von Neumann hierarchy

# Idea

The **von Neumann hierarchy** is a way of "building up" all [[pure set]]s recursively, starting with the [[empty set]], and indexed by the [[ordinal number]]s.

# Definition

Using [[transfinite recursion]], define a hierarchy of well-founded sets $V_\alpha$, where $\alpha\in\mathbf{Ord}$ is an [[ordinal number]], as follows:

* $V_0 = \emptyset$
* $V_{\alpha+1} = P(V_\alpha)$ (the [[power set]] of $V_\alpha$)
* $V_\alpha = \cup_{\beta\lt\alpha} V_\beta$ if $\alpha$ is a limit ordinal.

The formula for $0$ is actually a special case of the formula for a limit ordinal.  Alternatively, you can do them all at once:

* $V_\alpha = \cup_{\beta \lt \alpha} P(V_\beta)$

The [[axiom of foundation]] in [[ZFC]] is equivalent to the statement that every set is an element of $V_\alpha$ for some ordinal $\alpha$.  The **rank** of a set $x$ is defined to be the least $\alpha$ for which $x\in V_\alpha$ (this is well-defined since the ordinals are [[well-order|well-ordered]]).


von Neumann regular category

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#### Category theory
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#### Topos Theory
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# Contents
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[[!redirects Von Neumann regular categories]]
[[!redirects Von Neumann regular category]]
[[!redirects von Neumann regular categories]]

## Idea

**Von Neumann regular categories**, or _regular categories_ for short, are a class of [[small categories]] introduced by [[William Lawvere]] ([2002](#Law02)) in the context of his theory of dimension and [[unity of opposites]] in order to provide [[presheaf toposes]] with well-behaved [[level of a topos|levels]] which in particular admit the [[Aufhebung]] of each level.

As shown in ([Kelly-Lawvere 1989](#KL89)) essential subtoposes of presheaf toposes $\mathcal{S}^{\mathcal{C}^{op}}$ correspond to idempotent two-sided ideals $J$ of the underlying small category $\mathcal{C}$. What causes the problems for the general existence of Aufhebung at each level is the fact that an infinite intersection of such $J$ is not necessarily idempotent itself. The condition occurring in the definition of a von Neumann regular category is a simple way to enforce this.

## Definition

A small category $\mathcal{C}$ is called _von Neumann regular_ if all two-sided ideals are [[idempotent]].

## Remark

The terminology is presumably chosen in view of the concept of a _von Neumann regular ring_ $R$ i.e. one such that every $a \in R$ has a 'weak inverse' $\bar{a}$ with $a = a \bar{a} a$ as the following property illustrates.

## Properties

* $\mathcal{C}$ is von Neumann regular iff  for any morphism $a$ in $\mathcal{C}$ there exists a reverse morphism $\bar{a}$ and two endomorphisms $x,y$ with $a=y a \bar{a} a x$. ([Lawvere 2002](#Law02))

## Related entries

* [[Aufhebung]]

* [[graphic category]]

* [[level of a topos]]

## References

* [[Francis Borceux|F. Borceux]], [[Jiri Rosicky|J. Rosicky]], _On Von Neumann Varieties_ , TAC **13** no. 1 (2004) pp.5-26. ([pdf](http://www.tac.mta.ca/tac/volumes/13/1/13-01.pdf))

* {#KL89}[[G. M. Kelly]], [[F. W. Lawvere]], _On the Complete Lattice of Essential Localizations_ , Bull.Soc.Math. de Belgique **XLI** (1989) 261-299 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]

* {#Law89b} [[F. W. Lawvere]], _Display of graphics and their applications, as exemplified by 2-categories and the Hegelian "taco"_ , Proceedings of the first international conference on algebraic methodology and software technology University of Ioowa, May 22-24 1989, Iowa City, pp.51-74.

* {#Law02} [[F. W. Lawvere]], _Linearization of graphic toposes via Coxeter groups_ , JPAA **168** (2002) pp.425-436.


von Neumann–Bernays–Gödel set theory

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## Idea

NBG or von Neumann–Bernays–Gödel set theory is a [[material set theory]]. It is a conservative extension of [[ZFC]] and its ontology includes [[proper classes]], like [[MK]]. But unlike [[ZFC]] and [[MK]], NBG can be finitely axiomatized.

## History

[[Georg Cantor]] was well acquainted with the phenomenon that some collections are 'too big' to be sets and in his late years made a distinction between _consistent_ and _inconsistent multitudes_, the former comprising sets, but his ideas were confined to private letters to (Jourdain and) Dedekind that were not published until 1932.

[[John von Neumann]] began $NBG$ in the 1920s ([von Neumann 1925](#vN1925), [von Neumann 1928](#vN1928)), but his version was unwieldy.  [[Paul Bernays]] and [[Kurt Gödel]] simplified it later.  It also illustrates Gödel's theorem that any [[first-order theory]] has a [[conservative extension]] with a finite axiomatization.

## Axioms

NBG is a [[material set theory]], based on a global binary membership [[predicate]] $\in$. The objects of NBG are called [[classes]]. If a class $x$ is a member of another class $A$, i.e., $x \in A$, then $x$ is called [[set]]. A class which is not a set is called [[proper class]]. NBG can be presented as a two-sorted theory, with lowercase letters denoting sets and uppercase letters denoting classes.

1. [[axiom of extensionality|Extensionality]]: Two classes are equal if and only if the have the same members.

2. [[axiom of pairing|Pairing]], [[axiom of union|Union]], [[axiom of power sets|Power Sets]] and [[axiom of infinity|Infinity]]: regard sets and are identical to the [[ZFC]] counterparts.

3. [[axiom of foundation|Foundation]]: For each nonempty class $A$ there exists a set $x \in A$ such that $x \cap A = \varnothing$.

4. [[axiom of comprehension|Class Comprehension schema]]: For any formula $\phi$ containing no quantifiers over classes (it may contain quantifiers over sets and it may contain both class and set parameters), there is a class $A$ whose elements are precisely those sets $x$ satisfying $\phi(x)$.  In particular, taking $\phi(x)$ as $x = x$ it follows that there exists the class $V$ of all sets.

5. Limitation of Size: A class $A$ is a set if and only if there is no bijection between $A$ and the class $V$ of all sets.

From the axiom of Limitation of Size, it turn out that $V$ is not a set but a proper class, avoiding [[Russell's paradox]]. Moreover, since one can prove that the [[ordinal number|ordinal numbers]] form a proper class, there is a bijection beetween $V$ and $\mathrm{Ord}$, i.e., the class of all sets can be well-ordered which implies the [[axiom of global choice]].

## Finite axiomatization

Every instance of class comprehension can be built out of a few, based on the logical connnectives.  This is similar to the finite axiomatization of [[bounded separation|bounded]] $ZFC$ (or, in other language, [[ETCS]]); indeed, $NBG$ is essentially bounded [[MK]].

## References

* [[John von Neumann]], "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik 154 (1925) 219–240, doi:[10.1515/crll.1925.154.219](http://dx.doi.org/10.1515/crll.1925.154.219), [GDZ](http://gdz.sub.uni-goettingen.de/index.php?id=resolveppn&PPN=GDZPPN002169606)
{#vN1925}

* [[John von Neumann]], "Die Axiomatisierung der Mengenlehre", Mathematische Zeitschrift 27 (1928) 669–752, doi:[10.1007/bf01171122](http://dx.doi.org/10.1007/bf01171122), [GDZ](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0027&DMDID=DMDLOG_0042)
{#vN1928}

* [[Gerhard Osius]], _Kategorielle Mengenlehre: Eine Charakterisierung der Kategorie der Klassen und Abbildungen_ , Math. Ann. **210** (1974) pp.171-196. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002309939))

[[!redirects NBG]]
[[!redirects NBG set theory]]
[[!redirects Neumann-Bernays-Gödel set theory]]
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[[!redirects von Neumann-Bernays-Gödel set theory]]
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[[!redirects von Neumann--Bernays--Gödel set theory]]

[[!redirects NBG class theory]]
[[!redirects Neumann-Bernays-Gödel class theory]]
[[!redirects Neumann–Bernays–Gödel class theory]]
[[!redirects Neumann--Bernays--Gödel class theory]]
[[!redirects von Neumann-Bernays-Gödel class theory]]
[[!redirects von Neumann–Bernays–Gödel class theory]]
[[!redirects von Neumann--Bernays--Gödel class theory]]


von Staudt

__Karl Georg Christian von Staudt__ was one of the major [[geometry of 19th century|geometers of the 19th century]].

* Jeremy Gray, _World out of nothing. A course in history of geometry in the 19th century_, Springer Undergraduate Mathematics Series (2007)
* wikipedia [Karl Georg Christian von Staudt](http://en.wikipedia.org/wiki/Karl_Georg_Christian_von_Staudt)

[[!redirects Karl Georg Christian von Staudt]]


Vopěnka's principle

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# Vopěnka\'s principle
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## Idea

**Vopěnka's principle** is a [[large cardinal]] axiom which implies a good deal of simplification in the theory of [[locally presentable categories]].

It is fairly strong as large cardinal axioms go: Its consistency follows from the existence of [[huge cardinal]]s, and it implies the existence of arbitrarily large [[measurable cardinal]]s.

## Statements

### The Vopěnka principle

Vopěnka's principle has many equivalent statements.  Here are a few:

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

The VP is equivalent to the statement:

Every [[discrete category|discrete]] [[full subcategory]] of a [[locally presentable category]] is [[small category|small]].

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###### Theorem

The VP is equivalent to the statement:

For every [[proper class]] sequence $\langle M_\alpha | \alpha \in Ord\rangle$ of [[logic|first-order structures]], there is a pair of [[ordinals]] $\alpha\lt\beta$  for which $M_\alpha$ [[elementary embedding|embeds elementarily]] into $M_\beta$.

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

The VP is equivalent to the statement:

For $C$ a [[locally presentable category]],  every [[full subcategory]] $D \hookrightarrow C$ which is closed under [[colimit]]s is a [[coreflective subcategory]].

=--

This is ([AdamekRosicky, theorem 6.28](#AdamekRosicky)).

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###### Theorem

The VP is equivalent to the statement:

Every [[cofibrantly generated model category]] (in a slightly more general sense than usual) is a [[combinatorial model category]].

=--

This is in ([Rosicky](#Rosicky))

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###### Remark

If one insists on the traditional stricter definition of cofibrant generated model category, then the VP still implies that these are all combinatorial. But the VP is slightly stronger than this statement.

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###### Theorem
The VP is equivalent to both of the statements:

1. For every $n$, there exists a [[C(n)-extendible cardinal]].
1. For every $n$, there exist arbitrarily large [[C(n)-extendible cardinals]].
=--

This is in ([BCMR](#BagariaCasacubertaMathiasRosicky)).

### The weak Vopěnka principle

The Vopěnka principle implies the weak Vopěnka principle.

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###### Theorem

The weak VP is equivalent to the statement:

For $C$ a [[locally presentable category]],  every [[full subcategory]] $D \hookrightarrow C$ which is closed under [[limit]]s is a [[reflective subcategory]].

=--

This is [AdamekRosicky, theorem 6.22 and example 6.23](#AdamekRosicky)

### Relativized versions of Vopěnka's principle

Vopěnka's principle can be relativized to levels of the [[Lévy hierarchy]] by restricting the complexity of the (definable) classes to which it is applied.  The following theorems are from ([BCMR](#BagariaCasacubertaMathiasRosicky)).

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###### Theorem
For any $n\ge 1$, the following statements are equivalent.

1. There exists a [[C(n)-extendible cardinal]].
1. Every proper class of first-order structures that is defined by a conjunction of a $\Sigma_{n+1}$ formula and a $\Pi_{n+1}$ formula contains distinct structures $M$ and $N$ and an [[elementary embedding]] $M\hookrightarrow N$.
=--

The "$n=0$ case" of this is:

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###### Theorem
For any $n\ge 1$, the following statements are equivalent.

1. There exists a [[supercompact cardinal]].
1. Every proper class of first-order structures that is defined by a $\Sigma_2$ formula contains distinct structures $M$ and $N$ and an [[elementary embedding]] $M\hookrightarrow N$.
=--

Many more refined results can be found in ([BCMR](#BagariaCasacubertaMathiasRosicky)).

