quasi-topological space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--

# Quasi-topological spaces
* table of contents
{: toc}

## Idea

The [[category]] of *quasi-topological spaces* was proposed in [Spanier 1963](#Spanier63) as a substitute for the category [[Top]] of ordinary [[topological spaces]], in order to serve as a [[convenient category of topological spaces|convenient category]] for the purposes of [[algebraic topology]]. In particular, quasi-topological spaces form a [[complete category|complete]] and [[cocomplete category|cocomplete]] [[cartesian closed category]].

Today, quasi-topological spaces seem to be regarded mostly as a historical curiosity, perhaps because working topologists were never comfortable with the [[set theory|set-theoretic]] issues that accompany them. In retrospect, however, they are an impressive testament to the conceptual insight of [[Edwin Spanier|Spanier]] into ideas of [[topos theory]] which were at the time (early 1960's) barely in the air, and even not quite born yet (being an early example of [[quasitopos]], whose name perhaps derives from Spanier's notion, compare [Dubuc & Español 2006, p. 12](#DubucEspanol06)).

## Definition

Let $\mathcal{C H}$ be the [[category]] of [[compact Hausdorff spaces]]. This may be regarded as a (large) [[site]] when equipped with the [[Grothendieck topology]] of [[finite open covers]], in fact a [[concrete site]].

+-- {: .un_defn}
###### Definition
A **quasi-topological space** is a (small-set valued) [[concrete sheaf]] on $\mathcal{C H}$.
=--

The (super-large) category of quasi-topological spaces is a [[quasitopos]] (although this is not immediately obvious for [[size issues|size reasons]] --- in particular, it is probably not a [[Grothendieck quasitopos]]).  In particular, it is a [[locally cartesian closed category]].

## References

* {#Spanier63} [[Edwin Spanier]], _Quasi-topologies_, Duke Mathematical Journal **30**, number 1 (1963) pp 1–14 ([doi:10.1215/S0012-7094-63-03001-1](https://doi.org/10.1215/S0012-7094-63-03001-1))

* {#DubucEspanol06} [[Eduardo Dubuc]], [[Luis Español]], _Quasitopoi over a base category_  ([arXiv:math.CT/0612727](http://arxiv.org/abs/math.CT/0612727))

[[!redirects quasi-topological space]]
[[!redirects quasi-topological spaces]]
[[!redirects quasitopological space]]
[[!redirects quasitopological spaces]]


quasicategory > history

< [[quasicategory]]
[[!redirects quasicategory -- history]]


quasicategory of fractions

## Idea and definition

If $C$ is a [[quasicategory]] and $W$ is a class of morphisms in $C$, then the __quasicategory of fractions__ is a functor $L\colon C\to C[W^{-1}]$ that has the following universal property: precomposition with $L$ identifies the quasicategory of functors $C[W^{-1}]\to D$ with the full subquasicategory of functors $C\to D$ that sends elements of $W$ to invertible morphisms in $D$.

## Terminology

In the existing literature on [[quasicategories]], some sources (like Lurie's [[Higher Topos Theory]]) do not assign a specific name to this concept and simply talk about it by saying that “$S^{-1}C$ is obtained from $C$ by inverting the morphisms of $S$”.
(See Warning 5.2.7.3 in [[Higher Topos Theory]].)

Other sources, like [[Cisinski]]'s [[Higher Categories and Homotopical Algebra]], use the term “localization” to refer to this concept, and refer to localizations in the sense of Lurie as “left Bousfield localizations”.

## Related concepts

* [[category of fractions]], [[homotopy category]]

* [[reflective localization]]

* [[localization of an (infinity,1)-category]]

[[!redirects quasicategories of fractions]]
[[!redirects quasi-categories of fractions]]
[[!redirects quasi-category of fractions]]
[[!redirects ∞-category of fractions]]
[[!redirects ∞-categories of fractions]]
[[!redirects (∞,1)-category of fractions]]
[[!redirects (∞,1)-categories of fractions]]
[[!redirects infinity-category of fractions]]
[[!redirects infinity-categories of fractions]]
[[!redirects (infinity,1)-category of fractions]]
[[!redirects (infinity,1)-categories of fractions]]


quasicoherent infinity-stack

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _quasicoherent [[∞-stack]] of modules_ is the anlog in [[(∞,1)-topos theory]] of the notion of [[quasicoherent sheaf]] in [[topos theory]].

A quasicoherent sheaf on a [[scheme]] $X$ is is equivalently a morphism of [[stack]]s $X \mapsto Mod$ into the canonical stack [[Mod]] $ : Spec A \mapsto A Mod$ of [[module]]s, which corresponds to the [[bifibration]] $Mod \to CRing$ over the category of commutative rings/algebras: this is the [[tangent category]] of [[CRing]].

This general abstract description of quasi-coherent sheaves has a fairly direct generalization to [[(∞,1)-topos theory]] over arbitrary [[(∞,1)-site]]s:

for $C$ any [[(∞,1)-site]], the [[tangent (∞,1)-category]] $T (C^{op}) \to C^{op}$ is the [[Cartesian fibration|bifibration]] whose [[fiber]]s over an object $A \in C^{op}$ plays the role of the [[∞-groupoid]] of [[module]]s over $A$. Under the [[(∞,1)-Grothendieck construction]] this corresponds to an [[(∞,1)-presheaf]] $\infty Mod : C^{op} \to \infty Cat$

$$
  \infty Mod : Spec R \mapsto Stab( C^{op}/R )
$$

or directly in terms of test spaces $U$:

$$
  \infty Mod : U \mapsto Stab( U/C )
  \,.
$$

For the special case that $C = (sAlg^{op})^\circ$ [[opposite (∞,1)-category]] presented by the [[model structure on simplicial algebras]] over a characteristic 0 ground field we have (with example 8.6 of _[[Stable ∞-Categories]]_ ) this reproduces the notion of quasicoherent $\infty$-stack as considered in [[dg-geometry]].

## Definition

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

For $C$ an [[(∞,1)-site]] let the [[(∞,1)-functor]]

$$
  Mod_C : C^{op} \to (\infty,1)Cat
$$

be that corresponding under the [[(∞,1)-Grothendieck construction]] to the [[tangent (∞,1)-category]] $T C^{op} \to C^{op}$.

=--

Let $\mathbf{H} := Sh_{(\infty,1)}(C)$ be the [[(∞,1)-category of (∞,1)-sheaves]] over $C$. Notice that this sits inside $[C^{op}, (\infty,1)Cat]$.

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

For $X \in \mathbf{H}$, the **$(\infty,1)$-category of quasi-coherent $\infty$-stacks on $X$ is the [[(∞,1)-category of (∞,1)-functors]]

$$
  QC(X) := [C^{op}, (\infty,1)Cat](X, Mod)
  \,.
$$

=--

## Model category theoretic presentation

For [[derived geometry]] modeled on formal duals of [[algebras over an operad]], the [[model category]] presentation of quasi-coherent $\infty$-stacks is locally given by a [[model structure on modules over an algebra over an operad]].

The following considers the special case of the [[commutative operad]].

### For commutative monoids

Let $\mathcal{C}$ be a [[monoidal model category]]. Write $CMon(\mathcal{C})$ for the category of [[commutative monoid]]s in $\mathcal{C}$.

For instance

* for $\mathcal{C} = $ [[Ab]] a monoid object in $\mathcal{C}$ is an ordinary [[ring]] and its formal dual an ordinary [[affine scheme]];

* for $\mathcal{C} = sAb$, the category of abelian [[simplicial group]]s, a monoid in $\mathcal{C}$ is a [[simplicial ring]] and its formal dual is one notion of affine [[derived scheme]].

In ([ToënVezzosi, I](ToënVezzosiI), [ToënVezzosi, II](#ToënVezzosiII)) are discussed structures of a [[model site]]/[[sSet-site]] on $CMon(\mathcal{C})$.

For every commutative monoid $A \in \mathcal{C}$ There is naturally a [[model category]] structure on the category $A Mod$ of $A$-[[module]]s in $\mathcal{C}$.

(...)

Let $[C^{op},sSet]_{proj,loc}$ the local [[model structure on sSet-presheaves]] that [[presentable (∞,1)-category|presents]] the [[(∞,1)-category of (∞,1)-sheaves]]/[[∞-stack]]s on $C$

$$
  ([C^{op}, sSet]_{proj,loc})^\circ \simeq \mathbf{H} := Sh_{(\infty,1)}(C)
  \,.
$$

as described as [[models for ∞-stack (∞,1)-toposes]].

So in this model an $\infty$-stack is a [[simplicial presheaf]] $C^{op} \to SSet$ on a simplicial site $C$ that takes values in [[Kan complex]]es and satisfies [[descent]] with respect to [[hypercover]]s in the [[homotopy category]] of $C$.

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

The [[simplicial presheaf]] of **quasicoherent $\infty$-stacks** is

$$
  QC : C^{op} \to sSet
$$

is given by

$$
  QC : Spec A \mapsto N( A Mod_{cof, \sim} )
  \,,
$$

where on the right we have the [[nerve]] of the non-full [[subcategory]] of the model category $A Mod$ on cofibrant objects and weak equivalences between them.

=--

This appears as ([ToënVezzosiII, definition 1.3.7.1](#ToenVezzosiII).

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

This $QC$ is indeed an [[∞-stack]] on the [[model site]] $C := Comm(\mathcal{C})^{op}$

=--

This is ([ToënVezzosiII, theorem 1.3.7.2](#ToenVezzsoi).

> notice: the $QC$ defined this way is not yet stabilized

## Related concepts

* [[quasicoherent sheaf]]

* **quasicoherent $\infty$-stack**

## References

The general discussion of the [[tangent (∞,1)-category]] is in

* [[Jacob Lurie]], _[[Deformation Theory]]_ .

The model category theoretic presentation over model sites of commutative monoids is discussed in

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical Algebraic Geometry I: Topos theory_ ([arXiv:0207028](http://arxiv.org/abs/math/0207028))
{#ToenVezzosiI}

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical Algebraic Geometry II: geometric stacks and applications_ ([arXiv:0404373](http://arxiv.org/abs/math/0404373))
{#ToenVezzosiII}

A discussion specific to [[dg-geometry]] with an emphasis on the [[geometric ∞-function theory]] of quasicoherent $\infty$-stacks over [[perfect ∞-stack]]s is in

* [[David Ben-Zvi|Ben-Zvi]], [[John Francis]],  [[David Nadler]], _Integral transforms and Drinfeld centers in derived algebraic geometry_ ([arXiv](http://arxiv.org/abs/0805.0157))
{#Ben-ZviFrancisNadler}

category: algebraic geometry
[[!redirects quasicoherent ∞-stack]]
[[!redirects quasicoherent ∞-stacks]]
[[!redirects quasicoherent infinity-stacks]]
[[!redirects quasicoherent (∞,1)-sheaf]]
[[!redirects quasicoherent (∞,1)-sheaves]]
[[!redirects quasicoherent (infinity,1)-sheaf]]
[[!redirects quasicoherent (infinity,1)-sheaves]]


quasicoherent sheaf

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

#Contents#
* table of contents
{: toc}

## Idea {#Idea}

A _quasicoherent sheaf of modules_ (often just "quasicoherent sheaf", for short) is a [[sheaf of modules]]  over the [[structure sheaf]] of a [[ringed space]] that is _locally [[presentable module|presentable]]_ in that it is locally the [[cokernel]] of a morphism of [[free modules]].

For comparison, by the [[Serre-Swan theorem]] a [[vector bundle]] on a suitable [[ringed space]] is equivalently encoded in its [[sheaf]] of [[sections]] which is even locally [[free module|free]] and [[projective module|projective]]. In this sense quasicoherent sheaves of modules are a generalization of [[vector bundles]].
The [[category]] of vector bundles is too small to be closed under various natural operations like [[kernels]], [[direct images]] and alike. In particular, it is not an [[abelian category]]. The category of all $\mathcal{O}$-modules and especially its [[full subcategory]] of quasicoherent sheaves of $\mathcal{O}$-modules are better behaved in that respect.

There are several different but equivalent ways to define and think of quasicoherent sheaves.

A very concrete definition characterizes quasicoherent sheaves as those that are, while not locally free, locally the [[cokernel]] of a morphism of free module sheaves. This is the definition given in the section

* [As locally presentable modules](#LocPres)

below. It makes very manifest how passing from vector bundles to quasicoherent sheaves adds in the [[cokernel]]s that are missing in the category of [[vector bundle]]s.

