pretopos

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# Pretopos
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## Definitions

### General

A __pretopos__ is a [[category]] which is both [[exact category|exact]] and [[extensive category|extensive]].  (See [[familial regularity and exactness]] for why extensivity and exactness deserve to be considered together.)  This implies that it is a [[coherent category]].

Pretoposes are suitable as frameworks for [[finite mathematics|finitist]] [[predicative mathematics]], respectively with [[coherent logic]].

Frequently one is especially interested in pretoposes having additional properties, such as:

* A __Heyting pretopos__ is a pretopos which is also a [[Heyting category]]; a __Boolean pretopos__ is a pretopos which is also a [[Boolean category]].  These are suitable as frameworks for [[finite mathematics|finitist]] [[predicative mathematics]], respectively with [[intuitionistic logic|intuitionistic]] or [[classical logic|classical]] logic.

* A __pretopos with choice__ is a pretopos that satisfies the [[axiom of choice]] (that every [[epimorphism]] is [[split epic|split]]). Every pretopos with choice is automatically boolean.

* A __[[Π-pretopos]]__ is a pretopos which is also a [[locally cartesian closed category]].  (A $\Pi$-pretopos is automatically a Heyting pretopos.)  These are suitable as frameworks for finitist [[constructive mathematics]] which is 'weakly predicative'.

* An __[[arithmetic pretopos]]__ is a pretopos which has (locally) a parameterised [[natural numbers object]]. Arithmetic pretoposes are suitable as frameworks for non-finitist but strongly predicative mathematics with [[coherent logic]], while Heyting arithmetic pretoposes and Boolean arithmetic pretoposes are suitable as frameworks for non-finitist but strongly predicative mathematics with intuitionistic and classical logic, respectively.

* A __$W$-pretopos__ is a pretopos which has (locally) [[W-types]] ([[initial algebra|initial]] [[algebra for an endofunctor|algebras]] for [[polynomial endofunctors]]), most famously a [[natural numbers object]].  (Here, we take any [[exponentiable morphism]] to define a polynomial endofunctor.  This paragraph is original to this page and not in the literature; it may not be the best definition.)

* An __arithmetic $\Pi$-pretopos__ is a pretopos that is both a $\Pi$-pretopos and an arithmetic pretopos; it is sufficient that the $\Pi$-pretopos have a [[natural numbers object]], and so could be called __$\Pi$-pretopos with NNO__. These are suitable as frameworks for weakly predicative constructive mathematics that is not finitist.

* A __[[ΠW-pretopos]]__ is a pretopos that is both a $\Pi$-pretopos and a $W$-pretopos.  (Now every morphism defines a polynomial endofunctor, since every morphism is exponentiable.)

* A __[[predicative topos]]__ is a [[ΠW-pretopos]] that satisfies the [[axiom of multiple choice]].  (This now differs from a $W$-[[topos]] below mainly in still lacking [[power objects]], hence still a context for weakly predicative constructive mathematics.)

* A __[[topos]]__ is a pretopos that has [[power objects]].  A topos is automatically a $\Pi$-pretopos; conversely, a $\Pi$-pretopos is a topos iff it has a [[subobject classifier]], and a Boolean $\Pi$-pretopos is always a topos.  Toposes and [[Boolean toposes]] are suitable as frameworks for finitist (but otherwise impredicative) mathematics, with intuitionistic and classical logic respectively.

* A __$W$-topos__ is of course a topos that is a $W$-pretopos; it is sufficient that the topos have a [[natural numbers object]] (see van den Berg & Moerdijk), so this is often called a __topos with NNO__.  These are suitable as frameworks for (non-predicative, non-finitist) [[constructive mathematics]], while Boolean $W$-toposes are suitable as a framework for [[classical mathematics]] without the [[axiom of choice]].

* A __topos with choice__ is a topos that satisfies the [[axiom of choice]] (that every [[epimorphism]] is [[split epic|split]]).  Every topos with choice is automatically boolean, so $W$-toposes with choice are suitable as a framework for full classical mathematics.  (In fact, a [[well-pointed category|well-pointed]] $W$-topos with choice is precisely a model of [[ETCS]].)

