Q-category

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### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
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#Contents#
* table of contents
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## Idea

A _$Q$-category_ is nothing but a [[coreflective subcategory]] and a _$Q^\circ$-category_ is nothing but a [[reflective subcategory]]. Since both of these encode reflective [[localization]]s, following [Rosenberg](#Rosenberg) the "Q" is for _quotient_ and is to indicate that in this context one is interested in notions similar to, but different from, the standard notion of [[sheaves]]:

for $\mathbb{A} := (A \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}} \bar A)$ a $Q^\circ$-category there is canonically induced a quadruple of [[adjoint functor]]s between the corresponding [[presheaf categories]]

$$
  PSh(\bar A)
   \stackrel{\overset{u_!}{\to}}{\stackrel{}{\stackrel{\overset{u^*}{\leftarrow}}{\stackrel{\overset{u_*}{\to}}{\underset{u^!}{\leftarrow}}}}}
  PSh(A)
$$

and a presheaf $F \in PSh(A)$ is called an _$\mathbb{A}$-sheaf_ if the canonical morphism

$$
  u^* F \to u^! F
$$

is an [[isomorphism]] in $PSh(\bar A)$. More generally, there are generalizations of this condition where presheaves of sets can be replaced with presheaves with values in other categories, notably in [[abelian categories]].

In a central motivating class of examples $\mathbb{A}$ is a category of [[sieve]]s on objects in a [[small category]] $C$ that are regarded as being [[covering]] but which do not necessarily satisfy the axioms of a [[Grothendieck topology]] and not even of a [[coverage]].

## Motivation

The [[Yoneda embedding]] is continuous but not cocontinuous functor. Hence the Grothendieck topologies are used to define smaller codomain of Yoneda, a sheaf subcategory of the category of presheaves such that for that corestricted embedding a desired class of covering cones will stay covering cones. More general families of diagrams than the sieves of a Grothendieck topology may be involved. The important properties of the categories of diagrams for doing the sheaf theory can be expressed in terms of an adjoint pair of functors; this adjoint pair gives an example of a $Q$-category. This generalization of sheaf theory can rephrase categorically also properties like formal smoothness and formal etaleness of functors. The sheafification and the construction of a Gabriel localization of an Abelian category can in this formalism be seen as special cases of the same construction.

## Definition

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

An __almost quotient category__, or a __$Q$-category__ is

* a pair of [[adjoint functor]]s

  $$
    \mathbb{A} : (u^* \dashv u_*) :  \bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}} A
    \,,
  $$

  with $u^*$ the [[left adjoint]];

* such that $u^*$ is [[full and faithful functor|full and faithful]]

In other words, $A$ is equipped with an equivalence with a [[coreflective subcategory]] of $\bar{A}$.

=--

+-- {: .num_note #QTerminology}
###### Note

* In motivating classes of examples $\bar A$ and $A$ are [[topos]]es and $\mathbb{A}$ is a [[geometric morphism]] between them. Therefore one generally speaks of $u_*$ as the **[[direct image]]** and of $u^*$ as the **[[inverse image]]** of the Q-category.

* The definition \ref{QCategory} is nothing but the definition of a [[coreflective subcategory]]. However the term _$Q$-category_ is used when the pair is used in a specific meaning useful to constructions in (generalized) sheaf theory (similarly like presheaf of objects in $D$ and contravariant functor to $D$ are synonyms, but a different word refers to a different context and intuition).

* One sometimes write the above data as $\bar A \stackrel{\overset{u}{\leftarrow}}{\to} A$.

=--

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

A __[[morphism]] of Q-categories__ from
$\mathbb{A} : (u^* \dashv u_*) :  \bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}} A$ to $\mathbb{B} : (v^* \dashv v_*) :  \bar B \stackrel{\overset{v^*}{\leftarrow}}{\underset{v_*}{\to}} B$ is a triple $(\Phi,\bar{\Phi},\phi)$ where $\Phi : A\to B$, $\bar{\Phi}:\bar{A}\to\bar{B}$ are functors and $\phi:\Phi u_*\Rightarrow v_*\bar{\Phi}$ is a [[natural isomorphism]] of functors. The composition is given by
$$
(\Phi,\bar{\Phi},\phi)\circ(\Phi',\bar{\Phi}',\phi') = (\Phi\Phi',\bar{\Phi}\bar{\Phi}',\phi\bar{\Phi}' \circ \Phi\phi')
$$

A __transformation of morphisms of Q-categories__ is a pair $(\alpha,\bar{\alpha}):(\Phi,\bar{\Phi},\phi)\to (\Psi,\bar{\Psi},\psi)$ of [[natural transformation]]s $\alpha:\Phi\to\Psi$ and $\bar{\alpha}:\bar{\Phi}\to\bar{\Psi}$ such that the diagram
$$\array{
\Phi u_* & \stackrel{\phi}\longrightarrow & v_* \bar{\Psi}\\
\alpha u_*\downarrow && \downarrow v_*\bar{\alpha}\\
\Psi u_* &\stackrel{\psi}\longrightarrow& v_* \bar{\Psi}

}$$
[[commuting diagram|commutes]].

=--

Small Q-categories, morphisms of Q-categories and natural transformations of morphisms form a [[2-category]] of small Q-categories.

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

A $Q^\circ$-category is a pair of functors $Q: A\leftrightarrow\bar{A}: I$,
where $Q$ is fully faithful and right adjoint to $I$. In other words, $A$ is equipped with an equivalence with a [[reflective subcategory]] of $\bar{A}$.

=--

## Examples

### (Co)Presheaves on a $Q$-category
 {#PresheafQCategory}

The [[category of presheaves]] over any $Q$-category canonically inherits itself the structure of a Q-category.

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

Let $\mathbb{A} = \bar A \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} A$ be a $Q$-category.
Let $C$ be any category.

Then forming [[copresheaves]] with values in $C$ yields a Q-category of the form

$$
  [\mathbb{A},C] :
   [\bar A, C]
     \stackrel{\overset{C^R = (-)\circ R}{\leftarrow}}{\underset{C^L = (-)\circ L}{\to}}
   [A,C]
  \,.
$$

=--

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

The $(C^R \dashv C^L)$-[[unit of an adjunction|unit]] is $C^\eta$  induced by the original unit $\eta: 1_A\to R L$

$$
  C^{\eta} : Id_{C^A} \to C^L \circ C^R = C^{R L}
$$

and the counit $C^\epsilon$ is induced by the original counit $\epsilon: L R\to 1_{\bar{A}}$

$$
  C^\epsilon : C^R\circ C^L = C^{L R}\to Id_{C^{\bar{A}}}
  \,.
$$

The only thing is who is adjoint -- now $C^R$ is the left adjoint.

The triangle identities for $C^R\dashv C^L$ can be obtained by expanding.  For $R: \bar{A}\to A$, one has $C^R : C^A\to C^{\bar{A}}$ is given by $C^R : G\mapsto GR$, and for $L:A\to\bar{A}$ one has $C^L:F\mapsto F L$. Then $C^\eta : Id_{C^A}\to C^R C^L = C^{LR}$ has the components $(C^\eta)_G : (Id_{C^A})(G) \Rightarrow C^R C^L (G)$ given by $G \eta : G\to G L R$. Thus for each functor $G\in C^{\bar{A}}$, the composition

$$G R\stackrel{G\eta R}\longrightarrow G R L R \stackrel{G R \eta}\longrightarrow G R$$

is the identity by the triangle identity for $L\dashv R$, but this is precisely the $G$-component of the transformation

$$
C^R \stackrel{C^R C^\eta}\longrightarrow C^R C^L C^R \stackrel{C^\epsilon C^R}\longrightarrow C^R.
$$

Similarly the $F$-component of

$$
C^{L} \stackrel{C^\eta C^L}\longrightarrow C^L C^R C^L \stackrel{C^L C^\epsilon}\longrightarrow C^L,
$$

for a functor $F\in C^A$ reads

$$ F L \stackrel{F L \eta}\longrightarrow F L R L  \stackrel{F \epsilon L}\longrightarrow F L $$

and this composition equals $Id_{F L}$ by another triangle identity for $L\dashv R$.

It is clear that if $\eta$ is iso then the composition with $\eta$ is also iso. Thus we obtain a $Q$-categories.

In other words, since the [[left adjoint]] being a [[full and faithful functor]] is equivalent to the unit of the [[adjunction]] being an [[isomorphism]], it follows from $L$ being full and faithful that $C^R$ is full and faithful.

=--

This appears as ([Kontsevich-Rosenberg, 2.7](#KontsevichRosenbergSpaces)).

+-- {: .num_note #ExtraAdjointsForPresheafQCategories}
###### Note

Assume in the context of prop. \ref{PresheafQCategories} that

* $A$ and $\bar A$ are [[small categories]];

* $C$ has all small [[limit]]s and [[colimit]]s (for instance $C$ = [[Set]]).

Then there is an [[adjoint quadruple]]

$$
  (u_! \dashv u^* \dashv u_* \dashv u^!) :
  [\bar A, C]
    \stackrel{\overset{u_! = Lan R}{\to}}{\stackrel{\overset{u^* = C^R}{\leftarrow}}{\stackrel{\overset{u_* = C^L}{\to}}{\underset{u^! = Ran L}{\leftarrow}}}}
  [A,C]
  \,,
$$

where $Lan R$ denotes the left [[Kan extension]] along $R$ and $Ran L$ the right Kan extension along $L$.

=--

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

By general properties of [[Kan extension]]s.

=--

+-- {: .num_defn #SubcategoryOfPresheafQCategories}
###### Proposition

Any subcategory $B\subset C^{\bar{A}}$ containing $Im(C^R)$ determines a Q-subcategory $C^A\leftrightarrow B$.

=--

### Domain and codomain fibration
 {#DomainAndCodomainFibration}

+-- {: .num_defn #DomainAndCodomainFibration}
###### Proposition

For any [[category]] $A$, write $\bar A := A^I$ for its [[arrow category]].
This comes equipped with the [[codomain fibration]] $cod : \bar A \to A$ and the [[domain cofibration]] $dom : \bar A \to A$. Both have a joint [[section]] $\epsilon : A \to \bar A$ by a [[full and faithful functor]] that assigns identity morphisms. These form a triple of [[adjoint functor]]s

$$
  (codom \dashv \epsilon \dashv dom) :
  A^I
    \stackrel{\overset{codom}{\to}}{\stackrel{\overset{\epsilon}{\leftarrow}}{\underset{dom}{\to}}}
  A
  \,.
$$

Taking this apart, we have that

$$
  A^{dom} : A^I \stackrel{\overset{\epsilon}{\leftarrow}}{\underset{dom}{\to}}
  A
$$

is a $Q$-category and

$$
  A^{dom} : A^I
    \stackrel{\overset{cod}{\to}}{\underset{\epsilon}{\leftarrow}}
  A
$$

is a $Q^o$-category.

=--

This appears as ([Kontsevich-Rosenberg, 2.5](#KontsevichRosenbergSpaces)).

Several classes of examples of Q-categories of interest arise as sub-Q-categories of those of this form. For instance the Q-categories of infinitesimal thickenings [below](#InfinitesimalThickening).

+-- {: .num_note #PresheavesOnCodomainFibration}
###### Note

Passing to copresheaf Q-categories as in prop. \ref{PresheafQCategories}
we have the Q-category

$$
  [A^I,Set]
   \stackrel{\overset{u^* = (-)\circ dom}{\leftarrow}}{\stackrel{\overset{u_* = (-)\circ \epsilon}{\to}}{\overset{u^! = (-)\circ codom}{\leftarrow}}}
  [A,Set]
$$

where the extra [[right adjoint]] is given by precomposition with the domain evaluation.

=--

### Q-category of cones

Let $C$ be a category and $LC$ be the category whose objects are [[cones]] $\alpha: x\to d$ over (small) diagrams $d: D\to C$ where $D$ are variable small categories; and the morphisms from $x\stackrel{\alpha}\to d$ to $x'\stackrel{\alpha'}\to d'$ are triples of the form $(f,\rho,\bar{f})$ where $f:x\to x'$ is a morphism in $C$, $\rho : D' \to D$ is a diagram (= functor), and $\bar{f}:d \circ\rho \to d'$ is a morphism of diagrams (= natural transformation) such that
$$\array{
x & \stackrel{f}\to & x'\\
\alpha \star\rho \downarrow && \downarrow\alpha'\\
d \circ \rho&\stackrel{\bar{f}}\to & d'
}$$
commutes and $\alpha \star \rho$ denotes the horizontal composition (= [[Godement product]]) of natural transformations.

