quadratic algebra

## Definition ##

A **(graded) quadratic algebra** is an $\mathbb{N}$-graded algebra $A$ which as a [[graded algebra]] admits a presentation

$$A \cong T(V)/I$$

where $T(V)$ is the [[tensor algebra]] of a finite-dimensional [[vector space]] $V$ of degree 1, and $I$ is a ideal generated by a space $R \subseteq V \otimes V$ of homogeneous elements of degree 2 in $T(V)$. Observe that $V$ and $R$ are uniquely determined by $A$: $V = A_1$, and if $I$ is the kernel of the canonical algebra map $T(A_1) \to A$, then $R = I_2$. It is often convenient to identify quadratic algebras with such pairs $(V, R)$.

A morphism of quadratic algebras is just a morphism as graded algebras. Alternatively, a morphism of quadratic algebras $f: A \to B$ determines and is determined by a linear map $f_1: A_1 \to B_1$ such that $f_2(R_A) \subseteq R_B$.

## Examples

1. The tensor algebra $T(V)$ ($V$ finite-dimensional) is of course quadratic.

1. The [[symmetric algebra]] $S(V)$ is quadratic.

1. The [[Grassmann algebra]] $\Lambda(V)$ is quadratic.

1. Extrapolating from the first three examples, a [[Koszul algebra]] is quadratic.

1. For many examples of [[quantum group|quantum groups]], for example quantum $GL_2$, the underlying algebra is quadratic. See the reference by Manin for further examples.

1. The coproduct $A + B$ (in the category of graded algebras) of two quadratic algebras is quadratic, with generators $(A + B)_1 = A_1 \oplus B_1$ and relations $R_A \oplus R_B \subseteq (A_1 \oplus B_1) \otimes (A_1 \oplus B_1)$. The tensor product $A \otimes B$ is also quadratic, with $(A \otimes B)_1 = A_1 \oplus B_1$ and taking the span of the coproduct relations $R_A \oplus R_B$ together with the commutation relations $a \otimes b - b \otimes a$ for $a \in A_1, b \in B_1$.

There is also a filtered notion of quadratic algebra, where the homogeneity requirement $R \subseteq V \otimes V$ is relaxed to allow inhomogeneous degree 2 terms, i.e., $R \subseteq k \oplus V \oplus V^{\otimes 2}$ in the tensor algebra, and this allows one to include more examples like universal enveloping algebras of finite-dimensional Lie algebras, and also Clifford algebras. But below we discuss only the graded case.

## Quadratic dual and Manin's monoidal products

If $(V, i: R \hookrightarrow V \otimes V)$ defines a quadratic algebra, its **quadratic dual** is defined by the pair $(V^*, R^\perp)$, where $R^\perp$ is the kernel of the composite

$$V^* \otimes V^* \cong (V \otimes V)^* \stackrel{i^*}{\to} R^*$$

In the literature where it commonly appears, the dual of a quadratic algebra $A$ is usually denoted $A^!$. There is a canonical isomorphism $A \cong A^{!!}$. It was first observed by Yuri Manin that this is the duality operator for a $*$-[[star-autonomous category|autonomous]] structure on the category of quadratic algebras:

* The monoidal product of $(V, R_A)$ and $(W, R_B)$ is defined by $(V \otimes W, \sigma(R_A \otimes R_B))$ where $\sigma$ refers to the canonical interchange isomorphism
$$V \otimes V \otimes W \otimes W \cong V \otimes W \otimes V \otimes W$$
(writing as if the tensor product were strict, as justified by Mac Lane's coherence theorem). Manin's notation for this is $A \bullet B$. The monoidal unit is the free algebra on one generator in degree 1.

* The dual monoidal product, denoted $A \circ B$, may be defined by the formula
$$A \circ B = (A^! \bullet B^!)^!$$
Up to coherent isomorphism, this may be more explicitly defined by the pair

$$(V \otimes W, \sigma(R_A \otimes 1_{W \otimes W}) + \sigma(1_{V \otimes V} \otimes R_B)).$$

(To see this last more clearly, observe that for finite-dimensional $V$, the mapping

$$(-)^\perp: Sub(V) \to Sub(V^*)$$

is a [[Galois connection|Galois correspondence]], and hence a bijection that takes meets to joins and joins to meets. Now the meet of $\sigma(R_{A}^\perp \otimes 1_{W^* \otimes W^*})$ and $\sigma(1_{V^* \otimes V^*} \otimes R_{B}^\perp)$ is $\sigma(R_{A}^\perp \otimes R_{B}^\perp)$. Applying $(-)^\perp$ to this, one obtains the join of $\sigma(R_A \otimes 1_{W \otimes W})$ and $\sigma(1_{V \otimes V} \otimes R_B)$ which is $\sigma(R_A \otimes 1_{W \otimes W}) + \sigma(1_{V \otimes V} \otimes R_B)$.)

+-- {: .un_thm}
######Theorem
There is a natural isomorphism $QAlg(A \bullet B, C) \cong QAlg(A, B^! \circ C)$.
=--

+-- {: .proof}
######Proof
A preliminary comment is that the aforementioned Galois correspondence is induced by the equivalence

$$\frac{X \subseteq Y^\perp}{\langle X, Y \rangle_V = 0}$$

where $X \subseteq V^*$, $Y \subseteq V$ are subspaces and $\langle -, - \rangle_V: V^* \otimes V \to k$ is the usual pairing to the ground field $k$.

Let $(U, R_A)$, $(V, R_B)$, $(W, R_C)$ define the quadratic algebras, and suppose that $f: U \otimes V \to W$ and $g: U \to V^* \otimes W$ correspond to one another under the adjunction

$$Vect_k(U \otimes V, W) \cong Vect_k(U, V^* \otimes W).$$

It then suffices to observe that the following statements are equivalent (below, each instance of $\sigma$ denotes an appropriate middle-four interchange):

* $f$ induces a (unique) graded algebra map $A \bullet B \to C$;

* $(f \otimes f)\sigma(R_A \otimes R_B) \subseteq R_C$;

* $\langle (f \otimes f)\sigma(R_A \otimes R_B), R_{C}^\perp \rangle_{W^\ast \otimes W^\ast} = 0$;

* $\langle \sigma(g \otimes g)(R_A), R_B \otimes R_{C}^\perp \rangle_{V \otimes V \otimes W^\ast \otimes W^\ast} = 0$;

* $\sigma (g \otimes g)(R_A) \subseteq (R_B \otimes R_{C}^\perp)^\perp$;

* $g$ induces a (unique) graded algebra map $A \to B^! \circ C$.
=--

This result may be effectively summarized by saying that the category of quadratic algebras carries a [[star-autonomous category]] structure, i.e., a [[closed monoidal category|closed symmetric monoidal category]] structure equipped with a dualizing object $D$, i.e., an object for which the double dual embedding $\delta_A: A \to [[A, D], D]$ is a natural isomorphism. The monoidal unit is $k[x]^! = k[\varepsilon]/(\varepsilon^2)$, the [[Grassmann algebra|algebra of Grassmann numbers]], and the dualizing object $D$ is the polynomial algebra $k[x]$. We then have $A^! \cong [A, D]$ for any quadratic algebra $D$.

## References ##

Y. Manin, _Some remarks on Koszul algebras and quantum groups_, Annales de l'institut Fourier, 37 no. 4 (1987), p. 191-205 [(pdf)](http://archive.numdam.org/ARCHIVE/AIF/AIF_1987__37_4/AIF_1987__37_4_191_0/AIF_1987__37_4_191_0.pdf)


quadratic differential

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## Idea

A *quadratic differential* (see [Strebel 1984](#Strebel84)) on a [[Riemann surface]] $\Sigma$ is a [[section]] of the second [[symmetric power]] of the [[canonical line bundle]].

(...)

## Related concepts

* [[Hilbert uniformization]]

* [[Witten conjecture]]

* [[moduli space of Riemann surfaces]]

## References

Textbook account:

* {#Strebel84} [[Kurt Strebel]], _Quadratic differentials_, Springer Verlag, Berlin, 1984 ([doi:10.1007/978-3-662-02414-0](https://link.springer.com/book/10.1007/978-3-662-02414-0))

See also:

* Wikipedia, *[Quadratic differential](https://en.wikipedia.org/wiki/Quadratic_differential)*

* John Hubbard, Howard Masur, *Quadratic differentials and foliations*, Acta Math. 142: 221-274 (1979) ([doi:10.1007/BF02395062](https://projecteuclid.org/journals/acta-mathematica/volume-142/issue-none/Quadratic-differentials-and-foliations/10.1007/BF02395062.full))

Specifically on Strebel differentials:

* Philip Tynan, *Explicit Examples of Strebel Differentials* ([arXiv:0910.4752](https://arxiv.org/abs/0910.4752))

[[!redirects quadratic differentials]]

[[!redirects Strebel differential]]
[[!redirects Strebel differentials]]


quadratic form

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## Definition

For $V$ a [[vector space]] or more generally a $k$-[[module]], then a _quadratic form_ on $V$ is a [[function]]

$$
  q\colon V \to k
$$

which is [[homogeneous function|homogeneous]] of degree 2 in that for all $v \in V$, $t \in k$

$$
  q(t v) = t^2 q(v)
$$

and such that the _[[polarization identity|polarization]]_ of $q$

$$
  (v,w) \mapsto q(v+w) - q(v) - q(w)
$$

is a [[bilinear form]].

Written entirely in terms of $q$, the axioms for a quadratic form are:

* $q(t v) = t^2 q(v)$,
* $q(t v + w) + t q(v) + t q(w) = t q(v + w) + t^2 q(v) + q(w)$,
* $q(u + v + w) + q(u) + q(v) + q(w) = q(u + v) + q(u + w) + q(v + w)$.

(Besides the homogeneity, these come from two requirements of a bilinear form to preserve scalar multiplication and addition, respectively.)  So we may alternatively define a __quadratic form__ to be a map $q\colon V \to k$ satisfying these three axioms.

A more general __quadratic map__ (or _homogeneous_ quadratic map to be specific) between vector spaces $V$ and $W$ is a map $q\colon V \to W$ that satisfies the above three conditions.  (Then an _affine_ quadratic map is the sum of a homogeneous quadratic map, a [[linear map]], and a constant, just as an [[affine linear map]] is the sum of a linear map and a constant.)

From the converse point of view, $q$ is a [[quadratic refinement]] of the bilinear form $(-,-)$.  (This always exists uniquely if $2 \in k$ is invertible, but in general the question involves the [[characteristic elements of a bilinear form|characteristic elements]] of $(-u,-)$.  See there for more.)

Quadratic forms with values in the [[real numbers]] $k = \mathbb{R}$ are called _positive definite_ or _negative definite_ if $q(v) \gt 0$ or $q(v) \lt 0$, respectively, for all $v \neq 0$.  See [[definiteness]] for more options.

## Related concepts

* [[linear function]]

* [[polarization identity]]

* [[bilinear form]]

* [[sesquilinear form]]

* [[symplectic form]], [[Kähler form]], [[Hermitian form]]

* [[completing the square]]

* [[quadratic refinement]]

* [[signature of a metric]]

* [[polynomial observable]]

* [[quadratic function]]

## References
 {#References}

The theory of quadratic forms emerged as a part of (elementary) [[number theory]], dealing with quadratic [[diophantine equations]], initially over the rational integers

The terminology "form" possibly originated with:

* [[Leonhard Euler]],  _Novae demonstrations circa divisors numerorum formae $x x + n y y$..._, Acad. Petrop. recitata, Nov 20, 1775, published poshumously

(which is cited as such in [Gauss 1798, paragraph 151](#Gauss1798)).

First classification results for forms over the [[integers]] were due to:

* {#Gauss1798} [[Carl Gauss]], section V of _[[Disquisitiones Arithmeticae]]_, 1798

(which speaks of _formas secundi gradus_)

* [[Hermann Minkowski]], _Grundlagen für eine Theorie der quadratischen Formen mit ganzzahligen Koeffizienten, Mémoires présentés par divers savants a l'Acad´emie des Sciences de l'institut national de France_, Tome XXIX, No. 2. 1884.