### Infinitary pretoposes
{#infinitary}

An __infinitary pretopos__ is an [[infinitary coherent category]] which is both [[infinitary extensive category|infinitary extensive]] and [[exact category|exact]].  Giraud\'s theorem says that infinitary pretoposes with small generating sets are the same as [[Grothendieck toposes]], and in particular are [[toposes]] (although this last result is not valid in [[predicative mathematics]]).

An infinitary pretopos (perhaps with [[well-powered category|well-poweredness]] included) is sometimes called an "$\infty$-pretopos" (e.g. in the [[Elephant]]), but we avoid that term because of potential confusion with the notion of [[(∞,1)-pretopos]].

## Examples

* Of course, any [[topos]] is a [[pretopos]], so any example there is an example of a [[pretopos]].

* An example of a pretopos that is not a [[topos]] is the [[category]]
of [[compact Hausdorff spaces]] ([MarraReggio](#MarraReggio)).

* An example of a Boolean pretopos that is not [[cartesian closed category|cartesian closed]] is the category of finite or countably infinite sets (cf. [[Elephant]] p.54).

* The category of [[decidable objects]] of a [[topos]] is a Boolean pretopos.

* The [[exact completion]] of the category [[Ho(Top)]] of [[topological spaces]] and [[homotopy classes]] of [[continuous maps]] is a pretopos.

* In [[dependent type theory]], a [[univalent universe]] where every type in the universe satisfies [[axiom K]] and is closed under the [[empty type]], [[unit type]], [[sum types]], [[dependent sum types]], [[product types]], [[propositional truncations]], and [[quotient sets]] is a pretopos.
  * the universe becomes an [[arithmetic pretopos]] if it is closed under a parameterized [[natural numbers type]]
  * the universe becomes a Heyting pretopos if the universe is closed under [[dependent product types]] for families of [[mere propositions]], or equivalently, if it is closed under types of [[embeddings]]
  * the universe becomes a Boolean pretopos if it is a Heyting pretopos and additionally every proposition satisfies [[excluded middle]]
  * the universe becomes a Pi-pretopos if it the universe is closed under all [[dependent product types]], or equivalently, if the universe is closed under [[function types]]
  * the universe becomes an arithmetic Pi-pretopos if it is a Pi-pretopos and is closed under a [[natural numbers type]]
  * the universe becomes an elementary topos if it is a Pi-pretopos and has a [[type of all propositions]]
  * the universe becomes an Boolean topos if it is a Boolean Pi-pretopos

## Properties

### Internal logic and mathematics in pretopoi

Like any coherent (or Heyting) category, a (Heyting) pretopos has an [[internal logic]].  Extensivity and exactness make a Heyting pretopos a very set-like category.  One can say imprecisely that it has "all the good first-order properties of a [[topos]]", meaning not that it has those properties that can be expressed in elementary terms (which is false) but that it has those properties that (unlike exponential and power objects) correspond to first-order reasoning in ordinary mathematics.  Therefore, pretoposes (especially Heyting, $\Pi$, and/or $W$ ones) are related to predicative constructive mathematics in a way similar to how toposes are related to non-predicative constructive mathematics.

### Colimits

A pretopos is necessarily [[balanced category|balanced]], but while it has coproducts and coequalizers of equivalence relations, it need not have all finite colimits.  However, if it has countable pullback-stable unions of subobjects (what the [[Elephant]] calls a _$\sigma$-pretopos_; see [[infinitary coherent category]]), then any internal binary relation generates an equivalence relation and therefore has a quotient, so we can construct arbitrary coequalizers and thus arbitrary finite colimits.  And we can perform an "internal" version of this argument in a $\Pi$-pretopos with a [[natural numbers object|NNO]], such as a $\Pi$-$W$-pretopos.

### The precanonical topology

A pretopos, being a [[coherent category]], admits a [[subcanonical site|subcanonical]] [[Grothendieck topology]] called the [[coherent topology]].  In a pretopos, this topology is generated by finite jointly epimorphic families.  Since the [[canonical topology]] on a [[Grothendieck topos]] consists of all jointly epimorphic families, the coherent topology on a pretopos is sometimes called the __precanonical topology__.