## Motivation

From a category-theoretic perspective, Vopěnka's principle can be motivated by applications and consequences, but it can also be argued for somewhat *a priori*, on the basis that *large discrete categories* are rather pathological objects.  We can't avoid them entirely (at least, not without restricting the rest of mathematics fairly severely), but maybe at least we can prevent them from occurring in some nice situations, such as full subcategories of locally presentable categories.  See [this MO answer](http://mathoverflow.net/questions/29302/reasons-to-believe-vopenkas-principle-huge-cardinals-are-consistent/29473#29473).

## Consequences {#Consequences}

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

The VP implies the statement:

Let $C$ be a [[left proper model category|left proper]] [[combinatorial model category]] and $Z \in Mor(C)$ a [[class]] of [[morphism]]s. Then the [[Bousfield localization of model categories|left Bousfield localization]] $L_Z W$ exists.

=--

This is theorem 2.3 in ([RosickyTholen](#RosickyTholen))

+-- {: .num_corollary #ConsequenceForReflectiveInfCatLoc}
###### Corollary

The VP implies the statement:

Let $C$ be a [[locally presentable (∞,1)-category]] and $Z$ a class of morphisms in $C$. Then the reflective [[localization of an (∞,1)-category|localization]] of $C$ at $W$ extsts.

=--

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

By the facts discussed at [[locally presentable (∞,1)-category]] and [[combinatorial model category]] and [[Bousfield localization of model categories]] we have that every locally presentable $(\infty,1)$-category is presented by a combinatorial model category and that under this correspondence reflective localizations correspond to left Bousfield localizations. The claim then follows with the ([above theorem](#ConsequenceForBousfieldLoc)).

=--

## Set-theoretic notes

### First- versus second-order

As usually stated, Vopěnka's principle is not formalizable in first-order [[ZF]] set theory, because it involves a "second-order" [[quantifier|quantification]] over [[proper classes]] ("...there does not exist a large discrete subcategory...").  It can, however, be formalized in this way in a class-set theory such as [[NBG]].

On the other hand, it can be formalized in ZF as a first-order axiom schema consisting of one axiom for each class-defining formula $\phi$, stating that "$\phi$ does not define a class which is a large discrete subcategory..."  We might call this axiom schema the **Vopěnka axiom scheme**.  As in most situations of this sort, the first-order Vopěnka scheme is appreciably weaker than the second-order Vopěnka principle.  See, for instance, [this MO question](http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably) and answer.

### Vopěnka cardinals

Unlike some large cardinal axioms, Vopěnka's principle does not appear to be merely an assertion that "there exist very large cardinals" but rather an assertion about the precise size of the "universe" (the "boundary" between sets and proper classes).  In other words, the universe could be "too big" for Vopěnka's principle to hold, in addition to being "too small."

(The equivalence of Vopěnka's principle with the existence of [[C(n)-extendible cardinals]] may appear to contradict this.  However, the property of being $C(n)$-extendible itself "depends on the size of the whole universe" in a sense.)

More precisely, if $\kappa$ is a cardinal such that $V_\kappa$ satisfies ZFC + Vopěnka's principle, then knowing that $\lambda\gt\kappa$ does not necessarily imply that $V_\lambda$ also satifies Vopěnka's principle.  By contrast, if $V_\kappa$ satisfies ZFC + "there exists a [[measurable cardinal]]" (say), then there must be a measurable cardinal less than $\kappa$, and that measurable cardinal will still exist in $V_\lambda$ for any $\lambda\gt\kappa$.  On the other hand, large cardinal axioms such as "there exist arbitrarily large measurable cardinals" have the same property that Vopěnka's principle does: even if measurable cardinals are unbounded below $\kappa$, they will not be unbounded below $\lambda$ if $\lambda$ is the next greatest [[inaccessible cardinal]] after $\kappa$.

Relativizing Vopěnka's principle to cardinals also raises the same first- versus second-order issues as above.  We say that a **Vopěnka cardinal** is one where Vopěnka's principle holds "in $V_\kappa$" where the quantification over classes is interpreted as quantification over all subsets of $V_\kappa$.  By contrast, we could define an **almost-Vopěnka cardinal** to be one where $V_\kappa$ satisfies the first-order Vopěnka scheme.  Then one can show, using the Mahlo reflection principle (see [here](http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably/46538#46538) again), that every Vopěnka cardinal $\kappa$  is a limit of $\kappa$-many almost-Vopěnka cardinals, and in particular the smallest almost-Vopěnka cardinal cannot be Vopěnka.  Thus, being Vopěnka is much stronger than being almost-Vopěnka.

### Definable counterexamples

If Vopěnka's principle fails, then there exist counterexamples to all of its equivalent statements, such as a large discrete full subcategory of a locally presentable category.  If Vopěnka's principle fails but the first-order Vopěnka scheme holds, then no such counterexamples can be explicitly definable.

On the other hand, if the Vopěnka scheme also fails, then there will be explicit finite formulas one can write down which define counterexamples.  However, there is no "universal" counterexample, in the following sense: if Vopěnka's principle is consistent, then for any class-defining formula $\phi$, there is a model of set theory in which Vopěnka's principle fails (and even in which the first-order Vopěnka scheme fails), but in which $\phi$ does not define a counterexample to it.  See [here](http://mathoverflow.net/questions/45602/can-vopenkas-principle-be-violated-definably/46538#46538) yet again.

## References

The relation to the theory of [[locally presentable categories]] is the contents of chapter 6 of

* [[Jiří Adámek]], [[Jiří Rosický]], _[[Locally presentable and accessible categories]]_,  London Mathematical Society Lecture Note Series 189
{#AdamekRosicky}

The relation to [[combinatorial model categories]] is discussed in

* [[Jiří Rosický]], _Are all cofibrantly generated model categories combinatorial?_ ([ps](http://www.math.muni.cz/~rosicky/papers/cof1.ps))
{#Rosicky}

The implication of VP on [[homotopy theory]], [[model categories]] and [[cohomology localization]] are discussed in the following articles

* [[Jiří Rosický]], [[Walter Tholen]], _Left-determined model categories and universal homotopy theories_  Transactions of the American Mathematical Society
Vol. 355, No. 9 (Sep., 2003), pp. 3611-3623 ([JSTOR](http://www.jstor.org/stable/1194855)).
{#RosickyTholen}

* [[Carles Casacuberta]], Dirk Scevenels, [[Jeff Smith]], _Implications of large-cardinal principles in homotopical localization_  Advances in Mathematics
Volume 197, Issue 1, 20 October 2005, Pages 120-139

* Joan Bagaria, [[Carles Casacuberta]], Adrian Mathias, [[Jiří Rosicky]] _Definable orthogonality classes in accessible categories are small_, [arXiv](http://arxiv.org/abs/1101.2792)
{#BagariaCasacubertaMathiasRosicky}

* Giulio Lo Monaco, _Vopěnka's principle in ∞-categories_, [arxiv:2105.04251](https://arxiv.org/abs/2105.04251)

category: foundational axiom

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vortex

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## Idea

A kind of [[soliton]].

\begin{imagefromfile}
    "file_name": "VorticesSchematics.jpg",
    "width": 300,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": -10,
        "right": 0,
        "left": 10
    },
    "caption": "From [Thuneberg](#Thuneberg)"
\end{imagefromfile}

## Examples

[[!include flux quantization in superconductors -- section]]

### Vortex anyons

See *[Votex anyons](braid+group+statistics#VortexAnyons)* at *[[braid group statistics]]*.

## Related concepts

* [[instanton]]

* [[monopole]]

* [[vortex string]]

* [[Skyrmion]]

* [[domain wall]]

* [[cosmic string]]

## References

### General

Reviews and surveys include

* [[David Tong]], _TASI Lectures on Solitons_ ([web](http://www.damtp.cam.ac.uk/user/tong/tasi.html)), part 3: _Vortices_ ([pdf](http://www.damtp.cam.ac.uk/user/tong/tasi/vortex.pdf))

On water vortices forming [[knots]]:

* Dustin Kleckner and William T. M. Irvine, *Creation and dynamics of knotted vortices*, Nature Physics **9** (2013) 253–258 [[doi:10.1038/nphys2560](https://doi.org/10.1038/nphys2560)]

On quantum vortices as in [[superconductors]] or [[Bose-Einstein condensates]]/[[superfluids]]:

* Rufus Phillips, *[Quantum Vortices](https://www.physics.umd.edu/courses/Phys726/The_Quantum_Vortex.htm)*.

* {#Thuneberg} Erkki Thuneberg, *[Superfluidity and Quantized Vortices](http://ltl.tkk.fi/research/theory/vortex.html)*

* K. R. Sreenivasan, *Vortices, particles and superfluidity* ([pdf](https://higherlogicdownload.s3.amazonaws.com/APS/22d184d0-d6d2-4493-97da-847d341d3d82/UploadedImages/Documents/Sreenivasan_DFD07.pdf), [[SreenivasanVortices.pdf:file]])

For more see at _[[vortex string]]_.

A historical precursor with an abandoned but influential speculation is

* [[Lord Kelvin]], _[[On Vortex Atoms]]_

[[!include defect anyons -- references]]

[[!redirects vortices]]


vortex string

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## Idea

By a _vortex string_ one typically means a [[string]]-shaped [[soliton]] threading through a [[vortex]]-solution in a 4-dimensional [[gauge theory]].

The original example are the magnetic flux tubes through the [[Abrikosov vortices]] seen in [[type II superconductors]] at critical values of the [[magnetic field]]. The idea to think of these as [[string]]-like dynamical objects in themselves, much as in [[string theory]], is due to [Nielsen-Olesen 73](#NielsenOlesen73), hence one also speaks of _Nielsen-Oleson vortices_ or _Nielsen-Olesen strings_.

More generally, such string-shaped vortices are considered in [[Yang-Mills theories]] with [[non-abelian group|non-abelian]] [[gauge group]] (see e.g. [Tong 09](#Tong09)).

## Details

[[!include flux quantization in superconductors -- section]]

## Related cocepts

* [[Abrikosov vortex]]

* [[cosmic string]]

* [[defect brane]]

## References

### In electromagnetism

Vortex strings were originally considered in the abelian gauge theory of [[electromagnetism]] as [[Abrikosov vortices]] in [[type II superconductors]].

* {#Abrikosov57a} [[Alexei Abrikosov]], _On the Magnetic properties of superconductors of the second group_, Sov. Phys. JETP 5 (1957) 1174-1182, Zh. Eksp. Teor. Fiz. 32 (1957) 1442-1452 ([spire:9138](https://inspirehep.net/literature/9138))

The suggestion that these vortex flux tube could be thought of as dynamical [[strings]] much as in [[string theory]]:

* {#NielsenOlesen73} [[Holger Bech Nielsen]], [[Poul Olesen]], *Vortex-line models for dual strings*, Nuclear Physics B **61** (1973) 45-61 \[doi:10.1016/0550-3213(73)90350-7\]

* {#BeekmanZaanen11} [[Aron J. Beekman]], [[Jan Zaanen]], *Electrodynamics of Abrikosov vortices: the field theoretical formulation*, Front. Phys. **6** (2011) 357–369 [[doi:10.1007/s11467-011-0205-0](https://doi.org/10.1007/s11467-011-0205-0)]

* {#Polyakov08} [[Alexander M. Polyakov]], *From Quarks to Strings* [[arXiv:0812.0183](https://arxiv.org/abs/0812.0183)]

  published as: *Quarks, strings and beyond*, section 44 in: [[Paolo Di Vecchia]] et al. (ed.), *The Birth of String Theory*, Cambridge University Press (2012) 544-551 [[doi:10.1017/CBO9780511977725.048](https://doi.org/10.1017/CBO9780511977725.048)]

  > "I remember that in the late sixties to early seventies Tolya Larkin and I discussed (many times) whether Abrikosov’s vortices could be viewed as elementary particles. Nothing concrete came out of this at that time, but it helped me with my later work. With some imagination we could have related the vortex lines with strings but we missed it."