* [[Hermann Minkowski]], _Untersuchungen über quadratische Formen. Bestimmung der Anzahl verschiedener Formen, die ein gegebenes Genus enthält_, Königsberg 1885; Acta Mathematica 7 (1885), 201–258

Discussion in the generality of [[noncommutative ring|noncommutative]] [[ground rings]]:

* [[John Milnor]], [[Dale Husemöller]], *Quadratic forms*, Appendix 1 of: *Symmetric Bilinear Forms*,  Ergebnisse der Mathematik und ihrer Grenzgebiete **73**, Springer (1973) [[doi:10.1007/978-3-642-88330-9](https://doi.org/10.1007/978-3-642-88330-9)]

See also

* Rudolf Scharlau, _Martin Kneser's work on quadratic forms and algebraic groups_, 2007 ([pdf](http://inst-mat.utalca.cl/qfc2007/Talks/scharlau.pdf))

Textbook accounts:

* [[Richard Elman]], [[Nikita Karpenko]], [[Alexander Merkurjev]], *Algebraic and Geometric Theory of Quadratic Forms*, Colloquium Publication **56**, AMS (2008) [[ams:coll-56](https://bookstore.ams.org/coll-56), [pdf](https://www.math.ucla.edu/~rse/old_book/Kniga.pdf)]

* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: §6 in: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]

Course notes:

* _On the relation between quadratic and bilinear forms_ ([pdf](http://www.maths.manchester.ac.uk/~khudian/Etudes/Algebra/quadrform2.pdf))

* _Bilinear and quadratic forms_ ([pdf](http://buzzard.ups.edu/courses/2007spring/projects/ott-paper-revised.pdf))

* section 10 in _Analytic theory of modular forms_ ([pdf](http://www.math.harvard.edu/~jbland/ma259x_notes.pdf))

Quadratic refinements of [[intersection pairing]] in [[cohomology]] is a powerful tool in [[algebraic topology]] and [[differential topology]]. See

* [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_

See also

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

* Wikipedia, _[Definite quadratic form](https://en.wikipedia.org/wiki/Definite_quadratic_form)_

[[!redirects quadratic form]]
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[[!redirects quadratic map]]
[[!redirects quadratic maps]]

[[!redirects signature of a quadratic form]]
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quadratic formula

# The quadratic formula
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## Idea

Consider the [[quadratic function]]

\[ \label{eqn} f(x) \coloneqq a{x}^2 + b{x} + c \]

which we wish to find the elements of the [[zero set]]. In certain contexts, the elements of the zero set are given by one or more versions of the quadratic formula.

## Discussion

The coefficients $a, b, c$ of the quadratic function are commonly taken from an [[algebraically closed field]] $K$ of [[characteristic]] $0$, such as the field $\mathbb{C}$ of [[complex numbers]], although any quadratically closed field whose characteristic is not $2$ would work just as well.  Alternatively, the coefficients can be taken from a [[real closed field]] $K$, such as the field $\mathbb{R}$ of [[real numbers]]; then the solutions belong to $K[\mathrm{i}]$.  (Of course, $\mathbb{R}[\mathrm{i}]$ is simply $\mathbb{C}$ again.)  More generally, starting from any [[integral domain]] $K$ whose characteristic is not $2$, the solutions belong to some [[splitting field]] of $K$.  (Of course, there are solutions in *some* splitting field, regardless of the characteristic, but they are not given by the quadratic formula if the characteristic is $2$.)

### Forms of the formula

Explicitly, the zero set of (eq:eqn) may be given by the __usual quadratic formula__
\[ \label{usual} x_\pm = \frac{-b \pm \sqrt{b^2 - 4a{c}}}{2a} ,\]
which works as long as $a \ne 0$.  There is also an __alternate quadratic formula__
\[ \label{alt} x_\pm = \frac{2c}{-b \mp \sqrt{b^2 - 4a{c}}} ,\]
which may be obtained from (eq:usual) by rationalizing the numerator; this works as long as $c \ne 0$.  (Note that $\pm$ and $\mp$ appear here simply to indicate the two [[square roots]] of the determinant $b^2 - 4a{c}$ and how they correspond to the two solutions $x_\pm$; we do not need to have a [[function]] $\sqrt{}$ which always chooses a 'principal' square root.)

These two formulas are reconciled in the [[projective line]] of $K$.  As long as $(a, b, c) \ne (0, 0, 0)$, there are two solutions (which might happen to be equal) in the projective line.  If $a = 0$, then one of these solutions is $\infty$, and (eq:usual) correctly gives us that solution (as long as $b \ne 0$) for one choice of square root, although it gives $0/0$ for the other choice.  Similarly, (eq:alt) correctly gives us $x = 0$ when $c = 0$ and $b \ne 0$, but it does not give us the other root when $c = 0$.  Note that if $a, c = 0$ but $b \ne 0$, then (eq:usual) gives us one root ($\infty$) while (eq:alt) gives us the other ($0$).

So in general, we should be given $a \ne 0$, $b \ne 0$, or $c \ne 0$ for a nondegenerate equation (eq:eqn).  If $a \ne 0$, then we use (eq:usual); if $c \ne 0$, then we use (eq:alt).  Finally, if $b \ne 0$, then we use both; each root will be successfully given by at least one formula for some choice of square root of $b^2 - 4a{c}$.

When the coefficients come from an [[ordered field]] $K$ (which we assume real closed), then we can write down a formula specially for the case when $b \ne 0$.  This is the __numerical analysts\' quadratic formula__
\[ \label{numanal} \begin {gathered}
   \displaystyle x_{\hat{b}} = \frac{2c}{-b - \hat{b}\sqrt{b^2 - 4a{c}}} ;\\
   \displaystyle x_{-\hat{b}} = \frac{-b - \hat{b}\sqrt{b^2 - 4a{c}}}{2a} .\\
\end {gathered} \]
In this formula, $\hat{b}$ is the sign of $b$, that is $b/{|b|}$; also, we must choose a nonnegative principal square root, so that $\sqrt{b^2 - 4a{c}} \lt 0$ in $K$ is avoided (and thus the common denominator of $x_{\hat{b}}$ and numerator of $x_{-\hat{b}}$ is nonzero even if not imaginary).  Despite the name, this formula is not sufficient for all purposes in [[numerical analysis]]; one still needs all three formulas and chooses between them based on whether $a \ne 0$, $b \ne 0$, or $c \ne 0$ is best established.

### Constructive issues

Everything above is valid in weak forms of [[constructive mathematics]] if one takes $\mathbb{C} \coloneqq \mathbb{C}_C$ to be the [[modulated Cauchy real numbers|modulated Cauchy]] complex numbers, see [Ruitenberg (1991)](#Ruitenberg91).

However, if one takes $\mathbb{C} \coloneqq \mathbb{C}_D$ to be the [[Dedekind real numbers|Dedekind]] complex numbers, there is an interesting issue about whether $b^2 - 4a{c} \ne 0$. Everything above is valid in weak forms of [[constructive mathematics]] except for the statement that $\mathbb{C}_D$ is [[quadratic closure|quadratically closed]]. That claim follows from [[weak countable choice]] ($WCC$), which in turn will follow from either [[excluded middle]] or [[countable choice]]. Nevertheless, the statement
$$ \forall\, a, b, c\colon \mathbb{C}_D,\; \exists\, r\colon \mathbb{C}_D,\; r^2 = b^2 - 4a{c} $$
is false in (for example) the [[internal language]] of the [[sheaf topos]] over the [[real line]]. (Essentially, this is because there is no [[continuous map]] $\sqrt{}$ on any neighbourhood of $0$ in $\mathbb{C}_D$.)  If we are given that $a, b, c$ are real, or if we are given that $b^2 \ne 4a{c}$, then there is no problem.  But in general, we cannot define this square root, which appears in every version of the quadratic formula.

However, there is a more subtle sense in which $\mathbb{C}_D$ is algebraically closed even without $WCC$; essentially, this allows us to approximate the [[subset]] of $\mathbb{C}_D$ whose elements are the two solutions of (eq:eqn) (using two-element subsets of the field of, say, [[Gaussian numbers]]) even if we can\'t approximate any one solution (using individual, say, Gaussian numbers); see [Richman (1998)](#Richman) for details.  The quadratic formula can then be interpreted as indicating this approximated subset.

### Simplified formulas and characteristic $2$

Sometimes one considers the quadratic function
\[f(x) \coloneqq a{x}^2 + 2p{x} + c \]
instead of (eq:eqn); then (eq:usual) simplifies to
\[ \label{simpl} x_\pm = \frac{-p \pm \sqrt{p^2 - a{c}}}a \]
(and similarly for (eq:alt) and (eq:numanal)).

This is valid even in characteristic $2$, but unfortunately then it is fairly useless, since $b = 2p = 0$.  More precisely, if $b = 0$, then (eq:simpl) with $p = 0$ gives the roots $\pm\sqrt{-c/a}$ in any characteristic, but in that case the equation was easy to solve without any formula.  On the other hand, if $b \ne 0$ and $\char K = 2$, then no version of the quadratic formula is applicable, yet this gives no information as to whether the polynomial is [[solvable polynomial|solvable]] and what its roots are if it is.  For example, the quadratic function $f(x) \coloneqq x^2 + x$ has roots $0$ and $1$ in $\F_2$ (or $0$ and $-1$ in any field, as may be found by factoring), while $f(x) \coloneqq x^2 + x + 1$ is not solvable over $\F_2$, yet both have $b^2 - 4a{c} = 1$ and give $0/0$ in both (eq:usual) and (eq:alt) (while (eq:numanal) and (eq:simpl) are directly inapplicable).

## See also

* [[quadratic function]]

* [[zero locus]]

* [[completing the square]]

* [[continuous quadratic formula]]

## References

* {#Ruitenberg91} Wim Ruitenberg, Constructing Roots of Polynomials over the Complex Numbers, Computational Aspects of Lie Group Representations and Related Topics, CWI Tract, Vol. 84, Centre for Mathematics and Computer Science, Amsterdam, 1991, pp. 107–128. ([pdf](https://www.mscsnet.mu.edu/~wim/publica/roots_new.pdf))

* [[Fred Richman]], *The fundamental theorem of algebra: a constructive development without choice*. Pacific Journal of Mathematics **196** 1 (2000) 213–230 [[doi:10.2140/pjm.2000.196.213](http://dx.doi.org/10.2140/pjm.2000.196.213), [pdf](https://msp.org/pjm/2000/196-1/pjm-v196-n1-p10-p.pdf)]
{#Richman}

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quadratic function

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## Definition

### In commutative rings

For a [[commutative ring]] $R$, a **quadratic function** is a [[function]]  $f \colon R \to R$ with [[elements]] $a \in R$, $b \in R$, $c \in R$ such that for all $x \in R$,

$$f(x) = a \cdot x^2 + b \cdot x + c$$

where $x^2$ is the canonical [[square function]] of the [[multiplication|multiplicative]] [[monoid]].

### Between $R$-modules

Given a [[commutative ring]] $R$ and $R$-[[module | modules]] $M$ and $N$, an __$R$-quadratic function__ on $M$ with values in $N$ is a map $q: M \to N$ such that the following properties hold:

* (cube relation) For any $x,y,z \in M$,
$$q(x+y+z) - q(x+y) - q(x+z) - q(y+z) + q(x) + q(y) + q(z) = 0$$
* ([[homogeneous function | homogeneous]] of degree 2) For any $x \in M$ and any $r \in R$,
$$q(r x) = r^2 q(x)$$

### Between abelian groups

Given [[abelian groups]] $G$ and $H$, since we can regard $G$ and $H$ as $\mathbb{Z}$-[[module | modules]], the above definition specializes to this particular case. A __quadratic function__ on $G$ with values in $H$ is a map $q: G \to H$ such that the following properties hold:

* (cube relation) For any $x,y,z \in G$,
$$q(x+y+z) - q(x+y) - q(x+z) - q(y+z) + q(x) + q(y) + q(z) = 0$$
* ([[homogeneous function|homogeneous]] of degree 2) For any $x \in G$ and any $r \in \mathbb{Z}$,
$$q(r x) = \sum_{i=1}^{r^2} q(x)$$

## See also

* [[completing the square]]

* [[quadratic formula]]

* [[quadratic form]], [[quadratic refinement]]

* [[linear function]], [[polynomial function]]

* [[real quadratic function]]

* [[quadratic abelian group]]

* [[cubic function]]

## References

See also:

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

Discussion of quadratic functions in the form of [[quadratic refinements]] of [[intersection pairings]] (in [[cohomology]]), as a phenomenon in [[algebraic topology]], [[differential topology]] as well as in [[string theory]]:

* [[Mike Hopkins]], [[Isadore Singer]], *[[Quadratic Functions in Geometry, Topology, and M-Theory]]*, J. Diff. Geom. **70** (2005) 329-452 $[$[arXiv:math/0211216](https://arxiv.org/abs/math/0211216), [euclid.jdg/1143642908](https://projecteuclid.org/euclid.jdg/1143642908)$]$

[[!redirects quadratic functions]]


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This entry is about the article

* [[Michael Hopkins]], [[Isadore Singer]]:

  \linebreak

  **Quadratic Functions in Geometry, Topology, and M-Theory**

  \linebreak

  J. Differential Geom. **70** 3 (2005)  329-452.

  [arXiv:math.AT/0211216](http://arxiv.org/abs/math.AT/0211216)

  [doi:10.4310/jdg/1143642908](https://doi.org/10.4310/jdg/1143642908)

  [euclid:1143642908](https://projecteuclid.org/euclid.jdg/1143642908)

which discusses ([[ordinary differential cohomology|ordinary]]) [[differential cohomology]] refinements of [[generalized (Eilenberg-Steenrod) cohomology]] and uses it to study [[quadratic refinements]] (via [[characteristic element of a bilinear form|characteristic cohomology classes]]) of [[intersection pairings]] $(x,y) \mapsto \int_\Sigma x \cup y$ in [[ordinary cohomology]]. [[mathematics|Mathematically]] this refines the construction of [[Theta characteristics]] to [[ordinary differential cohomology]] and to higher [[intermediate Jacobians]]. [[physics|Physically]] it is motivated from and related to [[self-dual higher gauge theory]] (see there for more) appearing in [[string theory]] and the corresponding [[quantum anomaly|quantum anomalies]]. In particular it makes rigorous the construction ([Witten 96](#M5-brane#Witten96)) of the [[partition function]] of the [[self-dual higher gauge field|self-dual]] [[B-field]] in the [[6d (2,0)-superconformal QFT]] on the [[worldvolume]] of the [[M5-brane]] via [[geometric quantization]] of abelian [[7d Chern-Simons theory]].