The [[codomain fibration]] of a pretopos is always a [[stack]] for its precanonical topology.  Being a pretopos is stronger than necessary for this condition to hold in a coherent category, however; see [[coherent category]] for the necessary and sufficient conditions.

## Related concepts

* [[Π-pretopos]], [[ΠW-pretopos]], [[list-arithmetic pretopos]]

* [[2-pretopos]]

* [[(∞,1)-pretopos]]

* [[conceptual completeness]]

## References

*  [[Benno van den Berg]],  [[Ieke Moerdijk]] (2008-09-25); _$W$-types in sheaves_; [vdBM_Wtypes.ps/pdf](http://www.phil.cmu.edu/projects/ast/Papers/)

* {#MarraReggio} [[Vincenzo Marra]], [[Luca Reggio]], _A characterisation of the category of compact Hausdorff spaces_, Theory Appl. Categ. 35 (2020), Paper No. 51, 1871–1906. [pdf](http://www.tac.mta.ca/tac/volumes/35/51/35-51.pdf), [arXiv:1808.09738](https://arxiv.org/abs/1808.09738).

* {#Taylor} [[Paul Taylor]], [MathOverflow answer](https://mathoverflow.net/questions/382348/subobjects-in-the-category-of-compact-hausdorff-spaces/382351#382351).

* [[Marino Gran]], [[Enrico Vitale]], *On the exact completion of the homotopy category*, Cahiers de Topologie et Géométrie Différentielle Catégoriques **39** 4 (1998)  287-297 [[numdam:CTGDC_1998__39_4_287_0](http://www.numdam.org/item/CTGDC_1998__39_4_287_0)]

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pretopos completion

# Pretopos completion
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## Idea

Pretopos completion sends every [[coherent category]] into a pretopos, called its pretopos completion; the operation is idempotent.  It is a "higher-ary" version of the [[exact completion]] of a regular category.

More generally, [[pretoposes]] form a full reflective sub-2-category of the 2-category of [[finitary sites]], which in turn includes coherent categories (with their coherent topologies) as a full sub-2-category.

## Relation to model theory

According to the observation of [[Mihály Makkai]] from around 1980, the pretopos completion of the [[syntactic category]] of a [[first-order theory]] corresponds to the operation $T \mapsto T^{eq}$ invented by [[Saharon Shelah]]. $T^{eq}$ is a universal extension of $T$ which admits [[elimination of imaginaries]].

## Higher-ary versions

For any an [[arity class]] $\kappa$, there is a corresponding version of $\kappa$-pretopos completion; see ([Shulman](#Shulman)).  The version for $\kappa = \{1\}$ is the [[exact completion]], while classical pretopos completion is the version for $\kappa=\omega$.  The version for $\kappa=$ the size of the universe includes the [[topos of sheaves]].

## Related entries

* [[exact category]], [[regular and exact completions]]
* [[pretopos]], [[∞-ary exact category]]
* [[imaginary element]]

## References

* Panagis Karazeris, _Notions of flatness relative to a Grothendieck topology_,

* [[Mike Shulman]], "Exact completions and small sheaves".  *Theory and Applications of Categories*, Vol. 27, 2012, No. 7, pp 97-173.  [Free online](http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html)
{#Shulman}

* [[Mihály Makkai]], G. E. Reyes, _First order categorical logic_, Lecture Notes in Math. __611__, Springer 1977

* David Ballard, William Boshuck, _Definability and descent_, The Journal of Symbolic Logic __63__:2 (Jun., 1998), pp. 372-378, [jstor](http://www.jstor.org/stable/2586836)

* Victor Harnik, _Model theory vs. categorical logic: two approaches to pretopos completion (a.k.a. $T^{eq}$)_, in: Models, logics, and higher-dimensional categories, 79–106 (Makkai volume), CRM Proc. Lecture Notes __53__, Amer. Math. Soc. 2011; [gBooks](http://books.google.fr/books?id=-PZpEXuvvm4C&lpg=PA79&ots=uB-pruKFx8&dq=Victor%20Harnik%20pretopos%20completion&pg=PA79#v=onepage&q=Victor%20Harnik%20pretopos%20completion&f=false)

