measuring

Measuring of coalgebras of algebra morphisms and algebras was introduced by Sweedler in

* [[M. E. Sweedler]], __Hopf algebras__, Mathematics Lecture Note Series, Benjamin 1969.

Given a commutative ring $k$, a $k$-coalgebra $(C,\Delta_C,\epsilon)$ and $k$-algebras $(A',\mu',\eta')$ and $(A,\mu,\eta)$, a __$C$-measuring__ from $A'$ to $A$ is a $k$-linear map $f: C\to Mod_k(A',A)$ such that
$$
ev\circ (f\otimes\mu_{A'}) = \mu_A\circ (ev\otimes ev)\circ(f\otimes A'\otimes f\otimes A')\circ (C\otimes \tau_{A',C}\otimes A')\circ (\Delta_C\otimes A'\otimes A') : C\otimes A'\otimes A'\to A,
$$
and $f\circ \eta' = \eta\circ\epsilon_C:C\to A$ where $\tau$ is the transposition of factors and $ev$ the evaluation of a map. In Sweedler notation, the requirements on $f$ are
$f(c)(a\cdot_{A'} b) = f(c_{(1)})(a)\cdot_A f(c_{(2)})(b)$
and $f(c)(1_{A'}) = \epsilon(c)1_{A}$.

We can instead of $f:C\to Mod_k(A',A)$ consider its adjoint map $\tilde{f}:C\otimes A'\to A$ and the convolution algebra $Mod_k(C,A)$.
Then, $f$ is a measuring (or some say taht $\tilde{f}$ is a measuring iff the other adjoint $\tilde{f}^*: A'\to Mod_k(C,A)$ is a map of algebras.

As a special case, we can consider the case of the identity map $A\overset{A}\to A$ and talk about a $C$-measuring of a monoid $A$, including in the general case of a symmetric monoidal category $D$ with unit $\mathbf{1}$, symmetry $\tau$, left and right unit coherences $l$ and $r$,
a map $\triangleright : C\otimes A\to A$ of a (counital) is a __measuring__ of the comonoid $C$ on a (unital) monoid $(A,\mu_A,\eta_A)$, or we say that $C$ *measures* $A$ if
$$
C\triangleright \mu_A = \mu_A\circ(\triangleright \otimes \triangleright)\circ (C\otimes \tau\otimes A)\circ (\Delta_C\otimes A\otimes A) : C\otimes A\otimes A\to A,
$$
and
$$
\triangleright \circ (C\otimes \eta) = \eta\circ r_C\circ (\epsilon\otimes \mathbf{1}): C\otimes \mathbf{1}\to A.
$$
In Sweedler notation, we also write $c\triangleright (ab) =
\sum (c_{(1)}\triangleright a)(c_{(2)}\triangleright b)$ and
$c\triangleright 1_A = \epsilon(c) 1_A$. Note that $C$ is not a monoid so a measuring is not an action (although the notation may suggest this). However if $C$ is a bialgebra, we may consider when it is an action.
A [[Hopf action]] is a measuring which is also an action of a <em> bimonoid </em> $B=(C,\Delta_C,\epsilon,\mu_C,\eta_C)$. In other words, $A$ is a monoid in the monoidal category of $B$-modules (also called $B$-module monoid or $B$-[[module algebra]]).

Measurings of algebras by bialgebras are used to define the (cocycled) [[crossed product algebra]]s, see also [[cleft extension]] (of an algebra by a bialgebra). For measurings and module algebras see

* S. Montgomery, Hopf algebras and their actions on rings, CBMS 82, AMS 1993.

* A. Klimyk, K. Schmüdgen, _Quantum groups and their representations_, Springer, 1997;

and for (co)module (co)algebras and generalizations see also

* [[Shahn Majid]], _Foundations of quantum group theory_, Cambridge University Press 1995, 2000.

There are also more elaborate versions of measurings, which play role in a Galois theory, see

* [[Tomasz Brzeziński]], _On modules associated to coalgebra Galois extensions, J. Algebra, 215, no. 1, 1999, 290-317.

Other works include

* [[Martin Hyland|M. Hyland]], I. Lo'pez Franco, C. Vasilakopoulou, _Hopf measuring comonoids and enrichment_, Proc. Lond. Math. Soc. (3) 115:5 (2017) 1118--1148

If $U$ and $V$ are $k$-algebras measured by a $k$-coalgebra $C$ and $M$ is a $U$-$V$-bimodule, then we say that $M$ is measured by $C$, if there is a $k$-bilinear map $\triangleright_M:C\otimes M\to M$ such that for all $u\in U$, $v\in V$, $m\in M$, $c\in C$,

$$
c\triangleright (u m v) = \sum (c_{(1)}\triangleright_U u) (c_{(2)}
\triangleright_M m)(c_{(3)}\triangleright_V v)
$$

If $C$ is in addition a $k$-bialgebra and $\triangleright_U,\triangleright_V,\triangleright_M$ actions of $H$, we say that a measuring is __Hopf action__ of $C$ on the $U$-$V$-bimodule $M$.
For Hopf actions on bimodules, one can define a
bimodule version of a (Hopf) [[smash product algebra]], see [[zoranskoda:Hopf smash product bimodule]].

There is a dual notion of a comeasuring, see

* [[Mitsuhiro Takeuchi]], _Matched pairs of groups and bismash products of hopf algebras_, Comm. Alg. __9__:8 (1981) 841--882 [doi](https://doi.org/10.1080/00927878108822621)

See also related $n$Lab page [[measuring coalgebra]] about the related enrichment (in the cocommutative case) and wikipedia:[measuring coalgebra](https://en.wikipedia.org/wiki/Measuring_coalgebra) which has a bit of both.

Given an algebra $A$ and a coalgebra $C$, a map
$\rho:C\to A\otimes C$
is a left __$A$-comeasuring__ of $C$ if
$$
\sum c^A\otimes c^C_{(1)} \otimes c^C_{(2)}
= \sum (c_{(1)})^A (c_{(2)})^A\otimes c_{(1)}^C\otimes c_{(2)}^C
$$
and $(A\otimes\epsilon)\circ\rho = \epsilon$,
where we denote $\rho(c) = \sum c^A\otimes c^C$ and $\Delta(c) = \sum c_{(1)}\otimes c_{(2)}$.
If $A$ is a bialgebra then $A$-comeasurings of coalgebras (with additional data) appear in the definition of (cocycle) cross coproduct coalgebras.


measuring coalgebra

[[!redirects measure coalgebra]]

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# Measuring coalgebras
* table of contents
{: toc}

## Idea

_Measuring coalgebras_ are an [[enriched category|enrichment]] of the category of [[commutative rings]] (or commutative $\mathbb{Z}$-algebras) in the [[cartesian closed category]] $k$ [[Cocomm Coalg]] of [[cocommutative coalgebra|cocommutative]] [[coalgebras]] (which we will write simply as $Coalg$), given a [[ground field]] $k$.

The starting point is the observation that the category $Coalg$ acts on the category [[CommAlg|Alg]] of [[commutative algebras]]: there is a functor

$$\{-, -\}: Coalg^{op} \times Alg \to Alg$$

where, given a coalgebra $C$ and an algebra $A$, $\{C, A\}$ is the [[abelian group|abelian-group]] [[hom-object|hom]] of additive homomorphisms $f: C \to A$, made into an algebra whose multiplication $f \cdot g$ is given by

$$C \overset{d}{\to} C \otimes C \overset{f \otimes g}{\to} A \otimes A \overset{m}{\to} A$$

where $d$ is the coalgebra comultiplication and $m$ is the algebra multiplication. That this is an "[[action]]" here means that there is a
[[natural isomorphism]]

$$\{C \otimes D, A\} \cong \{C, \{D, A\}\}$$

of algebras; here $Alg$ is sometimes described as an [[actegory]] over $Coalg$.

## Definition

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

Given two algebras $A, B$, the **measuring coalgebra** $\mu(A, B)$ is by definition the [[representing object]] of the [[functor]]

$$Alg(A, \{-, B\}): Coalg^{op} \to Set$$

so that there is an [[isomorphism]], [[natural isomorphism|natural]] for coalgebras $C$, of the form

$$Coalg(C, \mu(A, B)) \cong Alg(A, \{C, B\})$$

=--

Assume the existence of [[equalizers]] in $Coalg$, and of a [[right adjoint]]

$$Cof: Vect \to Coalg$$

to the [[forgetful functor]] $U: Coalg \to Vect$ (the [[cofree object|cofree]] cocommutative coalgebra construction). We let

$$\pi: U \circ Cof \to 1_{Vect}$$

denote the [[counit of an adjunction|counit of the adjunction]] $U \dashv Cof$.

We construct $\mu(A, B)$ explicitly as the [[equalizer]] in $Coalg$ of a pair of maps of the form

$$Cof(B^A) \overset{\to}{\to} Cof(B^{A \otimes A}) \times Cof(B^k)$$

where we denote the [[internal hom]] in [[Vect]] by [[exponential object|exponentiation]] (and we recall here that the [[cartesian product]] in $Coalg$ is given by [[tensor product]] at the level of $Vect$). The first of these maps is

$$\langle Cof(B^{m_A}), Cof(B^{u_A}) \rangle: Cof(B^A) \to Cof(B^{A \otimes A}) \times Cof(B^k)$$

where $m_A: A \otimes A \to A$ is the multiplication on $A$ and $u_A: k \to A$ is the unit. The second is given by a pair of maps

$$\langle \Phi, \Psi \rangle$$

which we now describe separately.

The map $\Phi: Cof(B^A) \to Cof(B^{A \otimes A})$ is the unique coalgebra map such that $U \Phi$ lifts the map

$$U Cof(B^A) \overset{\delta}{\to} U Cof(B^A) \otimes U Cof(B^A) \overset{\pi \otimes \pi}{\to} B^A \otimes B^A \overset{\otimes_1}{\to} (B \otimes B)^{A \otimes A} \overset{m_{B}^{A \otimes A}}{\to} B^{A \otimes A}$$

through $\pi: U Cof(B^{A \otimes A}) \to B^{A \otimes A}$. Here $\delta$ denotes the comultiplication (same as the diagonal map as seen in $Coalg$), and $\otimes_1$ indicates the structure of enriched functoriality for $\otimes$.

The map $\Psi: Cof(B^A) \to Cof(B^k)$ is the unique coalgebra map such that $U \Psi$ lifts the map

$$U Cof(B^A) \overset{\varepsilon}{\to} k \overset{u_B}{\to} B \cong B^k$$

through $\pi: U Cof(B^A) \to B^A$. Here $\varepsilon$ denotes the counit (same as the unique map to the terminal object as seen in $Coalg$).

## Enrichment of algebras in coalgebras

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

The measure coalgebra $\mu(A, B)$ indeed gives an [[enriched category|enrichment]]

$$
  \mu(-, -): Alg^{op} \times Alg \to Coalg
  \,.
$$

=--

Here the composition law in $Coalg$

$$\mu(A_0, A_1) \times \mu(A_1, A_2) \to \mu(A_0, A_2)$$

(recalling that the [[product]] in $Coalg$ is the [[tensor product]] of the underlying additive groups) is derived by universality from a composition of maps:

$$\array{
Coalg(C, \mu(A_0, A_1) \times \mu(A_1, A_2)) & \cong & Coalg(C, \mu(A_0, A_1)) \times Coalg(C, \mu(A_1, A_2)) & (Coalg(C, -) preserves products)\\
 & \cong & Alg(A_0, \{C, A_1\}) \times Alg(A_1, \{C, A_2\}) & (definition of \mu) \\
 & \to & Alg(A_0, \{C, A_1\}) \times Alg(\{C, A_1\}, \{C, \{C, A_2\}\}) & (functoriality of \{C, -\})\\
 & \to & Alg(A_0, \{C, \{C, A_2\}\}) & (composition law)\\
 & \cong & Alg(A_0, \{C \otimes C, A_2\}) & (actegory constraint)\\
 & \to & Alg(A_0, \{C, A_2\}) & (using d: C \to C \otimes C)\\
 & \cong & Coalg(C, \mu(A_0, A_2)) & (definition of \mu)
}$$

## Literature

* M. E. Sweedler, _Hopf algebras_, Mathematics Lecture Note Series, W. A. Benjamin, 1969
* [[Mathieu Anel]], [[André Joyal]], _Sweedler Theory for (co)algebras and the bar-cobar constructions_, [arXiv:1309.6952](https://arxiv.org/abs/1309.6952)

A dual notion to measuring coalgebra is [[Tambara's universal coacting bialgebra]] from

* [[Daisuke Tambara]], _The coendomorphism bialgebra of an algebra_, J. Fac. Sci. Univ. Tokyo Sect. IA Math, __37__, 425-456, 1990 [pdf](https://repository.dl.itc.u-tokyo.ac.jp/record/39399/files/jfs370210.pdf)
* [[Martin Hyland]], [[Ignacio López Franco]], [[Christina Vasilakopoulou]], _Hopf measuring comonoids and enrichment_, Proc. London Math. Soc. __115__:5 (2017) 1118-1148, [doi](https://doi.org/10.1112/plms.12064)
* [[Christina Vasilakopoulou]], _Enrichment of categories of algebras and modules_, [arXiv:1205.6450](https://arxiv.org/abs/1205.6450)
* [[Marjorie Batchelor]], _Measuring coalgebras, quantum group-like objects, and non-commutative geometry_, In: Bartocci, C., [[Ugo Bruzzo|Bruzzo, U.]], Cianci, R. (eds) Differential Geometric Methods in Theoretical Physics. Lecture Notes in Physics __375__ [doi](https://doi.org/10.1007/3-540-53763-5_45); _Measuring comodules - their applications_, J.  Geom. Physics __36__:3-4 (2009) 251-269 (doi); _Difference operators, measuring coalgebras and quantum group like objects_, Adv. Math. __105__, 190-218 (1994) [pdf](https://core.ac.uk/download/pdf/81927027.pdf)
* Maximilien Péroux, _The coalgebraic enrichment of algebras in higher categories_, J. Pure Appl. Alg. __226__:3 (2022) 106849 [doi](https://doi.org/10.1016/j.jpaa.2021.106849)
* Paige Randall North, Maximilien Péroux, _Coinductive control of inductive data types_, [arXiv:2303.16793](https://arxiv.org/abs/2303.16793)

[[!redirects measure coalgebras]]
[[!redirects measuring coalgebra]]
[[!redirects measuring coalgebras]]


mechanical system

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* table of contents
{:toc}

## Idea

In [[physics]] there is the notion of a _physical system_, the totality of all the entities and processes -- its _mechanics_ -- regarded as being part of the physics under discussion.