Then one defines composition of morphisms by the formula
$$
(f_1, \rho_1,\bar{f_1})\circ(f_2, \rho_2,\bar{f_2})
\stackrel{def}{=}(f_1\circ f_2, \rho_2\circ\rho_1, \bar{f_1} \circ (\bar{f_2} \star \rho_1)).
$$

There is a fully faithful functor $Q_C:C\to LC$ that to any $x\in C$ assigns the trivial cone $id_x :x\to x$ and to any morphism the corresponding morphism of trivial cones. Its right adjoint is the morphism $I_C:LC\to C$ defined by sending the cone $\alpha: x\to d$ over a diagram $d:D\to C$ its vertex $x$ and to a cone morphism $(f,\rho,\bar{f})$, the morphism of vertices $f$. Then $I_C\circ Q_C = Id_C$. The identity transformation can be thus taken as the unit of the adjunction. The counit of the adjunction $\epsilon: Q_C\circ I_C \to Id_{LC}$ is constructed as follows: to a cone $\alpha:x\to d$ assign the morphism $(1_x, const, \alpha)$ where $const: D\to C$ is the constant diagram which is the unique diagram from $D = dom(d)$ to the final category $1=\{\star\}$. One can check that these data indeed define an adjoint pair $Q_C\dashv I_C$ of functors.
$Q_C: C\leftrightarrow I_C: LC$ is therefore a Q-category, and it is called the __Q-category of cones__.

If $\mathcal{L}\subset Cat$ is a family of small categories, then one considers the full subcategory $L_{\mathcal{L}}C$ of cones whose domains are in $\mathcal{L}$; the rest of the construction restricts to obtain a Q-category $Q^{\mathcal{L}}_C : C\leftrightarrow L_{\mathcal{L}}C : I_C^{\mathcal{L}}$.

The most classical case is when $\mathcal{L}$ is the (say skeletal) category $Dis$ of small discrete categories (=just identity morphisms), one obtains then the Q-category $Q^{Dis}_C: C\leftrightarrow L_{Dis}C$.
A __semicosite__ (or semicositus pl. semicositi) is a Q-category of the form $C\leftrightarrow \bar{C}$ where $\bar{C}$ is a full subcategory of $L_{Dis}C$ and the adjoint pair is obtained by the restriction. A semicosite is a __precosite__ (=Grothendieck precotopology) if

(i) $id_x\in Ob(\bar{C})$ whenever $x\in Ob(C)$.

(ii) $\{x\stackrel{\phi_i}\to x_i\}_{i\in I}\in Ob(\bar{C})$ and $\{x_i\stackrel{\phi_{ij}}\to x_{ij}\}_{j\in J_i}\in Ob(\bar{C})$ then $\{x\stackrel{\phi_{ij}\circ\phi_i}\to x_{ij}\}\in Ob(\bar{C})$

(iii) $\{x\to x_i\}_{i\in I}\in Ob(\bar{C})$ and $g\in C(x,y)$, then the family of pushouts $\{y\mapsto x_i\coprod_x y\}_{i\in I}$ exists and belongs to $Ob(\bar{C})$.

An example of a cosite is a cosite of closed sets of a topological space.

### Sieves

The $Q$-category of [[sieves]].

The $Q$-subcategory of the $Q$-category of (all) sieves corresponding to the subcategory of sieves corresponding to the Grothendieck topology...

(needs explanation)

### The Q-category factoring a fully faithful factor

Any fully faithful functor among small categories $F: A\to B$ factors canonically into the composition $A\stackrel{u^*}\to \bar{A}\hookrightarrow B$ where $\bar{A}\subset B$ is the full subcategory of $B$ whose objects are all $b$ in $Ob B$ such that $a\mapsto B(F(a),b)$ is a representable functor $A^{op}\to Set$, and $u^*$ is the corestriction of $F$ to $\bar{A}$. This corestriction makes sense: $F$ is fully faithful, hence $B(F(a),F(a)) = B(a,a)$, i.e. $F(a)\in \bar{A}$ for all $a$ in $Ob A$. For each $b\in \bar{A}$, define now $u_*(b)$ as the functor representing $B(F(-),b)$, i.e. by $\bar{A}(u^*(a),b) = B(F(a),b) \cong B(a,u_*(b))$
(KR NcSpaces A1.1.1). This relation on objects extends to an adjunction $u^*\dashv u_*$ with $u^*$ fully faithful.

### Quasi-(co)-sites

(...)

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

Let $A$ be a [[category]] and $\mathcal{T}$ a map that assigns to every [[object]] a collection of co[[sieve]]s on that object ([[subfunctor]]s of $A(x,-)$), which includes maximal cosieve (functor $A(x,-)$).

Write $\bar A_{\mathcal{T}}$ for the category whose objects are co[[sieve]]s on $A$, and whose morphisms are morphisms in $A$ that respect the corresponding cosieves. This yields a Q-category

$$
  \bar A_{\mathcal{T}}
   \stackrel{\leftarrow}{\to}
  A
  \,.
$$

Call this a **quasi-cosite** if

1. for any two cosieves in $\mathcal{T}$ their intersection is also in $\mathcal{T}$;

1. for any cosive in $\mathcal{T}$ any cosieve containing it is also in $\mathcal{T}$.

=--

This is ([KontsevichRosenberg, 2.2](#KontsevichRosenberg)).

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

For any Q-category $\mathbb{A}$, the **quasi-cosite associated** with $\mathbb{A}$ is the Q-category $\mathcal{T}\mathbb{A}$ defined by...

=--

The following is supposed to be the standard quasi-cosite for non-commutative geometry.

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

Let $k$ be a [[ring]] and $A := Alg_k$ the category of [[associative algebra]]s over $k$. Let $\bar A \subset A^I$ be the [[full subcategory]] of the [domain fibration](DomainAndCodomainFibration) whose objects are the **faithfully flat** morphisms, i.e. those morphisms $\phi : R \to T$ in $Alg_k$ such that the induced

$$
  \phi^* : R Mod \to T Mod
$$

is an [[exact functor|exact]] and [[full and faithful functor]]. This forms a Q-category.

Write $\mathcal{T}Alg_k$ for the quasi-cosite associated with this Q-category by def. \ref{QuasiCoSiteAssociatedToQCategory}. This is the **standard quasi-cosite for [[noncommutative geometry]]**.

=--

This is ([KontsevichRosenberg, A.1.9.2](#KontsevichRosenberg)).

### Infinitesimal thickenings
 {#InfinitesimalThickening}

Generally, [[infinitesimal object|infinitesimal thickenings]] are characterized by [[coreflective subcategory|coreflective embeddings]]:

A characteristic property of an [[infinitesimally thickened point]] $D$ is that for any object $X$ without infinitesimal thickening, there are in general nontrivial morphisms $D \to X$, but there is only a unique morphism $X \to D$, reflecting the fact that $D$ has only a single global point. Thus if by $i : C \to \bar C$ we denote the inclusion of objects $X$ without infinitesimal thickening into the collection of possibly infinitesimally thickened objects, and by $p : \bar C \to C$ the projection that contracts away the infinitesimal extension, we have indeed an [[adjunction]]

$$
  \bar C(i(X), D) \simeq C(X, p(D)) \simeq C(X,*) \simeq *
  \,.
$$

This general concept is described at infinitesimal neighbourhood site. See also the discussion below at [Relation to cohesive toposes](#RelationToCohesiveToposes).

The following is one realization of this general concept.

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

Let $k$ be a [[field]] and $A := CAlg_k$ be the category of commutative [[associative algebra]]s over $k$.

Let $\bar A \subset A^I$ be the [[full subcategory]] of the [[codomain fibration]] Q-category from prop. \ref{DomainAndCodomainFibration} on those morphisms of commutative algebras which are [[epimorphism]]s and whose [[kernel]] is nilpotent. Then

$$
  CAlg_k^{inf} = (\epsilon \dashv dom) : \bar A \stackrel{\overset{\epsilon}{\leftarrow}}{\underset{dom}{\to}} A
$$

is a Q-category.

The analogous statement is true with $A = Alg_k$ the category of all [[associative algebra]]s, not necessarily commutative.

=--

This appears as ([Kontsevich-Rosenberg, 2.6](#KontsevichRosenbergSpaces)).

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

Here we think of an algebra epimorphism $\phi : \mathbf{B} \to B$ with nilpotent kernel -- an [[infinitesimal ring extension]] -- as the infinitesimal thickening of $Spec B$ by $Spec ker \phi$ to $Spec \mathbf{B}$.

The functor $\epsilon$ builds the trivial (empty) infinitesimal thickenings

$$
  \epsilon B : B \stackrel{id_B}{\to} B
  \,.
$$

The functor $dom$ remembers the _thickened_ algebra

$$
  dom (\mathbf{B} \to B) = \mathbf{B}
  \,.
$$

But notice that we also have the codomain-functor, which is the functor that forgets the thickening

$$
  cod (\mathbf{B} \to B) = B
$$

and that the [[adjoint pair]] $(\epsilon \dashv dom)$ does extend to an [[adjoint triple]]

$$
  CAlg_k^{inf} : \bar A \stackrel{\overset{cod}{\to}}{\stackrel{\overset{\epsilon}{\leftarrow}}{\underset{dom}{\to}}} A
  \,.
$$

In the discussion at Infinitesimal cohesion it is the pair $(cod \dashv \epsilon)$ that appears in the axiomatization, or rather its version on the opposite categories

$$
  (\epsilon^{op} \dashv cod^{op}) : \bar A^{op} \stackrel{\overset{\epsilon^{op}}{\leftarrow}}{\underset{cod^{op}}{\to}} A^{op}
  \,,
$$

not the other adjoint pair $(\epsilon \dashv dom)$ used here.

This is the reason for the shift in adjoint triples that is mentioned in this remark over at _[[cohesive (infinity,1)-topos -- infinitesimal cohesion|infinitesimal cohesion]]_ .

=--

## $\mathbb{A}$-Sheaves
  {#ASheaves}

We discuss the notion of a _$\mathbb{A}$-sheaves_ on a Q-category $\mathbb{A}$.

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

Let $\mathbb{A} = (\bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}}A)$ be a [Q-category](#QCategory). An object $x \in A$ is called an **$\mathbb{A}$-sheaf** if for all $y \in \bar A$ the canonical morphism

$$
  \bar A(y, u^*(x)) \to A(u_*(y), x)
$$

is an [[isomorphism]] (in [[Set]], hence a [[bijection]]).

This morphism is given by

$$
  g \mapsto \eta_x^{-1} \circ u_*(g)
$$

where $\eta_x : x \to u_* u^*(x)$ is the $x$-component of the $(u^* \dashv u_*)$-[[unit of an adjunction|counit]] (which is an [[isomorphism]] because $u^*$ is a [[full and faithful functor]]).

=--

This appears as ([Kontsevich-Rosenberg, 3.1.1](#KontsevichRosenbergSpaces)).

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

Let $\mathbb{A}$ be a Q-category with an extra [[right adjoint]] $u^! : A \to \bar A$

$$
  \mathbb{A}
  :
  \bar A
    \stackrel{\overset{u^*}{\leftarrow}}{\stackrel{\overset{u_*}{\to}}{\underset{u^!}{\leftarrow}}}
  A
  \,.
$$

Then an object $x \in A$ is an $\mathbb{A}$-sheaf in the sense of def. \ref{BareSheafCondition} precisely if the canonical morphism

$$
  (u^*(x) \to u^!(x))
  :=
  u^*(X)
    \stackrel{}{\to}
  u^! u_* u^* (x)
    \stackrel{\simeq}{\to}
  u^! (x)
$$

is an [[isomorphism]] in $\bar A$.

=--

This appears as ([Kontsevich-Rosenberg, 3.1.3](#KontsevichRosenbergSpaces)).

For more details on the canonical morphism appearing here see the section Adjoint quadruples at [[cohesive topos]].

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

Using the $(u_* \dashv u^!)$-[[adjunction]]-[[isomorphism]] we have that the canonical morphism from def. \ref{BareSheafCondition} is isomorphic to

$$
  \phi : \bar A(y, u^* x) \to A(u_* y, x) \stackrel{\simeq}{\to}
   \bar A(y, u^! x)
  \,,
$$

for all $y \in \bar A$, where the second map sends every morphism to its [[adjunct]]. Using the definition of the first morphism from def. \ref{BareSheafCondition} and the expression of [[adjunct]]s (as discussed there) by composition with (co)-units, we find that the composite map here sends any morphism

$$
  g : y \to u^* x
$$

to the composite

$$
  \array{
     y
     \\
     \downarrow
     \\
     u^! u_* y &\stackrel{u^! u_* g}{\to}& u^! u_* u^* x
     \\
     && \downarrow
     \\
     && u^! x
  }
  \,.
$$

Using that [[unit of an adjunction|adjunction units]] are [[natural transformation]]s, we can complete this to a [[commuting diagram]]

$$
  \array{
     y &\stackrel{g}{\to}& u^* x
     \\
     \downarrow && \downarrow
     \\
     u^! u_* y &\stackrel{u^! u_* g}{\to}& u^! u_* u^* x
     \\
     && \downarrow
     \\
     && u^! x
  }
  \,.
$$

This shows that $\phi$ acts on any $g$ by postcomposition with the canonical morphism $u^* x \to u^! x$. By the [[Yoneda lemma]] it follows that $\phi$ is an [[isomorphism]] for all $y$ precisely if $u^* x \to u^!$ is an isomorphism.