* [[Saharon Shelah]], _Classification theory and the number of non-isomorphic models_, Studies in Logic and the Foundations of Mathematics __92__, North Holland, Amsterdam 1978

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pretriangulated A-∞-category

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## Idea

Just as [[pretriangulated dg-categories]] are a model for [[stable (∞,1)-categories|stable]] [[linear (∞,1)-categories]] in the setting of [[dg-categories]], _pretriangulated $A_\infty$-categories_ are designed to be a model for [[stable (∞,1)-categories|stable]] [[linear (∞,1)-categories]] in the setting of $A_\infty$-[[A-∞-categories|categories]].

## References

* Yuri Bespalov, [[Volodymyr Lyubashenko]], Oleksandr Manzyuk, _Pretriangulated $A_\infty$-categories_, Proceedings of the Institute of Mathematics of NAS of Ukraine, vol. 76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008, 599 p., [pdf](http://www.math.ksu.edu/~lub/cmcMonad.pdf).

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pretriangulated dg-category

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## Idea

_Pretriangulated dg-categories_ over a [[commutative ring]] $k$ are, roughly speaking, [[dg-categories]] whose [[homotopy category of a dg-category|homotopy category]] is canonically [[triangulated category|triangulated]].  These form a model for [[stable (∞,1)-category|stable]] [[k-linear (∞,1)-categories]], in a sense which is made precise below (at least in [[characteristic zero]]).  In other words pretriangulated dg-categories can be viewed as [[enhanced triangulated categories]].  For this reason some authors call them _stable dg-categories_.

## History

The notion of pretriangulated dg-category goes back to [(Bondal-Kapranov 1990)](#BondalKapranov90).

[[Goncalo Tabuada]] demonstrated the existence of a [[model structure]] on the category of small [[dg-categories]], the [[quasi-equiconic model structure on dg-categories]], where the [[fibrant objects]] are the pretriangulated dg-categories.  See [(Tabuada 07, Theorem 2.2 and Proposition 2.10)](#Tabuada07).  This model structure can be [[left Bousfield localization|Bousfield localized]] to the [[Morita model structure on dg-categories]], where the [[fibrant objects]] are the [[idempotent complete (infinity,1)-category|idempotent complete]] pretriangulated dg-categories.  In [(Cohn 13)](#Cohn13) it is shown that the associated [[(infinity,1)-category]] is equivalent to the [[(infinity,1)-category]] of [[stable (infinity,1)-category|stable]] [[k-linear (∞,1)-categories]].

## Definitions

### Pretriangulated dg-categories

Let $A$ be a [[dg-category]] and $P(A)$ the dg-category of [[dg-presheaves]] or right [[dg-modules]] over $A$.  The [[Yoneda embedding]] induces a [[fully faithful functor]] $h : ho(A) \hookrightarrow ho(P(A))$ on the [[homotopy category of a dg-category|homotopy categories]].  The category $ho(P(A))$ has a canonical [[triangulated structure]] (which can be written down directly).

+-- {: .un_defn}
###### Definition
The dg-category $A$ is called **pretriangulated** if the functor $h$ is stable under the [[suspension functor]] (and its [[loop space functor|inverse]]), and under taking [[mapping cones]] in $ho(P(A))$.
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###### Proposition
A dg-category $A$ is pretriangulated if and only if it is a [[fibrant object]] in the [[quasi-equiconic model structure on dg-categories]].
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See [(Tabuada 07, Proposition 2.10)](#Tabuada07).

### Strongly pretriangulated dg-categories

Let $A$ be a [[dg-category]].