## Types of mechanical systems

* [[classical mechanical system]]

* [[quantum mechanical system]]

* [[subsystem]], [[composite system]]

## Examples

* [[Harmonic oscillator]]

[[!redirects mechanical systems]]

[[!redirects physical system]]
[[!redirects physical systems]]


mechanics

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#Contents#
* table of contents
{:toc}

## Idea

In [[physics]] the word _mechanics_ refers essentially to every formalization of [[dynamics]]. Often the use of the word it may be restricted to mean the dynamics of point [[particle]]s or [[rigid body dynamics]].

Historically one distinguishes between _[[classical mechanics]]_ and _[[quantum mechanics]]_ . The full theory of dynamics is nature is given by quantum mechanics, but classical mechanics (named after its full development at the end of the 19th century, before the insights that led to its refinement to quantum mechanics) provides an accurate approximation for "macroscopic" or otherwise "large" systems.

## Related entries

* [[classical mechanics]], [[quantum mechanics]]

  * [[kinematics]], [[dynamics]]

  * [[rigid body dynamics]]

  * [[continuum mechanics]]

* [[Hamiltonian mechanics]], [[Lagrangian mechanics]]

* [[matrix mechanics]]


mediality

A [[binary algebraic structure]] $(A,\cdot)$ satisfies the medial law or __mediality__ if for all $a,b,c,d\in A$
$$
(a\cdot b)\cdot(c\cdot d) = (a\cdot c)\cdot (b\cdot d)
$$
Medial unital magmas are, by [[Eckmann-Hilton argument]], automatically Abelian [[monoid]]s. There is a relation between medial quasigroups and
Abelian groups, given by [[Bruck-Toyoda theorem]].

Every transpose (conjugate, in the sense of associated ternary relation) conditions to mediality is also the mediality condition.
An old term for mediality is also entropy condition/property ("entropic quasigroups").

* Maciej Niebrzydowski, Józef H Przytycki, _Entropic magmas, their homology and related invariants of links and graphs_, Algebraic & Geometric Topology __13__ (2013) 3223–3243 [doi](http://dx.doi.org/10.2140/agt.2013.13.3223)

category: algebra

[[!redirects medial law]]
[[!redirects medial quasigroup]]
[[!redirects medial loop]]
[[!redirects medial magma]]
[[!redirects entropic magma]]
[[!redirects entropic quasigroup]]
[[!redirects entropic loop]]


Mee Seong Im

* [personal page](https://sites.google.com/site/meeseongim/)

## Selected writings

Discussion of [[non-deterministic automata]] as 1-dimensional [[defect QFT|defect]] [[TQFTs]]:

* Paul Gustafson, [[Mee Seong Im]], Remy Kaldawy, [[Mikhail Khovanov]], Zachary Lihn, *Automata and one-dimensional TQFTs with defects* [[arXiv:2301.00700](https://arxiv.org/abs/2301.00700)]

Review:

* [[Mee Seong Im]], *Correspondence between automata and one-dimensional Boolean topological theories and TQFTs*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#ImMar2023), video:[YT](https://www.youtube.com/watch?v=q5kynYxwTmk)]

category: people


Meer Ashwinkumar

* [institute page](https://db.ipmu.jp/member/personal/6568en.html)

## Selected writings

On [[semi-holomorphic 4d Chern-Simons theory]] and relation to [[integrable systems]]:

* [[Meer Ashwinkumar]], _Integrable Lattice Models and Holography_ ([arXiv:2003.08931](https://arxiv.org/abs/2003.08931))

* [[Meer Ashwinkumar]], Jun-ichi Sakamoto, [[Masahito Yamazaki]], *Dualities and Discretizations of Integrable Quantum Field Theories from 4d Chern-Simons Theory* [[arXiv:2309.14412](https://arxiv.org/abs/2309.14412)]

On [[D-brane]]-realizations of [[semi-holomorphic 4d Chern-Simons theory]]:

* [[Meer Ashwinkumar]], [[Meng-Chwan Tan]], Qin Zhao, _Branes and Categorifying Integrable Lattice Models_ ([arXiv:1806.02821](https://arxiv.org/abs/1806.02821))

and specifically with relation to the [[quantum geometric Langlands correspondence]]:

* [[Meer Ashwinkumar]], [[Meng-Chwan Tan]], _Unifying Lattice Models, Links and Quantum Geometric Langlands via Branes in String Theory_ ([arXiv:1910.01134](https://arxiv.org/abs/1910.01134))

On regularized [[quantization]] of [[Green-Schwarz sigma-models]] for [[super p-branes]], specifically of the [[super 4-brane in 9d]], in generalization of the [matrix model of the M2-brane](M2-brane#QuantizationViaBFSSMatrixModel):

* [[Meer Ashwinkumar]], [[Lennart Schmidt]], [[Meng-Chwan Tan]], _Matrix Regularization of Classical Nambu Brackets and Super $p$-Branes_, J. High Energ. Phys. **2021** 172 (2021) [[arXiv:2103.06666](https://arxiv.org/abs/2103.06666), doi:10.1007/JHEP07(2021)172]

reviewed in:

* [[Meer Ashwinkumar]], *Matrix Quantization of Classical Nambu Brackets and Super $p$-Branes*, talk at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#2023)*, New York University, Abu Dhabi (2023) [[web](/nlab/show/M-Theory+and+Mathematics#Ashwinkumar2023)]

## Related $n$Lab entries

* [[semi-holomorphic 4d Chern-Simons theory]]

category: people


Mees de Roo

* [inspire page](https://inspirehep.net/authors/1012250)

## Selected writings

On [[black brane|black]] [[D7-branes]] and [[D8-branes]]:

* {#BergshoeffRooGreenPapadopoulosTownsend96} [[Eric Bergshoeff]], [[Mees de Roo]], [[Michael Green]], [[George Papadopoulos]], [[Paul Townsend]], _Duality of Type II 7-branes and 8-branes_, Nucl. Phys. B **470**  (1996) 113-135 [[arXiv:hep-th/9601150](https://arxiv.org/abs/hep-th/9601150)]

On [[double dimensional reduction]] of the [[M5-brane]] to the [[D4-brane]]:

* [[Eric Bergshoeff]], [[Mees de Roo]], [[Tomas Ortin]], *The Eleven-dimensional Five-brane*, Phys. Lett. B **386** (1996) 85-90 [[arXiv:hep-th/9606118](https://arxiv.org/abs/hep-th/9606118), doi:10.1016/0370-2693(96)00913-6]

Identification of the [[D8-brane]] as a [[black brane]] in [[Romans supergravity]]/[[massive type IIA string theory]]:

* {#BergshoeffdeBooGreenPapadopoulosTownsend96} [[Eric Bergshoeff]], [[Mees de Roo]], [[Michael Green]], [[George Papadopoulos]], [[Paul Townsend]], _Duality of Type II 7-branes and 8-branes_, Nucl. Phys. B **470** (1996) 113-135 [[arXiv:hep-th/9601150](https://arxiv.org/abs/hep-th/9601150)]

On [[intersecting branes|intersecting]] [[M-branes]]:

* [[Mees de Roo]], *Intersecting branes and Supersymmetry*, in: *Supersymmetry and Quantum Field Theory*, Proceedings of the D. Volkov Memorial Seminar Held in Kharkov, Ukraine, 5–7 January 1997, Springer (1998) [[arXiv:hep-th/9703124](https://arxiv.org/abs/hep-th/9703124), [doi:10.1007/BFb0105225](https://doi.org/10.1007/BFb0105225)]

* {#BREJS97} [[Eric Bergshoeff]], [[Mees De Roo]], [[Eduardo Eyras]], [[Bert Janssen]], [[Jan Pieter van der Schaar]], _Intersections involving waves and monopoles in eleven dimensions_, Class. Quantum Grav. **14** 2757 (1997) [[arXiv:hep-th/9704120](https://arxiv.org/abs/hep-th/9704120), [doi:0264-9381/14/10/005](http://iopscience.iop.org/0264-9381/14/10/005)]

On the [[non-abelian DBI action]]:

* [[Eric Bergshoeff]], [[Adel Bilal]], [[Mees de Roo]], A. Sevrin, _Supersymmetric non-abelian Born-Infeld revisited_, JHEP 0107, 029 (2001) [[arXiv:hep-th/0105274](https://arxiv.org/abs/hep-th/0105274)]

On [[D=8 supergravity]]:

* [[Mees de Roo]], [[Giuseppe Dibitetto]], Yihao Yin, _Critical points of maximal $D=8$ gauged supergravities_, JHEP 1201:029 (2012) [[arXiv:1110.2886](http://arxiv.org/abs/1110.2886)]

category: people

[[!redirects Mees De Roo]]


meet

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## Idea

In a [[preordered set]] or [[partially ordered set]] $P$, the _meet_ (or _infimum_, abbreviated _inf_, or _greatest lower bound_, abbreviated _glb_) of a [[subset]] $S$ of $P$ is, if it exists, the largest element of $P$ which is smaller or equal to all the elements in $S$. If this element is itself member of $S$, then it is also called the _[[minimum]]_ of that subset.

If we think of the pre-ordered set as a [[category]] (a [[(0,1)-category]]) then the meet is the [[limit]] over the given subset, if it exists, regarded as a [[diagram]]. Thus in a [[partially ordered set]] this is unique if it exists, otherwise it is unique up to [[isomorphism]].

## Definition

If $x$ and $y$ are elements of a [[partial order|poset]], then their **meet** is an element $x \wedge y$ of the poset such that:

* $x \wedge y \leq x$ and $x \wedge y \leq y$;
* if $a \leq x$ and $a \leq y$, then $a \leq x \wedge y$.

Such a meet may not exist; if it does, then it is unique.

In a [[preorder|proset]], a meet may be defined similarly, but it need not be unique.  (However, it is still unique up to the natural [[equivalence]] in the proset.)

The above definition is for the meet of two elements of a poset, but it can easily be generalised to any number of elements.  It may be more common to use 'meet' for a meet of finitely many elements and 'infimum' for a meet of (possibly) infinitely many elements, but they are the same concept.  The meet may also be called the **minimum** if it equals one of the original elements.

A poset that has all finite meets is a **meet-[[semilattice]]**.  A poset that has all infima is an **[[inflattice]]**.

A meet of [[subsets]] or [[subobjects]] is called an [[intersection]].

## Examples

### General

* A meet of no elements is a [[top]] element.

* Any element $a$ is a meet of that one element.

### Infimum of real numbers

Often one considers infima of subsets of the [[real numbers]] $\mathbb{R}$, regarded with their canonical [[preordering]], which in this case is in fact a [[total order]].

For $S \subset \mathbb{R}$ a subset, say that a _lower bound_ is an element $b \in \mathbb{R}$ such that $\underset{s \in S \subset \mathbb{R}}{\forall}( b \leq s )$.

Then the infimum of $S$ is, if it exists, that lower bound $inf(S)$ of $S$ such that for $b$ any other lower bound of $S$ then $b \leq inf(S)$.

See [[join#constructive]] for the case in [[constructive analysis]].

## Properties

As a poset is a special kind of [[category]], a meet is simply a [[product]] in that category.

## Related concepts

* [[join]]

* **meet**

[[!redirects meet]]
[[!redirects meets]]
[[!redirects infimum]]
[[!redirects infimums]]
[[!redirects infima]]
[[!redirects inf]]
[[!redirects infs]]
[[!redirects greatest lower bound]]
[[!redirects greatest lower bounds]]
[[!redirects glb]]
[[!redirects glbs]]
[[!redirects (0,1)-limit]]
[[!redirects (0,1)-limits]]


meet-semilattice object

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# Contents
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## Idea

The notion of a *meet-semilattice object* is the generalization of that of *[[meet-semilattice]]* as one passes from the ambient [[category of sets]] [[internalization|into]] more general ambient  [[categories]] with suitable properties.