See also:

* Wikipedia, _[Nielsen-Olesen vortex](https://en.wikipedia.org/wiki/Nielsen%E2%80%93Olesen_vortex)_

* Wikipedia, *[Quantum vortex](https://en.wikipedia.org/wiki/Quantum_vortex)*

### In Yang-Mills theory

A more general picture of vortex strings in [[Yang-Mills theory]] with [[non-abelian group|nonabelian]] [[gauge group]] (in fact in [[D=4 N=2 super Yang-Mills theory]]):

* [[Amihay Hanany]], [[David Tong]], _Vortices, Instantons and Branes_, JHEP 0307 (2003) 037 ([arXiv:hep-th/0306150](https://arxiv.org/abs/hep-th/0306150))

* [[Amihay Hanany]], [[David Tong]], _Vortex Strings and Four-Dimensional Gauge Dynamics_, JHEP 0404 (2004) 066 ([arXiv:hep-th/0403158](https://arxiv.org/abs/hep-th/0403158))

reviewed in:

* {#Tong09} [[David Tong]], _Quantum Vortex Strings: A Review_, Annals Phys. 324:30-52, 2009 ([arXiv:0809.5060](https://arxiv.org/abs/0809.5060))

* [[David Tong]], _Vortices, Strings, and Vortex Strings_ ([[TongVortexStrings.pdf:file]])

As [[probe brane|probe]] [[D1-branes]] of [[D1-D5 brane bound states]] in a context of [[AdS-QCD duality]]:

* [[Joseph Polchinski]], [[Matthew Strassler]], p. 34 of: _The String Dual of a Confining Four-Dimensional Gauge Theory_ ([arXiv:hep-th/0003136](https://arxiv.org/abs/hep-th/0003136))

* Roberto Auzzi, S. Prem Kumar, Section 5.2 in: _Non-Abelian k-Vortex Dynamics in $\mathcal{N} = 1^\ast$ theory and its Gravity Dual_, JHEP 0812:077, 2008 ([arXiv:0810.3201](https://arxiv.org/abs/0810.3201))

On [[vortex strings]] in $\mathcal{N}=2$ super QCD identified as [[superstrings]] in [[type II string theory]] [[KK-compactification|compactified]] on a [[conifold]]:

* [[Mikhail Shifman]], [[A. Yung]], *Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories*, Rev. Mod. Phys. **79** 1139 (2007) [[doi:10.1103/RevModPhys.79.1139](https://doi.org/10.1103/RevModPhys.79.1139), [arXiv:hep-th/0703267](https://arxiv.org/abs/hep-th/0703267)

* [[Mikhail Shifman]], [[A. Yung]], *Supersymmetric Solitons*, Cambridge University Press (2009) [[doi:10.1017/CBO9780511575693](https://doi.org/10.1017/CBO9780511575693)]

* [[Mikhail Shifman]], [[A. Yung]], *Critical String from Non-Abelian Vortex in Four Dimensions*,
Physics Letters B **750** (2015) 416-419
[[arXiv:1502.00683](https://arxiv.org/abs/1502.00683), [doi:10.1016/j.physletb.2015.09.045](https://doi.org/10.1016/j.physletb.2015.09.045)
]

* P. Koroteev, [[Mikhail Shifman]], [[A. Yung]], *Studying Critical String Emerging from Non-Abelian Vortex in Four Dimensions*, Phys.Lett. B **759** (2016) 154-158
[[arXiv:1605.01472](https://arxiv.org/abs/1605.01472), [doi:10.1016/j.physletb.2016.05.075](https://doi.org/10.1016/j.physletb.2016.05.075)
]

[[!redirects vortex strings]]

[[!redirects Abrikosov vortex]]
[[!redirects Abrikosov vortices]]

[[!redirects Abrikosov vortex string]]
[[!redirects Abrikosov vortex strings]]

[[!redirects Nielsen-Olesen vortex]]
[[!redirects Nielsen-Olesen vortices]]

[[!redirects Nielsen-Olesen string]]
[[!redirects Nielsen-Olesen strings]]

[[!redirects Nielsen-Olesen vortex string]]
[[!redirects Nielsen-Olesen vortex strings]]


vpatryshev > history

< [[vpatryshev]]


vukovinski

Spirited Systems Redesigner


Vyacheslav Soroka

* [memorial page](http://ivv5hpp.uni-muenster.de/u/douplii/soroka)

* [[Steven Duplij]], _Supergravity was discovered by D.V. Volkov and V.A. Soroka in 1973, wasn't it?_, East Eur. J. Phys., v3, p. 81-82 (2019) ([arXiv:1910.03259](https://arxiv.org/abs/1910.03259))

## Selected writings

Introducing [[supergravity]] ([[D=4 supergravity]]):

* [[Dmitry Volkov]], [[Vyacheslav Soroka]], _Higgs effect for Goldstone particles with spin 1/2_, ZhETF Pis. Red. (JETP Letters, AIP translation), 18, n.8 (1973) 529 ([pdf](https://www.jetpletters.ac.ru/ps/1568/article_24038.shtml))

On [[D=4 supergravity]] formulated in [[superspace]]:

* V. Akulov, [[Dmitry Volkov]] and [[Vyacheslav Soroka]], _Generally covariant theories of gauge fields on superspace_, Theor. Math. Phys. 31 (1977) 285 ([doi:10.1007/BF01041233](https://doi.org/10.1007/BF01041233))

On the history of [[supergravity]]:

* [[Vyacheslav Soroka]], _The Sources of Supergravity_ in [The Supersymmetric World. The Beginnings of the Theory, G. Kane and M. Shifman (Eds.), World Scientific, 2000](http://www.worldscientific.com/doi/10.1142/9789812385505_0011) or [arXiv:hep-th/0203171](https://arxiv.org/abs/hep-th/0203171)

category: people


Véronique Bernard

## Selected writings

On [[chiral perturbation theory]]:

* [[Véronique Bernard]], [[Ulf-G. Meissner]], _Chiral perturbation theory_, Ann. Rev. Nucl. Part. Sci.57:33-60, 2007 ([arXiv:hep-ph/0611231](https://arxiv.org/abs/hep-ph/0611231))

On [[baryon chiral perturbation theory]]:

* [[Véronique Bernard]], _Chiral Perturbation Theory and Baryon Properties_, Prog. Part. Nucl. Phys. 60:82-160, 2008 ([arXiv:0706.0312](https://arxiv.org/abs/0706.0312))

category: people


Věra Trnková

Věra Trnková (1934 - 2018) was a pure [[category theory|category theorist]]. She was a student of [[Eduard Čech]]. The list of her students includes [[Jiří Adámek]], Jiří Velebil, Libor Barto, Václav Koubek.

* [Věra Trnková's homepage (archived)](https://web.archive.org/web/20180325022120/http://www.karlin.mff.cuni.cz/~trnkova/)
* [short obituary (in Czech)](https://www.mff.cuni.cz/verejnost/konalo-se/2018-05-trnkova/)
* J. Adámek, M. Katětov: Věra Trnková's unbelievable 60, in Mathematica Bohemica [dml.cz](https://dml.cz/handle/10338.dmlcz/126082), 1994.
* [Věra Trnková's Wikipedia entry](https://en.wikipedia.org/wiki/V%C4%9Bra_Trnkov%C3%A1)

## related entries

* [[category theory]]

category: people

[[!redirects Vera Trnkova]]


W algebra

## Idea

A __W algebra__ is a higher spin extension of the [[Virasoro algebra]]. It is an extended symmetry algebra
in [[conformal field theory]].

## References

An early survey is

* [[Peter Bouwknegt]], Kareljan Schoutens, _W symmetry in conformal field theory_, Phys. Rep. __223__:4 (1993) 183--276 doi

* [[Peter Bouwknegt]], Jim McCarthy, [[Krzysztof Pilch]], _The $W_3$ algebra: modules, semi-infinite cohomology, and BV algebras_, Lec. Notes in Phys. __42__, Springer 1996 ([doi:10.1007/978-3-540-68719-1](https://link.springer.com/book/10.1007/978-3-540-68719-1))

* Wikipedia, _[W-algebra](https://en.wikipedia.org/wiki/W-algebra#CITEREFBouwknegt1993)_

Relation to [[Jordan algebra]] is discussed in

* L. J. Romans, _Realisations of classical and quantum $W_3$ symmetry_, Nuclear Physics B __352__:3 (1991) 829--848 doi

Relation to [[L-infinity algebras]] is discussed in

* {#BlumenhagenFuchsTraube17} [[Ralph Blumenhagen]], Michael Fuchs, Matthias Traube, _$\mathcal{W}$-Algebras are $L_\infty$-algebras_ ([arXiv:1705.00736](https://arxiv.org/abs/1705.00736))

category: physics, algebra

[[!redirects W algebra]]
[[!redirects W algebras]]
[[!redirects W-algebra]]
[[!redirects W-algebras]]


W*-category

+-- {: .rightHandSide}
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### Context
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of $W^\ast$-categories is the special case of that of [[C-star category|$C^\ast$-categories]] like [[von Neumann algebras]] (aka $W^\ast$-algebras) are a special case of [[C-star algebra|$C^\ast$-algebras]]. Hence a more systematic name for $W^\ast$-categories would be *$W^\ast$-algebroids* or *von Neumann algebroids*.

## Definition

A __W\*-category__ is a [[C*-category]] $C$
such that for any objects $A,B\in C$, the [[hom-object]] $Hom(A,B)$
admits a [[predual]] as a [[Banach space]].
That is, there is a [[Banach space]]
$Hom(A,B)_*$ such that $(Hom(A,B)_*)^*$ is isomorphic to $Hom(A,B)$
in the category of [[Banach spaces]] and [[contractive maps]] (alias _short maps_).

## W\*-functors

A __W\*-functor__ is a [[functor]] $F\colon C\to D$
such that $F(f^*)=F(f)^*$ for any [[morphism]] $f$ in $C$
and the map of [[Banach spaces]] $Hom(A,B)\to Hom(F(A),F(B))$
admits a [[predual]] for any objects $A$ and $B$ in $C$.

## Bounded natural transformation

The good notion of natural transformations between W\*-categories
is given by __bounded natural transformations__:
a [[natural transformation]] $t\colon F\to G$ between W\*-functors
is bounded if the norm of $t_X\colon F(X)\to G(X)$
is bounded with respect to the object $X$ of $C$.

## The bicategory of W\*-categories

W\*-categories, W\*-functors, and bounded natural transformations
form a [[bicategory]].

This [[bicategory]] is a good setting to work with objects
like [[Hilbert spaces]], [[Hilbert W*-modules]] over [[von Neumann algebras]], [[W*-representations]] of [[von Neumann algebras]], etc.

In particular, in this [[bicategory]],
the [[category of Hilbert spaces]] has infinite [[direct sums]] (generalizing the definition of a [[biproduct]] to infinite families of objects),
unlike in the usual [[bicategory]]
of [[categories]], [[functors]], and [[natural transformations]],
where it only has [[finite limits]] and [[finite colimits]].

The same is true for [[Hilbert W*-modules]] over [[von Neumann algebras]], [[W*-representations]] of [[von Neumann algebras]].

## Related concepts

* [[von Neumann algebra]]

* [[C-star category|$C^\ast$-category]]

## References

* [[P. Ghez]], [[Ricardo Lima]], [[John E. Roberts]], _W\*-categories_, Pacific Journal of Mathematics 120:1 (1985), 79–109 [[doi:10.2140/pjm.1985.120.79](https://doi.org/10.2140/pjm.1985.120.79)]

[[!redirects W*-categories]]

[[!redirects W-star category]]
[[!redirects W-star+categories]]


W*-representation

## Definition

A __W\*-representation__, or simply a __representation__
of a [[von Neumann algebra]] $A$ on a [[Hilbert space]] $H$
is a morphism of [[von Neumann algebras]] $A\to B(H)$,
where $B(H)$ is the [[von Neumann algebra]]
of [[bounded operators]] on the [[Hilbert space]] $H$.

## Warning

If the map $A\to B(H)$ is injective,
the resulting notion coincides with that of a “concrete von Neumann algebra”,
as opposed to an (abstract) [[von Neumann algebra]].
An isomorphism of representations is sometimes referred to as
a “spatial isomorphism of concrete von Neumann algebras”.

Such terminology is confusing, but is present in some sources, especially older ones.

## Properties

The category of [[W*-representations]] of $A$
is equivalent to the category of [[Hilbert W*-modules]] over $A$.