But it turns out that there is a more abstract, more [[sheaf and topos theory|sheaf theoretic]] reformulation of this definition: if we think of the underlying [[space]] as a ([[presheaf|pre]])[[sheaf]] (as motivated at [[motivation for sheaves, cohomology and higher stacks]]) we find that a quasicoherent sheaf on a space is given by an assignment of a module to each plot, such that the pullback of these modules is given, up to coherent isomorphism, by tensoring over the corresponding rings. This is described in the section

* [As sheaves over Aff/X](#AsSheaves).

and in more details in the section

* [Definition as sheaves over Aff](#AsSheavesII).

The tensoring operation appearing here is that defining the pullback operations in the [[stack]] that classifies the canonical [[bifibration]] $Mod \to CRing$ of [[module]]s over rings. In view of this, one finds that this definition, in turn, is equivalent to a very fundamental definition:

with $QC := (-)Mod : Ring \to Cat$ the functor that sends a ring to its category of modules, one finds that the category of quasicoherent sheaves on a [[space]] $X$ is simply the [[hom-object]]

$$
  QC(X) := Hom(X,QC)
$$

in the corresponding 2-category of category-valued (pre)sheaves, i.e (pre)[[stack]]s. This is the perspective described in

* [As hom objects](#AsCocycles)

below. By the equivalence between [[Grothendieck fibration]]s and [[pseudofunctor]]s this in turn is directly equivalent to the identification of $QC(X)$ with the category of cartesian functors between the [[category of elements]] of $X$ and $Mod$. This is described in

* [As cartesian morphisms of fibered categories](#AsFibHoms)

This definition, finally, provides a powerful [[nPOV]] on quasicoherent sheaves: all notions involved, sheaf, stack, morphism of stacks, have natural, immediate and well understood generalizations to [[higher category theory]]. Therefore this last definition immediately generalizes to a definition of quasicoherent $\infty$-sheaves or "derived" quasicoherent sheaves, such as they appear for instance in [[geometric ∞-function theory]]. This is discussed in the section

* [Higher quasicoherent sheaves](#Higher).

## Definition

### As locally presentable modules {#LocPres}

Let $(X,O_X)$ be a [[ringed space]] or, more generally, a [[ringed site]].

A __quasicoherent sheaf of $O_X$-modules__ on $X$ is a sheaf $\mathcal{E}$ of $O_X$-modules that is locally a [[cokernel]] of a [[morphism]] of [[free module]]s.

This means: there is a [[cover]] $\{U_\alpha\}_{\alpha\in A}$ of $X$ by open sets such that for every $\alpha$ there exist $I_\alpha$ and $J_\alpha$ (not necessarily finite) and an [[exact sequence]] of sheaves of $O_X$-modules of the form

$$
  O_X^{I_\alpha}|_{U_\alpha}
  \to
  O_X^{J_\alpha}|_{U_\alpha}
  \to
  \mathcal{E}|_{U_\alpha}\to 0,
$$

This should be viewed as a _local presentation_ of $\mathcal{E}$.

If $I_\alpha, J_\alpha$ can be chosen finite and $\mathcal{E}$ is of finite type then the quasicoherent sheaf is a **[[coherent sheaf]]**. (See there for details.)
However , coherent sheaves are ill-behaved for a general ringed space, and even general [[scheme]]s; they behave well on [[Noetherian scheme]]s.

Replacing covers by open sets, by covers of a terminal object in a site, the definition extends to [[ringed site]]s with a terminal object; the restrictions of $O_X$-modules should be replaced by pullbacks.
There are generalizations for [[algebraic stack]]s, ind-schemes, diagrams of schemes (for example [configuration schemes](http://arxiv.org/abs/math/0012061) of V. Lunts, obtained by gluing along closed embeddings of schemes; simplicial schemes) and so on.

### As sheaves on $Aff/X$ {#AsSheaves}

There is an equivalent reformulation of the above in terms of [[sheaf|sheaves]] of $\mathcal{O}$-modules on the [[site]] $Aff/X$ of [[affine scheme]]s over $X$.

This is the [[over category]] whose objects are morphism (of [[scheme]]s) of the form $Spec A \to X$ and whose morphisms are commuting triangles

$$
  \array{
    Spec A &&\stackrel{f}{\to}&& Spec B
    \\
    & {}_a\searrow && \swarrow_b
    \\
    && X
  }
  \,.
$$

Then: a quasicoherent sheaf on $(X, \mathcal{O}_X)$ is a [[sheaf]] $N$ of $\mathcal{O}_X$-[[module]]s on $Aff/X$  such that for each morphism $f$ as above the restriction morphism

$$
  N(f) : N(b) \to N(a)
$$

extends to an [[isomorphism]]

$$
  N(b) \otimes_{f^*} A \stackrel{\simeq}{\to} N(a)
$$

of $A$-[[module]]s.

For a very explicit statement of this see for instance [page 13](http://arxiv.org/PS_cache/arxiv/pdf/0910/0910.5130v1.pdf#page=13) of

* [[Paul Goerss]], _Topological modular forms (aftern Hopkins, Miller, and Lurie)_ ([arXiv](http://arxiv.org/abs/0910.5130))

See also a very precise and detailed treatment in

* [[Dmitri Orlov]], _Quasi-coherent sheaves in commutative and non-commutative geometry_, Izv. RAN. Ser. Mat., 2003, __67__:3, 119–138 (preprint versions: [pdf](http://web.archive.org/web/20210506231653/http://www.mi-ras.ru/~orlov/papers/NST.pdf) [dvi](http://www.mpim-bonn.mpg.de/preprints/send?bid=57), [ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=56))

###Direct definition for presheaves of sets on Aff {#AsSheavesII}

Here is a more detailed way to say again what the above paragraph said.

Let $Aff = CRing^{op}$; recall the fibered category $Mod\to CRing^{op}$ where for each $f:A\to B$ in $CRing$ the inverse image functor is $f^*=B\otimes_A - :{}_A Mod\to {}_B Mod$. Then the identity functor $CRing\to CRing$ can be interpreted as the presheaf of rings and is denoted by $O$ (the "structure sheaf"). An $O$-module is a presheaf of $O$-modules. Usually some Grothendieck topology is given and one asks for sheaves in fact. We can Yoneda extend $O$-modules to presheaves. We now define quasicoherent sheaves of $O$-modules on an arbitrary presheaf $X$ on $Aff$, viewed as a covariant functor on $CRing$.

A **quasicoherent sheaf of $O$-modules** on $X$ is a rule assigning to any $\phi\in X(A)$ an $A$-module $M_\phi = M_{A,\phi}$ and to any morphism $f:A\to B$ in $CRing$ an isomorphism $\theta_{f,\phi}:f^*(M_\phi)\to M_{X(f)(\phi)}$ such that for any [[composable pair]] $A\stackrel{f}\to B\stackrel{g}\to C$ and any $\phi\in X(A)$ the cocycle condition

$$
\theta_{g\circ f,\phi}\circ \alpha_{fg} = \theta_{g,X(f)(\phi)}\circ g^*(\theta_{f,\phi})\colon g^* f^*(M_\phi)\to M_{X(g\circ f)(\phi)}
$$

holds, where $\alpha_{fg}:g^* f^*(M_\phi)\to (g\circ f)^*(M_\phi)$ is the canonical isomorphism which is part of the data of the (covariant) pseudofunctor $A\to {}_A Mod$, $f\mapsto f^*$.

Notice that if $X = h^C = h_{Spec C}$ is (co)representable presheaf, then $\phi\in [A,X]_{Pshv(Aff)}=[C,A]_{CRing}$ is the same as a morphism $\phi^{op}:C\to A$ of rings; restricting the quasicoherent sheaf to $Spec A$ along $\phi:Spec A\to X$ and taking the global sections over $A$, would give the $A$-module $M_\phi$.

Clearly, $Aff$ and $O$ can be much generalized. For example, rings may be noncommutative or one can take category opposite to the category of monads in $Set$ and an arbitrary (not identity) presheaf $D$ of monads in $Set$; the extension of scalars for monads gives an inverse image functor for Eilenberg-Moore categories. Durov's construction of quasicoherent sheaves for monads in $Set$ is an example where commutative algebraic monads are used; the theory of quasicoherent sheaves of $D$-modules ("$O$-modules with integrable connection") is another. Instead setups involving operads, higher operads and alike can be used as well; commutativity condition is useful if one wants a monoidal category of quasicoherent sheaves.

### As homs into the stack of modules {#AsCocycles}

The above definition may be further re-interpreted as follows.

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

On the [[site]] $Aff = CRing^{op}$, let

$$
  QC : CRing \to Cat
$$

$$
  (Spec S \stackrel{f^{op}}{\to} Spec R)
  \mapsto (R Mod \stackrel{S \otimes_{f} -}{\to} S Mod)
$$

be the (pseudo)functor ([[stack]]) corresponding to the canonical [[Grothendieck fibration]] of [[module]]s $Mod \to CRing$. Its right [[Kan extension]] through the 2-[[Yoneda embedding]]  $Y : CRing^{op} \hookrightarrow [CRing,Cat]$ is given on a presheaf $X : CRing \to Set$ by the [[hom-object]]

$$
  QC(X) := (Ran_Y QC)(X) := [CRing,Cat](X,QC)
  \,.
$$

When $X$ is the functor [[representable functor|represented]] by
a scheme, then $QC(X)$ is the category of quasicoherent sheaves on
$X$, as defined above.

=--

We now explain the above statement in detail and thereby prove it.

Let $C = $[[Ring]]${}^{op}$ be the category of (commutative, unital) [[ring]]s.
For $R$ a [[ring]] write $Spec R$ for it regarded as an object of $C$.
Write $Spec f = f^{op} : Spec(S) \to Spec(R)$ for the morphism in $Ring^{op}$ corresponding to the map $f : R \to S$ of commutative rings.

Consider the [[2-category]] of (pre)[[stack]]s on
$C$. The canonical [[module]] [[bifibration]] $p : Mod \to Ring$ of the category of modules over all rings is the bifibration whose fibered part corresponds to the (pre)stack $QC \in [C^{op},Cat]$ given on objects by

$$
  QC : Spec R \mapsto R Mod
$$

and on morphisms by

$$
  QC : (Spec S \stackrel{f^{op}}{\to} Spec R)
         \mapsto
       (R Mod \stackrel{S\otimes_{f} }{\to} S Mod)
  \,,
$$

where on the right we have the functor that sends any $R$-module $N$ to the
[[tensor product]] over $S$ with the $R$-$S$-[[bimodule]]
$S = {}_S S_R$ with its canonical left $S$-action and with the right $R$-action induced by the ring homomorphism $f$.

One may think of this as the stack of generalized algebraic vector bundles:

the operation $S \otimes_{f} - : R Mod \to S Mod$ corresponds to the
[[pullback]] of [[bundle]]s along a morphism of the underlying spaces.
(See for instance the discussion of [[monadic descent]] at [[Sweedler coring]] for more on this.)

We may [[Kan extension|right Kan extend]] the 2-functor $QC : CRing^{op} \to Cat$ through the [[Yoneda embedding]] $CRing^{op} \hookrightarrow [CRing,Cat]$
to get a definition of $QC$ on arbitrary [[presheaf|presheaves]].

$$
  \array{
    CRing^{op} &\stackrel{QC}{\to}& Cat
    \\
    {}^{Y}\downarrow & \nearrow_{\mathrlap{Ran_Y QC}}
    \\
    [CRing,Cat]
  }
  \,.
$$

Consider $X \in [C^{op},Set]$ any ([[presheaf|pre]])[[sheaf]] on $C$.
This may be the presheaf [[representable functor|represented]]
by a [[scheme]], but for the purposes of the definition of $QC$ it may be much more generally any presheaf.

By the general formula for [[Kan extension]] in terms of a [[weighted limit]] given by an [[end]]
we have

$$
  Ran_Y QC : X \mapsto \int_{R \in Ring} ([Ring,Cat]^{op}(X,Y(R)))^{QC(R)}
$$

which using the [[Yoneda lemma]] is

$$
  \cdots = \int_{R \in CRing} [CRing,Cat](X(R), QC(R))
  \,.
$$

This is the [[end]]-formula for the [[hom-object]] in an [[enriched functor category]]
$[C^{op},Cat]$,
hence this is nothing but the category of (pseudo)[[natural transformation]]s between
the 2-functor $X$ and the 2-functor $QC$.

We write for short

$$
  QC(X) := (Ran_Y QC)(X) := [C^{op},Cat](X,QC)
  \,.
$$

This definition of "generalized vector bundles" on arbitrary presheaves is
entirely analogous to the definition of differential forms on arbitrary
presheaves, that is discussed in some detail for instance in the entry on
[[rational homotopy theory]].

