mass gap

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## Idea

The _mass gap problem_ is an open conceptual problem in the [[quantization of Yang-Mills theory]], closely related to what in the [[phenomenology]] of [[quantum chromodynamics]] is called the _[[confinement]]_ of [[quarks]], hence the existence of ordinary [[hadron|hadronic]], in particular [[baryon|baryonic]] [[matter]].

The [[Lagrangian]] for [[Yang-Mills theory]] coupled to [[fermion fields]] (such as for [[QCD]]) makes manifest the existence of [[mass]]-less [[quarks]] and [[gluons]]. Indeed, at very high [[temperature]] [[QCD]] is thought to exhibit a [[quark-gluon plasma]] well described by these degrees of freedoms.

But at comparatively smaller [[temperature]] it is observed in [[experiment]] as well as in [[lattice QCD]] computer experiment that QCD exhibits _[[confinement]]_, meaning that the low energy states of the theory are not free massless quarks and gluons anymore, but that these form [[bound states]] in the form of [[hadrons]] (including, notably, [[protons]] and [[neutrons]], and hence all the ordinary [[baryon|baryonic]] [[matter]] of the [[observable universe]]).

The existence of massive [[hadron]]-[[bound states]] in low-energy [[QCD]] is thus well-established [[phenomenology|phenomenologically]], but it is as yet lacking a conceptual theoretical explanation.

The _mass gap problem_ is the problem in [[mathematical physics]] to demonstrate theoretically (i.e. not just by [[lattice QCD|computer simulation]]) the existence of this mass gap/[[confinement]]-phenomenon in [[QCD]] and in [[Yang-Mills theory]] coupled to [[fermion fields]] in general.

This issue is well and widely known in the [[particle physics]]-community, see for instance [Kutschke 96, Section 3.1](#Kutschke96), [INFN 15](#INFN15). It gained more attention among the [[mathematics]]/[[mathematical physics]]-community when the [Clay Mathematics Institute](http://www.claymath.org/) declared the problem to be one in a list of "[Millennium Problems](http://www.claymath.org/millennium-problems)", see [Jaffe-Witten 2000](#JaffeWitten00), [Douglas 04](#Douglas04).

## Related pages

* [[confinement]]

* [[mass term]]

* [[gapped Hamiltonian]]

## References

### The mass gap problem
 {#ReferencesMassGapProblem}

A survey and problem description in [[mathematics]]/[[mathematical physics]]:

* [Clay Mathematics Institute](http://www.claymath.org/), _[Yang-Mills & The Mass Gap](https://www.claymath.org/millennium/yang-mills-the-maths-gap)_

* {#JaffeWitten00} [[Arthur Jaffe]], [[Edward Witten]], _Quantum Yang-Mills theory_ (2000) [[pdf](https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf), [pdf](https://www.arthurjaffe.com/Assets/pdf/QuantumYangMillsWebRevised.pdf), [[JaffeWitten-QuantumYangMills.pdf:file]]]

* {#Atiyah2000} [[Michael Atiyah]], *The millennium problems II*, talk at *[CMI Millennium Event at Collège de France](https://www.msri.org/realvideo/ln/hosted/cmi/2000/cmiparis/index-tate.html)* (24-25 May 2000) [[web](https://www.claymath.org/lectures/the-millennium-prize-problems-ii/), [YT](https://youtu.be/trxjCOmg8GY), discussion of Yang-Mills&MassGap starting at: [29:40](https://youtu.be/trxjCOmg8GY?t=1764)]

> [40:20](https://youtu.be/trxjCOmg8GY?t=2420) What is the future of this problem [Yang-Mills & mass-gap] in physics, what could happen?

> [40:24](https://youtu.be/trxjCOmg8GY?t=2424) This is now my personal guess or, if you like, hints [or] suggestions.

> [40:30](https://youtu.be/trxjCOmg8GY?t=2430) You can try a direct analytical assault. If you are a young man and brave and you have a long way into thre future, you can just get out your hammer and chisel and crack away.

> [40:47](https://youtu.be/trxjCOmg8GY?t=2447) Or you might think that you want something different.

> [40:50](https://youtu.be/trxjCOmg8GY?t=2450) You might find to get a better understanding of the formal structure of [[quantum field theory]] of this Yang-Mills type.

> [40:58](https://youtu.be/trxjCOmg8GY?t=2458) These Yang-Mills theories have very elaborate mathematical formal structure, which incorporate very mysterious [[duality in physics|dualities]] of various kinds and also a thing called *[[supersymmetry]]*, and many of the applications in mathematics incorporate all these.

> [41:51](https://youtu.be/trxjCOmg8GY?t=2475) So here there is a very elaborate structure, and it is just possible that if we get a better understanding of that structure then that might have given us a better way of trying to lay the foundations.

> [41:25](https://youtu.be/trxjCOmg8GY?t=2485) For example, a problem that can be described in two different ways, by duality, two dual pictures might look quite different, one of them might be practical and the other one might be intractable.

> [41:35](https://youtu.be/trxjCOmg8GY?t=2495) They are certainly not at all unreasonable to think this might be a way starting to understand better these formal operators[?], the formal operators here is very sophisticated.

> [41:47](https://youtu.be/trxjCOmg8GY?t=2507) You could of course sit back and wait, await development in string theory.

> [41:54](https://youtu.be/trxjCOmg8GY?t=2514) String theory and the latest things that follow string theory, called *[[M-theory]]* and every other new theory, develop [at] an enormous rate.

> [42:00](https://youtu.be/trxjCOmg8GY?t=2520) The theoretical physics community here is tremendously active, there are very beautiful things happening every week, new results come out, many of them will have mathematical overtones.

> [42:10](https://youtu.be/trxjCOmg8GY?t=2530) And it could be that if we wait a few years there'll be such a totally new  picture emerging from string theory that we'll get a better idea how to go about attacking the problem, so that's not an unreasonable expectation.

> [42:21](https://youtu.be/trxjCOmg8GY?t=2541) You see, trying to lay the foundations for physics is like an architect trying to lay the foundations for a building that's rapidly going up.

> [42:30](https://youtu.be/trxjCOmg8GY?t=2550) I mean, which do you do first, it's not quite clear: You want to see the design or you want to lay the foundations. Maybe you want the design first.

> [42:39](https://youtu.be/trxjCOmg8GY?t=2559) Then, of course, you might also sit back and say: Well, quantum electrodynamics was very hard, because we didn't incorporate extra things like proton and neutrons, we only worried about electrons. Maybe even that isn't enough perhaps we should have incorporated gravity, then we'll have all the forces under control and perhaps at that stage the fundamental theory will be even easier.

> [43:00](https://youtu.be/trxjCOmg8GY?t=2580) People in string theory aim here and this is the ultimate theory, and perhaps the ultimate theory will be easier than the transitional theories. But who knows.

> [43:09](https://youtu.be/trxjCOmg8GY?t=2589) The big challenge [for] mathematics and the physicists of the 21st century is to really make progress along this program by whatever method they can. Good luck to you.

Early report on the progress (essentially none):

* {#Douglas04} [[Michael Douglas]], _Report on the Status of the Yang-Mills Millennium Prize Problem_ (2004) [[[Douglas-StatusOfYMMillenniumProb.pdf:file]]]

Further comments:

* [[Ludvig Faddeev]], _Mass in Quantum Yang-Mills Theory_, Bull Braz Math Soc, New Series 33(2), 201-212 ([arXiv:0911.1013](https://arxiv.org/abs/0911.1013), [pdf](http://emis.impa.br/EMIS/journals/em/docs/boletim/vol332/v33-2-a4-2002.pdf))

Notes reviewing more technical details of the problem:

* Jay Yablon, _The Origins of QCD Confinement in Yang-Mills Gauge Theory_, January 2008 ([pdf](http://jayryablon.files.wordpress.com/2008/01/qcd-confinement-handout-10.pdf))

Problem description in terms of [[probability theory|probabilistic]] [[lattice gauge theory]]:

* [[Sourav Chatterjee]], _Yang-Mills for probabilists_, in: _Probability and Analysis in Interacting Physical Systems_, PROMS **283** (2019) Springer ([arXiv:1803.01950](https://arxiv.org/abs/1803.01950), [doi:10.1007/978-3-030-15338-0](https://link.springer.com/book/10.1007/978-3-030-15338-0))

See also

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

* Wikipedia, _[Yang-Mills existence and the mass gap](https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_existence_and_mass_gap)_

### In the nuclear physics literature
{#ExplicitMentioningInPhysicsLiterature}

In the [[nuclear physics]]/[[QCD]]-literature the mass gap problem is (and has long been) known as the _[confinement problem](confinement#OpenProblemConfinementReferences)_.
Explicit mentioning of the CMI's "mass gap" Millennium Problem in [[nuclear physics]]/[[phenomenology|phenomenological]] discussion of [[confinement]] includes the following:

* [[Igor Klebanov]], Oyvind Tafjord, "Can we quantitatively understand quark and gluon confinement in Quantum Chromodynamics and the existence of a mass gap?", 10th of *[10 Physics Problems for the Next Millennium](http://theory.caltech.edu/~preskill/millennium.html)* selected at [Strings 2000](https://inspirehep.net/conferences/972466)

* [[Yakov Shnir]], Section 9.1. of: *Magnetic Monopoles*, Springer 2005 ([ISBN:978-3-540-29082-7](https://www.springer.com/gp/book/9783540252771))

* [[Koji Hashimoto]], pp. 126 of: _D-Brane -- Superstrings and New Perspective of Our World_, Springer (2012) [[doi:10.1007/978-3-642-23574-0](https://link.springer.com/book/10.1007%2F978-3-642-23574-0), [spire:1188897](http://inspirehep.net/record/1188897)]

* [Barnaföldi & Gogokhia (2019), p. 2](#BarnafoldiGogokhia19)

* [Roberts & Schmidt (2020), p. 2](#RobertsSchmidt20)

* [Roberts (2021), footn. 2](#Roberts21)

* [Roberts (2022), p. 2](#Roberts22)

[[!include open problem of confinement -- references]]

### Approaches

#### Computer lattice QFT

Of course various partial approaches exist, notably computer-[[experiment]] in [[lattice QCD]]. (Such computer-checks of the mass-gap problem are analogous to computer checks of the [[Riemann hypothesis]], see [there](Riemann+hypothesis#ReferencesComputerChecks))
High-accuracy computation of [[hadron]]-[[masses]] in [[lattice QCD]]-simulations are reported here:

* S. Durr, Z. Fodor, J. Frison, C. Hoelbling, R. Hoffmann, S.D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, K.K. Szabo, G. Vulvert,

  _Ab-initio Determination of Light Hadron Masses_,

  Science 322:1224-1227,2008 ([arXiv:0906.3599](https://arxiv.org/abs/0906.3599))

* Zoltan Fodor, Christian Hoelbling, section V of _Light Hadron Masses from Lattice QCD_, Rev. Mod. Phys. 84, 449, ([arXiv:1203.4789](https://arxiv.org/abs/1203.4789))

* S. Aoki et. al. _Review of lattice results concerning low-energy particle physics_ ([arXiv:1607.00299](https://arxiv.org/abs/1607.00299))

\linebreak

* Yonggoo Heo, C. Kobdaj, M.F.M. Lutz, _Constraints from a large-$N_c$ analysis on meson-baryon interactions at chiral order $Q^3$_, Phys. Rev. D 100, 094035 (2019) ([arXiv:1908.11816](https://arxiv.org/abs/1908.11816))

> Still after many decades of vigorous studies the outstanding challenge of modern physics is to establish a rigorous link of QCD to low-energy hadron physics as it is observed in the many experimental cross section measurements.

#### Rigorous lattice QFT

But there is also an approach via rigorous analytic [[lattice gauge theory]]:

* {#Chatterjee20} [[Sourav Chatterjee]], _A probabilistic mechanism for quark confinement_, Comm. Math. Phys. 2021 ([arXiv:2006.16229](https://arxiv.org/abs/2006.16229))

(...)

### In solid state physics

Discussion in [[solid state physics]]:

* Alessandra N. Braga, Wagner P. Pires, Jeferson Danilo L. Silva, Danilo T. Alves, Van Sérgio Alves, *Renormalization of the band gap in 2D materials near an interface between two dielectrics* [[arXiv:2306.04759](https://arxiv.org/abs/2306.04759)]

[[!redirects mass gaps]]

[[!redirects mass gap problem]]
[[!redirects mass gap problems]]


mass term

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## Idea

In [[physics]], a *mass term* is, quite generally, the [[sum|summand]] in a [[Lagrangian density]], [[Hamiltonian]] or [[equation of motion]] which expresses the [[mass]] of a given (quasi-)[[particle]] or [[field (physics)|field]].