See the article [[Hilbert W*-module]] for more information.

## Related concepts

* [[Hilbert W*-module]]

* [[von Neumann algebra]]

* [[duality between geometry and algebra]]

[[!redirects W*-representations]]
[[!redirects representation of a von Neumann algebra]]
[[!redirects representations of von Neumann algebras]]


W-boson

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

#Contents#
* table of contents
{:toc}

## Idea

The W- and Z-[[bosons]] are the quanta of the [[electroweak field]],[[vector boson]] in the [[standard model of particle physics]].

A hypothetical high-[[mass]] cousin of the Z-boson is the [[Z'-boson]].

## Related concepts

* [[charged current]]

* [[electroweak symmetry breaking]], [[Higgs effect]]

* [[leptonic decay]]

## References

### General

(...)

### Mass discrepancy
 {#ReferencesMassDiscrepancy}

* {#CDF22} CDF Collaboration, *High-precision measurement of the W boson mass with the CDF II detector*, Science **376** (2022) 170-176 [[doi:10.1126/science.abk1781](https://www.science.org/doi/10.1126/science.abk1781)]

The following article claims that this mass discrepancy is compatible with (in fact predicted by, see footnote 3) the observed [[flavour anomalies]]:

* {#ACMM22} Marcel Algueró, [[Andreas Crivellin]], Claudio Andrea Manzari, Joaquim Matias, *Unified Explanation of the Anomalies in Semi-Leptonic $B$ decays and the $W$ Mass*, Physical Review D (2022) [[arXiv:2201.08170](https://arxiv.org/abs/2201.08170)]

>

[[!redirects W-bosons]]

[[!redirects Z-boson]]
[[!redirects Z-bosons]]

[[!redirects B-boson]]
[[!redirects B-bosons]]


w-contractible ring

+-- {: .rightHandSide}
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###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

A [[commutative ring]] $R$ is **w-contractible** if every [[faithfully flat morphism|faithfully flat]] [[pro-étale morphism]] $Spec A \to Spec R$ has a [[section]].

=--

([Bhatt-Scholze 13, def. 2.4.1](#BhattScholze13))

## Properties

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

For every [[commutative ring]] $R$, there is a w-contractible $A$, def. \ref{wContractible},
equipped with a [[faithfully flat morphism|faithfully flat]] [[pro-étale morphism]] $Spec A \to Spec R$.

=--

([Bhatt-Scholze 13, lemma 2.4.9](#BhattScholze13))

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

For $R$ w-contractible, the [[profinite set]] $\pi_0(Spec R)$
is an [[extremally disconnected profinite set]].

=--

part of ([Bhatt-Scholze 13, theorem 1.8](#BhattScholze13))

## Related concepts

* [[pro-étale site]]

## References

* [[Bhargav Bhatt]], [[Peter Scholze]], _The pro-étale topology for schemes_ ([arXiv:1309.1198](http://arxiv.org/abs/1309.1198))
 {#BhattScholze13}

[[!redirects w-contractible rings]]


W-star category > history

see at *[[W-star category]]*


W-suspension

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

\tableofcontents

## Idea

A W-suspension is similar to a [[W-type]], except instead of being inductively generated by [[term]] constructors and [[function]] constructors, W-suspensions are inductively generated by term constructors and [[identification]] constructors.

## Definition

There are multiple possible definitions of a (directed, pseudo-) [[graph]] in dependent type theory:

* a graph consists of a type $A$ and a binary type family $R(x, y)$ indexed by $x:A$ and $y:A$. $A$ is the type of vertices of a graph and each $R(x, y)$ is the type of edges of a graph.

* a graph consists of types $A$ and $B$, a type family $R(z)$ indexed by $z:B$, and functions $s:B \to A$ and $t:B \to A$. $A$ is the type of vertices of a graph, $B$ is the type of endpoint configurations of edges in the graph, and $R(x)$ is the type of edges in the graph given an endpoint configuration $x:B$. Given elements $x:A$ and $y:A$, the type of edges between $x:A$ and $y:A$ in the graph is the type

$$\sum_{z:B} R(z) \times (s(z) =_A x) \times (t(z) =_A y)$$

* a graph consists of a type $A$ and a type $R$ and two functions $s:R \to A$ and $t:R \to A$. $A$ is the type of vertices in $A$, and the type of edges the type of edges between $x:A$ and $y:A$ in the graph is the type

$$\sum_{r:R} (s(r) =_A x) \times (t(r) =_A y)$$

These definitions of graph result in different possible definitions of a W-suspension, whose points are generated by the vertices of the graph and whose paths are generated by the edges of the graph.

### W-suspensions of binary type families

The first notion of graph leads to the following higher inductive type:

Given a graph $(A, R)$, one could construct the **W-suspension** $\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y)$ which is generated by the graph in the following sense: there are point constructors
$$\mathrm{points}:A \to \underset{x:A,y:A}{\mathrm{Wsus}} R(x, y)$$
and for each $x:A$ and $y:A$, path constructors
$$\mathrm{paths}(x, y):R(x, y) \to \left(\mathrm{points}(x) =_{\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y)} \mathrm{points}(y)\right)$$

The recursion principle of $\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y)$ says that given a type $C$ together with

* a function $c_\mathrm{points}:A \to C$
* a dependent function $c_\mathrm{paths}:\prod_{x:A} \prod_{y:A} R(x, y) \to (c_\mathrm{points}(x) =_C c_\mathrm{points}(y))$

there exists a function $c:\left(\underset{x:A, y:A}{\mathrm{Wsus}} R(x, y)\right) \to C$ such that

* for all $x:A$, $c(\mathrm{points}(x)) \equiv c_\mathrm{points}(x)$, and
* for all $x:A$, $y:A$, and $r:R(x, y)$, $\mathrm{ap}_c(\mathrm{paths}(x,y,r)) \equiv c_\mathrm{paths}(x, y, r)$

Similarly, the induction principle of $\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y)$ says that given a type family $C(z)$ indexed by $z:\underset{x:A, y:A}{\mathrm{Wsus}} R(x, y)$ together with

* a dependent function $c_\mathrm{points}:\prod_{x:A} C(\mathrm{points}(x))$
* a dependent function $c_\mathrm{paths}:\prod_{x:A} \prod_{y:A} \prod_{r:R(x, y)} c_\mathrm{points}(x) =_C^{\mathrm{paths}(x,y,r)} c_\mathrm{points}(y)$

there exists a dependent function $c:\prod_{z:\underset{x:A, y:A}{\mathrm{Wsus}} R(x, y)} C(z)$ such that

* for all $x:A$, $c(\mathrm{points}(x)) \equiv c_\mathrm{points}(x)$, and
* for all $x:A$, $y:A$, and $r:R(x, y)$, $\mathrm{apd}_c(\mathrm{paths}(x,y,r)) \equiv c_\mathrm{paths}(x, y, r)$

The large recursion principle of $\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y)$ says that given

* a type family $x:A \vdash C(x)$
* a dependent function $c_\mathrm{equiv}:\prod_{x:A} \prod_{y:A} R(x, y) \to (C(x) \simeq C(y))$

there exists a type family $z:\left(\underset{x:A, y:A}{\mathrm{Wsus}} R(x, y)\right) \vdash D(z)$ such that

* for all $x:A$, $D(c(x)) \equiv C(x)$, and
* for all $x:A$, $y:A$, and $r:R(x, y)$, $\mathrm{tr}_C(\mathrm{paths}(x,y,r)) \equiv c_\mathrm{equiv}(x, y, r):C(x) \simeq C(y)$

### Sojakova W-suspensions

The second notion of graph leads to the following higher inducive type defined in [Sojakova15](#Sojakova15):

Given types $A$ and $B$, a type family $R(x)$ indexed by $x:B$, and functions $s:B \to A$ and $t:B \to A$, one could construct the **W-suspension** $\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)$ which is generated by the graph in the following sense:
there are point constructors
$$\mathrm{points}:A \to \underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)$$
and for each endpoint configuration $x:B$, path constructors
$$\mathrm{paths}(x):R(x) \to \left(\mathrm{points}(s(x)) =_{\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)} \mathrm{points}(t(x))\right)$$

The recursion principle of $\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)$ says that given a type $C$ together with

* a function $c_\mathrm{points}:A \to C$
* a dependent function $c_\mathrm{paths}:\prod_{x:B} R(x) \to (c_\mathrm{points}(s(x)) =_C c_\mathrm{points}(t(x)))$

there exists a function $c:\left(\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)\right) \to C$ such that

* for all $x:A$, $c(\mathrm{points}(x)) \equiv c_\mathrm{points}(x)$, and
* for all $x:B$, and $r:R(x)$, $\mathrm{ap}_c(\mathrm{paths}(s,r)) \equiv c_\mathrm{paths}(s, r)$

Similarly, the induction principle of $\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)$ says that given a type family $C(z)$ indexed by $z:\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)$ together with

* a dependent function $c_\mathrm{points}:\prod_{x:A} C(\mathrm{points}(x))$
* a dependent function $c_\mathrm{paths}:\prod_{x:B} \prod_{r:R(x)} c_\mathrm{points}(s(x)) =_C^{\mathrm{paths}(x,r)} c_\mathrm{points}(t(x))$

there exists a dependent function $c:\prod_{z:\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)} C(z)$ such that

* for all $x:A$, $c(\mathrm{points}(x)) \equiv c_\mathrm{points}(x)$, and
* for all $x:B$, and $r:R(x)$, $\mathrm{apd}_c(\mathrm{paths}(x,r)) \equiv c_\mathrm{paths}(x, r)$

### W-suspensions of parallel functions

For the final notion of graph, the resulting definition of W-suspension is just the [[coequalizer type]] of $s:R \to A$ and $t:R \to A$.

$$\mathrm{Wsus}_{R,A}(s, t) \coloneqq \mathrm{coeq}_{R,A}(s, t)$$

### Relation between the definitions

W-suspensions in the first sense could be constructed as the coequalizer of the left and middle dependent pair projections $\pi_{\sum}^A$ and $\pi_{\sum}^A \circ \pi_{\sum}^{\sum_{y:A} R(-, y)}$ of the [[dependent pair type]] $\sum_{x:A} \sum_{y:A} R(x, y)$, both of which have [[domain]] $\sum_{x:A} \sum_{y:A} R(x, y)$ and [[codomain]] $A$:

$$\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y) \coloneqq \mathrm{coeq}_{\sum_{x:A} \sum_{y:A} R(x, y),A}(\pi_{\sum}^A, \pi_{\sum}^A \circ \pi_{\sum}^{\sum_{y:A} R(-, y)})$$

Alternatively, given two [[parallel morphisms|parallel]] functions $s:B \to A$ and $t:B \to A$, the coequalizer of $s$ and $t$ could be defined as the W-suspension (in the first sense) generated by the graph defined by the family of dependent sum types $\sum_{z:B} (s(z) =_A x) \times (t(z) =_A y)$

$$\underset{x:A,y:A}{\mathrm{Wsus}} \sum_{z:B} (s(z) =_A x) \times (t(z) =_A y)$$

## Related concepts

* [[coequalizer type]]

* [[W-type]]

## References

* {#Sojakova15} [[Kristina Sojakova]], *Higher Inductive Types as Homotopy-Initial Algebras*, ACM SIGPLAN Notices **50** 1 (2015) 31–42 [[arXiv:1402.0761](http://arxiv.org/abs/1402.0761), [doi:10.1145/2775051.2676983](https://doi.org/10.1145/2775051.2676983)]

[[!redirects W-suspension]]
[[!redirects W-suspensions]]


W-topical dagger 2-poset

[[!redirects W-topical dagger 2-posets]]

+-- {: .rightHandSide}
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###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

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

## Idea ##

A W-topical dagger 2-poset is a dagger 2-poset whose [[category of maps]] is a [[W-topos]].

## Definition ##

A **W-topical dagger 2-poset** $C$ is an [[elementarily topical dagger 2-poset]] with an object $\mathbb{N} \in Ob(C)$ and [[map in a dagger 2-poset|maps]] $0 \in Map_C(\mathcal{P}(0),\mathcal{N})$ and $s \in Map_C(\mathbb{N},\mathbb{N})$, such that for every object $A$ with maps $0_A \in Map_C(\mathcal{P}(0),A)$ and $s_A \in Map_C(A,A)$, there is a map $f \in Map_C(\mathbb{N},A)$ such that $f \circ 0 = 0_A$ and $f \circ s = s_A \circ f$.

