ of a [[spinor bundle]] [[associated bundle|associated]] to this [[principal bundle]]. Therefore there is an induced [[connection on a vector bundle]] $\nabla$ on this spinor bundle.

Let $D_\nabla$ be the [[Dirac operator]] of the given [[Riemannian metric]] and this conneciton $\nabla$. The **minimal coupling** term in the [[action functional]] on the space of these sections is

$$
  S_{gc}(\nabla, \psi) = \int_\Sigma \langle \psi, D_\nabla \psi\rangle
  \,.
$$

## Examples

All the couplings appearin in the [[standard model of particle physics]] are "minimal" in this sense.

## Related concepts

* [[spinors in Yang-Mills theory]]

* [[Yukawa coupling]]

[[!include gauge field - table]]


minimal dendroidal fibration

## Idea

[[minimal fibrations]] for [[dendroidal sets]] (i.e. in the [[model structureon dendroidal sets]])

## References

* [[Ieke Moerdijk]], [[Joost Nuiten]], _Minimal fibrations of dendroidal sets_ ([arXiv:1509.01073](http://arxiv.org/abs/1509.01073))

[[!redirects minimal dendroidal fibrations]]


minimal dg-module

[[!redirects KS-model]]
[[!redirects minimal KS-extension]]

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###Context###
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _minimal dg-module_ ([Roig 92](#Roig92), [Roig 94, section 1](#Roig94)) is a [[minimal model]] in the context of the [[homotopy theory]] of [[dg-modules]].

Hence over [[dgc-algebras]] in non-positve degree, minimal dg-modules are the analogue of [[minimal Sullivan model]] as one passes from [[dg-algebras]] to (just) [[dg-modules]]. Minimal KS-extensions hence  play the role of formal duals of [[minimal fibration]] in some applications of [[rational homotopy theory]].

(In [Halperin 83](#Halperin83) it has "Koszul-Sullivan extensions" for [[relative Sullivan algebras]], and "KS" in "KS-models" refers to that usage.)

## Definition

Throughout, let $A$ be a [[dg-algebra]]. Eventually this is thought of as being a [[Sullivan model]] for the [[rationalization]] of the [[quotient]] of a [[topological space]] by a circle action.

We take all [[differentials]] to have degree +1. For $V$ a [[vector space]] and $n$ a [[natural number]], we write $V[n]$ for the [[chain complex]] concentrated on $V$ in degree $n$.

+-- {: .num_defn #HirschExtension}
###### Definition
**(Hirsch extension of dg-modules)**

Let $N$ be a [[dg-module]] over $A$ and let $n \in \mathbb{N}$ be a [[natural number]]. Then a degree $n$ **[[Hirsch extension]]** of $N$  is
a [[monomorphisms]] of [[dg-modules]] of the form

$$
  N \hookrightarrow (N \oplus (A \otimes V[n]), d_{\phi})
$$

given by a choice of

1. a [[vector space]] $V$

1. a [[linear map]] $d_V \;\colon\; V \to N_{n+1}$

where the differential $d_{\phi}$ is $d_N$ on $N$, is $d_A$ on $A$, and is given on $V$ by $\phi$ followed by the [[action]] $\rho$ of $A$ on $V$:

$$
  d_\phi (a \otimes v) = (d_A a) \otimes v + (-1)^{\vert a \vert}  \rho(a,\phi(v))
  \,.
$$

=--

([Roig 94, def. 1.8](#Roig94))

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

It follows that for $N_1, N_2$ two $A$-[[dg-modules]] then homomorphisms $f$ out of a Hirsch extension of the former (def. \ref{HirschExtension})

$$
  f
   \;\colon\;
  (N_1 \olpus (A \otimes V[n]), d_\phi)
   \longrightarrow
  N_2
$$

is equivalently

1. a homomorphism of dg-modules $h \colon N_1 \to N_2$;

1. a [[linear map]] $g \colon V \to (N_2)_{n}$

such that

* $h \circ d_{\phi} = d_{N_2} \circ g$.

=--

The following defines a kind of [[minimal cofibrations]] of dg-modules.

+-- {: .num_defn #KSFactorization}
###### Definition
**(minimal KS-extension)**

For $N$ a [[dg-module]] over $A$, then a **[[minimal KS-extension]]** of $N$ is a certain [[transfinite composition]] of Hirsch extensions, namely a [[monomorphism]]

$$
  N \hookrightarrow \hat N
$$

equipped with an [[exhaustive filtration]] $\{\hat N(n,q)\}_{n,q \in \mathbb{N}}$ such that:

1. $\hat N(0,0) \simeq N$;

1. the inclusions $\hat N(n,q) \hookrightarrow \hat N(n,q+1)$ are Hirsch extensions (def. \ref{HirschExtension}).

1. $N(n+1,0) = \underset{\longrightarrow}{\lim}_q \hat N(n,q)$ ([[colimit]] over the sequence of Hirsch extensions in the previous degree).

Accordingly, a **minimal KS-factorization** of a morphism $N_1 \to N_2$ of $A$-dg-modules is a factorization as a minimal KS-extension followed by a [[quasi-isomorphism]]

$$
  \array{
    N _1 && \longrightarrow && N_2
    \\
    & {}_{\mathllap{\text{minimal} \atop \text{KS-extension}}}\searrow && \nearrow_{\mathrlap{\text{quasi-iso}}}
    \\
    && N_1 \oplus (A \otimes V)
  }
  \,.
$$

Finally a **minimal KS-model** is a [[dg-module]] $N$ such that $0 \hookrightarrow N$ is a minimal KS-fibration.

=--

([Roig 94, def. 1.9](#Roig94))

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

Let the [[ground field]] be of [[characteristic zero]].

Let $f \colon N_1 \longrightarrow N_2$ be a morphism of $A$-[[dg-modules]] such that it induces a [[monomorphism]] in degree-0 [[cochain cohomology]], $H^0(f) \colon H^0(N_1) \hookrightarrow H^0(N_2)$ then it admits a minimal KS-factorization (def. \ref{KSFactorization}).

In particular, every [[dg-module]] has a minimal KS-model (def. \ref{KSFactorization}).

=--

[Roig & Saralegi-Aranguren 00, theorem 1.3.1](#RoigSaralegiAranguren00)

## Examples

* [4-sphere -- circle action](4-sphere#CircleAction)

## Related entries

* [[circle action]]

## References
 {#References}

* {#Roig92} [[Agustí Roig]], _Alguns punts d'àlgebra homotòpica_, Barcelona (1992)

* {#Roig93} [[Agustí Roig]], _Minimal resolutions and other minimal models_, Publicacions Matemàtiques (1993) Volume: 37, Issue: 2, page 285-303 ([web](https://eudml.org/doc/41535))

* {#Roig94} [[Agustí Roig]], _Formalizability of dg modules and morphisms of cdg algebras_, Volume 38, Issue 3 (1994), 434-451 ([euclid](http://projecteuclid.org/euclid.ijm/1255986724))

* [[Igor Kriz]], [[Peter May]], section IV.3 of _Operads, Algebras, Modules and Motives_, 1994 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/kmbooklatex.pdf))

* {#RoigSaralegiAranguren00} [[Agustí Roig]], [[Martintxo Saralegi-Aranguren]], _Minimal Models for Non-Free Circle Actions_, Illinois Journal of Mathematics, volume 44, number 4 (2000) ([arXiv:math/0004141](https://arxiv.org/abs/math/0004141))

See also

* {#Halperin83} [[Steve Halperin]], _Lectures on minimal models_, Mem. Soc. Math. Franc. no 9/10 (1983) ([web](https://eudml.org/doc/94833))

* {#Silveira84} [[Flavio da Silveira]], _Rational homotopy theory of fibrations_, Pacific Journal of Mathematics, Vol. 113, No. 1 (1984) ([pdf](http://msp.org/pjm/1984/113-1/pjm-v113-n1-p01-s.pdf))

[[!redirects minimal dg-modules]]

[[!redirects Hirsch extension]]
[[!redirects Hirsch extensions]]

[[!redirects minimal Hirsch cofibration]]
[[!redirects minimal Hirsch cofibrations]]

[[!redirects minimal KS-extension]]
[[!redirects minimal KS-extensions]]

[[!redirects minimal KS-model]]
[[!redirects minimal KS-models]]


minimal fibration

> This entry is about the concept in [[homotopy theory]]. For the concept if [[conformal field theory]] see at _[[minimal model CFT]]_.

***

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#Contents#
* table of contents
{:toc}

## Idea

In models of [[homotopy theory]] by [[homotopical categories]] it is a common concept that [[morphisms]] may be replaced by [[fibrant resolutions]] or [[cofibrant resolutions]]. These are in general far from being unique. Moreover, generic ways of constructing them, notably via a [[small object argument]], yield resolutions which are "very large" in that they are [[relative cell complexes]] (or co-cell complexes, respectively) with large transfinite [[cardinality]] of the [[set]] of cell attachments.

In contrast, a _minimal (co-)fibration_ is one which is "as small as possible" in some precise sense. Typically this minimality is witnessed by the property that any [[weak equivalence]] between minimal (co-)fibrations is already an [[isomorphism]].

If one thinks of [[objects]] in the given [[homotopical category]] as "models" (namely as models for [[homotopy types]] as in the usage of "[[model category]]") then one often speaks of "minimal models".

The discussion of the concept of minimal (co-)fibrations is often restricted to particular choices of ambient [[homotopical categories]] (see the list of [Examples](#Examples) below), prominent examples being the [[minimal Kan fibrations]] in the [[classical model structure on simplicial sets]] and the [[minimal Sullivan algebras]] in the [[projective model structure on dgc-algebras]]. A general concept of minimal models is considered in [Roig 93](#Roig93).

## Examples
 {#Examples}

* [[minimal Kan fibration]]

* [[minimal Joyal fibration]]

* [[minimal dendroidal fibration]]

* [[minimal dg-module]]

## References

A general concept of minimal fibrations and minimal models (in general [[model categories]]) is discussed in

* {#Roig93} [[Agustí Roig]], _Minimal resolutions and other minimal models_,  Publicacions Matemàtiques Vol. 37, No. 2 (1993), pp. 285-303 ([JSTOR](http://www.jstor.org/stable/43736447))

and with an eye specifically towards [[minimal dg-modules]] in

* {#Roig94} [[Agustí Roig]], section 1 of _Formalizability of dg modules and morphisms of cdg algebras_, Volume 38, Issue 3 (1994), 434-451 ([euclid](http://projecteuclid.org/euclid.ijm/1255986724))

For discussion in [[rational homotopy theory]] see at _[[Sullivan minimal model]]_.