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###### Definition
The **$n$-translation** of an object $X \in A$ is an object $X[n] \in A$ representing the functor
  $$ \Hom(\cdot, X)[n]. $$
The **cone** of a closed morphism $f : X \to Y$ of degree zero is an object $\Cone(f) \in A$ representing the functor
  $$ \Cone(\Hom(\cdot, X) \stackrel{f_*}{\to} \Hom(\cdot, Y)), $$
which is a [[mapping cone]] in [[category of chain complexes|chain complexes]].
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###### Definition
The dg-category $A$ is called **strongly pretriangulated** if it admits a [[zero object]], all translations of all objects, and all cones of all morphisms.
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###### Definition
Let $A$ be a dg-category.  A **strongly pretriangulated envelope** of $A$ is the data of a strongly pretriangulated dg-category $tri(A)$ and a [[fully faithful]] functor $A \hookrightarrow tri(A)$ such that any functor $u: A \to B$ to a strongly pretriangulated dg-category $B$ factors uniquely through a functor $tri(A) \to B$.
=--

A strongly pretriangulated envelope $A \hookrightarrow tri(A)$ always exists, and may be constructed by taking $tri(A)$ to be the full dg-subcategory of $P(A)$ spanned by the objects of the [[full subcategory|full]] [[triangulated subcategory]] of $ho(P(A))$ generated by the [[representable presheaves]], and $A \hookrightarrow tri(A)$ to be the functor induced by the Yoneda embedding.  There is also another construction using twisted complexes, see [Bondal-Kapranov](#BondalKapranov90).

Now we have the following characterization of pretriangulated dg-categories.

+-- {: .num_prop}
###### Proposition
Let $A$ be a dg-category and $A \hookrightarrow tri(A)$ be a strongly pretriangulated envelope of $A$.
$A$ is pretriangulated if and only if the induced fully faithful functor $ho(A) \hookrightarrow ho(tri(A))$ is [[essentially surjective]] (and hence an [[equivalence of categories]]).
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As an immediate corollary, note that for a pretriangulated dg-category $A$, its [[homotopy category of a dg-category|homotopy category]] $ho(A)$ inherits a canonical triangulated structure.

## Properties

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###### Proposition
Let $u : A \to B$ be a functor between two dg-categories.  If $A$ and $B$ are pretriangulated then the induced functor $ho(u): ho(A) \to ho(B)$ is [[triangulated functor|triangulated]].  Further, $u$ is a [[quasi-equivalence of dg-categories|quasi-equivalence]] if and only if $ho(u)$ is a triangulated equivalence.
=--

## Applications

The use of pretriangulated dg-categories as [[enhanced triangulated categories]] has been especially successful in the study of the various [[triangulated categories of sheaves]] on [[algebraic varieties]].

In the below paper it is shown that the triangulated categories of [[quasicoherent sheaf|quasicoherent sheaves]] on quasiprojective varieties and some of their cousins (like categories of perfect complexes on quasiprojective varieties) have unique dg-enhancements. [[Fernando Muro]] has developed an obstruction theory for the existance and measuring non-uniqueness of dg-enhancements in more general settings (unpublished).

* [[Valery Lunts|V. A. Lunts]], [[Dmitri Orlov|D. O. Orlov]], _Uniqueness of enhancement for triangulated categories_, J. Amer. Math. Soc. __23__ (2010), 853-908, [journal](http://www.ams.org/journals/jams/2010-23-03/S0894-0347-10-00664-8/home.html), [arXiv:0908.4187](http://arxiv.org/abs/0908.4187).

* [[Valery Lunts]], Olaf M. Schnuerer, _New enhancements of derived categories of coherent sheaves and applications_, 2014, [arXiv](http://arxiv.org/abs/1406.7559).

Similarly, pretriangulated dg-categories have proven to give a good model for [[derived noncommutative algebraic geometry]] in the sense of [[Maxim Kontsevich]].  See there for relevant references.  In this connection see also the work of [[Goncalo Tabuada]] who has developed a theory of [[noncommutative motives]] in this framework.

## Related concepts

* [[triangulated category]]

* [[enhanced triangulated category]]

* [[stable derivator]]

* [[stable (∞,1)-category]]

## References

* {#BondalKapranov90} [[Alexei Bondal]], [[Mikhail Kapranov]], _Enhanced triangulated categories_, Матем. Сборник, Том 181 (1990), No.5, 669–683 (Russian); transl. in USSR Math. USSR Sbornik, vol. 70 (1991), No. 1, pp. 93–107, (MR91g:18010) ([[bondalKaprEnhTRiangCat.pdf:file]])

The [[model structure]] presenting pretriangulated dg-categories is discussed in

* {#Tabuada07} [[Goncalo Tabuada]], _Theorie homotopique des DG-categories_, Ph.D. thesis, Universite Denis Diderot - Paris 7, [arXiv:0710.4303](http://arxiv.org/abs/0710.4303).