## Examples ##

The dagger 2-poset [[Rel]] of [[sets]] and [[relations]] is a W-topical dagger 2-poset.

## See also ##

* [[dagger 2-poset]]
* [[elementarily topical dagger 2-poset]]


W-type

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

# W-types
* table of contents
{: toc}

## Idea
 {#Idea}

In [[type theory]], by a *$\mathcal{W}$-type* [[Martin-Löf (1982), pp. 171](#Martin-Löf82), [(1984)](#Martin-Löf84), [pp. 43](https://ncatlab.org/nlab/files/MartinLofIntuitionisticTypeTheory.pdf#page=49)] one means a [[type]] which is defined [[induction|inductively]] in a *[[well-founded relation|$\mathcal{W}$ell-founded]]* way based on a type $C$ of "*c*onstructors" and a type of $A$ of "*a*rities". As such, $\mathcal{W}$-types are special kinds of [[inductive types]] (see [below](#WTypesInTypeTheory)).

The same terminology "$\mathcal{W}$-type" is used [[Moerdijk & Palmgren (2000)](MoerdijkPalmgren00)] for [[objects]] in (suitable [[pretopos|pre]]-)[[toposes]] which provide [[categorical semantics]] for the type-theoretic notion (see [further below](#WTypesInCategories)).

Concretely, the [[terms]]/[[elements]] of a $\mathcal{W}$-type may be thought of as "rooted well-founded [[trees]]" with a certain branching type; different $\mathcal{W}$-types are distinguished by their branching signatures: In [[categorical semantics]] in [[Sets]], a branching signature is represented by a [[family of sets]] $\{A_c\}_{c \in C}$ such that

* each *node* of the tree is labeled with an element $c \colon C$ -- referring to a *constructor*;

* if a node is labeled by $c$ then it has exactly ${|A_c|}$ outgoing edges, each labeled by some $a \colon A_c$ -- the *arity* of the constructor $c$.

If one discards the requirement that the trees be well-founded, then the notion of $\mathcal{W}$-type becomes that of a [[coinductive type]] called an *[[M-type]]* (presumably since "M" is like a "W" upside down).

In practice, $\mathcal{W}$-types are used to:

1. provide a [[constructive mathematics|constructive]] counterpart of the [[classical logic|classical]] notion of a [[well-ordering]]

1. uniformly define a variety of well-behaved [[inductive types]].

More complex [[inductive types]], with multiple constructors that are assumed only to be strictly [[positive type|positive]], can be reduced to $\mathcal{W}$-types, at least in the presence of other structure such as [[sum types]] and [[function extensionality]]; see for instance [Abbott, Altenkirch & Ghani (2004)](#AbbottAltenkirchGhani04). This can even be extended to [[inductive families]].

{#InMost} In most [[set theory|set theoretic]] [[mathematical foundations]], $\mathcal{W}$-types can be proven to exist, but in [[predicative mathematics]] and in [[type theory]], where this is not the case, they are often introduced [[axiom|axiomatically]] (as usual for [[inductive types]] more generally).

## Definition
 {#Definition}

There are two slightly different formulations of W-types:

1. [$\mathcal{W}$-types in type theory](#WTypesInTypeTheory)

1. [$\mathcal{W}$-types in categories](#WTypesInCategories)

### $\mathcal{W}$-types in type theory
 {#WTypesInTypeTheory}

\begin{definition}\label{WTypeInferenceRules}
**([[inference rules]] for $\mathcal{W}$-types)**
\linebreak

(1) **[[type formation rule]]:**

$$
  \frac
    {
      C \,\colon\, Type
      \;;
      \;\;\;
      c \,\colon\, C
      \;\;\vdash\;\;
      A(c) \,\colon\,Type
      \mathclap{\phantom{\vert_{\vert}}}
    }
    {
      \mathclap{\phantom{\vert^{\vert}}}
      \underset{c \colon C}{\mathcal{W}}\, A(c)
        \,\colon\,
      Type
    }
$$

(2) **[[term introduction rule]]:**

$$
  \frac{
    \vdash\; root \,\colon\, C
    \;;
    \;\;
    subtr
      \,\colon\,
    A(root) \to \underset{c \colon C}{\mathcal{W}}\, A(c)
  }{
    \mathclap{\phantom{\vert^{\vert}}}
    tree\big(root ,\, subtr\big)
      \,\colon\,
    \underset{c \colon C}{\mathcal{W}}\, A(c)
  }
$$

(3) **[[term elimination rule]]:**

$$
  \frac{
    \begin{array}{l}
      t \,\colon\, \underset{c \colon C}{\mathcal{W}}\, A(c)
      \;\vdash\;
      D(t) \,\colon\, Type
      \;;
      \\
      root \,\colon\, C
      \,,
      \;
      subt
        \,\colon\,
      A(root) \to \underset{c \colon C}{\mathcal{W}}\, A(c)
      \,,
      \;
      subt_D
        \,\colon\,
      \underset{a \colon A(root)}{\prod}
        D\big(subt(a)\big)
      \\
      \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
      \;\;\vdash\;\;
      tree_D\big(root,\,subt ,\, subt_D\big)
        \,\colon\,
      D\big(tree(c,\, subt)\big)
      \mathclap{\phantom{\vert_{\vert}}}
    \end{array}
  }{
    \mathclap{\phantom{\vert^{\vert}}}
    t
      \,\colon\,
    \underset{c \colon C}{\mathcal{W}}\, A(c)
    \;\vdash\;
    wrec_{(D,tree_D)}(t) \,\colon\, D(t)
  }
$$

(4) **[[computation rule]]:**

$$
  \frac{
    \begin{array}{l}
      t
        \,\colon\,
      \underset{c \colon C}{\mathcal{W}}\, A(c)
      \;\vdash\;
      D(t) \,\colon\, Type
      \;;
      \\
      root \,\colon\, C
      \,,\;
      subt
        \,\colon\,
      A(root) \to \underset{c \colon C}{\mathcal{W}}\, A(c)
      \,,
      \;
      subt_D
        \,\colon\,
      \underset{a\colon A(c)}{\Pi} D\big(subt(a)\big)
      \\
      \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
      \;\;\vdash\;\;
      tree_D\big(root,\,subt,\,subt_D\big)
        \,\colon\,
      D\big(tree(root,\, subt)\big)
      \mathclap{\phantom{\vert_{\vert}}}
    \end{array}
  }{
    \begin{array}{l}
    \mathclap{\phantom{\vert^{\vert}}}
    root \,\colon\, C
    \,,
    \;\;
    subtr
      \,\colon\,
    A(root) \to \underset{c \colon C}{\mathcal{W}}\, A(c)
    \\
    \;\;\;\;\;\;\;\;\;\;\;
    \;\;\vdash\;\;
    wrec_{(D,tree_D)}\big( tree(root, \, subtr) \big)
    \;=\;
    tree_D
    \Big(
      root
      ,\,
      subt
      ,\,
      \lambda a . wrec_{(D,tree_D)}\big(subtr(a)\big)
    \Big)
    \end{array}
  }
$$

\end{definition}

(due to [Martin-Löf (1984)](#Martin-Löf84), [pp. 43](https://ncatlab.org/nlab/files/MartinLofIntuitionisticTypeTheory.pdf#page=49); streamlined review e.g. in [Awodey, Gambino & Sojakova (2012, §2)](#AwodeyGambinoSojakova12))

\begin{remark}
 If the type theory is *[[extensional type theory|extensional]]*, i.e. [[identity type|identities]] have unique proofs and (dependent) [[function types]] are [[function extensionality|extensional]] ($f=g$ if and only if $f(x)=g(x)$ for all $x$), then this is more or less equivalent to the [[1-category]]-[[category theory|theoretic]]  definition [below](#WTypesInCategories). The type theories that are the [[internal logic]] of familiar kinds of categories are all extensional in this sense.

The main distinction from the naive categorical theory below is that the map $f$ (eq:TheDisplayMap) must be assumed to be a [[display map]], i.e. to exhibit $A$ as a [[dependent type]] over $B$, in order that the dependent product $\Pi_f$ be defined.

In the case of dependent polynomial functors, it seems that $q$ must also be a display map, in order to define $\Sigma_q$.  However, using [[adjunction|adjointness]], one can still define the W-type even if $q$ is not a display map.  This more general version is what in type theory is called an *indexed W-type*; if $q$ is a display map then one sometimes refers to *non-uniform parameters* instead of *indices*.  (By contrast, *uniform parameters* are the kind discussed above where $g f = h$, so that the entire construction takes place in a single slice category  This is an instance of the [[red herring principle]], since non-uniform parameters are not really parameters at all, but a special kind of indices.)  For example, [[identity type]]s are indexed W-types but not parametrized ones (even non-uniformly); see [this blog post](http://homotopytypetheory.org/2011/04/18/whats-special-about-identity-types/).)

\end{remark}

\begin{remark} In [[intensional type theory]], a $\mathcal{W}$-type is only an initial algebra with respect to [[propositional equality]], not [[definitional equality]].  In particular, the constructors are injective only propositionally, not definitionally.  This applies already for the [[natural numbers type]] (Exp. \ref{NaturalNumbersAsWType}).

\end{remark}

### $\mathcal{W}$-types in categories
 {#WTypesInCategories}

#### Plain version

We discuss the [[categorical semantics]] of $\mathcal{W}$-types, due to [Moerdijk & Palmgren (2000)](#MoerdijkPalmgren00).

Here one describes a $\mathcal{W}$-type as an [[initial algebra|initial]] [[algebra for an endofunctor]] on an ambient category $\mathcal{C}$. The family $\{A_c\}_{c\in C}$ can be thought of as a [[morphism]]

\[
  \label{TheDisplayMap}
  \array{
    A
    \\
    \big\downarrow\mathrlap{^{f}}
    \\
    C
  }
\]

in some [[category]] $\mathcal{C}$ (such that the [[fiber]] over $c\in C$ is $A_c$), and the [[endofunctor]] in question is the [[composition|composite]]

$$
  \mathcal{C}
    \overset{A^*}{\longrightarrow}
  \mathcal{C}_{/A}
    \overset{\Pi_f}{\longrightarrow}
  \mathcal{C}_{/C}
    \overset{\Sigma_C}{\longrightarrow}
  \mathcal{C}
  \,,
$$

where

* $A^*$ denotes [[context extension]], hence the [[pullback]] functor (a.k.a. $A\times -$);

* $\Pi_f$ denotes the interpretation of the [[dependent product]], i.e. the right [[base change]] along $f$;

* $\Sigma_C$ the denotes the interpretation of the [[dependent sum]], i.e. the left [[base change]] given by the  [[forgetful functor]] from $\mathcal{C}_{/C}$ to $\mathcal{C}$.

Equivalently, it is the composite

$$
  \mathcal{C}
    \overset{C^*}{\longrightarrow}
  \mathcal{C}_{/C}
    \overset{\bigcirc_f}{\longrightarrow}
  \mathcal{C}_{/C}
    \overset{\Sigma_C}{\longrightarrow}
  \mathcal{C}
  \,,
$$

where $\bigcirc_f$ is the [[reader monad]] $\Pi_f \circ f^*$. In other words, the dependent product is not actually dependent.

Such a composite is called a _[[polynomial endofunctor]]_. Explicitly, it is the functor $X \mapsto \sum_{c : C} X^{A_c}$.

This definition makes sense in any [[locally cartesian closed category]], although the W-type (the initial algebra) may or may not exist in any given such category.  (A non-elementary construction of them is given by the [[transfinite construction of free algebras]].)