Discussion in [[equivariant rational homotopy theory]] is in

* Georgia Triantafillou, _Equivariant minimal models_, Trans. Amer. Math. Soc. vol 274 pp 509-532 (1982) ([jstor](http://www.jstor.org/stable/1999119 ))

[[!redirects minimal fibrations]]

[[!redirects minimal cofibration]]
[[!redirects minimal cofibrations]]

[[!redirects minimal model]]
[[!redirects minimal models]]


minimal ideal

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## Idea

A minimal [[ideal]] is a proper ideal which doesn't contain any other proper ideal.
Formally, $J$ is minimal iff for every ideal $I$, $I \leq J$ implies $I=(0)$ or $I=J$.

This notion is the dual to the more popular one of [[maximal ideal]].

## Related concepts

* [[minimal object]]
* [[maximal ideal]]

## References

See also

* Wikipedia, *[Minimal ideal](https://en.wikipedia.org/wiki/Minimal_ideal)*

[[!redirects minimal ideals]]


minimal inner fibration

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## Idea

The condition that a [[quasicategory]] has "no non-trivial cells" above degree $n$ (which makes it a particularly strict model of an [[(n,1)-category]]) is not invariant under [[Joyal model structure|categorical equivalence]]. Hence there is no intrinsic characterization of the class of the simplicial sets which are "(n+1)-[[simplicial skeleton|coskeletal]]" in this sense.

(Warning: in [Lurie, Def. 2.3.4.1](#Lurie) such an "(n+1)-coskeletal" quasi-category is called an "$n$-category", but this is not the intrinsic notion of [[(n,1)-category]].)

However there is such a description of the class of quasi-categories which are _equivalent_ to such $(n+1)$-coskeletal quasicategories. To make this more concrete the notion of a *minimal inner fibration* can be used (a quasi-categorical analog of [[minimal Kan fibrations]], see also at _[[minimal fibration]]_). This is an [[inner fibration]] of simplicial sets satisfying a relative homotopy condition and that of a *minimal quasi-category* .

Every quasi-category is equivalent to a minimal quasi-category.

## Definition

+-- {: .num_defn #homotopic relative}
###### Definition
Let
$$\array{
A&\stackrel{u}{\to}&X
\\
\downarrow^i&&\downarrow^p
\\
B&\stackrel{v}{\to}&S}$$

denote a lifting problem. Then putative solutions $f,g$ of this lifting problem are called *homotopic relative $A$ over $S$* if they are equivalent as objects in the fiber of the map

$$X^B\to X^A\times_{S^A}S^B$$

Equivalently $f,g$ are homotopic relative $A$ over $B$ if there is a map

$$F:B\times \Delta[1]\to X$$

such that

$F|B\times\{0\}=f$

$F|B\times\{1\}=g$

$p\circ F=v\circ \pi_B$

$F\circ(i\times id_{\Delta[1]})=u\circ\pi_A$

$F|\{b\}\times \Delta[1]$

and $F|\{b\}\times\Delta[1]$ is an equivalence in the $(\infty,1)$-category $X_{v(b)}$ for every vertex $b$ of $B$.

=--
+-- {: .un_defn}
###### Definition 2.3.3.1

Let $p : X \to S$ be an inner fibration of simplicial sets. $p$ is called *minimal inner fibration*  if $f = f^\prime$ for every pair of maps $f , f ^\prime : \Delta[n] \to X$ which are homotopic relative to $\partial \Delta[n]$ over $S$ .

An $(\infty,1)$-category $C$ is called *minimal $(\infty,1)$-category* if $C\to *$ is minimal.
=--

+-- {: .un_prop #prop2.3.4.18}
###### Proposition 2.3.4.18
Let $C$ be an $(\infty,1)$-category and let $n\ge -1$. The the following statements are equivalent:

1. There exists a minimal model $C^\prime\subseteq C$ such that $C^\prime$ is an $(n+1)$-coskeletal quasi-category.

1. There exists a categorical equivalence $D\to C$, where $D$ is an $(n+1)$-coskeletal quasi-category.

1. For every pair of objects $X,Y\in C$, the mapping space $Map_C(X,Y)\in H$ is $(n-1)$-truncated.
=--

+-- {: .un_cor #cor2.3.4.19}
###### Corollary 2.3.4.19

Let $X$ be a [[Kan complex]]. Then is is equivalent to an $(n+1)$-coskeletal quasi-category  iff it is $n$[[truncation|-truncated]].

=--

## Related concepts

* [[minimal Kan fibration]]

* [[inner fibration]], [[left fibration]], [[right fibration]]

## References

Section 2.3.3 and section 2.3.4 of

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

[[!redirects minimal Joyal fibration]]
[[!redirects minimal Joyal fibrations]]

[[!redirects minimal inner fibrations]]


minimal Kan fibration

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## Idea

The concept of _minimal Kan fibration_ is the specialization of the concept of [[minimal fibrations]] to the [[Kan fibrations]] in the [[classical model structure on simplicial sets]].
Hence a _minimal Kan fibration_ is a [[Kan fibration]] whose [[fibers]] are, in some sense, as small as possible in its homotopy class. Moreover, every [[Kan fibration]] has a [[strong deformation retract]] to a minimal Kan fibration (prop. \ref{KanFibrationHasMinimalStrongDeformationRetract} below).
Hence minimal [[Kan complexes]] (i.e. minimal fibrations over the point) are the analogue in [[simplicial sets]] of [[minimal Sullivan models]] in [[rational homotopy theory]].

Minimal Kan fibrations over connected bases happen to be [[fiber bundles]], locally trivial over each simplex (lemma \ref{MinimalKanFibrationAreFiberBundles} below). This implies that their [[geometric realization]] into any [[convenient category of topological spaces]] is also a fiber bundle and hence in particular a [[Serre fibration]].  This is what makes minimal fibrations play a key role in all available proofs of the [[Quillen equivalence]] between the [[model structure on topological spaces]] and the standard [[model structure on simplicial sets]] (see at _[homotopy hypothsis -- for Kan complexes](homotopy+hypothesis#ForKanComplexes)_).

## Definition

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

A [[Kan fibration]] $\phi \colon S \longrightarrow T$, is called a **[[minimal Kan fibration]]** if for any two cells in the same fiber with the same [[boundary]] if they are homotopic relative their boundary, then they are already equal.

More formally, $\phi$ is minimal precisely if for every [[commuting diagram]] of the form

$$
  \array{
    (\partial \Delta[n]) \times \Delta[1]
    &\stackrel{p_1}{\longrightarrow}&
    \partial \Delta[n]
    \\
    \downarrow && \downarrow
    \\
    \Delta[n] \times \Delta[1]
    &\stackrel{h}{\longrightarrow}&
    S
    \\
    \downarrow^{\mathrlap{p_1}} && \downarrow^{\mathrlap{\phi}}
    \\
    \Delta[n] &\longrightarrow& T
  }
$$

then the two composites

$$
  \Delta[n]
  \stackrel{\overset{d_0}{\longrightarrow}}{\underset{d_1}{\longrightarrow}}
  \Delta[n] \times \Delta[1]
  \stackrel{h}{\longrightarrow}
  S
$$

are equal.

=--

## Properties

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

The [[pullback]] (in [[sSet]]) of a [[minimal Kan fibration]], def. \ref{MinimalKanFibration}, along any morphism is again a mimimal Kan fibration.

=--

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

For every [[Kan fibration]], there exists a fiberwise [[strong deformation retract]] to a [[minimal Kan fibration]], def. \ref{MinimalKanFibration}.

=--

(e.g. [Goerss-Jardine 96, chapter I, prop. 10.3](#GoerssJardine96), [Joyal-Tierney 05, theorem 3.3.1, theorem 3.3.3](#JoyalTierney05)).

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

Choose representatives by [[induction]], use that in the induction step one needs lifts of [[anodyne extensions]] against a [[Kan fibration]], which exist.

=--

+-- {: .num_lemma #FiberwiseHomotopyEquivalenceOfMinimalFibrationsIsIso}
###### Lemma

A morphism between [[minimal Kan fibrations]], def. \ref{MinimalKanFibration}, which is fiberwise a [[homotopy equivalence]], is already an [[isomorphism]].

=--

(e.g. [Goerss-Jardine 96, chapter I, lemma 10.4](#GoerssJardine96))

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

Show the statement degreewise. In the [[induction]] one needs to lift [[anodyne extensions]] agains a [[Kan fibration]].

=--

+-- {: .num_lemma #MinimalKanFibrationAreFiberBundles}
###### Lemma

Every [[minimal Kan fibration]], def. \ref{MinimalKanFibration}, over a [[connected]] base is a simplicial [[fiber bundle]], locally trivial over every simplex of the base.

=--

(e.g. [Goerss-Jardine 96, chapter I, corollary 10.8](#GoerssJardine96))

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

By assumption of the base being connected, the classifying maps for the fibers over any two vertices are connected by a [[zig-zag]] of [[homotopies]], hence by [this lemma](Kan+fibration#PullbackOfKanFibrationSendsLeftHomotopyToFiberwiseHomotopyequivalence) the fibers are connected by [[homotopy equivalences]] and then by prop. \ref{PullbackPreservesMinimalFibration} and lemma \ref{FiberwiseHomotopyEquivalenceOfMinimalFibrationsIsIso} they are already isomorphic. Write $F$ for this [[typical fiber]].

Moreover, for all $n$ the morphisms $\Delta[n] \to \Delta[0] \to \Delta[n]$ are [[left homotopy|left homotopic]] to $\Delta[n] \stackrel{id}{\to} \Delta[n]$ and so applying [this lemma](Kan+fibration#PullbackOfKanFibrationSendsLeftHomotopyToFiberwiseHomotopyequivalence) and prop. \ref{FiberwiseHomotopyEquivalenceOfMinimalFibrationsIsIso} once more yields that the fiber over each $\Delta[n]$ is [[isomorphism|isomorphic]] to $\Delta[n]\times F$.

=--

## Related concepts

[[minimal fibration]]

* [[minimal Joyal fibration]]

* [[minimal dendroidal fibration]]

##References

* {#GabrielZisman67} [[Pierre Gabriel]], [[Michel Zisman]], chapter VI.5 of _[[Calculus of fractions and homotopy theory]]_, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967)  ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf))

* E. Curtis, _Simplicial Homotopy Theory_, Advances in Math., 6, (1971), 107 – 209.

* {#GoerssJardine96} [[Paul Goerss]], [[Rick Jardine]], chapter I, section 10 of _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1996)

* [[André Joyal]], [[Myles Tierney]], section 3.3 of _Notes on simplicial homotopy theory_, ([pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern47.pdf))

[[!redirects minimal Kan fibrations]]

[[!redirects minimal Kan complex]]
[[!redirects minimal Kan complexes]]


minimal logic

\tableofcontents

## Idea

Minimal logic, introduced by I. Johansson in 1936, is [[intuitionistic logic]] without the _ex falso quodlibet_ rule $\bot \vdash A$.  It may also be defined by starting with Gentzen\'s [[sequent calculus]] for [[classical logic]] with $\bot$ but not $\neg$, and restricting to sequents $\Gamma \vdash \Delta$ where $\Delta$ must contain *exactly one* formula. The same is not true if you start with Gentzen\'s sequent calculus with $\neg$ but not $\bot$; for example, the sequent $ p, \neg p \vdash \neg q$ is valid in minimal logic, but it is not derivable in that sequent calculus if each succedent is required to consist of exactly one formula.