We claim that the category $QC(X)$ is the category of quasicoherent sheaves on
$X$ as defined by other means above, whenever that other definition applies to $X$.

To see this, straightforwardly unwrap the definition:
an object $N$ in $QC(X) = [C^{op},Cat](X,QC)$
is a [[transfor|pseudonatural transformation]] of 2-functors $N : X \to QC$,
where $X$ is regarded as a 2-functor by the canonical embedding
$disc : Set \hookrightarrow Cat$ that regards a [[set]] as a [[discrete category]].

The components of $N$ are

* for each $Spec R \in Ring^{op}$ a [[functor]] $N|_{Spec R} : X(R) \to QC(R) = R Mod$:

  this functor picks one $R$-module $N(r) \in R Mod$ for each plot $(r : Spec R \to X) \in X(Spec R)$;

* for each morphism $f : Spec A \to Spec B$
  a pseudonaturality square

  $$
    \array{
      X(Spec B)
         &\stackrel{X(f)}{\to}&
      X(Spec A)
      \\
      {}^{\mathllap{N|_{Spec B}}}\downarrow
      &{}^{\Gamma(f)}\swArrow_{\simeq}&
      \downarrow^{\mathrlap{N|_{Spec A}}}
      \\
      B Mod &\underset{- \otimes_{f^*} A}{\to}& A Mod
    }
  $$

  in [[Cat]] (these are subject to coherence conditions). This unwraps to the following data:

  * the component functors $N|_{Spec A}$ provide an
    assignment $a \mapsto N(a)$ of modules $N(a)$ to each
    plot $(a : Spec A \to X) \in X(Spec A)$;

  * these assignments form a presheaf on the [[overcategory]]
    $Aff/X$ by taking the restriction morphism

    $$
      N(f) : N(b) \to N(a)
    $$

    to be that underlying the components of the
    natural isomorphism in the above diagram

    $$
      N(b)\otimes_{f^*} A \stackrel{\simeq}{\to} N(a)
      \,,
    $$

    i.e. the restriction of this morphism to $(n,1)$.

* for each tuple of composable morphisms

  $$
    \array{
      && Spec B
      \\
      & {}^f\nearrow && \searrow^g
      \\
      Spec A &&\to&& Spec C
    }
  $$

  a pseudo-naturality prism equation relating, $N(f)$, $N(g)$ and $N(g\circ g)$.  The present author is too lazy to write out the diagram in detail, but it is of precisely the kind described in great detail for instance in the entry on [[group cohomology]]. Under the above identification, this yields the cocycle condition mentioned  in the above definitions.

This way, the transformation $N : X \to QC$ defines
manifestly a quasicoherent sheaf on $Aff/X$ in the sense of the
definition in the above section [As sheaves on Aff/X](#AsSheaves). Conversely, every quasicoherent sheaf according to that definition gives rise to a transformation $N : X \to QC$ under this prescription.

### As cartesian morphisms of fibrations {#AsFibHoms}

By the equivalence between [[pseudofunctor]]s $Ring \to Cat$
and [[Grothendieck fibration]]s $F \to Ring^{op}$ induced by the
[[Grothendieck construction]], the above may equivalently be
reformulated as follows.

Recall from the discussion at [[Grothendieck fibration]] that the equivalence in question is between the following two [[bicategory|bicategories]]:

* on the one hand the bicategory whose objects are [[pseudofunctors]]
  $Ring \to Cat$, whose morphisms are pseudonatural
  transformations, and whose 2-morphisms are modifications of these

* on the other hand the bicategory whose objects are [[Grothendieck fibration]]s $F \to Ring^{op}$, whose morphism are **cartesian functors**

  $$
    \array{
      F_1 &&\to&& F_2
      \\
      & \searrow && \swarrow
      \\
      && Ring^{op}
    }
  $$

  and whose 2-morphisms are [[natural transformation]]s between these.

Recall furthermore that for $X : Ring \to Cat$ an ordinary [[presheaf]], i.e. a pseudofunctor that factors through an ordinary functor $Ring \to Set$ via the inclusion $Set \to Cat$, the [[Grothendieck fibration]] associated with $X$ is the [[category of elements]] $Ring^{op}/X$ of $X$.

Recall furthermore that by definition, the pseudofunctor $QC : Ring Cat$ is the one corresponding to the [[Grothendieck fibration]] $Mod^{op} \to Ring^{op}$.

Therefore, by the above equivalence of 2-categories, we find that the category of functors $[Ring,Cat](X,QC)$ is equivalent to the category of cartesian functors over $Ring^{Op}$, $CartFunc_E(Ring^{op}/X,Mod^{op})$

$$
  QC(X) \simeq [Ring,Cat](X,QX) \simeq CartFunc(Ring^{op}/X, Mod^{op})
  \,.
$$

In this form quasicoherent sheaves on $X$ are conceived for instance in paragraph 1.1.5 of

* [[Maxim Kontsevich]], [[Alexander Rosenberg]], _Noncommutative stacks_ ([ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=2333))

Here, as in the above discussion, the [[fibered category]] of modules can be replaced by a more general fibered category $\pi: \mathcal{F}\to\mathcal{B}$. Then the __category of quasicoherent modules in__ this __fibered category__ is the category opposite to the category of cartesian sections of $\pi$. This viewpoint is used by Rosenberg-Kontsevich in their preprint on noncommutative stacks ([dvi](http://www.mpim-bonn.mpg.de/preprints/send?bid=2305), [ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=2333)).

Given a category $\mathrm{Aff}$ of affine schemes (opposite to the category of rings) equipped with some [[subcanonical coverage|subcanonical pretopology]] one considers the [[stack]] of $O$-modules over $\mathrm{Aff}$: the fiber over a ring $R$, it assigns the category $Qcoh(\mathrm{Spec}\,R)$. Now given any stack on a subcanonical site, one defines the fiber over a sheaf on it so that the fiber over a representable sheaf is equivalent to the fiber over its representing object. There is a canonical way to do this (will write later about it -- Zoran); this is in particular a source of a definition $Qcoh$ on an ind-scheme. On ind-schemes Beilinson and Drinfel'd in

* A. Beilinson, V. Drinfel'd, _Quantization of Hitchin's integrable system and Hecke eigensheaves on Hitchin system_, preliminary version ([pdf](http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf))

consider two variants: a less important variant of quasicoherent $O_X^p$-modules (existing in bigger generality) and more delicate variant of quasicoherent $O^!_X$-modules defined for "reasonable ind-schemes"; one of the differences is which functors play the role of pullbacks.  In particular, these notions apply for a rather general variant of the category of formal schemes.

### Quasicoherent modules in higher/derived geometry
 {#Higher}

The last definition has a  straightforward generalization to various [[higher geometry]] setups, such as [[derived schemes]] and other [[generalized schemes]].

#### By maps into the stack $QCoh$

For instance the notion of
quasicoherent sheaves generalized to [[derived stack]]s on the
site of [[simplicial ring]]s as described at
[[geometric ∞-function theory]] is obtained, we claim,
simply by taking
$QC : SRing \to (\infty,1)Cat$ to be the functor that assigns the
[[(∞,1)-category]] for modules over a simplicial ring to any
simplicial ring, and then setting for any derived stack $X$

$$
  QC(X) := Hom(X,QC)
  \,.
$$

Moreover, using the theorem described at [[tangent (∞,1)-category]],
that the [[bifibration]] of modules over simplicial rings is nothing
but the [[tangent (∞,1)-category]] of $SRing$, one sees that
all this is a special case of an even much more general abstract nonsense:

for any presentable [[(∞,1)-category]] [[site]] $C$ whatsoever, we have
the [[tangent (∞,1)-category]] fibration $T_C \to C$. With the [[(∞,1)-functor]] classifying it denoted
$QC : C^{op} \to (\infty,1)Cat$ we may adopt for any
[[∞-stack]] $X : C^{op} \to \infty Grpd$ the definition

$$
  QX(X) := [C^{op},\infty Grpd](X,QC)
$$

as a definition of generalized $\infty$-vector bundles on $X$. This general nonsense is considered further at [[schreiber:∞-vector bundle]]. Concrete realizations are discussed at [[quasicoherent ∞-stack]].

#### As extensions of the structure sheaf
 {#HigherGeometryAsExtensionsOfStructureSheaf}

In ([LurieQC, section 2.2, section 2.3](#LurieQC)) the following definition is given.

Let $\mathcal{G}$ be a [[geometry (for structured (∞,1)-toposes)]]. Let

$$
  \mathcal{G}^{mod} \coloneqq (T (\mathcal{G}^{op}))_{cpt}^{op}
$$

be the opposite of the [[full sub-(∞,1)-category]] on the [[compact objects]] of the [[tangent (∞,1)-category]] of its [[opposite (∞,1)-category]].

For instance for [[E-∞ geometry]] we have $\mathcal{G} = CRing_\infty$ is the [[(∞,1)-category]] of [[E-∞ rings]] with [[etale morphisms]] as admissible maps.

([LurieQC, above Notation 2.2.4](#LurieQC))

Then the canonical  [[(∞,1)-functor]]

$$
  \mathcal{G} \longrightarrow \mathcal{G}^{mod}
$$

is a morphism of discrete [[geometry (for structured (∞,1)-toposes)|geometries]].

For $\mathcal{X}$ an [[(∞,1)-topos]], a [[left exact (∞,1)-functor]]

$$
  \mathcal{O} \colon \mathcal{G} \longrightarrow \mathcal{X}
$$

constitutes a $\mathcal{G}$-[[structure sheaf]] and makes $(\mathcal{X}, \mathcal{O})$ be a $\mathcal{G}$-[[structured (∞,1)-topos]]. A left exact extension of this

$$
  \array{
    \mathcal{G} &\stackrel{\mathcal{O}}{\longrightarrow}& \mathcal{X}
    \\
    \downarrow & \nearrow_{\mathrlap{(\mathcal{O}, \mathcal{F})}}
    \\
    \mathcal{G}^{mod}
  }
$$

exhibits a sheaf $\mathcal{F}$ of $\mathcal{O}$-[[modules]] on $\mathcal{X}$.

([LurieQC, notation 2.2.4](#LurieQC))

Now if $(\mathcal{X},\mathcal{O})$ is [[locally representable structured (infinity,1)-topos]] then such an $\mathcal{O}$-module $\mathcal{F}$ is _quasi-coherent_ if also $(\mathcal{X}, (\mathcal{O}, \mathcal{F}))$ is locally representable.

([LurieQC, def. 2.3.6](#LurieQC))

### Synthetic definition using the internal language {#SyntheticDescription}

Let $X$ be a [[scheme]]. Recall that the [[big Zariski topos]] of $X$ is the topos of sheaves over $Sch/X$ (or, more precisely, the affine schemes over $X$ which are locally of finite presentation). In this topos, there is a [[local ring]] $\mathbb{A}^1$, the sheaf mapping an $X$-scheme $T$ to $\mathcal{O}_T(T)$.

A sheaf $N$ of modules over $\mathbb{A}^1$ is quasicoherent if and only if, from the [[internal language|internal point of view]] of the big Zariski topos, the canonical map
$$ N \otimes_{\mathbb{A}^1} A \longrightarrow Hom(Hom_{\mathbb{A}^1-Alg}(A, \mathbb{A}^1), N) $$
is bijective for all finitely presented $\mathbb{A}^1$-algebras $A$. The outer Hom set is the set of all maps from the set $Hom_{\mathbb{A}^1-Alg}(A, \mathbb{A}^1)$ to the (underlying set of) $N$.

This characterization has a geometric interpretation. The set $Hom_{\mathbb{A}^1-Alg}(A, \mathbb{A}^1)$ deserves the name "spectrum of $A$", since it consists of what classically is known as the ($\mathbb{A}^1$-)rational points of $A$. Furthermore, if $A$ is induced from a sheaf of $\mathcal{O}_X$-modules, then the object of the Zariski topos which is described by this set-theoretical expression coincides with the [[functor of points]] of the [[relative spectrum]] of that sheaf.

The set $Hom(Hom_{\mathbb{A}^1-Alg}(A, \mathbb{A}^1), N)$ is therefore the set of all $N$-valued functions on the spectrum of $A$. An element of $N \otimes_{\mathbb{A}^1} A$ gives rise to such a function: associate to a pure tensor $x \otimes f$ the function $\varphi \mapsto f(\varphi) x$.

In a [[synthetic differential geometry|synthetic]]/algebraic context, there should be no more functions than those which result from this construction. This is what the characterization expresses.