The above definition is most useful when the category $\mathcal{C}$ is not just
[[locally cartesian closed category|locally cartesian closed]] but is a [[Π-pretopos]], since often we want to use at least [[coproducts]] in constructing $A$ and $C$.  For example, a [[natural numbers object]] (Exp. \ref{NaturalNumbersAsWType}) is a $\mathcal{W}$-type specified by one of the coproduct inclusions $1\to 1+1$, and the [[list object]] $L X$ is a $\mathcal{W}$-type specified by $X\to X+1$.  More generally, endofunctors that look like [[polynomials]] in the traditional sense:
$$
F(Y) = A_n \times Y^{\times n}  + \dots + A_1 \times Y  + A_0
$$
can be constructed as [[polynomial endofunctors]] in the above sense for any [[lextensive category]], and hence in any $\Pi$-pretopos.  A $\Pi$-pretopos in which all W-types exist is called a **[[ΠW-pretopos]]**.

In a [[topos]] with a [[natural numbers object]] (NNO), W-types for any polynomial endofunctor can be constructed as certain sets of well-founded trees; thus every topos with a NNO is a [[ΠW-pretopos]].  This applies in particular in the topos [[Set]] (unless one is a [[predicative mathematics|predicativist]], in which case $Set$ is not a topos and W-types in it must be postulated explicitly).

#### Dependent $\mathcal{W}$-types
 {#DependentWTypesCategoricalSemantics}

The above has a natural generalization to [[dependent type|dependent]] or *indexed* W-types ([Gambino & Hyland (2004)](#GambinoHyland04)) with a type $C$ of *indices*:

given a [[diagram]] of the form

$$
  \array{
    A &\stackrel{f}{\longrightarrow}& B
    \\
    \downarrow^{\mathrlap{h}} && \downarrow^{\mathrlap{g}}
    \\
    C && C
  }
$$

there is the _[[dependent polynomial functor]]_

$$
  \mathcal{C}_{/C} \overset{h^\ast}{\longrightarrow}
  \mathcal{C}_{/A}
    \overset{\Pi_f}{\longrightarrow}
  \mathcal{C}_{/B}
    \overset{\Sigma_g}{\longrightarrow}
  \mathcal{C}_{/C}
  \,,
$$

This reduces to the above for $C = \ast$ the [[terminal object]].

Notice that we do not necessarily have $g f = h$, so this is not just a [[polynomial endofunctor]] of $\mathcal{C}/_{C}$ considered as a lccc in its own right.  If we *do* have $g f = h$, then $C$ is called a type of *parameters* instead of indices.

## Examples

\begin{example}\label{NaturalNumbersAsWType}
**([[natural numbers type]] as a [[W-type|$\mathcal{W}$-type]])**
\linebreak
The [[natural numbers type]] $(\mathbb{N},\, 0,\, succ)$ is equivalently the [[W-type|$\mathcal{W}$-type]] (Def. \ref{WTypeInferenceRules}) with

* $C \,\coloneqq\, \{0, succ\} \,\simeq\, \ast \sqcup \ast$;

* $A_0 \,\coloneqq\, \varnothing$ ([[empty type]]);

  $A_{succ} \,\coloneqq\, \ast$ ([[unit type]])

\end{example}
[[Martin-Löf (1984)](#Martin-Löf84), [pp. 45](/nlab/files/MartinLofIntuitionisticTypeTheory.pdf#page=51), [Dybjer (1997, p. 330, 333)](#Dybjer97)]

## Properties
 {#Properties}

\begin{proposition}
**([Danielsson (2012)](#Danielsson12))**
\linebreak
In [[homotopy type theory]], if $C$ has [[h-level]] $n\geq -1$, then any $\mathcal{W}$-type of the form $\underset{C}{\mathcal{W}} B$ has h-level $n$ (as it should be for [[(infinity,1)-colimits|$\infty$-colimits]]).
\end{proposition}

## Related concepts

* [[M-type]]

* [[predicative topos]]

[[!include notions of type]]

## References

### General

The original definition in [[type theory]] is due to

* {#Martin-Löf82} [[Per Martin-Löf]], pp. 171 of: *Constructive Mathematics and Computer Programming*, in: *Proceedings of the Sixth International Congress of Logic, Methodology and Philosophy of Science (1979)*, Studies in Logic and the Foundations of Mathematics **104** (1982) 153-175 $[$doi:10.1016/S0049-237X(09)70189-2, [ISBN:978-0-444-85423-0](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/104/suppl/C)$]$

* {#Martin-Löf84} [[Per Martin-Löf]] (notes by [[Giovanni Sambin]]), [pp. 43](https://ncatlab.org/nlab/files/MartinLofIntuitionisticTypeTheory.pdf#page=49) of: _Intuitionistic type theory_, Lecture notes Padua 1984, Bibliopolis, Napoli (1984) [[pdf](https://archive-pml.github.io/martin-lof/pdfs/Bibliopolis-Book-retypeset-1984.pdf), [[MartinLofIntuitionisticTypeTheory.pdf:file]]]

In [[homotopy type theory]]:

* [[Jasper Hugunin]], *Why Not W?*, Leibniz International Proceedings in Informatics (LIPIcs) **188** (2021) [[doi:10.4230/LIPIcs.TYPES.2020.8](https://doi.org/10.4230/LIPIcs.TYPES.2020.8), [pdf](https://jashug.github.io/papers/whynotw.pdf)]

On the [[h-level]] of W-types:

* {#Danielsson12} Nils Anders Danielsson, _Positive h-levels are closed under W_ (2012) [[web](https://homotopytypetheory.org/2012/09/21/positive-h-levels-are-closed-under-w/)]

* [[Jasper Hugunin]], _IWTypes_, <https://github.com/jashug/IWTypes>

which also computes the [[identity types]] of W-types (and more generally [[indexed W-types]]).

W-types in [[Coq]]: [wiki](https://coq.inria.fr/cocorico/WTypeInsteadOfInductiveTypes)

[[!include semantics of W-types -- references]]

[[!redirects W-types]]


W. F. Newns

W. F. Newns was a professor at the University of Liverpool.

## Selected writings

On [[vector fields as derivations]]:

* [[W. F. Newns]], [[A. G. Walker]],  _Tangent Planes To a Differentiable Manifold_.  Journal of the London Mathematical Society s1-31:4 (1956), 400–407 ([doi:10.1112/jlms/s1-31.4.400](https://doi.org/10.1112/jlms/s1-31.4.400))

category: people


W. Forrest Stinespring

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

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=6468)

## Selected writings

Introducing what later came to be known as the [[Kraus decomposition]] of [[quantum operations]]:

* {#Stinespring55} [[W. Forrest Stinespring]], *Positive functions on $C^\ast$-algebras*, Proc. Amer. Math. Soc. **6** 2 (1955) 211-216 [[doi:2032342](https://www.jstor.org/stable/2032342), [doi:10.2307/2032342](https://doi.org/10.2307/2032342)]

category: people


W. Hugh Woodin

William Hugh Woodin is a Professor of Mathematics at Harvard University
and a Professor Emeritus of Mathematics at the University of California, Berkeley.

[Website](https://www.math.harvard.edu/people/woodin-hugh/)

[Wikipedia page](https://en.wikipedia.org/wiki/W._Hugh_Woodin)

## Related concepts

* [[Woodin cardinal]]
* [[ultimate-L]]
* [[Omega conjecture]]
* [[axiom of determinacy]]

[[!redirects William Hugh Woodin]]
[[!redirects Hugh Woodin]]
[[!redirects Woodin]]


W. Stephen Wilson

W. Stephen Wilson is an American [[homotopy theory|homotopy theorist]] based at John Hopkins University. He specializes 'in the use and development of [[MU|complex (co)bordism]], [[BP|Brown-Peterson (co)homology]] and [[Morava K-theory]]'.

* [Mathematical Home Page](http://www.math.jhu.edu/~wsw/)

## Selected writings

Introducing the [[Johnson-Wilson spectrum]] (see at _[[Morava K-theory]]_):

* [[David Copeland Johnson]], [[W. Stephen Wilson]], _BP operations and Morava's extraordinary K-theories_, Math. Z. 144 (1): 55−75 (1975) ([pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/jw-morava.pdf))

On unstable [[cohomology operations]] (hence: in [[non-abelian cohomology]]):

* [[John Michael Boardman]], [[David Copeland Johnson]], [[W. Stephen Wilson]], _Unstable Operations in Generalized Cohomology_ ([pdf](https://hopf.math.purdue.edu/Boardman-Johnson-Wilson/bjw.pdf)), in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford 1995 ([doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

On [[Hopf rings]] in [[algebraic topology]]:

* [[Douglas Ravenel]], [[W. Stephen Wilson]], _The Hopf ring for complex cobordism_,  Bull. Amer. Math. Soc. 80 (6) 1185 - 1189, November 1974 (doi:10.1016/0022-4049(77)90070-6, [euclid](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-80/issue-6/The-Hopf-ring-for-complex-cobordism/bams/1183536024.full?tab=ArticleLink), [pdf](https://people.math.rochester.edu/faculty/doug/mypapers/hopfring.pdf))

  > (on [[MU]])

* [[W. Stephen Wilson]], _Hopf rings in algebraic topology_, Expositiones Mathematicae, 18:369–388, 2000 ([pdf](https://math.jhu.edu/~wsw/papers2/math/39-hopf-rings-expo-2000.pdf))

category: people

[[!redirects W.S. Wilson]]
[[!redirects W. S. Wilson]]


w4-orientation of EO(2)-theory

#Contents#
* table of contents
{:toc}

## Idea

The real form $EO(2)$ of [[ER-theory]] (i.e. [[real-oriented cohomology theory|real oriented]] [[Morava E-theory]]) is [[orientation in generalized cohomology|oriented]] by [[Spin structure]] combined with [[w4-structure]], hence by trivialization of the [[Stiefel-Whitney classes]] $w_1$, $w_2$ and $w_4$. ([Kriz-Sati 04, section 5.2, esp. p. 29](#KrizSati04)).

## Related concepts

[[!include genera and partition functions - table]]

## References

* {#KrizSati04} [[Igor Kriz]], [[Hisham Sati]], _M Theory, Type IIA Superstrings, and Elliptic Cohomology_, Adv.Theor.Math.Phys. 8 (2004) 345-395 ([arXiv:hep-th/0404013](http://arxiv.org/abs/hep-th/0404013))


w4-structure

## Idea

[[c-structure]] for $c$ the 4th [[Stiefel-Whitney class]] $w_4$

I.e. trivialization of $w_4$

## Related concepts

* [[p1-structure]]

* [[w4-orientation of EO(2)-theory]]


Wagner-Preston theorem

#Contents#
* table of contents
{:toc}

## Idea

The Wagner-Preston theorem is a generalization of [[Cayley's theorem]] to [[inverse semigroups]]. The same idea, when sets have additional local structure, is responsible for the appearance of [[pseudogroup]]s of symmetries in geometry.

## Statement

Every [[inverse semigroup]] has an injective homomorphism into the inverse semigroup of partial bijections of some set.

## Literature

* V. V. Wagner, _Generalised groups_. Proceedings of the USSR Academy of Sciences (in Russian) 84 (1952), 1119–1122.

* V. V. Wagner, _The theory of generalised heaps and generalised groups_, Matematicheskii Sbornik. Novaya Seriya (in Russian). 32 (74) (1953): 545–632.

* G. B. Preston, _Representation of inverse semi-groups_, J. London Math. Soc. 29 (1954) 411-419

* [planetmath](https://www.planetmath.org/WagnerPrestonRepresentationTheorem)
* wikipedia: [inverse semigroup](https://en.wikipedia.org/wiki/Inverse_semigroup)

category: algebra


Waldhausen (infinity,1)-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The analogue of [[Waldhausen category]] for [[(infinity,1)-categories]].

## Examples

* Any ordinary [[Waldhausen category]] is a [[Waldhausen (infinity,1)-category]].  See ([Barwick 12](#Barwick12), Example 2.12).

* Any [[stable (infinity,1)-category]] has a canonical Waldhausen structure.  See ([Barwick 12](#Barwick12), Example 2.11).

## Related entries

* [[algebraic K-theory]]

## References

* {#Barwick12} [[Clark Barwick]], _On the algebraic K-theory of higher categories_, [arXiv:1204.3607](http://arxiv.org/abs/1204.3607)

[[!redirects Waldhausen (infinity,1)-categories]]


Waldhausen category

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

#Contents#
* table of contents
{:toc}

## Idea

A _Waldhausen category_ $C'$ is a [[homotopical category]] equipped with a bit of extra structure that allows us to consider it as a presentation (via [[simplicial localization]]) of an [[(infinity,1)-category]] $C$ such that the extra structure allows us to conveniently compute the [[K-theory]] [[Grothendieck group]] $\mathbf{K}(C)$ of $C$.