## Definition

In a [[finitely complete category]] $C$, a **meet-semilattice object** is a [[cartesian monoidal preordered object]] that is also a [[partially ordered object]].

## See also

* [[cartesian monoidal preordered object]]

* [[partially ordered object]]

* [[join-semilattice object]]

* [[lattice object]]

* [[semilattice object]]

* [[meet-semilattice]]

[[!redirects meet-semilattice objects]]


Megan Shulman

* [webpage](http://www.ugcs.caltech.edu/~meg/)

## related $n$Lab entries

* [[equivariant cohomology theory]]

* [[Mackey functor]]

category: people


Mehran Jalali Farahani

* [institute page](https://www.maxwell.ac.uk/person/jalali-farahani-mehran/)

* [InSpire page](https://inspirehep.net/authors/2659192)

## Selected writings

On [[higher gauge theory]]:

* [[Leron Borsten]], [[Mehran Jalali Farahani]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Jiří Nárožný]], [[Dominik Rist]], [[Christian Saemann]], [[Martin Wolf]], *Higher Gauge Theory*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2401.05275](https://arxiv.org/abs/2401.05275)]

category: people

[[!redirects Mehran J. Farahani]]
[[!redirects Mehran Farahani]]


Mehrdad Shahshahani

* [personal page](http://math.ipm.ac.ir/shahshahani/)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=32512)

## Selected writings

On [[Cayley graph spectra]]:

* {#DiaconisShahshahani81} [[Persi Diaconis]], [[Mehrdad Shahshahani]], *Generating a random permutation with random transpositions*, Z. Wahrscheinlichkeitstheorie verw Gebiete 57, 159–179 (1981) ([doi:10.1007/BF00535487](https://doi.org/10.1007/BF00535487))

category: people


Mehrnoosh Sadrzadeh

* [webpage](https://msadrzadeh.com/)

category:people


Mei Chee Shum

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=32934)

## Selected writings

On [[tortile category|tortile]]/[[ribbon category|ribbon]] [[tensor categories]]:

* [[Mei Chee Shum]], *Tortile tensor categories*, Journal of Pure and Applied Algebra **93** 1 (1994) 57-110 [10.1016/0022-4049(92)00039-T]

category: people


Meister Eckhart

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## About

Meister Eckhart was a medieval christian [[mystic]].

> The German monk Meister Eckhart (1260-1327) was the first to develop the terminology of philosophy in German, translating and adapting Latin terms. For him, God becomes conscious of himself only within his creation. Eckhart also argues that Divine Knowledge is 'the negation of negation'. As with other mystics, Eckhart's aim was the unification of the soul with God. Christ is continually born within each believing soul.  ([source](https://www.marxists.org/reference/archive/smith-cyril/works/articles/interim.htm))

## Quotes

> {#UnverhuellteWahrheit} Denn, solange der Mensch dieser Wahrheit nicht gleicht, solange wird er diese Rede nicht verstehen; denn dies ist eine unverhüllte Wahrheit, die da gekommen ist aus dem Herzen Gottes unmittelbar.

> {#TheEye} The eye with which God sees me is the eye with which I see Him; my eye and His eye are one and the same. In righteousness I am weighed in God and He in me. If God did not exist nor would I; if I did not exist, nor would He. But there is no need to know this, for there are things that are easily misunderstood (and that can be grasped only in the concept). (quoted in [Hegel 1832, p 249](#Hegel1832))

> {#DiesZuWissenIstichtNot} Nun bitte ich euch, ebenso zu sein, auf daß ihr diese Rede verstehet; denn ich sage euch bei der ewigen Wahrheit: Wenn ihr dieser Wahrheit, von der wir nun sprechen wollen, nicht gleicht, so könnt ihr mich nicht verstehen. $[$...$]$ So denn sagen wir, daß der Mensch so arm dastehen müsse, daß er keine Stätte sei noch habe, darin Gott wirken könne. Wo der Mensch (noch) Stätte (in sich) behält, da behält er noch Unterschiedenheit. Darum bitte ich Gott, daß er mich Gottes quitt mache; denn mein wesentliches Sein ist oberhalb von Gott, sofern wir Gott als Beginn der Kreaturen fassen. In jenem Sein Gottes nämlich, wo Gott über allem Sein und über aller Unterschiedenheit ist, dort war ich selber, da wollte ich mich selber und erkannte mich selber (willens), diesen Menschen (= mich) zu schaffen. Und darum bin ich Ursache meiner selbst meinem Sein nach, das ewig ist, nicht aber meinem Werden nach, das zeitlich ist. Und darum bin ich ungeboren, und nach der Weise meiner Ungeborenheit kann ich niemals sterben. Nach der Weise meiner Ungeborenheit bin ich ewig gewesen und bin ich jetzt und werde ich ewiglich bleiben. Was ich meiner Geborenheit nach bin, das wird sterben und zunichte werden, denn es ist sterblich; darum muß es mit der Zeit verderben. In meiner (ewigen) Geburt wurden alle Dinge geboren, und ich war Ursache meiner selbst und aller Dinge; und hätte ich gewollt, so wäre weder ich noch wären alle Dinge; wäre aber ich nicht, so wäre auch »Gott« nicht: daß Gott »Gott« ist, dafür bin ich die Ursache; wäre ich nicht, so wäre Gott nicht »Gott«. Dies zu wissen ist nicht not. (Eckehart, Predigten _[Selig sind die Armen im Geiste](http://www.teeweg.de/de/varia/selig-die%20armen.html)_)

## Relation to Hegel
 {#RelationToHegel}

On the relation to [[Hegel]]:

> The final "Hermetic" period of Hegel's life is his time in Berlin, from 1818 until his death on November 14, 1831. This is contrary to what one might expect. It might be assumed that Hegel's "Hermeticism" was merely an aberration of youth, which the "arch rationalist" moved away from as he matured. Surprisingly, precisely the reverse seems to be the case. In Berlin, Hegel developed a friendship with Franz von Baader, the premiere occultist and mystic of the day. Together they studied Meister Eckhart. ([Magee 01, Introduction, section 1](#Magee01))

> Another Hegelian, Hans Martensen, author of one of the first scholarly studies of Meister Eckhart, remarked that "German mysticism is the first form in which German philosophy revealed itself in the history of thought" ("philosophy" for Hegelians generally means Hegel's Philosophy). Wilhelm Dilthey noted the same continuity between German mysticism and speculative philosophy. ([Magee 01, Introduction, section 1](#Magee01))

see also ([Hacket 13, pp. 629 "Eckhart reception in the 19th century"](#Hacket13))

## Related concepts

* [[mysticism]], [[hermeticism]], [[gnosticism]]

## References

* Wikipedia, _[Meister Eckhart](http://en.wikipedia.org/wiki/Meister_Eckhart)_

* {#Hegel1832} [[Georg Hegel]], _Lectures on the philosophy of religion_, Volume 1, 1832 ([pdf](http://hegel.net/hegelwerke/Hegel_Werke_Bd11_Religion_Bd1_1aufl_1832.pdf))

* {#Magee01} [[Glenn Alexander Magee]], _Hegel and the Hermetic Tradition_, Cornell University Press, 2001 ([introduction html](https://www.marxists.org/reference/subject/philosophy/works/en/magee.htm))

* {#Hacket13} Jeremiah Hackett, _A companion to Meister Eckhart_, 2013

The YouTube channel *Let's Talk Religion* offers a discussion of Eckharts philosophy and of contemporaries related to him, like Ibn Arabi and Marguerite Porete:

* Let's Talk Religion, _Meister Eckhart & Christian Mysticism_. YouTube. [Video link](https://www.youtube.com/watch?v=Hg0c32KbA-g).

* Let's Talk Religion, _Ibn 'Arabi & The Unity of Being_ YouTube. [Video link](https://www.youtube.com/watch?v=-bgWnzjONXE).

* Let's Talk Religion, _Marguerite Porete & The Mirror of Simple Souls_ YouTube. [Video link](https://www.youtube.com/watch?v=ZMFzvsZrrIo).

category: people


Melanie Becker

* [webpage](http://people.physics.tamu.edu/mbecker/)

## Selected writings

Textbook account of [[perturbative string theory]] and [[M-theory]]/[[F-theory]]:

* [[Katrin Becker]], [[Melanie Becker]], [[John Schwarz]], _String Theory and M-Theory: A Modern Introduction_, Cambridge University Press 2006 ([doi:10.1017/CBO9780511816086](https://doi.org/10.1017/CBO9780511816086))

On the [[BFSS matrix model]]:

* [[Katrin Becker]], [[Melanie Becker]], _A Two-Loop Test of M(atrix) Theory_, Nucl.Phys. B506 (1997) 48-60 ([arXiv:hep-th/9705091](https://arxiv.org/abs/hep-th/9705091))

* [[Katrin Becker]], [[Melanie Becker]], [[Joseph Polchinski]], [[Arkady Tseytlin]], _Higher Order Graviton Scattering in M(atrix) Theory_, Phys.Rev.D56:3174-3178,1997 ([arXiv:hep-th/9706072](https://arxiv.org/abs/hep-th/9706072))

On [[M-theory on 8-manifolds]] and [[C-field tadpole cancellation]]:

* [[Katrin Becker]], [[Melanie Becker]], _M-Theory on Eight-Manifolds_, Nucl.Phys. B477 (1996) 155-167 ([arXiv:hep-th/9605053](https://arxiv.org/abs/hep-th/9605053))

* [[Melanie Becker]], Dragos Constantin, [[Sylvester James Gates Jr.]], William D. Linch III, Willie Merrell, J. Phillips, _M-theory on Spin(7) Manifolds, Fluxes and 3D, N=1 Supergravity_, Nucl.Phys. B683 (2004) 67-104 ([arXiv:hep-th/0312040](https://arxiv.org/abs/hep-th/0312040))

On [[M5-branes]], [[M2-branes]] and [[non-perturbative effects]] ([[M5-brane instantons]], [[M2-brane instantons]]) in [[string theory]]/[[M-theory]]:

* [[Katrin Becker]], [[Melanie Becker]], [[Andrew Strominger]], section 2.1 of _Five-branes, membranes and nonperturbative string theory_, Nucl. Phys. B 456, 130 (1995) ([hep-th/9507158](http://arxiv.org/abs/hep-th/9507158))

Discussion of [[Cayley forms]] and [[wrapped branes]] on [[associative submanifolds]]:

* [[Katrin Becker]], [[Melanie Becker]], D.R.Morrison, [[Hirosi Ooguri]], Y.Oz, Z.Yin, _Supersymmetric Cycles in Exceptional Holonomy Manifolds and Calabi-Yau 4-Folds_, Nucl.Phys.B480:225-238,1996 ([arXiv:hep-th/9608116](http://arxiv.org/abs/hep-th/9608116))

On [[moduli stabilization]] [[non-geometric string vacua]]:

* [[Katrin Becker]], [[Melanie Becker]], [[Cumrun Vafa]], [[Johannes Walcher]], _Moduli Stabilization in Non-Geometric Backgrounds_,  	Nucl.Phys.B770:1-46,2007 ([arXiv:hep-th/0611001](https://arxiv.org/abs/hep-th/0611001))

On [[supersymmetry breaking]] in [[flux compactifications]] of [[M-theory]]:

* [[Katrin Becker]], [[Melanie Becker]], _Supersymmetry Breaking, M-Theory and Fluxes_, JHEP 0107:038,2001 ([arXiv:hep-th/0107044](https://arxiv.org/abs/hep-th/0107044))

On [[D=5 supergravity]] formulated in $\mathcal{N} = 1/2$ [[superspace]]:

* [[Katrin Becker]], [[Melanie Becker]], Daniel Butter, William D. Linch III, Stephen Randall, _Five-dimensional Supergravity in N = 1/2 Superspace_ ([arXiv:1909.09208](https://arxiv.org/abs/1909.09208))

## Related entries

* [[non-perturbative effect]]

* [[BFSS matrix model]]

category: people


Melanie Swan

* [personal page](https://www.melanieswan.com/)

## Selected writings

On [[quantum computation]], [[blockchains]] and [[machine learning]]:

* [[Melanie Swan]], [[Renato P dos Santos]], [[Frank Witte]], Between Science and Economics, Volume 2: *Quantum Computing Physics, Blockchains, and Deep Learning Smart Networks*, World Scientific 2020 ([doi:10.1142/q0243](https://doi.org/10.1142/q0243))

  * On [[holographic quantum error correction]]:

    Part 5 (Ch. 13, 14): *The AdS/CFT Correspondence and Holographic Codes* ([doi:10.1142/9781786348210_0013](https://doi.org/10.1142/9781786348210_0013), [doi:10.1142/9781786348210_0014](https://www.worldscientific.com/doi/abs/10.1142/9781786348210_0014))

category: people


Mellin transform

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## Definition

For $\tau \mapsto f(\tau)$ a suitably well-behaved [[function]] of a single [[variable]], its _Mellin transform_ is the function given by the [[integral]] expression

$$
  s \mapsto \int_0^\infty \tau^{s-1} f(\tau) \; d\tau
 \,.
$$

## Examples

### Of the exponential function

For the [[exponential function]] $\exp(-(-)A)$ the Mellin transform is proportional to the inverse [[power]] $A^{-(-)}$

$$
  \int_0^\infty \tau^{s-1} \exp(-\tau A) \,d\tau
  =
  (s-1)! \, A^{-s}
  \,.
$$

For $s = 1$ this has a [[complex number|complex]] analougue given by the [[Fourier transform]] of the [[Heaviside distribution]] ([this example](Cauchy+principal+value#RelationToFourierTransformOfHeavisideDistribution)).