## Properties

### Quasicoherent sheaves over affine schemes

Given an affine scheme $X=\mathrm{Spec}\,R$ (where $R$ is a commutative unital ring), the affine Serre theorem establishes the equivalence of the category $Qcoh(\mathrm{Spec}\,R)$ of quasicoherent sheaves (in Zariski topology) and the category of $R$-modules. Similarly on a projective scheme of the type $Proj(A)$ where
$A$ is a nonnegatively graded ring, the (projective) Serre theorem establishes the equivalence of $Qcoh(\mathrm{Proj}\,(A))$ and
the localization of the category of graded $A$-modules by the subcategory of modules of finite length (and similarly, of coherent sheaves and graded $A$-modules of finite type modulo finite-length). These theorems are among basic motivating theorems for [[noncommutative algebraic geometry]]. An interesting in-depth comparison of the notions of quasi-coherent sheaves in commutative and noncommutative context are also in Orlov's article quoted above.

### The category of quasicoherent sheaves

In the case of general (commutative) schemes, every presheaf of $O_X$-modules which is quasicoherent in the sense of having local presentation as above, is in fact a sheaf. It is known that the category of quasicoherent sheaves of $O_X$-modules over any [[quasicompact]] quasiseparated scheme is a [[Grothendieck category]] and in particular has enough [[injective object]]s.

## Related concepts

### General

* [[coherent sheaf]]

* [[modules over a ring are equivalent to quasicoherent sheaves over its spectrum]]

* [[Bondal-Orlov reconstruction theorem]]

* [[module over a derived stack]]

### D-Modules

The category of [[D-module]]s on a [[space]] $X$ is equivalently that of quasicoherent sheaves on the corresponding [[deRham space]].

## References

Quasicoherent sheaves in [[E-∞ geometry]] (on "[[Spectral Schemes]]" over [[E-∞ rings]]) are discussed in

* [[Jacob Lurie]], _[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]_
 {#LurieQC}

Their [[descent]] properties are discussed in

* [[Jacob Lurie]], _[[Descent Theorems]]_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XI.pdf))

and a [[Grothendieck existence theorem]] for [[coherent sheaves]] in this higher context is discussed in

* [[Jacob Lurie]], _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XII.pdf))

On a geometric stack

* Leovigildo Alonso Tarrío, Ana Jeremías López, Marta Pérez Rodríguez, María J. Vale Gonsalves, _A functorial formalism for quasi-coherent sheaves on a geometric stack_, Expositiones Mathematicae __33__:4 (2015) 452-501 [doi](https://doi.org/10.1016/j.exmath.2014.12.007)

category: algebraic geometry

[[!redirects quasicoherent sheaves]]
[[!redirects quasicoherent module]]
[[!redirects quasicoherent modules]]
[[!redirects quasi-coherent sheaves]]
[[!redirects quasi-coherent module]]
[[!redirects quasi-coherent modules]]
[[!redirects quasi-coherent sheaf]]
[[!redirects quasicoherent sheaf of modules]]
[[!redirects quasicoherent sheaves of modules]]


quasicoherent ∞-stack > history

< [[quasicoherent ∞-stack]]


quasicompact > history

< [[quasicompact]]


quasicompact morphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Compact objects
+--{: .hide}
[[!include compact object - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The modifier __'quasi-compact'__ (or simply __'quasicompact'__) is used to denote a compactness property in the relative setup (that is, for morphisms) and in the setups emphasising non-Hausdorff topology. For example, schemes over complex numbers have both the [[complex analytic topology]] and the [[Zariski topology]]; we use "quasicompact" for Zariski and "compact" for complex topology in that setup.

For many topologists, analysts and so on, the word 'compact' means a [[compact Hausdorff space]]. Some topological schools distinguish a "[[compact space]]" (which is not necessarily [[Hausdorff space|Hausdorff]]) and a "[[compactum]]" (which is Hausdorff); and similarly a "[[paracompact space]]" and a "[[paracompactum]]". In [[algebraic geometry]], in contrast, one usually says __quasicompact space__ to denote a topological space which is compact but not necessarily Hausdorff. For example, the [[Zariski topology]] on an [[algebraic variety]] and the topology of an étalé space of a [[sheaf]] (even over a Hausdorff topological space) are typically not Hausdorff.

## Definition

A [[scheme]] is quasicompact iff it has a Zariski cover by [[finite number|finitely many]] open affine subschemes. In particular, any affine scheme is quasicompact.

Most important is the relative version of this concept. A morphism $f\colon X \to Y$ of schemes is a __quasicompact morphism__ if the inverse image of a quasicompact Zariski open subset of $Y$ is quasicompact (EGAI6.6.1).

It is straightforward to show [EGAI6.6.4] that it is enough to require this for affine subsets of $Y$, or even to require the existence of a single covering $Y = \cup_i U_i$ of $Y$ by open affine subschemes $U_i$, such that the inverse image $U_i \times_Y X$ of $U_i$ in $X$ is quasicompact. A scheme $X$ over a base scheme $S$ is quasicompact if the canonical morphism $X \to S$ is quasicompact. This is consistent, because if $X$ is a usual scheme (over the spectrum of integers $S = \mathbb{Z}$) or, more generally, a relative scheme over an _affine_ scheme $S$, quasicompactness of the canonical morphism $X\to S$ is by the above criteria clearly equivalent to the usual quasicompactness of $X$.

## Properties

A composition of quasicompact morphisms is quasicompact, and the pullback of quasicompact morphisms is quasicompact. This enables the definition for [[algebraic stack]]s: a morphism of algebraic stacks is quasicompact if the pullback of that morphism to some atlas is quasicompact.

Algebraic geometers sometimes (but more rarely) also talk about __quasicompact objects__ in more general categories, meaning [[compact object]]s (object which corepresent covariant functors commuting with filtered colimits); with or without a modifier denoting a cardinal ($\kappa$-quasicompact objects).

## Related concepts

* [[finitely presented morphism]]

## References

An old discussion on the terminological aspects (Mike, Zoran, Toby) is at $n$Forum [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2680&Focus=22689#Comment_22689).

category: algebraic geometry
[[!redirects quasicompact]]
[[!redirects quasi-compact]]

[[!redirects quasicompact morphism]]
[[!redirects quasicompact morphisms]]
[[!redirects quasi-compact morphism]]
[[!redirects quasi-compact morphisms]]

[[!redirects quasicompact scheme]]
[[!redirects quasicompact schemes]]
[[!redirects quasi-compact scheme]]
[[!redirects quasi-compact schemes]]

[[!redirects quasicompact map]]
[[!redirects quasicompact maps]]
[[!redirects quasi-compact map]]
[[!redirects quasi-compact maps]]


quasicompact quasiseparated morphism

## Idea

A morphism of [[schemes]] (and more general objects, e.g., stacks) that is [[quasicompact]] and [[quasiseparated]].

## An analogue of good covers for quasicompact quasiseparated schemes

Quasicompact quasiseparated schemes are characterized by a property that they admit an analogue (of sorts) of a [[good cover]]: such schemes are precisely those schemes that can be covered by finitely many affine opens such that the intersection of any two opens can itself be covered by finitely many affine opens.

## Related concepts

* [[quasicompact morphism]], [[quasiseparated morphism]]

* [[good cover]]

## References

* [[Ravi Vakil]], [[The Rising Sea]], 5.1.2.


quasicrystal

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Properties

### Noncommutative Brillouin zone

The [[Brillouin zone]] of quasicrystals may be understood as having [[noncommutative topology]]/[[noncommutative geometry]] ([Bellissard 03](#Bellissard03), [Ypma 05, p. 8](#Ypma05), [Putnam 10](#Putnam10)).

For more see at *[[noncommutative topology of quasiperiodicity]]*.

## References

### General

Survey:

* Christian Janot, *The Properties and Applications of Quasicrystals*, Europhysics News 27 (1996) ([pdf](https://www.europhysicsnews.org/articles/epn/pdf/1996/02/epn19962702p60.pdf))

* Christian Janot, *Quasicrystals -- A Primer*, Oxford Classic Texts in the Physical Sciences, Oxford University Press, 2012 ([ISBN:978-0-19-965740-7](https://global.oup.com/academic/product/quasicrystals-9780199657407?cc=ae&lang=en&))

See also

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

### As topological phases

As or in relation to [[topological phases of matter]]:

* Yao Wang et al., *Quantum Topological Boundary States in Quasi-crystal*,  Adv Mater 2019 Dec;31(49):e1905624 ([doi:10.1002/adma.201905624](https://doi.org/10.1002/adma.201905624))

* Dominic V. Else, Sheng-Jie Huang, Abhinav Prem, Andrey Gromov, *Quantum many-body topology of quasicrystals* ([arXiv:2103.13393](https://arxiv.org/abs/2103.13393))

* Oded Zilberberg, *Topology in quasicrystals*, Optical Materials Express Vol. 11, Issue 4, pp. 1143-1157 (2021) ([doi:10.1364/OME.416552](https://doi.org/10.1364/OME.416552))

As a substrate for [[topological quantum computation]]:

* Marcelo Amaral, [[David Chester]], Fang Fang, Klee Irwin, *Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing*, Symmetry **14** 9 (2022) 1780 [[arXiv:2207.08928](https://arxiv.org/abs/2207.08928), [doi:10.3390/sym14091780](https://doi.org/10.3390/sym14091780)]

### Mathematical description

Mathematical disucssion:

* A. Hof, *On diffraction by aperiodic structures*, Commun. Math. Phys., 169 (1995), p. 25-43.

* J-B. Gouere, *Quasicrystals and almost periodicity*, Commun. Math. Phys., 255 (2005), p. 655-681.

Discussion of the [[noncommutative topology]]/[[KK-theory]] of the Brillouin zone in the spirit of the [[K-theory classification of topological phases of matter]]:

* {#Bellissard03} Jean Bellissard, *The Noncommutative Geometry of Aperiodic Solids*, in: Geometric and Topological Methods for Quantum Field Theory, pp. 86-156 (2003) ([pdf](http://people.math.gatech.edu/~jeanbel/Publi/ncg02.pdf), [doi:10.1142/9789812705068_0002](https://doi.org/10.1142/9789812705068_0002))

* {#Ypma05} Fonger Ypma, *Quasicrystals, $C^\ast$-algebras and K-theory*, 2005 ([pdf](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.214.4414&rep=rep1&type=pdf))

* {#Putnam10} Ian F. Putnam, *Non-commutative methods for the K-theory of $C^\ast$-algebras of aperiodic patterns from cut-and-project systems*,  Commun. Math. Phys. 294, 703–729 (2010) ([pdf](https://www.math.uvic.ca/faculty/putnam/r/0802_CMP.pdf), [doi:10.1007/s00220-009-0968-0](https://doi.org/10.1007/s00220-009-0968-0))

* Hervé Oyono-Oyonoa, Samuel Petite, *$C^\ast$-algebras of Penrose hyperbolic tilings*, Journal of Geometry and Physics Volume 61, Issue 2, February 2011, Pages 400-424 ([doi:10.1016/j.geomphys.2010.09.019](https://doi.org/10.1016/j.geomphys.2010.09.019))

For more see at *[[noncommutative topology of quasiperiodicity]]*.

### Phenomenology

Analysis of possible quasicrystal configurations:

via [[machine learning]]:

* Roman Eremin, Pavel Zolotarev, Tilmann Leisegang, Pavlo Solokha,
*A machine learning approach for predicting formation enthalpy: A case study of Mackay-type approximants of icosahedral quasicrystals*, AIP Conference Proceedings 2163, 020003 (2019) ([doi:10.1063/1.5130082](https://doi.org/10.1063/1.5130082))

via [[topological data analysis]]:

* Pavlo Solokha et al., *New Quasicrystal Approximant in the Sc–Pd System: From Topological Data Mining to the Bench*, Chem. Mater. 2020, 32, 3, 1064–1079 ([doi:10.1021/acs.chemmater.9b03767](https://doi.org/10.1021/acs.chemmater.9b03767))

* Søren S. Sørensen, *Revealing hidden medium-range order in amorphous materials using topological data analysis*, Science Advances  09 Sep 2020:
Vol. 6, no. 37, eabc2320 ([doi:10.1126/sciadv.abc2320](https://doi.org/10.1126/sciadv.abc2320))

### In AdS/CFT

Discussion of [[asymptotic boundaries]] of [[hyperbolic space|hyperbolic]] [[tensor networks]] as conformal [[quasicrystals]] (see also at [[AdS/CFT in solid state physics]]):

* Latham Boyle, Madeline Dickens, Felix Flicker, _Conformal Quasicrystals and Holography_, Phys. Rev. X 10, 011009 (2020) ([arXiv:1805.02665](https://arxiv.org/abs/1805.02665))

* [[Alexander Jahn]], [[Zoltán Zimborás]], [[Jens Eisert]], _Central charges of aperiodic holographic tensor network models_, Phys. Rev. A 102, 042407 ([arXiv:1911.03485](https://arxiv.org/abs/1911.03485))

  > (via [[Majorana dimer]] [[Majorana dimer code|codes]])

[[!redirects quasicrystals]]

[[!redirects quasi-crystal]]
[[!redirects quasi-crystals]]


quasideterminant

In the noncommutative case, [[determinant]]s are not useful invariants of [[matrix|matrices]] (in fact, various classical formulas for determinants mutually disagree over noncommutative [[ring]]s) and other [[polynomial]] suggestions were not of much success (in some cases the [[superdeterminant]] and [[Dieudonné determinant]] are of use, but they can be easily expressed in terms of quasideterminants anyway). Quasideterminants will be [[noncommutative rational function]]s, rather than polynomial, expressions.