=--

+-- {: .num_def #MonoAndEpiPresheaves}
###### Definition

Let $\mathbb{A} = (u^* \dashv u_* \dashv u^!): \bar A \to A$ be a  Q-category with an extra right adjoint as in prop. \ref{SheafConditionByExtraRightAdjoint}.

We say

* an object $x \in A$ is **$\mathbb{A}$-monopresheaf** if $u^* x \to u^! x$ is a [[monomorphism]] in $\bar A$.

* an object $x \in A$ is **$\mathbb{A}$-epipresheaf** if $u^* x \to u^! x$ is an [[strict epimorphism]] in $\bar A$.

=--

This appears as [Kontsevich-Rosenberg, 3.1.2, 3.1.4](#KontsevichRosenbergSpaces).

+-- {: .num_note #PresheafSheafCondition}
###### Note

Let $\mathbb{A} : \bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}} A$ be a Q-category where $A$ is a [[small category]], and let $C$ be a category with all small [[limit]]s. Then the Q-category of presheaves from prop \ref{PresheafQCategories} has an extra right adjoint $u^!_C := Ran_{u^*}$

$$
  C^{\mathbb{A}} :
   C^{\bar A}
    \stackrel{\overset{f^* := C^{u_*}}{\leftarrow}}{\stackrel{\underset{f_* := C^{u^*}}{\to}}{\underset{f^! := u^!_C}{\leftarrow}}}
   C^{A}
$$

given by the right [[Kan extension]] along $u^*$, which exists by the assumption that $C$ has all small limits.

Therefore by prop. \ref{SheafConditionByExtraRightAdjoint} a presheaf $F \in C^{A}$ is a $C^{\mathbb{A}}$-sheaf, def. \ref{BareSheafCondition}, precisely if the canonical morphism

$$
  f^* F \to f^! F
$$

is an [[isomorphism]].

=--

This appears as ([Kontsevich-Rosenberg, 3.5](#KontsevichRosenbergSpaces)).

## Examples

### Formal smoothness and $Alg_k^{inf}$-sheaves
 {#FormalSmoothness}

Let

$$
  CAlg_k^{inf} : \bar A \stackrel{\overset{i}{\leftarrow}}{\underset{p}{\to}}
   CAlg_k
$$

be the Q-category of infinitesimal thickenings from prop. \ref{InfinitesimalThickening}. Write

$$
  [CAlg_k^{inf},Set] : [\bar A,Set]
    \stackrel{\overset{u^*}{\leftarrow}}{\stackrel{\overset{u_*}{\to}}{\underset{u^!}{\leftarrow}}}
  [CAlg_k,Set]
$$

be the corresponding Q-category of copresheaves from prop. \ref{PresheafQCategories}. Notice that $[CAlg_k, Set]$ is the [[presheaf topos]] that contains [[scheme]]s over $k$. Recall the notion of
[[formally smooth scheme]], [[formally étale morphism]] and
[[formally unramified morphism]] of schemes.

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

An object $X \in [CAlg_k, Set]$ is

* [[formally étale morphism|formally étale]] precisely if it is an $CAlg_k^{inf}$-sheaf in the sense of def. \ref{BareSheafCondition}, hence precisely if the canonical morphism

  $$
    u^* F \to u^! F
  $$

  from prop. \ref{SheafConditionByExtraRightAdjoint} is an [[isomorphism]];

* [[formally unramified]] precisely if it is a $CAlg_k^{inf}$ monopresheaf, def. \ref{MonoAndEpiPresheaves}, hence precisly if $u^* F \to u^! F$ is a [[monomorphism]];

* [[formally smooth]] precisely if it is a strict epipresheaf, def. \ref{MonoAndEpiPresheaves}, hence precisely if $u^* F \to u^! F$ is a [[strict epimorphism]].

=--

This appears as ([Kontsevich-Rosenberg, 4.1](#KontsevichRosenbergSpaces)).

This [[category theory|category theoretic]] reformulation of these three properties therefore admits straightforward generalization of these notions to other contexts. See the section Infinitesimal paths at [[cohesive (∞,1)-topos]].

For instance we have the following direct generalization is of interest in [[noncommutative geometry]].

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

Let $Alg_k$ be the full category of [[associative algebra]]s over $k$, not necessarily commutative. Write $Alk_k^{inf} : \bar A \to Alg_k$ for the Q-category of infinitesimal thickenings as in def. \ref{InfinitesimalThickening}. Notice that $[Alg_k, Set]$ is the [[presheaf topos]] that contains [[noncommutative scheme]]s.

We then say an object $X \in [Alg_k, Set]$ is

* [[formally étale morphism|formally étale]] precisely if it is an $Alg_k^{inf}$-sheaf in the sense of def. \ref{BareSheafCondition}, hence precisely if the canonical morphism

  $$
    u^* F \to u^! F
  $$

  from prop. \ref{SheafConditionByExtraRightAdjoint} is an [[isomorphism]];

* [[unramified|formally unramified]] precisely if it is a $Alg_k^{inf}$ monopresheaf, hence precisly if $u^* F \to u^! F$ is a [[monomorphism]];

* [[formally smooth]] precisely if $u^* F \to u^! F$ is a [[strict epimorphism]].

=--

This appears as ([Kontsevich-Rosenberg, section 4.2](#KontsevichRosenbergSpaces)).

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

Let $R \in Alg_k$ and write $Spec R \in [Alg_k, Set]$ for the corresponding [[representable functor]]. We have that

1. $Spec R$ is an $Alg_k^{inf}$ epipresheaf (formally smooth) precisely if $R$ is Quillen-Cuntz quasi-free: the $R \otimes_k R^{op}$-[[module]] $\Omega^1_{R|k}$, being the [[kernel]] of the multiplication morphism

   $$
     \Omega^1_{R|k} := ker(R \otimes_k R \stackrel{mult}{\to} R)
     \,,
   $$

   is a [[projective object|projective]] in $R \otimes R^{op}$[[Mod]];

1. $Spec R$ is an $Alg_k^{inf}$-monopresheaf (formally unramified) precisely if $\Omega^1_{R|k} = 0$.

=--

This appears as ([Kontsevich-Rosenberg, prop. 4.3](#KontsevichRosenbergSpaces)).

## Relation to other concepts

### Relation to cohesive toposes
  {#RelationToCohesiveToposes}

If a Q-category $\mathbb{A}$ has the extra [[right adjoint]] of prop. \ref{SheafConditionByExtraRightAdjoint} and in addition an extra left adjoint to a total of a quadruple of [[adjoint functor]]s

$$
  \mathbb{A} :
   \bar A
     \stackrel{\overset{u_!}{\to}}{\stackrel{\overset{u^*}{\leftarrow}}{\stackrel{\overset{u_*}{\rightarrow}}{\underset{u^!}{\leftarrow}}}}
   A
$$

then essential axioms characterizing a [[cohesive topos]] are satisfied, in particular if for instance $\bar A$ and $A$ are [[presheaf topos]]es as in \ref{PresheafQCategories} (this is considered around [Kontsevich-Rosenberg, 3.5.1](#KontsevichRosenbergSpaces)).

Notably in this case the canonical [[natural transformation]]

$$
  u^* \to u^!
$$

from prop. \ref{SheafConditionByExtraRightAdjoint} is the one appears in the axioms of a [[cohesive topos]]: if this transformation is a [[monomorphism]] in a cohesive topos -- hence if in the language of Q-categories all objects are monopresheaves -- one says that _discrete objects are concrete_ in the cohesive topos.

Moreover, due to the extra left adjoint $u_!$ there is a canonical dual morphism

$$
  u_* \to u_!
  \,.
$$

In ([Lawvere](#Lawvere)) is the suggestion that it is interesting to consider the full subcategory of $\bar A$ on which $u_* \to u_!$ is an isomorphism. This is dual to the statement of the above section on [A-Sheaves](#ASheaves) which asserts that it is interesting to consider the full subcategory of $A$ on which $u^* \to u^!$ is an isomorphism.

More concretely, the axioms of Infinitesimal cohesion are an abstraction of the situation of prop. \ref{InfinitesimalThickening}. In every cohesive $(\infty,1)$-topos equipped with an _infinitesimal neighbourhood_ there is an analog of the characterization of formal smoothness from prop. \ref{FormalSmoothnessByASheaves}. See the section Infinitesimal paths and de Rham spaces.

### Sheafification versus the Gabriel localization $G_{\mathcal{F}} = H^2_{\mathcal{F}}$

(...)

## References

The term _$Q$-category_ originally was short for _almost quotient category_ , presumably because of its role in the theory of [[Gabriel localization]]. The sheaf formalism in terms of $Q$-categories has been introduced in the mimeographed notes

* [[Alexander Rosenberg]], _Q-categories, sheaves and localization_, (in Russian) Seminar on supermanifolds __25__, Leites ed. Stockholms Universitet 1988 (there is a recent partial English translation)
 {#Rosenberg}

The formalism is used (and briefly surveyed) in

* [[Maxim Kontsevich]], [[Alexander Rosenberg]], _Noncommutative spaces_, preprint MPI-2004-35 ([[KontsevichRosenbergNCSpaces.pdf:file]], [ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=2331), [dvi](http://www.mpim-bonn.mpg.de/preprints/send?bid=2303))
 {#KontsevichRosenbergSpaces}

and also used in the general definition of "noncommutative" stacks in

* [[Maxim Kontsevich]], [[Alexander Rosenberg]], _Noncommutative stacks_, preprint MPI-2004-37 ([ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=2333), [dvi](http://www.mpim-bonn.mpg.de/preprints/send?bid=2305))
 {#KontsevichRosenbergStacks}

The [[epipresheaf]] condition for the Q-category of nilpotent (infinitesimal) thickenings is in the Kontsevich-Rosenberg paper interpreted as [[formally smooth morphism|formal smoothness]] what is further studied in

* [[T. Brzezi?ski]], _Notes on formal smoothness_, *in*: Modules and Comodules (series _Trends in Mathematics_). T Brzeziński, JL Gomez Pardo, I Shestakov, PF Smith (eds), Birkhäuser, Basel, 2008, pp. 113-124 ([doi](http://dx.doi.org/10.1007/978-3-7643-8742-6), [arXiv:0710.5527](http://arxiv.org/abs/0710.5527))
 {#Brzezinski}

The condition that $u_* x \to u_! x$ is an isomorpophism , dual to the condition for $\mathbb{A}$-sheaves considered above, has been considered in

* [[Bill Lawvere]], _Axiomatic cohesion_ Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. ([pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf))
{#Lawvere}

[[!redirects Q-categories]]


Q/Z

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

#Contents#
* table of contents
{:toc}

## Idea

The [[quotient group]] $\mathbb{Q}/\mathbb{Z}$ of the additive [[abelian group]] of [[rational numbers]] by its [[subgroup]] of [[integers]].

It is also the [[coequalizer]] of the [[identity function]] of the rational numbers and the function $x \mapsto x + 1$ on the rational numbers:

$$
\mathbb{Q}\underoverset{\quad x \mapsto x + 1 \quad}{\mathrm{id}_{\mathbb{Q}}}{\rightrightarrows}\mathbb{Q} \to \mathbb{Q}/\mathbb{Z}
$$

## Properties

### Relation to cyclic groups

For each [[natural number]] $n \in \mathbb{N}$ the [[cyclic group]] $\mathbb{Z}/n\mathbb{Z}$ is a [[subgroup]]

$$
  \array{
    \mathbb{Z}/n\mathbb{Z}
    &\overset{\;\;\;\;\;\;}{\hookrightarrow}&
    \mathbb{Q}/\mathbb{Z}
    \\
    [k]
    &\mapsto&
    [k/n]
  }
$$

In fact $\mathbb{Q}/\mathbb{Z}$ is the [[filtered colimit]] of a [[diagram]] of cyclic groups and [[inclusions]] between them. In more detail, whenever $m|n$, multiplication by $n/m$ induces an inclusion

$$\mathbb{Z}/m\mathbb{Z} \hookrightarrow \mathbb{Z}/n\mathbb{Z}$$

and this gives a [[functor]] to [[abelian groups]] from the [[lattice]] of [[natural numbers]] ordered by divisibility. The [[colimit]] of this functor is the [[abelian group]] $\mathbb{Q}/\mathbb{Z}$.

### Relation to Prüfer groups

For each [[prime number|prime]] $p$ the $p$-[[Prüfer group]] $\mathbb{Z}[1/p]/\mathbb{Z}$, similarly formed as the colimit of groups $\mathbb{Z}/p^k\mathbb{Z}$ and inclusions between them, embeds in $\mathbb{Q}/\mathbb{Z}$. Furthermore, it follows from the [[Chinese remainder theorem]] that the induced map

$$\bigoplus_p \mathbb{Z}[1/p]/\mathbb{Z} \to \mathbb{Q}/\mathbb{Z}$$

is an [[isomorphism]].