Often the terminology is used specifically for masses of [[spinors]]/[[Dirac fields]], since here a mass term is subject to subtle constraints which may or may not exclude the mere existence of admissible mass terms.

Schematically, if a massless relativistic [[Dirac equation]] (here in [[Minkowski spacetime]]) is written in the form

$$
  \partial_0 \psi
  \;=\;
  k\!\!\!/ \cdot \psi
  \,,
$$

where $k\!\!\!/$ are the [[Feynman slash|Clifford momentum]]-[[linear operator|operator]], then for a mass term addition

$$
  \partial_0 \psi
  \;=\;
  k\!\!\!/ \cdot \psi
  \;+\;
  m \gamma_0
  \,,
$$

to lead to the required [[special relativity|relativistic]] [[dispersion relation]]

$$
  E \,=\, \sqrt{k^2 + m^2}
$$

one needs that the new [[Clifford algebra|Clifford]]-generator $\gamma_0$ exists as an operator on the given [[Hilbert space]] of [[spinor]] $\psi$ such that it skew-commutes with all the Clifford momenta

$$
  \gamma_0 \cdot k\!\!\!/
  \;+\;
  k\!\!\!/\cdot \gamma_0
  \;=\;
  0
  \,.
$$

This is the kind of condition that also appears in [[Karoubi K-theory|Karoubi]]'s formlation of [[topological K-theory]]-classes as [[equivalence classes]] of  [[Clifford modules]] (see [Freed & Hopkins 2021, Thm. 9.63](#FreedHopkins21)).

In this guise, mass terms for [[electron]]-excitations in [[semi-metals]] play a role in the [[K-theory classification of topological phases of matter]] (see [below](#ExamplesInTopologicalPhasesOfMatter)).

## Examples

### In particle physics

In [[particle physics]] the (non-)existence of mass terms for [[fermions|fermionic]] [[fundamental particles]] goes along with their characterization as [[Dirac spinors]], [[Weyl spinors]] and [[Majorana spinors]].

This plays a central role notably in discussion of the (still hypothetical) detailed nature of [[neutrinos]].

### In topological phases of matter
 {#ExamplesInTopologicalPhasesOfMatter}

In [[solid state physics]] of [[crystals]], [[electrons]] are typically well-approximated by the non-relativistic Dirac equation, but around nodal points in the [[Brillouin torus]], where [[electronic band structure|electron bands]] cross, the effective [[dispersion relation]] is again of form of a relativistic Dirac- or Weyl-equation (whence one speaks of *Dirac points* or *Weyl points*).

\begin{imagefromfile}
    "file_name": "MassTermsForSemiMetalNodalPoints-220517.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "(graphics from [SS22](#SS22))"
\end{imagefromfile}

This phenomenon appears in particular in [[topological semi-metals]], in which case the existence of a mass term, whose addition to the [[dispersion relation]] will remove the band crossing by "opening an energy gap", is thought to signal the decay of the [[semi-metal]]-phase into a [[topological insulator]]-phase.

See for instance the example of the *[[Haldane model]]*.

## Related concepts

* [[mass]]

* [[mass gap]]

* [[Karoubi K-theory]]

* [[K-theory classification of topological phases of matter]]

## References

A careful discussion of relativistic mass terms is in:

* {#FreedHopkins21} [[Daniel S. Freed]], [[Michael J. Hopkins]], *Reflection positivity and invertible topological phases*, Geom. Topol. **25** (2021) 1165-1330 $[$[arXiv:1604.06527](https://arxiv.org/abs/1604.06527), [doi:10.2140/gt.2021.25.1165](https://doi.org/10.2140/gt.2021.25.1165)$]$

Discussion aimed at the description of [[topological semi-metal]]-phases in [[solid state physics]] includes:

* {#MorimotoFurusaki13} Takahiro Morimoto and Akira Furusaki, Sec. V of: *Topological classification with additional symmetries from Clifford algebras*, Phys. Rev. B **88** (2013) 125129 ([arXiv:1306.2505](https://arxiv.org/abs/1306.2505),  [doi:10.1103/PhysRevB.88.125129](https://doi.org/10.1103/PhysRevB.88.125129))

* {#ChiuSchnyder14} [[Ching-Kai Chiu]], [[Andreas P. Schnyder]], Section A.2 of: *Classification of reflection-symmetry-protected topological semimetals and nodal superconductors*, Phys. Rev. B **90** 205136 (2014) $[$[doi:10.1103/PhysRevB.90.205136](https://doi.org/10.1103/PhysRevB.90.205136)$]$

* {#CTSR15} [[Ching-Kai Chiu]], [[Jeffrey C.Y. Teo]], [[Andreas P. Schnyder]], [[Shinsei Ryu]], Section III.C of *Classification of topological quantum matter with symmetries*, Rev. Mod. Phys. **88** (2016) 035005 ([arXiv:1505.03535](https://arxiv.org/abs/1505.03535), [doi:10.1103/RevModPhys.88.035005](https://doi.org/10.1103/RevModPhys.88.035005))

See also discussion in the context of [[differential K-theory|differential]] [[KO-theory]]:

* [[Kiyonori Gomi]], [[Mayuko Yamashita]], *Differential KO-theory via gradations and mass terms* [[arXiv:2111.01377](https://arxiv.org/abs/2111.01377)]

Some of the above material is taken from:

* {#SS22} [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Topological Quantum Computation in TED-K]]*

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Massey product

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## Idea

The _Massey product_ of length $n$ is a certain $n$-ary products on the [[cohomology ring]] of an [[A-infinity algebra]] (in particular a [[dg-algebra]]).

### In components

Roughly, Massey Products are to [[cohomology]] as [[Toda Brackets]] are to [[homotopy]].

Somewhat more fully, while Toda brackets are relations between mapping space groups $Map_* (\Sigma^n A_0, A_{n+2}) $ and chains of maps $ A_0 \to \cdots \to A_{n+2} $, and generalizing nullhomotopy of composition, Massey products are a relation between cohomology groups
$ H^{p_0 + \cdots + p_k - k + 1}(X) $ and $ H^{p_0} (X) \otimes \cdots \otimes H^{p_k}(X) $, generalizing the vanishing of pairwise [[cup product]]s.

The case $k=2$ is straight-forward enough: given three homogeneous classes $ [u],[v],[w] $ such that $ [u]\smile[v] = [v]\smile[w] = 0$, there are (various) choices of cochains $ s , t $ with $ d s = u \cdot v $ and $ d t = v \cdot w $.  The Massey triple product is the set of sums $ [ u \cdot t \pm s \cdot w ] $,
where the sign is chosen for cocyclicity.

## Properties

### Relation to Steenrod squares
 {#RelationToSteenrodSquares}

Let $\omega, \omega_1, \omega_2 \in H^\bullet(X,\mathbb{Z}/2)$ such that their triple Massey product exists.
Then the [[cup product]] of $\omega$ with the triple Massey product is independent of the ambiguity in the Massey products and equals the cup product of $\omega_1$ with $\omega_2$ and with the [[Steenrod square]] of $\omega$ of degree $deg(\omega)-1$:

$$
  \omega \cup
  \left\langle
    \omega_1, \omega, \omega_2
  \right\rangle
  \;=\;
  \omega_1 \cup \omega_2 \cup Sq^{ deg(\omega) -1 }( \omega )
  \,.
$$

([Taylor 11, slide 10](#Taylor11), following [Milgram 68](#Milgram68))

### Relation to $A_\infty$-algebra

For $A$ a [[dg-algebra]], its [[chain homology]] $H_\bullet(A)$ inherits an [[A-infinity algebra]] structure by [[Kadeishvili's theorem]]. Then for every $n \in \mathbb{N}$ the $n$-ary $A_\infty$-product on elements $ (a_1, \cdots, a_n) \in H_\bullet(A)^n$ is given, up to a sign, by the Massey product $\langle a_1, \cdots, a_n\rangle$.

For $n = 3$ this is due to ([Stasheff](#Stasheff)). For general $n$  this appears as ([LPWZ, theorem 3.1](#LPWZ)).

## Related concepts

* [[cup product]]

* [[cohomology operation]]

## References
 {#References}

### General

* {#Kraines66} [[David Kraines]], _Massey higher products_,  Transactions of the American Mathematical Society Vol. 124, No. 3 (Sep., 1966), pp. 431-449 ([jstor](http://www.jstor.org/stable/1994385 ))

* Edward J. O'Neill, _On Massey products_,  Pacific J. Math. Volume 76, Number 1 (1978), 123-127. ([EUCLID](http://projecteuclid.org/euclid.pjm/1102807031))

* {#Kochmann96} [[Stanley Kochmann]], section 5.4 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Milgram68} [[R. James Milgram]], _Steenrod squares and higher Massey products_, Bol. Soc. Mat. Mexicana (2) 13 (1968), 32–57.MR0263074 ([web](https://www.researchgate.net/publication/268753745_Steenrod_squares_and_higher_Massey_products))

* {#Taylor11} [[Laurence Taylor]], _Massey Triple Products_, Princeton 2011 ([pdf](https://www3.nd.edu/~taylor/talks/2011-03-22-Princeton.pdf))

See also

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

### Relation to $A_\infty$-algebra
 {#ReferencesRelationToAInfinity}

The relation of Massey products to [[A-infinity algebra]] structures is in Chapter 12 of

* {#Stasheff} [[Jim Stasheff]], _H-spaces from a homotopy point of view_

for $n = 3$, and for general $n$ in Theorem 3.1 and Corollary A.5 of

* {#LPWZ} D.-M. Lu, J. H. Palmieri, Q.-S. Wu, J. J. Zhang, _$A_\infty$-structures in Ext algebras_, J. Pure Appl. Alg. 213 (2009), 2017--2037 (Theorem 3.1 and Corollary A.5) ([arXiv:math/0606144](http://arxiv.org/abs/math/0606144))

as well as from item 1.4 on in

* [[Bruno Vallette]], _Algebra+Homotopy=Operad_ ([arXiv:1202.3245](http://arxiv.org/abs/1202.3245))

and sections 9.4.10 to 9.4.12 of

* {#ValetteLoday} [[Bruno Vallette]], [[Jean-Louis Loday]], _Algebraic Operads_ ([pdf](http://math.unice.fr/~brunov/Operads.pdf))

Notice that the definition of Massey product on top of p.282 of [Vallette-Loday](#ValetteLoday), $\langle x,y,z\rangle$ depends on choices of $a,b$ which don't appear in the notation. Then lemma 9.4.11 talks about a particular choice of $a,b$ which is made in the body of the proof. The actual statement of the lemma only can be deduced after reading the proof. It then says that for these particular choices of a,b the said equality holds. (See [this MO discussion](http://mathoverflow.net/questions/92315/massey-products-vs-a-infty-structures)).

### In ordinary differential cohomology
 {#ReferencesInOrdinaryDifferentialCohomology}

Massey products in [[ordinary differential cohomology]]/[[Deligne cohomology]] are discussed in

* Wenger, _Massey products in Deligne cohomology_.

* C. Deninger, _Higher order operations in Deligne cohomology_, Inventiones
Math. 122 N1 (1995).

* Alexander Schwarzhaupt, _Massey products in Deligne-Beilinson cohomology_ ([web](http://www.researchgate.net/publication/29799379_Massey_products_in_Deligne-Beilinson_cohomology), [[SchwarzhauptMasseyDeligne.pdf:file]]).

* [[Daniel Grady]], [[Hisham Sati]], _Massey products in differential cohomology via stacks_, J. Homotopy Relat. Struct. 13 (2017) 169-223 ([arXiv:1510.06366](http://arxiv.org/abs/1510.06366), [doi:10.1007/s40062-017-0178-y](https://doi.org/10.1007/s40062-017-0178-y)).

[[!redirects Massey products]]


Massimo Porrati

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

## Selected writings

On the [[Schwinger effect]] seen in [[open string|open]] [[string theory]]:

* [[Constantin Bachas]], [[Massimo Porrati]], _Pair Creation of Open Strings in an Electric Field_, Phys. Lett. B296 (1992) 77-84 ([arXiv:hep-th/9209032](https://arxiv.org/abs/hep-th/9209032))

On adjustments of the [[action functional]] for [[spin]] [[mass|massive]] $3/2$-particles ([[Rarita-Schwinger field]]) by non-[[minimal coupling]] that evades the [[Velo-Zwanziger problem]]:

* [[Massimo Porrati]], Rakibur Rahman, _Causal Propagation of a Charged Spin 3/2 Field in an External Electromagnetic Background_,  	Phys. Rev. D80:025009, 2009 ([arXiv:0906.1432](http://arxiv.org/abs/0906.1432))

Discussion of the [[Velo-Zwanziger problem]] for the higher [[string]] excitations that appear in [[string theory]]:

* [[Massimo Porrati]], Rakibur Rahman, [[Augusto Sagnotti]], _String Theory and The Velo-Zwanziger Problem_, Nucl. Phys. B846:250-282, 2011 ([arXiv:1011.6411](http://arxiv.org/abs/1011.6411))

category: people


massive type IIA string theory

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## Idea

What is called _massive type IIA string theory_ is a deformation of [[type IIA string theory]] which contains the [[RR-field]] [[flux forms]] $F_0$ and/or [[Hodge duality|Hodge dually]] $F_{10}$, which couple to [[D8-branes]].