## Definition

Let $A = (a^i_j)\in M_n(R)$ be an $n\times n$ matrix
over an arbitrary noncommutative (but unital and associative) [[ring]] $R$. In fact it makes sense to work with many objects (see [[horizontal categorification]]): having, say, an [[abelian category]] where $a^i_j$ is a morphism from the object $i$ to the object $j$.
Let us choose a row label $i$ and a column label $j$.
By $A^{\hat{i}}_{\hat{j}}$ we'll denote the $(n-1)\times(n-1)$ matrix obtained
from $A$ by removing the $i$-th row and the $j$-th column.
The $(i,j)$-th **quasideterminant** $|A|_{ij}$ is
\[ |A|_{ij} = a^i_j -
\sum_{k \neq i, l\neq j} a^i_l  (A^{\hat{i}}_{\hat{j}})^{-1}_{lk} a^k_j
\]
provided the right-hand side is defined (the corresponding inverses exist).

## Properties

Up to $n^2$ quasideterminants of a given $A \in M_n(R)$
may be defined. If all the $n^2$ quasideterminants $|A|_{ij}$ exist
and are invertible then the inverse $A^{-1}$ of $A$ exists in
$M_n(R)$ and

$$
(|A|_{ji})^{-1} = (A^{-1})^i_j.
$$

Quasideterminants for a matrix with entries in a commutative ring $R$ are, up to a sign,
ratios of the determinant of the matrix and the determinant
of the appropriate $(n-1)\times (n-1)$ submatrix, as one can see by remembering the formula for matrix inverse in terms of the cofactor matrices. For systems of linear equations with coefficients (from one side) in a noncommutative ring, there are solution formulas involving quasideterminants and generalizing Cramer's rule (hence left and right Cramer's rule). Quasideterminants with generic entries (entries in a free skewfield) satisfy generalizations of many classical identities: Muir's law of extensionality, Silvester's law, etc., and a new "heredity" principle and so-called "homological identities". Furthermore, for polynomial equations over noncommutative rings, noncommutative Viete's formulas have been found and used in applications.

## References

Quasideterminants were introduced by I. Gel'fand and V. Retakh around 1990.

* [[I. M. Gel'fand]], [[V. Retakh|V. S. Retakh]], _Determinants of matrices over noncommutative rings_, Funct.Anal.Appl. __25__ (1991), no.2, pp. 91--102.
engl. transl. __21__ (1991), pp. 51--58.

* I.M. Gel'fand, V.S. Retakh, A theory of noncommutative determinants and characteristic functions of graphs, Funct.Anal.Appl. __26__ (1992), no.4, pp. 231--246.

* I.M. Gel'fand, V.S. Retakh, _Quasideterminants I_, Selecta Mathematica, N. S. 3 (1997) no.4, pp. 517--546; [doi](https://doi.org/10.1007/s000290050019)

* D.Krob, B.Leclerc, _Minor identities for quasi-determinants and quantum determinants_, Comm.Math.Phys. 169 (1995), pp. 1--23 [doi](https://doi.org/10.1007/BF02101594)

* Chapter 16: Quasideterminants and Cohn localization in [[Z. Škoda]], _Noncommutative localization in noncommutative geometry_, London Math. Society Lecture Note Series 330, ed.  A. Ranicki; pp. 220--313, [arXiv:math.QA/0403276](http://arxiv.org/abs/math/0403276))

* [[V. Retakh]], R. Wilson, _Advanced course on quasideterminants and universal localization_ (2007) [[[RetakhWilson-Quasideterminants.pdf:file]]]

category: algebra
[[!redirects quasideterminants]]


quasifibration

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

#Contents#
* table of contents
{:toc}

## Definition

A **quasifibration** is a kind of [[fibration]], a [[morphism]] in [[homotopy theory]] that acts like a [[fibration]] in that the actual fibres are, via the canonical inclusions, [[weak homotopy equivalence|weakly homotopy equivalent]] to the [[homotopy fiber|homotopy fibres]] (note that no [[lifting properties]] are used in the definition). As a result the [[long exact sequence]] in homotopy for the replacement of the map by a fibration becomes an long exact sequence for the map itself.

## Related concepts

* [[Hopf construction]]

## References

For variations on the definition and some history, see

* [[Tom Goodwillie]], email to Don Davis and others, 16 May 2001 ([txt](http://www.lehigh.edu/~dmd1/tg516.txt))

Also see

* [[Peter May]], _Weak Equivalences and Quasifibrations_, in: Groups of Self-Equivalences and Related Topics, Lecture Notes in Mathematics Volume 1425, 1990, pp 91-101 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/67.pdf), doi: 10.1007/BFb0083834)

[[!redirects quasifibrations]]
[[!redirects quasi-fibration]]
[[!redirects quasi-fibrations]]


quasifield

## Outline

A quasifield (earlier also called a Veblen-Wedderburn system) is an algebraic structure with two [[binary operation]]s $+$ and $\cdot$ which is weaker than a [[division ring]] and which is motivated by [[synthetic projective geometry]]. There are the
left and right versions. Associative right quasifield is the same as a [[near-field]].

## Definition

A __left quasifield__ is an algebraic structure $(A,+,\cdot)$ with two binary operations $+$ and $\cdot$ such that

* $(A,+)$ is a group with neutral element $0$
* $(A\backslash \{0\},\cdot)$ is a [[loop (algebra)|loop]]
* left distributivity $a\cdot(b+c) = a\cdot b + a\cdot c$ for all $a,b,c\in A$
* if $a,b,c\in A$, $a\neq b$ then the equation $a\cdot x = b \cdot x + c$ has a unique solution for $x\in A$

A left quasifield is Abelian if the underlying group $(A,+)$ is [[Abelian group|Abelian]].

## Literature

* wikipedia [quasifield](http://en.wikipedia.org/wiki/Quasifield)
* O. Veblen, J.H.M. Wedderburn, _Non-Desarguesian and non-Pascalian geometries_, Trans AMS __8__, 379–388 (1907) [pdf](http://www.ams.org/tran/1907-008-03/S0002-9947-1907-1500792-1/S0002-9947-1907-1500792-1.pdf)
* Marshall Hall Jr., _Projective planes_, Trans. Amer. Math. Soc. __54__: 229–277 (1943) [pdf](http://www.ams.org/journals/tran/1943-054-02/S0002-9947-1943-0008892-4/S0002-9947-1943-0008892-4.pdf)
* [[Charles Weibel]], _Survey of non-Desarguesian planes_, Notices of the American Mathematical Society __54__ (10): 1294–1303 (2007) [pdf](http://www.ams.org/notices/200710/tx071001294p.pdf)

category: algebra
[[!redirects quasifields]]


quasigroup

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

#Contents#
* table of contents
{:toc}

## Idea

A *quasigroup* is a binary algebraic structure (a [[magma]]) in which one-sided multiplication is a [[bijection]] in that all [[equations]] of the form

$$
  \begin{aligned}
    a \cdot x & = b
    \\
    x \cdot a & = b
  \end{aligned}
$$

have a unique solution for $x$.

The notion of _quasigroup_ is hence a generalization of the notion of _[[group]]_, in that it does not require the [[associativity]] law nor the existence of an [[identity element]].

If a quasigroup does have a two-sided [[identity element]] then it is called a _[[loop (algebra)|loop]]_ (French _la boucle_, Russian лупа) and a *[[Moufang loop]]* if some further relations are satisfied.

Note that, in the absence of associativity, it is not enough (even for a [[loop (algebra)|loop]]) to say that every element has an [[inverse element]] (on either side); instead, you must say that division is always possible.  This is because the definition $x/y = x y^{-1}$ won\'t work right without associativity.

Some consider the concept of quasigroup to be an example of [[centipede mathematics]], see more at [[historical notes on  quasigroups]].

## Definitions

The usual definition is this:
+-- {: .un_defn}
###### Definition

A __quasigroup__ is a [[set]] $G$ equipped with a binary operation $G \times G \to G$ (which we will write with concatenation) such that:

*  for all $x$ and $y$, there exist unique $l$ and $r$ such that $l x = y$ and $x r = y$.

Then $l$ is called the __left quotient__ $y / x$ ($y$ divided by $x$, $y$ over $x$) and $r$ is called the __right quotient__ $x \backslash y$ ($x$ dividing $y$, $x$ under $y$).
=--
Note that we must specify, in the definition, that $l$ and $r$ are unique; without associativity, we cannot prove this.

As with the inverse elements of a group, we can make the quotients into operations so that all axioms are equations:
+-- {: .un_defn}
###### Definition

A __quasigroup__ is a [[set]] $G$ equipped with three binary operations (product, left quotient, and right quotient) such that these equations always hold:

*  $(x / y) y = x$,
*  $x (x \backslash y) = y$,
*  $(x y) / y = x$,
*  $x \backslash (x y) = y$.

=--

Also, without the right quotient we have left quasigroups, and without the left quotient the right quasigroups.
Thus quasigroups are described by a [[Lawvere theory]] and can therefore be [[internalization|internalized]] into any [[cartesian monoidal category]]. There are weaker structures, say left and right quasigroups in which either $\backslash$ or $/$ is well defined.

## Properties

Every left quasigroup $G$ has a function $id_L:G\to G$ defined as $id_L(x)=x\backslash x$ for all $x$ in $G$, and every right quasigroup $G$ has a function $id_R:G\to G$ defined as $id_R(x)=x/x$ for all $x$ in $G$. Every left quasigroup additionally has a function $inv_L:G\to G$ defined as $inv_L(x)=x\backslash id_L(x)$ for all $x$ in $G$, and every right quasigroup has a function $inv_R:G\to G$ defined as $inv_R(x)=id_L(x)/x$ for all $x$ in $G$.

A quasigroup is a __possibly empty left loop__ if $id_L(x) = id_L(y)$ for all $x,y$ in $G$, and it is a __possibly empty right loop__ if $id_R(x) = id_R(y)$ for all $x,y$ in $G$. A quasigroup is a __[[possibly empty loop]]__ if additionally $id_L(x) = id_R(y)$ for all $x,y$ in $G$, and it is an __[[invertible quasigroup]]__ if $inv_L(x) = inv_R(x)$ for all $x$ in $G$. If the quasigroup is commutative, then the left and right quotients are [[opposite magma]]s of each other.

The left (resp. right) quasigroups that are also [[possibly empty loop|possibly empty left (right) loops]] can be explicitly defined through the left (right) quotient operation itself, with multiplication being defined from the left (right) quotient.

Let us define a __left quotient__ on a set $G$ as a binary operation $(-)\backslash(-):G\times G\to G$ and functions $id_L(a) = a\backslash a$ and $inv_L(a) = a\backslash id_L(a)$ such that:

  * For all $a$ and $b$ in $G$, $id_L(a)=id_L(b)$
  * For all $a$ in $G$, $inv_L(a)\backslash id_L(a)=a$

For any element $a$ in $G$, the element $id_L(a) = a\backslash a$ is called a __right identity element__, and the element $inv_L(a) = a\backslash id_L(a)$ is called the __right inverse element__ of $a$. For all elements $a$ and $b$ in $G$, __left multiplication__ of $a$ and $b$ is defined as $a \cdot_L b = inv_L(a) \backslash b$.

The multiplication operation is __associative__ if for all $a$, $b$, and $c$ in $G$, $(a\backslash b)\backslash c= b\backslash (a \cdot_L c)$, __unital__ if there exists an element $1$ in $G$ such that for all $a$ in $G$, $id_L(a) = 1$, and __commutative__ if for all $a$ and $b$ in $G$, $a \cdot_L b = b \cdot_L a$.