In [[perturbative quantum field theory]], especially in the computation of [[propagators]] such as the [[Feynman propagator]], this formula is known as the _[[Schwinger parameterization]]_ for $A^{-s}$, leading to the "[[worldline formalism]]". See below at _[1-Loop amplitudes](#1LoopVacuumAmplitudes)_.

### Zeta functions

A [[zeta function]]/[[L-function]] is the [[analytic continuation]] of the Mellin transform of the corresponding [[theta function]].

In particular it sends the [[Jacobi theta function]] to the (completed) [[Riemann zeta function]]:

$$ \hat \zeta(s) = \int_0^\infty t^{s-1} \hat \theta(t) \, d t $$

More generally, the Mellin transform appears as a stage in the expression of [[zeta functions]] as [[adelic integrals]] in [[Iwasawa-Tate theory]].

### 1-loop vacuum amplitudes
 {#1LoopVacuumAmplitudes}

[[1-loop vacuum amplitudes]] in [[quantum field theory]] are analytically continued Mellin transforms of [[partition functions]]. Here the parameter $\tau$ is called the _[[Schwinger parameter]]_ and the Mellin transform turns the [[worldline formalism]]-picture into the [[Feynman propagator]]-picture.

$$ Tr H^{-s} = \int_0^\infty t^{s-1} Tr\, \exp(- t H) \, d t \,. $$

## Examples

### Zeta functions

* [[zeta function of an elliptic differential operator]]

* [[Riemann zeta function]]

## Related concepts

* [[Laplace transform]]

* [[Fourier transform]]

* the generalization of Mellin transforms from [[automorphic forms]] to [[automorphic representation]] is the concept of [[automorphic L-function]]

## References

* Wikipedia, _[Mellin transform](http://en.wikipedia.org/wiki/Mellin_transform)_

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

* [[Joel Shapiro]], _Schwinger trick and Feynman parameter_, 2007 ([pdf](https://www.physics.rutgers.edu/grad/615/lects/schwingertrick.pdf))

For the [[adelic integral]]-version see at _[[Iwasawa-Tate theory]]_.

For [[function fields]]:

* [[David Goss]], _A formal Mellin transform in the arithmetic of function fields_, Transactions of the AMS, volume 327, Number 2, October 1991 ([pdf](http://www.ams.org/journals/tran/1991-327-02/S0002-9947-1991-1041048-5/S0002-9947-1991-1041048-5.pdf))

For the appearance in  [[physics]] as integrals over [[Schwinger parameters]] producing [[Feynman propagators]] see

* [[Joel Shapiro]], _Schwinger trick and Feynman parameters_ ([pdf](https://www.physics.rutgers.edu/grad/613/615lects/schwingertrick.pdf))

* Stefan Weinzierl, section 4.2.1 of _Mathematical aspects of particle physics_, 2010 ([pdf](http://wwwthep.physik.uni-mainz.de/~stefanw/download/script_thep2.pdf))

[[!redirects Mellin transforms]]

[[!redirects Schwinger parameter]]
[[!redirects Schwinger parameters]]

[[!redirects Schwinger parameterization]]
[[!redirects Schwinger parameterizations]]

[[!redirects Schwinger parameterisation]]
[[!redirects Schwinger parameterisations]]


Melvin Fitting

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

## Selected writings

On [[proof theory]] for [[modal logic]]:

* {#Fitting07} [[Melvin Fitting]], *Modal proof theory*, Ch. 2 in: *The Handbook of Modal Logic*, Studies in Logic and Practical Reasoning **3** (2007) 85-183 [[doi:10.1016/S1570-2464(07)80005-X](https://doi.org/10.1016/S1570-2464(07)80005-X), [book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/)]

## Related entries

* [[epistemic logic]]

* [[S5 modal logic]]

category: people


Melvin Hochster

* [webpage](http://www.math.lsa.umich.edu/~hochster/)

## Selected writings

Introducing [[Hochster duality]]:

* [[M. Hochster]], _Prime ideal structure in commutative rings_, Transactions of the American Mathematical Society, 142, (1969), 43–60

category:people

[[!redirects M. Hochster]]


Melvin Rothenberg

* [Mathematics Genealogy page](https://genealogy.math.ndsu.nodak.edu/id.php?id=6677)

* [institute page](https://mathematics.uchicago.edu/people/emeriti-faculty/)

## Selected writings

On [[equivariant differential topology]] and [[equivariant bundles]]:

* {#LashofRothenberg78} [[Richard Lashof]], [[Melvin Rothenberg]], Section 1 of: _$G$-Smoothing theory_, p. 211-266 in: _Algebraic and Geometric Topology_, Proceedings of Symposia in Pure Mathematics, Vol. XXXII, Part 1, American Mathematical Society 1978 ([doi:10.1090/pspum/032.1](https://doi.org/10.1090/pspum/032.1), [[LashofRothenbergGSmoothingTheory.pdf:file]])

category: people


membership relation

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## Idea

A membership relation is a [[binary relation]] $\in$ found in [[simple type theory|simply sorted]] [[set theory]] between terms which can be elements and terms which can be sets. Membership relations are found both [[material set theory]] and [[structural set theory]].

This is in contrast to [[dependent type theory|dependently sorted]] set theories, where membership $\in$ is a typing [[judgment]] rather than a binary relation.

## Properties

Membership relations have different properties in different set theories which can be distinguished.

### Homogeneous and heterogeneous membership relations

Recall that a [[binary relation]] $R$ takes values in sorts $A$ and $B$. A binary relation $R$ is **homogeneous** or an **endorelation** if it takes values in the same sort $A$, and a binary relation is **heterogeneous** if it takes values in two distinct sorts $A$ and $B$.

The membership relation $\in$ is homogeneous if both sets and elements are terms of the same sort, while $\in$ is heterogeneous if sets and elements are terms of different sorts. For example, in [[ZFC]], [[ZFA]], and [[fully formal ETCS]], the membership relations $\in$ are homogeneous, while in the traditional presentation of [[ETCS]] as a two-sorted theory, the membership relation $\in$ is heterogeneous.

### Material and structural membership relations

Another important difference is the distinction between the membership relations in material and structural set theories, in what we could call **material membership relation** and **structural membership relation**. This has been proposed to distinguish between [[material set theory]] and [[structural set theory]] in set theories with [[homogeneous membership relations]]:

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]].

A few definitions have been proposed to distinguish between homogeneous membership relations in material and structural set theories:

#### Structural membership relations without Quine atoms

Both material set theories like [[ZFC]] and [[ZFA]] and structural set theories like [[fully formal ETCS]] have a homogeneous membership relation on the theory (and fully formal ETCS has multiple homogeneous membership relations). This means that merely having a homogeneous membership relation is not sufficient for the membership relation to be a material membership relation.

One proposed difference between material set theory and structural set theory is that in material set theory, elements have internal structure and/or sets are the internal structure of some other objects, while in structural set theory, elements do not have internal structure and sets are not the internal structure of other objects.

Thus, an element $e$ **has internal structure** if there exists an object $a$ such that $a \in e$, and an set $S$ **is an internal structure** if there exists an object $a$ such that $S \in a$.

$$\mathrm{hasInternalStructure}(e) \coloneqq \mathrm{isElement}(e) \wedge \exists a.a \in e$$
$$\mathrm{isInternalStructure}(S) \coloneqq \mathrm{isSet}(S) \wedge \exists a.S \in a$$

If an set theory only has one homogeneous membership relation, then the membership relation is a **material membership relation** if there exists an element $e$ which has internal structure or a set $S$ which is an internal structure, and the membership relation is a **structural membership relation** if for all elements $e$, $e$ does not have internal structure, and for all sets $S$, $S$ is not an internal structure.

$$\mathrm{isMaterial} \coloneqq \exists a.(\mathrm{hasInternalStructure}(a) \vee \mathrm{isInternalStructure}(a))$$

$$\mathrm{isStructural} \coloneqq \forall a.(\neg\mathrm{hasInternalStructure}(a) \wedge \neg\mathrm{isInternalStructure}(a))$$

Thus, the membership relation in a theory like [[ZFC]] is a material membership relation because sets and elements are the same, and thus all elements have internal structure and all sets are internal structure. The same is true of any theory of [[pure sets]], such as [[New Foundations]]. In a theory with non-set [[urelements]] such as [[ZFA]], the membership relation is still a material membership relation, because every object in the theory is an element, and while urelements do not have internal structure, there exist other elements which do have internal structure, namely those elements which are sets.

In contrast, the homogeneous membership relation in [[fully formal ETCS]], defined by
$$a \in b \coloneqq s(a) = 1 \wedge s(b) = 0 \wedge t(a) = t(b)$$
where the constant primitive $1$ is the identity morphism of the terminal object and the constant primitive $0$ is the identity morphism of the initial object, and where sets and elements are defined as
$$\mathrm{isSet}(a) \coloneqq s(b) = 0$$
$$\mathrm{isElement}(a) \coloneqq s(b) = 1$$
is a structural membership relation, because it can be proven that every element does not have internal structure and every set is not internal structure.

#### Structural membership relations with Quine atoms

The above definition states that in a structural set theory with a homogeneous membership relation, [[Quine atoms]] cannot exist. However, this means that in [[fully formal ETCS]] when sets are defined as identity functions or as functions with singleton codomain, the defined  membership relation $\in$ is a material membership relation, as in both cases the singleton is a [[Quine atom]] with respect to $\in$. From one point of view, one could argue that those given definitions of set are incorrect, and one should instead use a definition of set which results in a structural homogeneous membership relation.

However, since the subset of [[identity functions]], functions with [[singleton]] [[codomain]], and functions with [[empty set]] [[domain]] are all in [[bijection]] with each other, it should not matter which subset of function to designate as the sets. This motivates an alternate definition of structural and material membership relations, which allow for Quine atoms to exist with respect to a structural membership relation.

A membership relation is a **structural membership relation** if each connected component of the respective membership graph is either an edgeless vertex, a rooted [[tree]] whose children are all leaves, or possibly a vertex with a single loop:

* the edgeless vertex represents the [[empty set]], as well as any non-set non-element objects in the set theory

* the root of the rooted tree represents a set with more than one element, as well as the [[singleton]] in presentations of unsorted structural set theories where the singleton is different from the element in the singleton. The children leaves of the root represent the elements of the set.

* the vertex with a single loop represents a [[Quine atom]] with respect to the membership relation $\in$, which occur in some definitions of sets in structural set theories, such as defining sets as [[identity functions]] or as functions with [[singleton]] [[codomain]], where the [[singleton]] is always a [[Quine atom]].

A **material membership relation** is then a membership relation whose membership graphs have more complex structure than those of structural membership relations.

## See also

* [[relation]]
* [[set theory]]

[[!redirects membership relation]]
[[!redirects membership relations]]

[[!redirects homogeneous membership relation]]
[[!redirects homogeneous membership relations]]

[[!redirects heterogeneous membership relation]]
[[!redirects heterogeneous membership relations]]

[[!redirects material membership relation]]
[[!redirects material membership relations]]

[[!redirects structural membership relation]]
[[!redirects structural membership relations]]


membrane

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## Idea

The next higher [[dimension|dimensional]] generalization of the concept of a _[[relativistic particle]]_ and a _[[relativistic string]]_ is called the relativistic _membrane_.

The concept of a [[bosonic modality|bosonic]] membrane was considered already by [Dirac 62](#Dirac62) as a hypothetical [[model (in theoretical physics)|model]] for the [[electron]], then abandoned and eventually re-incarnated as the "super-membrane in 11d" due to [Bergshoeff-Sezgin-Townsend 87](#BergshoeffSezginTownsend87) (whose "m" became the "M" in "[[M-theory]]"), namely the [[Green-Schwarz sigma-model for super p-branes]] corresponding to the [[brane scan]]-entry with $p=2$, $D=11$, now commonly known as the _[[M2-brane]]_.

In fact, according to the [[brane scan]], a [[super p-brane|super 2-brane]] [[sigma model]] exists on [[superspacetimes]] of dimension 4, 5, 7, and 11:

* 4: [[super 2-brane in 4d]]

* 5: [[super 2-brane in 5d]]

* 7: ...