This is the [[UV-completion]] of [[massive type IIA supergravity]] in analogy to how plain [[type IIA string theory]] is the [[UV-completion]] of plain [[type IIA supergravity]].

## Properties

### Black D8-branes

It is (only) massive type IIA string theory where [[D8-branes]] exist as actual [[black branes]] ([BdRGPT96 96](#BergshoessdeRooGreenPapadopoulosTownsend96), [Chamblin-Perry 97](#ChamblinPerry97), [Janssen-Meessen-Ortin 99](#JanssenMeessenOrtin99))
Similarly for [[D6-D8-brane bound states]] ([Singh 02a](#Singh02a), [Singh 02b](#Singh02b)).

This is because the [[normal bundle|normal]] [[n-sphere]] around a [[D8-brane]] (with its 9-dimensional [[worldvolume]]) in 10-dimensional [[spacetime]] is a [[0-sphere]], [[D-brane charge|D8-brane charge]] is measured by the [[RR-field]] 0-[[flux form]] $F_0$, or else by its [[Hodge duality|Hodge dual]] 10-form $F_{10}$. These behave like a [[cosmological constant]] in the corresponding [[D=10 supergravity]] ("[[Romans supergravity]]") and causes the mass term for the [[B-field]].

### Lift to M-theory?

The massive version of the [[duality between type IIA string theory and M-theory]] is more subtle, since [[D=11 supergravity]] does not admit a corresponding mass deformation ([BDHS 97](#BDHS97), [Deser 97](#Deser97), [Tsimpis 05](#Tsimpis05)).

In [AJTZ10](#AJTZ10) it was argued that a massive strong-coupling limit may just not exist.

But in [Hull 98](#Hull98) an embedding of massive IIA into [[M-theory]] was claimed, and a corresponding [[BFSS matrix model]] compactification was claimed in [Lowe-Nastae-Ramgoolam 03](#LoweNastaeRamgoolam03)).

## Related concepts

* [[massive type IIA supergravity]]

* [[D8-brane]], [[D(-2)-brane]]

## References

### General

An early hint is in

* [[Joseph Polchinski]], p. 8,9 of: _Dirichlet-Branes and Ramond-Ramond Charges_, Phys. Rev. Lett. **7** 5 (1995) 4724-4727 [[arXiv:hep-th/9510017](https://arxiv.org/abs/hep-th/9510017)]

Serious development in

* {#BergshoeffdeBooGreenPapadopoulosTownsend96} [[Eric Bergshoeff]], Mees de Roo, [[Michael Green]], [[George Papadopoulos]], [[Paul Townsend]], _Duality of Type II 7-branes and 8-branes_, Nucl. Phys. B470:113-135, 1996 ([arXiv:hep-th/9601150](https://arxiv.org/abs/hep-th/9601150))

See also:

* Michael Haack, [[Jan Louis]], [[Harvendra Singh]], _Massive Type IIA Theory on K3_, JHEP 0104:040, 2001 ([arXiv:hep-th/0102110](https://arxiv.org/abs/hep-th/0102110))

* [[Harvendra Singh]], _Romans type IIA theory and the heterotic strings_, Phys.Lett.B545:403-410, 2002 ([arXiv:hep-th/0201206](https://arxiv.org/abs/hep-th/0201206))

* [[Tom Banks]], K. van den Broek, _Massive IIA flux compactifications and U-dualities_, JHEP 0703:068, 2007 ([arXiv:hep-th/0611185](https://arxiv.org/abs/hep-th/0611185))

### Black branes

#### D8-branes

As a [[black brane]] the [[D8-brane]] was identified as a solution to [[Romans supergravity]]/[[massive type IIA string theory]] in

* [Bergshoeff-de Boo-Green-Papadopoulos-Townsend 96](#BergshoeffdeBooGreenPapadopoulosTownsend96)

See also

* {#ChamblinPerry97} A. Chamblin, [[Malcolm Perry]], _Dynamic D8-branes in IIA string theory_ ([arXiv:hep-th/9712112](https://arxiv.org/abs/hep-th/9712112))

* {#JanssenMeessenOrtin99} [[Bert Janssen]], [[Patrick Meessen]], [[Tomas Ortin]], _The D8-brane tied up: String and brane solutions in massive type IIA supergravity_, Phys. Lett. B453 (1999) 229-236 ([spire:494174](http://inspirehep.net/record/494174), doi:10.1016/S0370-2693(99)00315-9)

#### D6-D8-brane bound states
 {#D6D8BoundStates}

On [[D6-D8-brane bound states]] in [[massive type IIA string theory]]:

* [[Harvendra Singh]], _Duality symmetric massive type II theories in $D=8$ and $D=6$_, JHEP 0204:017, 2002 ([arXiv:hep-th/0109147](https://arxiv.org/abs/hep-th/0109147))

* [[Harvendra Singh]], _Note on (D6,D8) Bound State, Massive Duality and Non-commutativity_, Nucl. Phys. B661 (2003) 394-408 ([arXiv:hep-th/0212103](https://arxiv.org/abs/hep-th/0212103))

* Fabio Apruzzi, [[Marco Fazzi]], Dario Rosa, [[Alessandro Tomasiello]],  _All $AdS_7$ solutions of type II supergravity_, JHEP 04 (2014) 064 ([arxiv:1309.2949](https://arxiv.org/abs/1309.2949))

* [[Marco Fazzi]], _Higher-dimensional field theories from type II supergravity_ ([arxiv:1712.04447](https://arxiv.org/abs/1712.04447))

With emphasis on [[charge quantization]] of the [[RR-field]] [[flux]] forms via [[bundle gerbes]]:

* [[Jussi Kalkkinen]], _Gerbes and Massive Type II Configurations_, JHEP 9907  :002, 1999 ([arXiv:hep-th/9905018](https://arxiv.org/abs/hep-th/9905018))

#### D6-D8-brane bound states with D2-D4-brane defects
 {#D6D8BoundStatesWithDefects}

On [[black brane|black]]$\;$[[D6-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D4-brane bound states]] inside them realizing [[AdS3-CFT2]] "inside" [[AdS7-CFT6]]:

* {#DibitettoPetri17} [[Giuseppe Dibitetto]], [[Nicolò Petri]], _6d surface defects from massive type IIA_, JHEP 01 (2018) 039 ([arxiv:1707.06154](https://arxiv.org/abs/1707.06154))

* [[Nicolò Petri]], section 6.5 of: _Supersymmetric objects in gauged supergravities_ ([arxiv:1802.04733](https://arxiv.org/abs/1802.04733))

* {#Petri18} [[Nicolò Petri]], _Surface defects in massive IIA_, talk at [Recent Trends in String Theory and Related Topics](http://physics.ipm.ac.ir/conferences/stringtheory3/) 2018 ([pdf](http://physics.ipm.ac.ir/conferences/stringtheory3/note/N.Petri.pdf))

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_3$ vacua and surface defects in massive IIA_, PoS CORFU2018 (2019) 142 ([arxiv:1904.02455](https://arxiv.org/abs/1904.02455), [doi:10.22323/1.347.0142](https://doi.org/10.22323/1.347.0142))

* [[Yolanda Lozano]], Niall T. Macpherson, Carlos Nunez, [[Anayeli Ramirez]], $1/4$ BPS $AdS_3/CFT_2$ ([arxiv:1909.09636](https://arxiv.org/abs/1909.09636))

* [[Yolanda Lozano]], Niall T. Macpherson, Carlos Nunez, [[Anayeli Ramirez]], _Two dimensional $N=(0,4)$ quivers dual to $AdS_3$ solutions in massive IIA_ ([arxiv:1909.10510](https://arxiv.org/abs/1909.10510))

* [[Yolanda Lozano]], Niall T. Macpherson, Carlos Nunez, [[Anayeli Ramirez]], _$AdS_3$ solutions in massive IIA, defect CFTs and T-duality_ ([arxiv:1909.11669](https://arxiv.org/abs/1909.11669))

* Kostas Filippas, _Non-integrability on $AdS_3$ supergravity_ ([arxiv:1910.12981](https://arxiv.org/abs/1910.12981))

#### D4-D8-brane bound states with D2-D6-brane defects

On [[black brane|black]]$\;$[[D4-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D6-brane bound states]] inside them realizing [[AdS3-CFT2]] "inside" [[AdS7-CFT6]]:

* {#DibitettoPetri18} [[Giuseppe Dibitetto]], [[Nicolò Petri]], _Surface defects in the D4 − D8 brane system_, JHEP 01 (2019) 193 ([arxiv:1807.07768](https://arxiv.org/abs/1807.07768))

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_3$ vacua and surface defects in massive IIA_ ([arxiv:1904.02455](https://arxiv.org/abs/1904.02455))

#### KK-Monopole

On the [[KK-monopole]] in massivel IIA and its relation to some form of massive [[D=11 supergravity]]:

* Eduardo Eyras, [[Yolanda Lozano]], *The Kaluza-Klein Monopole in a Massive IIA Background*, Nucl. Phys. B**546** (1999) 197-218 [[arXiv:hep-th/9812188](https://arxiv.org/abs/hep-th/9812188), doi:10.1016/S0550-3213(99)00098-X]

### M-Theory/Strong coupling limit

Discussion of impossibility of a mass-deformation of [[D=11 supergravity]]

* {#BDHS97} K. Bautier, S. Deser, [[Marc Henneaux]], D. Seminara, _No Cosmological $D=11$ Supergravity_, Phys. Lett. B406:49-53, 1997 ([arXiv:hep-th/9704131](https://arxiv.org/abs/hep-th/9704131))

* {#Deser97} S. Deser, _Uniqueness of $D=11$ Supergravity_ ([arXiv:hep-th/9712064](https://arxiv.org/abs/hep-th/9712064))

Speculation that strong-coupling limit of massive type IIA does not exist:

* {#AJTZ10} [[Ofer Aharony]], [[Daniel Jafferis]], [[Alessandro Tomasiello]], [[Alberto Zaffaroni]], _Massive type IIA string theory cannot be strongly coupled_, JHEP 1011:047, 2010 ([arXiv:1007.2451](https://arxiv.org/abs/1007.2451))

### Lift to M-theory
 {#ReferencesLiftToMTheory}

Discussion of possible lift of [[massive type IIA string theory]] to [[M-theory]] (in view of the [[duality between type IIA string theory and M-theory]]):

* [[Paul S. Howe]], [[Neil D. Lambert]], [[Peter C. West]], *A New Massive Type IIA Supergravity From Compactification*, Phys. Lett. B **416** (1998) 303-308 [[arXiv:hep-th/9707139](https://arxiv.org/abs/hep-th/9707139), doi:10.1016/S0370-2693(97)01199-4]

* [[Eric Bergshoeff]], [[Yolanda Lozano]], [[Tomas Ortin]], *Massive Branes*, Nucl. Phys. B **518** (1998) 363-423 [[arXiv:hep-th/9712115](https://arxiv.org/abs/hep-th/9712115), doi:10.1016/S0550-3213(98)00045-5]

* {#Hull98} [[Chris Hull]], *Massive String Theories From M-Theory and F-Theory*, JHEP 9811:027 (1998) [[arXiv:hep-th/9811021](https://arxiv.org/abs/hep-th/9811021), [doi:10.1088/1126-6708/1998/11/027](https://iopscience.iop.org/article/10.1088/1126-6708/1998/11/027)]

> Our purpose here is to argue that although the Romans supergravity theory may not be derivable from 11-dimensional supergravity, or any covariant massive deformation thereof, the massive IIA superstring, whose low energy limit is the Romans theory, can be obtained from M-theory.