The article introduces a systematic general definition for the refinement of any [[generalized (Eilenberg-Steenrod) cohomology]] theory to [[differential cohomology]] (the context for higher [[gauge field]]s in physics) in terms of [[differential function complexes]].
In this construction continuous classifying maps from a [[smooth manifold]] into a [[spectrum]] [[Brown representability theorem|representing]] the given [[cohomology]] are equipped with smooth [[differential form]]s that have under the [[de Rham theorem]] the same class in [[real cohomology]] as the pullback along the classifying map of a collection of given real cocycles on the spectrum.

For a somewhat streamlined account see at at _[[differential cohomology hexagon]]_ the section _[Hopkins-Singer coefficients](differential%20cohomology%20diagram#HopkinsSingerCoefficients)_.

A review talk is recorded at

* [[Michael Hopkins]], _Differential cohomology for general cohomology theories and one physical motivation, talk at Simons Center Workshop on Dierential Cohomology (2011) ([video](http://scgp.stonybrook.edu/archives/851))

This states that the article "should have been written" in terms of [[smooth infinity-stacks]]. This essentially happened later in

* {#HopkinsQuick14} [[Michael J. Hopkins]], [[Gereon Quick]], *Hodge filtered complex bordism*, Journal of Topology **8** 1 (2014) 147-183 [[arXiv:1212.2173](https://arxiv.org/abs/1212.2173), [doi:10.1112/jtopol/jtu021](https://doi.org/10.1112/jtopol/jtu021)]

now in the context of [[Hodge-filtered differential cohomology]].

(For other references that adopt this perspective see at _[[differential cohomology hexagon]]_ the [list of references](differential+cohomology+diagram#References).)

The connection of this work to the [[physics]] of the [[electromagnetic field]] and of  [[higher gauge field]] in  [[string theory]] was later further advertized in

* [[Daniel Freed]], _[[Dirac charge quantization and generalized differential cohomology]]_ ([arXiv:hep-th/0011220](http://arxiv.org/abs/hep-th/0011220)).

To this date generalized [[differential cohomology]] theories keep being studied mostly with motivation from [[string theory]], but the work of Hopkins and Singer has put this subject on solid mathematical ground, and an independent mathematical field of differential cohomology is developing since then. See the list of references at _[differential cohomology -- References](differential+cohomology#References)_ and at _[differential cohomology hexagon -- References](differential+cohomology+diagram#References)_.

#Some linked keywords#
* table of contents
{:toc}

## Differential cohomology

* [[differential function complex]]

* [[differential cohomology]]

  * [[generalized (Eilenberg-Steenrod) cohomology]]

    * [[ordinary differential cohomology]]

    * [[differential K-theory]]

  * [[Chern character]]

    * [[curvature]]

* [[differential orientation]]

  * [[differential Thom class]]

* [[fiber integration in differential cohomology]]

  * [[fiber integration in ordinary differential cohomology]]

  * [[fiber integration in differential K-theory]]

## Algebraic topology

* [[intersection pairing]], [[quadratic refinement]]

* [[signature genus]], [[Kervaire invariant]]

* [[Wu class]], [[integral Wu structure]]

## Gauge fields

* [[gauge field]]

  * [[field strength]]

  * [[quantum anomaly]]

    * [[Green-Schwarz mechanism]]

* [[higher dimensional Chern-Simons theory]]

* [[self-dual higher gauge theory]]

category: reference

[[!redirects Quadratic Functions in Geometry, Topology,and M-Theory]]


quadratic Hamiltonian

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#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Geometric quantization
+--{: .hide}
[[!include geometric quantization - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Definition

Given a [[symplectic vector space]] $(V,\omega)$, then a [[function]] $H \colon V \to \mathbb{R}$ regarded as a [[Hamiltonian]] is called _quadratic_ if in terms of linear [[coordinates]] it is a degree-2 [[polynomial]]. If it is in fact a [[quadratic form]] then it is called a _homogeneous quadratic Hamiltonian_.

## Properties

### Quantization

Quadratic Hamiltonians enjoy particularly nice properties under [[quantization]]. In particular [[Weyl quantization]] restricts on them to a [[homomorphism]] of [[Lie algebras]] from the [[Poisson bracket]] to the [[commutator]] (e.g. [Robbin-Salamon 93](#RobbinSalamon93)).

### Relation to metaplectic group and extended affine symplectic group

The [[quantomorphisms]] of the [[symplectic manifold]] $(V,\omega)$ which come from paths of quadratic Hamiltonians form the [[extended affine symplectic group]] $ESp(V,\omega)$. The further [[subgroup]] of those coming from homogeneous quadratic Hamiltonians form the [[metaplectic group]] ([Robbin-Salamon 93](#RobbinSalamon93))

## References

* D. G. Currie, E. J. Saletan, _Canonical transformations and quadratic hamiltonians
_, II Nuovo Cimento B Series 11 , Volume 9, Issue 1, pp 143-153,  1972

* {#Leray81} [[Jean Leray]], section 1.1 of _Lagrangian analysis and quantum mechanics_, MIT press 1981 [pdf](http://www.maths.ed.ac.uk/~aar/papers/leraybook.pdf)

* {#RobbinSalamon93} [[Joel Robbin]],  [[Dietmar Salamon]], _Feynman path integrals on phase space and the metaplectic representation_  Math. Z. __221__ (1996), no. 2, 307–335, ([MR98f:58051](http://www.ams.org/mathscinet-getitem?mr=98f:58051), [doi](http://dx.doi.org/10.1007/BF02622118), [[RobbinSalamonMetaplectic.pdf:file]]), also  in [[Dietmar Salamon]] (ed.), _Symplectic Geometry_, LMS Lecture Note series 192 (1993)

* Monique Combescure, Didier Robert, _Quadratic Quantum Hamiltonians revisited_ ([arXiv:math-ph/0509027](http://arxiv.org/abs/math-ph/0509027))

[[!redirects quadratic Hamiltonians]]

[[!redirects homogeneously quadratic Hamiltonian]]
[[!redirects homogeneously quadratic Hamiltonians]]


quadratic invariants - table

| [[manifold]] [[dimension]] | invariant | [[quadratic form]] | [[quadratic refinement]] |
|--|--|--|--|
| $4k$ | [[signature genus]] | [[intersection pairing]] | [[integral Wu structure]]  |
| $4k+2$ | [[Kervaire invariant]] |  | [[framed manifold|framing]] |


quadratic irrational number

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

#Contents#
* table of contents
{:toc}

## Definition

A __quadratic number__ is a [[number]] that is the [[root]] of some quadratic [[polynomial]] with [[rational number|rational]] [[coefficients]]. A __quadratic irrational number__ is a __quadratic number__ that is also [[irrational number|irrational]]. The __real quadratic numbers__  are quadratic numbers without a square root of $-1$ ([[imaginary unit]]), while __complex quadratic numbers__ are quadratic numbers with a square root of $-1$. __Real quadratic irrational numbers__ are real quadratic numbers that are irrational, while __complex quadratic irrational numbers__ are complex quadratic numbers that are irrational.

## Properties

Every quadratic irrational number $x:K$ could be expressed as

$$x = a + b\cdot\sqrt{c}$$

where $a:\mathbb{Q}\subseteq K$, $b:\mathbb{Q}\backslash\{0\}\subseteq K$, and $c:\mathbb{Z}\backslash\{0\}\subseteq K$, and where the [[principal square root]] $\sqrt{c}$ is not a positive integer or $i$ times a positive integer. $x$ is a real quadratic irrational number if $0\lt c$ and $x$ is a complex quadratic irrational number if $c\lt 0$.

Every quadratic [[number field]] is a [[subfield]] of the complex quadratic numbers.

## Related entries

* [[quadratic formula]]

* [[rational number]]

* [[irrational number]]

* [[number field]]

## References

* Wikipedia, _[Quadratic irrational number](https://en.wikipedia.org/wiki/Quadratic_irrational_number)_

[[!redirects quadratic irrational numbers]]

[[!redirects quadratic irrational]]
[[!redirects quadratic irrationals]]

[[!redirects quadratic number]]
[[!redirects quadratic numbers]]


quadratic operad

${^d\bigwedge}$ can be intuitively understood as the space where the canonical vertex differential coalgebra exists over some division algebra.

this, of course is the generalization of quadratic operad.
the way d acts on 5 can be numeric or '5' can be a variable holding a polynomial... this, is one view as to what the true canonical [[quadratic operad]] is.

the quadratic operation for an operad is the tensor product effectively.


quadratic reciprocity law

# Contents
* table of contents
{: toc}

## Idea

In [[number theory]], the quadratic reciprocity law determines a precise relationship between the truth of $p R q$ and the truth of $q R p$, where $p R q$ is this [[binary relation]] on odd [[prime numbers]]: "$p$ is a square [[modular arithmetic|modulo]] $q$".

The law of quadratic reciprocity is a celebrated result due to [[Gauss]]; the ideas required to prove it helped to inaugurate modern [[number theory]]. It is today largely considered within the context of [[class field theory]], involving how [[prime number|primes]] decompose in [[Galois extension|abelian extensions]] of [[number fields]].

## Statement

For $p \geq 1$ an odd [[prime number]], and for any [[integer]] $a$, define the __Legendre symbol__ (or _quadratic reciprocity symbol_) $\left(\frac{a}{p}\right)$ to be the unique element in $\{-1, 0, 1\}$ for which

$$
\left(\frac{a}{p}\right) \equiv a^{\frac{p-1}{2}} \; mod \; p
$$

Notice that the Legendre symbol defines a surjective [[homomorphism]] on a [[cyclic group]]

$$(\mathbb{Z}/p)^\ast \cong C_{p-1} \to \{-1, 1\}$$

whose [[kernel]] is the set of squares in the multiplicative group $(\mathbb{Z}/p)^\ast$, so that $\left(\frac{a}{p}\right) = 1$ means $a$ is a square modulo $p$, and $\left(\frac{a}{p}\right) = -1$ means $a$ is a non-square modulo $p$.

+-- {: .num_theorem}
###### Theorem (Quadratic Reciprocity)
For $p$, $q$ distinct odd primes,

$$\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}.$$

$$\,$$

$$\left(\frac{2}{p}\right) = (-1)^{\frac{p^2-1}{8}}$$

i.e., $2$ is a square modulo $p$ if and only if $p \equiv \pm 1 \; mod \; 8$.
=--

The case for odd primes $p$, $q$ can be restated as:

* if $p,q$ are odd and $p \equiv 1 \; mod \; 4$, then $\left(\frac{p}{q}\right) = \left(\frac{q}{p}\right)$

* if $p,q \equiv 3 \; mod \; 4$ then  $\left(\frac{p}{q}\right) = -\left(\frac{q}{p}\right)$

The law of quadratic reciprocity was first fully proven by [Gauss](#Gauss), although special cases were proven by [[Fermat]] (who effectively realized when $-1$ was a square modulo $p$ based on his [[two squares theorem]] -- well before Legendre introduced his eponymous symbol), and [[Euler]] (who proved the case for $q = 2$).

None of these early authors were in full possession of modern notations such as the congruence symbol or the Legendre symbol, which greatly economize the amount of thought needed to prove this theorem. Recognition of quadratic reciprocity was likely rooted in empirical observations, for example the study of periods of expansions of $1/q$ in base $p$ (source?).