* 11: [[M2-brane]]

## Related concepts

* ´[[membrane instanton]]

## References

### Pre-history

Early speculations trying to unify the [[electron]] and the [[muon]] as two excitations of a single relativistic membrane:

* {#Dirac62} [[Paul Dirac]], *An Extensible Model of the Electron*, Proc. Roy. Soc. A **268** (1962) 57-67 [[jstor:2414316](https://www.jstor.org/stable/2414316)]

  > (also proposing the [[Dirac-Born-Infeld action]])

* [[Paul Dirac]], _The motion of an Extended Particle in the Gravitational Field_, in: [[Leopold Infeld|L. Infeld]] (ed.), *Relativistic Theories of Gravitation*, Proceedings of a Conference held in Warsaw and Jablonna, July 1962, P. W. N. Publishers, 1964, Warsaw, 163-171; discussion 171-175 [[spire:1623740](http://inspirehep.net/record/1623740), article:[[Dirac-MotionOfAnExtendedParticle.pdf:file]], full proceedings:[pdf](https://cds.cern.ch/record/2282975/files/warsaw-1962.pdf)]

* [[Paul Dirac]], _Particles of Finite Size in the Gravitational Field_, Proc. Roy. Soc. A **270** (1962) 354-356 [[doi:10.1098/rspa.1962.0228](https://doi.org/10.1098/rspa.1962.0228)]

[[!include fundamental M2-brane sigma-model -- references]]

[[!include quantization of M2-brane to matrix model -- references]]

[[!redirects membranes]]

[[!redirects relativistic membrane]]
[[!redirects relativistic membranes]]

[[!redirects super-membrane]]
[[!redirects super membrane]]
[[!redirects supermembrane]]
[[!redirects super-membranes]]
[[!redirects super membranes]]
[[!redirects supermembranes]]


membrane instanton

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

#Contents#
* table of contents
{:toc}

## Idea

Membrane [[instantons]] are constituted by [[supermembranes]] ([[M2-branes]])  [[wrapped brane|wrapped]] on [[spacelike]] (and [[supersymmetric cycle|supersymmetric]]) [[cycles]]. Also known as _[[SM2-branes]]_.

These appear as [[non-perturvative effects]] in [[M-theory]] [[model (physics)|model building]]. For instance they generate the [[hierarchy problem|gauge hierarchy]] in constructions of [[M-theory on G2-manifolds]] such as the [[G2-MSSM]].

Basically the effect is that the [[action functional]] of the given [[11-dimensional supergravity]] is multiplied by the exponentiated [[Nambu-Goto action]] of the [[wrapped brane|wrapping]] membrane configurations $\phi \colon \Sigma \to X$. These are of the form (e.g. [Harvey-Moore 99, (4.13)](#HarveyMoore99), [Atiyah-Witten 01 (4.17)](#AtiyahWitten01), [Curio 02, (2.15)](#Curio02))

$$
  \exp\left(
    \int_{\Sigma}
    \left(
      \ell_P^{-3} vol(\phi^\ast g) + i \phi^\ast C
    \right)
  \right)
  \,,
$$

where $\ell_P$ is the [[Planck length]] of [[11-dimensional supergravity]], $T = 1/\ell^3$ is the membrane [[tension]], $C$ is the [[background field|background]] [[supergravity C-field]], $g$ the background [[metric]] and $vol$ its [[volume form]], .

## Properties

### Relation to worldsheet and D-brane instantons

Under the [[duality between M-theory and type IIA string theory]], membrane instantons become string [[worldsheet instantons]] or [[D-brane]] instantons, depending on whether they [[wrapped brane]] wrape the M-theory circle fiber or not. See at _[[non-perturbative effect]]_ the section _[Worldsheet and brane instantons](https://ncatlab.org/nlab/show/non-perturbative+effect#WorldsheetAndBraneInstantons)_ for more.

### Relation to complex volume

When here $(\Sigma, \phi^\ast g)$ is a [[hyperbolic 3-manifold]] and $\phi^\ast C = CS(\omega)$ is the [[Chern-Simons form|Chern-Simons invariant]] then this is the exponentiated "[[complex volume]]" of $\Sigma$[^footnote].

## Related concepts

* [[hierarchy problem]]

* [[non-perturbative effect]]

* [[complex volume]]

* [[associative submanifold]]

* [[D3-brane instanton]]

* [[M5-brane instanton]]

[[!include table of branes]]

## References

* [[Katrin Becker]], [[Melanie Becker]], [[Andrew Strominger]], section 2.1 of _Five-branes, membranes and nonperturbative string theory_, Nucl. Phys. B 456, 130 (1995) ([hep-th/9507158](http://arxiv.org/abs/hep-th/9507158))

* {#HarveyMoore99} [[Jeff Harvey]], [[Greg Moore]], _Superpotentials and Membrane Instantons_ ([arXiv:hep-th/9907026](http://arxiv.org/abs/hep-th/9907026), [spire:503172](http://inspirehep.net/record/503172))

Discussion specifically for [[M-theory on G2-manifolds]] includes

* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]], around (4.17) of _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))

* {#Curio02} [[Gottfried Curio]], _Superpotentials for M-theory on a $G_2$ holonomy manifold and Triality symmetry_, JHEP 0303:024,2003 ([arXiv:hep-th/0212211](http://arxiv.org/abs/hep-th/0212211))

Discussion related to the [[cosmological constant]] in models of [[M-theory on G2-manifolds]] includes

* {#CarlosLukasMorris04} Beatriz de Carlos, Andre Lukas, Stephen Morris, _Non-perturbative vacua for M-theory on G2 manifolds_, JHEP 0412:018, 2004 ([arxiv:hep-th/0409255](https://arxiv.org/abs/hep-th/0409255))

which concludes that with taking [[non-perturbative effects]] from membrane instantons into account one gets 4d vacua with vanishing and negative [[cosmological constant]] ([[Minkowski spacetime]] and [[anti-de Sitter spacetime]]) but not with positive [[cosmological constant]] ([[de Sitter spacetime]]). They close by speculating that [[M5-brane]] instantons might yield [[de Sitter spacetime]].

[^footnote]: This relation was pointed out by [[Hisham Sati]].

[[!redirects membrane instantons]]

[[!redirects M2-instanton]]
[[!redirects M2-instantons]]

[[!redirects M2-brane instanton]]
[[!redirects M2-brane instantons]]


membrane matrix model

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###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Applying a certain regularization prescription to the [[action functional]] of the [[superstring]] [[sigma-model]] it turns into a [[theory (physics)|theory]] whose [[field (physics)|fields]] are [[Lie algebra]] elements -- [[matrices]] for the case of [[matrix Lie algebras]] -- known as the [[IKKT matrix model]].

A similar regularization to the [[action functional]] of the [[membrane]] (the [[M2-brane]]) would have to involve objects replacing matrices which instead of a binary [[Lie bracket]] $[-,-]$ have a trinary [[Nambu bracket|Nambu]]-like bracket $[-,-,-]$. ([Park-Sochichiu 08](#ParkSochichiu08), [Sato09](#Sato09), [DeBellis-Saemann-Szabo 11](#DeBellisSaemannSzabo11)).

Given the relation of the [[IKKT matrix model]] to [[type IIB string theory]] it is naturally and often expected that making sense of this might give a way to understand [[M-theory]] (whence the title for instance of ([Sato 09](#Sato09))).

For this to work a clear understanding of the nature of these trinary brackets is required. After the success of the [[BLG model]] for the [[M2-brane]] several authors proposed that these are to be thought of as Flippov "[[3-Lie algebras]]" ([Park-Sochichiu 08](#ParkSochichiu08), [Sato09](#Sato09), [DeBellis-Saemann-Szabo 11](#DeBellisSaemannSzabo11)). However, by the discussion there, following ([MFFMER 08](n-Lie+algebra#MFFMER08)) it seems natural to think of "3-Lie algebras" as really being [[Lie 2-algebras]] equipped with a binary [[invariant polynomial]].

Accordingly, a membrane matrix model should be a generalization of the [[IKKT matrix model]] from [[Lie algebras]] to [[Lie 2-algebras]]. This proposal is explored in ([Ritter-Saemann 13](#RitterSaemann13)).

## Related concepts

* [[M2-brane]], [[M-theory]]

[[!include brane matrix models -- content]]

## References

* {#Sato09} Matsuo Sato, _Covariant formulation of M-theory I_, Int. J. Mod. Phys. A 24 (2009) 5019 ([arXiv:0902.1333](http://arxiv.org/abs/0902.1333))

* {#ParkSochichiu08} J.-H. Park, C. Sochichiu, _Taking  the square root of Nambu-Goto action and obtaining Filippov-Lie algebra gauge theory action_, Eur. Phys. J. C 64 (2009) 161 ([arXiv:0806.0335](http://arxiv.org/abs/0806.0335))

* {#DeBellisSaemannSzabo11} J. DeBellis, [[Christian Saemann]], [[Richard Szabo]], _Quantized Nambu-Poisson manifolds in a 3-Lie algebra reduced model_, JHEP 1104 (2011) 075

* {#RitterSaemann13} [[Patricia Ritter]], [[Christian Saemann]], _Lie 2-algebra models_ ([arXiv:1308.4892](http://arxiv.org/abs/1308.4892))


Meng Cheng

* [personal page](https://meng-cheng.github.io/)

## Selected writings

Introducing [[Majorana dimers]] for [[Ising anyons]]:

* Brayden Ware, Jun Ho Son, [[Meng Cheng]], Ryan V. Mishmash, [[Jason Alicea]], and Bela Bauer, *Ising anyons in frustration-free Majorana-dimer models*, Phys. Rev. B 94, 115127 2016 ([arXiv:10.1103/PhysRevB.94.115127](https://doi.org/10.1103/PhysRevB.94.115127))

On the [[adiabatic quantum computation|adiabaticity]] of [[topological quantum computation]] by [[braid group|braiding]] of [defect anyons](braid+group+statistics#AsBraidingOfDefects):

* {#ChengGalitskiDasSarma11} [[Meng Cheng]], [[Victor Galitski]], [[Sankar Das Sarma]], *Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors*, Phys. Rev. B **84** (2011) 104529 $[$[arXiv:1106.2549](https://arxiv.org/abs/1106.2549), [doi:10.1103/PhysRevB.84.104529](https://doi.org/10.1103/PhysRevB.84.104529)$]$

On [[symmetry protected topological phases]] and their [[topological orders]]:

* {#BBCW19} [[Maissam Barkeshli]], [[Parsa Bonderson]], [[Meng Cheng]], [[Zhenghan Wang]], *Symmetry Fractionalization, Defects, and Gauging of Topological Phases*, Phys. Rev. B **100** 115147 (2019) $[$[arXiv:1410.4540](https://arxiv.org/abs/1410.4540), [doi:10.1103/PhysRevB.100.115147](https://doi.org/10.1103/PhysRevB.100.115147)$]$

category: people


Meng-Chwan Tan

* [webpage](http://www.physics.nus.edu.sg/staff/tanmc.html)

* [institute page](https://www.physics.nus.edu.sg/~strings/members.htm)

## Selected writings

On realization of [[geometric Langlands correspondence]] in [[M-theory]]:

* [[Meng-Chwan Tan]], _M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems_ ([arXiv:1301.1977](http://arxiv.org/abs/1301.1977))

On [[D-brane]]-realizations of [[semi-holomorphic 4d Chern-Simons theory]]:

* [[Meer Ashwinkumar]], [[Meng-Chwan Tan]], Qin Zhao, _Branes and Categorifying Integrable Lattice Models_ ([arXiv:1806.02821](https://arxiv.org/abs/1806.02821))

and specifically with relation to the [[quantum geometric Langlands correspondence]]:

* [[Meer Ashwinkumar]], [[Meng-Chwan Tan]], _Unifying Lattice Models, Links and Quantum Geometric Langlands via Branes in String Theory_ ([arXiv:1910.01134](https://arxiv.org/abs/1910.01134))

On regularized [[quantization]] of [[Green-Schwarz sigma-models]] for [[super p-branes]], specifically of the [[super 4-brane in 9d]], in generalization of the [matrix model of the M2-brane](M2-brane#QuantizationViaBFSSMatrixModel):

* [[Meer Ashwinkumar]], [[Lennart Schmidt]], [[Meng-Chwan Tan]], Section 2 of: _Matrix Regularization of Classical Nambu Brackets and Super $p$-Branes_ ([arXiv:2103.06666](https://arxiv.org/abs/2103.06666))

On [[Vafa-Witten theory]]:

* Zhi-Cong Ong, [[Meng-Chwan Tan]], *Vafa-Witten Theory: Invariants, Floer Homologies, Higgs Bundles, a Geometric Langlands Correspondence, and Categorification* [[arXiv:2203.17115](https://arxiv.org/abs/2203.17115)]

## Related $n$Lab entries

* [[geometric Langlands duality]]

* [[4d Chern-Simons theory]]

category: people


Mengyang Zhang

* [institute page](https://phy.princeton.edu/people/mengyang-zhang)

## Selected writings

On 3-dimensional ([[quantum gravity|quantum]]) [[gravity]] ([[general relativity]]) with [[cosmological constant]], and its (non-)relation to [[Chern-Simons theory]] with non-[[compact Lie group|compact]] [[gauge groups]]:

* [[Scott Collier]], [[Lorenz Eberhardt]], [[Mengyang Zhang]], *3d gravity and Chern-Simons theory -- references* [[arXiv:2304.13650](https://arxiv.org/abs/2304.13650)]

category: people


Mentor Stafa

* [institute page](http://www.math.tulane.edu/~mstafa/)

## Selected writings

On [[mapping spaces]] of [[group homomorphisms]] from [[finitely generated group|finitely generated]] [[discrete groups]] to [[Lie groups]]:

* [[Frederick R. Cohen]], [[Mentor Stafa]], *A survey on spaces of homomorphisms to Lie groups*,  In: Callegaro F., Cohen F., De Concini C., Feichtner E., Gaiffi G., Salvetti M. (eds.) *Configuration Spaces*, INdAM Series, vol 14, Springer 2016 ([arXiv:1412.5668](https://arxiv.org/abs/1412.5668), [doi:10.1007/978-3-319-31580-5_15](https://doi.org/10.1007/978-3-319-31580-5_15))

category: people


Mercator series

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

#Contents#
* table of contents
{:toc}

## Idea

The "Mercator series" (so named after its appearance in [Mercator 1667](#Mercator1667)) is the [[Taylor series]] of the [[natural logarithm]] around 1.