> The type IIB supergravity theory also cannot be obtained from 11-dimensional supergravity, but the type IIB string theory can be obtained from M-theory by compactifying on a 2-torus and taking a limit

* [[Maria P. Garcia del Moral]], [[Alvaro Restuccia]], *10D Massive Type IIA Supergravities as the uplift of Parabolic M2-brane Torus bundles*, Forthsch. d. Phys. **64** 4-5 (2016) [[arXiv:1511.04784](https://arxiv.org/abs/1511.04784), [doi:10.1002/prop.201500087](https://doi.org/10.1002/prop.201500087)]

* {#LoweNastaeRamgoolam03} D. A. Lowe, [[Horatiu Nastase]], [[Sanyaje Ramgoolam]], _Massive IIA String Theory and Matrix Theory Compactification_, Nucl. Phys. B 667 (2003) 55-89 ([cern:609880](http://cds.cern.ch/record/609880), doi:10.1016/S0550-3213(03)00547-9)

  > (corresponding [[BFSS matrix model]])

* [[Maria P. Garcia del Moral]], P. Leon, [[Alvaro Restuccia]], *Wordsheet description of a* massive *type IIA superstring in 10D*, J. High Energ. Phys. **2023** 104 (2023) [[arXiv:2306.16620](https://arxiv.org/abs/2306.16620), doi:10.1007/JHEP11(2023)104]


massive type IIA supergravity

+-- {: .rightHandSide}
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###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called _massive type IIA supergravity_ or _Romans supergravity_ (due to [Romans 86](#Romans86)) is a deformation of plain [[type IIA supergravity]] in which the [[B-field]] acquires a [[mass]]-term.

## Properties

### Lift to string theory

The lift of Romans supergravity to to [[string theory]] is known as _[[massive type IIA string theory]]_, where the mass term is identified with the [[RR-field]] [[flux form]] of degree 0 (or of degree 10 by [[Hodge duality]]) which reflects the presence of [[D8-branes]].

### Black D8-brane solutions

The mass term makes Romans supergravity have [[black brane]]-solutions corresponding to [[D8-branes]]

([Bergshoess-de Roo-Green-Papadopoulos-Townsend 96](#BergshoessdeRooGreenPapadopoulosTownsend96), [Chamblin-Perry 97](#ChamblinPerry97), [Janssen-Meessen-Ortin 99](#JanssenMeessenOrtin99))

and to [[D6-D8-brane bound states]] ([Singh 02a](#Singh02a), [Singh 02b](#Singh02b))

## Related concepts

* [[massive type IIA string theory]]

## Related concepts

* [[massive type IIA string theory]]

* [[D8-brane]]

## References

### General

* {#Romans86} [[L. J. Romans]], _Massive $N = 2a$ supergravity in ten dimensions_ , Physics Letters B Volume 169, Issue 4, 3 April 1986, Pages 374-380 (doi:10.1016/0370-2693(86)90375-8, [spire:17727](http://inspirehep.net/record/17727))

* {#Tsimpis05} [[Dimitrios Tsimpis]], _Massive IIA supergravities_, JHEP 0510:057, 2005 ([arXiv:hep-th/0508214](https://arxiv.org/abs/hep-th/0508214))

### Black branes

#### D8-branes

As a [[black brane]] the [[D8-brane]] was identified as a solution to [[Romans supergravity]]/[[massive type IIA string theory]] in

* {#BergshoeffdeBooGreenPapadopoulosTownsend96} [[Eric Bergshoeff]], Mees de Roo, [[Michael Green]], [[George Papadopoulos]], [[Paul Townsend]], _Duality of Type II 7-branes and 8-branes_, Nucl. Phys. B470:113-135, 1996 ([arXiv:hep-th/9601150](https://arxiv.org/abs/hep-th/9601150))

See also

* {#ChamblinPerry97} A. Chamblin, [[Malcolm Perry]], _Dynamic D8-branes in IIA string theory_ ([arXiv:hep-th/9712112](https://arxiv.org/abs/hep-th/9712112))

* {#JanssenMeessenOrtin99} [[Bert Janssen]], Patrick Meessen, [[Tomas Ortin]], _The D8-brane tied up: String and brane solutions in massive type IIA supergravity_, Phys. Lett. B453 (1999) 229-236 ([spire:494174](http://inspirehep.net/record/494174), doi:10.1016/S0370-2693(99)00315-9)

#### D6-D8-brane bound states
 {#D6D8BoundStates}

On [[D6-D8-brane bound states]] in [[massive type IIA string theory]]:

* [[Harvendra Singh]], _Duality symmetric massive type II theories in $D=8$ and $D=6$_, JHEP 0204:017, 2002 ([arXiv:hep-th/0109147](https://arxiv.org/abs/hep-th/0109147))

* [[Harvendra Singh]], _Note on (D6,D8) Bound State, Massive Duality and Non-commutativity_, Nucl. Phys. B661 (2003) 394-408 ([arXiv:hep-th/0212103](https://arxiv.org/abs/hep-th/0212103))

* Fabio Apruzzi, [[Marco Fazzi]], Dario Rosa, [[Alessandro Tomasiello]],  _All $AdS_7$ solutions of type II supergravity_, JHEP 04 (2014) 064 ([arxiv:1309.2949](https://arxiv.org/abs/1309.2949))

* [[Marco Fazzi]], _Higher-dimensional field theories from type II supergravity_ ([arxiv:1712.04447](https://arxiv.org/abs/1712.04447))

With emphasis on [[charge quantization]] of the [[RR-field]] [[flux]] forms via [[bundle gerbes]]:

* [[Jussi Kalkkinen]], _Gerbes and Massive Type II Configurations_, JHEP 9907  :002, 1999 ([arXiv:hep-th/9905018](https://arxiv.org/abs/hep-th/9905018))

#### D6-D8-brane bound states with D2-D4-brane defects
 {#D6D8BoundStatesWithDefects}

On [[black brane|black]]$\;$[[D6-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D4-brane bound states]] inside them realizing [[AdS3-CFT2]] "inside" [[AdS7-CFT6]]:

* {#DibitettoPetri17} [[Giuseppe Dibitetto]], [[Nicolò Petri]], _6d surface defects from massive type IIA_, JHEP 01 (2018) 039 ([arxiv:1707.06154](https://arxiv.org/abs/1707.06154))

* [[Nicolò Petri]], section 6.5 of: _Supersymmetric objects in gauged supergravities_ ([arxiv:1802.04733](https://arxiv.org/abs/1802.04733))

* {#Petri18} [[Nicolò Petri]], _Surface defects in massive IIA_, talk at [Recent Trends in String Theory and Related Topics](http://physics.ipm.ac.ir/conferences/stringtheory3/) 2018 ([pdf](http://physics.ipm.ac.ir/conferences/stringtheory3/note/N.Petri.pdf))

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_3$ vacua and surface defects in massive IIA_ ([arxiv:1904.02455](https://arxiv.org/abs/1904.02455))

* [[Yolanda Lozano]], Niall T. Macpherson, Carlos Nunez, [[Anayeli Ramirez]], $1/4$ BPS $AdS_3/CFT_2$ ([arxiv:1909.09636](https://arxiv.org/abs/1909.09636))

* [[Yolanda Lozano]], Niall T. Macpherson, Carlos Nunez, [[Anayeli Ramirez]], _Two dimensional $N=(0,4)$ quivers dual to $AdS_3$ solutions in massive IIA_ ([arxiv:1909.10510](https://arxiv.org/abs/1909.10510))

* [[Yolanda Lozano]], Niall T. Macpherson, Carlos Nunez, [[Anayeli Ramirez]], _$AdS_3$ solutions in massive IIA, defect CFTs and T-duality_ ([arxiv:1909.11669](https://arxiv.org/abs/1909.11669))

* Kostas Filippas, _Non-integrability on $AdS_3$ supergravity_ ([arxiv:1910.12981](https://arxiv.org/abs/1910.12981))

#### D4-D8-brane bound states with D2-D6-brane defects

On [[black brane|black]]$\;$[[D4-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D6-brane bound states]] inside them realizing [[AdS3-CFT2]] "inside" [[AdS7-CFT6]]:

* {#DibitettoPetri18} [[Giuseppe Dibitetto]], [[Nicolò Petri]], _Surface defects in the D4 − D8 brane system_, JHEP 01 (2019) 193 ([arxiv:1807.07768](https://arxiv.org/abs/1807.07768))

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_3$ vacua and surface defects in massive IIA_ ([arxiv:1904.02455](https://arxiv.org/abs/1904.02455))

[[!redirects Romans supergravity]]

[[!redirects Romans massive type IIA supergravity]]


massive Yang-Mills theory

#Contents#
* table of contents
{:toc}

## Idea

While [[Yang-Mills theory]] with a fundamental [[mass]]-term (as opposed to a [[spontaneous symmetry breaking|dynamical]] mass via a [[Higgs mechanism]]) cannot be both [[renormalizability|renormalizable]] and unitary ([Delbourgo, Twisk & Thompson 1988](#DelbourgoTwiskThompson88)), it does appear in certain [[effective field theories]], notably in [[quantum hadrodynamics]].

## Related conceps

* [[Yang-Mills theory]]

* [[Higgs mechanism]]

## References

### General

* {#DelbourgoTwiskThompson88} Robert Delbourgo, S. Twisk, [[George Thompson]], _Massive Yang-Mills theory: Renormalizability versus unitarity_, Int. J. Mod. Phys. A **3** 2 (1988) 435-449 [[doi:10.1142/S0217751X88000163](https://doi.org/10.1142/S0217751X88000163)]

* U. Ellwanger, N. Wschebor, _Massive Yang-Mills Theory in Abelian Gauges_, Int. J. Mod. Phys. A18 (2003) 1595-1612 ([arXiv:hep-th/0205057](https://arxiv.org/abs/hep-th/0205057))

### In quantum hadrodynamics

Equivalence between [[hidden local symmetry]]- and [[massive Yang-Mills theory]]-description of [[Skyrmion]] [[quantum hadrodynamics]]:

* [[Atsushi Hosaka]], H. Toki, [[Wolfram Weise]], _Skyrme Solitons With Vector Mesons: Equivalence of the Massive Yang-Mills and Hidden Local Symmetry Scheme, 1988, Z. Phys. A332 (1989) 97-102 ([spire:24079](http://inspirehep.net/record/24079))

More on relation to [[Skyrmions]]:

* Marcelo Ipinza, Patricio Salgado-Rebolledo, _Meron-like topological solitons in massive Yang-Mills theory and the Skyrme model_ ([arXiv:2005.04920](https://arxiv.org/abs/2005.04920))


Massoud Amini

* [personal page](https://modares.ac.ir/~MAMINI)

* [GoogleScholar page](https://scholar.google.com/citations?user=fOFui3kAAAAJ&hl=en)

## Selected writings

On [[C-star algebras|$C^\ast$-algebras]]:

* [[Massoud Amini]], *Locally Compact Pro-$C^\ast$-Algebras*, Canadian Journal of Mathematics **56** 1 (2004) 3-22  [[arXiv:math/0205253](https://arxiv.org/abs/math/0205253), [doi:10.4153/CJM-2004-001-6](https://doi.org/10.4153/CJM-2004-001-6)]

category: people


Master course on algebraic stacks

This page is dedicated to the notes

* [[Bertrand Toen]], _Master course on algebraic stacks_, lectures notes, Toulouse, 2005, [web](http://perso.math.univ-toulouse.fr/btoen/videos-lecture-notes-etc/).

on [[algebraic stacks]] and [[functor of points|functorial]] [[algebraic geometry]].

## Contents

* Lecture 1: _Reflexions sur la notion d'espace I_, [pdf](http://perso.math.univ-toulouse.fr/btoen/files/2015/02/cours1.pdf).
* Lecture 2: _Reflexions sur la notion d'espace II_, [pdf](http://perso.math.univ-toulouse.fr/btoen/files/2015/02/cours2.pdf).
* Lecture 3: _Schémas et espaces algébriques I_, [pdf](http://perso.math.univ-toulouse.fr/btoen/files/2015/02/cours3.pdf).
* Lecture 4: _Schémas et espaces algébriques II_, [pdf](http://perso.math.univ-toulouse.fr/btoen/files/2015/02/cours4.pdf).
* Lecture 4 1/2: _Compléments sur les schémas et espaces algébriques_, [pdf](http://perso.math.univ-toulouse.fr/btoen/files/2015/02/cours4-2.pdf).
* Lecture 5: _Champs I_, [pdf](http://perso.math.univ-toulouse.fr/btoen/files/2015/02/cours5.pdf).
* Lecture 6: _Champs II_, [pdf](http://perso.math.univ-toulouse.fr/btoen/files/2015/02/cours6.pdf).
* Lecture 7: _Champs III_, [pdf](http://perso.math.univ-toulouse.fr/btoen/files/2015/02/cours7.pdf).
* Lecture 8: _Champs IV_, [pdf](http://perso.math.univ-toulouse.fr/btoen/files/2015/02/cours81.pdf).
* Lecture 9: _Quelques exercices_, [pdf](http://perso.math.univ-toulouse.fr/btoen/files/2015/02/cours92.pdf)

## English translation

* Ruìān Chén, [PDF 1](https://web.archive.org/web/20190617183638/http://www-personal.umich.edu/~ruchen/articles/stacks.pdf), [[toen-master-course.pdf:file]]


master equation

Especially in [[physics]] there are some types of [[equations]] that in their subfield are called _master equations_. Examples include the following:

1. in [[BV-BRST formalism]] the _classical master equation_ is the condition that the refinement of an [[action functional]] to a function on its [[derived critical locus]] squares to 0 with respect to a certain graded Poisson-bracket. The _quantum master equation_ is a deformation of this equation. See at _[[BV-BRST formalism]]_ for details on all this.