## Sample classroom application

Consider the problem of computing the length of the period of $1/65537$ in its decimal expansion. This period is the same as the least positive exponent $n$ such that $10^n \equiv 1 \; mod \; 65537$, and is a divisor of $65536$ (thus, a power of $2$). Indeed, $65537$ is a [[Fermat prime]]: $2^{2^4} + 1$.

The period is a proper divisor of $65536$ if and only if $10$ is a square modulo $p = 65537$, leading one to contemplate

$$\left(\frac{10}{p}\right) = \left(\frac{2}{p}\right) \left(\frac{5}{p}\right)$$

where the first factor is pretty clearly $1$ (already $2^{32} \equiv 1 \; mod \; p$, so certainly $2^{\frac{p-1}{2}} \equiv 1 \; mod \; p$). The other factor is

$$\left(\frac{5}{p}\right) = \left(\frac{p}{5}\right) = \left(\frac{2}{5}\right) = -1$$

and we therefore conclude $10$ is a non-square modulo $65537$, or in other words that the length of the period of $1/65537$ is $65536$.

## Proofs

There are today several hundred published proofs of the quadratic reciprocity laws; it is rightly regarded as a capstone result in introductions to number theory. We give two proofs here.

### Proof via Gauss sums

This proof, following [Lang](#Lang) (pp. 76--78), is middling in sophistication but at least indicates the relevance of [[cyclotomic extension]]s of the rationals $\mathbb{Q}$ and [[quadratic extension]]s therein. Let $p$ be a fixed odd prime, and let $\zeta$ be a primitive $p^{th}$ [[root of unity]]. Using the Legendre symbol, we introduce a particular [[character]] sum called a **[[Gauss sum]]**,

$$S \coloneqq \sum_a \left(\frac{a}{p}\right) \zeta^a,$$

where the sum is over non-zero elements $a$ of $\mathbb{Z}/p$.

+-- {: .num_lemma}
###### Lemma
$S^2 = \left(\frac{-1}{p}\right) p$.
=--

+-- {: .proof}
###### Proof
We have $S^2 = \sum_{a, b} \left(\frac{a b}{p}\right) \zeta^{a + b}$. For any fixed non-zero $a$, as $b$ ranges over all non-zero residue classes, so does $a b$. We may therefore replace $b$ by $a b$ to get

$$\array{
S^2 & = & \sum_{a, b} \left(\frac{a^2 b}{p}\right) \zeta^{a + a b} \\
 & = & \sum_{a, b} \left(\frac{b}{p}\right) \zeta^{a(1 + b)} \\
 & = & \sum_a \left(\frac{-1}{p}\right) \zeta^0 + \sum_{b \neq -1} \left(\frac{b}{p}\right) \sum_a \zeta^{a(1 + b)}
}.$$

Now, for $b \neq -1$, the character sum $\sum_a \zeta^{a(1+b)}$ evaluates to $-1$ since $1 + \zeta + \ldots + \zeta^{p-1} = 0$. Thus the calculation continues:

$$\array{
S^2 & = & \left(\frac{-1}{p}\right) (p-1) + (-1)\sum_{b \neq -1} \left(\frac{b}{p}\right) \\
 & = & \left(\frac{-1}{p}\right) p - \sum_b \left(\frac{b}{p}\right) \\
 & = & \left(\frac{-1}{p}\right) p
}$$

which completes the proof.
=--

Thus $S$ belongs to a quadratic extension of $\mathbb{Q}$, either $\mathbb{Q}(\sqrt{p})$ or $\mathbb{Q}(\sqrt{-p})$ depending on the Legendre symbol. In particular, the prime $p$ is [[ramification|ramified]] in $\mathbb{Q}(\zeta)$.

+-- {: .proof}
###### Proof of quadratic reciprocity law
First let $p$, $q$ be two distinct odd primes. With $S$ defined as above, we calculate $S^q \; mod \; q$, where $mod \; q$ indicates the quotient of the [[ideal]] generated by $q$ in the above-mentioned quadratic extension of $\mathbb{Q}$, in two ways.

On the one hand, by the preceding lemma,

$$\array{
S^q & = & S(S^2)^{\frac{q-1}{2}} \\
 & = & S\left(\frac{-1}{p}\right)^{\frac{q-1}{2}} p^{\frac{q-1}{2}} \\
 & = & S(-1)^{\frac{p-1}{2}\frac{q-1}{2}} p^{\frac{q-1}{2}} \\
 & \equiv & S (-1)^{\frac{p-1}{2}\frac{q-1}{2}} \left(\frac{p}{q}\right) \; mod \; q
.}$$

On the other hand, since raising to the power $q$ is an [[isomorphism|automorphism]] on the [[quotient]] [[ring]] obtained modulo $q$ (either $\mathbb{F}_q \times \mathbb{F}_q$ or $\mathbb{F}_{q^2}$, depending on whether $p$ is a square modulo $q$ or not), we have

$$\array{
S^q & \equiv & \sum_a \left(\frac{a}{p}\right) \zeta^{a q} \; mod \; q \\
 & \equiv & \left(\frac{q}{p}\right) \sum_a \left(\frac{a q}{p}\right) \zeta^{a q} \; mod \; q \\
 & \equiv & \left(\frac{q}{p}\right) S \; mod \; q
}$$

and now we can cancel $S$ (which is invertible modulo $q$, since $S^2$ is) to arrive at

$$\left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}} \left(\frac{p}{q}\right)$$

which gives the first clause of the reciprocity law.

As for the second clause, we can give a similar analysis. The prime $q = 2$ is ramified in the cyclotomic extension $\mathbb{Q}[i]$ since $(1 + i)^2 = 2 i$, and for any prime $p$ we may calculate $(1 + i)^p \; mod \; p$ in two ways. On the one hand,

$$\array{
(1 + i)^p & = & (1+i)(2 i)^{\frac{p-1}{2}} \\
 & \equiv & (1+i)i^{\frac{p-1}{2}} \left(\frac{2}{p}\right) \; mod \; p
}$$

and on the other hand,

$$\array{
(1+i)^p & \equiv & 1^p + i^p \; mod \; p
}$$

and by examining the cases $p \equiv 1, 3, 5, 7 \; mod \; 8$ separately, we easily deduce the second clause.
=--

### Zolotarev's proof

The (in hindsight obvious!) structural meaning of the Legendre symbol was first given by [Zolotarev](#Zolotarev):

+-- {: .num_lemma}
###### Lemma (Zolotarev)
For $p$ an odd prime and $a$ relatively prime to $p$, $\left(\frac{a}{p}\right)$ is the sign or [[signature of a permutation]] on $(\mathbb{Z}/p)^\ast$ given by multiplying by $a$.
=--

+-- {: .proof}
###### Proof
Both the Legendre symbol $\left(\frac{ }{p}\right) \colon (\mathbb{Z}/p)^\ast \to \{-1, 1\}$ and

$$(\mathbb{Z}/p)^\ast \stackrel{Cayley}{\to} Perm((\mathbb{Z}/p)^\ast) \stackrel{sign}{\to} \{-1, 1\}$$

are homomorphisms [whose domain is a cyclic group](http://ncatlab.org/nlab/show/root#roots_of_unity_in_fields_11), so it suffices to show they agree on a generator $g$. But $Cayley(g)$ is a cyclic [[permutation]] on $p-1$ elements, hence has sign $(-1)^{p-2} = -1$, which agrees with $\left(\frac{g}{p}\right)$.
=--

We turn now to Zolotarev's  proof of the "hard" case of quadratic reciprocity where $p$ and $q$ are distinct odd primes. What is interesting is that from this point forward, we don't use primality of $p$ and $q$ at all, i.e., the remainder of the argument carries over if $p$ and $q$ are replaced by odd, relatively prime integers $m$ and $n$, and we simply define $\left(\frac{a}{n}\right)$ as a sign of a permutation of multiplying by $a$ on $(\mathbb{Z}/n)^\ast$. Indeed, one often defines the **Jacobi symbol** by

$$\left(\frac{a}{n}\right) \coloneqq \prod_{i} \left(\frac{a}{p_i}\right)$$

where $n = p_1 \ldots p_r$ is the prime factorization of an odd integer, and what we really do is generalize quadratic reciprocity from the Legendre symbol to the Jacobi symbol. Thus, the primality assumption is concentrated purely in the preceding lemma.

+-- {: .proof}
###### Proof of quadratic reciprocity
For $m \gt 1$ an odd number, we consider the set $\mathbb{Z}/m = \{0, 1, \ldots, m-1\}$ as carrying on the one hand a ring structure, and on the other as carrying a structure of linear order (no compatibility between these structures!). Now suppose $m$ and $n$ are odd and relatively prime; then by the [[Chinese remainder theorem]], there is a unique ring isomorphism

$$\mathbb{Z}/m n \to \mathbb{Z}/m \times \mathbb{Z}/n.$$

If we endow $\mathbb{Z}/m \times \mathbb{Z}/n$ with the dictionary or [[lexicographic order]], then there is also a unique isomorphism of linear or [[total orders]]

$$(\mathbb{Z}/m \times \mathbb{Z}/n)_{Dict} \to \mathbb{Z}/m n$$

which takes an element $(x, y) \in \mathbb{Z}/m \times \mathbb{Z}/n$ to $n x + y \in \{0, 1, \ldots, m n - 1\}$. (Intuitively, imagine assembling a rectangular array of cards, with $m$ columns and $n$ cards in each column, into a single stack of $m n$ cards, by gathering up the first column, followed by gathering up the second column and placing it underneath, etc.)

The composite $\alpha$ of the functions

$$(\mathbb{Z}/m \times \mathbb{Z}/n)_{Dict} \stackrel{ord \cong}{\to} \mathbb{Z}/m n \stackrel{ring \cong}{\to} \mathbb{Z}/m \times \mathbb{Z}/n$$

is thus $(x, y) \mapsto n x + y \mapsto (n x + y, y)$. The composite is thus a juxtaposition of $n$ row permutations, one for each fixed $y$, where $x$ in row $y$ is taken to $n x + y$ in row $y$. The sign of such a permutation $x \mapsto n x + y$ is

$$sign((-) + y) \cdot sign(n(-)) = sign(n(-)) = \left(\frac{n}{m}\right)$$

(since the permutation $(-) + y$ is a cyclic permutation on an odd number of elements, hence is even). Thus

$$sign(\alpha) = \left(\frac{n}{m}\right)^n = \left(\frac{n}{m}\right)$$

where the second equation follows since $n$ is odd.

Similarly, consider $\mathbb{Z}/m \times \mathbb{Z}/n$ with the _reverse_ dictionary order, where $(x, y) \lt (x', y')$ if $y \lt y'$ or $y = y'$ and $x \lt x'$. We again have a unique order isomorphism and unique ring isomorphism, and we can analyze their composition $\beta$,

$$(\mathbb{Z}/m \times \mathbb{Z}/n)_{RevDict} \stackrel{ord \cong}{\to} \mathbb{Z}/m n \stackrel{ring \cong}{\to} \mathbb{Z}/m \times \mathbb{Z}/n,$$

as taking $(x, y) \mapsto (x, x + m y)$. We similarly calculate

$$sign(\beta) = \left(\frac{m}{n}\right).$$

Finally, $\beta^{-1} \circ \alpha$ is the unique order-preserving isomorphism

$$(\mathbb{Z}/m \times \mathbb{Z}/n)_{Dict} \to (\mathbb{Z}/m \times \mathbb{Z}/n)_{RevDict}$$

which gives a permutation on the underlying set $\mathbb{Z}/m \times \mathbb{Z}/n$, and it remains to show that its sign is $(-1)^{\frac{m-1}{2}\frac{n-1}{2}}$.

But this is easy to see. Recall that given a permutation $\sigma$ on a linearly ordered set (such as $(\mathbb{Z}/m \times \mathbb{Z}/n)_{Dict}$), an _inversion_ is a pair $(x \lt y)$ such that $\sigma(x) \gt \sigma(y)$, and if $I(\sigma)$ is the total number of inversions, then

$$\sign(\sigma) = (-1)^{I(\sigma)}.$$

In the present case $\sigma = \beta^{-1} \alpha$, we see $I(\sigma)$ counts pairs of pairs $(i, j)$, $(i', j')$ where $(i, j) \lt (i', j')$ in dictionary order but $(i, j) \gt (i', j')$ in reverse dictionary order; in other words when $i \lt i'$ but $j' \lt j$. The number of such occurrences is

$$I(\sigma) = \frac{m(m-1)}{2} \frac{n(n-1)}{2}$$

whence (again since $m, n$ are odd)

$$(-1)^{I(\sigma)} = (-1)^{\frac{m-1}{2}\frac{n-1}{2}}$$

which completes the proof.
=--

## Other reciprocity laws

As mentioned earlier, quadratic reciprocity law is due to Gauss and is the first of a number of reciprocity laws in number theory. The Wikipedia article lists a bevy of reciprocity laws, gradually increasing in modernity and abstraction (and power), and culminating in [[Artin reciprocity]], a capstone of the classical [[class field theory]].