Notably a Waldhausen category provides the notion of co[[fibration sequence]]s, which are crucial structures controlling $\mathbf{K}(C)$. Dual to the discussion at [[homotopy limit]] and [[homotopy pullback]], ordinary [[pushout]]s in Waldhausen categories of the form

$$
 \array{
  A &\hookrightarrow& B
  \\
  \downarrow && \downarrow
  \\
  0 &\to& B//A
 }
$$

with $A \hookrightarrow B$ a special morphism called a Waldhausen cofibration compute _homotopy pushout_s and hence coexact sequences in the corresponding [[stable (infinity,1)-category]].

Using this, the [[Waldhausen S-construction]] on $C'$ is an algorithm for computing the [[K-theory]] spectrum of $C$.

## Definition

Waldhausen in his work in [[K-theory]] introduced the notion of a category with cofibrations and weak equivalences, nowadays known as _Waldhausen category_. As the original name suggests, this is a category $C$ with zero object $0$, equipped with a choice of two classes of maps $\mathrm{cof}$ of the cofibrations and $w.e.$ of weak equivalences such that

* (C1) all isomorphisms are cofibrations

* (C2) there is a zero object $0$ and for any object $a$ the unique morphism $0\to a$ is a cofibration

* (C3) if $a\hookrightarrow b$ is a cofibration and $a\to c$ any morphism then the pushout $c\to b\cup_a c$ is a cofibration

* (W1) all isomorphisms are weak equivalences

* (W2) weak equivalences are closed under composition (make a subcategory)

* (W3) "glueing for weak equivalences": Given any
commutative diagram of the form
  $$\array{
  D &\leftarrow& A &\hookrightarrow &B\\
  \downarrow^\sim&& \downarrow^\sim &&\downarrow^\sim\\
  D' &\leftarrow &A' &\hookrightarrow &B'
  }$$
  in which the vertical arrows are weak equivalences and right horizontal maps cofibrations, the induced map
$B\cup_A D\hookrightarrow B'\cup_{A'} D'$
is a weak equivalence.

The axioms imply that for any cofibration $A\hookrightarrow B$ there is a cofibration sequence $A\hookrightarrow B\to B/A$ where $B/A$ is the choice of the cokernel $B\cup_A 0$.

Given a Waldhausen category $C$ whose weak equivalence classes from a set, one defines $K_0(C)$ as an abelian group whose elements are the weak equivalence classes modulo the relation $[A]+[B/A]=[B]$ for any cofibration sequence $A\hookrightarrow B\to B/A$.

Waldhausen then devises the so called S-construction $C\mapsto S_\bullet C$ from Waldhausen categories to simplicial categories with cofibrations and weak equivalences (hence one can iterate the construction producing multisimplicial categories).

The [[K-theory space]] of a Waldhausen construction is
given by formula $\Omega\mathrm{hocolim}_{\Delta^{\mathrm{op}}}([n]\mapsto N_\bullet(w.e.(S_n C)))$, where $\Omega$ is the loop space functor, $N$ is the simplicial [[nerve]], w.e. takes the (simplicial) subcategory of weak equivalence and $[n]\in\Delta$. This construction can be improved (using iterated [[Waldhausen S-construction]]) to the [[K-theory]] $\Omega$-[[spectrum]] of $C$; the K-theory space will be just the one-space of the K-theory spectrum.

Then the K-groups are the [[homotopy group]]s of the K-theory space.

## Remarks

* The axioms of a Waldhausen category $C$ are very similar to the axioms of a [[category of fibrant objects]] on the [[opposite category]] $C^{op}$ in which the initial object is also terminal. One difference is that the weak equivalences in a Waldhausen category are not required to satisfy [[2-out-of-3]]. For example, Waldhausen gives an example of a Waldhausen category where the weak equivalences are [[simple homotopy equivalences]]. Another difference is in axiom W3, whose analog in a [[category of fibrant objects]] is the axioms that every object has a [[path object]]. It still follows that one has fibration sequences in a [[category of fibrant objects]].

## Examples

### Waldhausen category of a small abelian category

For $C$ a [[small category|small]] [[abelian category]] the [[category of chain complexes|category of bounded chain complexes]] $Ch^b(C)$ becomes a Waldhausen category by taking

* a [[weak equivalence]] is a [[quasi-isomorphism]] of chain complexes;

* a cofibration $f : A_\bullet \to X_\bullet$ is a chain morphism that is a [[monomorphism]] in $C$ in each degree $f_n : A_n \to X_n$.

### Waldhausen category of a small exact category

For $C$ just a [[Quillen exact category]] with ambient [[abelian category]] $\hat C$ there is an analogous, slightly more sophisticated construction of a Waldhausen category structure on $Ch^b(C)$:

* weak equivalences are the morphisms that are [[quasi-isomorphism]]s when regarded as morphisms in $\hat C$;

* the cofibrations are the degreewise _admissible morphisms_, i.e. those morphisms $A \to X$ such that the pushout $A \to X \to A/X$ computed in the ambient [[abelian category]] $\hat C$ is in $C$.

## Related concepts

* [[category with weak equivalences]]

* [[category of fibrant objects]], [[category of cofibrant objects]]

* [[model category]]

## References

Waldhausen categories are discussed with an eye towards their application in the computation of [[Grothendieck group]]s in [chapter 2](http://www.math.rutgers.edu/~weibel/Kbook/Kbook.II.pdf) of

* [[Charles Weibel]], _The K-book: An introduction to algebraic K-theory_ ([web](http://www.math.rutgers.edu/~weibel/Kbook.html))

Section 1 of

* [[R. W. Thomason]], Thomas Trobaugh, _Higher algebraic K-theory of schemes and of derived categories_, _The Grothendieck Festschrift_, 1990, 247-435.([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/tt.pdf))

[[!redirects Waldhausen categories]]


Waldhausen K-theory of a dg-category

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## Idea

The **Waldhausen K-theory** of a [[dg-category]] is defined as the [[Waldhausen K-theory]] of a [[Waldhausen category]] associated to it.  It is an _additive invariant_ in the sense of [[noncommutative motives]].  There is a [[non-connective algebraic K-theory|nonconnective]] variant studied by [[Marco Schlichting]], which is a _localizing invariant_ (again in the sense of [[noncommutative motives]]).

On the other hand, one could define the K-theory of a [[pretriangulated dg-category]] as the [[algebraic K-theory]] of its [[dg-nerve]], which is a [[stable infinity-category]].  This should coincide with the Waldhausen K-theory (presumably).

## Definition

Let $A$ be a [[dg-category]].  Consider the [[triangulated category]] $D(A)$ of [[dg-presheaves]] on $A$, and let $perf(A) \subset D(A)$ denote its [[full subcategory]] of [[compact objects]].  There is the structure of a [[Waldhausen category]] on $perf(A)$ where the [[weak equivalences]] are objectwise [[quasi-isomorphisms]] and the [[cofibrations]] are degreewise [[split monomorphisms]].  The **Waldhausen K-theory** of $A$ is the [[Waldhausen K-theory]] of $perf(A)$ with this Waldhausen structure.

## References

* [[Bernhard Keller]], _On differential graded categories_, International Congress of Mathematicians. Vol. II, 151--190, Eur. Math. Soc., Zürich, 2006, [pdf](https://www.imj-prg.fr/~bernhard.keller/publ/dgcatX.pdf).

* [[Marco Schlichting]], _Negative K-theory of derived categories_, Math. Z. 253 (2006), no. 1, 97 - 134, [pdf](http://homepages.warwick.ac.uk/~masiap/research/frob5.pdf).

* [[Goncalo Tabuada]], _Higher K-theory via universal invariants_, Duke Mathematical Journal, 145 (2008), no. 1, 121-206, [arXiv](http://arxiv.org/abs/0706.2420).


Waldhausen S-construction

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## Idea ##

The Waldhausen $S_\bullet$-construction is a procedure that produces [[algebraic K-theory]] (as an [[infinite loop space]] or [[connective spectrum]]) from a category or [[(infinity,1)-category]] equipped with a [[Waldhausen structure]].

The construction can take as input any of the following:

* [[Waldhausen category]]
* [[Waldhausen (infinity,1)-category]]
* [[stable (infinity,1)-category]]
* [[stable derivator]]
* ...

## Definition ##

### For Waldhausen categories

Recall from the definition at [[K-theory]] that the K-theory spectrum $K(C)$ of the [[(∞,1)-category]] $C$ is the diagonal simplicial set on the [[bisimplicial set]] $Core(Func(\Delta^n,C))$ of sequences of morphisms in $C$ and equivalences between these (the [[core]] of the [[Segal space]] induced by $C$).

The Waldhausen S-construction mimics precisely this: for $C'$ a [[Waldhausen category]] for every integer $n$ define a simplicial set $S_n C'$ to be the [[nerve]] of the category whose

* objects are sequences $0 \hookrightarrow A_{0,1} \hookrightarrow \cdots \hookrightarrow A_{0,n}$ of Waldhausen cofibrations;

  * together with choices of quotients $A_{i j} = A_{0, j}/ A_{0,i}$, i.e.  cofibration sequences $A_{0,i} \to A_{0,j} \to A_{i j}$

* morphisms are collections of morphisms $\{A_{i,j} \to B_{i,j}\}$ that commute with all diagrams in sight.

Then one finds that the [[nerve and realization|realization]] of the bisimplicial set $S_\bullet C'$ with respect to one variable is itself naturally a topological [[Waldhausen category]]. Therefore the above construction can be repeated to yield a sequence of topological categories $S^{(n)}_\bullet C'$. The corresponding sequence of thick [[geometric realization|topological realization]]s is a [[spectrum]]

$$
  \mathbf{K}(C)_n = |S^{(n)}_\bullet C'|
$$

this is the **S-construction** of the  **Waldhausen [[K-theory]] spectrum** of $C'$.

(... roughly at least, need to polish this, see link below meanwhile...)

## Related

* [[algebraic K-theory]]
* [[Quillen Q-construction]]
* [[K-theory of a stable (infinity,1)-category]]
* [[K-theory of a Waldhausen (infinity,1)-category]]

* [[categorified Dold-Kan correspondence]]

## References ##

### For Waldhausen categories

* {#Waldhausen85} [[F. Waldhausen]], _Algebraic K-theory of spaces_, Alg. and Geo. Top., Springer Lect. Notes Math. 1126 (1985), 318-419, [pdf](http://www.maths.ed.ac.uk/~aar/surgery/rutgers/wald.pdf).

The Waldhausen S-construction is recalled for instance in section 1 of

* {#ThomasonTrobaugh90} [[R. W. Thomason]], Thomas Trobaugh, _Higher algebraic K-theory of schemes and of derived categories_, _The Grothendieck Festschrift_, 1990, 247-435.

or in section 1 of

* Paul D. Mitchener, _Symmetric Waldhausen K-theory spectra of topological categories_ ([pdf](http://bib.mathematics.dk/unzip.php?filename=DMF-2001-05-001-v1.pdf.gz))

### For Waldhausen (infinity,1)-categories and stable (infinity,1)-categories

* [[C. Barwick]], _On the algebraic K-theory of higher categories_, [arXiv:1204.3607](http://arxiv.org/abs/1204.3607).

* [[Andrew J. Blumberg]], [[David Gepner]], [[Goncalo Tabuada]], _A universal characterization of higher algebraic K-theory_, [arXiv:1001.2282](http://arxiv.org/abs/1001.2282v4).

### Other

A combinatorial construction of symmetries due to Nadler has a relation to the S-construction in a special case:

* [[David Nadler]], _Cyclic symmetries of $A_n$-quiver representations_, [arxiv/1306.0070](http://arxiv.org/abs/1306.0070)

> This short note contains a combinatorial construction of symmetries arising in symplectic geometry (partially wrapped or infinitesimal Fukaya categories), algebraic geometry (derived categories of singularities), and K-theory (Waldhausen's S-construction). Our specific motivation (in the spirit of expectations of Kontsevich, and to be taken up in general elsewhere) is a combinatorial construction of quantizations of Lagrangian skeleta (equivalent to microlocal sheaves in their many guises). We explain here the one dimensional case of ribbon graphs where the main result of this paper gives an immediate solution.