1. in [[statistical mechanics]], Langevin equation and generalizations

1. linear evolutionary differential equations (often for operator functions) of the form $d y/ d t = L y$ where $L$ is a linear operator not involving differentiating with respect to time is sometimes called a master equation in some contexts

[[!redirects master equations]]


master Ward identity > history

> see at _[[Ward identity]]_


Matan Prasma

[[!redirects Matan Prezma]]

NB. Used to write his last name *Prezma*.

* [GoogleScholar page](https://scholar.google.ch/citations?user=nNnFyUgp35EC&hl=en)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=175574)

## Selected writings

* _Homotopy Normal Maps_ (2010) ([arXiv:1011.4708](http://arxiv.org/abs/1011.4708))

* _Descent data of cosimplicial 2-groupoids_, [arXiv:1112.3072](http://arxiv.org/abs/1112.3072)

* with [[Tomer Schlank]], _Sylow theorems for ∞-groups_, [arXiv:1602.04494](http://arxiv.org/abs/1602.04494)

On the [[Grothendieck construction for model categories]]:

* {#HarpazPrasma15} [[Yonatan Harpaz]], [[Matan Prasma]], _The Grothendieck construction for model categories_, Advances in Mathematics **281** (2015) 1306-1363 [[arXiv:1404.1852](https://arxiv.org/abs/1404.1852), [10.1016/j.aim.2015.03.031](https://doi.org/10.1016/j.aim.2015.03.031)]

category:people

[[!redirects MatanP]]
[[!redirects Matan Prezma]]


matching family

+-- {: .rightHandSide}
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###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _matching family_ of elements is an explicit component-wise characterizaton of a morphism from a [[sieve]] into a general [[presheaf]].

Since such morphisms govern the [[sheaf]] property and the operation of [[sheafification]], these can be discussed in terms of matching families.
As such, the set of matching families for a given [[covering]] and [[presheaf]] is the corresponding [[descent object]].

## Definition

Let $(C,\tau)$ be a [[site]] and $P:C^{\mathrm{op}}\to\mathrm{Set}$ a [[presheaf]] on $C$. Let $S\in \tau(c)$ be a [[cover]]ing [[sieve]] on [[object]] $c\in C$ (in particular a subobject of the [[representable functor|representable]] presheaf $h_c$).

A __matching family__ for $S$ of elements in $P$ is a rule assigning to each $f:d\to c$ in $S$ an element $x_f$ of $P(d)$ such that
for all $g:e\to d$
$$ P(g)(x_f) = x_{f\circ g}. $$
Notice that $f\circ g\in S$ because $S$ is a sieve, so that the condition makes sense; furthermore the order of composition and the contravariant nature of $P$ agree. If we view the sieve $S$ as a subobject of the representable $h_c$, then a matching family $(x_f)_{f\in S}$ is precisely a [[natural transformation]] $x:S\to P$, $x: f\mapsto x_f$.

An __amalgamation__ of the matching family $(x_f)_{f\in S}$ for $S$ is an element $x\in P(c)$ such that $P(f)(x) = x_f$ for all $f\in S$.

## Properties

### Characterization of sheaves

$P$ is a [[sheaf]] for the [[Grothendieck topology]] $\tau$ iff for all $c$, for all $S\in\tau(c)$ and every matching family $(x)_{f\in S}$ for $S$, there is a unique amalgamation. Equivalently $P$ is a sheaf if any natural transformation $x:S\to P$ has a unique extension to $h_C\to P$ (along inclusion $S\hookrightarrow h_c$); or to phrase it differently, $P$ is a sheaf (resp. [[separated presheaf]]) iff the precomposition with the inclusion $i_S : S\hookrightarrow h_C$ is an [[isomorphism]] (resp. [[monomorphism]]) $i_S:\mathrm{Nat}(h_C,P)\to \mathrm{Nat}(S,P)$.

Suppose now that $C$ has all [[pullback]]s.  Let $R = (f_i:c_i\to c)_{i\in I}$ be any [[cover]] of $c$ (i.e., the smallest [[sieve]] containing $R$ is a [[covering]] sieve in $\tau$) and let $p_{ij}:c_i\times_c c_j\to c_i$, $q_{ij}:c_i\times_c c_j\to c_j$ be the two projections of the pullback of $f_j$ along $f_i$. A __matching family__ for $R$ of elements in a presheaf $P$ is by definition a family $(x_i)_{i\in I}$ of elements $x_i\in P(c_i)$, such that for all $i,j\in I$, $P(p_{ij})(x_i) = P(q_{ij})(x_j)$.

### Sheafification

Let $\mathrm{Match}(R,P)$ be the set of matching families for $R$ of elements in $P$. Sieves over $c$ form a [[filtered category]], where partial ordering is by reverse inclusion ([[refinement]] of sieves).  There is an endofunctor $()^+ : PShv(C,\tau)\to PShv(C,\tau)$ given by
$$ P^+(c) := \mathrm{colim}_{R\in\tau(C)} \mathrm{Match}(R,P)$$
In other words, elements in $P^+(c)$ are matching families $(x^R_f)_{f\in R}$ for all covering sieves modulo the equivalence given by agreement $x^R_f = x^{R'}_f$, for all $f\in R''$, where $R''\subset  R\cap R'$ is a common refinement of $R$ and $R'$. This is called the [[plus construction]].

Endofunctor $P\mapsto P^+$ extends to a presheaf on $C$ by $P^+(g:d\to c) : (x_f)_{f\in R}\mapsto (x_{g\circ h})_{h\in g^*R}$ where $g^* R = \{h:e\to d | e\in C, g\circ h\in R\}$ (recall that by the stability axiom of Grothendieck topologies, $g^*(d)\in \tau(d)$ is a covering sieve over $d$).

The presheaf $P^+$ comes equipped with a canonical natural transformation $\eta:P\to P^+$ which to an element $x\in P(c)$ assigns the equivalence class of the matching family $(P(f)(x))_{f\in Ob(C/c)}$ where the [[maximal sieve]] $Ob(C/c)$ is the class of objects of the slice category $C/c$.

$\eta$ is a [[monomorphism]] (resp. [[isomorphism]]) of presheaves iff the presheaf $P$ is a [[separated presheaf]] (resp. [[sheaf]]); moreover any morphism $P\to F$ of presheaves, where $F$ is a sheaf, factors uniquely through $\eta:P\to P^+$. For any presheaf $P$, $P^+$ is separated presheaf and if $P$ is already separated then $P^+$ is a sheaf. In particular, for any presheaf $P^{++}$ is a sheaf. A fortiori, $P^+(\eta)\circ\eta:P\to P^{++}$ realizes [[sheafification]].

### Co-representation by Cech groupoids
 {#CoRepresentationByCechGroupoid}

When [[presheaves]] of sets are regarded a [[presheaves of groupoids]], the [[Cech groupoid]] serves to co-represent matching families, hence serves as the _[[codescent object]]_ of the given covering and presheaf. See [there](Čech+groupoid#Codescent) for more.

## References

A standard reference is

* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_, Springer 1992. (chapter III)

[[!redirects matching families]]


mate

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition
 {#Definition}

\begin{proposition}\label{MateBijection}
**(mate bijection)**
\linebreak
Given a [[2-category]] $K$, [[adjunction|adjoint pairs]] $(\eta,\epsilon) \colon f \dashv u \colon b \to a$ and $(\eta',\epsilon') \colon f' \dashv u' \colon b' \to a'$ , and [[1-morphisms|1-cells]] $x \colon a \to a'$ and $y \colon b \to b'$, there is a [[bijection]]
$$ K(a,b')(f' x,y f) \cong K(b,a')(x u,u' y) $$
given by [[pasting]] with the [[unit of an adjunction|unit]] of one adjunction and the [[counit of an adjunction|counit]] of the other, i.e.:

\begin{tikzcd}[
   sep = 30pt
  ]
    a
    \ar[r, "{ x }"{description}]
    \ar[d, "{ f }"{description}]
    &
    a'
    \ar[
      d,
      "{ f' }"{description}
    ]
    \ar[
      dl,
      Rightarrow,
      shorten=10pt,
      "{ \lambda }"
    ]
    \ar[
      rr,
      phantom,
      shift right=20pt,
      "{ \mapsto }"
    ]
    &&
    b
    \ar[d, equals]
    \ar[r, "{ u }"{description}]
    &
    a
    \ar[
      dl,
      Rightarrow,
      shorten=10pt,
      "{ \epsilon }"
    ]
    \ar[r, "{ x }"{description}]
    \ar[d, "{ f }"{description}]
    &
    a'
    \ar[
      d,
      "{ f' }"{description}
    ]
    \ar[
      dl,
      Rightarrow,
      shorten=10pt,
      "{ \lambda }"
    ]
    \ar[r, equals]
    &
    a'
    \ar[
      dl,
      Rightarrow,
      shorten=10pt,
      "{ \eta' }"
    ]
    \ar[
      d,
      equals
    ]
    \\
    b
    \ar[r, "{ y }"{description}]
    &
    b'
    &&
    b
    \ar[r, equals]
    &
    b
    \ar[r, "{ y }"{description}]
    &
    b'
    \ar[r, "{ u' }"{description}
    ]
    &
    a'
\end{tikzcd}

and

\begin{tikzcd}[
   sep = 30pt
  ]
    b
    \ar[r, "{ y }"{description}]
    \ar[d, "{ u }"{description}]
    &
    b'
    \ar[
      d,
      "{ u' }"{description}
    ]
    \ar[
      from=dl,
      Rightarrow,
      shorten=10pt,
      "{ \mu }"{swap}
    ]
    \ar[
      rr,
      phantom,
      shift right=20pt,
      "{ \mapsto }"
    ]
    &&
    a
    \ar[d, equals]
    \ar[r, "{ f }"{description}]
    &
    b
    \ar[
      from=dl,
      Rightarrow,
      shorten=10pt,
      "{ \eta }"{swap}
    ]
    \ar[r, "{ y }"{description} ]
    \ar[d, "{ u }"{description}]
    &
    b'
    \ar[
      d,
      "{ u' }"{description}
    ]
    \ar[
      from=dl,
      Rightarrow,
      shorten=10pt,
      "{ \mu }"{swap}
    ]
    \ar[r, equals]
    &
    b'
    \ar[
      from=dl,
      Rightarrow,
      shorten=10pt,
      "{ \epsilon' }"{swap}
    ]
    \ar[
      d,
      equals
    ]
    \\
    a
    \ar[r, "{ x }"{description}]
    &
    a'
    &&
    a
    \ar[r, equals]
    &
    a
    \ar[r, "{ x }"{description}]
    &
    a'
    \ar[r, "{ f' }"{description}
    ]
    &
    b'
\end{tikzcd}

\end{proposition}

\begin{proof}\label{ProofOfMateBijection}
That this is a bijection follows easily from the [[triangle identities]], which say that the gray-shaded cells in the following [[pasting diagram]] cancel out;

\begin{tikzcd}[
   sep = 30pt
  ]
    \color{gray}
    a
    \ar[
      d,
      gray,
      "{ f }"{description}
    ]
    \ar[r, equals, gray]
    &
    \color{gray}a
    \ar[
      dl,
      gray,
      Rightarrow,
      shorten=10pt,
      "{ \eta }"
    ]
    \ar[d, equals, gray]
    \\
    \color{gray}
    b
    \ar[d, equals, gray]
    \ar[r, gray, "{ u }"{description}]
    &
    a
    \ar[
      dl,
      gray,
      Rightarrow,
      shorten=10pt,
      "{ \epsilon }"{gray}
    ]
    \ar[r, "{ x }"{description}]
    \ar[d, "{ f }"{description}]
    &
    a'
    \ar[
      d,
      "{ f' }"{description}
    ]
    \ar[
      dl,
      Rightarrow,
      shorten=10pt,
      "{ \lambda }"
    ]
    \ar[r, gray, equals]
    &
    \color{gray}
    a'
    \ar[
      dl,
      gray,
      Rightarrow,
      shorten=10pt,
      "{ \eta' }"
    ]
    \ar[
      d,
      gray,
      equals
    ]
    \\
    \color{gray}
    b
    \ar[r, gray, equals]
    &
    b
    \ar[r, "{ y }"{description}]
    &
    b'
    \ar[r, gray, "{ u' }"{description}
    ]
    \ar[
      d, gray, equals
    ]
    &
    \color{gray}
    a'
    \ar[
      dl,
      gray,
      Rightarrow,
      shorten=10pt,
      "{ \epsilon' }"
    ]
    \ar[d, gray, "{ f' }"]
    \\
    & &
    \color{gray}
    b'
    \ar[r, equals, gray]
    &
    \color{gray}
    b'
\end{tikzcd}

\end{proof}

\begin{definition}\label{TerminologyOfMates}
The 2-cells $\lambda$ and $\mu$ in prop. \ref{MateBijection} are called **mates** [[Kelly & Street (2006) p. 87](#KellyStreet06); [Leinster (2004), pp. 150](#Leinster04)]
 (earlier: *[[conjugate transformation of adjoints|conjugates]]*, [MacLane (1971), p. 98](#MacLane71), see Exp. \ref{ConjugateTransformationOfAdjoints} below) with respect to the adjunctions $f \dashv u$ and $f' \dashv u'$ (and to the 1-cells $x$ and $y$).
\end{definition}

## Properties

### General

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

Strict [[2-functors]] preserve adjunctions and [[pasting diagrams]], so that if $F \colon K \to J$ is a 2-functor and if $\lambda$ and $\mu$ are mates wrt $f \dashv u$ and $f' \dashv u'$ in $K$, then $F \lambda$ and $F \mu$ are mates wrt $F f \dashv F u$ and $F f' \dashv F u'$ in $J$.