A __right quotient__ on a set $G$ is a binary operation $(-)/(-):G\times G\to G$ such that:

  * For all $a$ and $b$ in $G$, $a/a=b/b$
  * For all $a$ in $G$, $(a/a)/((a/a)/a)=a$

For any element $a$ in $G$, the element $a/a$ is called a __left identity element__, and the element $(a/a)/a$ is called the __left inverse element__ of $a$. For all elements $a$ and $b$, __right multiplication__ of $a$ and $b$ is defined as $a/((b/b)/b)$.

The multiplication operation is __associative__ if for all $a$, $b$, and $c$ in $G$, $a/(b/c)=(a/((c/c)/c)/b$, __unital__ if there exists an element $1$ in $G$ such that for all $a$ in $G$, $a/a=1$, and __commutative__ if for all $a$ and $b$ in $G$, $a/((b/b)/b) = b/((a/a)/a)$.

A quasigroup that is left invertible and right invertible can be defined as a set with a left and right quotient $(G,\backslash,/)$ where left and right multiplications are equal (i.e. $a/((b/b)/b) = (a\backslash (a\backslash a))\backslash b$) for all $a$ and $b$ in $G$. If additionally, there exists a element $1$ in $G$ such that $a\backslash a = 1$ and $a/a = 1$, then the quasigroup is a [[loop (algebra)|loop]]. If the condition is relaxed to the requirements that left and right identity elements are equal (i.e. $a/a = a \backslash a$) for all $a$ in $G$ and the element $1$ is not required to be in $G$, then the loop might [[possibly empty loop|possibly be empty]]. If the associativity requirement is added to the left and right quotients of the quasigroup, then it becomes an [[associative quasigroup]], where equality of left and right identity and inverse elements can be derived.

## Examples

*  Any group is an associative quasigroup with [[identity elements]].
* Every [[associative quasigroup]], every [[nonassociative group]], and every [[loop (algebra)|loop]] is a quasigroup.
* Every [[invertible quasigroup]] is a quasigroup.
*  Any [[abelian group]] is a quasigroup in two other ways: the product switches places with one of the quotients.  (The other quotient remains a quotient.)
* [[H-space]]s are (homotopy-) loops --- this is because the [[shearing maps]] $(x,y)\mapsto (x,x y)$ and $(x,y)\mapsto (x y, y)$ are equivalences.  This generalizes the octonion examples.  Note that a [[loop space]] is always equivalent to a *group*, hence not all homotopy loops are loop spaces.  In particular ...
*  The nonzero elements of a (not necessarily associative) [[division algebra]] (such as the [[octonion]]s) form a quasigroup; this fact is basically the definition of 'division algebra'.

## Applications

There are interesting subvarieties of quasigroups (which are still not associative). Also, left [[racks]] (and quandles in particular) are precisely left distributive left quasigroups, with abundance of recent applications in the study of knots and links. Finite racks have been studied in the connection to classification of finite dimensional pointed Hopf algebras. Local augmented Lie racks appeared as integration objects in the local integration theory of [[Leibniz algebra]]s.

TS-quasigroups are related to [[Steiner triple system]]s.

Cayley multiplication tables of finite quasigroups are [[Latin square]]s (basically the 'sudoku squares' from the quotation [[historical notes on quasigroups|here]]).

As a sample of centipede mathematics, we have the following result on smooth quasigroups, i.e., quasigroups internal to the category of [[smooth manifolds]]:

+-- {: .num_prop}
###### Proposition
The tangent bundle of a smooth quasigroup $Q$ is trivial.
=--

+-- {: .proof}
###### Proof
Suppose WLOG that $Q$ is inhabited by an element $x$, and let $V = T_x(Q)$ be the tangent space at $x$. Define a map

$$\phi \colon Q \times V \to T Q: (y, v) \mapsto (d (K \circ L_y))(v)$$

where $L_y: Q \to Q$ is the smooth map $z \mapsto y z$ and $K: Q \to Q$ is the map $z \mapsto z/x$. The map $\phi$ commutes with the bundle projections $\pi_Q: Q \times V \to Q$, $\pi: T Q \to Q$. The map $K$ has an inverse $J: z \mapsto z x$ and each map $L_y$ has an inverse $M_y: z \mapsto y \backslash z$. We may therefore write down an inverse to $\phi$:

$$T Q \to Q \times V: w \mapsto (\pi(w), d(M_{\pi(w)} \circ J))(w)).$$

This shows $T Q$ is isomorphic to the product bundle $Q \times V$.
=--

## Related concepts

* [[loop (algebra)]], [[Moufang loop]]

* [[n-ary quasigroup]]

[[!include oidification - table]]

## Literature

* [[eom]]: [quasi-group](http://www.encyclopediaofmath.org/index.php/Quasi-group), [Webs, geometry of](http://www.encyclopediaofmath.org/index.php/Webs,_geometry_of), [Net (in differential geometry)](http://www.encyclopediaofmath.org/index.php/Net_%28in_differential_geometry%29)
* R.H. Bruck, _A survey of binary systems_, Springer-Verlag 1958
* [[Kenneth Kunen]], _Quasigroups, loops, and associative laws_, J. Algebra __185__ (1) (1996), pp. 194–204
* Momo Bangoura, _Bigèbres quasi-Lie et boucles de Lie_,  Bull. Belg. Math. Soc. Simon Stevin __16__:4 (2009), 593-616 [euclid](http://projecteuclid.org/euclid.bbms/1257776236) [arXiv:math.SG/0607662 ](http://arxiv.org/abs/math/0607662); _Quasi-bigèbres de Lie et cohomologie d'algèbre de Lie_, [arxiv/1006.0677](http://arxiv.org/abs/1006.0677)
* Lev Vasilʹevich Sabinin, _Smooth quasigroups and loops: forty-five years of incredible growth_, Commentationes Mathematicae Universitatis Carolinae __41__ (2000), No. 2, 377--400 [cdml](http://dml.cz/dmlcz/119171) [pdf](http://dml.cz/dmlcz/119171)

[[!redirects quasigroup]]
[[!redirects quasigroups]]

[[!redirects left quasigroup]]
[[!redirects left quasigroups]]

[[!redirects right quasigroup]]
[[!redirects right quasigroups]]


quasigroupoid

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

#Contents#
* table of contents
{:toc}

## Idea

Just as a [[groupoid]] is the [[oidification]] of a [[group]] and a [[ringoid]] is the oidification of a [[ring]], a [[quasigroupoid]] should be the oidification of a [[quasigroup]].

## Definition

A __quasigroupoid__ is a [[magmoid]] $Q$ such that for every [[span]]

$$
  \array{
     && s
      \\
      & {}^{f}\swarrow
      && \searrow^{g}
     \\
     x
     &&&&
     y
  }
$$

in $Q$, there exists morphisms $i:x\to y$ and $j:y \to x$ such that $i \circ f = g$ and $j \circ g = f$, and for every [[cospan]]

$$
  \array{
     && a &&&& b
      \\
      &
      && {}_{f}\searrow
       &
       & \swarrow_g
      &&
     \\
     &&&&
     c
     &&&&
  }
$$

in $Q$, there exists morphisms $d:a\to b$ and $e:b \to a$ such that $g \circ d = f$ and $f \circ e = g$.

## Examples

* Every [[groupoid]] is a quasigroupoid.

* Every [[loopoid]] and [[associative quasigroupoid]] is a quasigroupoid.

* A one-object quasigroupoid is a [[quasigroup]].

* A quasigroupoid [[enriched magmoid|enriched]] in [[truth values]] is an [[equivalence relation]].

## Related concepts

[[!include oidification - table]]


quasiminimal structure

## Idea

A quasiminimal structure is a particular class of [[pregeometries]] (infinite finitary matroids). Quasiminimal excellent classes were defined by Zilber as useful axiomatic setup satisfying [[categoricity]].

## Literature

Introduced in

* [[Boris Zilber]], _A categoricity theorem for quasi-minimal excellent classes_, Logic and its applications, Contemporary Mathematics 380 (Amer. Math. Soc. 2005) 297--306.

In original treatments, among the axioms, excellence was hard/technical to prove. A breakthrough is a result that it is essentially automatic:

* Martin Bays, Bradd Hart, Tapani Hyttinen, Meeri Kesälä, Jonathan Kirby, _Quasiminimal structures and excellence_, Bull. London Math. Soc. __46__ (2014) 155--163 [doi](https://doi.org/10.1112/blms/bdt076)
* Levon Haykazyan, _Categoricity in quasiminimal pregeometry classes_,  J. Symb. Logic 81(1), 56--64 (2014) [doi](https://doi,org/10.1017/jsl.2014.36) [arXiv:1308.1892](https://arxiv.org/abs/1308.1892)

Applications include

* B. Zilber, _Covers of the multiplicative group of an algebraically closed field of characteristic zero_,
J. London Math. Soc. (2) 74 (2006) 41--58
* B. Zilber, _Pseudo-exponentiation on algebraically closed fields of characteristic zero_, Ann. Pure Appl.
Logic 132 (2005) 67--95

Other

* John T. Baldwin, _Notes on quasiminimality and excellence_, The Bulletin of Symbolic Logic __10__:3 (2004) 334--366

category: model theory


quasinormal mode

## Idea

__Quasinormal modes__ are analogues of normal modes for physical systems in the presence of dissipation. In other words, these are the eigenstates of linearized equations of motion, e.g. of nonlinear wave equations. The eigenvalues become complex, signifying nonconservation of energy. This notion is especially prevalent recently in [[black hole]] physics.

## References

* [wikipedia](http://en.wikipedia.org/wiki/Quasinormal_mode)

* [[Frederik Denef]], Sean A. Hartnoll, Subir Sachdev, _Black hole determinants and quasinormal modes_, Class.Quant.Grav. __27__, 125001 (2010), [arxiv/0908.2657](http://arxiv.org/abs/0908.2657); _Quantum oscillations and black hole ringing_, Physical Review D __80__, 126016 (2009), [arxiv/0908.1788](https://arxiv.org/abs/0908.1788), [journal](http://link.aps.org/doi/10.1103/PhysRevD.80.126016)

* Sayan K. Chakrabarti, [[Kumar S. Gupta]], _Asymptotic quasinormal modes of d-dimensional Schwarzschild black hole with Gauss-Bonnet correction_, Int. J. Mod. Phys. A **21** (2006) 3565-3574 [[hep-th/0506133](https://arxiv.org/abs/hep-th/0506133), [doi:10.1142/S0217751X06031612](https://doi.org/10.1142/S0217751X06031612)]

* Shouvik Datta, Justin R. David, _Higher spin quasinormal modes and one-loop determinants in the BTZ black hole_, JHEP 2012:79 [arxiv/1112.4619](https://arxiv.org/abs/1112.4619)

[[!redirects quasinormal modes]]


quasiparticle

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

#Contents#
* table of contents
{:toc}

## Idea

Where a *[[fundamental particle]]* is the [[quantum]] of a [[field (physics)|field]] which is thought to be part of a "fundamental" [[quantum field theory]], a *quasi-particle* is the [[quantum]] of a [[field (physics)|field]] as it appears in some *[[effective quantum field theory|effective]]* quantum field thoery: A quasi-particle may behave much as a fundamental particle, but on closer resolution it appears as a collective excitement of more fundamental particles.

For an archetypical example in [[condensed matter theory]]: [[phonons]] in a [[crystal]] are quasi-particle excitations, underlying which are collective [[wave]]-oscillations of the more fundamental [[electrons]] and [[nuclei]] that constitute the underlying [[crystal lattice]].

## Examples

* [[phonon]]

* [[Cooper pair]]

## Related concepts

* [[anyon]]

## References

See also

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

[[!redirects quasiparticles]]

[[!redirects quasi-particle]]
[[!redirects quasi-particles]]


quasiseparated morphism

#Contents#
* table of contents
{:toc}

## Definition

Let $f : X \to Y$ be a morphism of [[scheme]]s. Let $\Delta_f: X \to X \times_Y X$ be the diagonal map. We say that $f$ is __quasi-separated__ if $\Delta_f$ is a [[quasicompact morphism]].

A scheme $X$ is quasi-separated if the morphism $X \to Spec\, \mathbf{Z}$ is quasi-separated, i.e. $\Delta:X\to X\times X$ is quasicompact. Every quasi-separated scheme is [[semiseparated scheme|semiseparated]].