## Related concepts

* [Jacobi theta function -- Functional equation and Reciprocity](Jacobi+theta+function#FunctionalEquation)

## References

* wikipedia: [reciprocity laws (mathematics)](http://en.wikipedia.org/wiki/Reciprocity_law_%28mathematics%29)

* Serge Lang, _Algebraic number theory_, Addison-Wesley (1970).
{#Lang}

* {#Gauss} [[Carl Friedrich Gauss]], _Disquisitiones arithmeticae_ (1801), Article IV.

* E.I. Zolotareff, _Nouvelle démonstration de la loi de de réciprocité de Legendre_, Nouvelles Annales de Mathématiques. 2e série 11 (1872) 354–362.
{#Zolotarev}

[[!redirects quadratic reciprocity]]
[[!redirects quadratic reciprocity law]]
[[!redirects Legendre symbol]]
[[!redirects Legendre symbols]]


quadratic refinement

+-- {: .rightHandSide}
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### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

Let

$$
  \langle -,-\rangle
  \colon
  V \otimes V \to k
$$

be a [[bilinear form]]. A ([[quadratic function|quadratic]]) [[function]]

$$
  q \colon V \to k
$$

is called a __quadratic refinement__ of $\langle -,-\rangle$ if

$$
  \langle v,w\rangle =
  q(v + w) - q(v) - q(w) + q(0)
$$

for all $v,w \in V$.

If such $q$ is indeed a [[quadratic form]] in that $q(t v) = t^2 q(v)$ then $q(0) = 0$ and

$$
  \langle v , v \rangle = 2 q(v)
  \,.
$$

This means that a quadratic refinement by a [[quadratic form]] always exists when $2 \in k$ is invertible. Otherwise its existence is a non-trivial condition. One way to express quadratic refinements is by [[characteristic elements of a bilinear form]]. See there for more.

## Related concepts

* [[cubical structure on a line bundle]]

## References

Quadratic refinements of [[intersection pairing]] in [[cohomology]] is a powerful tool in [[algebraic topology]] and [[differential topology]]. See:

* [[Mike Hopkins]], [[Isadore Singer]], *[[Quadratic Functions in Geometry, Topology, and M-Theory]]*, J. Diff. Geom. **70** (2005) 329-452 $[$[arXiv:math/0211216](https://arxiv.org/abs/math/0211216), [euclid.jdg/1143642908](https://projecteuclid.org/euclid.jdg/1143642908)$]$

[[!redirects quadratic refinement]]
[[!redirects quadratic refinements]]


quadrature

## Related concepts

* [[integrable system]]

* [[integrable PDE]]

## References

*  Eric Weisstein, _[Quadrature](http://mathworld.wolfram.com/Quadrature.html)_

[[!redirects quadratures]]


quadric

#Contents#
* table of contents
{:toc}

## Idea

A _quadric_ is an [[algebraic variety]] defined as the [[zero locus]] of a [[quadratic form|quadratic]] [[function]].

## Examples

* The quadrics in the [[Euclidean space|Euclidean]] [[plane]] are the [[conic sections]].

* [[Klein quadric]]

## References

Textbook account:

* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: §11 in: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]

See also:

* Wikipedia, _[Quadric](http://en.wikipedia.org/wiki/Quadric)_, _[Quadric (Projective geometry)](http://en.wikipedia.org/wiki/Quadric_(projective_geometry))_

[[!redirects quadrics]]


quadruple

A **quadruple** is an [[n-tuple]] for $n = 4$.

[[!redirects quadruples]]


quadrupole moment

## Related entries

* [[dipole moment]]

* [[gravitational wave]]

## References

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

[[!redirects quadrupole moments]]

[[!redirects quadrupole]]
[[!redirects quadrupoles]]


qualitative quantity

> Please do not use the $n$Lab to dump your articles. If you would like to participate in editing the $n$Lab, please participate in the discussion on the [nForum](http://nforum.ncatlab.org/) for a while to get a feeling for how we go about things.


quality type

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesion
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

A **quality type** is a key concept in [[W. Lawvere]]'s axiomatic approach to [[cohesion]] which permits to analyze a [[space]] understood as a domain of _quantitative_ variation via its _qualitative_ aspects. This can be viewed as an [[axiom|axiomatisation]] of the commonly encountered situation in [[geometry]] or [[dynamics]] that problems permit (only) a qualitative analysis.

A primary example for that kind of deferred analysis is the study of [[topological spaces]] via the [[homotopy category]] (hence via the [[homotopy types]] which they represent). So in a very broad sense, quality types are intended as ingredients to a _synthetic homotopy theory_[^SHT] where the (homotopy) contraction of a space (the collapsing of a [[cylinder object|cylinder]]/two [[idempotents]]) is extracted as essence of the concept of a spatial 'attribute'.

Technically, a quality type amounts to a special sort of [[essential geometric morphism|essential localization]] and is therefore called a **quintessential localization**  in ([Johnstone 1996](#JS96)).

In typical (topos) cases these come in the form of [[adjoint modalities|adjoint cylinders]] $\Pi\dashv\Delta\dashv\Gamma$ (with $\Delta$ [[fully faithful functor|fully faithful]]) such that the [[connected components]] and the [[global section|global sections functor]] coincide (="collapse"): $\Pi\cong\Gamma$ (cf. [[points-to-pieces transform]]). This implies for spaces in the domain of the functors that every connected component contains exactly one point. Spaces with this property are called **infinitesimal spaces**[^Inf] in ([Lawvere 2008](#Law08)).

Hence from a more _geometrical point of view_, an object in a quality type is a particular simple kind of space with 'degenerate' components, or, if you prefer, a space with 'thick' or 'coarse' points which in turn can be viewed as a minimal vestige of cohesion: when a set is a space with no cohesion, an object in a quality type is a space with _almost_ no cohesion.[^GoC] (For an elaboration of this perspective in the context of [[cohesive toposes]] see at [[infinitesimal cohesion]].)

Quality types together with the **continuity axiom** are an essential ingredient to Lawvere's ([2007 axioms](#Law07)) for geometry:

>That Continuity axiom (preservation of infinite products by $\pi_0$) was introduced in order to obtain homotopy types that are "qualities" in an intuitive sense[^qual] (as they should be automatically in the continuous case). Lawvere ([message to catlist oct 26th 2011](https://github.com/punkdit/categories/blob/master/gmane/science/mathematics/categories/7016))

([Lawvere 1998](#Law98), [Lawvere 2008](#Law08), [Marmolejo-Menni 2016](#MarmoMenni16)) also point to the role of continuity in ensuring that the Hurewicz homotopy category becomes a quality type. The definition of quality type yields important contrasts with _pure variation_ and _sufficient cohesion_ permitting to attend to the _fine structure_ of the [[gros topos|petit-gros]] landscape. So we can say with only slight exaggeration that the 2007 axioms revolve around the concept of a quality type and that cohesive spaces are those that admit qualitative (homotopical) analysis!

In the following we discuss the case for ordinary categories, for the higher order generalization see [[infinitesimal cohesive (infinity,1)-topos]].

## Definition

Let $\mathcal{S},\mathcal{F}$ be [[extensive categories]]. A [[fully faithful functor]] $q^*:\mathcal{S}\to\mathcal{F}$ is said to exhibit $\mathcal{F}$ as a **quality type** over $\mathcal{S}$ if $q^*$ has both a [[left adjoint]] and a [[right adjoint]] which moreover coincide (an [[ambidextrous adjoint]]): $q_!\dashv q^*\dashv q_*$ and $q_!\cong q_*$.

Let $\mathcal{E}$ be a (pre)cohesive category over $\mathcal{S}$ with adjoint string $p_!\dashv p^* \dashv p_* \dashv p^!:\mathcal{S}\to\mathcal{E}$ and $q^*:\mathcal{S}\to\mathcal{F}$ a quality type over $\mathcal{S}$. A functor $s:\mathcal{E}\to\mathcal{F}$ is called an **intensive quality** on $\mathcal{E}$ if $s$ preserves finite products and finite coproducts and $q_*\circ s =p_*$.

Dually, a finite coproduct preserving functor $h:\mathcal{E}\to\mathcal{F}$ with $q_!\circ h=p_!$ is called an **extensive quality** on $\mathcal{E}$.

####Remark

Intuitively, an intensive quality is compatible with the points of its domain spaces and an extensive quality with the connected components.

Note that a quality type yields itself a (pre)cohesive category with (degenerate) adjoint string $q_!\dashv q^*\dashv q_*\dashv q^*$. This situations obtains when the induced transformation $p_*\to p_!$ of the [[Nullstellensatz]] is not only an epimorphism but a natural isomorphism (cf. [Lawvere 2007](#Law07)).

## Examples

> If we divest smooth spaces of all global cohesion, keeping only the jets (on which the Thom-Mather singularities depend), we obtain a category in which every connected component of any object has exactly one point, so that the natural map between those functors is an isomorphism.  Lawvere (2004, p.108)

* Kan complexes over $Set$: The role of Kan complexes in this context is discussed in Marmolejo-Menni ([2016](#MarmoMenni16)).

* Galois theory - presheaves on the opposite of the category of finite-dimensional local k-algebras (Lawvere 2004).

####Remark

The properties of (pre)cohesive categories are fine-tuned to yield a canonical extensive quality via the Hurewicz construction $\pi (X^Y)$. In the blueprint for this, namely the combinatorial homotopy theory in Gabriel-Zisman ([1967](#GabrielZisman67), chap. III), the category of spaces has objects [[compactly generated space|Kelley spaces]]. In a sense, the concept of continuity can be viewed as a mean to bypass the construction of the [[category of fractions]] necessary in this case.

### The topos of $\mathbb{F}_1$-actions
 {#TheToposOfF1Actions}

Lawvere's work on the petit-gros topos distinction ([Lawvere 1976](#Law76)) started with the examination of two toposes of graphs ([Lawvere 1986](#Law86)). The first one of _"irreflexive" [[quiver|graphs]]_ is given by the presheaf topos of actions of the small diagram category $E\rightrightarrows V$ on $Set$ and is an [[étendue]] hence petit.

Whereas the second one of _reflexive graphs_ is given by the actions of the [[graphic monoid]] $\Delta _1=\{1,\partial_0,\partial_1\}$ with $\partial_i\partial_j=\partial_i$ for $i,j=0,1$. $\Delta _1$ is [[Morita equivalence|Morita equivalent]] to the diagram category $E\stackrel{\leftrightarrows}{\leftarrow} V$ and consists entirely of [[idempotents]]. Its topos of actions is gros and when taken over $FinSet$ satisfies the axioms for a (sufficiently) [[cohesive topos]]. The 'abelianization' of $\Delta _1$ is $\mathbb{F}_1=\{1,e\}$, the multiplicative monoid with a generic (nontrivial) idempotent $e$ which incidentally is also the monoid underlying the [[blueprint]] for the [[field with one element]].

As observed in ([Lawvere 1989](#Law89), p.277) $\mathcal{S}^{\mathbb{F}_1^{op}}$ is a quality type over $\mathcal{S}$. It was probably the first one to arise and its status of being neither petit nor gros being commented on. Some of the details are spelled out in the following section.

The surjective homomorphism $q:\Delta_1\to\mathbb{F}_1$ induces an adjoint string $q_!\dashv q^*\dashv q_*:\mathcal{S}^{\Delta_1^{op}}\to\mathcal{S}^{\mathbb{F}_1^{op}}$ with $q_*$ discarding all non-loops in a reflexive graph.

For details how the [[zeta function]] arises via the Burnside ring of $\mathcal{S}^{\mathbb{F}_1^{op}}$ in this context see ([Lawvere 1989](#Law89), pp.291-292). $\mathbb{F}_1$-torsors are discussed in Johnstone ([2002](#J02a), p.380).

## Quality types as localizations

Properties of quality types were first studied by P. Johnstone ([1996](#JS96)) using the language of [[localization|localizations]].

+-- {: .num_defn #quintessentialloc}
###### Definition
Let $\mathcal{C}$ be a finitely complete category. An [[level|essential localization]] $l\dashv r\dashv i:\mathcal{L}\to\mathcal{C}$ is called **quintessential** if $l$ is naturally isomorphic to $i$.
=--

A trivial example of a quintessential localization is provided by $id_\mathcal{C}$.