Because it is not the case here that
anything follows from a falsehood, minimal logic is a
[[paraconsistent logic]]. However, it still satisfies the weaker law
$$ A,\, \neg A \;\vdash\; \neg B $$
for any $A$ and $B$.  This is a weaker form of _ex contradictione quodlibet_, which is still strong enough to be undesirable if one wants to do serious things with a paraconsistent logic.

Models of minimal logic are cartesian closed categories, optionally
with finite $n$-ary coproducts for $n\gt 1$.  Under the [Curry-Howard
correspondence](http://en.wikipedia.org/wiki/Curry-Howard_correspondence) minimal logic corresponds to [simply-typed lambda
calculus](http://en.wikipedia.org/wiki/Simply-typed_lambda_calculus), which is roughly speaking another way of saying that both are forms of the [[internal logic]] of [[cartesian closed category|cartesian closed]] categories.

## See also

* [[cartesian closed category]]

## References

* Wikipedia, [Minimal logic](https://en.wikipedia.org/wiki/Minimal_logic)

For a definition of the minimal logic, see Chapter 2 in

* [[Sergei P. Odintsov]], _Constructive Negations and Paraconsistency_, Trends in Logic 26, Springer, 2008.  [doi](http://dx.doi.org/10.1007/978-1-4020-6867-6).


minimal model CFT

> This entry is about the concept in [[conformal field theory]]. For the conceopt of _[[minimal models]]_ in [[homotopy theory]] see at _[[minimal fibration]]_.

***

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## Idea

In [[conformal field theory]] a _minimal model_ is a [[2d CFT]] or [[2d SCFT]], whose field content consists of a [[finite number]] of [[irreducible representations]] of the (super-)[[Virasoro algebra]].

## Related concepts

* [[Gepner model]]

* [[rational conformal field theory]]

## References

See also

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

[[!redirects minimal model CFTs]]


minimal negation operator

The material previously on this page is now located at the [Online Encyclopedia of Integer Sequences Wiki](https://oeis.org/wiki/Minimal_negation_operator).

[[!redirects minimal negation]]
[[!redirects minimal negations]]
[[!redirects Minimal Negation Operator]]
[[!redirects Minimal Negation Operators]]
[[!redirects minimal negation operators]]
[[!redirects minimal negation operation]]
[[!redirects minimal negation operations]]
[[!redirects logical boundary operator]]
[[!redirects logical boundary operators]]
[[!redirects logical boundary operation]]
[[!redirects logical boundary operations]]


minimal object

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## Idea

The notion of  **minimal objects** generalizes the notion of minimal elements of [[preorders]]. Intuitively, minimal elements are ones that don't have anything below them, even though they might not be _the minimum_. A 'minimum object' would be instead an [[initial object]].

Minimal objects have been first defined in [(Adámek et al. '12)](#ABHKMS) in the context of [[coalgebra]] theory, relative to an [[orthogonal factorization system]]. Trivializing the [[factorization system]] gives a general notion of minimality and maximality, though these are quite a bit stronger in more general [[categories]].

## Definition
### In preorders
\begin{definition}
In a [[preorder]] $(C, \leq)$, a minimal element $m \in C$ is one such that if $a \leq m$, then $m \leq a$. Dually, a maximal element is one such that $m \leq a$ implies $a \leq m$, i.e. which is minimal in $C^{op}$.
\end{definition}

\begin{remark}
If $C$ is [[bounded lattice|bounded below]], meaning there  is an element $\bot \in C$ such that $\bot \leq a$ for all $a \in C$ ($\bot$ is [[initial object|initial]]), then $\bot$ is the only minimal element in $C$.
Hence in many cases (chiefly, when $C$ is the preorder of [[ideals]] of a ring), one speaks of minimality in $C\setminus\{\bot\}$, that is, one defines a minimal element to be one for which $a \leq m$ implies $a=m$ _or $a=\bot$.

This is not the definition we take here!
\end{remark}

### In categories
Let $(\mathcal{E}, \mathcal{M})$ be an [[orthogonal factorization system]] on a [[category]] $\mathcal{C}$. This means:

0. $\mathcal{E}$ and $\mathcal{M}$ are [[wide subcategories]] of $\mathcal{C}$ closed under composition and each containing all isomorphisms of $\mathcal{C}$.
1. Every morphism in $\mathcal{C}$ factors uniquely as a morphism in $\mathcal{E}$ followed by one in $\mathcal{M}$:
\[
  a \overset{f}\to b = a \overset{e_f}\twoheadrightarrow \mathrm{im}(f) \overset{m_f}\rightarrowtail b.
\]
2. The morphisms in $\mathcal{E}$ are exactly the ones having the [[left lifting property]] against the morphisms in $\mathcal{M}$, and the morphisms in $\mathcal{M}$ are exactly the ones having the [[right lifting property]] against the morphisms in $\mathcal{E}$.
3. The lifts are unique.

Here we'll tend to denote morphisms in $\mathcal{E}$ with $\twoheadrightarrow$ and morphisms in $\mathcal{M}$ with $\rightarrowtail$. In fact, a common source of examples for these factorization systems is given by the systems of [[strong epimorphism|strongly epic]] and [[monic|monomorphism]] arrows.

\begin{definition}
An **$\mathcal{M}$-minimal object** is an object $M:\mathcal{C}$ such that every morphism $h : A \rightarrowtail M$ in $\mathcal{M}$ is an isomorphism.
\end{definition}

\begin{definition}
An **$\mathcal{E}$-minimal object** is an object $M:\mathcal{C}$ such that every morphism $h : M \twoheadrightarrow A$ in $\mathcal{E}$ is an isomorphism.
\end{definition}

Clearly the two definitions above are dual to each other, since $\mathcal{C}^{op}$ is equipped with the orthogonal factorization system $(\mathcal{M}^{op}, \mathcal{E}^{op})$. Indeed, one might call $\mathcal{E}$-minimal objects **$\mathcal{M}$-maximal**.

\begin{remark}
Instantiating this definition for the trivial factorization systems gives us a general definition of minimal and maximal objects in a category:

In fact, every category admits a trivial orthogonal factorization systems, namely $(\mathcal{I}, \mathcal{C})$, where $\mathcal{I}$ is the [[core|wide subcategory of isomorphisms]] of $\mathcal{C}$. In this instance, every object is $\mathcal{I}$-minimal (since every isomorphism to it is already an iso) but an $\mathcal{I}$-maximal object is one such that every morphism out of it is invertible.
We call these simply **maximal objects**.

Every category also admits $(\mathcal{C}, \mathcal{I})$ as a trivial orthogonal factorization system. In this case, $\mathcal{C}$-minimal objects are the ones for which every morphism into them is invertible, whereas every object is trivially $\mathcal{C}$-maximal. We call these simply **minimal objects**.

These notions recover the order-theoretic ones when $\mathcal{C}$ is a preorder.
\end{remark}

\begin{proposition}
The following are equivalent:

1. An object $M : \mathcal{C}$ is $\mathcal{M}$-minimal
2. Every morphism $h:A \to M$ is in $\mathcal{E}$.
3. (if $\mathcal{E}$=epimorphisms and $\mathcal{C}$ has [[weak equalizers]]) All parallel pairs of arrows $A \rightrightarrows M$ are equal.

\end{proposition}

\begin{corollary}

Dually, the following are equivalent:

1. An object $M : \mathcal{C}$ is $\mathcal{M}$-maximal
2. Every morphism $h:M \to A$ is in $\mathcal{M}$.
3. (if $\mathcal{M}$=monomorphisms and $\mathcal{C}$ has [[weak coequalizers]]) All parallel pairs of arrows $M \rightrightarrows A$ are equal (i.e. $M$ is [[subterminal]])

\end{corollary}

## Relation with initial objects

Like for preorders, the only minimal objects in a category having initial objects are the latter (up to isomorphism, of course). In fact, suppose $\mathcal{C}$ has initial object $0$ and suppose $M : \mathcal{C}$ is minimal. Then the initial map $?_M : 0 \to M$ has to be an isomorphism, so that $M \cong 0$.

However, unlike in preorders, not all initial objects are minimal! Notice, in fact, that the argument above shows $0$ is minimal under the assumption a minimal object already exists, but this doesn't have to be true.
A **minimal initial object** is known as [[strict initial object]].

Everything just said dualizes: every maximal object in a category with terminal object is terminal, but not every terminal object is maximal. A **maximal terminal object** is known as [[strict terminal object]].

## Minimization
Let $\mathcal{C}$ be a category equipped with an $(\mathcal{E}, \mathcal{M})$-factorization system.

\begin{definition}
An **$\mathcal{M}$-minimization** of $A: \mathcal{C}$ is a morphism $m:M \to A$ in $\mathcal{M}$ where $M:\mathcal{C}$ is $\mathcal{M}$-minimal.
Dually, an **$\mathcal{E}$-minimization** of $A: \mathcal{C}$ is a morphism $e:A \to E$ in $\mathcal{E}$ where $E:\mathcal{C}$ is $\mathcal{E}$-minimal.
\end{definition}

With abuse of notation, we call 'minimization of $A$' both the morphism and the minimal object.

## Examples

1. Let $\mathcal{C} = \mathbf{Coalg}_*(F)$ is the category of [[pointed object|pointed]] [[coalgebras]] for some endofunctor $F:\mathbf{Set} \to \mathbf{Set}$. This category inherits the epi-mono factorization system from $\mathbf{Set}$. Then a mono-minimal object in $\mathbf{Coalg}_*(F)$ is a _reachable coalgebra_, that is, one that has no non-trivial pointed subcoalgebra. Intuitively, this means every state of a reachable coalgebra is reachable in finitely many transitions from its distinguished point (playing the role of 'initial state').

2. In $\mathcal{C} = \mathbf{Coalg}(F)$, an epi-maximal object is a _simple coalgebra_, that is, one that has no non-trivial quotient. Intuitively, this means that there are no states in the coalgebra with the same behaviour.

3. In the same setting as (2), when $F$ preserves monos, then a quotient $m : S \to S'$ where $S'$ is epi-maximal is called a _minimization_ for $S$. Concretely, it exhibits a way to reduce the automaton $S$ to a smaller automaton $S'$ in which no states are duplicated (where we consider duplicated states with the same behaviour).