## Properties

Every [[separated morphism of schemes]] is quasi-separated; every monomorphism of schemes is separated hence also quasi-separated.

## Related concepts

* [[quasicompact morphism]]

* [[quasicompact quasiseparated morphism]]

* [[morphism of finite presentation]]

## References

* MathOverflow [why-does-finitely-presented-imply-quasi-separated](http://mathoverflow.net/questions/36737/why-does-finitely-presented-imply-quasi-separated)
* [[Daniel Murfet]], _Concentrated schemes_, [pdf](http://therisingsea.org/notes/ConcentratedSchemes.pdf), an expositional digest from EGA

category: algebraic geometry

[[!redirects quasi-separated morphism]]
[[!redirects quasi-separated morphisms]]
[[!redirects quasiseparated morphisms]]
[[!redirects quasiseparated]]
[[!redirects quasi-separated]]
[[!redirects quasiseparated scheme]]
[[!redirects quasi-separated schemes]]
[[!redirects quasi-separated scheme]]
[[!redirects quasi-separated map]]
[[!redirects quasi-separated maps]]


quasisimple group

A [[group]] $H$ is called _quasisimple_ if and only if $H=[H,H]$ and $H/Z(H)$ is [[simple group|simple]]. The [[finite group|finite]] quasisimple groups have been classified, as a consequence of the classification of finite [[simple group]]s and the calculation of the [[Schur multiplier]] of each finite simple group.


quasitopological groupoid

A stub for now.

A quasitopological groupoid is a generalisation of a [[internal category|topological groupoid]] in that composition is not necessarily continuous, but only left and right composition by each arrow (when this is defined). It is the [[horizontal categorification]] of a quasitopological group.

More generally, one can talk about a [[quasi-C groupoid]], where $C$ is a (topological?) concrete category or a site (Warning: the last object is current research by [[David Roberts|me]], and may only be a curiosity).

More later...


quasitopos

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

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

## Idea

A quasitopos is a particular kind of [[category]] that has properties similar to that of a [[topos]], but is not quite a [[topos]]. A major difference is that it need not be [[balanced category|balanced]]: a [[morphism]] that is both [[monomorphism|monic]] and [[epimorphism|epic]] is not necessarily invertible. A quasitopos that is balanced is a topos.

Instead of the usual [[subobject classifier]], it has a classifier only for *[[strong monomorphism|strong]]* [[subobject]]s. It satisfies the uniqueness, but not the existence, part of the sheaf axioms ([[Elephant]] A2.6).

Note that some of the literature definitions use the notion of a [[regular monomorphism]].  Since every regular monomorphism is a strong one, this article only uses  [[strong monomorphism]].

## Definition

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

A **quasitopos** is a [[finitely complete category|finitely complete]], [[finitely cocomplete category|finitely cocomplete]], [[locally cartesian closed category]] $E$ in which there exists an [[object]] $\Omega$ that classifies [[strong monomorphism|strong monomorphisms]].

=--

In particular, this means

* Every finite [[limit]] and [[colimit]] exists;
* For each morphism $f: A \to B$, the [[base change|pullback functor]] between [[over category|slice quasitoposes]],
$$f^*: E/B \to E/A,$$
admits a [[adjoint functor|right adjoint]];
* There is a map $t: 1 \to \Omega$ such that every [[strong monomorphism]] $i: A \to X$ occurs as the [[pullback]] of $t$ along some unique morphism $\chi_i: X \to \Omega$:
$$\array{
A & \to & 1\\
i \downarrow & & \downarrow t\\
X & \overset{\chi_i}{\to} & \Omega
}$$

The object $\Omega$ above is sometimes called a **strong-subobject classifier**, since it classifies strong subobjects, but also sometimes called a **weak subobject classifier**, since it satisfies a weaker property than an ordinary [[subobject classifier]].

+-- {: .un_remark}
###### Remark

Equivalently, in addition to finite limits and colimits and local cartesian closure, one may ask only that there exists a classifier $t\colon 1\to\Omega$ as above for *some* class $\mathcal{M}$ of [[monomorphisms]] which contains the [[regular monomorphisms]] and is closed under composition and pullback.  From this one can show that every morphism factors as an epimorphism followed by a regular monomorphism (see [Wyler, proposition 12.5](#Wyler)).  It then follows that every strong monomorphism is regular, and therefore $\mathcal{M}$ is precisely the class of strong monomorphisms.

=--

+-- {: .un_prop}
###### Definition/Proposition

Let $C$ be a category with two [[Grothendieck topologies]] $J$ and $K$ such that $J\subseteq K$.  The [[full subcategory]] $BiSep(C,J,K) \hookrightarrow PSh(C)$ of the [[category of presheaves]] over $C$ consisting of the [[sheaves]] for $J$ that are also [[separated presheaves|separated]] for $K$ is a quasitopos.  A category equivalent to such a category is called a **Grothendieck quasitopos**, by analogy with the notion of [[Grothendieck topos]].

=--

In particular, this includes the category of separated presheaves on a given site (if we take $J$ to be the trivial topology), and also includes all Grothendieck toposes (if we take $K=J$).

Equivalently, a Grothendieck quasitopos is a category of the form $Sep_k(\mathbf{E})$, the category of $k$-separated objects for a [[Lawvere-Tierney topology]] $k$ on a Grothendieck topos $\mathbf{E}$.

## Properties

### General

+-- {: .num_lemma #PushoutOfStrongMonos}
###### Lemma
**(pushout of strong monos)

In a quasitopos the [[pushout]] of a [[strong monomorphism]] is again a strong mono, and the resulting square is also a [[pullback]] square.

=--

This appears as [Elephant, Lemma A.2.6.2](#Elephant), using the synonym _cocover_ for _strong monomorphism_. Since a [[topos]] is a quasitopos in which all monomorphisms are strong, this implies that the pushout of a mono in a topos is again a mono and that the resulting square is a pullback. Together with the fact that colimits are universal in a topos, this implies that a topos is an [[adhesive category]].

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

A quasitopos that is also a [[balanced category]] is a [[topos]].

=--

This is [Elephant, corollary 2.6.3](#Elephant).

+-- {: .num_lemma}
###### Lemma

A quasitopos has [[disjoint coproduct]]s precisely if the unique morphism $\emptyset \to *$ from the [[initial object]] to the [[terminal object]] is a [[strong monomorphism]].

=--

This is [Elephant, corollary 2.6.5](#Elephant).

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

An [[object]] $C$ in a quasitopos is called **coarse** if for every [[bimorphism|monic epic]] morphism $f : A \to B$ every morphism $A \to C$ factors uniquely through $f$.

=--

So the coarse objects are those that see monic epic morphisms as [[isomorphism]]s, hence that do not see the failure of the quasitopos to be a [[balanced category]].

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

In a quasitopos $\mathcal{E}$ the [[full subcategory]]
on coarse objects is a [[topos]] and a [[reflective subcategory]]

$$
  Coarse(\mathcal{E}) \stackrel{\leftarrow}{\hookrightarrow} \mathcal{E}
  \,.
$$

=--

This is [Elephant, prop 2.6.12](#Elephant).

+-- {: .num_lemma}
###### Lemma

If $\mathcal{E} \simeq SepPSh(C)$ is a Grothendieck quasitopos of [[separated presheaves]] on a [[site]] $C$, then $Coarse(\mathcal{E}) \simeq Sh(C)$ is the [[sheaf topos]] on $C$.

=--

This is in [Elephant, section A4.4](#Elephant).

### Characterization

There is a [[Giraud theorem]] characterizing Grothendieck quasitoposes:

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

Grothendieck quasitoposes are those quasitoposes which are [[locally small category|locally small]], [[cocomplete category|cocomplete]], and have a [[generating set]], or equivalently as the [[locally presentable categories]] which are locally cartesian closed and in which every *strong* [[congruence]] has a [[effective quotient]].

=--

see C2.2.13 of the ([Elephant](#Elephant))

### Extensivity and exactness

A topos is always [[extensive category|extensive]] and [[exact category|exact]], but this is not the case for quasitopoi.

A quasitopos is a [[coherent category]], since it has finite colimits which are stable under pullback (since it is locally cartesian closed), and so in particular its initial object is [[strict initial object|strict]], and it has finite coproducts which are pullback-stable, but they need not be disjoint: for objects $A$ and $B$, in the pullback
$$\array{P & \overset{}{\to} & B\\
  \downarrow && \downarrow\\
  A & \underset{}{\to} & A+B}$$
the object $P$ need not be initial.  This is easy to see from the fact that any Heyting category is a quasitopos, since then $A+B$ is the join $A\vee B$, and so the pullback is the meet $A\wedge B$, which is not in general the bottom element.

It is true, however, that such a $P$ is always a *quotient* of the initial object, i.e. the unique map $0\to P$ is epic.  If the map $0\to 1$ is strong monic, as it is in the "topological" examples, then $0$ can have no proper epimorphic images, and so coproducts are disjoint.  The converse also holds, since if coproducts are disjoint then $0\to 1$ is an equalizer of the two injections $1\rightrightarrows 1+1$.  A quasitopos with this property is sometimes called **solid**.

More generally, in any quasitopos $E$, we can factor $0\to 1$ into an epic followed by a strong monic, $0\to \bar{0} \to 1$.  One can prove that then the [[slice category]] $E/\bar{0}$ is a Heyting algebra (i.e. a posetal quasitopos), while the [[co-slice category]] $\bar{0}/E$ is a solid quasitopos, and moreover $E$ itself is recoverable via [[Artin gluing]] from a particular functor $E/\bar{0} \to \bar{0}/E$.  Thus, to a certain extent, the only interest in the theory of quasitoposes, above and beyond the theory of Heyting algebras, is in the solid ones.

By contrast, if a solid quasitopos is additionally [[exact category|exact]], and hence a [[pretopos]], then in particular it is [[balanced category|balanced]], which implies that it is in fact a topos.  One can prove, however, that a quasitopos is always *quasi-exact*, meaning that every *strong* [[congruence]] has an [[effective quotient]].

### Internal logic

Like a topos, a quasitopos has an [[internal logic]], for which the usual choice is to represent propositions by *strong* subobjects.  The resulting internal logic fails to satisfy the [[function comprehension|function comprehension principle]], forcing one to distinguish between [[functions]] and [[anafunctions]].

## Examples ##

* Any (elementary) [[topos]] is a quasitopos.  The first two properties are known, and in a topos every monomorphism is strong, so the ordinary subobject classifier works.

  Conversely, if a quasitopos is also a [[balanced category]], then it is also a topos.

* Any [[Heyting algebra]] is a quasitopos.  This is in notable contrast to the case of topoi, since no nontrivial poset is a topos.  The crucial distinction is that every morphism in a poset is both monic and epic, but only the identities are strong monic or strong epic.

* The category of [[pseudotopological spaces]] is a quasitopos, as is the category of [[subsequential spaces]].  (The latter is Grothendieck, but not the former.)  The category of [[topological spaces]] fails only to be locally cartesian closed.  In such "topological" quasitopoi, the strong monics are the "subspace inclusions" (i.e. those monics whose source has the topology induced from the target), and the strong-subobject classifier is the two-point space with the indiscrete topology.  (In particular, we cannot demand any sort of [[separation axiom]] and still have a quasitopos.)

* The category of [[marked simplicial set]]s.

* A category of [[concrete sheaves]] on a [[concrete site]] is a Grothendieck quasitopos. See [[local topos]].

  This includes the following examples:

  * The category of [[simplicial complex|simplicial complexes]].

  * The category of [[diffeological space|diffeological spaces]].

* {#exampsep} The following examples are categories of separated presheaves for the $\neg\neg$-topology on various presheaf toposes:

  * {#mono} The category [[M-category#def|Mono]] of [[monomorphisms]] between sets (as presheaves on the [[interval category]] - the [[Sierpinski topos]]).

  * {#endorel} The category [[relation#endorel|EndoRel or Bin]] of sets equipped with a [[relation]] (as presheaves on [[Quiv]]
    $$G_1 = (0 \stackrel{\overset{s}{\to}}{\underset{t}{\to}} 1),$$
    a truncation of the [[globular category]]).

  * {#endoref} The category of sets equipped with a reflexive relation (as presheaves on a truncated reflexive globular category).

  * {#endosym} The category of sets equipped with a symmetric relation (as presheaves on the full subcategory of finite sets and injections consisting of just the objects $1$, $2$).

  * {#endorefsym} The category of sets equipped with a reflexive symmetric relation (as presheaves on the full subcategory of finite sets consisting of just the objects $1$, $2$). See [[category of simple graphs]].