### Relation to the circle group

We have a canonical [[subgroup]] inclusion into the [[circle group]]: If the latter is identified as the canonical subgroup $U(1) \;\subset\; \mathbb{C}^\times $ in the [[group of units]] of the [[complex numbers]], this inclusion is given by

$$
  \array{
    \mathbb{Q}/\mathbb{Z}
    &\overset{\;\;\;\;\;\;}{\hookrightarrow}&
    U(1)
    \\
    [q] &\mapsto& \exp\big( 2\pi\mathrm{i} q\big)
    \,.
  }
$$

From this point of view, $\mathbb{Q}/\mathbb{Z}$ is the [[torsion subgroup]] of [[U(1)]], whose elements are precisely the [[roots of unity]].

### Further properties

* The group $\mathbb{Q}/\mathbb{Z}$ is an [[injective object]] in the [[category]] [[Ab]] of [[abelian groups]].

* It is also a [[cogenerator]] in the category of abelian groups.

  Proof: let $A$ be an abelian group. It suffices to check that for every $a \in A$ there exists $f: A \to \mathbb{Q}/\mathbb{Z}$ such that $f(a) \neq 0$. But if $\langle a \rangle$ is the cyclic subgroup generated by $a$, then it is easy to find a map $g: \langle a \rangle \to \mathbb{Q}/\mathbb{Z}$ such that $g(a) \neq 0$, and then we can extend $g$ to a map $f: A \to \mathbb{Q}/\mathbb{Z}$ using injectivity of $\mathbb{Q}/\mathbb{Z}$.

This means every abelian group embeds into an [[injective object|injective]] abelian group,

$$A \to \prod_{f: A \to \mathbb{Q}/\mathbb{Z}} \mathbb{Q}/\mathbb{Z},$$

and into an [[algebraic dual|algebraic double dual]], $A \hookrightarrow \hom(\hom(A, \mathbb{Q}/\mathbb{Z}), \mathbb{Q}/\mathbb{Z})$. The algebraic double dual of $\mathbb{Z}$ is its [[profinite completion]]

$$\prod_{p} \mathbb{Z}_p$$

where $\mathbb{Z}_p$ is the group of $p$-[[p-adic number|adic integers]] (this is connected with the direct sum decomposition of $\mathbb{Q}/\mathbb{Z}$ into [[Prüfer group|Prüfer]] $p$-groups).

* For any [[ring]] $R$, its [[algebraic dual]] $\hom(R, \mathbb{Q}/\mathbb{Z})$, equipped with the left $R$-module structure defined by $r \cdot f: s \mapsto f(s r)$, is an injective cogenerator in the category of left $R$-modules. This follows easily from the corresponding property for $\mathbb{Q}/\mathbb{Z}$.

## Related concepts

* [[cyclic group]], [[multiplicative group of integers modulo n]]

* [[Prüfer group]], [[root of unity]]

* [[Brown-Comenetz duality]]

* [[e-invariant]]


qbit

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Computation
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum information theory]] and [[quantum computing]], by a *q-bit* (or *qubit*) one means a [[quantum state]] in a 2-[[dimension of a vector space|dimensional]] [[complex vector space|complex]] [[Hilbert space]] [[space of quantum states|of states]].

Hence the quantum [[data type]] $QBit$ is the 2-[[dimension of a vector space|dimensional]] [[complex vector space]] equipped with its canonical [[quantum measurement]]-[[linear basis|basis]]

$$
  \mathbb{C}^2
  \,\simeq\,
  \mathbb{C} \cdot \vert 0 \rangle \oplus \mathbb{C} \cdot \vert 1 \rangle
  \,.
$$

Analogous higher- but still finite- $d$-dimensional quantum data (types) are called *[[qudit|qdits]]* ("qtrits" for $d = 3$).

## Properties

### In terms of geometric quantization

In [[geometric quantization]] qbits are naturally understood as the states given by the [[geometric quantization of the 2-sphere]] for [[prequantum line bundle]] (plus [[metaplectic correction]]) being of unit [[first Chern class]]. See at _[geometric quantization of the 2-sphere -- The space of quantum states](geometric+quantization+of+the+2-sphere#TheSpaceOfQuantumStates)_.

## Related concepts

* [[data type]]

* [[spin resonance qbit]]

* [[one clean qbit]]

* [[Pauli gate]]

* [[bit]], [[truth value]]

* [[bit flip channel]], [[bit flip code]]

* [[quantum logic gate]], [[quantum circuit]]

* [[quantum logic]], [[quantum computing]], [[quantum error correction]]

* [[dimer]]

* [[quantum set]]

## References

### General

The term *q-bit* goes back to

* [[Benjamin Schumacher]], *Quantum coding*, Phys. Rev. A **51** (1995) 2738 $[$[doi:10.1103/PhysRevA.51.2738](https://doi.org/10.1103/PhysRevA.51.2738)$]$

and was popularized by early adoption such as in

* {#Shor95} [[Peter W. Shor]], *Scheme for reducing decoherence in quantum computer memory*, Phys. Rev. A 52, R2493(R) 1995 ([doi:10.1103/PhysRevA.52.R2493](https://doi.org/10.1103/PhysRevA.52.R2493))

Textbook account:

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], §1.2 in: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

See also:

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

Laboratoy-realizations of qbits for use in [[quantum computers]]:

* *[Qubit Zoo](https://qubitzoo.org/qubitzoo/)*

[[!include spin resonance qbits -- references]]

[[!include superconducting qbits -- references]]

[[!redirects Qbit]]
[[!redirects qbits]]
[[!redirects Qbits]]
[[!redirects quantum bit]]
[[!redirects quantum bits]]

[[!redirects QBit]]

[[!redirects qubit]]
[[!redirects qubits]]

[[!redirects q-bit]]
[[!redirects q-bits]]


QCD

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Quantum chromodynamics_ ("QCD") is the [[quantum field theory]] of [[Yang-Mills theory]]: it describes the [[quantum theory]] of  [[gluons]] and [[quarks]].

The corresponding [[effective field theory]] that describes [[confinement|confined]] [[bound states]] such as [[protons]] is [[quantum hadrodynamics]].

## Properties

### Confinements

See at _[[confinement]]_.

### Asymptotic freedom

See at _[[asymptotic freedom]]_.

### Trace anomaly

See at _[[QCD trace anomaly]]_.

### Phase diagram
 {#PhaseDiagram}

[[QCD]] has an intricate [[phase diagram]] (e.g. [Shuryak 96](#Shuryak96) [Hands 01](#Hands01), [Schaefer 05](#Schaefer05)), phases including

* [[quark-gluon plasma]],

* [[colour superconductivity]],

* ...

## Related concepts

* [[D=2 QCD]]

* [[lattice QCD]]

* [[super QCD]]

* [[confinement]]

  * [[chiral perturbation theory]], [[quantum hadrodynamics]]

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

* [[AdS/QCD correspondence]]

* [[Gross-Neveu model]]

* [[QCD cosmology]]

* [[quantum field theory]]

  * [[gauge theory]]

    * [[Yang-Mills theory]]

      * [[quantization of Yang-Mills theory]]

    * [[QED]]

    * **QCD**

      * [[quark]], [[gluon]]

      * [[bound states]]:

        [[glueballs]], [[hadrons]] ([[mesons]], [[baryons]])

      * [[colour charge]], [[flavour (particle physics)]]

      * [[instanton in QCD]], [[caloron]], [[skyrmion]]

      * [[confinement]]

      * [[mass gap]]

      * [[quark-gluon plasma]]

      * [[colour superconductivity]]

    * [[electroweak field]]

* [[string theory results applied elsewhere]]

## References

### General

Introduction and review:

* [[Jiri Chyla]], _Quarks, partons and Quantum Chromodynamics_, ([spire:674163/)](http://inspirehep.net/record/674163/), [[ChylaQuarksQCD.pdf:file]])

* Peter Skands, _Introduction to QCD_ ([arXiv:1207.2389](https://arxiv.org/abs/1207.2389), [doi:10.1142/9789814525220_0008](https://doi.org/10.1142/9789814525220_0008))

* Y. Kurihara, _QCD at LHC for beginners_ Lesson 1 ([pdf](http://www.iop.vast.ac.vn/theor/conferences/vsop/18/files/QCD_Kurihara_1.pdf)), Lesson 2 ([pdf](http://www.iop.vast.ac.vn/theor/conferences/vsop/18/files/QCD_Kurihara_2.pdf)) Lesson 3 ([pdf](http://www.iop.vast.ac.vn/theor/conferences/vsop/18/files/QCD_Kurihara_3.pdf))

* [[Particle Data Group]], _Quantum Chromodynamics_ ([pdf](http://pdg.lbl.gov/2015/reviews/rpp2015-rev-qcd.pdf))

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

Textbooks:

* [[Robert E. Marshak]], Chapter 5 of: *Conceptual Foundations of Modern Particle Physics*, World Scientific 1993 ([doi:10.1142/1767](https://doi.org/10.1142/1767))

* B. L. Ioffe , V. S. Fadin, L. N. Lipatov, _Quantum Chromodynamics -- Perturbative and Nonperturbative Aspects_, Cambridge University Press (2010) ([doi:10.1017/CBO9780511711817](https://doi.org/10.1017/CBO9780511711817))

with emphasis on [[phenomenology]]:

* {#HalzenMartin84} [[Francis Halzen]], [[Alan Martin]], _Quarks and Leptons: An Introductory Course in Modern Particle Physics_, Wiley 1984 ([pdf](http://ajbell.web.cern.ch/ajbell/Documents/eBooks/Quarks%20&%20Leptons.pdf))

On QCD's [[running coupling constant]]:

* [[Alexandre Deur]], [[Stanley Brodsky]], [[Guy de Teramond]], *The QCD Running Coupling*, Prog. Part. Nuc. Phys. 90 1 (2016) ([arXiv:1604.08082](https://arxiv.org/abs/1604.08082), [doi:10.1016/j.ppnp.2016.04.003](https://doi.org/10.1016/j.ppnp.2016.04.003))

Rigorous construction as a [[perturbative quantum field theory]] via [[causal perturbation theory]] is discussed in

* {#Scharf01} [[Günter Scharf]], chapter 3 of _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001

Survey with emphasis on [[non-perturbative effects]]:

* {#Brambilla14} Brambilla et al., _[[QCD and strongly coupled gauge theories - challenges and perspectives]]_, Eur Phys J C Part Fields. 2014; 74(10): 2981 ([doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5))

* Giorgio Comitini, *Perturbative methods in non-perturbative Quantum Chromodynamics* [[arXiv:2306.13624](https://arxiv.org/abs/2306.13624)]

Discussion of on-shell methods in QCD [[perturbation theory]] includes

* [[Zvi Bern]], [[Lance Dixon]], [[David Kosower]], _On-Shell Methods in Perturbative QCD_ ([arXiv:0704.2798](http://arxiv.org/abs/0704.2798))

See also

* Bo-Lun Du, Xing-Gang Wu, Jian-Ming Shen, Stanley J. Brodsky, _Extending the Predictive Power of Perturbative QCD_ ([arXiv:1807.11144](https://arxiv.org/abs/1807.11144))

* Christian Drischler, Wick Haxton, Kenneth McElvain, Emanuele Mereghetti, Amy Nicholson, Pavlos Vranas, André Walker-Loud, _Towards grounding nuclear physics in QCD_ ([arxiv:1910.07961](https://arxiv.org/abs/1910.07961))

As [[thermal field theory]]:

* Jacopo Ghiglieri, Aleksi Kurkela, Michael Strickland, Aleksi Vuorinen, _Perturbative Thermal QCD: Formalism and Applications_ ([arXiv:2002.10188](https://arxiv.org/abs/2002.10188))

### Phase diagram
 {#Phases}

On the [[phase diagram]] of [[QCD]]:

* {#Shuryak96} [[Edward Shuryak]], _The Phases of QCD_ ([arXiv:hep-ph/9609249](https://arxiv.org/abs/hep-ph/9609249))

* {#Hands01} Simon Hands, _The Phase Diagram of QCD_, Contemp. Phys. 42:209-225, 2001 ([arXiv:physics/0105022](https://arxiv.org/abs/physics/0105022))

* {#Schaefer05} [[Thomas Schaefer]], _Phases of QCD_ ([arXiv:hep-ph/0509068](https://arxiv.org/abs/hep-ph/0509068))

Via [[lattice QCD]]:

* Jana N. Guenther, _Overview of the QCD phase diagram -- Recent progress from the lattice_ ([arXiv:2010.15503](https://arxiv.org/abs/2010.15503))

### History

On the early history of the development of quantum chromodynamics (the [[quark]]-model):

* [[George Zweig]], _Origins of the quark model_, 1980 ([pdf](https://authors.library.caltech.edu/18969/1/Origins_of_the_Quark_Model_Final_Zweig%5B1%5D.pdf), [[ZweigQuarkModel1980.pdf:file]])

* [[Mikhail Shifman]], *OPE-based Methods in Nonperturbative QCD*, Eur. Phys. J. C (2022) [[arXiv:2208.10600](https://arxiv.org/abs/2208.10600)]

### Lattice QCD

Due to _[[confinement]]_, the fundamental degrees of freedom in terms of which QCD is formulated, namely the [[quarks]], are actually not the low-energy [[bound states]] of the theory, which instead are the [[hadrons]]. This leaves room to speculate that QCD is not really the fundamental theory of the [[strong nuclear force]].