=--

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

If $\alpha \colon F \Rightarrow G$ is a [[2-natural transformation]], then the naturality identities $\alpha_b \circ F f = G f \circ \alpha_a$ and $\alpha_a \circ F u = G u \circ \alpha_b$ are mates wrt $F f \dashv F u$ and $G f \dashv G u$.

=--

### Naturality

There are two [[double categories]] with objects those of $K$, vertical arrows [[adjoint pairs]] in $K$ and horizontal arrows 1-cells of $K$.  In one the 2-cells are those of the form $\lambda$ above, while in the other they are those of the form $\mu$.  It is easily shown, as in Kelly--Street, that the triangle identities and the definition of composition of adjoints make these two double categories isomorphic.  So for any $K$ there is a double category $Adj(K)$, defined up to isomorphism as above but with mate-pairs in $K$ as 2-cells.

What this means is that, for example, the mate of a square coming from a [[pasting diagram]] is given by pasting the mates of the individual 2-cells (whenever this makes sense).

In the double category $Adj(K)$, every vertical arrow has both a [[companion]] (the left adjoint) and a [[conjoint]] (the right adjoint).  (In fact, in some sense it is the universal double category constructed from $K$ with this property.)  Therefore, it is equivalent to a [[2-category equipped with proarrows]].  More explicitly, there is a forgetful functor $L \colon Adj_V(K) \to K$ from the 2-category of objects, adjunctions and mate-pairs in $K$ to $K$ that sends an adjunction $f \dashv u$ to $f$.  It is [[locally fully faithful 2-functor|locally fully faithful]], and moreover every $L f$ has a right adjoint in $K$ by definition; this gives the more traditional definition of a proarrow equipment.

## Examples
 {#Examples}

\begin{example}\label{ConjugateTransformationOfAdjoints}
**([[conjugate transformation of adjoints]])**
\linebreak
Let $F \dashv U \colon D \to C$ be an [[adjunction]] in the [[2-category]] [[Cat]], i.e. a pair of [[adjoint functors]], and $A \colon * \to C$ and $X \colon * \to D$ be [[objects]] of $C$ and $D$ considered as [[functors]] out of the [[terminal category]] $*$.  Then taking mates with respect to $1 \dashv 1 \colon * \to *$ and $F \dashv U$ yields the [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism)
$$
  D(F A,\, X) \;\cong\; C(A,\, U X)
$$
and the pasting operations as above yield the notion of [[conjugate transformation of adjoints]]. (This is the original notion, due to [MacLane (1971), p. 98](#MacLane71))

Moreover, the naturality of the mate correspondence yields [[natural isomorphism|naturality]] of the bijection.
\end{example}

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

If the ambient [[2-category]] $K$ is the [[delooping]] of a [[monoidal category]] $(\mathcal{C}, \otimes)$ in that

$$
  K \simeq \mathbf{B}_\otimes \mathcal{C}
$$

then an [[adjunction]] in $K$ is a pair of [[dual objects]] and the mate-construction is the construction of [[dual morphisms]] between [[dualizable objects]].

=--

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

Suppose given a [[commutative square]] (up to [[isomorphism]]) of [[functor]]s:
$$\array{ & \overset{f^*}{\to} & \\
  ^{g^*}\downarrow && \downarrow^{k^*}\\
  & \underset{h^*}{\to} & }$$
in which $f^*$ and $h^*$ have [[left adjoint]]s $f_!$ and $h_!$, respectively. (The classical example is a [[Wirthmüller context]].)  Then the [[natural isomorphism]] that makes the square commute

$$
  k^* f^* \to h^* g^*
$$

has a mate

$$
  h_! k^* \to g^* f_!
$$

defined as the composite

$$
  h_! k^* \overset{\eta}{\to} h_! k^* f^* f_! \overset{\cong}{\to} h_! h^* g^* f_! \overset{\epsilon}{\to} g^* f_!
  \,.
$$

One says that the original square satisfies the **[[Beck-Chevalley condition]]** if this mate is an [[equivalence]].

=--

## Multi-variable mates

There is a version of the mate correspondence that applies to [[two-variable adjunctions]] and $n$-variable adjunctions; see [Cheng-Gurski-Riehl](#CGR).

The relationship between two of the adjoints in a multivariable adjunction can be described as a **parametrized adjunction**: fixing the variables in each of the categories that appear in the domains of both adjoints, the pair of functors define an adjunction between the remaining two categories. Relative to the parametrized adjunctions that define a multivariable adjunction, the multivariable mates can be understood as [parametrized mates](https://golem.ph.utexas.edu/category/2012/11/parametrized_mates_and_multiva.html).

## References

The example of [[conjugate transformation of adjoints]] (but without the terminology of "mates")

* {#MacLane71} [[Saunders MacLane]], p. 98 of: _[[Categories Work|Categories for the working mathematician]]_, Graduate texts in mathematics, Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

The explicit notion of mates may be officially due to

* {#KellyStreet06} [[Max Kelly]], [[Ross Street]], §2.2 (esp. p. 87) of: *Review of the elements of 2-categories*, in: G.M. Kelly (ed.), Category Seminar, Lecture Notes in Mathematics **420** (1974) [[doi:10.1007/BFb0063101](https://doi.org/10.1007/BFb0063101)]

and is reviewed in:

* {#Leinster04} [[Tom Leinster]], Section 6.1, pp. 150 in: *Higher operads, higher categories*, London Math. Soc. Lec. Note Series **298**, Cambridge University Press (2004) [[math.CT/0305049](http://arxiv.org/abs/math.CT/0305049), [doi:10.1017/CBO9780511525896](https://doi.org/10.1017/CBO9780511525896)]

Further review and discussion:

* {#CGR} [[Eugenia Cheng]], [[Nick Gurski]], [[Emily Riehl]], _Multivariable adjunctions and mates_, J. K-Theory **13** (2014), 337–396, [doi:10.1017/is013012007jkt250](https://doi.org/10.1017/is013012007jkt250), [arXiv:1208.4520](http://arxiv.org/abs/1208.4520).

* [[Emily Riehl]], _Parametrized mates and multivariable adjunctions_ [blog post](https://golem.ph.utexas.edu/category/2012/11/parametrized_mates_and_multiva.html).

{#ReferencesGeneralizationtoBicategories} Discussion in the generalization of [[bicategories]]:

* [[Aaron Lauda]], §3 of _Frobenius algebras and ambidextrous adjunctions_, Theory and Applications of Categories, **16** 04 (2006) 84-122 &lbrack[tac:16-04](http://www.tac.mta.ca/tac/volumes/16/4/16-04abs.html)]

* [[Richard Garner]], [[Michael Shulman]], around 13.7 of _Enriched categories as a free cocompletion_, Advances in Mathematics **289** (2016) Pages 1-94, [doi:10.1016/j.aim.2015.11.012](https://doi.org/10.1016/j.aim.2015.11.012), [arXiv:1301.3191](https://arxiv.org/abs/1301.3191).

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], Def. 6.1.12 in: *2-Dimensional Categories*, Oxford University Press (2021) [[arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378)]

[[!redirects mates]]
[[!redirects mate correspondence]]
[[!redirects mates correspondence]]


material set theory

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

# Material set theory
* table of contents
{: toc}

## Overview

__Material set theory__ (also called _membership-based set
theory_) is a style of [[set theory]] with a primitive global membership relation "$\in$" where sets are characterized only by "$\in$" and [[propositional equality]] of sets. (The terminology 'material', or at least 'materialistic', goes back at least to [Friedman 1997](#Friedman1997).) Material set theory contrasts with *[[structural set theory]]* (cf. *[[material versus structural set theory]]*), as well as with set theories which are neither structural nor material. Material set theories are [[simple type theory|simply sorted]] [[set theory]].

## Related concepts

* [[cumulative hierarchy]]

* [[unsorted set theory]]

* [[two-sorted set theory]]

* [[set-theoretic multiverse]]

* [[material versus structural set theory]]

## References

Relation to [[structural set theory]] is discussed in

* {#Shulman18} [[Michael Shulman]], _Comparing material and structural set theories_ ([arXiv:1808.05204](https://arxiv.org/abs/1808.05204))

See also

* {#Friedman1997} [[Harvey Friedman]] (1997). [Foundational Completeness](https://cs.nyu.edu/pipermail/fom/1997-November/000143.html). FOM, 1997 November.

[[!redirects material set theory]]
[[!redirects material set theories]]


material versus structural set theory

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

\tableofcontents

## Overview

This article is about attempting to identify what makes a [[set theory]] a [[material set theory]] or a [[structural set theory]], and whether certain types of set theories should be considered material set theories, structural set theories, or neither material nor structural.

We consider only [[simple type theory|simply sorted]] [[set theories]], which all have [[membership relations]]. All [[dependently sorted set theories]] are structural set theories with its membership being a typing judgment rather than a binary relation.

### Homogeneous and heterogeneous membership relations

One difference between set theories is between the set theories where sets and elements are in two different sorts and the membership relation is a [[heterogeneous membership relation]], and the set theory where sets and elements are in the same sort and the membership relation is a [[homogeneous membership relation]].

One proposal is to define [[structural set theory]] to be the former and [[material set theory]] to be the latter.

Having a homogeneous membership relation is not sufficient for a set theory to be a material set theory, since unsorted set theories such as [[fully formal ETCS]] also have a homogeneous membership relation $\in$, but should be a structural set theory, since it is based in [[category theory]]. (Though see below at the section titled propositional equality of sets for an argument that any set theory like [[fully formal ETCS]] with a homogeneous membership relation has the wrong notion of sameness of sets to be a structural set theory.)

Nor is having a heterogeneous membership relation sufficient for defining structural set theories. There exist two-sorted presentations of [[pure set]] theories and material set theories with [[urelements]], with separate sorts $Set$ and $Element$ representing all sets and all elements respectively, as well as a global membership relation $\in$ defined as
$$a:Element, A:Set \vdash a \in A prop$$
and functions $\mathrm{asElem}$ and $\mathrm{asSet}$ which [[element reflection|reflect]] a set $A$ over to an element $\mathrm{asElem}(A)$, and for [[pure set]] theory, [[set reflection|reflect]] an element $a$ over to a set $\mathrm{asSet}(a)$.

As neither having a homogeneous nor heterogeneous membership relation isn't sufficient for defining both material and structural set theories, it isn't necessary for defining either notion, and thus whether a membership relation is homogeneous or heterogeneous is orthogonal to the set theory being material or structural.

### Primitive and derived membership relations

Another proposed difference between material set theory and structural set theory is that in material set theory, the membership relation is a primitive. However, making the membership relation $\in$ primitive is not sufficient to make a set theory a material set theory, since one could make the membership relation a primitive in [[fully formal ETCS]] as well, and replace the definition with an axiom stating a logical equivalence between the membership relation and what would have been its definition:

* for every function $f$ and $g$, a relation $f \in g$,
* axiom: for every function $f$ and $g$, $f \in g$ if and only if $s(f) = 1$, $s(g) = 0$, and $t(f) = t(g)$.