## Properties
\begin{proposition}
In a category $\mathcal{C}$ with products, minimal objects, if they exists, are unique up to isomorphism.
\end{proposition}
\begin{proof}
Let $M,M'$ be minimal in $\mathcal{C}$. Then $\pi_M : M \times M' \to M$ and $\pi_{M'} : M \times M' \to M'$ are both invertible, proving $M \cong M \times M' \cong M'$.
\end{proof}

\begin{proposition}
Let $\mathcal{C}$ be a category with finite products.
Then if $M$ is minimal it is weakly initial.
\end{proposition}
\begin{proof}
Let $M:\mathcal{C}$ be such minimal object. Let $A$ be any other object in $\mathcal{C}$. Then, by hypothesis on $M$, the projection $\pi_M : M \times A \to M$ is invertible. Therefore $M$ admits a morphism into every object of $\mathcal{C}$, given by $w_A := M \overset{\pi^{-1}_M}\to M \times A \overset{\pi_A}\to A$.
\end{proof}

Notice we can't say anything about the uniqueness of these morphisms $w_A : M \to A$. In general, they are not even functorially chosen.

## Related concepts

* [[minimal ideal]], [[maximal ideal]]
* [[strict initial object]],  [[strict terminal object]]
* [[simple coalgebra]]
* [[reachability]]

## References

* {#ABHKMS} [[Jiri Adamek|J. Adámek]], [[Filippo Bonchi|F. Bonchi]], M. Hülsbusch, B. König, S. Milius, and [[Alexandra Silva|A. Silva]], _A coalgebraic perspective on minimization and determinization_, in Proceedings of the 15th international conference on Foundations of Software Science and Computational Structures, 2012, pp. 58–73.

* {#Weissmann22} Thorsten Weissmann, _Minimality Notions via Factorization Systems and Examples_ Logical Methods in Computer Science **18** (2022) [doi:10.46298/lmcs-18(3:31)2022]

* {#BKP} Nick Bezhanishvili, Clemens Kupke, [[Prakash Panangaden]], *Minimization via Duality*, in: *Logic, Language, Information and Computation. WoLLIC 2012*, Lecture Notes in Computer Science **7456**, Springer (2012) [[doi:10.1007/978-3-642-32621-9_14](https://doi.org/10.1007/978-3-642-32621-9_14), [pdf](https://staff.science.uva.nl/n.bezhanishvili/Papers/Wollic.pdf)]

[[!redirects minimal objects]]
[[!redirects maximal object]]
[[!redirects maximal objects]]


minimal simplicial circle

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

#Contents#
* table of contents
{:toc}

## Idea

The smallest [[simplicial set]] whose [[homotopy type]] in the [[classical homotopy category]] is that of the [[circle]] has precisely two non-degenerate [[simplices]], one in degree 0 -- a single [[vertex]] -- and one in degree 1 -- a single [[edge]] which is a [[loop]] on that vertex:

$$
  S \;\coloneqq\; \Delta[1]/\partial \Delta[1]
  \,.
$$

Despite or maybe because its simplicity, the minimal simplicial circle plays a central role in many constructions, notably in the context of [[cyclic homology]] (e.g. [Loday 1992, 7.1.2](#Loday92)).

## Properties

\begin{example}
  The [[normalized chain complex]] of the [[free simplicial abelian group]] of the minimal simplicial circle $S$ has the group of [[integers]] in degrees 0 and 1, and all [[differentials]] are [[zero morphism|zero]]:
$$
  N_\bullet \circ \mathbb{Z}(S)
  \;\simeq\;
  \left[
  \array{
    \vdots
    \\
    \big\downarrow
    \\
    0
    \\
    \big\downarrow
    \\
    \mathbb{Z}
    \\
    \big\downarrow {}^{\mathrlap{ 0 }}
    \\
    \mathbb{Z}
  }
  \;\;
  \right]
  \;\simeq\;
  \mathbb{Z} \oplus \mathbb{Z}[1]
  \,.
$$
\end{example}

## Related concepts

* [[circle]], [[unit circle]], [[circle type]], [[circle group]]

* [[cyclic category]]

  [[cyclic object]], [[cyclic set]], [[cyclic space]]

  [[cyclic homology]], [[cyclic cohomology]]

* [[inertia groupoid]]

## References

Discussion in relation to the [[cyclic category]] and [[cyclic sets]]/[[cyclic spaces]]:

* {#Loday92} [[Jean-Louis Loday]], *Cyclic Spaces and $S^1$-Equivariant Homology* ([doi:10.1007/978-3-662-21739-9_7](https://link.springer.com/chapter/10.1007/978-3-662-21739-9_7))

  Chapter 7 in: *Cyclic Homology*, Grundlehren **301**, Springer 1992 ([doi:10.1007/978-3-662-21739-9](https://link.springer.com/book/10.1007/978-3-662-21739-9))


minimal surface

#Contents#
* table of contents
{:toc}

## Idea

A *minimal surface* is an [[embedding|embedded]] [[Riemannian manifold|Riemannian]] [[surface]] which *locally* [[minimum|minimizes]] its [[area]].

## Related concepts

* [[Polyakov action]], [[Nambu-Goto action]]

* [[Ryu-Takayanagi formula]], [[ER = EPR]]

## References

See also:

* Wikipedia, *[Minimal surface](https://en.wikipedia.org/wiki/Minimal_surface)*

Discussion of complex minimal surfaces via the [[Oka principle]]:

* [[Antonio Alarcón]], [[Franc Forstnerič]], *New complex analytic methods in the theory of minimal surfaces: a survey*,  Journal of the Australian Mathematical Society, 106(3), 287-341 ([arXiv:1711.08024](https://arxiv.org/abs/1711.08024), [doi:10.1017/S1446788718000125](https://doi.org/10.1017/S1446788718000125))

[[!redirects minimal surfaces]]

[[!redirects minimal area]]


Minkowski metric

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Gravity
+-- {: .hide}
[[!include gravity contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Minkowski metric_ is a [[pseudo-Riemannian metric]] which is completely _flat_ in that its [[Riemann curvature]] vanishes.

For $p \in \mathbb{N}$, a [[Cartesian space]] $\mathbb{R}^{p+1}$ equipped with the Minkowski metric $\eta$ is called _[[Minkowski spacetime]]_ of [[dimension]] $p + 1$, denoted $\mathbb{R}^{p,1} \coloneqq (\mathbb{R}^{p+1}, \eta)$.

## Definition

In terms of the standard [[coordinates]] $(x^\mu)_{\mu = 0}^p$ on $\mathbb{R}^{p+1}$ the Minkowski metric is the [[invariant differential form|constant]] rank 2-[[tensor]] which is given by the [[diagonal matrix]] $\eta = diag(-1,+1,+1, \cdots , +1)$ (or else $\eta = diag(+1,-1,-1, \cdots, -1)$).

This means that for $v = (v^\mu)_{\mu = 0}^p \in \mathbb{R}^{p,1}$ then

$$
  \eta(v_1, v_2)
    \;=\;
  \eta_{\mu \nu} v_1^\mu v_2 \mu
  \coloneqq
  - (v^0)^2 + \underoverset{j = 1}{p}{\sum} (v^j)^2
  \,.
$$

## Properties

### In terms of hermitian matrices.

In the dimensions $p+1 \in \{3,4,6,10\}$ the [[quadratic form]] corresponding to the Minkowski metric may be identified simply with the [[determinant]] function on $2 \times 2$ [[hermitian matrices]] with [[coefficients]] in one of the four [[real numbers|real]] [[normed division algebra]] $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}\}$.

For details see at _[[geometry of physics -- A first idea of quantum field theory]]_ the chapter _[spacetime](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#Spacetime)_.

## Related concepts

* [[shell]]

* [[super-Minkowski spacetime]]

## References

Named after _[[Hermann Minkowski]]_.

[[!redirects Minkowski metrics]]

[[!redirects Lorentz metric]]
[[!redirects Lorentz metrics]]


Minkowski space

+-- {: .rightHandSide}
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### Context
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Gravity
+-- {: .hide}
[[!include gravity contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

For $d-1 \in \mathbb{N}$, $d$-[[dimension]]al _Minkowski space_ is the [[Lorentzian manifold]] whose underlying [[smooth manifold]] is the [[Cartesian space]] $\mathbb{R}^d$ and whose [[pseudo-Riemannian metric]] is at each point the [[Minkowski metric]].

This is naturally a [[spacetime]].

## Properties

### Isometries

The [[isometry group]] of Minkowski space is the [[Poincaré group]].
The study of Minkowski spacetime with its isometries is also called _[[Lorentzian geometry]]_. This is the context of the [[theory of special relativity]].

### Gravitational stability

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

Minkowski spacetimes is a [[gravitational stability|stable]] solution of the vacuum [[Einstein equations]].

=--

This is due to ([ChristodoulouKlainerman 1993](#ChristodoulouKlainerman)).

[[!include Hodge star operator on Minkowski spacetime -- section]]

## Related entries

* [[Minkowski metric]]

* [[spacetime]]

  * [[super Minkowski space]]

  * [[Schwarzschild spacetime]]

* [[light cone]], [[celestial sphere]]

* [[twistor space]]

* [[piecewise flat spacetime]]

* [[flat space holography]]

## References

The notion of Minkowski spacetime originates with

* {#Minkowski07} [[Hermann Minkowski]], _Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern_,  Math. Ann. (1910) 68: 472, reprinted from: Nachrichten der Kgl. Ges. d. Wiss. zu Göttingen, Math.-phys. Kl., Sitzung vom 21. Dezember 1907 ([doi:10.1007/BF01455871](https://doi.org/10.1007/BF01455871))

> {#TheViewsOfSpaceAndTime} The views of space and time that I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of both will retain an independent reality.

> (Address to the 80th Assembly of German Natural Scientists and Physicians, (Sep 21, 1908), see [WikiQuote](https://en.wikiquote.org/wiki/Hermann_Minkowski))

See also

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

Gravitational stability of Minkowski space is proven in

* {#ChristodoulouKlainerman} Demetrios Christodoulou, [[Sergiu Klainerman]], _The global nonlinear stability of the Minkowski space_ Princeton University Press (1993)

[[!redirects Minkowski space]]
[[!redirects Minkowski spaces]]

[[!redirects Minkowski spacetime]]
[[!redirects Minkowski space-time]]

[[!redirects Minkowski spacetimes]]
[[!redirects Minkowski space-times]]


Minkowski's inequality

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### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

What is commonly known as _Minkowski's inequality_ is the statement that the [[p-norm]] ${\Vert f\Vert_p} \coloneqq \root{p}{\int_X {\vert f\vert^p} d\mu}$ on [[Lebesgue spaces]] indeed satisfies the [[triangle inequality]].

## Proofs

Our proof of Minkowski's inequality is broken down into a few simple lemmas. The plan is to boil it down to two things: the scaling axiom, and convexity of the function $x \mapsto {|x|^p}$ (as a function from real or complex numbers to nonnegative real numbers).