However, brute-force computation in [[lattice QCD]] shows that the [[quark]]-model does reproduce these [[hadron]] [[bound states]] _somehow_ (even if the real understanding of how it does so remains open, this is the _[[mass gap problem]]_):

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

  _Ab-initio Determination of Light Hadron Masses_,

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

  conclusion on p. 4:

> our study strongly suggests that [[QCD]] is the theory of the [[strong nuclear force|strong interaction]], at low energies as well

* {#FodorHoelbling12} Zoltan Fodor, Christian Hoelbling, _Light Hadron Masses from Lattice QCD_, Rev. Mod. Phys. 84, 449, ([arXiv:1203.4789](https://arxiv.org/abs/1203.4789))

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

On [[string diagram]]-calculus ([Cvitanović 2008](string+diagram#Cvitanovic08): "bird tracks") for the [[representation theory]] involved in [[quantum chromodynamics]]

* [[Stéphane Peigné]], *Introduction to color in QCD: Initiation to the birdtrack pictorial technique* [[arXiv:2302.07574](https://arxiv.org/abs/2302.07574)]

### Light-cone quantization

Non-perturbative QCD and [[hadron]] bound states by [[light-cone quantization]] (see there for more):

* [[G. Peter Lepage]], [[Stanley J. Brodsky]], *Exclusive processes in perturbative quantum chromodynamics*, Phys. Rev. D **22** (1980) 2157 [[doi:10.1103/PhysRevD.22.2157](https://doi.org/10.1103/PhysRevD.22.2157)]

* {#PauliBrodsky85b} [[Hans-Christian Pauli]], [[Stanley J. Brodsky]], *Discretized light-cone quantization: Solution to a field theory in one space and one time dimension*, Phys. Rev. D **32** (1985) 2001 [[doi:10.1103/PhysRevD.32.2001](https://doi.org/10.1103/PhysRevD.32.2001)]

* [[Stanley J. Brodsky]], Gary McCartor, [[Hans-Christian Pauli]], [[Stephen S. Pinsky]], *The challenge of light-cone quantization of gauge field theory*, Particle World **3** 3 (1993) 109-124  [[cds:240388](http://cds.cern.ch/record/240388), [inspire:335247](https://inspirehep.net/literature/335247)]

* {#BrodskyPauliPinsky98} [[Stanley Brodsky]], [[Hans-Christian Pauli]], [[Stephen S. Pinsky]], *Quantum Chromodynamics and Other Field Theories on the Light Cone*, Phys. Rept. **301** (1998) 299-486 [[arXiv:hep-ph/9705477](https://arxiv.org/abs/hep-ph/9705477), doi:10.1016/S0370-1573(97)00089-6]

* [[Stanley Brodsky]], [[Thomas Heinzl]] et al. *Light-Front Quantum Chromodynamics: A framework for the analysis of hadron physics*, White Paper of International Light Cone Advisory Committee, Nuclear Physics B - Proceedings Supplements **251**, **252** (2014) 165-174 [[doi:10.1016/j.nuclphysbps.2014.05.004](https://doi.org/10.1016/j.nuclphysbps.2014.05.004), [arXiv:1309.6333](https://arxiv.org/abs/1309.6333)]

[[!redirects quantum chromodynamics]]


This page collects material related to

* {#Brambilla14} Brambilla et al.,

  **QCD and strongly coupled gauge theories - challenges and perspectives**

  Eur Phys J C Part Fields. 2014; 74(10): 2981

  [arXiv:1404.3723](https://arxiv.org/abs/1404.3723)

  [doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5)

on open problems, [[theory (physics)|theoretically]] and [[experiment|experimentally]], in [[quantum chromodynamics]] ([[QCD]]), particularly those related to [[non-perturbative effects]] and hence notably to [[confinement]].

#Contents#
* table of contents
{:toc}

## Overview

(...)

## The nature of QCD

>  The success of the technique does not remove the challenge of understanding the non-perturbative aspects of the theory. The two aspects are
deeply intertwined. The Lagrangian of QCD is written in terms of quark and gluon degrees of freedom which become apparent at large energy but remain hidden inside hadrons in the low-energy regime. This confinement property is related to the increase of $\alpha_s$ at low energy, but it has never been
demonstrated analytically.

> We have clear indications of the confinement of quarks into hadrons from both experiments and lattice QCD. Computations of the heavy quark–antiquark potential, for example, display a linear behavior in the quark–antiquark distance, which cannot be obtained in pure perturbation theory. Indeed the two main characteristics of QCD: confinement and the appearance of nearly massless pseudoscalar mesons, emergent from the spontaneous breaking of chiral symmetry, are non-perturbative phenomena whose precise understanding continues to be a target of research.

> Even in the simpler case of gluodynamics in the absence of quarks, we do not have a precise understanding of how a gap in the spectrum is formed and the glueball spectrum is generated.

## Light quarks

### Introduction

### Hadron structure

* [[light quarks]]

* [[chiral perturbation theory]]

* [[form factors]]

* [[pion]]

### Hadron spectroscopy

* [[lattice gauge theory]]

### Chiral dynamics

* [[chiral perturbation theory]]

### Low-energy porecision observables

### Future directions

## Heavy quarks

* [[heavy quarks]]

* [[semileptonic decays]]

* [[quarkonium]]

## Searching for new physics with precision measurement

* [[precision measurement]]

* [[flavour anomalies]]

## Deconfinement

* [[quark-gluon plasma]]

* [[holographic QCD]]

## Nuclear physics and dense QCD in colliders and compact stars

* [[neutron stars]]

## Vacuum structure and infrared QCD

* [[confinement]]

* [[chiral symmetry breaking]]

## Strongly coupled theories and conformal symmetry

* [[integrability]]

* [[supersymmetry]]

* [[super Yang-Mills theory]]

* [[electroweak symmetry breaking]]

* [[AdS/QCD correspondence]]

category: reference


QCD cosmology

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

#Contents#
* table of contents
{:toc}

## Idea

The evolution of the early [[observable universe]] must have been crucially affected by the properties (e.g. [[equation of state]]) of the primordial [[quark-gluon plasma]], and then, as the [[temperature]] decreased, by its [[confinement]] [[phase transition]] during [[nucleosynthesis]] to a [[hadron]] [[gas]]. The study of the properties of [[QCD]] relevant for [[cosmology]] this way is called _QCD cosmology_.

The crucial effects in QCD cosmology are thus controlled by [[quantum chromodynamics]] ([[QCD]]) and specifically in its [[non-perturbative effect|non-perturbative]] [[confinement|confined]] [[state of matter|phase]], hence are subject to the [confinement problem](confinement#OpenProblem).

## Related concepts

* [[structure formation]]

## References

### General

* Paolo Castorina, Vincenzo Greco, Salvatore Plumari, _QCD Equation of State and Cosmological Parameters in Early Universe_, Phys. Rev. D 92, 063530 (2015) ([arXiv:1505.07655](https://arxiv.org/abs/1505.07655))

### Via lattice QCD

Application of [[lattice QCD]] to [[QCD cosmology]]:

* Michael McGuigan, Wolfgang Söldner, _QCD Cosmology from the Lattice Equation of State_ ([arXiv:0810.0265](https://arxiv.org/abs/0810.0265))

* Sz. Borsanyi et al. _Lattice QCD for Cosmology_ ([arXiv:1606.07494](https://arxiv.org/abs/1606.07494))


QCD trace anomaly

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#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Algebraic Qunantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

While [[quantum chromodynamics]] (QCD) expressed via its [[Lagrangian density]] for [[quark]]-[[field (physics)|physics]] looks [[classical field theory|clasically]] like a [[conformal field theory]], a [[quantum anomaly]] in the [[trace]] of its [[energy momentum tensor]] ([[conformal anomaly]]) breaks [[conformal invariance]] as a  [[quantum field theory]]. This is called the _QCD trace anomaly_.

This is relevant in particular for computation of the [[equation of state]] of the [[quark-gluon plasma]].

## References

### General

Original articles:

* R. J. Crewther, _Nonperturbative  evaluation of the anomalies in low-energy  theorems,_  Phys. Rev. Lett. 281421 (1972) ([doi:10.1103/PhysRevLett.28.1421](https://doi.org/10.1103/PhysRevLett.28.1421))

* M. S. Chanowitz, [[John Ellis]], _Canonical Anomalies and Broken Scale Invariance_, Phys. Lett. 40B397 (1972) (doi:10.1016/0370-2693(72)90829-5)

* M. S. Chanowitz, [[John Ellis]], _Canonical Trace Anomalies_, Phys. Rev.D72490 (1973) ([doi:10.1103/PhysRevD.7.2490](https://doi.org/10.1103/PhysRevD.7.2490))

* J. C. Collins, A. Duncan, S. D. Joglekar, _Trace and Dilatation Anomalies in Gauge Theories_, Phys. Rev.D16438 (1977) ([doi:10.1103/PhysRevD.16.438](https://doi.org/10.1103/PhysRevD.16.438))

Further discussion:

* Jens O. Andersen, Lars E. Leganger, Michael Strickland, _The QCD trace anomaly_, Phys.Rev. D84 (2011) 087703 ([arXiv:1106.0514](https://arxiv.org/abs/1106.0514))

### In geometric engineering of QCD on branes
 {#ReferencesHolographicQCD}

Discussion via [[geometric engineering of QFTs|geometric engineering]] of [[QCD]] on [[intersecting branes]] ("[[holographic QCD]]"):

* Jose L. Goity, Roberto C. Trinchero, _Holographic models and the QCD trace anomaly_, Phys. Rev. D 86, 034033 – 2012 ([arXiv:1204.6327](https://arxiv.org/abs/1204.6327))

* Aalok Misra, Charles Gale, _The QCD Trace Anomaly at Strong Coupling from M-Theory_ ([arXiv:1909.04062](https://arxiv.org/abs/1909.04062))

The [[QCD trace anomaly]] affects notably the [[equation of state]] of the [[quark-gluon plasma]], see there at _[References -- Holographic description of quark-gluon plasma](quark-gluon+plasma#ReferencesViaAdSCFT)_

[[!redirects trace anomaly in quantum chromodynamics]]
[[!redirects trace anomaly of quantum chromodynamics]]

[[!redirects conformal anomaly in quantum chromodynamics]]
[[!redirects conformal anomaly of quantum chromodynamics]]


QFT on non-commutative spacetime

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#### Quantum field theory
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=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The idea of generalizing [[quantum field theory]] from ordinary [[spacetimes]] to spacetimes in [[noncommutative geometry]].

## References

A review is in

* [[Richard Szabo]], _Quantum Field Theory on Noncommutative Spaces_ ([arXiv:hep-th/0109162](http://arxiv.org/abs/hep-th/0109162))

[[!redirects QFT on noncommutative spacetime]]


QFT with defects

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=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

A ([[prequantum field theory|pre]]-)[[quantum field theory|quantum]] [[field theory]] _with defects_ is, roughly a [[field theory]] that assigns data not just to plain [[manifolds]]/[[cobordisms]], but to spaces that may carry certain singularities and/or colorings. At the locus of such a singularity the [[bulk]] field theory may then undergo transitions.

Such defects are known by many names. In [[codimension]] 1 they are often called [[domain walls]]. If they are [[boundaries]] they are often called _[[branes]]_, the corresponding domain walls are then sometimes called [[bi-branes]]. Examples of Dimension-1 defects are [[Wilson lines]] and [[cosmic strings]] (at least in [[gauge theory]]) and dimension-0 defects are often called [[monopoles]].

## Definition

### General
 {#DefinitionGeneral}

A plain $n$-dimensional local [[FQFT]] (a [[bulk]] field theory) is a [[monoidal (∞,n)-functor|symmetric monoidal]] [[(∞,n)-functor]] from the [[(∞,n)-category of cobordisms]].

$$
  Z : Bord_n \to \mathcal{C}^\otimes
  \,.
$$

If one replaces plain [[cobordisms]] here with cobordisms $Bord_n^{Def}$ "with singularities" including [[boundaries]] and [[corners]] but also partitions labeled in a certain index set $Def$, one calls a morphism

$$
  Z : Bord_n^{Def} \to \mathcal{C}
$$

a TQFT _with defects_. A general formalization is in ([Lurie 09, section 4.3](#Lurie09)), see at _[Cobordism theorem -- For cobordisms with singuarities (boundaries/branes and defects/domain walls)](cobordism+hypothesis#ForCobordismsWithSingularities)_. See also ([Davydov, Runkel & Kong 2011](#DavydovRunkelKong11) and [Carqueville-Runkel-Schaumann](#CarquevilleRunkelSchaumann)).

Such a morphism carries data as follows:

* for each label in $Def$ of codimension 0 there is an ordinary [[bulk]] field theory;

* for each label in $Def$ of codimension 1 data on how to "connect" the two TQFTs on both sides

* etc.

So one may think of the codimension $k$ colors as _defects_ where the TQFT that one is looking at changes its nature.