See also paragraph 2.3 of

* [[Dmitri Orlov]], _Smooth and proper noncommutative schemes and gluing of DG categories_, [arXiv:1402.7364](http://arxiv.org/abs/1402.7364).

For a summary of the various [[model structures on dg-categories]], see Section 2 of the paper

* {#CisinskiTabuada11} [[Denis-Charles Cisinski]], [[Goncalo Tabuada]], _Non-connective K-theory via universal invariants_, Compositio Math. 147 (2011), 1281-1320, [arXiv:0903.3717](http://arxiv.org/abs/0903.3717), [pdf](http://www.math.univ-toulouse.fr/~dcisinsk/Non-connective-K-theory.pdf).

The relation to [[stable (infinity,1)-categories]] is discussed in

* {#Cohn13} [[Lee Cohn]], _Differential Graded Categories are k-linear Stable Infinity Categories_ ([arXiv:1308.2587](http://arxiv.org/abs/1308.2587))

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pretriangulated envelope of a dg-category

## Idea

Any [[dg-category]] admits a pretriangulated envelope, which is a [[fully faithful functor|fully faithful]] embedding into a [[strongly pretriangulated dg-category]]. When the [[dg-category]] is itself [[pretriangulated dg-category|pretriangulated]], this is in fact an [[equivalence of dg-categories]].

## Definition

Let $T$ be a [[dg-category]].

+-- {: .un_defn}
###### Definition
The **pretriangulated envelope** (or _pretriangulated hull_, _pretriangulated completion_) of $T$, denoted $tri(T)$, is the full sub-[[dg-category]] of the [[derived dg-category of a dg-category|derived dg-category]] $D(T)$ generated by the representable [[dg-modules]] under [[homotopy fibres]] and [[homotopy cofibres]].
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Here, the notions of [[homotopy fibre]] and [[homotopy cofibre]] can be taken in a [[dg-model category]] [[dg-category presented by a dg-model category|presenting]] $D(T)$. Explicitly $tri(T)$ can be described as follows.

+-- {: .un_lemma}
###### Lemma
The pretriangulated envelope $tri(T)$ coincides with the full sub-dg-category of $D(T)$ spanned by the finitely generated semi-free [[dg-modules]].
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## Related concepts

* [[pretriangulated dg-category]]
* [[quasi-equiconic model structure on dg-categories]]

## References

Paragraph 2.3 of

* [[Dmitri Orlov]], _Smooth and proper noncommutative schemes and gluing of DG categories_, [arXiv:1402.7364](http://arxiv.org/abs/1402.7364).

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preuniform locale

\tableofcontents

## Definition

The functor $\mathcal{O}:OLoc^\op \to Frm$ from the [[opposite category]] of [[overt locales]] to the [[category]] of [[frames]] is a [[regular hyperdoctrine]]. This means that it has an [[internal logic]], where given an overt locale $L$, we can define the composition of open relations $E, F \in \mathcal{O}(L \times L)$ as

$$F \circ E \coloneqq \{(x, z):L \times L \vert \exists y:L.(x, y) \in E \wedge (y, z) \in F\}$$

A **preuniform locale** is a [[overt locale]] $L$ equipped with a [[filter]] $\mathcal{E}$ on the frame of opens $\mathcal{O}(L \times L)$ such that for each $E \in \mathcal{E}$,

* $(x, x) \in E$

* there exists an $E^o \in \mathcal{E}$ such that $(x, y) \in E$ entails and is entailed by $(y, x) \in E^o$

* there exists an $F \in \mathcal{E}$ such that $F \circ F \leq E$

## See also

* [[locale]]

* [[overt locale]]

* [[preuniform locale]], [[uniform locale]]

* [[premetric locale]], [[metric locale]]

## References

* Graham Manuell, *Uniform locales and their constructive aspects*, ([arXiv:2106.00678](https://arxiv.org/abs/2106.00678))


prevector space

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## Idea ##

In the same vein that [[commutative rings]] are to [[integral domains]] and [[GCD rings]] are to [[GCD domains]], [[prevector spaces]] are to [[vector spaces]].