* The category of [[bornological set|bornological sets]].

* The category of assemblies of a [[partial combinatory algebra]].

* The category of Spanier's [[quasi-topological space|quasi-topological spaces]], the category of concrete sheaves on the site consisting of compact Hausdorff spaces with the finite covering topology. See [Dubuc-Espanol](#DE).

## Related concept

* **quasitopos**

* [[(∞,1)-quasitopos]]

## References ##

Original articles include

* [[Jacques Penon]], _Quasi-topos_ , C.R. Acad. Sci. Paris **276** Série A (1973) pp.237–240. ([gallica](http://gallica.bnf.fr/ark:/12148/bpt6k6217213f/f251.image))

* [[Jacques Penon]], _Sur le quasi-topos_ , Cahiers Top. Géom. Diff. 18 (1977), 181–218.

Textbook accounts:

* {#Wyler} [[Oswald Wyler]], _Lecture Notes on Topoi and Quasitopoi_ , World Scientific Singapore 1991 ([doi:10.1142/1047](https://doi.org/10.1142/1047))

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_, Oxford UP 2002.
{#Elephant} (section A2.6)

* [[Jiří Adámek]], [[Horst Herrlich]], [[George Strecker]], *Abstract and Concrete Categories*, Wiley 1990, reprinted as: Reprints in Theory and Applications of Categories, No. 17 (2006) pp. 1-507 ([tac:tr17](http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html))

Quasi-toposes of [[concrete sheaves]] are considered in

* [[Eduardo Dubuc]], _Concrete quasitopoi_ , Lecture Notes in Math. 753 (1979), 239–254

* {#DE} [[Eduardo Dubuc]], [[Luis Español]], _Quasitopoi over a base category_  ([arXiv:math.CT/0612727](http://arxiv.org/abs/math.CT/0612727))

A review is in

* [[John Baez]], [[Alex Hoffnung]], _Convenient categories of smooth spaces_, Trans. Amer. Math. Soc., Vol. 363 No. 11 (2011), 5789-5825, ([arXiv](http://arxiv.org/abs/0807.1704))

More generally, quasi-[[sheaf toposes]] are discussed in

* [[Richard Garner]], [[Stephen Lack]], _Grothendieck quasitoposes_ , arXiv:1106.5331 (2012). ([abstract](http://arxiv.org/abs/1106.5331))
 {#GarnerLack}

[[!redirects quasitopoi]]
[[!redirects quasitoposes]]
[[!redirects quasi-topos]]
[[!redirects quasi-topoi]]
[[!redirects quasi-toposes]]
[[!redirects quasi-exact category]]
[[!redirects strong-subobject classifier]]
[[!redirects weak subobject classifier]]
[[!redirects Grothendieck quasitopos]]
[[!redirects Grothendieck quasitoposes]]
[[!redirects Grothendieck quasitopoi]]


quasitoric manifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[toric variety]]

## References

See also:

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

Discussion in context with [[MSp]] and [[quaternionic oriented cohomology theory]]:

* {#Laughton08} [[Craig Laughton]], _Quasitoric manifolds and cobordism theory_, Manchester 2008 ([pdf](http://www.ma.man.ac.uk/~nige/craiglphd.pdf), [[LaughtonQuasitoric.pdf:file]])

[[!redirects quasitoric manifolds]]


quasitriangular bialgebra

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

#Contents#
* table of contents
{:toc}

## Idea

A _quasi-triangular bialgebra_ / _triangular bialgebra_ is a [[bialgebra]] equipped with just the right structure such as to make its [[category of modules]] into a [[braided monoidal category]]/[[symmetric monoidal category]].

## Definition

Let $A$ be an algebra in a symmetric [[monoidal category]] $C$ with symmetry $\tau$; fix $m,l$ and $D\in A^{\otimes k}$ and let $1\leq i_r\leq l$ for $1\leq r\leq m$ be different. Then denote $D_{i_1,\ldots,i_m}\in A^{\otimes n}$ as the image of $R\otimes 1^{\otimes (l-m)}$ under the permutation which is the composition of the $m$ transpositions $(r,i_r)$ of tensor factors interchanging $r$ and $i_r$. In the following $C$ is the monoidal category of $k$-vector spaces.

A $k$-[[bialgebra]] (in particular $k$-[[Hopf algebra]]) is __quasitriangular__ if there is an invertible element $R\in H\otimes H$ such that for any $h\in H$
$$
\tau\circ\Delta(h) = R\Delta(h)R^{-1}
$$
where $\tau=\tau_{H,H}:H\otimes H\to H\otimes H$ and

$$
(\Delta\otimes id)(R)=R_{13} R_{23}
$$

$$
(id\otimes\Delta)(R)=R_{13} R_{12}
$$

An invertible element $R$ satisfying these 3 properties is called the __universal $R$-element__. As a corollary

$$
(\epsilon\otimes id) R = 1,\,\,\,\,\,(id\otimes\epsilon)R = id
$$

and the [[quantum Yang-Baxter equation]] holds in the form

$$
R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}
$$

A quasitriangular $H$ is called __triangular__ if $R_{21}:=\tau(R) = R^{-1}$.

The category of representations of a quasitrianguar bialgebra is a _braided_ monoidal category. If $R$ is a universal $R$-element, then $R_{21}^{-1}$ is as well. If $H$ is quasitriangular, $H^{cop}$ and $H_{op}$ are as well, with the universal $R$-element being $R_{21}$, or instead, $R_{12}^{-1}$. Any twisting of a quasitriangular bialgebra by a [[bialgebra cocycle|bialgebra 2-cocycle]] twists the universal $R$-element as well; hence the twisted bialgebra is again quasitriangular. Often the $R$-element does not exist as an element in $H\otimes H$ but rather in some completion of the tensor square; we say that $H$ is essentially quasitriangular, this is true for quantized enveloping algebras $U_q(G)$ in the rational form. The famous Sweedler's Hopf algebra has a 1-parameter family of universal $R$-matrices.

## Properties

### Tannaka duality

A quasitriangular structure on a [[bialgebra]] corresponds to a [[braided monoidal category]] structure on the [[category of modules]] of the underlying algebra. (For instance chapter 1, section 2 of ([Carroll](#Carroll))).

[[!include structure on algebras and their module categories - table]]

## Related concepts

* [[quasitriangular Hopf algebra]]

## References

* V. G. Drinfel'd, _Quantum groups_, Proc. ICM 1986, Vol. 1, 2 798–820, AMS 1987.

* S. Majid, _Quasitriangular Hopf algebras and Yang-Baxter equations_, Int. j. mod. physics A, __5__, 01, pp. 1-91 (1990) [doi:10.1142/S0217751X90000027](http://dx.doi.org/10.1142/S0217751X90000027)

* S. Majid, _Foundations of quantum group theory_, Cambridge University Press 1995, 2000.

* A. U. Klymik, K. Schmuedgen, _Quantum groups and their representations_, Springer 1997.

* V. Chari, A. Pressley, _A guide to quantum groups_, Camb. Univ. Press 1994

* Robert Carroll, _Calculus revisited_
 {#Carroll}

[[!redirects quasitriangular bialgebras]]

[[!redirects triangular bialgebra]]
[[!redirects triangular bialgebras]]

[[!redirects universal R-element]]


quasitriangulated quasi-Hopf algebra

#Contents#
* table of contents
{:toc}

## Idea

A [[quasi-Hopf algebra]] with the structure analgous to a [[quasitriangular Hopf algebra]].

## Examples

The [[Drinfeld double]] of a [[quasi-Hopf algebra]] is quasi-triangulated.

[[!redirects quasitriangulated quasi-Hopf algebras]]

[[!redirects quasi-triangulated quasi-Hopf algebra]]
[[!redirects quasi-triangulated quasi-Hopf algebras]]

[[!redirects quasitriangulated quasiHopf algebra]]
[[!redirects quasitriangulated quasiHopf algebras]]


quaternion

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

A _quaternion_ or _Hamilton number_ is a kind of [[number]] similar to the [[complex numbers]] but with three instead of one [[square root]] of $(-1)$ adjoined, satisfying certain relations.

The __quaternions__ form the largest [[associative algebra|associative]] [[normed division algebra]], usually denoted  $\mathbb{H}$ after [[William Rowan Hamilton]] (since $\mathbb{Q}$ is taken for the [[rational numbers]]).

## Properties

### Normed division algebra structure

The structure of $\mathbb{H}$ as an $\mathbb{R}$-[[associative unital algebra|algebra]] is given by a basis $\{1, i, j, k\}$ of the underlying [[vector space]] of $\mathbb{H}$, equipped with a multiplication table where $1$ is the [[identity element]] and otherwise uniquely specified by the equations

$$i^2 = j^2 = k^2 = i j k = -1,$$

and extended by $\mathbb{R}$-linearity to all of $\mathbb{H}$. The norm on $\mathbb{H}$ is given by

$${\|\alpha\|}^2 = \alpha \widebar{\alpha}$$

where given an $\mathbb{R}$-[[linear combination]] $\alpha = a 1 + b i + c j + d k$, we define the _conjugate_ $\widebar{\alpha} \coloneqq a 1 - b i - c j - d k$. A simple calculation yields

$${\|\alpha\|}^2 = a^2 + b^2 + c^2 + d^2$$

whence for $\alpha \neq 0$, the multiplicative [[inverse]] is

$$\alpha^{-1} = \frac1{{\|\alpha\|}^2} \widebar{\alpha}.$$

In this way $\mathbb{H}$ is a normed division algebra.

### Modules and bimodules

We have canonical left and right [[module]] structures on $\mathbb{H}^n$, but as $\mathbb{H}$ is not commutative, if we want to talk about tensor products of modules, we need to consider [[bimodules]]. This also means that ordinary [[linear algebra]] as is used over a field is not quite the same when dealing with quaternions. For instance, one needs to distinguish between _left_ and _right_ [[eigenvalues]] of [[matrices]] in $M_n(\mathbb{H})$ (using the left and right module structures on $\mathbb{H}^n$ respectively), and only left eigenvalues relate to the [[spectrum]] of the associated linear operator.

Using the conjugation operation one can define an inner product $\langle q,p\rangle := \overline{q} p$ on $\mathbb{H}^n$ so that the corresponding [[orthogonal group of an inner product space|orthogonal group]] is the [[compact symplectic group]].

### Automorphisms

+-- {: .num_prop #AutomorphismsOfQUatrnionsAlgebraIsSO3}
###### Proposition

The [[automorphism group]] of the quaternions, as a real algebra, is [[special orthogonal group|SO(3)]], [[action|acting]] canonically  on their imaginary part (in generalization of how the product of [[complex numbers]] respects the [[complex conjugation]] action)

=--

See also at _[normed division algebra -- automorphism](normed+division+algebra#Automorphisms)_

(e.g. [Klimov-Zhuravlev, p. 85](#KlimovZhuravlev))

## Related concepts

* [[quaternionic vector space]]

* [[quaternionic vector bundle]]

* [[Dieudonné determinant]]

* [[SL(2,H)]]

[[!include exceptional spinors and division algebras -- table]]

* [[supersymmetry and division algebra]]

* [[quaternionic quantum mechanics]]

* [[quaternion group]], [[quaternionic unitary group]]

* [[tessarines]]

## References

Monographs:

* {#KlimovZhuravlev} D. M. Klimov, V. Ph. Zhuravlev, *Group-Theoretic Methods in Mechanics and Applied Mathematics*, Routledge (2004, 2020) [[ISBN:9780367446987](https://www.routledge.com/Group-Theoretic-Methods-in-Mechanics-and-Applied-Mathematics/Klimov-Zhuravlev/p/book/9780367446987)]

* [[Ernst Binz]], Sonja Pods, Ch 1 in: *The geometry of Heisenberg groups --- With Applications in Signal Theory, Optics, Quantization, and Field Quantization*, Mathematical Surveys and Monographs **151**, American Mathematical Society (2008) [[ams:surv-151](https://bookstore.ams.org/surv-151)]

* [[Tevian Dray]], [[Corinne Manogue]], Section 3.1 of: _The Geometry of Octonions_, World Scientific (2015) [[doi:10.1142/8456](https://doi.org/10.1142/8456)]

See also:

* T. Y. Lam, _Hamilton's Quaternions_ ([ps](http://math.berkeley.edu/~lam/quat.ps))

[[!redirects quaternion]]
[[!redirects quaternions]]
[[!redirects quaternionic]]

[[!redirects quaternion number]]
[[!redirects quaternion numbers]]

[[!redirects Hamilton number]]
[[!redirects Hamilton numbers]]