This is similar to how in [[ZFC]] one could take [[equality]] to be a primitive or a defined concept via the [[axiom of extensionality]]. Thus, having a primitive or a derived membership relation is neither sufficient nor necessary for [[structural set theories]].

However, having a primitive membership relation does seem to be a necessary condition for a set theory to be a [[material set theory]].

### Elements with "internal structure"

Another proposed difference between material set theory and structural set theory is that in material set theory, [[elements]] have some notion of "internal structure" and/or [[sets]] are the "internal structure" of some other object in the theory, while in structural set theory, elements do not have internal structure and sets are not the internal structure of other objects. Both could be defined by the membership relation of the theory and distinguished via structure of the membership [[graphs]] of the membership relations, and thus one distinguishes between [[material membership relations]] and [[structural membership relations]], as detailed in the article on [[membership relation]].

A set theory with a [[homogeneous membership relation]] is a **[[material set theory]]** if it has a material membership relations, and it is a **[[structural set theory]]** if it has a structural membership relation. If the set theory has multiple homogeneous membership relations, the set theory is a **structural set theory** if there exists a structural membership relation, and the set theory is a **material set theory** if every homogeneous membership relation $\in$ is a [[material membership relation]].

However, there are multiple distinct proposed definitions of material and structural membership relations. In addition, this proposal only works if the set theory has a [[homogeneous membership relations]], and there exist both structural and material set theories which have [[heterogeneous membership relations]], where sets and elements are in fundamentally different sorts.

### Primitive sorts: sets/elements vs functions/relations

Another proposed difference between material set theory and structural set theory is the difference between the primitives of the two theories. Material set theory takes [[sets]] or [[elements]] to be the primitive concepts, while structural set theory takes [[functions]] or [[relations]] to be the primitive concepts.

### Propositional equality of sets

The article on [[structural set theory]] states the following:

> *Structural set theory* provides a [[foundation of mathematics]] which is free of this "superfluous baggage" attendant on theories such as ZF, in which there is lots of information such as whether or not $3\in 17$ (yes, says von Neumann; no, says Zermelo) which is never used in mathematics. In a structural set theory, the elements (such as $3$) of a set (such as $\mathbb{N}$) *have* no identity apart from their existence as elements of that set, and whatever structure is given to that set by the functions and relations placed upon it.

There is an argument following the [[principle of equivalence]] that the structural notion of sameness of elements is [[propositional equality]], while the structural notion of sameness of sets is [[bijection]]. From this perspective, [[propositional equality]] of sets is "superfluous baggage" attendant on theories which has propositional equality of sets, and thus that any set theory with propositional equality of sets is not structural. This includes set theories like [[fully formal ETCS]] where sets and elements are in the same sort and the set theory has a [[homogeneous membership relation]].

## See also

* [[material set theory]]
* [[structural set theory]]
* [[set theory versus dependent type theory]]

[[!redirects material versus structural]]
[[!redirects material versus structural set theories]]


material-structural adjunction

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### Context
#### Foundations
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[[!include foundations - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Any model of [[material set theory]] gives rise to a [[category]] of [[sets]], which is a model of [[structural set theory]].  Conversely, from any [[structural set theory]] we can construct a model of material set theory by using [[graphs]] to model hereditary membership relations, as described at _[[pure set]]_.  Viewed in the correct context, these two constructions are a pair of [[adjoint functors]].  Moreover, the adjunction is [[idempotent adjunction|idempotent]], and thus induces an equivalence between models of the two types of set theory which satisfy appropriate "foundation-type" axioms.

## Definition

One has to be a little careful in defining the relevant categories of models in order to make this precise.  Of course, there are also many different variants depending on how strong or weak the set theories in question are.  There is also an additional subtlety in that models of material set theory naturally form a [[1-category]], while models of structural set theory naturally form a [[2-category]].

...

## Idempotence

The adjunction is [[idempotent adjunction|idempotent]], and therefore restricts to an equivalence between the full images of the two functors involved.  In the well-founded case, we can characterize these as follows:

* A model of material set theory is in the fixed subcategory if it satisfies the [[axiom of foundation]], it has [[transitive closure|transitive closures]], and it satisfies [[Mostowski's principle]].

* A model of structural set theory is in the fixed subcategory if every set can be embedded into an extensional well-founded relation.

If we use a non-well-founded construction of pure sets as the right adjoint, then the axiom of foundation must be replaced by an appropriate [[axiom of anti-foundation]].  Note that axioms of anti-foundation usually include appropriate ill-founded generalizations of Mostowski's principle and the existence of transitive closures.  Similarly, on the structural side the necessary axiom becomes embeddability of all sets into a suitable type of [[extensional relation]].

## References

The material-structural adjunction was first observed, for a particular pair of theories, in:

* William Mitchell 1973, _Categories of Boolean topoi_, Journal of Pure and Applied Algebra 3(3), pp. 193-201. ([link](http://dx.doi.org/10.1016/0022-4049%2873%2990009-1))


Math > history

< [[Math]]
[[!redirects Math -- history]]


math archives

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### Context
#### Mathematics
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For a separate list of math blogs and forums see [[math blogs]]. Main wikis should belong to both lists. For math societies and institutes see [[math institutions]]. There is also page of [[math resources by individuals]]. The level up is [[math resources]]. See also [[list of mathematical software]].

We list archives of **online** books and videos, not the catalogues of physical libraries. We shall not list commercial subscription lists (especially of current journals) except semi-commercial jstor and few publication sites by the main mathematical societies.

#Contents#
* table of contents
{:toc}

## Archives of articles, books, proceedings, videos

### Math and physics only (or mainly)

* [mathseminars.org](https://mathseminars.org)
* [arXiv](http://arxiv.org) archive of free preprints (math, physics, computational linguistics)

   * [front](http://front.math.ucdavis.edu) alternative user interface for arXiv

* [spires](http://www.slac.stanford.edu/spires) high energy physics publication database

* The SAO/[NASA Astrophysics Data System](http://www.adsabs.harvard.edu)   (it also contains abstracts, scans, citations of many papers in mathematical physics)

* [numdam](http://numdam.org) old issues of many French math journals including [[Cahiers]]

* [math-net.ru](http://www.mathnet.ru), [English version](http://www.mathnet.ru/index.phtml?&option_lang=eng) -- past years of main Russian math journals; tracks citations within the system; searchable also by name of a mathematician

* [Hopf archive](http://hopf.math.purdue.edu) topology and K-theory

* [K-theory](http://www.math.uiuc.edu/K-theory) archive

* [ICM archive](http://www.mathunion.org/ICM) past ICM proceedings

* project [euclid](http://projecteuclid.org/DPubS?Service=UI&version=1.0&verb=Display&handle=euclid&page=browse) list of journals (colors denote levels of free access)

* [journal list](http://www.ams.org/dmr/JournalList.html) at Amer. Math. Soc. digital mathematical registry; and [source list](http://www.ams.org/dmr/SourceList.html) (publishers and organizations behind sci journals)

* [Matematicheskie Zametki](http://www.mathnet.ru/php/journal.phtml?jrnid=mzm&option_lang=eng): Mathematical Journal published by Russian Academy of Sciences containing research work discussing [vedic maths tricks] (http://mental-maths-trick.blogspot.com/2011/10/do-super-quick-maths-calculation-using.html) and [whitepaper] (http://magazine.amstat.org/blog/2011/05/01/white-paper/).

* Polish virtual [math library](http://matwbn.icm.edu.pl/index.php?jez=en) (cf. semi-commercial [impan journals](http://journals.impan.gov.pl)) -- Polish journals

* *huge* MPIM list of [electronic journals](http://rzblx1.uni-regensburg.de/ezeit/fl.phtml?bibid=MPIMA&colors=3&lang=en&notation=SA-SP) -- most are  available only locally, but many free ones listed as well

* [lib.mexmat.ru](http://lib.mexmat.ru) Moscow MexMat electronic library: most items available only locally, however some items and the indexes are available

* [ELibM](http://www.emis.de/ELibM.html) -- open access Electronic Library of Mathematics at [emis](http://www.emis.de)

* [archive of category list](http://blog.gmane.org/gmane.science.mathematics.categories)
* [tac](http://www.tac.mta.ca/tac) -- Theory and Application of Categories, free journal and a category [reprint](http://www.tac.mta.ca/tac/reprints/index.html) collection
* [MSRI books](http://library.msri.org/books)
* [eprint network](http://www.osti.gov/eprints/pathways/mathematics.shtml)
* [Aarhus](http://math.au.dk/forskning/publikationer/instituttets-serier/year/2010/series/pp/) math institute, preprints and dissertations
* encyclopaedias in Russian online: [dic.academic.ru](http://dic.academic.ru)

### General archives: not math/physics centered

* [JSTOR](http://www.jstor.org) -- a big repository of past issues of main, mainly American and English journals -- requires institution subscription
* [DigiZeitschrift](http://www.digizeitschriften.de), Das Deutsche Digitale Zeitschriftenarchiv; includes [39 math journals](http://www.digizeitschriften.de/main/searchcol/?tx_goobit3_search[extquery]=DOCSTRCT%3Aperiodical&DC=510.mathematics)
* [Gallica](http://gallica.bnf.fr) repository of free French books (in general area); includes old issues of Comptes Rendus
* [Goettingen digital library](http://gdz.sub.uni-goettingen.de) -- mainly German journals
* [Archive for Mathematical Sciences & Philosophy](http://www.archmathsci.org) -- video archive

## General lists of math resources
* [Math on the Web](http://www.mathontheweb.org/mathweb/index.html) links maintained by American Mathematical Society

* [fsu vmlib](http://www.math.fsu.edu/Virtual) Florida State University Virtual Math Library -- long list of resources

## Major individual handbooks and tables online

* [Abramowitz-Stegun](http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP), Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables
* [Handbook of K-theory](http://www.math.uiuc.edu/K-theory/handbook)

## Reviews and abstracts

* [mathscinet](http://www.ams.org/mathscinet/search)

* [Zentralblatt](http://www.zentralblatt-math.org/zmath/en/advanced)

* [MSC2010](http://msc2010.org/Default.html) classification

* [Referativnii Zhurnal Matematika](http://www.viniti.ru/pro_ref_el.html)

## Wikis and encyclopedia-like collections

* [eom.springer.de](http://eom.springer.de)  encyclopaedia of math articles with approximately 8000 entries

* [[HomePage|nlab]] this wiki on higher category theory and applications
* Joyal's CatLab: <http://www.ncatlab.org/joyalscatlab/published/HomePage>
[[!redirects math archive]]
[[!redirects math arxive]]
[[!redirects math arxives]]


math blogs

* [Joel David Hamkins](http://jdh.hamkins.org) "mathematics and philosophy of the infinite", foundations

* [Logic ForAll](http://logic-forall.blogspot.com) by [[Valeria de Paiva]]

* [Lovely little lemmas] (http://lovelylittlelemmas.rjprojects.net/) by Remy van Dobben de Bruyn

* [Low Dimensional Topology](http://ldtopology.wordpress.com) group blog by Ryan Budney, Nathan Dunfield, Jesse Johnson, Daniel Moskovich and Henry Wilton

* [mathbabe](http://mathbabe.org/) Blog on mathematical finance by former academic

* [Mathblogging.com](http://www.mathblogging.org/) A compilation of math blogs sorted according to genre including the "weekly pics" of the hosts: Felix Breuer, Frederik von Heymann, and Peter Krautzberger

* [math for love](http://mathforlove.com/) Blog of mathematics education consultants includes puzzles, teaching tips, and philosophical musings

* [Mathematics and Computation](http://math.andrej.com) (Andrej Bauer, includes also topology and foundations--last update July 2011)

* [Mathematics Under the Microscope](http://micromath.wordpress.com) (Alexandre Borovik)

* [Mathematics without Apologies](https://mathematicswithoutapologies.wordpress.com) (Michael Harris)

* [mathlight's blog](http://mathlight.wordpress.com) ([[Zoran Škoda]], noncommutative and derived geometry, mathematical physics, sci publishing)

* [Mathlinks / Art of Problem Solving](http://www.artofproblemsolving.com/Forum/portal.php?ml=1) community, math olympiad style problems discussed

* [Mathlog](http://www.scienceblogs.de/mathlog) (in German, variety of topics from Open Access journals to Klein groups and geometry)

* [Math Vault](http://mathvault.ca) (Everything higher mathematics: college math, calculus, linear algebra, logic, probability, statisitcs, proof-based math, educational technology and learning)

* [neverendingbooks](http://www.neverendingbooks.org) (Lieven le Bruyn)

* [Noncommutative Geometry](http://noncommutativegeometry.blogspot.com) (Connes, Khalkhali...)