## Definition ##

A $R$-[[module]] $V$ is a **prevector space** if $R$ is a [[prefield ring]]. The elements of $V$ are called **vectors**.

## Examples ##

* Every $\mathbb{Q}$-vector space is a $\mathbb{Q}$-prevector space.

* Every classical vector space is a $F$-prevector space for a classical [[field]] $F$ (defined using [[denial inequality]]).

* Every Heyting vector space is a $F$-prevector space for a [[Heyting field]] $F$.

* Every discrete vector space is a $F$-prevector space for a [[discrete field]] $F$.

* Every residue vector space is a $F$-prevector space for a [[residue field]] $F$.

## See also ##

* [[module]]

* [[vector space]]

* [[super prevector space]]

* [[N-graded prevector space]]

* [[prefield ring]]


primary homotopy operation

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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

The _primary homotopy operations_ are the operations generated by

* the [[Whitehead product]]s;

and

* the [[composition operation]]s.

## Properties

The primary homotopy operations encode via [[Pi-algebras]] into an algebraic structure akin to models for a [[Lawvere theory]].  (This is discussed more fully in the entry on [[Pi-algebras]].)

[[!redirects primary homotopy operations]]


primary homotopy operations > history

< [[primary homotopy operations]]
[[!redirects primary homotopy operations>history]]


prime element

An element of a [[ring]] (or [[rig]]) or [[lattice]] (or [[proset]]) is called __prime__ if the [[ideal]] that it generates is a [[prime ideal]] (thus a [[principal ideal|principal]] prime ideal).

Note that by this definition [[zero]] is a prime element of the [[rig]] of [[natural numbers]], although it is not a [[prime number]].  Prime numbers as such correspond more closely to [[maximal ideals]] or [[irreducible ideals]] than to prime ideals.

Classically (using [[excluded middle]]), a prime element of a [[frame]] corresponds precisely to a [[point of a locale|point]] of the corresponding [[locale]].  For a [[constructive mathematics|constructive]] treatment, however, one must use the [[completely prime filters]] of the frame instead.

[[!redirects prime element]]
[[!redirects prime elements]]


prime factor

#Contents#
* table of contents
{:toc}

## Idea

For $n \in \mathbb{N}$ a [[natural number]] a _prime factor_ is a [[prime number]] $p \in \mathbb{N}$ which divides $n$, hence such that there is $k \in \mathbb{N}$ with $k \cdot p = n$.

The _prime factorization_ of a natural number is its expression entirely as a product of prime factors.

## Related concepts

* [[coprime integers]]

* [[fundamental theorem of finitely generated abelian groups]]

## References

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

[[!redirects prime factors]]

[[!redirects prime factorization]]
[[!redirects prime factorizations]]


prime field

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#Contents#
* table of contents
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## Definition

For $p \geq 2$ a [[prime number]], the [[quotient]] $\mathbb{Z}/p\mathbb{Z}$ of the [[ring]] of [[integers]] by $p$ is a [[finite field]] denoted $\mathbb{F}_p$, a _prime field_.

Prime fields have [[positive characteristic]].

Every prime field is a [[prime power local ring]] with [[trivial nilradical]].

## Properties

### Analogy with knots/prime geodesics

The [[spectrum of a commutative ring|spectra]] of prime fields inside [[Spec(Z)]] are [[analogy|analogous]] to [[knots]]/[[prime geodesics]] inside a ([[hyperbolic manifold|hyperbolic]]) 3-dimensional space, see at
_[Spec(Z) -- As a 3-dimensional space containing knots](http://ncatlab.org/nlab/show/Spec+Z#As3dSpaceContainingKnots)_

### Function field analogy

[[!include function field analogy -- table]]

## Related concepts

* [[prime power local ring]]

* [[integers modulo n]]

* [[field with one element]]

* [[linear code]]

[[!redirects prime fields]]

[[!redirects Fp]]


prime geodesic

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#Contents#
* table of contents
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## Idea

A _prime geodesic_ is a primitive closed [[geodesic]] in a [[hyperbolic manifold]].