## Statement

\begin{proposition}\label{MercatorSeries}
The [[Taylor series]] of the [[natural logarithm]] around $1 \in \mathbb{R}$ is the following [[series]]:
\[
  \label{MercatorTaylorSeriesForNaturalLogarithm}
  \begin{aligned}
  \underoverset{n = 0}{\infty}{\sum}
  \tfrac{1}{n!}
  \left(
    \frac{d^n}{ d x^n} ln(1 + x)
  \right)_{\vert x = 0}
  x^n
  &
  \;\;
  =
  \;\;
  \underoverset{n = 1}{\infty}{\sum}
  \frac
    {(-1)^{n+1}}
    {n}
  x^n
  \\
  &
  \;\;
  =
  \;\;
  x
  - \tfrac{1}{2} x^2
  + \tfrac{1}{3} x^3
  - \tfrac{1}{4} x^4
  + \cdots
  \,.
  \end{aligned}
\]
\end{proposition}
\begin{proof}
For the first two terms notice that
$$
  ln(1 + x)
  \;\xrightarrow{x \to 0}\;
  ln(1)
  \,=\,
  0
$$
and that the [[derivative]] of the natural logarithm is:
$$
  \frac{d}{d x} \ln(1 + x)
  \;=\;
  \tfrac{1}{1+x}
  \;\xrightarrow{ x \to 0 }\;
  1
  \,.
$$
From here on, noticing for $k \in \mathbb{N}_+$ that:
$$
  \frac{d}{d x}
  \left(
  \frac{1}{(1 + x)^k}
  \right)
  \;=\;
  - k
  \frac{1}{(1 + x)^{k+1}}
  \;\xrightarrow{x \to 0}\;
  - k
$$
we obtain for $n \in \mathbb{N}_+$, by [[induction]]:
$$
  \begin{aligned}
  \frac{d^n}{d x^n}
  ln(1 + x)
  &
  \;=\;
  \frac{d^{n-1}}{d x^{n-1}}
  \left(
    \frac{1}{1 + x}
  \right)
  \\
  &
  \;=\;
  (n-1)! \cdot (-1)^{n-1}
  \frac{1}{(1 + x)^{n-1}}
  \\
  &
  \;\xrightarrow{ x \to 0 }\;
  (n-1)! \cdot (-1)^{n+1}
  \end{aligned}
  \,.
$$
Plugging this into the defining equation on the left of (eq:MercatorTaylorSeriesForNaturalLogarithm) and using
$$
  \frac{(n-1)!}{n!} = \frac{1}{n}
$$
yields the claim.
\end{proof}

## Related concepts

* [[Chern form]]

* [[geometric series]]

## References

Apparently first published in:

* {#Mercator1667} [[Nicholas Mercator]], *Logarithmotechnia: Sive Methodus constuendi Logrithmos*, London 1667  ([GoogleBooks](https://books.google.de/books?hl=en&lr=&id=HFl1JqyXpSoC&oi=fnd&pg=PP6&dq=Logarithmotechnia.&ots=2xDsxWNWwO&sig=PfSwMZjoMnW7EAtH79uxX3ATsqc&redir_esc=y#v=onepage&q=Logarithmotechnia.&f=false))

but will have been known before that.

See also:

* Wikipedia, *[Mercator series](https://en.wikipedia.org/wiki/Mercator_series)*


mere proposition

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

# Mere Propositions
* table of contents
{: toc}

## Idea

In [[homotopy type theory]], the notion of **mere proposition** is an internalization of the notion of [[proposition]] / [[(-1)-truncated]] object.  Mere propositions are also called **h-propositions**, types of [[h-level]] $1$, $(-1)$-[[truncated types]], or even just **propositions** if no ambiguity will result.  Semantically, mere propositions correspond to [[(-1)-groupoids]] or more generally [[(-1)-truncated objects]].

Mere propositions are one way to embed [[logic]] into homotopy type theory, via the [[propositions as some types]] / [[bracket types]] approach.  For many purposes, this is the "correct" way to embed logic (as opposed to the [[propositions as types]] approach --- for instance, the PAT version of [[excluded middle]] contradicts the [[univalence axiom]], whereas the mere-propositional version does not).

## Definition

Consider [[intensional type theory|intensional]] [[type theory]] with [[dependent sums]], [[dependent products]], and [[identity types]].

There are many equivalent definitions of a [[type]] being an _h-proposition_.  Perhaps the simplest is as follows.

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

For $A$ a type, let $isProp(A)$ denote the [[dependent product]] of the [[identity types]] for all pairs of [[terms]] of $A$:

$$ isProp(A) \coloneqq \prod_{x\colon A} \prod_{y\colon A} (x=y) $$

We say that $A$ is an h-proposition if $isProp(A)$ is an [[inhabited type]].

=--

In [[propositions as types]] language, this can be pronounced as "every two elements of $A$ are equal".

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

The following are provably [[equivalence in homotopy type theory|equivalent]] definitions of $isProp(A)$:

* $isProp(A) \coloneqq \prod_{x\colon A} \prod_{y\colon A} isContr(x=y) $
* $ isProp(A) \coloneqq (A \to isContr(A)) $
* $isProp(A) \coloneqq \mathrm{isEquiv}(\mathrm{const}_{\mathbb{2}, A})$
* $isProp(A) \coloneqq \prod_{f:\mathbb{2} \to A} f(0) = f(1)$
* $isProp(A) \coloneqq \prod_{f:\mathbb{2} \to A} isContr(f(0) = f(1))$
* $isProp(A) \coloneqq \prod_{f:\mathbb{2} \to A} \prod_{x:\mathbb{2}} \prod_{y:\mathbb{2}} f(x) = f(y)$
* $isProp(A) \coloneqq \prod_{f:\mathbb{2} \to A} \sum_{p:\mathbb{I} \to A} \prod_{x:\mathbb{2}} p(\mathrm{rec}_\mathbb{2}^\mathbb{I}(0_\mathbb{I}, 1_\mathbb{I})(x)) = f(x)$

=--

The first fits into the general [[inductive definition]] of [[n-groupoid]]: an $\infty$-groupoid is an $n$-groupoid if all its hom-groupoids are $(n-1)$-groupoids.

The second says that being a proposition is equivalent to being "[[contractible]] if [[inhabited]]", and when [[isContr]] is expanded out to $\sum_{x:A} \prod_{y:A} x =_A y$, the resulting type $A \to \sum_{x:A} \prod_{y:A} x =_A y$ is just the type of the [[graph of a dependent function|graphs of dependent functions]] of $\prod_{x:A} \prod_{y:A} x =_A y$.

The third states that being a proposition is the same as being a $\mathrm{bool}$-[[localization of a type|local type]]: the function
$$\mathrm{const}_{\mathbb{2}, A} \equiv \lambda x:A.\lambda b:v.x:A \to (\mathrm{bool} \to A)$$
which takes elements $x:A$ to [[constant functions]] from the [[boolean domain]] with value $x$ is an [[equivalence of types]].

The fourth states that being a proposition is equivalent to the condition that for every function from the [[boolean domain]], $f(0) = f(1)$, and is interderivable from the usual definition of isProp via the recursion principle of the boolean domain.

The fifth states that being a proposition is equivalent to the condition that for every function from the [[boolean domain]], $f(0) = f(1)$ is contractible, and is interderivable from the first via the recursion principle of the boolean domain.

The sixth states that being a proposition is equivalent to the condition that every function from the [[boolean domain]] is a [[weakly constant function]], and can be proven to be equivalent to the fourth via double induction on the boolean domain and the properties of identity types.

The seventh states that being a proposition is equivalent to the condition that every function from the [[boolean domain]] to $A$ factors through the [[interval type]] via the function $\mathrm{rec}_\mathbb{2}^\mathbb{I}(0_\mathbb{I}, 1_\mathbb{I}):\mathbb{2} \to \mathbb{I}$ defined by the recursion principle of the boolean domain.

### Rules for isProp

The usual definition of isProp is given by the following rules:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{isProp}(A) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{isProp}(A) \equiv \prod_{x:A} \prod_{y:A} x =_A y \; \mathrm{type}}$$

If the [[dependent type theory]] doesn't have [[dependent function types]], but still has [[identity types]] one could still define isProp by adding the formation, introduction, elimination, computation, and uniqueness rules for isProp:

Formation rules for isProp types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A, y:A \vdash x =_A y \; \mathrm{type}}{\Gamma \vdash \mathrm{isProp}(A) \; \mathrm{type}}$$

Introduction rules for isProp types:
$$\frac{\Gamma, x:A, y:A \vdash p(x, y):x =_A y}{\Gamma \vdash \lambda(x).\lambda(y).p(x, y):\mathrm{isProp}(A)}$$

Elimination rules for isProp types:
$$\frac{\Gamma \vdash p:\mathrm{isProp}(A) \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A}{\Gamma \vdash p(a, b):a =_A b}$$

Computation rules for isProp types:
$$\frac{\Gamma, x:A, y:A \vdash p(x, y):x =_A y \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A}{\Gamma \vdash \beta_\mathrm{isProp}:(\lambda(x).\lambda(y).p(x, y))(a, b) =_{a =_A b} p(a, b)}$$

Uniqueness rules for isProp types:
$$\frac{\Gamma \vdash p:\mathrm{isProp}(A)}{\Gamma \vdash \eta_\mathrm{isProp}:p =_{\mathrm{isProp}(A)} \lambda(x).\lambda(y).p(x, y)}$$

## Properties

* We can prove that for any $A$, the type $isProp(A)$ is an h-proposition, i.e. $isProp(isProp(A))$.  Thus, any two inhabitants of $isProp(A)$ (witnesses that $A$ is an h-proposition) are "equal".

* Every function between h-propositions $A$ and $B$ is an [[embedding of types]].

* If $A$ and $B$ are h-props and there exist maps $A\to B$ and $B\to A$, then $A$ and $B$ are [[equivalence of types|equivalent]]. This follows from the fact that if $A$ and $B$ are types and there exists embeddings $A\to B$ and $B\to A$, then $A$ and $B$ are equivalent.

## Categorical semantics

We discuss the [[categorical semantics]] of h-propositions.

Let $\mathcal{C}$ be a [[locally cartesian closed category]] with [[categorical semantics of homotopy type theory|sufficient structure]] to interpret all the above type theory.  This means that $C$ has a [[weak factorization system]] with [[stable path objects]], and that [[trivial cofibrations]] are preserved by pullback along fibrations between fibrant objects.  (We ignore questions of [[coherence]], which are not important for this discussion.)

Then for a fibrant object $A$, the object $isProp(A)$ defined above (with the first definition) is obtained by taking [[dependent product]] of the path-object $A^I \to A\times A$ along the projection $A\times A\to 1$.  By [[adjunction]], a [[global element]] of $isProp(A)$ is therefore equivalent to a [[section]] of the path-object, which exactly means a (right) [[homotopy]] between the two projections $A\times A\;\rightrightarrows\; A$.  The existence of such a homotopy, in turn, is equivalent to saying that *any two maps $X\;\rightrightarrows\; A$ are homotopic*.

More generally, we may apply this locally.  Suppose that $A\to B$ is a fibration, which we can regard as a dependent type
$$x\colon B \vdash A(x)\colon Type.$$
Then we have a dependent type
$$x\colon B \vdash isProp(A(x))\colon Type$$
represented by a fibration $isProp(A)\to B$.  By applying the above argument in the [[slice category]] $\mathcal{C}/B$, we see that $isProp(A)\to B$ has a [[section]] exactly when any two maps in $\mathcal{C}/B$ into $A\to B$ are homotopic over $B$.  We can also construct the type
$$\prod_{x\colon B} isProp(A(x))$$
in global context, which has a global element precisely when $isProp(A)\to B$ has a section.

## Related concepts

* [[subsingleton]], [[subterminal object]]

* [[propositional truncation]]

* [[isEquiv]]

* [[isContr]]

* **isProp**

* [[center of contraction]]

* [[propositional logic as a dependent type theory]]

* [[coinduction]]

* [[strict proposition]]

[[!include homotopy n-types - table]]

## References

* [section 3.3](http://planetmath.org/33merepropositions) of [[Homotopy Type Theory -- Univalent Foundations of Mathematics]]

* [HoTT repository](https://github.com/HoTT/HoTT/blob/master/Coq/HLevel.v)

[[!redirects mere propositions]]
[[!redirects h-proposition]]
[[!redirects h-propositions]]
[[!redirects h-prop]]
[[!redirects h-props]]
[[!redirects hProp]]
[[!redirects hProps]]
[[!redirects h-level 1]]
[[!redirects h-level 1 type]]
[[!redirects h-level 1 types]]
[[!redirects isProp]]
[[!redirects (-1)-truncated type]]
[[!redirects (-1)-truncated types]]

[[!redirects merely]]


mere proposition > history

< [[mere proposition]]


mereology

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###Context###
#### Philosophy
+--{: .hide}
[[!include philosophy - contents]]
=--
#### Foundations
+--{: .hide}
[[!include foundations - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

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

[[!redirects Mereology]]

##Idea##
The term **mereology** was coined by the logician Stanislaw Lesniewski from greek 'meros'-part to designate the _study of the parthood relation_.