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

Another simple _example_ of a quintessential localization is given by the category $\mathcal{C}$ with objects pairs $(X, e)$ where $X$ is a set and $e=e^2$ an idempotent map $X\to X$. A morphism $f:(X_1, e_1)\to (X_2, e_2)$ is a function $f:X_1\to X_2$ with $f\cdot e_1=e_2\cdot f$. These equivariant morphisms are bound to preserve fixpoints: when $e_1(x)=x$ then $f(e_1(x))=f(x)=e_2(f(x))$. Then the fixpoint set functor $r:\mathcal{C}\to Set$ with $r(X, e)=\{x\in X | e(x)=x \}$ is left as well as right adjoint to $i(X)=(X, id_X)$ since an equivariant morphism $f:(X,e)\to (Y,id_Y)$ is uniquely determined by its restriction to the fixpoints of $e$ and its values are given by $f(e(x))$. The [[adjoint modalities]] $i\cdot r\dashv i\cdot r:\mathcal{C}\to\mathcal{C}$ corresponding to $i\dashv r\dashv i:Set\to\mathcal{C}$ map $(X,e)$ to $(r(X),id_{r(X)})$. Of course, $\mathcal{C}$ is up to equivalence just the category $\mathcal{S}^{\mathbb{F}_1^{op}}$ from above!

=--

+-- {: .num_example}
###### Example
Consider the monoid $(\overline{\mathbb{N}}, \cdot , 0)$ consisting of the natural numbers in their usual order with a point $\infty$ attached at infinity such that $\forall n\in \mathbb{N}: n$<$\infty$ with multiplication $m\cdot n=max\{m,n\}$. $\overline{\mathbb{N}}$ is commutative and satisfies the [[graphic category|graphic identity]] $x\cdot y\cdot x = x\cdot y$. Then $i:Set\hookrightarrow Set^{\overline{\mathbb{N}}}$ with $i(X)=(X, \pi_2:\overline{\mathbb{N}}\times X\to X)$ exhibits $Set^{\overline{\mathbb{N}}}$ as a quality type over $Set$ (Cf. Freyd et al. ([1999](#FOPTST99)) and O'Hearn et al. ([1995](#OPTT95))).

This can be viewed as a more elaborate version of the preceding example where $\infty$ corresponds to $e$ upon identifying $\mathcal{C}$ as the category $\mathcal{S}^{\mathbb{F}_1^{op}}$.

(Caveat: this needs checking!)

=--

To say that $l\dashv r\dashv i:\mathcal{L}\to\mathcal{C}$ is a quintessential localization amounts to say that $i:\mathcal{L}\to\mathcal{C}$ exhibits $\mathcal{C}$ as a quality type over $\mathcal{L}$ with $r$ providing the right adjoint to $i\simeq l$ (provided $\mathcal{L}$, $\mathcal{C}$ are extensive).

Note that a quintessential subtopos is [[dense subtopos|dense]] since $i$ is up to natural isomorphism a left adjoint whence preserves all colimits and the initial object in particular!

Since the [[adjoint modality|adjoint modalities]] $l\cdot r\dashv i\cdot r$ corresponding to a quintessential subtopos as a [[level of a topos|level]] coincide up to natural isomorphism, one sees that quintessential levels are their own [[Aufhebung]].

+-- {: .num_defn #persistentloc}
###### Definition
Let $\mathcal{C}$ be a finitely complete category. A [[localization]]  $r\dashv i:\mathcal{L}\to\mathcal{C}$ is called **persistent** if $\mathcal{L}$ is closed under subobjects in $\mathcal{C}$.
=--

Quintessential localizations of toposes are persistent ([Johnstone 1996](#JS96), p.94). Since [[separated object|separated objects]] for a [[Lawvere-Tierney topology]] $j$ are precisely the subobjects of $j-sheaves$ it follows that a persistent (and, in particular, a quintessential) subtopos coincides with its [[quasitopos]] of $j$-separated objects. Conversely, a subtopos with the property that it coincides with the corresponding quasitopos is persistent.

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

Let $\mathcal{E}$ be a [[Cauchy completion|Cauchy- complete category]]. The quintessential localizations of $\mathcal{E}$ correspond precisely to idempotent endomorphisms of the identity functor on $\mathcal{E}$. Moreover, quintessential subcategories of $\mathcal{E}$ form a semilattice under intersection.
=--

For the proof cf. ([Johnstone 1996](#JS96), p.93).

## Related entries

* [[cohesive topos]]

* [[infinitesimal cohesive (infinity,1)-topos]]

* [[compactly generated topological space]]

* [[essential localization]]

* [[bireflective subcategory]]

* [[ambidextrous adjunction]]

* [[Nullstellensatz]]

* [[Aufhebung]]

## Remarks on the literature

The definition of quality types occurs in ([Lawvere 2007](#Law07)) with important additional remarks in ([Lawvere 2008](#Law08)). The first occurrence of a quality type is in the work of the 1980s in the context of the gros topos of graphs: a discussion of this example is in ([Lawvere 1989](#Law89a),  p.277) where already the contrast to sufficient cohesion is observed. ([Lawvere 1991](#Law91a), pp.9-10) has a short but suggestive discussion in the context of [[synthetic differential geometry]]. The short abstract ([Lawvere 1998](#Law98)) anticipates the 2007 axioms as 'fine-tuning' to accomodate the homotopy category as a quality type. Quality types resurface without an explicit definition in the [2004 paper](#Law04) on data types.

Under the name of quintessential localization they are the focus of ([Johnstone 1996](#JS96)) where the characterization via central idempotents is given. Some additional results occur in the context of the [[Nullstellensatz]] in ([Johnstone 2011](#JS11)). ([Menni 2014](#Menni14b)) attends to the contrast between quality types and sufficient cohesion. A variant is studied as the concept of a [[bireflective subcategory]] in Freyd et al. ([1999](#FOPTST99)).

## References

* {#FOPTST99}[[Peter Freyd|P. J. Freyd]], P. W. O'Hearn, [[John Power|A. J. Power]], M. Takeyama, [[Ross Street|R. Street]], R. D. Tennent, _Bireflectivity_ , Theor. Comp. Sci. **228** (1999) pp.49-76.

* [[Pierre Gabriel|P. Gabriel]], M. Zisman, _Calculus of Fractions and Homotopy Theory_ , Springer Heidelberg 1967.{#GabrielZisman67}

* {#JS96}[[Peter Johnstone|P. Johnstone]], _Remarks on Quintessential and Persistent Localizations_ , TAC **2** no.8 (1996) pp.90-99. ([pdf](http://www.tac.mta.ca/tac/volumes/1996/n8/n8.pdf))

* {#JS11} [[Peter Johnstone|P. Johnstone]], _Remarks on Punctual Local Connectedness_ , TAC **25** no.3 (2011) pp.51-63. ([pdf](http://www.tac.mta.ca/tac/volumes/25/3/25-03abs.html))

* {#J02} [[Peter Johnstone|P. Johnstone]], _Sketches of an Elephant vol. 1_ , Cambridge UP 2002. (pp.202,380)

* {#RRZ04} M. La Palme Reyes, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, _Generic Figures and their Glueings_ , Polimetrica Milano 2004.

* {#Law76}[[F. W. Lawvere]], _Variable quantities and variable structures in topoi_ , pp.101-131 in Heller, Tierney (eds.), Algebra, Topology and Category Theory, Academic Press New York 1976.

* {#Law86} [[F. W. Lawvere]], _Categories of Spaces may not be Generalized Spaces as Exemplified by Directed Graphs_, Revista Colombiana de Matemáticas **XX** (1986) pp.179-186. Reprinted with commentary in TAC **9** (2005) pp.1-7. ([pdf](http://tac.mta.ca/tac/reprints/articles/9/tr9.pdf))

* {#Law89a} [[F. W. Lawvere]], _Qualitative Distinctions between some Toposes of Generalized Graphs_ , Cont. Math. **92** (1989) pp.261-299.

* {#Law91a} [[F. W. Lawvere]], _Some Thoughts on the Future of Category Theory_ , pp.1-13 in Springer LNM **1488** (1991).

* {#Law92} [[F. W. Lawvere]], _Categories of Space and Quantity_, pp.14-30 in: J. Echeverria et al (eds.), _The Space of Mathematics_ , de Gruyter Berlin 1992.

* {#Law96} [[F. W. Lawvere]], _Unity and Identity of Opposites in Calculus and Physics_ , App. Cat. Struc **4** (1996) pp.167-174.

* {#Law98} [[F. W. Lawvere]], _Are Homotopy Types the same as Infinitesimal Skeleta ?_ , abstract (1998). ([link](http://cms.math.ca/Events/summer98/s98-abs/node17.e))

* {#Law99} [[F. W. Lawvere]], _Kinship and Mathematical Categories_ , pp.411-425 in: R. Jackendoff, P. Bloom, K. Wynn (eds), _Language, Logic, and Concepts - Essays in Memory of John Macnamara_, MIT Press 1999.

* {#Law04} [[F. W. Lawvere]], _Left and right adjoint operations on spaces and data types_ , Theor. Comp. Sci. **316** (2004) pp.105-111.

* {#Law07} [[F. W. Lawvere]], _Axiomatic cohesion_ , TAC **19** no.3 (2007) pp.41–49. ([pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf))

* {#Law08} [[F. W. Lawvere]], _Cohesive Toposes: Combinatorial and Infinitesimal Cases_, Como Ms. 2008. ([pdf](http://comocategoryarchive.com/Archive/temporary_new_material/FWLawvere-Cohesive-Toposes-Como-January-2008.pdf))

* {#MarmoMenni16}F. Marmolejo, [[Matías Menni|M. Menni]], _On the relation between continuous and combinatorial_ , arXiv:1602.02826 (2016). ([abstract](http://arxiv.org/abs/1602.02826))

* {#Menni14a} [[Matías Menni|M. Menni]], _Sufficient Cohesion over Atomic Toposes_ , Cah. Top. Géom. Diff. Cat. **LV** (2014). ([preprint](https://sites.google.com/site/matiasmenni/SufCohesion12.pdf?attredirects=0))

* {#Menni14b} [[Matías Menni|M. Menni]], _Continuous Cohesion over Sets_ , TAC **29** no.20 (2014) pp.542-568. ([pdf](http://www.tac.mta.ca/tac/volumes/29/20/29-20.pdf))

* {#OPTT95} P. W. O'Hearn, A. J. Power, M. Takeyama, R. D. Tennent, _Syntactic Control of Interference Revisited_ , Electronic Notes in Theor. Comp. Sci. **1** (1995) pp.1-40.

* {#PRW89} [[Robert Paré|R. Paré]], [[Bob Rosebrugh|R. Rosebrugh]], R. J. Wood, _Idempotents in Bicategories_ , Bull. Austr. Math. Soc. **39** (1989) pp.421-434.

* C. Ruiz S., R. Ruiz, _Conditions for a Realization Functor to Commute with Finite Products_ , Revista Colombiana de Matemáticas **XV** (1981) pp.113-146. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002540479))

[^SHT]: The term _synthetic homotopy theory_ is slightly misleading here as Lawvere's ideas don't provide a fully fledged homotopy theory but are rather observations on the role of the cylinder configuration in the analysis of space and spatial dimension. Important facets of his views are documented in Lawvere [1992](#Law92), [1994](#Law94), [1999](#1999) and a more thorough discussion of the homotopical apects of the concept appears in Marmolejo-Menni ([2016](#MarmoMenni16)).
In a complementary direction, _[[homotopy type theory]]_ provides a synthetic formulation of [[homotopy theory]] proper, not just of the [[homotopy category]], and _[[cohesive homotopy type theory]]_ implements [[cohesion]] _in_ that natively homotopy-theoretic context.

[^qual]: The claim that _quality types_ intend to model the philosophical concept of _quality_ reoccurs at several places in Lawvere's writings though the concrete connection still needs to be spelled out. Some motivation is provided in [Lawvere (1992)](#Law92) where it is linked to the negation of _quantity_ as a logical category of being that is indifferent to non-being (cf. Hegel's [[Science of Logic]]) e.g. whereas the temperature varies continuously or "indifferently" below and above zero degree, the same transition makes a crucial difference for the phase or "qualitative being" of water. Note that Lawvere sticks here to the traditional view that quantitative being comes before qualitative being whereas Hegel in his _Logic_ reversed the traditional hierarchy of the categories.

[^Inf]: A motivation for this terminology from [[synthetic differential geometry|SDG]] can be found in a remark in ([Lawvere 1991](Law91a), pp.9-10).

[^GoC]: From another perspective, one could view an object in a quality type as a set with _germs of cohesion_.