First, some generalities. Let $V$ be a (real or complex) vector space equipped with a function ${\|(-)\|}\colon V \to [0, \infty]$ that satisfies the scaling axiom: ${\|t v\|} = {|t|} \, {\|v\|}$ for all scalars $t$, and the separation axiom: ${\|v\|} = 0$ implies $v = 0$. As usual, we define the [[unit ball]] in $V$ to be $\{v \in V \;|\; {\|v\|} \leq 1\}.$

+-- {: .num_lemma #conditions}
###### Lemma
Given that the scaling and separation axioms hold, the following conditions are equivalent:

1. The triangle inequality is satisfied.
1. The unit ball is [[convex set|convex]].
1. If ${\|u\|} = {\|v\|} = 1$, then ${\|t u + (1-t)v\|} \leq 1$ for all $t \in [0, 1]$.
=--

+-- {: .proof}
###### Proof
Since conditions 2. and 3. are pretty obviously equivalent, we just prove 1. and 3. are equivalent. Condition 1. implies condition 3. easily: if $\|u\| = 1 = \|v\|$ and $0 \leq t \leq 1$, we have

$$\array{
{\|t u + (1-t)v\|} & \leq & {\|t u\|} + {\|(1-t)v\|} \\
 & = & t {\|u\|} + (1-t) {\|v\|} \\
 & = & t + (1-t) = 1.}$$

Now we prove that 3. implies 1.  Suppose ${\|v\|}, {\|v'\|} \in (0, \infty)$. Let $u = \frac{v}{{\|v\|}}$ and $u' = \frac{v'}{{\|v'\|}}$ be the associated unit vectors.
Then
$$\array{
\frac{v + v'}{{\|v\|}+{\|v'\|}} & = & \left(\frac{{\|v\|}}{{\|v\|}+{\|v'\|}}\right)\frac{v}{{\|v\|}} + \left(\frac{{\|v'\|}}{{\|v\|}+{\|v'\|}}\right)\frac{v'}{{\|v'\|}} \\
 & = & t u + (1-t)u'}$$
where $t = \frac{{\|v\|}}{{\|v\|} + {\|v'\|}}$. If condition 3. holds, then
$${\|t u + (1-t)u'\|} \leq 1$$
but by the scaling axiom, this is the same as saying
$$\frac{{\|v + v'\|}}{{\|v\|} + {\|v'\|}} \leq 1,$$
which is the triangle inequality.
=--

Consider now $L^p$ with its $p$-norm ${\|f\|} = {|f|_p}$.  By Lemma \ref{conditions}, the Minkowski inequality is equivalent to

* **Condition 4:** If ${|u|_{p}^{p}} = 1$ and ${|v|_{p}^{p}} = 1$, then ${|t u + (1-t)v|_{p}^{p}} \leq 1$ whenever $0 \leq t \leq 1$.

This allows us to remove the cumbersome exponent $1/p$ in the definition of the $p$-norm.

+-- {: .num_lemma #convex}
###### Lemma
Define $\phi\colon \mathbb{C} \to \mathbb{R}$ by $\phi(x) = |x|^p$.  Then $\phi$ is convex, i.e., for all $x, y$,
$${|t x + (1-t)y|^p} \leq t{|x|^p} + (1-t){|y|^p}$$
for all $t \in [0, 1]$.
=--

+-- {: .proof}
###### Proof
The function $g: x \mapsto {|x|}$ is convex, and for $1 \lt p$ the function $f: t \mapsto t^p$ for $t \geq 0$ is monotone increasing and convex, by the first and second derivative tests. Thus $g(t x + (1-t)y) \leq t g(x) + (1-t)g(y)$ and then

$$\array{
f(g(t x + (1-t)y)) & \leq & f(t g(x) + (1-t)g(y)) \\
 & \leq & t f(g(x)) + (1-t)f(g(y))
}$$

so $f \circ g: x \mapsto {|x|^p}$ is convex.
=--

+-- {: .proof}
###### Proof of Minkowski's inequality
Let $u$ and $v$ be unit vectors in $L^p$. By condition 4, it suffices to show that ${|t u + (1-t)v|_p^p} \leq 1$ for all $t \in [0, 1]$. But
$$\int_X {|t u + (1-t)v|^p} \,d\mu \leq \int_X t{|u|}^p + (1-t){|v|}^p \,d\mu$$
by Lemma \ref{convex}. Using $\int {|u|^p} = 1 = \int {|v|^p}$, we are done.
=--

Another commonly seen proof of Minkowski's inequality derives it with the help of [[Hölder's inequality]]; see there for some commentary on this. But this is probably not the first thing one would think of unless one knows the trick, whereas the alternative proof given above seems geometrically motivated and fairly simple.

## Related concepts

* [[Hölder's inequality]]

* [[inequality]]

## References

* [[Todd Trimble]], _[[toddtrimble:p-norms]]_

* Wikipedia, _[Minowski's inequality](https://en.wikipedia.org/wiki/Minkowski_inequality)_

* [[eom]], _[Minkowski inequality](https://www.encyclopediaofmath.org/index.php/Minkowski_inequality)_

[[!redirects Minkowski inequality]]

[[!redirects Minkowski's inequality]]
[[!redirects Minkowski\'s inequality]]
[[!redirects Minkowski/'s inequality]]


minor

Minor is used in two meanings

* a submatrix of a [[matrix]]

* the determinant of a square submatrix of a matrix with entries in a commutative [[ring]]

where the second meaning is a bit more standard. There is a noncommutative analogue, namely the quasiminor, which is the quasideterminant of a square submatrix of a matrix with entries in a noncommutative ring.

[[!redirects minors]]


minor scale

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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The idea of a minor scale comes from [[Peter Freyd]], in his attempt to give an algebraic (in the sense of universal algebra) description of the real [[interval]], which he believed to be more fundamental than the [[real numbers]] themselves in [[analysis]].

Certain minor scales are models of [[multiplicative linear logic]].

Minor scales are an example of [[centipede mathematics]]; a lot of this page will be determining which of Freyd's results in _Algebraic Real Analysis_ still applies in minor scales and possible weakenings of the scale identity axiom.

+-- {: .standout}
Note: The terms _multiplicative conjunction_, _multiplicative disjunction_, _linear implication_, _reverse linear implication_, _additive conjunction_, and _additive disjunction_ used in this page come from [[scales]], which are algebraic models of multiplicative-additive [[linear logics]]. While the same operations may be defined in the same way in minor scales as in scales, the operations may no longer correspond to any linear logic at all, and are simply defined to reduce syntactical complexity in equational sentences and axioms.
=--

## Definition

### In terms of multiplicative conjunction and disjunction

A __minor scale__ is a [[symmetric closed midpoint algebra]] $(M,\vert, \odot, (-)^\bullet, \bot, \top)$ with two binary operations $(-)\otimes(-):M\times M\to M$ called __[[multiplicative conjunction]]__ and $(-)\oplus(-):M\times M\to M$ called __[[multiplicative disjunction]]__ such that for all $a$ in $M$,

$$a \otimes b = b \otimes a$$

$$\bot \otimes a = \bot$$

$$a \otimes \top = a$$

$$a \oplus b = b \oplus a$$

$$\bot \oplus a = a$$

$$a \oplus \top = \top$$

### In terms of linear implication

+-- {: .standout}
Note: linear implication is usually denoted using $\multimap$. We use $\rightarrowtail$ instead because there is no left-pointing $\multimap$ in [[LaTeX]] for the dual linear implication, while there is $\leftarrowtail$
=--

From $\rightarrowtail$ one could define the zoom operators.

__$\bot$-zooming__ can be defined as

$a^\vee \coloneqq a^\bullet \rightarrowtail a$

__$\top$-zooming__ can be defined as

$a^\wedge \coloneqq (a \rightarrowtail a^\bullet)^\bullet$

### In terms of zoom operations

A __minor scale__ is a [[symmetric closed midpoint algebra]] $(M,\vert, \odot, (-)^\bullet, \bot, \top)$ with two functions $(-)^\wedge:M\to M$ called __$\top$-zooming__ and $(-)^\vee:M\to M$ called __$\bot$-zooming__ such that for all $a$ in $M$,

$$(\bot \vert a)^\wedge = \bot$$

$$(\bot \vert a)^\vee = a$$

$$(a \vert \top)^\wedge = a$$

$$(a \vert \top)^\vee = \top$$

### In terms of central dilatations

A __minor scale__ is a [[symmetric closed midpoint algebra]] $(M,\vert, \odot, (-)^\bullet, \bot, \top)$ with a function $(-)^\dagger:M\to M$ called __central dilatation__ such that for all $a$ in $M$,

$$(\odot \vert a)^\dagger = a$$

$$(\top \vert (\top \vert a))^\dagger = \top$$

$$(\bot \vert (\bot \vert a))^\dagger = \bot$$

__$\bot$-zooming__ can be defined as

$$a^\vee \coloneqq ((a \vert (\odot \vert \top))^\dagger)^\dagger$$

__$\top$-zooming__ can be defined as

$$a^\wedge \coloneqq (((\bot \vert \odot) \vert a)^\dagger)^\dagger$$

__Multiplicative conjunction__ can be defined as

$$a \otimes b \coloneqq (a \vert b)^\wedge$$

__Multiplicative disjunction__ can be defined as

$$a \oplus b \coloneqq (a \vert b)^\vee$$

__Linear implication__ can be defined as

$$a \multimap b \coloneqq a^\bullet \oplus b$$

### Other operations

The binary operation __additive [[conjunction]]__ is defined as

$$a \wedge b \coloneqq a \otimes (a^\bullet \oplus b)$$

and __additive [[disjunction]]__ is defined as

$$a \vee b \coloneqq a \oplus (a^\bullet \otimes b)$$

The binary operation __dilatation__ is defined as

$$a\vartriangleleft b \coloneqq ((a^\bullet \vert \bot)\oplus b)^\wedge$$

or as

$$a\vartriangleleft b \coloneqq ((a^\bullet \vert \top)\otimes b)^\vee$$

### Defining operations in terms of other operations

Multiplication conjunction, multiplicative disjunction, $\top$-zooming, $\bot$-zooming, linear implication, and dual linear implication can all be defined in terms of each other, $\vert$, and $(-)^\bullet$.

operation	$a \otimes b$	$a \oplus b$	$a^\wedge$	$a^\vee$	$a \leftarrowtail b$	$a \rightarrowtail b$

$a \otimes b$	$a \otimes b$	$(a^\bullet \oplus b^\bullet)^\bullet$	$(a\vert b)^\wedge$	$((a^\bullet \vert b^\bullet)^\vee)^\bullet$	$a^\bullet \leftarrowtail b$	$(a \rightarrowtail b^\bullet)^\bullet$

$a \oplus b$	$(a^\bullet \otimes b^\bullet)^\bullet$	$a \oplus b$	$((a^\bullet \vert b^\bullet)^\wedge)^\bullet$	$(a \vert b)^\vee$	$(a \leftarrowtail b^\bullet)^\bullet$	$a^\bullet \rightarrowtail b$