Traditionally a 'part' of philosophy, the modern mathematical theory started with the work of the _Polish school of logic_ in the early 20th century in the context of the [[foundations of mathematics]]. These early approaches went hand in hand with systems of [[pointless topology]].

More recently, the main impact for the field comes from computer science in the context of _logics for spatial reasoning_ and computer vision.

##Remark
With the inception of topos theory [[William Lawvere]] has conceived of logic from the perspective of mereology by making the role of the [[subobject classifier]] manifest.

More specifically, he has pointed out in the 1980s the relevance of [[co-Heyting algebra|co-Heyting and bi-Heyting structures]] to mereology in this context and proposed to view logical theories from the point of view of a 'mereodynamics'.

##Related entries

* [[co-Heyting algebra]]
* [[co-Heyting boundary]]
* [[closed subspace]]
* [[Alfred Tarski]]
* [[William Lawvere]]

##Links

* [wikipedia entry](http://en.wikipedia.org/wiki/Mereology)
* [Stanford encyclopedia of philosophy entry](http://plato.stanford.edu/entries/mereology/)

##References

* L. Champollion, M. Krifka, _Mereology_ , pp.512-541 in Dekker, Aloni (eds.), _Cambridge Handbook of Formal Semantics_ , Cambridge UP 2016. ([draft](https://ling.auf.net/lingbuzz/002099))

* P. Forrest, _Nonclassical Mereology and Its Application to Sets_ , Notre Dame J. Form. Logic **43** no.2 (2003) pp.79-94. ([pdf](http://projecteuclid.org/download/pdfview_1/euclid.ndjfl/1071509430))

* [[Kazimierz Kuratowski|C. Kuratowski]], _Sur l'opération $\bar{A}$ de l'analysis situs_ , Fund. Math. **III** (1922) pp.192-195. ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3121.pdf))

* {#Law86} [[William Lawvere|F. W. Lawvere]], _Introduction_ , pp.1-16 in Lawvere, Schanuel (eds.), _Categories in Continuum Physics_ , Springer LNM **1174** 1986.

* {#Law91a} F. W. Lawvere, _Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes_ , pp.279-281 in Carboni, Pedicchio, Rosolini (eds.) , _Category Theory Como Conference_, Springer LNM **1488** (1991).

* S. Lesniewski, _Grundzüge eines neuen Systems der Grundlagen der Mathematik_ , Fund. Math. **XIV** pp.1-81 (1929).[^Lesn] ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm14/fm1411.pdf))

* T. Mormann, _Heyting Mereology as a Framework for Spatial Reasoning_ , Axiomathes **23** no.1 (2013) pp.237-264. ([draft](http://philpapers.org/go.pl?id=MORCMA-3&proxyId=&u=http%3A%2F%2Fphilpapers.org%2Farchive%2FMORCMA-3.pdf))

* W. Noll, _The Geometry of Contact, Separation and Reformation of Continuous Bodies_ , Research report no.92-NA-029 Carnegie Mellon University 1992.  ([pdf](http://repository.cmu.edu/cgi/viewcontent.cgi?article=1387&context=math))

* W. Noll, _The geometry of contact, separation and reformation of continuous bodies_ , Arch. Rat. Mech. Anal. **122** (1993) pp.97-112.

* [[Dana Scott|Dana S. Scott]], _Geometry without Points_ , Talk Vienna 2014. ([slides](http://www.logic.at/latd2014/2014%20Vienna%20Scott.pdf))

* J.G. Stell, M.F. Worboys, _The algebraic structure of sets of regions_ , pp.163-174 in Hirtle, Frank (eds.), _Spatial Information Theory_, Springer LNCS **1329** (1997).

[^Lesn]: Caveat: Lesniewski conceives his foundations as layered into a _Protothetik_ , roughly the deductive system or 'logistics', an _ontology_ , roughly the theory of classes, and, _mereology_. This paper that is one in row that stretches from 1927 to 1938 is concerned with the first two layers.


meridian

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#### Spheres
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=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given an [[n-sphere]] $S^n$ for $n \,\geq 1\,$, regarded under its standard [[unit sphere]] [[embedding]] into [[Cartesian space]]

$$
  S^n
  \;\subset\;
  \mathbb{R}^{n+1} \,\simeq\, \mathbb{R} \times \mathbb{R}^n
  \,,
$$

then a _meridian_ is any [[geodesic]] (great half-circle) from the "north pole" $\big(1,\vec 0\big)$ to the "south pole" $\big(-1,\vec 0\big)$ (or more generally between any [[pair]] of [[antipodes]]).

Hence the $n$-sphere may be regarded as the [[union]] of

1. its north pole

1. its south pole

1. all meridians.

This decomposition exhibits the $n$-sphere as the (un-reduced) [[suspension]] of the $(n-1)$-sphere which is its [[equator]].

## Related concepts

* [[suspension]], [[suspension type]]

* [[equator]]

* [[hemisphere]], [[antipode]]

* [[stereographic projection]]

* [[clutching construction]]

## References

See also:

* Wikipedia, *Meridian_(geography)*

[[!redirects meridians]]


Merlin Carl

## Selected publications

* [[Merlin Carl]], *Formal and Natural Proof: A Phenomenological Approach*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019)  [[doi:10.1007/978-3-030-15655-8_14](https://doi.org/10.1007/978-3-030-15655-8_14), [draft](https://kops.uni-konstanz.de/server/api/core/bitstreams/96c96cc1-322e-430c-b4d0-63f2a7abd7fe/content), [pdf](https://www.math.uni-konstanz.de/~carl/Paper/FormalNaturalProofFinal2015.pdf)]

category: people


meromorphic connection

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

## Idea

The theory of **meromorphic connections** is a modern viewpoint on local behaviour of a class of [[ordinary differential equation|systems of ODE]]-s with meromorphic coefficients in a complex domain.

## Definition

### 1-Dimensional case

Consider the field $\mathcal{K}$ of meromorphic functions in a neighborhood of $0\in \mathbb{C}$ with possible pole at $0$ and a finite dimensional $\mathcal{K}$-module $M$. A **meromorphic** connection at $x=0$ is a $\mathbb{C}$-linear operator $\nabla:M\to M$ satisfying

$$
\nabla(f u) = \frac{d f}{d x}u + f\nabla(u),\,\,\,\,f\in\mathcal{K}, u\in M
$$

In fact, it is customary, in modern literature to consider just a germ: the connections on two different neighborhoods agreeing on the intersection are identified. This way $\mathcal{K}$ is isomorphic to the field of formal [[Laurent series]] $\mathbb{C}[ [u] ][u^{-1}]$.

There is a natural tensor product on the category of $\mathcal{K}$-modules with meromorphic connections. Namely $(M,\nabla_M)\otimes (N\nabla_N) = (M\otimes N,\nabla)$ where

$$
\nabla(u\otimes v) = \nabla_M (u)\otimes v + u\otimes\nabla_N(v)
$$

There is also an inner hom, namely $HOM((M,\nabla_M),(P,\nabla_P))$ is $Hom_{\mathcal{K}}(M,P)$ with a meromorphic connection

$$
\nabla(\phi)(u) = \nabla_P (\phi(u)) - \phi(\nabla_M (u)),\,\,\,\,u\in M, \phi:M\to N.
$$

## References

* chapter 5, _Theory of meromorphic connections_, from R. Hotta, K. Takeuchi, T. Tanisaki, _D-modules, perverse sheaves, and representation theory_, Progress in Mathematics __236__, Birkhäuser

* [[Claude Sabbah|C. Sabbah]], _Isomonodromic deformations and Frobenius manifolds_, Springer 2007, [doi](http://dx.doi.org/10.1007/978-1-84800-054-4), [errata](http://www.math.polytechnique.fr/cmat/sabbah/coursP6errata.pdf)

* P. Maisonobe, C. Sabbah, _D-modules cohérents et holonomes_, Hermann, Paris 1993.

* L.Katzarkov, [[Maxim Kontsevich|M.Kontsevich]], T.Pantev, _Hodge theoretic aspects of mirror symmetry_, [arxiv/0806.0107](http://arxiv.org/abs/0806.0107)

* D. Babbitt, V.S. Varadarajan, _Deformations of nilpotent matrices over rings and reduction of analytic families of meromorphic differential equations_, Mem. Amer. Math. Soc. __55__ (325), iv+147, 1985; _Local moduli for meromorphic differential equations_, Astérisque 169-170 (1989), 1–217.

* V.S. Varadarajan, _Linear meromorphic differential equation: a modern point of view_, Bull. AMS __33__, n. 1, 1996, [pdf](http://www.ams.org/bull/1996-33-01/S0273-0979-96-00624-6/S0273-0979-96-00624-6.pdf), [citeseer:pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.104.8107&rep=rep1&type=pdf).

* [[Pierre Deligne]], _Équations différentielles à points singuliers réguliers_, Lect. Notes in Math. __163__, Springer-Verlag (1970)

A discussion of meromorphic connections on the [[complex projective line]]:

* Daniel S. Sage, _Meromorphic connections on the projective line with specified local behavior_ ([arXiv:2212.14108](https://arxiv.org/abs/2212.14108))


meromorphic function

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#### Analytic geometry
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#Contents#
* table of contents
{:toc}

## Idea

Let $\Omega$ be a domain in a [[complex manifold]] and let $P \subset \Omega$ be a (complex-) analytic subset which is [[empty subset|empty]] or of [[codimension]] one. A [[holomorphic function]] $f$ defined on the [[complement]] $\Omega \setminus P$ is called a _meromorphic function_ in $\Omega$ if for every point $p \in P$ one can find an arbitrarily small [[neighbourhood]] $U$ of $p$ in $\Omega$ and functions $\phi$, $\psi$ holomorphic in $U$ without common non-invertible factors in $Int(U)$, such that $f = \phi/\psi$ in $U \setminus P$.

In one complex dimension (one complex variable), hence on a [[Riemann surface]], a _meromorphic function_ is a [[complex numbers|complex]]-[[analytic function]] which is defined away from a set of isolated points. Equivalently this is a [[holomorphic function]] with values in the [[Riemann sphere]].  Compare a [[holomorphic function]], which is valued in the [[complex plane]] (the Riemann sphere minus a point).

## Related concepts

* [[entire function]]

* [[sheaf of meromorphic functions]]

* [[function field of a variety]]

## References

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

* [[eom]], _[Meromorphic function](https://www.encyclopediaofmath.org/index.php/Meromorphic_function)_

category: analysis
[[!redirects meromorphic function]]
[[!redirects meromorphic functions]]
[[!redirects meromorphic map]]
[[!redirects meromorphic maps]]


meros

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#### Foundations
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=--
=--
=--

# Meros
* this block creates the table of contents, leave as is
{: toc}

## Idea

The concept of **meros** (plural **meroi** or **meroses**) is a [[relation|relational]] analogue of that of a *[[topos]]*; where the [[morphisms]] in a topos are like [[functions]], the morphisms in a meros are like [[relations]]. Like [[Set]] is the canonical [[base topos]], so [[Rel]] is the archetypical meros. More generally, like topoi provide a context for generalized [[constructive set theory]], so meroi provide a context for generalized [[relational set theory]].

Where the [[boolean]] [[truth values]] of [[set theory]] are identified/generalized to [[elements]] of the [[subobject classifier]] of a topos, a meros instead represents Boolean values as degrees of inclusion between relations: between any two objects, there is a false relation (a [[zero morphism]]) and a true relation.