[[!redirects quality]]
[[!redirects Quality type]]

[[!redirects extensive quality]]
[[!redirects intensive quality]]

[[!redirects quintessential localization]]
[[!redirects quintessential localizations]]

[[!redirects quintessential localisation]]
[[!redirects quintessential localisations]]

[[!redirects persistent localization]]
[[!redirects persistent localizations]]

[[!redirects persistent localisation]]
[[!redirects persistent localisations]]

[[!redirects bireflective subcategory]]


quandle

# Quandles
* table of contents
{: toc}

##The Idea##

A quandle is a set equipped with a binary operation satisfying axioms analogous to the three [[Reidemeister moves]] in knot theory.  A quandle is a special case of a [[rack]].

While mainly used to obtain invariants of [[knot|knots]], quandles are interesting algebraic structures in their own right. In particular, the definition of a quandle axiomatizes the properties of conjugation in a group.  More abstractly, we can say that a quandle is an algebraic structure where every element acts as an automorphism of that structure, fixing that element.

##Definition##

A __quandle__ is a selfdistributive idempotent right [[quasigroup]].
In more detail, a **quandle** is a [[rack]] obeying the law

$$ a \triangleright a = a $$

or equivalently

$$a \triangleleft a = a \, .$$

In other words, a quandle is a set $Q$ equipped with two binary operations, $\triangleright$ and $\triangleleft$, obeying the laws:

\[ a \triangleright (b \triangleright c) = (a \triangleright b)\triangleright (a \triangleright  c)   \]

\[ (c \triangleleft b) \triangleleft a = (c \triangleleft a)\triangleleft (b \triangleleft a)  \]

\[ (a \triangleright b)\triangleleft a = b \]

\[ a \triangleright (b \triangleleft a) = b \]

\[    a \triangleright a = a \]

\[  a \triangleleft a = a  \]

Given laws 3 and 4, the operation $\triangleright$ determines the operation $\triangleleft$, and vice versa, and then law 1 is equivalent to law 2, while law 5 is equivalent to law 6.  So, this definition has a certain redundancy built in.  See [[rack]] for more discussion of related points.

##Examples##

Every [[group]] gives a quandle where the operations come from conjugation:

$$ a \triangleright b = a b a^{-1}$$
$$ b \triangleleft a = a^{-1} b a $$

In fact, every equational law satisfied by [[conjugation]] in a group follows from the quandle axioms.  So, one can think of a quandle as what is left of a group when we forget multiplication, the identity, and inverses, and only remember the operation of conjugation.

Every tame knot in $\mathbb{R}^3$ has a "fundamental quandle".  To define this, one can note that the [[fundamental group]] of the knot complement, or [[knot group]], has a presentation (the [[Wirtinger presentation]]) in which the relations only involve conjugation.  So, this presentation can also be used as a presentation of a quandle.  The fundamental quandle is a very powerful invariant of knots.  In particular, if two knots have [[isomorphism|isomorphic]] fundamental quandles then there is a [[homeomorphism]] of $\mathbb{R}^3$, possibly orientation reversing, taking one knot to the other.

Less powerful but more easily computable invariants of knots may be obtained by counting the homomorphisms from the knot quandle to a fixed quandle $Q$.  Since the Wirtinger presentation has one generator for each strand in a [[knot diagram]], these invariants can be computed by counting ways of labelling each strand by an element of $Q$, subject to certain constraints easily read off from a diagram of the knot.  More sophisticated invariants of this sort can be constructed with the help of quandle [[cohomology]].

The Alexander quandles are also important, since they can be used to compute the [[Alexander polynomial]] of a knot.  Let $A$ be a module over the ring  $\mathbb{Z}[t, t^{-1}]$ of [[Laurent polynomial|Laurent polynomials]] in one variable. Then the **Alexander quandle** consists of $A$ made into a quandle with the left action given by

$$a \triangleright b = t a + (1-t)b $$

Analogous to how evaluating the Alexander polynomial at $t = -1$ (and then taking absolute value) defines the _determinant_ of a knot, similarly, instantiating the Alexander quandle at $t = -1$ gives rise to the **dihedral quandle**

$$a \triangleright b = 2b - a $$

which, when interpreted as an action on the ring $\mathbb{Z}_n$ of integers modulo $n$, may be used to define the classical notion of [[colorable knot|n-colorability]] of a knot.

[[rack|Racks]] are a useful generalization of quandles in topology, since while quandles can represent knots on a round linear object (such as rope or a thread), racks can represent ribbons, which may be twisted as well as knotted.

A quandle $Q$ is said to be **involutory** if it obeys the law

$$ a \triangleright (a \triangleright b) = b $$

or equivalently

$$ (b \triangleleft a) \triangleleft a = b$$

Any [[symmetric space]] gives an involutory quandle, where $a \triangleright b$ is the result of 'reflecting $b$ through $a$'.
In fact this leads to an elegant definition of symmetric spaces.  Note that involutory quandles are algebras of a certain [[Lawvere theory]], since [[rack|racks]] are already algebras of a Lawvere theory, and involutory quandles are racks obeying some extra equational laws.  We may thus define involutory quandle objects in any [[category]] with finite [[products]], such as the category of [[smooth manifolds]].  Loos has shown that a connected symmetric space is the the same as an involutory quandle object $Q$ in the category of smooth manifolds with the additional properties that:

* each point $a$ is an _isolated_ fixed point of the operation $a \triangleright -$.

* $Q$ is connected.

This fact is Theorem I.4.3. in:

* Wolgang Bertram, _The Geometry of Jordan and Lie Structures_, Lecture Notes in Mathematics **1754**, Springer, Berlin, 2000.

He attributes this result to:

* Ottmar Loos, _Symmetric Spaces I_, Chapter II, Benjamin, New York, 1969.

## Related entries

* [[rack]]

* [[shelf]]

## Links

* Wikipedia: [Racks and quandles](http://en.wikipedia.org/wiki/Racks_and_quandles)

* Gavin Wraith, [A personal story about knots](http://www.wra1th.plus.com/gcw/math/Rack.html).

## References ##

Surveys are

* S. Nelson, _What is a Quandle?_ , Notices AMS **63** no.4 (2016) pp.378-380. ([pdf](http://www.ams.org/publications/journals/notices/201604/rnoti-p378.pdf))

* J. Scott Carter, _A survey of quandle ideas_, ([arxiv](http://arxiv.org/abs/1002.4429)).

A monograph is

* M. Elhamdadi, S. Nelson, _Quandles: An Introduction to the Algebra of Knots_ , AMS Providence 2015. ([link](http://bookstore.ams.org/stml-74))

The paper by Crans makes it clear that quandles are algebras of a [[Lawvere theory]], so that quandles may be defined in any [[cartesian monoidal category]] (a category with finite [[products]]).  It also shows that any Lie algebra gives a quandle in the category of cocommutative [[coalgebra|coalgebras]].

* Alissa Crans, _Shelves, racks, spindles and quandles_, ([arXiv](http://arxiv.org/PS_cache/math/pdf/0409/0409602v1.pdf#page=56), in _Lie 2-Algebras_).

Other research papers:

* David Joyce, _A classifying invariant of knots; the knot quandle_, J. Pure Appl. Alg. __23__ (1982), 37-65, doi, [MR83m:57007](http://www.ams.org/mathscinet-getitem?mr=638121),  ([pdf](http://www.sciencedirect.com/science/article/pii/0022404982900779/pdf?md5=cc81b2cf5a01afc9277d58d10128878a&pid=1-s2.0-0022404982900779-main.pdf)).

* David Joyce, _Simple quandles_,  J. Algebra __79__ (1982), no. 2, 307–318, doi, [MR84d:20078](http://www.ams.org/mathscinet-getitem?mr=682881)

* Seiichi Kamada, _Knot invariants derived from quandles and racks_, ([arXiv](http://arxiv.org/abs/math/0211096)).

* J. Scott Carter, Masahico Saito, _Quandle homology theory and cocycle knot invariants_, ([arXiv](http://arxiv.org/abs/math/0112026)).

* Michael Eisermann, _Quandle coverings and their Galois correspondence_, ([pdf](http://www.igt.uni-stuttgart.de/eiserm/publications/qcovering.pdf)).

* Aaron Kaestner, Sam Nelson, Leo Selker, _Parity biquandle invariants of virtual knots_,  [arxiv/1507.05583](http://arxiv.org/abs/1507.05583)

* Dominique Bourn, _Partial Mal'tsevness and category of quandles_, Talk at CT2015, [abstract](http://ct2015.web.ua.pt/abstracts/bourn_d.pdf), [slides pdf](http://ct2015.web.ua.pt/slides/Bourn.pdf)
[[!redirects quandle]]
[[!redirects quandles]]


quantale

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# Contents
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## Definition

A **quantale** is a [[closed monoidal category|closed monoidal]] [[suplattice]].  Equivalently, it is a [[monoid object]] in the [[closed symmetric monoidal category]] of suplattices where the [[morphisms]] are the [[maps]] of [[sets]] that [[preserved colimit|preserve]] arbitrary [[joins]].

In most recent works, quantales are defined more generally as [[semigroup|semi]]-[[group objects]] [[internalization|in]] the [[monoidal category]] of suplattices.

In more detail, a quantale is a [[poset]] having all [[joins]] and an associative, [[tensor product]] $\otimes$ which [[distributive category|distributes]] over joins (the internal-homs then come automatically by the [[adjoint functor theorem]]); this tensor product is unital if we understand quantales in a narrower sense as monoids.

The internal-homs in a quantale are sometimes called _residuations_ and written $x\backslash y$ and $y/x$.  Unitality is
skipped by some authors; in that case we can talk about subclass of unital quantales.

As a semigroup (monoid if unital) in suplattices, a quantale is essentially the same thing as an 1-object [[quantaloid]], i.e., an 1-object category enriched in suplattices.

## Quantales and Frames

Additional conditions often imposed on a quantale include:

* Commutativity: $x\otimes y = y\otimes x$
* Idempotence: $x\otimes x = x$
* Affineness: the unit for $\otimes$ is the top element: $1=\top$.

(On affineness: see also [[semicartesian monoidal category]].) If idempotence and affineness are assumed, then $\otimes$ is forced to be the [[meet]] (whence commutativity is automatic) and the quantale is thereby a [[frame]]; see Proposition \ref{meet} below.  General quantales are sometimes considered to be a "noncommutative" version of a frame, whose [[opposite category]] would be a category of "noncommutative [[locale]]s."

(This is the origin of the name "quantale," a [portmanteau](http://en.wikipedia.org/wiki/Portmanteau) of "quantum" and "locale".  Note, though, that quantales seem to be generally treated in the literature more as "quantum frames" than "quantum locales," and in particular their morphisms usually go in the "frame direction."  Possibly this can be explained by the fact that in the past, it was common to use the word "locale" for what we now call a "frame" and simply distinguish between "locale homomorphisms" (now called "frame homomorphisms") and "continuous maps."  The name "quantale" was introduced by C.J. Mulvey.)

+-- {: .num_prop #meet}
###### Proposition
If a [[monoid object]] in the [[cartesian monoidal category]] of [[posets]] is idempotent and affine, then the monoid multiplication is the meet operation.
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+-- {: .proof}
###### Proof
Let $(M, \otimes, 1)$ be the monoid. If $a \leq x$ and $a \leq y$, then $a = a \otimes a \leq x \otimes y$. On the other hand, we have $x \otimes y \leq x \otimes 1 = x$ and similarly $x \otimes y \leq y$, so $a \leq x \otimes y$ implies $a \leq x$ and $a \leq y$ by transitivity. Thus $a \leq x \otimes y$ iff $a \leq x$ and $a \leq y$, i.e., $x \otimes y$ satisfies the defining property of the meet $x \wedge y$.
=--

Along similar lines, the following construction provides passage from commutative affine quantales to frames:

+-- {: .num_lemma}
###### Lemma
Let $(Q, \cdot, 1)$ be a commutative affine quantale, and let $Idem(Q)$ be the subposet of elements $x \cdot x = x$. Then $Idem(Q)$ is a frame, where the meet operation is given by multiplication in $Q$. The functor $Idem$ is right adjoint to the forgetful functor from commutative affine quantales to frames.
=--

+-- {: .proof}
###### Proof
If $x, y$ are idempotent, then so is $x \cdot y$ using the fact that $\cdot$ is commutative. Thus $Idem(Q)$ is an idempotent affine submonoid of $Q$, which by Proposition \ref{meet} forces $\cdot$ to be the meet. Next, we show that $Idem(Q)$ is closed under taking joins in $Q$: if $x_i$ is a collection of idempotents, we have

$$x_i \leq x_i x_i \leq (\bigvee_i x_i) (\bigvee_i x_i)$$

for all $i$, whence

$$\bigvee_i x_i \leq (\bigvee_i x_i) (\bigvee_i x_i),$$

which is all we need (the opposite inequality is automatic since $a \cdot a \leq a \cdot 1 = a$ for all $a \in Q$). Since joins in $Idem(Q)$ are calculated just as they are in $Q$, and since multiplication in $Q$ distributes over arbitrary joins, we have that binary meets distribute over arbitrary joins in $Idem(Q)$.