$a^\wedge$	$a \otimes a$	$(a^\bullet \oplus a^\bullet)^\bullet$	$a^\wedge$	$((a^\bullet)^\vee)^\bullet$	$a^\bullet \leftarrowtail a$	$(a \rightarrowtail a^\bullet)^\bullet$

$a^\vee$	$(a^\bullet \otimes a^\bullet)^\bullet$	$a \oplus a$	$((a^\bullet)^\wedge)^\bullet$	$a^\vee$	$(a \leftarrowtail a^\bullet)^\bullet$	$a^\bullet \rightarrowtail a$

$a \leftarrowtail b$	$a^\bullet \otimes b$	$(a \oplus b^\bullet)^\bullet$	$(a^\bullet \vert b)^\wedge$	$((a \vert b^\bullet)^\vee)^\bullet$	$a \leftarrowtail b$	$(a^\bullet \rightarrowtail b^\bullet)^\bullet$

$a \rightarrowtail b$	$(a \otimes b^\bullet)^\bullet$	$a^\bullet \oplus b$	$((a \vert b^\bullet)^\wedge)^\bullet$	$(a^\bullet \vert b)^\vee$	$(a^\bullet \leftarrowtail b^\bullet)^\bullet$	$a \rightarrowtail b$

## Additional possible equational axioms

Here is a list of other possible equational axioms that a minor scale might satisfy:

* the __scale identities__: for all $a$ and $b$ in $M$,

$$a \otimes b = (a^\vee \otimes b^\wedge) \vert (a^\wedge \otimes b^\vee)$$

$$a \oplus b = (a^\wedge \oplus b^\vee) \vert (a^\vee \oplus b^\wedge)$$

The rest of the axioms follow from the scale identities:

* $\top$ is a [[fixed point]] of $(-)^\vee$:

$$\top^\vee = \top$$

* $\bot$ is a fixed point of $(-)^\wedge$:

$$\bot^\wedge = \bot$$

* the __axiom of compensation__: for all $a$ in $M$,

$$a = a^\vee \vert a^\wedge$$

* the __axioms of central distributivity__: for all $a$ in $M$,

$$a \otimes \odot = a^\wedge \vert \bot$$

$$a \oplus \odot = a^\vee \vert \top$$

* the __axiom of balance__: for all $a$ and $b$ in $M$,

$$a \vert (a \multimap b) = b \vert (b \multimap a)$$

* the __disjunction property__: for all $a$ and $b$ in $M$,

$$(a \rightarrowtail b) \vee (b \rightarrowtail a) = \top$$

* the __coalgebra equation__: for all $a$ in $M$,

$$a^\vee \vee (a^\wedge)^\bullet = \top$$

* __associativity of multiplicative conjunction__: for all $a$, $b$, and $c$ in $M$

$$a \otimes (b \otimes c) = (a \otimes b) \otimes c$$

* __associativity of multiplicative disjunction__: for all $a$, $b$, and $c$ in $M$

$$a \oplus (b \oplus c) = (a \oplus b) \oplus c$$

* the various __[[lattice]] axioms__ for $(M,\top,\bot,\wedge,\vee)$: for all $a$, $b$, and $c$ in $M$,

$$a \wedge a = a$$

$$a \wedge b = b \wedge a$$

$$a \wedge (b \wedge c) = (a \wedge b) \wedge c$$

$$\top \wedge a = a$$

$$a \vee a = a$$

$$a \vee b = b \vee a$$

$$a \vee (b \vee c) = (a \vee b) \vee c$$

$$\bot \vee a = a$$

$$a \wedge (a \vee b) = a$$

$$a \vee (a \wedge b) = a$$

* for all $a$ and $b$ in $M$,

$$(a \otimes b) \rightarrowtail (a^\wedge \vert b^\wedge) = \top$$

* for all $a$ and $b$ in $M$,

$$ (a \oplus b) \leftarrowtail (a^\vee \vert b^\vee) = \bot$$

## Properties

Every minor scale with $\bot = \top$ is [[trivial object|trivial]].

$\bot$ is a [[fixed point]] of $(-)^\vee$ and $\top$ is a fixed point of $(-)^\wedge$.

As a minor scale is a [[closed midpoint algebra]], a minor scale has a [[partial order]] $\leq$. If $a \leq b$, then $a \multimap b = \top$.

The [[currying]] of $\vartriangleleft$ results in a __dilatation at an element__ $(-)\vartriangleleft:M\to (M\to M)$. For every element $a$ in $M$, dilatation at $a$ is a [[retraction]] of contraction at $a$: $(a\vartriangleleft) \circ (a\vert) = id_M$, where $id_M$ is the [[identity function]] on $M$.

## Examples

The [[unit interval]] with $a \vert b \coloneqq \frac{a + b}{2}$, $\odot = \frac{1}{2}$, $a^\bullet = 1 - a$, $\bot = 0$, $\top = 1$, $a \otimes b = max(a + b - 1,0)$, $a \oplus b = min(a + b,1)$, $a^\wedge = max(2a - 1,0)$ and $a^\vee = min(2a,1)$ is an example of a minor scale.

The set of truth values in Girard's [[linear logic]] is a minor scale.

## Related concepts

* [[symmetric closed midpoint algebra]]

* [[scale]]

* [[linear logic]]

* [[centipede mathematics]]

## References

* [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))


Minoru Wakimoto

* [inspire entry](https://inspirehep.net/authors/1956048)

## Selected writings

On non-integrable but "admissible" [[irreps]] of [[affine Lie algebras]]:

* [[Victor G. Kac]], [[Minoru Wakimoto]], *Modular invariant representations of infinite-dimensional Lie algebras and superalgebras*,  PNAS **85** 14 (1988)  4956-4960 [[doi:10.1073/pnas.85.14.4956](https://doi.org/10.1073/pnas.85.14.4956)]

* [[Victor G. Kac]], [[Minoru Wakimoto]], *Classification of modular invariant representations of affine algebras*, p. 138-177 in V. G. Kač (ed.): *Infinite dimensional Lie algebras and groups*, Advanced series in Mathematical physics **7**, World Scientific (1989) [[pdf](https://math.mit.edu/~kac/not-easily-available/admissible.pdf),  [cds:268092](https://cds.cern.ch/record/268092)]

* [[Victor G. Kac]], [[Minoru Wakimoto]], *Modular and conformal invariance constraints in representation theory of affine algebras*, Advances in Mathematics **70** 2 (1988) 156-236 [doi:10.1016/0001-8708(88)90055-2, [spire:275458](https://inspirehep.net/literature/275458)]

Lecture notes on [[affine Lie algebras]]:

* [[Minoru Wakimoto ]], *Lectures on Infinite-Dimensional Lie Algebra*, World Scientific (2001) [[doi:10.1142/4269](https://doi.org/10.1142/4269)]

category: people


Miodrag Iovanov

[[!redirects Iovanov]]
Miodrag C Iovanov is a Romanian mathematician, working at the university of Iowa.

His research interests are in: Noncommutative Algebra and Ring Theory, Hopf algebras and quantum Groups, representation theory.

* [webpage](http://homepage.divms.uiowa.edu/~miovanov/)

* [ArXiv articles](http://arxiv.org/find/all/1/au:+iovanov/0/1/0/all/0/1)

category:people


MIP-star=RE

* Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, Henry Yuen, _MIP${}^\ast$=RE_ ([arxiv/2001.04383](https://arxiv.org/abs/2001.04383))

> We show that the class MIP* of languages that can be decided by a classical verifier interacting with multiple all-powerful quantum provers sharing entanglement is equal to the class RE of recursively enumerable languages.
 Our proof builds upon the quantum low-degree test of (Natarajan and Vidick, FOCS 2018) and the classical low-individual degree test of (Ji, et al., 2020) by integrating recent developments from (Natarajan and Wright, FOCS 2019) and combining them with the recursive compression framework of (Fitzsimons et al., STOC 2019).

> An immediate byproduct of our result is that there is an efficient reduction from the Halting Problem to the problem of deciding whether a two-player nonlocal game has entangled value 1 or at most 1/2. Using a known connection, undecidability of the entangled value implies a negative answer to Tsirelson's problem: we show, by providing an explicit example, that the closure Cqa of the set of quantum tensor product correlations is strictly included in the set Cqc of quantum commuting correlations. Following work of (Fritz, Rev. Math. Phys. 2012) and (Junge et al., J. Math. Phys. 2011) our results provide a refutation of [[Connes' embedding conjecture]] from the theory of von Neumann algebras.

* Henry Yuen, _A tale of Turing machines, quantum-entangled particles, and operator algebras_, [yt](https://www.youtube.com/watch?v=XReR87kFrS4) zoom presentation
* William Slofstra (University of Waterloo), _The algebraic side of MIP* = RE_, [yt](https://www.youtube.com/watch?v=vkrWHo8JYJg) zoom presentation, Simons Institute


Miranda Cheng

* [webpage](https://sites.google.com/site/mcheng0606/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Miranda_Cheng)

## Selected writings

On [[elliptic genera]]:

* {#Cheng13} [[Miranda Cheng]], _(Mock) Modular Forms in String Theory and Moonshine_, lecture notes 2016 ([[ChengModularForms16.pdf:file]])

On [[moonshine]] around the [[duality between heterotic and type II string theory]]:

* {#ChengHarrisonVolpatoZimet16} [[Miranda Cheng]], Sarah M. Harrison, [[Roberto Volpato]], Max Zimet, _K3 String Theory, Lattices and Moonshine_, Research in the Mathematical Sciences 2018 5:32 ([arXiv:1612.04404](https://arxiv.org/abs/1612.04404), [doi:10.1007/s40687-018-0150-4](https://doi.org/10.1007/s40687-018-0150-4))

## Related entries

* [[automorphism of a vertex operator algebra]]

* [[Moonshine]], [[umbral moonshine]]

* [[string theory]] on [[K3-surfaces]]

* [[duality between heterotic and type II string theory]]

* [[elliptic genus]]

category: people

[[!redirects Miranda C. N. Cheng]]


Mirjam Cvetič

[[!redirects Mirjam Cvetic]]

* [webpage](http://www.physics.upenn.edu/people/standing-faculty/mirjam-cvetic)

## Selected writings

On [[intersecting D-brane models]]:

* [[Ralph Blumenhagen]], [[Mirjam Cvetic]], [[Paul Langacker]], [[Gary Shiu]], *Towards Realistic Intersecting D-Brane Models*, Ann. Rev. Nucl. Part. Sci. **55** (2005) 71-139 [[arXiv:hep-th/0502005](http://arxiv.org/abs/hep-th/0502005), [doi:10.1146/annurev.nucl.55.090704.151541](https://doi.org/10.1146/annurev.nucl.55.090704.151541)]

On [[string phenomenology]] with [[F-theory]]:

* {#CLLO18} [[Mirjam Cvetič]], Ling Lin, Muyang Liu, Paul-Konstantin Oehlmann, _An F-theory Realization of the Chiral MSSM with $\mathbb{Z}_2$-Parity_ ([arXiv:1807.01320](https://arxiv.org/abs/1807.01320))

* {#CveticHalversonLinLiuTian19} [[Mirjam Cvetic]], [[James Halverson]], Ling Lin, Muyang Liu, Jiahua Tian, _A Quadrillion Standard Models from F-theory_ ([arXiv:1903.00009](https://arxiv.org/abs/1903.00009))

  (realizations of the [[MSSM]] in [[F-theory]])

On [[near horizon geometries]] of [[black branes]] which [[KK-compactification|KK-compactify]] to [[black holes]]:

* [[Mirjam Cvetič]], Paulo J. Porfírio, Alejandro Satz, _Gaussian null coordinates, near-horizon geometry and conserved charges on the horizon of extremal non-dilatonic black p-branes_ ([arXiv:2003.09304](https://arxiv.org/abs/2003.09304))

Relating [[M-theory on Spin(7)-manifolds]] with [[F-theory on Spin(7)-manifolds]] via [[Higgs bundles]]:

* {#CHRTZ20} [[Mirjam Cvetic]], [[Jonathan Heckman]], Thomas B. Rochais, Ethan Torres, [[Gianluca Zoccarato]], _Geometric Unification of Higgs Bundle Vacua_ ([arXiv:2003.13682](https://arxiv.org/abs/2003.13682))

## Related $n$Lab entries

* [[string phenomenology]]

* [[F-theory]]

* [[intersecting D-brane model]]

* [[MSSM]]

* [[M-theory on G2-manifolds]]

* [[enhanced gauge symmetry]]

* [[T-duality]]

category: people


Mirna Džamonja

* [webpage](https://www.logiqueconsult.eu/)

## Selected publications

* [[Mirna Džamonja]], [[Deborah Kant]], *Interview With a Set Theorist*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019) [[doi:10.1007/978-3-030-15655-8_1](https://doi.org/10.1007/978-3-030-15655-8_1), [pdf](https://hal.archives-ouvertes.fr/hal-01858063/document), [pdf](https://deborahkant.org/wp-content/uploads/2018/07/PhilTalk0716.pdf)]

* [[Mirna Džamonja]], *A New Foundational Crisis in Mathematics, Is It Really Happening?*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019) [[doi:10.1007/978-3-030-15655-8_11](https://doi.org/10.1007/978-3-030-15655-8_11), [arXiv:1802.06221](https://arxiv.org/abs/1802.06221)]

category: people


mirror map

#Contents#
* table of contents
{:toc}

## Idea

[[mirror symmetry|Mirror symmetry]] is in fact a [[duality in string theory|duality]] involving families of varieties. Dependence on the moduli is defined using the mirror map, which can therefore be described using corresponding [[Gauss-Manin connection]] and corresponding  [[Picard-Fuchs equations]].

## References

* S. Hosono, [[Albrecht Klemm]], [[Stefan Theisen]], [[Shing-Tung Yau]], _Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces_, Commun. Math. Phys. __167__ (1995) 301-350 [[doi:10.1007/BF02100589](https://doi.org/10.1007/BF02100589), [arXiv:hep-th/9308122](https://arxiv.org/abs/hep-th/9308122)]

* [[Bong H. Lian]], [[Shing-Tung Yau]], _Arithmetic properties of mirror map and quantum coupling_, Commun. Math. Phys. __176__ (1996) 163--191 [doi](https://doi.org/10.1007/BF02099367)

* B. Lian, S-T. Yau, _Mirror symmetry, rational curves on algebraic manifolds and hypergeometric series_, 163--183 in: XIth International Congress of Mathematical Physics (Paris 1994), Daniel Iagolnitzer (eds.), Intern. Press 1995
* Bong H. Lian, [[Shing-Tung Yau]], _Mirror maps, modular relations and hypergeometric series I_, arXiv:[hep-th/9507151](https://arxiv.org/pdf/hep-th/9507151)

* Bong H. Lian, [[Shing-Tung Yau]], _Mirror maps, modular relations and hypergeometric series II_, arXiv:[hep-th](https://arxiv.org/abs/hep-th/9507153) Nuclear Physics B - Proc. Suppl. __46__:1--3 (1996) 248--262 doi

Number theoretic examples are studied in

* Gert Almkvist, Wadim Zudilin, _Differential equations, mirror maps and zeta values_, arXiv:[math.NT](https://arxiv.org/abs/math/0402386)

A criterium when a mirror map is in fact a elliptic modular function is in

* [[Charles Doran|Charles F. Doran]], _Picard-Fuchs uniformization: modularity of the mirror map and mirror-moonshine_, arxiv:[math.AG/9812162](https://arxiv.org/pdf/math/9812162)

A special case of mirror maps is discussed in video

* Julien Roques, _Hypergeometric mirror maps_, [yt](https://www.youtube.com/watch?v=xT22fZCvF4w)

category: algebraic geometry


mirror symmetry

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

#Contents#
* table of contents
{:toc}

## Idea

To every complex 3-dimensional [[Calabi-Yau variety]] $X$ are associated two similar but differing types of [[sigma-model]] $N=2$-supersymmetric [[2d CFTs]]. There is at least for some CY $X$ a map $X \mapsto \hat X$ which exchanges the [[Hodge numbers]] $h^{1,1}$ and $h^{1,2}$ such that $SCFT_A(X)$ is expected to be equivalent to $SCFT_B(\hat X)$.

$$
  SCFT_A(X) \simeq SCFT_B(\hat X)
  \,.
$$

This is called _mirror symmetry_. At least in some cases this can be understood as a special case of [[T-duality]] ([Strominger-Yau-Zaslow 96](#StromingerYauZaslow96)).

In this form mirror symmetry remains a conjecture, not the least because for the moment there is no complete construction of these SCFTs. But to every such $SCFT(X)$ one can associate two [[TCFT]]s, $A(X)$ and $B(X)$, the [[A-model]] and the [[B-model]]. These $N=1$ supersymmetric field theories were obtained by [[Edward Witten]] using a "topological twist". The topological [[A-model]] can be expressed in terms of [[symplectic geometry]] of a variety and the topological B-model can be expressed in terms of the [[algebraic geometry]] of a variety.

These topological theories are easier to understand and do retain a little bit of the information encoded in the full SCFTs. In terms of these the statement of mirror symmetry says that passing to mirror CYs _exchanges_ the [[A-model]] with the $B$-model and conversely:

$$
  A(X) \simeq B(\hat X),\,\,\,\,\,\,\,B(X)\simeq A(\hat X)
  \,.
$$

By a version of the [[cobordism hypothesis]]-theorem, these [[TCFT]]s (see there) are encoded by [[A-∞ categories]] that are [[Calabi-Yau categories]]: the [[A-model]] by the [[Fukaya category]] $Fuk(X)$ of $X$ which can be understood as a [[stable (∞,1)-category]] representing the Lagrangian intersection theory on the underlying [[symplectic manifold]]; and the [[B-model]] by an [[enhanced triangulated category|enhancement]] of the [[derived category]] of [[coherent sheaves]] $D^b_\infty(\hat X)$ on $\hat X$.

In terms of this data, mirror symmetry is the assertion that these [[A-∞ categories]] are equivalent and simultaneously the same under exchange $X\leftrightarrow \hat{X}$:

$$
  Fuk(X) \simeq D^b_\infty(\hat{X}),
\,\,\,\, and \,\,\,\,
Fuk(\hat{X}) \simeq D^b_\infty(X).
$$

This categorical formulation was introduced by [[Maxim Kontsevich]] in 1994 under the name **homological mirror symmetry**. The equivalence of the categorical expression of mirror symmetry to the SCFT formulation has been proven by [[Maxim Kontsevich]] and independently by [[Kevin Costello]], who showed how the datum of a topological conformal field theory is equivalent to the datum of a [[Calabi-Yau category|Calabi-Yau A-∞-category]](see [[TCFT]]).

The mirror symmetry conjecture roughly claims that every Calabi-Yau 3-fold has a mirror. In fact one considers (mirror symmetry for) degenerating families for Calabi-Yau 3-folds in large volume limit (which may be expressed precisely via the Gromov-Hausdorff metric). The appropriate definition of (an appropriate version of) the [[Fukaya category]] of a symplectic manifold is difficult to achieve in desired generality.
Invariants/tools of Fukaya category include symplectic [[Floer homology]] and Gromov-Witten invariants (building up the [[quantum cohomology]]).

Mirror symmetry is related to the [[T-duality]] on each fiber of an associated Lagrangian fibration [Strominger-Yau-Zaslow 96](#StromingerYauZaslow96).

Although the non-Calabi-Yau case may be of lesser interest to physics, one can still formulate some mirror symmetry statements for, for instance, Fano manifolds. The mirror to a Fano manifold is a [[Landau-Ginzburg model]] (see [Hori-Vafa 00](#HoriVafa00); see also work of Auroux for an explanation via the Strominger-Yau-Zaslow T-duality philosophy). Then the statements are: the [[A-model]] of the Fano (given by the Fukaya category) is equivalent to the B-model of the [[Landau-Ginzburg model]] (given by the category of matrix factorizations); and the B-model of the Fano (given by the derived category of sheaves) is equivalent to the [[A-model]] of the [[Landau-Ginzburg model]] (given by the Fukaya-Seidel category). A few of the relevant names: Kontsevich, Hori-Vafa, Auroux, Katzarkov, Orlov, Seidel, ...

## Properties

### Relation to tropical geometry.

Close relation to [[tropical geometry]], see e.g. [Gross 11](#Gross11).

## Related concepts

* [[mirror map]]

* [[moduli space of Calabi-Yau spaces]]

* [[3d mirror symmetry]]

* [[duality in physics]], [[duality in string theory]]

* [[geometric Langlands duality]]

* [[Landau-Ginzburg model]]

## References

### General

The original statement of the homological mirror symmetry conjecture is in

* [[Maxim Kontsevich]], _Homological algebra of mirror symmetry_, Proc. ICM Zürich 1994, [alg-geom/9411018](http://arxiv.org/abs/alg-geom/9411018)

See also:

* [[Paul Aspinwall]], [[Brian Greene]], [[David Morrison]], _Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory_, Nucl.Phys. B416 (1994) 414-480 ([arXiv:hep-th/9309097](https://arxiv.org/abs/hep-th/9309097))

* [[Philip Candelas]], Xenia C. de la Ossa, Paul S. Green, Linda Parkes, _A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory_, Nuclear Phys. B 359 (1):21--74, 1991

Review:

* {#HoriVafa00} [[Kentaro Hori]], [[Cumrun Vafa]], _Mirror Symmetry_ ([arXiv:hep-th/0002222](https://arxiv.org/abs/hep-th/0002222))

* M. Ballard, _Meet homological mirror symmetry_ ([arxiv:0801.2014](http://arxiv.org/abs/0801.2014))

* A. Port, _An introduction to homological mirror symmetry and the case of elliptic curves_ ([arXiv:1501.00730](http://arxiv.org/abs/1501.00730))

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 10.1.2 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

History:

* [[Maxim Kontsevich]], *On the History of quantum cohomology and homological mirror symmetry* (2021) [video:[YT](https://youtu.be/Ml2x5NnEQ1I)]

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

* {#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))

Other reviews include

* [[Paul Aspinwall]], [[Tom Bridgeland]], [[Alastair Craw]], [[Michael Douglas]], Mark Gross, [[Anton Kapustin]], [[Gregory Moore]], [[Graeme Segal]], [[Balázs Szendrői]], P. Wilson,

  _Dirichlet branes and mirror symmetry_,

  Clay Mathematics Monograph Volume 4, Amer. Math. Soc. Clay Math. Institute 2009

  ([pdf](http://www.claymath.org/library/monographs/cmim04c.pdf), [pdf](http://www2.maths.ox.ac.uk/cmi/library/monographs/cmim04.pdf))  (very readable!)