* [Nuit Blanche](http://nuit-blanche.blogspot.com) (data/imager/sensory compression, pattern recognition, networking, internet, AI/machine learning, neuroscience)

* [Oscar Cunningham's Blog](http://oscarcunningham.com/) (Category theory, nontrivial trivialities)

* [Peter Cameron's Blog] (http://cameroncounts.wordpress.com/page/6/) notes on group theory, graph theory, number theory, and combinatorics

* [Phd plus Epsilon](http://blogs.ams.org/phdplus/) Adriana Salerno reflects on her life as a junior faculty member at a Liberal Arts college

* [Proof Math is Beautiful](http://proofmathisbeautiful.tumblr.com) Beautiful images of mathematical interest.

* [Quomodocumque](http://quomodocumque.wordpress.com/) University professor Jordan Ellenberg's blog

* [Secret Blogging Seminar](http://sbseminar.wordpress.com) run by 8 recent Berkeley Ph.D. graduates. "Sort of like a seminar, but with (even) more rude commentary from the audience."

* [Shadows of Simplicity](http://tcjpn.wordpress.com) - notes on analytic, algebraic, geometric, and topological enumerative combinatorics, fractional and operational calculus, and umbral calculus

* [SymOmega](http://symomega.wordpress.com) (root systems, Weyl groups, buildings, algebraic groups...)

* [Tanya Khovanova's math blog](http://blog.tanyakhovanova.com/) Light musings, puzzles and anecdotes concerning mathematics

* [The accidental mathematician](http://ilaba.wordpress.com) (Izabella Laba: personal blog of a math professor, various topics, rarely even math)

* [theHigherGeometer](https://thehighergeometer.wordpress.com/) ([[David Roberts]])

* [The Math Less Travelled](http://mathlesstraveled.com/) Expository writing aimed to expose the beauty of mathematics to a general audience

* [The n-Category Café](http://golem.ph.utexas.edu/category) A group blog on math, physics and philosophy written by (currently) seven people and affiliated with this wiki

* [The Polymath Blog](http://polymathprojects.wordpress.com) (Tim Gowers "polymath" group projects: number theory, polynomials, algorithms, complexity, ...--last updated July 2011)

* [Three-Toed Sloth](http://bactra.org/weblog)  (Cosma Shalizi: books, science, automata, complex networks, stochastic processes, numerics and simulation, arts, literature, politics)

* [Transcendence](http://whzecomjm.com/) A math blog by whzecomjm covering Algebra, Number Theory etc. in Chinese.

* [Vi Hart's blog](http://vihart.com/) Great videos and photos of mathematical art and story-telling

* [Victor Porton's math blog](http://portonmath.wordpress.com) (filters, posets, funcoids and reloids), abundant with self-centric claims of importance

* [What's new](http://terrytao.wordpress.com) (Terence Tao)

* [XOR's hammer](http://xorshammer.wordpress.com) (logics and foundations)

### Older Blogs (last entry is over a year old)

* [Annoying Precision](http://qchu.wordpress.com) by Qiaochu Yuan (algebra, representation theory, combinatorics, abstract nonsense, arithmetics)

* [Ars Mathematica](http://arsmathematica.net) (analysis, probability...)  - last updated in 2015

* [category theory](http://community.livejournal.com/category_theory) at community.livejournal (in Russian)

* [Climbing Mount Bourbaki](http://amathew.wordpress.com) (Akhil Mathew: Lie algebras, differential and algebraic geometry...)  - last updated in 2013

* [Concrete Nonsense](http://concretenonsense.wordpress.com) A group blog on combinatorics, [[Lie algebras]], [[representation theory]], [[algebraic geometry]] -- last update 2015

* [CSE Blog - quant, math, computer science puzzles](http://pratikpoddarcse.blogspot.com) ~ 250 Quant, Math & Computer Science Puzzles for Interview Preparation & Brain Teasing

* [Delta Epsilons](http://deltaepsilons.wordpress.com) A group blog focusing on [[algebra]], representation theory, and Olympiad-style problem-solving --last update August 2010

* [degrees of freedom](http://blogs.scientificamerican.com/degrees-of-freedom/) Math and Physics blog of Scientific American written by  Davide Castelvecchi - last updated 2012

* [Embûches tissues](http://embuchestissues.wordpress.com) (computational algebraic geometry, commutative algebra) -- last update March 2010

* [Evolutionblog](http://scienceblogs.com/evolutionblog/) Mathematics and Politics blog by a college professor

* [FFF](http://ffbandf.wordpress.com) Forking, Forcing and back&Forthing -- a blog by 10 model theorists - last updated in 2013

* [Geometric Group Theory](http://392c.wordpress.com) (structured text with theorems, proofs and exercises in GGT) -- last updated August 2009

* [Geometry and combinatorics](http://konradswanepoel.wordpress.com) (Konrad Swanepoel) - last updated 2014

* [Geometry and the Imagination](http://lamington.wordpress.com) (Danny Calegari: hyperbolic geometry, geometric group theory...)

* [God plays dice](https://gottwurfelt.com/) and [previous location](http://godplaysdice.blogspot.com) (probability, combinatorics, puzzles, financial math)

* [Höchste Zeit, unsern Kurs zu ändern](http://shenme.de/blog) -- model theory of fields and commutative algebra

* [IMU - Journals](https://blog.wias-berlin.de/imu-journals) blog on mathematical journals

* Konrad [Volkel](http://blog.konradvoelkel.de) blog (cohomology, geometry, topology); has also nice [math resource list](http://blog.konradvoelkel.de/2012/05/mathematics-resources) - blog last updated in 2015

* [Le Petit Chercheur Illustré](http://yetaspblog.wordpress.com) (Signal processing and applied math, compressed sensing and inverse problems, random embeddings)

* [Martin's Blog](http://www.martinorr.name/blog) Martin Orr: algebraic geometry especially abelian varieties, Galois representations

* [Math-Blog](http://math-blog.com) (6 people blog: math books, university math, popular math etc.)

* [Mathcraft](http://mathcraft.wonderhowto.com/) Math-inspired artwork and how-to info

* [Math Goes Pop!](http://www.mathgoespop.com/) UCLA grad student writes about math in the media and pop culture

* [Math Monday](http://momath.org/home/math-monday/) The Museum of Mathematics will open in 2012 -- this blog details its progress and fun math projects

* [Math Section](http://mathsection.com/) ETH Zurich Master's degree student writes about applications of mathematics in everyday life

* [Mathematical Formalities](http://categorymath.wordpress.com/) This blog is written by [[Sanath Devalapurkar]], presenting some of his ideas and thoughts on (higher) category theory and $K$-theory.

* [Mathematical Musings](http://matthewkahle.wordpress.com/)
(Interactions of probability, statistical physics, topology, and combinatorics. Matthew Kahle)

* [Maxwell's Demon](http://www.maxwelldemon.com) (Edmund Harriss: vizualizing math and math art, building math sculptures)

* [Misapplied Math](http://www.klittlepage.com) (Math (mainly statistics, probability, and stochastic control), data science, and CS)

* [Motivic Stuff](http://homotopical.wordpress.com) (Andreas Holmstrom writes on Cohomology, homotopy theory, and arithmetic geometry; see also the related [Cohomology Theory Database](http://notes.andreasholmstrom.org/index.php) by the same author)

* [Musings Mathematical and Otherwise](http://edwardfhughes.wordpress.com) (Edward Hughes - Algebraic Geometry and String Theory)

* [My research in geometry](http://alekskleyn.blogspot.com) (Aleks Kleyn)

* [NAPmath - math and technology in education] (http://NAPmath.wordpress.com)

* [Nathaniel Johnston's Blog](http://www.nathanieljohnston.com) (PhD student in quantum information -- last update July 2011)

* [Nikita Markarian's mathblog](http://nikitamarkarian.wordpress.com) (derived algebraic geometry, dg-, $L_\infty, A_\infty, G_\infty$-algebras, operads, TQFT, BV formalism, Hochschild and cyclic (co)homology...--last updated July 2011)

* [Noncommutative geometry with point](http://noncommutativeag.wordpress.com), by a grad student at Kansas--last updated September 2010

* [NUMBERPLAY](http://wordplay.blogs.nytimes.com/category/numberplay/) A New York Times blog on math/numeric puzzles by Pradeep Mutalik

* [Paramanand's Math Notes] (http://paramanands.blogspot.com/) A math blog by Paramanand Singh covering Algebra, Number Theory, Mathematical Analysis, Geometry among other topics

* [Political Math](http://www.politicalmathblog.com/) Visualization of political information

* [Pronunciation of Mathematicians' Names] (http://pronouncemath.blogspot.com) A blog about how to pronounce famous mathematicians' names.

* [QED Insight](http://qedinsight.wordpress.com) (by Santo D'Agostino; mathematics, physics, education, pitched mainly to high school and undergraduate university students and teachers, with some material for lay people as well)

* [qrazydragon](http://qrazydraqon.livejournal.com) (math, books, internet, in Russian)

* [Reverberations in Mathematics](http://julianrachman.wordpress.com) (My Journey into Higher Mathematics - [Julian Rachman](http://jrmathematics.x10host.com)

* [Rigorous Trivialities](http://rigtriv.wordpress.com) (a group blog - mainly algebraic geometry, including well-organized introductory notes [algebraic geometry from the beginning](http://rigtriv.wordpress.com/ag-from-the-beginning) -- last updated 2015)

* [Sketches of Topology](http://sketchesoftopology.wordpress.com) (colorful pictures, "visualizations of low dimensional topology")

* [The Everything Seminar](http://cornellmath.wordpress.com) ("Geometry, topology, categories, groups, physics"; recently lots about experimental particle physics: LHC and CMS)

* [The Numbers Guy](http://blogs.wsj.com/numbersguy/) Wall Street Journal's math/economics blog by Carl Bialik

* [The Unapologetic Mathematician](http://unapologetic.wordpress.com) (An expository "blath" by former academic John Armstrong) last updated in 2012

* [The Rising Sea](http://therisingsea.org) (Daniel Murfet, previous blog moved to more of a homepage format, with arXiv of neat LaTeX notes on algebraic geometry, triangulated categories, sheaves...)

* [Theoretical atlas](http://theoreticalatlas.wordpress.com) (Jeffrey Morton: groupoids, groupoidification, stacks, quantum physics, noncommutative geometry)

* [Things of interest](http://blog.matthen.com/) (Mathematica visualisations, puzzles)

* [Todd and Vishal's blog](http://topologicalmusings.wordpress.com)--last updated January 2011

* [Uncover a few](http://uncoverafew.wordpress.com/) New blog of problems and teaching ideas

* [Vivatsgasse 7](http://vivatsgasse7.wordpress.com) (Max Planck Bonn graduate students-- last update November 2007)

* [Wiskundemeisjes](http://www.wiskundemeisjes.nl) (A blog about mathematics for a general audience, written by Jeanine Daems and Ionica Smeets, in Dutch)

## Probability and Statistics

### Blogs

* [Statistical Modeling, Causal Inference and Social Science](http://www.stat.columbia.edu/~gelman/blog)

* [The Endeavour](http://www.johndcook.com/blog)

### Wikis

* [Wikitistics - Statistics Wiki](http://statistics.wikia.com/wiki/Main_Page)

## Math-related

### Blogs

* [Mathematics in Australia](http://austmaths.wordpress.com)

* [Out of print math](http://outofprintmath.blogspot.com)

* [Wild About Math](http://wildaboutmath.com/)

* [Mental Maths Games For 11 Year Olds](http://mental-maths-trick.blogspot.com/2014/11/mental-maths-aged-11-years-old-fun.html)

* [Sage: Open Source Mathematical Software](http://sagemath.blogspot.com)

* [Transient Dynamic](http://robclewley.github.io/) (A blog on next-gen computational tools for mathematical modeling in complex systems, especially dynamical systems models)

## Technical help with math blogs

* Luca Trevisan's [LaTeX to Wordpress converter](http://lucatrevisan.wordpress.com/latex-to-wordpress)

* This nifty [script] (http://www.junkdrawerblog.com/2008/01/cant-read-light-text-on-dark-backgrounds.html) fixes the eyestrain caused by poorly designed blog [contrasts](http://blog.teleonetics.com/2008/02/28/concept-identity-vs-concept-similarity-2/). 15 seconds to install, works for Firefox only.

* [Detexify](http://detexify.kirelabs.org/classify.html) for finding LaTeX symbols by drawing them.

category: meta, reference

[[!redirects Online]]
[[!redirects Online resources]]
[[!redirects online resources]]
[[!redirects Online Resources]]
[[!redirects math blogs and wikis]]
[[!redirects mathematical blogs]]
[[!redirects Math Blogs]]
[[!redirects Math blogs]]

[[!redirects Online Resources]]