In particular, when the QFT on one side of the defect is trivial, then the defect behaves like a _boundary condition_ for the remaining QFT. Since at least for $n=2$ QFT such boundary conditions are also called _[[branes]]_, defects are also called _[[bi-branes]]_.

The statement of the **cobordism theorem with singularities** ([Lurie 09, theorem 4.3.11](#Lurie09)) is essentially the following:

Given a [[symmetric monoidal (∞,n)-category]] $\mathcal{C}^\otimes$, then for every choice of [[pasting diagram]] of [[k-morphisms]] for all $k$, there is a type of manifolds with singularity $Def$, such that $Bord_n^Def$ is the free symmetric monoidal $(\infty,n)$-category on this data, hence such that TFTs with defects $(Bord_n^Def)^\otimes \to \mathcal{C}^\otimes$ are equivalently given by realizing such a pasting diagram in $\mathcal{C}$, where each of the given [[k-morphism]] appears as the value of a codimension $(n-k)$-defect. (See also [Lurie 09, remark 4.3.14](#Lurie09)).

### Topological defects from spontaneously broken symmetry
 {#DefectsFromBrokenSymmetry}

> under construction

An old notion of _defects_ in field theory -- well preceeding the [above](#DefinitionGeneral) general notion in the context of [[FQFT]] -- is that of **[[topological defects]] in the [[vacuum]] structure of [[gauge theories]] that exhibit [[spontaneous symmetry breaking]] (such as a [[Higgs mechanism]]).

A comprehensive review in in ([Vilenkin-Shellard 94](#VilenkinShellard94). Steps towards conceptually systematizing these broken-symmetry defects and their interaction are made in [Preskill-Vilenkin 92](#PreskillVilenkin92). We now discuss this may be translated to and formalized in the general [[FQFT]] definition [above](#DefinitionGeneral) along the lines of ([Fiorenza-Valentino](#FiorenzaValentino), [FSS](#FSS)).

Let $G$ be a [[Lie group]], to be thought of as the (local or global) [[gauge group]] of some [[gauge theory]]. Let $H \hookrightarrow G$ a [[subgroup]], to be thought of as the subgroup of global symmetries preserved by some [[vacuum]] configuration (which "[[spontaneous symmetry breaking|spontaneously breaks]]" the symmetry from $G$ to $H$, the archetypical example is the [[Higgs mechanism]]).

Then the space of such vacuum configurations is the [[coset]] space $G/H$. So given a [[manifold]] $X$ ([[spacetime]]), [[vacuum]] configuratons are given by [[functions]] $X \to G/H$. Hence $G/H_1$ is the "[[moduli space]] of vacua" or "vacuum space" in this context.

The functions $X \to G/H$ are to be [[smooth functions]] in the [[bulk]] of spacetime. If they are allowed to be non-smooth or even non-[[continuous function|continuous]] along given strata of $X$, then these are called **defects** in the sense of broken gauge symmetry.

In particular (with counting adapted to $dim X = 4$)

* if $X \to G/H$ is not smooth but is smooth on the pre-image of each element of $\pi_0(G/H)$ and becomes a smooth function on $X-S_1$ where $S_1 \hookrightarrow X$ is a [[codimension]]-1 [[submanifold]], then $S_1$ is said to be a [[domain wall]] for vacuum configurations.

* if $X \to G/H$ is not smooth but becomes smooth on $X-S_2$, where $S_2$ is a codimension-2 submanifold, then $S_2$ is calld a [[cosmic string]]-defect of the vacuum configurations;

* if $X \to G/H$ is not smooth but becomes smooth on $X-S_3$, where $S_3$ is a codimension-3 submanifold, then $S_3$ is calld a [[monopole]]-defect of the vacuum configurations.

> hm, need to fine-tune the technical conditions here, to make the following statement come out right...

So

* [[domain walls]] can appear when $\pi_0(G/H)$ is non-trivial;

* [[cosmic strings]] can appear when $\pi_1(G/H)$ is non-trivial;

* [[monopoles]] can appear when $\pi_2(G/H)$ is non-trivial.

Next consider a sequence of [[subgroups]]

$$
  H_2 \hookrightarrow H_1 \hookrightarrow H_0 \coloneqq G
$$

to be thought of as coming from two consecutive steps of [[spontaneous symmetry breaking]], the first one down to $H_1$ at some [[energy]]-scale $E_1$, and the second at some lower energy scale $E_2 \lt E_1$.

Then we say that vacuum defects at energy $E_2$ of codimension-$k$ which wind around an element $\pi_k(H_1/H_2)$ are **metastable** if they become unstable at energy $E_1$, hence if their image in $\pi_k(H_0/H_2)$ is trivial.

So if we add to the singular cobordism category the $k$-morphism which is the $k$-dimenional unit cube with an open $k$-[[ball]] removed, then the boundary field data for metastable codimension $n-k$-defects is

$$
  \array{
     [0,1]^k - D^k &\to & \Pi(H_1/H_2)
     \\
     \downarrow &\swArrow& \downarrow
     \\
     [0,1]^k &\to& \Pi(H_0/H_2) &\to& \Pi(H_0/H_1)
  }
$$

we have a [[homotopy fiber sequence]]

$$
  \Pi(H_1/H_2) \to \Pi(H_0/H_2) \to \Pi(H_0/H_1)
  \,.
$$

This induces a [[long exact sequence of homotopy groups]]

$$
   \cdots \to \pi_{k+1}(H_0/H_1) \to \pi_k(H_1/H_2) \to \pi_k(H_0/H_2) \to \pi_k(H_0/H_1)
  \to
  \pi_{k-1}(H_1/H_2) \to \cdots
  \,.
$$

So for every metastable defect of codimension $n-k$ given by $c \in ker(\pi_k(H_1/H_2) \to \pi_k(H_0/H_2))$ there is an element in $\pi_{k+1}(H_0/H_1)$ of one codimension higher. One says ([Preskill-Vilenkin 92](#PreskillVilenkin92)) that the codimenion $(n-k)$-defect may end on that codimension $(n-k-1)$-defect.

(...)

In order to formalize this we introduce, following the [[cobordism theorem]] with singularities, cells in $Span_n(\mathbf{H})$ which label spontaneous-symmetriy-breaking defects as well as their defects-of-defects which exhibit their decay by higher codimension defects.

Consider a span of the form

$$
  \array{
    [\Pi(S^{k-1}), \Pi(H_1/H_2)]
    &\leftarrow&
    [\Pi(S^{k-1}), \Pi(H_1/H_2)]
    &\rightarrow&
    \ast
  }
  \,.
$$

Comparing this to the span that comes from the "cap" $S^{k-1} \to D^{k} \leftarrow \emptyset$ in the theory at energy level $E_2$, which is just

$$
  \array{
    [\Pi(S^{k-1}), \Pi(H_1/H_2)]
    &\leftarrow&
    [\Pi(D^{k}), \Pi(H_1/H_2)]
    &\rightarrow&
    \ast
  }
$$

shows that the former models a $k$-disk which is not filled with spacetime, but nevertheless closes the disk. This is the defect given as a removal of a piece of spacetime.

In order to formalize how these defects may decay at higher energy, consider next  a [[span]] of [[field (physics)|field]] configurations of the form

$$
  \array{
    [\Pi(S^{k-1}), \Pi(H_1/H_2)]
    &\leftarrow&
    [\Pi(S^{k}), \Pi(H_0/H_1)]
    &\rightarrow&
    [\ast, \Pi(H_0/H_2)]
  }
  \,.
$$

Comparing again to the span that comes from the "cap" $S^{k-1} \to D^{k} \leftarrow \emptyset$ in the theory at energy level $E_2$, which is just

$$
  \array{
    [\Pi(S^{k-1}), \Pi(H_1/H_2)]
    &\leftarrow&
    [\Pi(D^{k}), \Pi(H_1/H_2)]
    &\rightarrow&
    \ast
  }
$$

shows that the former models a $k$-disk whose center point carries a singularity: the fields at the bounding $S^{k-1}$ take values in the moduli space of the ambient theory $\Pi(H_1/H_2)$, but then at the tip of the "cap" there is a "field insertion" of a field with values in $\Pi(H_0/H_2)$. Hence this labels a defect of codimension $k$.

To construct such a decay-process span that captures the above story from [Preskill-Vilenkin 92](#PreskillVilenkin92), consider the following diagram:

$$
  \array{
     && [\Pi(S^{k}), \Pi(H_0/H_1)]
     \\
     && \downarrow
     \\
     && [\Pi(S^{k-1}), \Omega\Pi(H_0/H_1)]
     \\
     & \swarrow && \searrow
     \\
     [\Pi(S^{k-1}), \Pi(H_1/H_2)] && (pb) && [\Pi(D^k), \Pi(H_0/H_2)] & \simeq & [\ast, \Pi(H_0/H_2)]
     \\
     & \searrow & & \swarrow
     \\
     && [\Pi(S^{k-1}), \Pi(H_0/H_2)]
  }
$$

This may be read as follows:

1. on the far left $[\Pi(S^k), \Pi(H_1/H_2)]$ is the space of fields of the ambient theory at energy scale $E_2$ around the defect;

1. the bottom left map is the "fluctuation" map that sends these fields to fields at the higher energy scale $E_1$;

1. the bottom right map exhibt the possible "decays": a lift through this map takes a field configuration that winds around a $k$-ball and contracts it through that $k$-ball, hence going forth and back through the bottom two maps corresponds to carrying a defect over the energy barrier from $E_2$ to $E_1$ and there having it decay away.

1. these decaying configurations are therefore given by the [[homotopy fiber product]] of the bottom two functions, which is $[\Pi(S^{k-1}), \Omega\Pi(H_0/H_1)]$, as indicated. But this space is really given by field configurations at energy scale $E_1$ that wind around a $(k+1)$-ball, as shown at the very top.

Hence the top part of this diagram is a span that exhibits a defect-of-defects which tells just the story that ([Preskill-Vilenkin 92](#PreskillVilenkin92)) is telling: a codimension-$k$ defect of the low energy theory decays at higher energy, and the decay is witnessed by the appearance of a codimension $(k+1)$-defect of the high energy theory.

$$
  \array{
     && {high\;energy \atop codim-(k+1)\;defects}
     \\
     && \downarrow
     \\
     && {low\;energy\;codim-k\;defects \atop with\;their\;decay\;processes}
     \\
     & \swarrow && \searrow
     \\
     {low\;energy \atop codim-k\;defects} && (pb) && {high\;energy \atop decay\;processes}
     \\
     & {}_{\mathllap{tunnel}}\searrow & & \swarrow_{\mathrlap{apply}}
     \\
     && {codim-k\;defects \atop raised\;to\;higher\;energy}
  }
$$

(...)

## Related concepts

* [[defect brane]]

* [[anyon]], [[braid group statistics]], [[loop braid group]]

* [[generalized global symmetry]]

\linebreak

[[!include field theory with boundaries and defects - table]]

## References

### General

A general formulation via an [[(∞,n)-category of cobordisms]] with defects:

* {#Lurie09} [[Jacob Lurie]], section 4.3 of: _[[On the Classification of Topological Field Theories]]_,  Current Developments in Mathematics Volume 2008 (2009) 129-280 [[arXiv:0905.0465](http://arxiv.org/abs/0905.0465), [doi:10.4310/CDM.2008.v2008.n1.a3](https://dx.doi.org/10.4310/CDM.2008.v2008.n1.a3)]

Defect TQFTs as 1-functors on stratified decorated bordisms are discussed in

* {#DavydovRunkelKong11} [[Alexei Davydov]], [[Ingo Runkel]], [[Liang Kong]], _Field theories with defects and the centre functor_ [[arXiv:1107.0495](https://arxiv.org/abs/1107.0495)], in [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ AMS (2011) [[ams:pspum-83](https://bookstore.ams.org/pspum-83)]

* {#CarquevilleRunkelSchaumann} [[Nils Carqueville]], [[Ingo Runkel]], [[Gregor Schaumann]], _Orbifolds of $n$-dimensional defect TQFTs_, Geom. Topol. **23** (2019) 781-864 *lbrack;[arXiv:1705.06085](http://arxiv.org/abs/1705.06085), [doi:10.2140/gt.2019.23.781](https://doi.org/10.2140/gt.2019.23.781)]

Details in dimension 2 and 3 are discussed in

* [[Chris Schommer-Pries]], _Topological Defects and Classification of Local TQFTs in Low Dimension_, [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]] ([pdf](https://ncatlab.org/nlab/files/SchommerPriesDefects.pdf))

* {#FuchsSchweigertValentino13} [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Alessandro Valentino]], *Bicategories for boundary conditions and for surface defects in 3-d TFT*, Commun. Math. Phys. **321**  (2013) 543–575 [[arXiv:1203.4568](https://arxiv.org/abs/1203.4568), [doi:10.1007/s00220-013-1723-0](https://doi.org/10.1007/s00220-013-1723-0)]

Review:

* [[Andrew Neitzke]], *Some uses of defects in quantum field theory* (2016?) [[html](https://gauss.math.yale.edu/~an592/talks/html/defects-aspen/talk.html)]