The relation to [[T-duality]] was established in

* {#StromingerYauZaslow96} [[Andrew Strominger]], [[Shing-Tung Yau]], [[Eric Zaslow]], _Mirror Symmetry is T-Duality_, Nucl.Phys.B479:243-259,1996 (DOI 10.1016/0550-3213(96)00434-8) [hep-th/9606040](http://arxiv.org/abs/hep-th/9606040)

Further references include

* [[Cumrun Vafa]], [[Shing-Tung Yau]] (eds.), _Winter school on mirror symmetry, vector bundles, and Lagrangian submanifolds_, Harvard 1999, AMS, Intern. Press (includes A. Strominger, S-T. Yau, E. Zaslow, _Mirror symmetry is $T$-duality_  as pages  333--347; ).

* K. Hori, S. Katz, A. Klemm et al. _Mirror symmetry I_, AMS, Clay Math. Institute 2003.

* Paul Seidel, _Fukaya categories and Picard-Lefschetz theory_, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich, 2008. viii+326 pp

* Mark Gross, Bernd Siebert, _Mirror symmetry via logarithmic degeneration data I_, [math.AG/0309070](http://arxiv.org/abs/math/0309070), _From real affine geometry to complex geometry_, [math.AG/0703822](http://arxiv.org/abs/math/0703822), _Mirror symmetry via logarithmic degeneration data II_, [arxiv/0709.2290](http://arxiv.org/abs/0709.2290)

* [[Anton Kapustin]], [[Dmitri Orlov]], _Lectures on mirror symmetry, derived categories, and D-branes_,  Uspehi Mat. Nauk __59__  (2004),  no. 5(359), 101--134;  translation in  Russian Math. Surveys __59__  (2004), no. 5, 907--940, [math.AG/0308173](http://arxiv.org/abs/math/0308173)

* [[Maxim Kontsevich]], [[Yan Soibelman]], _Homological mirror symmetry and torus fibrations_, [math.SG/0011041](http://arxiv.org/abs/math/0011041)

* Yong-Geun Oh, Kenji Fukaya, _Floer homology in symplectic geometry and mirror symmetry_, Proc. ICM 2006, [pdf](http://www.math.wisc.edu/~oh/Oh-icm2006.pdf)

* wikipedia: [mirror symmetry (string theory)](http://en.wikipedia.org/wiki/Mirror_symmetry_%28string_theory%29), [homological mirror symmetry](http://en.wikipedia.org/wiki/Homological_mirror_symmetry)

* partial notes from Miami 08 workshop: [miami08-notes](http://www-math.mit.edu/~auroux/frg/miami08-notes) and abstracts from [miami09](http://www-math.mit.edu/~auroux/frg/miami09-abstracts.html), [miami10](http://www-math.mit.edu/~auroux/frg/miami10-abstracts.html)

* {#Gross11} [[Mark Gross]], _Tropical geometry and mirror symmetry_, CBMS regional conf. ser. 114 (2011), based on the CBMS course in Kansas, [AMS book page] (http://www.ams.org/bookstore-getitem/item=CBMS-114), [pdf](http://www.math.ucsd.edu/~mgross/kansas.pdf)

Discussion in the context of [[derived Morita equivalence]] includes

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

### Complete proofs {#CompleteProofs}

Here is a list with references that give complete proofs of *homological* mirror symmetry on certain (types of) spaces.

* M. Abouzaid, I. Smith, _Homological mirror symmetry for the four-torus_, Duke Math. J. 152 (2010), 373–440, [arXiv:0903.3065](http://arxiv.org/abs/0903.3065)

* A. Polishchuk and E. Zaslow, _Categorical mirror symmetry: the elliptic curve_, Adv. Theor. Math. Phys. 2:443470, 1998.

* V. Golyshev, V. Lunts, D. Orlov, _Mirror symmetry for abelian varieties_, J. Alg. Geom. __10__ (2001), no. 3, 433--496, [math.AG/9812003](http://arxiv.org/abs/math/9812003)

* P. Seidel, _Homological mirror symmetry for the quartic surface_, [arXiv:0310414](http://arxiv.org/abs/math/0310414)

* Alexander I. Efimov, _Homological mirror symmetry for curves of higher genus_, Inventiones Math. __166__ (2006), 537–582, [arXiv:0907.3903](http://arxiv.org/abs/0907.3903)

* [[D. Auroux]], [[L. Katzarkov]], [[D. Orlov]], _Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves_, []();  _Mirror symmetry for weighted projective planes and their noncommutative deformations_, Ann. Math. __167__ (2008), 867–943, [math.AG/0404281](http://arxiv.org/abs/math/0404281)

* [[Mohammed Abouzaid]], Denis Auroux, Alexander I. Efimov, Ludmil Katzarkov, Dmitri Orlov, _Homological mirror symmetry for punctured spheres_, [arxiv/1103.4322](http://arxiv.org/abs/1103.4322)

* [[Paul Seidel]], _Homological mirror symmetry for the genus two curve_, J. Algebraic Geometry, to appear, [arXiv:0812.1171](http://arxiv.org/abs/0812.1171)

### Computation via topological recursion

Computation via [[topological recursion]] in [[matrix models]] and all-[[genus of  a surface|genus]] proofs of mirror symmetry is due to

* {#BouchardKlemmMarinoPasquetti09} [[Vincent Bouchard]], [[Albrecht Klemm]], [[Marcos Marino]], [[Sara Pasquetti]], _Remodeling the B-model_, Commun.Math.Phys.287:117-178, 2009 ([arXiv:0709.1453](https://arxiv.org/abs/0709.1453))

* [[Bertrand Eynard]], [[Amir-Kian Kashani-Poor]], Olivier Marchal, _A matrix model for the topological string I: Deriving the matrix model_ ([arXiv:1003.1737](https://arxiv.org/abs/1003.1737))

* [[Bertrand Eynard]], [[Amir-Kian Kashani-Poor]], Olivier Marchal, _A matrix model for the topological string II: The spectral curve and mirror geometry_ ([arXiv:1007.2194](https://arxiv.org/abs/1007.2194))

* {#EynardOrantin12} [[Bertrand Eynard]], [[Nicolas Orantin]], _Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture_ ([arXiv:1205.1103](https://arxiv.org/abs/1205.1103))

* {#FangLiuZong13} Bohan Fang, Chiu-Chu Melissa Liu, Zhengyu Zong, _All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds_ ([arXiv:1310.4818](https://arxiv.org/abs/1310.4818))

[[!redirects homological mirror symmetry]]


Misha Gavrilovich

* [webpage](http://mishap.sdf.org/)

## Selected writings

On [[separation axioms in terms of lifting properties]]:

* {#Gavrolovich14} [[Misha Gavrilovich]], *Point set topology as diagram chasing computations*,  The De Morgan Gazette. 2014. Vol. 5. No. 4. P. 23-32.  ([arXiv:1408.6710](https://arxiv.org/abs/1408.6710), [pdf](http://mishap.sdf.org/mints-lifting-property-as-negation-DMG_5_no_4_2014.pdf))

* [[Misha Gavrilovich]], *The unreasonable power of the lifting property in elementary mathematics*, 2017 ([arXiv:1707.06615](https://arxiv.org/abs/1707.06615),  [pdf](http://mishap.sdf.org/by:gavrilovich/expressive_power_of_the_lifting_property.pdf))

category: people


Misha Verbitsky

* [webpage](https://en.wikipedia.org/wiki/Misha_Verbitsky)

## Selected writings

On [[symplectic orbifolds]]:

* [[Misha Verbitsky]], _Holomorphic symplectic geometry and orbifold singularities_, Asian J. Math. 4 (2000), no. 3, 553-563 ([arXiv:math/9903175](https://arxiv.org/abs/math/9903175))

On [[hyperkähler manifolds]]:

* {#Verbitsky} [[Misha Verbitsky]], _Hyperkähler manifolds with torsion, supersymmetry and Hodge theory_, Asian J. Math, V. 6 No. 4, pp. 679-712, Dec. 2002.

## Related $n$Lab entries

* [[quaternion-Kähler manifold]]

category: people


Mishchenko-Fomenko index theorem

+-- {: .rightHandSide}
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###Context###
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
#### Operator algebra
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Mishchenko-Fomenko index theorem_ is a generalization of the [[Atiyah-Singer index theorem]] to [[differential operators]] on [[Hilbert module]]-bundles over some [[C*-algebra]].

## References

The original English version of the original work is

* A.S. Mishchenko, A.T. Fomenko, _The index of elliptic operators over C*-algebras., Math. USSR, Izv. 15 (1980), 87–112.

A more explicit rederivation is in section 6.1 of

* [[Thomas Schick]], _$L^2$-index, KK-theory, and connections_, New York J. Math. 11 (2005) ([arXiv:math/0306171](http://arxiv.org/abs/math/0306171))


Mitchell Buckley

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=205791)

* [MO page](https://mathoverflow.net/users/75189/mitchell-buckley)

## Selected writings

On [[fibrations of 2-categories]]:

* {#Buckley} [[Mitchell Buckley]], *Fibred 2-categories and bicategories*, Journal of Pure and Applied Algebra **218** 6 (2014) 1034-1074 [[arXiv:1212.6283](https://arxiv.org/abs/1212.6283), [doi:10.1016/j.jpaa.2013.11.002](https://doi.org/10.1016/j.jpaa.2013.11.002)]

* Mitchell Buckley, [[Timmy Fieremans]], [[Christina Vasilakopoulou]], [[Joost Vercruysse]], _A Larson-Sweedler theorem for Hopf V-categories_, [arXiv:1908.02049](https://arxiv.org/abs/1908.02049)

category: people


Mitchell Faulk

* [personal page](https://mitchellfaulk.github.io/)

## Selected writings

On [[fully dualizable object|fully]] [[dualizable objects]]:

* [[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)]

category: people