Interpretation in terms of a [[categorified Dold-Kan correspondence]] is discussed in

* {#Dyckerhoff17} [[Tobias Dyckerhoff]], _A categorified Dold-Kan correspondence_ ([arXiv:1710.08356](https://arxiv.org/abs/1710.08356))


Walecka model

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## Idea

The _Walecka model_ (also: _QHD-I model_) is an [[effective field theory]] related to [[chiral perturbation theory]], that describes the residual [[strong nuclear force]] between [[baryons]] and specifically between [[nucleons]], via exchange of [[sigma-mesons]] and [[omega-mesons]], with the [[nucleons]] appearing as explicit [[effective field theory|effective]] [[field (physics)|fields]] (as opposed to emergent [[Skyrmion]] fields), as is more generally the case in [[baryon chiral perturbation theory]]. Some authors use the term _[[quantum hadrodynamics]]_ specifically for the Walecka model of [[nuclear physics]].

The inclusion also of [[pions]] and of [[rho-mesons]] into the Walecka model came to be known as _[[quantum hadrodynamics]]_, specifically _QHD-II_. See there for more.

## Related concepts

[[!include effective field theories of nuclear physics -- contents]]

## References

[[!include quantum hadrodynamics -- references]]

[[!include baryon chiral perturbation theory -- references]]

[[!redirects baryon chiral perturbation theory]]


Walker coordinates

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## Idea

Adapted [[coordinates]] for [[smooth manifolds]] with [[special holonomy]] in which the [[Riemannian metric]] takes a nice form.

## Related concepts

* [[Riemann normal coordinates]]

## References

The original article is

* A. G. Walker, _Canonical form for a Riemannian space with a parallel field of null planes_, Quart. J. Math., Oxford Ser. (2) 1 (1950), 69–79. MR MR0035085(11,688d)

Review includes

* [[Helga Baum]], _Holonomy groups of Lorentzian manifolds --  A status report_, 2011 ([pdf](http://www.mathematik.hu-berlin.de/~baum/publikationen-fr/Publikationen-ps-dvi/Baum-Holonomy-report-final.pdf))

[[!redirects Walker coordinate chart]]
[[!redirects Walker coordinate charts]]


Walker-Wang model

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#Contents#
* table of contents
{:toc}

## Idea

The _Walker-Wang model_ is a [[model (in theoretical physics)]] which is a [[4d TQFT]] analog of the [[Levin-Wen model]] [[3d TQFT]]. It is the [[Hamiltonian]] formulation of the TQFT whose [[partition function]] formulation is the [[Crane-Yetter model]]. It is speculated to have some application to [[topological insulators]] in [[solid state physics]].

## References

* [[Kevin Walker]], [[Zhenghan Wang]], _(3+1)-TQFTs and Topological Insulators_, Frontiers of Physics volume 7, pages 150–159 (2012) ([arXiv:1104.2632](http://arxiv.org/abs/1104.2632), [doi:10.1007/s11467-011-0194-z](https://doi.org/10.1007/s11467-011-0194-z))

Discussion of [[defect QFT|line defects]] and the [[loop braid group]] statistics, with application to [[topological phases of matter]] using [[higher gauge theory]]/[[higher parallel transport]]:

* C. W. von Keyserlingk, F. J. Burnell, Steven H. Simon,  *Three-dimensional topological lattice models with surface anyons*, Phys. Rev. B 87, 045107 ([arXiv:1208.5128] (http://arxiv.org/abs/1208.5128), [doi:10.1103/PhysRevB.87.045107](https://doi.org/10.1103/PhysRevB.87.045107))

* Alex Bullivant, Marcos Calcada, Zoltán Kádár, [[João Faria Martins]], Paul Martin, *Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1) D with higher gauge symmetry*, Reviews in Mathematical Physics, Vol. 32, No. 04, 2050011 (2020) ([arXi:1702.00868](https://arxiv.org/abs/1702.00868))

* AtMa P.O. Chan, Peng Ye, Shinsei Ryu, *Braiding with Borromean Rings in (3+1)-Dimensional Spacetime*, Phys. Rev. Lett. 121, 061601 (2018) [[arXiv:1703.01926](https://arxiv.org/abs/1703.01926)]

* Tian Lan, [[Liang Kong]], [[Xiao-Gang Wen]], *A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons*, Phys. Rev. X 8, 021074 (2018) ([arXiv:1704.04221](https://arxiv.org/abs/1704.04221))

* [[Clement Delcamp]], *Excitation basis for (3+1)d topological phases*, Journal of High Energy Physics **2017** (2017) 128 ([arXiv:1709.04924](https://arxiv.org/abs/1709.04924) doi:10.1007/JHEP12(2017)128)

* Tian Lan, [[Xiao-Gang Wen]], *A classification of 3+1D bosonic topological orders (II): the case when some point-like excitations are fermions*,  	Phys. Rev. X 9, 021005 (2019) ([arXiv:1801.08530](https://arxiv.org/abs/1801.08530))

* Alex Bullivant, [[João Faria Martins]], Paul Martin, *Representations of the Loop Braid Group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory*, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number 7 ([arXiv:1807.09551](https://arxiv.org/abs/1807.09551))

* Qing-Rui Wang, Meng Cheng, Chenjie Wang, Zheng-Cheng Gu, *Topological Quantum Field Theory for Abelian Topological Phases and Loop Braiding Statistics in (3+1)-Dimensions*, Phys. Rev. B 99, 235137 (2019) ([arXiv:1810.13428](https://arxiv.org/abs/1810.13428))

* [[Clement Delcamp]], Apoorv Tiwari, *On 2-form gauge models of topological phases*, JHEP 05 (2019) 064 ([arXiv:1901.02249](https://arxiv.org/abs/1901.02249))

With application to [[quantum computation]]:

* Sam Roberts, Stephen D. Bartlett, *Symmetry-protected self-correcting quantum memories*, Phys. Rev. X 10, 031041 (2020) ([arXiv:1805.01474](https://arxiv.org/abs/1805.01474))

* Sam Roberts and Stephen D. Bartlett, *Symmetry-Protected Self-Correcting Quantum Memories*, Phys. Rev. X 10, 031041 2020 ([doi:10.1103/PhysRevX.10.031041](https://doi.org/10.1103/PhysRevX.10.031041))

* Sam Roberts, Dominic J. Williamson, *3-Fermion topological quantum computation* ([arXiv:2011.04693](https://arxiv.org/abs/2011.04693))

* Charles Stahl, Rahul Nandkishore, *Symmetry protected self correcting quantum memory in three space dimensions* ([arXiv:2103.08622](https://arxiv.org/abs/2103.08622))

[[!redirects Walker-Wang models]]


\tableofcontents

\section{Idea}

The _walking 2-isomorphism with trivial boundary_ is, roughly speaking, the minimal [[2-category]] which contains a [[2-isomorphism]] between identity [[1-morphism|1-arrows]]. It is in fact a [[2-groupoid]], and a model of the [[homotopy type]] of the 2-truncation of [[2-sphere]].

It is an example of a [[walking structure]], and can be compared for example with the [[walking 2-isomorphism]].

\section{Definition and elementary observations}

\begin{defn} Let $F$ be the free strict 2-category on the 2-truncated [[reflexive globular set]] with exactly one object $\bullet$, no non-identity 1-arrows, a 2-arrow $\iota: id(\bullet) \rightarrow id(\bullet)$, and a 2-arrow $\iota^{-1}: id(\bullet) \rightarrow id(\bullet)$. The _walking 2-isomorphism with trivial boundary_ is the strict 2-category obtained as the quotient of $F$ by the equivalence relation on 2-arrows generated by forcing the equations $\iota \circ \iota^{-1} = id$ and $\iota^{-1} \circ \iota = id$ to hold. \end{defn}

\begin{rmk} \label{RemarkCharacterisationOf2Arrows} There are exactly $\mathbb{Z}$ 2-arrows $\id\left( \bullet \right) \rightarrow id\left( \bullet \right)$, namely one for each possible string of compositions of $\iota$ and $\iota^{-1}$, taking into account (strict) associativity. Here $\mathbb{Z}$ is of course the [[integer|integers]]. This amounts to a computation of $\pi_{2}\left(S^{2}\right)$, the second homotopy group of the [[2-sphere]]. \end{rmk}

\begin{rmk} Let $\cdot$ denote horizontal composition. By the interchange law, we have that

$$
\begin{aligned}
\left( \iota^{-1} \cdot \iota \right) \circ \iota^{-1} &= \left( \iota^{-1} \cdot \iota \right) \circ \left( id \cdot \iota^{-1} \right) \\
  &= \left(\iota^{-1} \circ id \right) \cdot \left( \iota \circ \iota^{-1} \right) \\
  &= \iota^{-1} \circ id \\
  &= \iota^{-1}.
\end{aligned}
$$

The only possibility, given Remark \ref{RemarkCharacterisationOf2Arrows}, is then that $\iota^{-1} \cdot \iota = id$.

An entirely analogous argument demonstrates that $\iota \circ \iota^{-1} = id$. Thus horizontal composition in the walking 2-isomorphism with trivial boundary is trivial.
\end{rmk}

\section{Representing of 2-isomorphisms with trivial boundary}

\begin{prpn} Let $\mathcal{I}$ denote the walking 2-isomorphism with trivial boundary. Let $\mathcal{A}$ be a [[2-category]], and let $\phi$ be a 2-isomorphism $id(a) \rightarrow id(a)$ in $\mathcal{A}$, for some object $a$ of $\mathcal{A}$. Then there is a unique [[2-functor|functor]] $\mathcal{I} \rightarrow \mathcal{A}$ such that $\iota$ maps to $\phi$. \end{prpn}

\begin{proof} Immediate from the definitions. \end{proof}


walking adjoint equivalence

\tableofcontents

\section{Idea}

The _walking adjoint equivalence_ (as in "[[walking structure]]") or _free-standing adjoint equivalence_ is the [[2-category]] (in fact a [[(2,1)-category]]) which 'represents' [[adjoint equivalence|adjoint equivalences]] in a 2-category. It is a [[categorification]] of the [[free-standing adjoint isomorphism]].

\section{Definition}

\begin{defn} The _free-standing adjoint equivalence_ is the unique (up to [[isomorphism]]) [[2-category]] $\mathcal{AE}$ with exactly two [[objects]] $0$ and $1$, [[1-morphisms]] freely [[generators and relations|generated]] by an arrow $i: 0 \rightarrow 1$, and an arrow $j: 1 \rightarrow 0$, and [[2-morphisms]] generated by [[2-isomorphisms]] $\iota_{0}: j \circ i \cong id(0)$ and $\iota_{1}: i \circ j \cong id(1)$, subject to the equalities
$$id(i) = (\iota_1 i)\circ (i \iota_0^{-1}), id(j) =(\iota_0^{-1} j) \circ  (j \iota_1)$$
\end{defn}

\begin{rmk} The [[1-morphism|arrow]] $i: 0 \rightarrow 1$ is an [[adjoint equivalence]].\end{rmk}

\begin{rmk} The free-standing adjoint equivalence is a [[(2,1)-category]]. \end{rmk}

\section{Representing of adjoint equivalences}

$\mathcal{AE}$ is the model for all adjoint equivalences in all 2-categories. In other words, any adjoint equivalence in a 2-category $\mathcal{A}$ is just a [[2-functor]] from $\mathcal{AE}$:

\begin{prpn} Let $\mathcal{C}$ be a 2-category (weak or strict). Let $f$ be a 1-arrow of $\mathcal{C}$ which is part of an [[adjoint equivalence]]. Then there is a [[2-functor]] $F: \mathcal{AE} \rightarrow \mathcal{C}$ such that the arrow $i:0 \rightarrow 1$ of $\mathcal{I}$ maps under $F$ to $f$. \end{prpn}

\begin{proof} Immediate from the definitions. \end{proof}

\section{Related entries}

* [[Adj]]
* [[walking equivalence]]

[[!redirects walking adjoint equivalences]]
[[!redirects free-standing adjoint equivalence]]
[[!redirects free-standing adjoint equivalences]]