Finally, if $A$ is a frame and $Q$ is a commutative affine quantale, it is clear that a quantale map $f \colon A \to Q$ takes elements in $A$ (which are idempotent under meet) to idempotents in $Q$. Hence $f$ factors uniquely through $Idem(Q) \hookrightarrow Q$, and the map $A \to Idem(Q)$ is a frame map. This shows that $Idem$ is the right adjoint as claimed.
=--

In fact, we may also observe that the forgetful functor from commutative affine quantales to commutative quantales also has a right adjoint, just by passing from a commutative quantale to the principal downset given by the quantale unit. (However, the forgetful functor from commutative quantales to quantales does _not_ have a right adjoint.)

On very general grounds, the forgetful functor from the category of frames to the category of quantales has a left adjoint (see [here](/nlab/show/colimits+in+categories+of+algebras#relatively_free_functors)). This forgetful functor is full and faithful, and as a result the unit of the adjunction is a regular epi described by a suitable quantale congruence; see the next section.

## Quantale congruences and nuclei

A *quantale congruence* is simply an [[equivalence relation]] $({\equiv}) \hookrightarrow Q \times Q$ on a quantale $Q$ that respects the quantale multiplication and the taking of sups: if $a \equiv b$ and $x \equiv y$, then $a x \equiv b y$, and if $x_i \equiv y_i$ for all $i \in I$, then $\bigvee_{i \in I} x_i \equiv \bigvee_{i \in I} y_i$.

Consequently, $Q/{\equiv}$ defines a quantale $\tilde{Q}$ with operations inherited along a quantale quotient map $q: Q \to \tilde{Q}$. Since $q$ preserves arbitrary sups, it has a right adjoint $r: \tilde{Q} \to Q$ by the [[adjoint functor theorem]] applied to posets. Thus we have a [[monad]] or [[closure operator]] $r q: Q \to Q$, and the algebras/fixpoints of the monad/closure operator are identified with the elements of $\tilde{Q}$, i.e., $r: \tilde{Q} \to Q$ is isomorphic to the inclusion $Fix(r q) \hookrightarrow Q$.

The monad $r q$ is a quantale nucleus in the sense of the following definition:

+-- {: .num_defn}
###### Definition
A function $j: Q \to Q$ on a quantale $Q$ is a *nucleus* if it is a [[monad]] ($x \leq y$ implies $j(x) \leq j(y)$, $x \leq j(x)$, $j j(x) = j(x)$ for all $x, y \in Q$) and is [[monoidal functor|lax monoidal]]: $1 \leq j(1)$ and $j(x) \cdot j(y) \leq j(x \cdot y)$ for all $x, y \in Q$.
=--

There is a natural bijective correspondence between congruences on a quantale $Q$ and nuclei on $Q$.

## Enrichment over quantales

Aside from being "noncommutative frames", a different way of thinking about quantales views them as a [[(0,1)-category|(0,1)-categorical]] analogue of a [[cosmos]] (in the sense of Benabou).  In particular, one can then study [[enriched categories]] over a quantale. A classic example is Lawvere [[metric space]]s, seen as categories enriched in the quantale $([0, \infty], \geq)$ with $+$ taken as tensor product.

Enrichment is often particularly interesting for $*$-quantales (see below), where one can study $*$-enriched categories.

## Examples ##

{#compsci}Quantales are a surprisingly commonplace structure in computer science. See the [automata references](#automata). A very simple example is the power set of strings (i.e., the [[power set]] of the [[free monoid]] over some set of characters $\Sigma$). The order is the inclusion order on sets, and meet and join are just intersection and union, respectively. Taking $\epsilon$ to be the empty string, and $a \cdot b$ to be the join of two strings, the quantalic operations are then:

* $1 = \{\epsilon\}$
* $I \otimes J = \{ i\cdot j \mid i \in I, j \in J \}$
* $K/J = \{ i \mid \forall_{j\in J} i\cdot j \in K \}$
* $I\backslash K = \{ j \mid \forall_{i\in I} i\cdot j \in K \}$

This example generalizes as follows: given any [[monoidal category|monoidal]] [[preorder]] $M$ (for instance, a monoid equipped with the discrete order, as in the previous example), the collection of down-closed subsets of $M$ carries a quantale structure given by [[Day convolution]] with respect to categories enriched in $\mathbf{2} = TV$, the [[Heyting algebra]] of truth values. Explicitly, if $e$ denotes the unit of $M$ and $\cdot$ the multiplication, then

* $1 = \{x\in M \mid x \leq e\}$
* $I \otimes J = \{x\in M \mid \exists_{i \in I} \exists_{j \in J} x \leq i \cdot j\}$
* $K/J = \{i \in M \mid \forall_{j\in J} i\cdot j \in K \}$
* $I\backslash K = \{j \in M \mid \forall_{i\in I} i\cdot j \in K \}$

Another class of examples: internal homs $\hom_{sLat}(X, X)$ in the closed monoidal category of suplattices. For example, when the suplattice $X$ is a power set $P(S)$, one may identify $\hom_{sLat}(P(S), P(S))$ with the poset of binary relations $P(S \times S)$, ordered by inclusion and where the quantalic multiplication is relational composition.

An example in algebra is given by the lattice of ideals of a commutative ring, with the tensor product given by ideal multiplication, which makes it into a commutative affine quantale. Residuation in this case is *ideal division* $(\mathfrak{a} : \mathfrak{b}) = \{x | x\mathfrak{b} \subseteq \mathfrak{a}\}$.

Quantales, as monoids in the symmetric monoidal category $sLat$, can be tensored to produce new quantales.

## $*$-quantales

A $*$-quantale is a quantale $Q$ equipped with an additional structure of an involution

$$ * : Q \to Q$$

for which $(x \otimes y)^* = y^* \otimes x^*$ and $1^* = 1$, where $1$ denotes the monoidal unit. (The operator is assumed to be covariant with respect to the poset structure.)

An example of a $*$-quantale is the quantale of binary relations on a set $S$, where the $*$-operation is relational opposite:

* $R^* = \{(y, x): (x, y) \in R\}$

Another example is obtained by taking the quantale of down-closed subsets of a $*$-monoidal poset $M$ (which is the same thing as a $*$-[[star-monoid|monoid]] in the [[cartesian monoidal category]] of posets), with the quantale structure given by Day convolution as described above, and the $*$-operator obtained by cocontinuously extending the $*$-operator on $M$. Explicitly,

* $L^* = \{x^*: x \in L\}$

A _$*$-enriched category_ over a $*$-quantale $Q$ is a category $(X, d: X \times X \to Q)$ enriched in the underlying quantale, such that

$$d(y, x) = d(x, y)^*$$

This notion can also be expressed in terms of lax morphisms of $*$-quantales; see below.

## Relation to linear logic
 {#RelationToLinearLogic}

A commutative quantale is in particular a [[symmetric monoidal category]] (a symmetric monoidal [[(0,1)-category]]). As such it may be thought of as a model for [[linear logic]] in the general sense. Precisely if it has a [dualizing object](dualizing+object+in+a+closed+category) then it is a [[star-autonomous category]] and hence a model for [[linear logic]] in the original sense. (see e.g. [Yetter 90, page 43](#Yetter90)). Indeed, quantales have been argued to provide models for [[quantum logic]], see there for more.

## Morphisms of quantales ##

There is a variety of notions of morphism of quantale, just as there is a variety of notions of morphism between closed monoidal categories. All the notions considered here are morphisms between the underlying sup-lattices, in other words preserve arbitrary joins, hence are left adjoints as functors between the underlying categories.

* At the weak end of the scale, one may consider _lax morphisms_ of quantales, i.e., (lax) [[monoidal functor]]s of quantales seen as monoidal categories.

  * An important example of this is that [[enriched category|categories enriched]] in a monoidal poset $M$, such as Lawvere [[metric space]]s, amount to the same thing as lax quantale morphisms of the form $2^d: 2^{M} \to 2^{X \times X}$ where the domain is the quantale of upward-closed subsets of $M$ with the Day convolution structure, and the codomain is the quantale of binary relations on $X$, with multiplication being relational composition.

* A stronger notion is of _strong morphisms_ of quantales seen as monoidal categories. As noted above, all quantale morphisms considered here are already left adjoints in $Cat$, and if the adjunction lifts to $MonCat$ (the 2-category of monoidal categories, lax monoidal functors, and monoidal transformations), then the left adjoint is strong monoidal. This often occurs in practice.

* An even stronger notion is where the morphisms also strongly preserve the closed structure, i.e., the internal homs or residuations.

* There are corresponding notions of morphisms of $*$-quantales, where in each case morphisms strongly respect the $*$ operations. For instance, the notion of $*$-enriched category over a $*$-monoidal poset $M$ can be equivalently recast as a lax morphism between $*$-quantales, $2^d: 2^M \to 2^{X \times X}.$

##Related entries:

* [[locale]],

* [[quantaloid]], [[semiquantale]]

* [[module over a quantale]]

## References

The initial paper to use the term `quantale' was

*  [[Christopher J. Mulvey]], _["&"](http://www.maths.sussex.ac.uk/Staff/CJM/research/pdf/&.pdf)_, Rend. Circ. Mat. Palermo, II. Ser., Suppl. 12, 99-104 (1986) [Zbl 0633.46065](http://zbmath.org/?q=an:0633.46065)

* [[eom]]: [quantale](http://www.encyclopediaofmath.org/index.php/Quantale)

Discussion of quantales as a model for [[linear logic]] and [[quantum logic]]:

* {#Yetter90} [[David Yetter]], *Quantales and (noncommutative) linear logic*, Journal of Symbolic Logic **55** (1990) 41-64 [[doi:10.2307/2274953](https://doi.org/10.2307/2274953)]

A monograph on quantales:

* Kimmo I. Rosenthal, _Quantales and their applications_, Pitman Res. Notes in Math. Series 234, Longman 1990

{#automata}Relations with automata and process semantics

* [[Samson Abramsky]],  [[Steve Vickers]], _Quantales, observational logic and process semantics_, Math. Struct. in Comput. Sci. __3__  (1993) 161-227. [doi](https://doi.org/10.1017/S0960129500000189), [CiteSeerX](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.210.1485)

Connections to operator algebras and etale groupoids is discussed in

* [[Pedro Resende]], _Étale groupoids and their quantales_, Adv. Math. __208__ (2007) 147-209; also published electronically: [doi](http://dx.doi.org/10.1016/j.aim.2006.02.004); [math/0412478](http://arxiv.org/abs/math/0412478)
* M.C. Protin, P. Resende, _Quantales of open groupoids_, J. Noncommut. Geom. __6__ (2012) 199–247.
* [[Pedro Resende]], Lectures on étale groupoids, inverse semigroups and quantales, Lecture Notes for the GAMAP IP Meeting, Antwerp, 4–18 September, 2006, 115 pp.; [pdf](http://www.math.ist.utl.pt/%7Epmr/poci55958/gncg51gamap-version2.pdf)
* [[Pedro Resende]], _Groupoid sheaves as quantale sheaves_, J. Pure Appl. Algebra 216 (2012), 41–70; [arxiv/0807.4848](http://arxiv.org/abs/0807.4848v3)
[doi](http://dx.doi.org/10.1016/j.jpaa.2011.05.002)
* D. Kruml, J.W. Pelletier, P. Resende, [[J. Rosický]], _On quantales and spectra of $C^\ast$-algebras_, Appl. Categ. Structures __11__ (2003) 543–560.
* D. Kruml, P. Resende, _On quantales that classify $C^\ast$-algebras_, Cah. Topol. Geom. Differ. Categ. __45__ (2004) 287–296.
* [[F. Borceux]], [[J. Rosický]], G. Van den Bossche, _Quantales and $C^\ast$-algebras, J. London Math. Soc. __40__ (1989) 398–404 [doi](http://dx.doi.org/10.1112/jlms/s2-40.3.398)

A connection between topoi, "Grothendieck" quantales and $C^\ast$-algebras

* Simon Henry, _Toposes, quantales and C* algebras in the atomic case_, [arxiv/1311.3451](http://arxiv.org/abs/1311.3451)

 Sheaves on a quantale

* [[Francis Borceux]], Rosanna Cruciani, Sheaves on a quantale, Cahiers de Topologie et Géométrie Différentielle Catégoriques (1993) 34:3, page 209-228 [pdf](http://archive.numdam.org/article/CTGDC_1993__34_3_209_0.pdf)

category: order, noncommutative geometry
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