* [[Nils Carqueville]], [[Michele Del Zotto]], [[Ingo Runkel]], *Topological defects*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]* [[arXiv:2311.02449](https://arxiv.org/abs/2311.02449)]

Discussion of defects in [[prequantum field theory]], hence for [[coefficients]] in an [[(∞,n)-category of spans]] is in

* {#FiorenzaValentino} [[Domenico Fiorenza]], [[Alessandro Valentino]], *Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors*,  Commun. Math. Phys. **338** (2015) 1043–1074  [[arXiv:1409.5723](https://arxiv.org/abs/1409.5723), [doi:10.1007/s00220-015-2371-3](https://doi.org/10.1007/s00220-015-2371-3)]

* {#FSS} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]] et. al, _[[schreiber:Higher Chern-Simons local prequantum field theory]]_

DIscussion with emphasis on [[Koszul duality]] such as in [[holography as Koszul duality]]:

* [[Natalie Paquette]], Brian R. Williams, *Koszul duality in quantum field theory* ([arXiv:2110.10257](https://arxiv.org/abs/2110.10257))

### Examples

#### General

Examples in physics of interaction of defects of various dimension is discussed in

* Muneto Nitta, _Defect formation from defect--anti-defect annihilations_,  Phys. Rev. D85:101702,2012 ([arXiv:1205.2442](http://arxiv.org/abs/1205.2442))

#### In 2d field theory

Defects in [[2-dimensional conformal field theory]] have a long history in real-world application, for instance in [[Kramers-Wannier duality]]. Formalization by [[higher algebra|higher]] [[categorical algebra]] via the [[FRS theorem on rational 2d CFT]] is due to:

* {#FRS04} [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], *Kramers-Wannier duality from conformal defects*, Phys. Rev. Lett. **93** (2004) 070601 [[doi:10.1103/PhysRevLett.93.070601](https://doi.org/10.1103/PhysRevLett.93.070601), [arXiv:cond-mat/0404051](http://arxiv.org/abs/cond-mat/0404051)]

with a comprehensive review in

* {#FFRS07} [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], *Duality and defects in rational conformal field theory*, Nucl. Phys. B **763** (2007) 354-430 [[arXiv:hep-th/0607247](http://arxiv.org/abs/hep-th/0607247), [doi:10.1016/j.nuclphysb.2006.11.017](https://doi.org/10.1016/j.nuclphysb.2006.11.017)]

Realization in [[lattice field theory]]:

* David Aasen, Roger S. K. Mong, Paul Fendley, *Topological Defects on the Lattice I: The Ising model*, J. Phys. A: Math. Theor. **49** (2016) 354001 [[arXiv:1601.07185](https://arxiv.org/abs/1601.07185), [doi:10.1088/1751-8113/49/35/354001](https://doi.org/10.1088/1751-8113/49/35/354001)]

* David Aasen, Paul Fendley, Roger S. K. Mong, *Topological Defects on the Lattice: Dualities and Degeneracies* [[arXiv:2008.08598](https://arxiv.org/abs/2008.08598), [inspire:1812559](https://inspirehep.net/literature/1812559)]

* Mao Tian Tan, Yifan Wang, Aditi Mitra, *Topological Defects in Floquet Circuits* [[arXiv:2206.06272](https://arxiv.org/abs/2206.06272)]

and simulation on a [[quantum computer]]:

* Sutapa Samanta, Derek S. Wang, Armin Rahmani, Aditi Mitra, *Isolated Majorana mode in a quantum computer from a duality twist* [[arXiv:2308.02387](https://arxiv.org/abs/2308.02387)]

Defects in 2-dimensional [[topological field theory]] have been studied a lot in the context of genus-0 TFT, where they are described using the language of [[planar algebras]].
See discussion at *[Planar Algebras, TFTs with Defects](http://golem.ph.utexas.edu/category/2008/09/planar_algebras_tfts_with_defe.html)*.

Lecture notes:

* [[Nils Carqueville]], _Lecture notes on 2-dimensional defect TQFT_ [[arXiv:1607.05747](http://arxiv.org/abs/1607.05747)]

#### In Chern-Simons theory

* An old example is the class of [[Turaev-Reshetikhin TQFT]], which is a functor on 3-dimensional [[cobordisms]] with codimension 3 and 2 defects. This is supposed to be the would-be result of [[Chern-Simons theory]], where the defect lines are the original [[Wilson lines]] in this context.

Defects in [[Chern-Simons theory]] and related systems are discussed in

* [[Davide Gaiotto]], [[Edward Witten]], _S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory_ ([arXiv:0807.3720](http://arxiv.org/abs/0807.3720))

* [[Anton Kapustin]], Mikhail Tikhonov, _Abelian duality, walls and boundary conditions in diverse dimensions_, JHEP 0911:006,2009 ([arXiv:0904.0840](http://arxiv.org/abs/0904.0840))

* [[Anton Kapustin]], [[Natalia Saulina]], _Surface operators in 3d TFT and 2d Rational CFT_ in [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ AMS, 2011

In the context of the [[3d-3d correspondence]]:

* Dongmin Gang, Nakwoo Kim, Mauricio Romo, Masahito Yamazaki, _Aspects of Defects in 3d-3d Correspondence_, J. High Energ. Phys. (2016) ([arXiv:1510.05011](https://arxiv.org/abs/1510.05011))

Defects in [[higher dimensional Chern-Simons theory]] on [[manifolds with corners]] are discussed in

* [[Hisham Sati]], _Corners in M-theory_, J.Phys.A44:255402,2011 ([arXiv:1101.2793](http://arxiv.org/abs/1101.2793))

#### Topological defects in gauge theories with broken symmetry
 {#TopologicalDefectsInGaugeTheories}

The following references discuss the traditional notion of _[[topological defects]] in the [[vacuum]] structure of gauge theory with [[spontaneous symmetry breaking]]_ such as [[domain walls]], [[cosmic strings]] and [[monopoles]].

Discussion of "topological defects in [[gauge theory]]" in higher codimension is in

* {#PreskillVilenkin92} [[John Preskill]], [[Alexander Vilenkin]], _Decay of Metastable Topological Defects_, Phys. Rev. D47 : 2324-2342 (1993) ([arXiv:hep-ph/9209210](http://arxiv.org/abs/hep-ph/9209210))

* {#VilenkinShellard94} [[Alexander Vilenkin]], E.P.S. Shellard, _Cosmic strings and other topological defects_, Cambridge University Press (1994)

#### In solid state physics

Defects field theory motivated from [[solid state physics]] is discussed in

* {#KitaevKong11} [[Alexei Kitaev]], [[Liang Kong]], _Models for gapped boundaries and domain walls_ Commun. Math. Phys. 313 (2012) 351-373  ([arXiv:1104.5047](http://arxiv.org/abs/1104.5047))

[[!include defect anyons -- references]]

[[!redirects QFTs with defects]]

[[!redirects defect]]
[[!redirects defects]]

[[!redirects field theory with singularities]]

[[!redirects defect field theory]]
[[!redirects defect field theories]]

[[!redirects defect QFT]]
[[!redirects defect QFTs]]


Qi Zhu

I am a master's student at the University of Bonn.

I obtained my undergraduate degree at the Technical University of Munich. In Fall 2019, I was an exchange student at the University of Minnesota.

Pursuing ([[higher category theory|higher]]) [[category theory]] and [[homotopy theory]].

See [JourneyInMath](https://journeyinmath.wordpress.com) for my blog and [QiZhuMath](https://qizhumath.wixsite.com/math) for my website.

## Selected writings

On [[fractured (infinity,1)-toposes|fractured $\infty$-toposes]] in [[condensed mathematics]] (cf. [[condensed local contractibility]]):

* [[Qi Zhu]], *Fractured Structure on Condensed Anima*, MSc thesis (2023) [[pdf](https://1429cecd-24a0-4223-8b7c-1ebf47aa12e2.filesusr.com/ugd/8e912a_fe1c2f8e90094d0fa065fb8129687963.pdf), [[QiZhu-FracturedCondensed.pdf:file]]]

exposition:

* [[Qi Zhu]], *Fractured structure on condensed spaces*, talk notes (2023) [[pdf](https://1429cecd-24a0-4223-8b7c-1ebf47aa12e2.filesusr.com/ugd/8e912a_8d2a90176ed14969a877e77ec1b787da.pdf), [[QiZhuFracturedCondensed.pdf:file]]]

category: people


Qiaochu Yuan

* [webpage](https://math.berkeley.edu/people/grad/qiaochu-yuan)

## Selected writings

On [[split epimorphisms]]/[[split monomorphisms]]:

* {#Yuan12} [[Qiaochu Yuan]], *[Split epimorphisms and split monomorphisms](https://qchu.wordpress.com/2012/10/01/split-epimorphisms-and-split-monomorphisms/)*, blog post (2012)

On the [[ordinary cohomology]] of topological [[tori]]:

* [[Qiaochu Yuan]], *[The cohomology of the $n$-torus](https://qchu.wordpress.com/2013/10/12/the-cohomology-of-the-n-torus/)* (2013)

## Related entries

* [[rigid object]]

* [[quantum probability]]

* [[simple ring]]

category: people


Qingtao Chen

* [Math Genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=154309)

## Selected writings

On the [[Witten genus]] in the case of [[string^c structure]]:

* {#ChenHanZhang10} [[Qingtao Chen]], [[Fei Han]], [[Weiping Zhang]], _Generalized Witten Genus and Vanishing Theorems_, Journal of Differential Geometry 88.1 (2011): 1-39. ([arXiv:1003.2325](http://arxiv.org/abs/1003.2325))

* [[Qingtao Chen]], [[Fei Han]], _Elliptic Genera, Transgression and Loop Space Chern-Simons Forms_, Communications in Analysis and Geometry Volume 17 (2009) Number 1 ([arXiv:math/0611104](http://arxiv.org/abs/math/0611104), [doi:10.4310/CAG.2009.v17.n1.a4]( https://dx.doi.org/10.4310/CAG.2009.v17.n1.a4))

On the [[volume conjecture]]:

* {#ChenYang15} [[Qingtao Chen]], [[Tian Yang]], _A volume conjecture for a family of Turaev-Viro type invariants of 3-manifolds with boundary_ ([arXiv:1503.02547](http://arxiv.org/abs/1503.02547))

* {#ChenLiuZhu15} [[Qingtao Chen]], [[Kefeng Liu]], Shengmao Zhu, _Volume conjecture for $SU(n)$-invariants_ ([arXiv:1511.00658](https://arxiv.org/abs/1511.00658))

## Related $n$Lab entries

* [[Witten genus]]

* [[volume conjecture]]

category: people


QML

+-- {: .rightHandSide}
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### Context
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

*QML* is a [[functional programming language|functional]] [[quantum programming language]].

## Related languages

* [[Quipper]]

## References

Original description:

* {#AltenkirchGrattage05} [[Thorsten Altenkirch]], [[Jonathan Grattage]], _A functional quantum programming language_, Logic in Computer Science (2005) 249-258 [[arXiv:quant-ph/0409065](https://arxiv.org/abs/quant-ph/0409065), [doi:10.1109/LICS.2005.1](https://doi.org/10.1109/LICS.2005.1), [pdf](http://www.cs.nott.ac.uk/~txa/publ/qml.pdf)]

* [[Jonathan Grattage]], *A functional quantum programming language*, PhD thesis, Nottingham (2006) [[eprints:10250](http://eprints.nottingham.ac.uk/10250), [pdf](https://eprints.nottingham.ac.uk/10250/1/thesis.pdf)]

* [[Alexander Green]], *Towards a formally verified functional quantum programming language* (2010) [[eprints:11457](http://eprints.nottingham.ac.uk/11457)]

  > (emphasis on [[formal software verification]])

and with emphasis on the [[quantum IO monad]]:

* [[Thorsten Altenkirch]], [[Alexander Green]], *The quantum IO monad*, Ch. 5 of: Simon Gay, Ian Mackie (eds.): *Semantic Techniques in Quantum Computation* (2010) 173-205 [[pdf](http://www.cs.nott.ac.uk/~txa/publ/qio-chapter.pdf), [talk slides](http://www.cs.nott.ac.uk/~txa/talks/qnet06.pdf), [doi:10.1017/CBO9781139193313.006](https://doi.org/10.1017/CBO9781139193313.006)]

Review:

* [[Jonathan Grattage]], _An overview of QML with a concrete implementation in Haskell_, Quantum Physics and Logic 2008 [[arXiv:0806.2735](https://arxiv.org/abs/0806.2735)]

* [[Jonathan Grattage]] with [[Thorsten Altenkirch]], *Compiling a functional quantum programming language* (talk slides) [[pdf](http://www.cs.nott.ac.uk/~pszgmh/bctcs-slides/grattage.pdf), [[Grattage-CompilingFunctionalQPL.pdf:file]]]


QPMC

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### Context
#### Computation
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

`QPMC` is a [[software verification]]-tool  for [[quantum programming languages]] (a "Quantum Protocol Model Checker").