## Definition

Following [Kawahara 1995](#Kawahara95), we define an auxiliary venue for meroi first. An *I-category* is a [[dagger category]] where every [[hom-set]] has [[lattice]] structure. Specifically,

\begin{defn} \label{ICategory}

An *I-category* is a [[dagger category]] where, for all objects $A, B$, there is a [[partial order]] $\sqsubseteq$ on the morphisms $A \to B$, a least [[relation]] $0_{A,B}$ and greatest relation $\Theta_{A,B}$ such that

$$
  \underset{
     A \to B
  }{
    \forall
  }
  \;\;
  0_{A,B}
    \sqsubseteq
  \alpha
    \sqsubseteq
  \Theta_{A,B}
$$

and the partial order commutes with the dagger structure:

$$
  \underset{
    \alpha, \beta
    \colon
    A \to B,
  }{
    \forall
  }
  \;\;
  \alpha
    \sqsubseteq
  \beta
    \;\;\iff\;\;
  \alpha^\dagger \sqsubseteq \beta^\dagger
  \,.
$$

\end{defn}

I-categories can have a [[zero object]] like typical dagger categories, but also a [[terminal object]].

\begin{defn} \label{TerminalObject}
When it exists, a *terminal object* in an I-category (Def. \ref{ICategory}) is an [[object]] $1$ such that $0_{1,1} \neq id_1 = \Theta_{1,1}$ and for all objects $A, B$, $\Theta_{1,B} \circ \Theta_{A,1} = \Theta_{A,B}$.
\end{defn}

We next highlight specific morphisms within I-categories.

\begin{defn} \label{PartialFunction}
A morphism $f \colon A \to B$ in an I-category (Def. \ref{ICategory}) is called a *[[partial function]]*, or *univalent*, when $f \circ f^\dagger \sqsubseteq id_B$.
\end{defn}

\begin{defn} \label{Function}
A partial function $f \colon A \to B$ (Def. \ref{PartialFunction}) is a *total function*, or function, when $id_A \sqsubseteq f^\dagger \circ f$.
\end{defn}

\begin{defn} \label{Rationality}
A morphism $\alpha \colon A \to B$ in an I-category (Def. \ref{ICategory}) is *rational* when there exist functions  $f \,\colon\, X \to A$ and $g \,\colon\, X \to B$ (in the sense of Def. \ref{Function}) such that $\alpha = g \circ f^\dagger$ and $f^\dagger \circ f \sqcap g^\dagger \circ g = id_X$.
\end{defn}

\begin{defn} \label{Quotient}
For all $\alpha \,\colon\, A \to B, \beta \,\colon\, A \to C, \gamma \,\colon\, B \to C$ there exists a *quotient* $\beta \div \gamma \,\colon\, A \to B$ such that $\gamma \circ \alpha \sqsubseteq \beta \iff \alpha \sqsubseteq \beta \div \gamma$.
\end{defn}

We also need the Dedekind formula, which relates any three morphisms between three objects.

\[
  \label{DedekindFormula}
  \underset{
    {
      \alpha \colon A \to B
    }
    \atop
    {
    { \beta \colon B \to C }
    \atop
    { \gamma \colon A \to C }
    }
  }{
    \forall
  }
  \;\;
   \beta \circ \alpha \sqcap \gamma
     \,\sqsubseteq\,
  ( \beta \sqcap \gamma \circ \alpha^\dagger ) \circ \alpha
\]

We are now ready to define meroi. Instead of a lattice, we will now expect a [[complete Heyting algebra]] for each [[hom-set]].

\begin{defn} \label{Meros}

A *meros* is an I-category (Def. \ref{ICategory}) where:

* The partial order $\sqsubseteq$ is not just a lattice, but a [[complete Heyting algebra]].

* Every relation is rational (in the sense of Def. \ref{Rationality}).

* The Dedekind formula (eq:DedekindFormula) holds whenever possible.

* There is a [[terminal object]] (in the sense of Def. \ref{TerminalObject}).

* All quotients (in the sense of Def. \ref{Quotient}) exist.

\end{defn}

## Properties

\begin{prop} \label{Zero relations}
For all $\alpha \,,\, A \to B$
we have  $0 \circ \alpha = \alpha \circ 0 = 0$.
\end{prop}
([Kawahara 1995](#Kawahara95))
\begin{proof}
$
  0 \sqsubseteq 0 \div \alpha
  \;\;\iff\;\;
  \alpha \circ 0 \sqsubseteq 0
  \;\;\iff\;\;
  \alpha \circ 0 = 0
  \,.
$
\end{proof}

## Examples

\begin{example}
[[Rel]] is the classic example of a meros, analogous to how [[Set]] is the archetypical [[topos]].

[Kawahara 1995](#Kawahara95) uses also the example of $Rel \times Rel$.
\end{example}

## Related Concepts

* [[SEAR]]

## References

The notion of meroi was introduced in:

* {#Kawahara95} [[Kawahara Yasuo]], _Relational Set Theory_, in Proceedings of the 6th International Conference on Category Theory and Computer Science, 1995 ([doi:10.1007/3-540-60164-3_19](http://dx.doi.org/10.1007/3-540-60164-3_19), [pdf](https://www.researchgate.net/profile/Yasuo-Kawahara/publication/226561376_Relational_set_theory/links/569f960908ae21a564270542/Relational-set-theory.pdf))

[[!redirects meros]]
[[!redirects meroi]]
[[!redirects meroses]]


meson

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### Context
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[QCD]] a _meson_ is a [[bound state]] of two [[quarks]] via the [[strong nuclear force]].

\linebreak

[[!include flavors of fundamental particles -- table]]

## Examples

### First-generation mesons
 {#FirstGenerationMesons}

The mesons in the first [[generation of fermions]], being [[bound states]] of [[up quarks]] and [[down quarks]] are

* [[π-meson]] (pion)

* [[ρ-meson]]

* [[ω-meson]]

{#WignerClassification} As [[linear representations]] of the [[Lorentz group]] (or rather of the[[Pin group]]) and [[isospin]] [[flavour (particle physics)|flavour]] group (under [[Wigner classification]]) these first-generation meson [[field (physics)|fields]] are given as in the first of the following two tables:

\begin{imagefromfile}
    "file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": 0,
        "right": 10,
        "bottom": 0,
        "left": 20
    }
\end{imagefromfile}

Here $\mathbf{n}$ denotes an [[irreducible representation]] of [[dimension]] $n$, and the subscript ${}_{\mathrm{sgn}}$ indicates the one of the two irreps of that dimension which pick up an extra sign under [[orientation]]-reversal ($\mathbf{4}_{{}_{sgn}} = \mathbf{4} \otimes \mathbf{1}_{{}_{sgn}}$ for $\mathbf{1}_{{}_{sgn}}$ the [[sign representation]]).

So the [[pion]] is a [[Lorentz group|Lorentz]]-[[pseudoscalar]], and the [[omega-meson]] and [[rho-meson]] are [[Lorentz group|Lorentz]] [[pseudovector]]  [[field (physics)|fields]].

As bilinears in the [[up-quark]] [[Weyl spinor]] [[field (physics)|field]] $u$ and [[down-quark]] [[Weyl spinor]] [[field (physics)|field]]  $d$ these [[meson]] [[field (physics)|fields]] are given as follows (with $(\sigma^0, \sigma^1, \sigma^2, \sigma^3)$ the [[Pauli matrices]]):

$$
  \sigma^0
  \underset{
    {\color{blue}omega}
    \atop
    {\color{blue}meson}
  }{
    \underbrace{
      \omega^\mu
    }
  }
  +
  \sigma^i
  \underset{
    {\color{blue}rho}
    \atop
    {\color{blue}meson}
  }{
    \underbrace{
      \rho^\mu_i
    }
  }
  \;=\;
  \left(
  \array{
    \bar u \gamma^\mu \gamma^5 u
    &
    \bar u \gamma^\mu \gamma^5 d
    \\
    \bar d \gamma^\mu \gamma^5 u
    &
    \bar d \gamma^\mu \gamma^5 d
  }
  \right)
$$

### Second-generation mesons

Mesons containing [[strange quarks]] (the first [[heavy mesons]] with respect to 2-flavor [[chiral perturbation theory]]) from the second [[generation of fermions]]:

* [[eta-meson]]

* [[kaon]], [[K*-meson]]

* [[phi-meson]]

### Third-generation mesons

[[heavy meson|Heavy mesons]] containing [[bottom quarks]] from the second [[generation of fermions]]:

* [[B meson]]

### See also

* [[quarkonium]]

## Properties

### Conceptualization and computation in AdS/QCD
 {#ConceptualizationAndComputationInAdSQCD}

In the [[Witten-Sakai-Sugimoto model]] for [[non-perturbative effect|strongly coupled]] [[QCD]] via an [[intersecting D-brane model]], the [[hadrons]] in [[QCD]] correspond to [[string theory|string-theoretic]]-phenomena in an ambient [[bulk field theory]] on an approximately [[anti de Sitter spacetime]]:

1. the [[mesons]] ([[bound states]] of 2 [[quarks]]) correspond to [[open strings]] in the bulk, whose two endpoints on the [[asymptotic boundary]] correspond to the two [[quarks]];

1. [[baryons]] ([[bound states]] of $N_c$ [[quarks]]) appear in two different but equivalent ([Sugimoto 16, 15.4.1](#Sugimoto16)) guises:

   1. as [[wrapped brane|wrapped]] [[D4-branes]] with $N_c$ [[open strings]] connecting them to the [[D8-brane]]

      ([Witten 98b](#Witten98b), [Gross-Ooguri 98](#GrossOoguri98))

   1. as [[skyrmions]]

      ([Sakai-Sugimoto 04, section 5.2](#SakaiSugimoto04), [Sakai-Sugimoto 05, section 3.3](#SakaiSugimoto05), see [Bartolini 17](#Bartolini17)).

For review see [Sugimoto 16](#Sugimoto16), also [Rebhan 14, around (18)](#Rebhan14).

> graphics grabbed from [Sugimoto 16](#Sugimoto16)

This produces [[baryon]] [[mass]] spectra with moderate quantitative agreement with [[experiment]] ([HSSY 07](#HSSY07)):

> graphics grabbed from [Sugimoto 16](#Sugimoto16)

## Related concepts

* [[scalar meson]], [[vector meson]], [[tensor mesons]]

* [[exotic meson]]

  * [[XYZ particle]]

  * [[tetraquark]]

* [[hadron]], [[baryon]]

* [[Skyrmion]]

* [[hadron supersymmetry]], [[hadron Kaluza-Klein theory]]

  * [[flavor brane]]

* [[confinement]]

  * [[chiral perturbation theory]]

  * [[quark bag model]], [[Cheshire cat principle]]

  * [[vector meson dominance]]

## References

### General

* Eef van Beveren, George Rupp, _Scalar and axial-vector mesons_, Eur. Phys. J. A31:468-473, 2007 ([arXiv:hep-ph/0610199](https://arxiv.org/abs/hep-ph/0610199))

Origin of [[effective field theory]] of [[mesons]] in [[nuclear physics]], via [[hidden local gauge symmetry]]:

* [[Jun John Sakurai]], _Theory of strong interactions_, Annals Phys. 11 (1960) 1-48 ([spire:3218](http://inspirehep.net/record/3218), doi:10.1016/0003-4916(60)90126-3)

Introduction and survey:

* [[Jun John Sakurai]], _Currents and Mesons_, Chicago Lectures in Physics, based on notes by George Barry, University of Chicago Press (1969) ([ISBN: 9780226733838](https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3622598.html))

* [[Rupert Machleidt]], _The Meson Theory of Nuclear Forces and Nuclear Structure_, In: Negele J.W., Vogt E. (eds.) _Advances in Nuclear Physics_ Advances in Nuclear Physics, vol 19. Springer 1989 ([doi:10.1007/978-1-4613-9907-0_2](https://doi.org/10.1007/978-1-4613-9907-0_2))

* {#MachleidtEntem11} [[Rupert Machleidt]], [[David Rodríguez Entem]], _Chiral effective field theory and nuclear forces_, Phys. Rept. 503:1-75, 2011 ([arXiv:1105.2919](https://arxiv.org/abs/1105.2919), [doi:10.1016/j.physrep.2011.02.001](https://doi.org/10.1016/j.physrep.2011.02.001))

* Francesco Giacosa, *Conventional mesons below 2 GeV* [[arXiv:2211.10092](https://arxiv.org/abs/2211.10092)]

* [[Franz Gross]], [[Eberhard Klempt]] et al., Section 8 of: *50 Years of Quantum Chromodynamics*, EJPC [[arXiv:2212.11107](https://arxiv.org/abs/2212.11107)]

See also

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

* Yi Zhang, _[Mesons](http://snst-hu.lzu.edu.cn/zhangyi/ndata/Meson.html)

The [[pole approximation]]:

* [[Murray Gell-Mann]] and Fredrik Zachariasen, _Form Factors and Vector Mesons_, Phys. Rev. 124, 953 (1961) ([doi:10.1103/PhysRev.124.953](https://doi.org/10.1103/PhysRev.124.953))

### In chiral perturbation theory

See the references at _[[chiral perturbation theory]]_.

[[!include quantum hadrodynamics -- references ]]

[[!include Skyrme hadrodynamics with vector mesons -- references]]

[[!include Skyrme hadrodynamics with heavy mesons -- references]]

[[!include WZW term of QCD chiral perturbation theory -- references]]

[[!include hadrons as KK-modes of 5d Yang-Mills theory -- references]]

### In the large $N$ limit

In the [[large N limit]]:

* {#Witten79} [[Edward Witten]], _Baryons in the $1/n$ Expansion_, Nucl. Phys. B160 (1979) 57-115 ([spire:140391](http://inspirehep.net/record/140391), doi:10.1016/0550-3213(79)90232-3)

* Jan Albert, Johan Henriksson, [[Leonardo Rastelli]], Alessandro Vichi, *Bootstrapping mesons at large $N$: Regge trajectory from spin-two maximization* [[arXiv:2312.15013](https://arxiv.org/abs/2312.15013)]

[[!redirects mesons]]

[[!redirects meson field]]
[[!redirects meson fields]]

[[!redirects meson field]]
[[!redirects meson fields]]


mesoscopic physics

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#### Solid state physics
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=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

By _mesoscopic physics_ one mostly means that part of [[solid state physics]] which applies to material on the [[scale]] between [[nanometers]] and [[micrometers]].

This is about the scale where "macroscopic" bulk properties well-described by [[thermodynamics]] (see [[thermodynamic limit]]) begin to transition to "microscopic" behaviour (say of single [[molecules]]) well-described by [[quantum mechanics]].

## Related concepts

* [[solid state physics]]

* [[nuclear physics]]

* [[high energy physics]]

## References

See also

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