## Related entries

* [[QWIRE]]

* [[Quipper]]

## References

Introducing `QPMC`:

* [[Yuan Feng]], Ernst Moritz Hahn, Andrea Turrini, Lijun Zhang, *`QPMC`: A Model Checker for Quantum Programs and Protocols*, in *Formal Methods. FM 2015*, Lecture Notes in Computer Science **9109**, Springer (2015)  [[doi:10.1007/978-3-319-19249-9_17](https://doi.org/10.1007/978-3-319-19249-9_17)]

  > "In practice, however, security analysis of [[quantum cryptography|quantum cryptographic]] protocols is notoriously difficult; for example, the manual proof of BB84 in [15] contains about 50 pages. It is hard to imagine such an analysis being carried out for more sophisticated quantum protocols. Thus, techniques for automated or semi-automated verification of these protocols will be indispensable."

Background and review:

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

  > "But to check whether a quantum system satisfies a certain property at a time point, one has to perform a quantum. measurement on the system, which can change the state of the system. This makes studies of the long-term behaviours of quantum systems much harder than that of classical system."

  > "The state spaces of the classical systems that model-checking algorithms can be applied to are usually finite or countably infinite. However, the state spaces of quantum systems are inherently continuous even when they are finite-dimensional. In order to develop algorithms for model-checking quantum systems, we have to exploit some deep mathematical properties of the systems so that it suffices to examine only a finite number of (or at most countably infinitely many) representative elements, e.g. those in an orthonormal basis, of their state spaces."

On [[software verification|verification]] of [[Quipper]]-programs  with QPMC:

* Linda Anticoli, Carla Piazza, Leonardo Taglialegne, Paolo Zuliani, _Towards Quantum Programs Verification: From Quipper Circuits to QPMC_, In: Devitt S., Lanese I. (eds.) *Reversible Computation. RC 2016*, Lecture Notes in Computer Science **9720** Springer (2016) [[arXiv:1708.06312](https://arxiv.org/abs/1708.06312), [doi:10.1007/978-3-319-40578-0_16](https://doi.org/10.1007/978-3-319-40578-0_16)]

category: people


QRAM

> This entry is about *QRAM* in the sense of [Giovanetti et al. 2008](#GiovanettiLloydMaccone08). For the other sense of *[[classically controlled quantum computation]]* see there.

***

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#### Quantum systems
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[[!include quantum systems -- contents]]
=--
#### Computation
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum computing]] and [[quantum information theory]], the notion of *quantum random access memory* (QRAM, due to [Giovanetti, Lloyd & Maccone 2008](#GiovanettiLloydMaccone08)) is meant to be the [[quantum physics|quantum]]-analog of the [[classical physics|classical]] notion of [[random access memory]]; the key point being that QRAM may be addressed by *[[quantum superpositions]]* of address values and then reads/writes the corresponding [[quantum superposition]] of quantum data ([GLM08, (1)](#GiovanettiLloydMaccone08)).

## Circuit model
 {#CircuitModel}

The *circuit model* for QRAM essentially identifies a quantum program

* of nominal type $\mathscr{H} \longrightarrow \mathscr{H}$

* but with access to a QRAM space $QMem$

with a  a [[quantum circuit]] of type

$$
  QMem \otimes \mathscr{H}
    \longrightarrow
  QMem \otimes \mathscr{H}
  \,.
$$

In the existing literature this is typically stated in terms of concrete examples rather than in abstract generality, see for instance [[Arunachalam et al 2015, Fig. 9](#ArunachalamEtAl15)] or [[Park, Petruccione & Rhee 2019, Fig 1](#ParkPetruccioneRhee19)] with review in [[Phalak, Chatterjee & Ghosh 2023, Fig 4](#PhalakChatterjeeGhosh23)]. A nicely transparent example is given in by [quantumcomputinguk.org](#quantumcomputinguk):

(Here the block "Memory Cells" is our "$QMem$", the rest is our $\mathscr{H}$.)

This [[quantum circuit]] is such that (assuming $q_2 =\cdots = q_6 = 0$)

* if the Read/Write flag is 0 then

  it keeps all qbits (except for "Routing" and) except that the "Readout QBit" is set to one of the four "Memory values" depending on the four possible values of the "Addressing QBits"

* if the Read/Write flag is 1 then

  it keeps all qbits (except for "Routing" and) except that the "Memory QBit" addressed by the value of the "Addressing QBits" is set to the value of $q_{10}$.

* for a [[superposition]] of these inputs it yields the corresponding superposition of outputs.

## Via linear state-effects

Recall that a classical random access memory of [[data type]] $Mem$ is modeled (in terms of [[monads in computer science]]) by the [[state monad]] induced from the *[[cartesian monoidal category|cartesian]]* [[internal hom]]-[[adjoint functor|adjunction]]

* [[product]]$\;$ $Mem \times (-) \;\;\dashv\;\; Maps(Mem, -)$ $\;$[[function set]]

namely:

$$
  RAM(Mem,A)
  \;\coloneqq\;
  Maps
  \big(
    Mem
    ,\,
    Mem
    \times
    A
  \big)
  \,.
$$

(Notice that in concrete applications people often insist that  $Mem =$ [[Bool]]${}^{\times^n}$, but the conceptual nature of RAM is indifferent to this choice.)

Now, with classical data types (such as [[bits]]) replaced by quantum data types (such as [[qbits]]), namely by *[[linear types]]*, the analogous [[internal hom]]-[[adjoint functor|adjunction]] is

* [[tensor product]]$\;$ $ QMem \otimes (-) \;\;\dashv\;\; LinMaps(QMem, - )$ $\;$[[linear space]]-of-[[linear maps]]

and the corresponding [[monad]] is

$$
  QRAM(Mem, \mathscr{H})
  \;\coloneqq\;
  LinMaps
  \big(
    QMem
    ,\,
    QMem
    \otimes
    \mathscr{H}
  \big)
$$

and it seems clear that this does correspond to the intended behaviour of QRAM (where we are free to set $QMem \coloneqq $ [[QBit]]${}^{\otimes^n}$, if desired, which is typically the case in the literature).

In particular, the intended equivalence between the "QRAM model" and the "[[quantum circuit]]-model" of [[quantum computation]] is just the [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) of the linear [[internal hom]]-[[adjunction]]

$$
  \Big\{
  In
  \longrightarrow
  LinMaps
  \big(
    QMem
    ,\,
    QMem
    \otimes
    Out
  \big)
  \Big\}
  \;\;\;\;\;\;\;
  \leftrightarrow
  \;\;\;\;\;\;\;
  \Big\{
  In \otimes QMem
  \longrightarrow
  Out \otimes QMem
  \Big\}
  \,.
$$

## References

The terminology "quantum random access memory" is due to

* {#GiovanettiLloydMaccone08} [[Vittorio Giovannetti]], [[Seth Lloyd]], [[Lorenzo Maccone]], *Quantum random access memory*, Phys. Rev. Lett. **100** 160501 (2008) [[doi:10.1103/PhysRevLett.100.160501](https://doi.org/10.1103/PhysRevLett.100.160501), [arXiv:0708.1879](https://arxiv.org/abs/0708.1879)]

* {#GiovanettiLloydMaccone08b} [[Vittorio Giovannetti]], [[Seth Lloyd]], [[Lorenzo Maccone]], *Architectures for a quantum random access memory*, Phys. Rev. A **78** 052310 (2008) [[doi:10.1103/PhysRevA.78.052310](https://doi.org/10.1103/PhysRevA.78.052310), [arXiv:0807.4994](https://arxiv.org/abs/0807.4994)]

Further development:

* {#ArunachalamEtAl15} S. Arunachalam et al., *On the robustness of bucket brigade quantum RAM*, New Journal of Physics, **17** 12  (2015) 123010 [[arXiv:1502.03450](https://arxiv.org/abs/1502.03450), [doi:10.1088/1367-2630/17/12/123010](https://doi.org/10.1088/1367-2630/17/12/123010)]

* {#ParkPetruccioneRhee19} Park, Petruccione, Rhee, *Circuit-Based Quantum Random Access Memory for Classical Data*, Scientific Reports **9** 3949 (2019) [[doi:10.1038/s41598-019-40439-3](https://doi.org/10.1038/s41598-019-40439-3)]

Review:

* [[Alessandro Luongo]], *[The QRAM](https://quantumalgorithms.org/chap-classical-data-quantum-computers.html#the-qram)*, §3.1.1. in: *[Quantum algorithms for data analysis](https://quantumalgorithms.org/)*

* {#PhalakChatterjeeGhosh23} Koustubh Phalak, Avimita Chatterjee, Swaroop Ghosh, *Quantum Random Access Memory For Dummies* [[arXiv:2305.01178](https://arxiv.org/abs/2305.01178)]

* Samuel Jaques, Arthur G. Rattew, *QRAM: A Survey and Critique* [[arXiv:2305.10310](https://arxiv.org/abs/2305.10310)]

With emhasis on laboratory realization:

* Chenxu Liu, et al., III.C (pp. 18) of:  *Quantum Memory: A Missing Piece in Quantum Computing Units* [[arXiv:2309.14432](https://arxiv.org/abs/2309.14432)]

With Qiskit:

* {#quantumcomputinguk} *[Implementing QRAM in Qiskit](https://quantumcomputinguk.org/tutorials/implementing-qram-in-qiskit-with-code)*

[[!redirects QRAMs]]

[[!redirects quantum RAM]]
[[!redirects quantum RAMs]]

[[!redirects quantum random access memory]]
[[!redirects quantum random access memories]]

[[!redirects quantum memory]]


quadrability

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

## Idea

A [[cospan]]

$$
  \array{
    F_B &&&& F_C
    \\
    & \searrow && \swarrow
    \\
    && F_A
  }
$$

in a category $\mathcal{C}$ is called _quadrable_ , if there exists a  [[cone]]

$$
  \array{
    && N
    \\
    & \swarrow && \searrow
    \\
    F_B &&&& F_C
    \\
    & \searrow && \swarrow
    \\
    && F_A
  }
  \,.
$$

## Definition

Let $Sp = \left\{\array{ B &&&& C \\ & \nwarrow && \nearrow \\ && A}\right\}$ denote the [[span]] [[diagram|diagram category]], that is, the category with three objects $A,B,C$ and two non-identity morphisms $A\to B$ and $A\to C$.

Let $\mathcal{C}$ be any category and let $\Delta$ denote the [[diagonal]] $\mathcal{C}\to [Sp^{op},\mathcal{C}]$ into the [[functor category]] sending an object $X\mapsto c_X$, where $c_X$ is the constant functor sending all objects and all morphisms of $Sp^{op}$ to $X$ and $id_X$ respectively.

For $F\in [Sp^{op}, \mathcal{C}]$ let $*_F$ denote the unique functor $*\to [Sp^{op},\mathcal{C}]$ from the [[terminal category]] such that the unique object of $*$
maps to $F$.

We say that a [[cospan]] $F$ in a category $\mathcal{C}$, that is, an object of the [[functor category]] $[Sp^{op},\mathcal{C}]$ is **quadrable** if there exists a [[cone]] $N$ for $F$, that is, an object $N$ in the [[comma category]] $(\Delta \downarrow *_F)$.

Dually, we say that a [[span]] $G$ in a category $C$, that is, an object of the functor category $[Sp,\mathcal{C}]$ is **coquadrable** if there exists a [[cocone]] $N$ for $G$, that is, an object $N$ in the comma category $(*_G \downarrow \Delta)$.

We say that a category $\mathcal{C}$ is quadrable (resp. coquadrable) if all cospans (resp. spans) in $C$ are quadrable (resp. coquadrable).

## Note on terminology

The term _quadrable_ is supposed to be a translation of the French  _carrable_ , whose use is more wide-spread. It appears for instance in

* [[Bertrand Toen]], _A master course on algebraic stacks (Toulouse 2005)_ cours 2 ([web](https://perso.math.univ-toulouse.fr/btoen/files/2015/02/cours2.pdf))

* [[André Joyal]], _Notes on Clans and Tribes_, [arxiv:1710.10238](https://arxiv.org/abs/1710.10238)

[[!redirects quadrable cospan]]
[[!redirects carrable cospan]]
[[!redirects coquadrable span]]
[[!redirects quadrable span]]

[[!redirects quadrable cospans]]
[[!redirects coquadrable spans]]
[[!redirects quadrable spans]]


quadratic abelian group

+-- {: .rightHandSide}
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#### Algebra
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=--
#### Group theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

A **quadratic abelian group** is an [[abelian group]] $G$ with a function $q: G \to \mathbb{Z}$ 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 of degree 2) For any $x \in G$ and any $r \in \mathbb{Z}$,
$$q(r x) = r^2 q(x)$$

## Examples

* Every [[ring]] is a quadratic abelian group.

* Every [[inner product abelian group]] is a quadratic abelian group with $q(x) \coloneqq \langle x, x \rangle$.

## See also

* [[quadratic function]]

* [[abelian group]]

* [[ring]]

* [[Clifford ring]]

* [[inner product abelian group]]

[[!redirects quadratic abelian groups]]