## Properties

### Analogy with prime numbers
 {#AnalogyWithPrimeNumbers}

Prime geodesics in hyperbolic geometry are in several ways [[analogy|analogous]] to [[prime numbers]] and [[prime ideals]] in [[arithmetic geometry]].

Notably they appear in the [[Selberg zeta function]] (see there) in the same way as prime ideals appear in the [[Artin L-function]], and moreover their [[asymptotics]] is controled in an analogous way by this function via the [[prime geodesic theorem]] as that of prime numbers is by the [[prime number theorem]] and the [[Riemann zeta function]].

A table of analogies is in ([Brown 09, p. 9](#Brown09)). Some are also mentioned in [Wikipedia, prime geodesic -- Number theory](http://en.wikipedia.org/wiki/Prime_geodesic#Number_theory).

### Prime geodesic theorem

The [[prime geodesic theorem]] (e.g. [Soundararajan-Young 13](#SoundararajanYoung13)) describes the [[asymptotic]] distribution of prime geodesics in [[analogy]] to the [[prime number theorem]].

## Related concepts

* [[prime number]], [[prime knot]]

* The [[Selberg zeta function]] of a hyperbolic manifold is an [[infinite product]] over prime geodesics.

## References

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

* {#Brown09} Darin Brown, _Lifting properties of prime geodesics_, Rocky Mountain J. Math. Volume 39, Number 2 (2009), 437-454 ([euclid](http://projecteuclid.org/euclid.rmjm/1239113439))

On the [[prime geodesic theorem]] (see there for more):

* {#SoundararajanYoung13} K. Soundararajan, Matthew P. Young, _The Prime Geodesic Theorem_, J. Reine Angew. Math. 676 (2013), 105-120 ([arXiv:1011.5486](http://arxiv.org/abs/1011.5486))

* Maki Nakasuji, _Prime Geodesic Theorem for Higher-Dimensional Hyperbolic Manifold_, Transactions of the AMS, Vol. 358, No. 8

[[!redirects prime geodesics]]


prime geodesic theorem

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#Contents#
* table of contents
{:toc}

## Idea

What is known as the _prime geodesic theorem_ is a result that descrives the [[asymptotic]] distribution of [[prime geodesics]] on a [[hyperbolic manifold]].

The prime geodesic theorem is [[analogy|analogous]] to the [[prime number theorem]]. Just as the zeros of the [[Riemann zeta function]] control the error term in the prime number theorem, so the zeros of the [[Selberg zeta function]] control the error term in the prime geodesic theorem.

## Statement

Let $X$ be a [[hyperbolic manifold]] of [[dimension]] $d + 1$. Let

$$
  \Gamma \coloneqq \pi_1(X)
$$

be its [[fundamental group]].

For each element $\gamma \in \Gamma$ there exists one closed [[geodesic]] in $X$ representing it. By standard convenient abuse of notation we write $\gamma$ also for that geodesic. Write

* $l(\gamma)$ for the [[length]] of the geodesic $\gamma$;

* $N(\gamma) \coloneqq e^{l(\gamma)}$ for what is called its _norm_;

* $\pi_\Gamma(x)$ for the number of primitive elements $\gamma \in \Gamma$ such that $N(\gamma) \leq x$.

The _prime geodesic theorem_ is a statement of the form that this number $\pi_\Gamma(x)$ satisfies

$$
  \pi_\Gamma(x)
  =
  li(x^d)
  +
  \underoverset{n = 0}{N}{\sum} li(x^{s_n}) + (error\;term)
$$

where

* $li(x) = \underoverset{0}{x}{\int}\frac{d t}{ln t}$ is the [[logarithmic integral function]];

* $\{s_1, \cdots, s_N\}$ are the zeros of the [[Selberg zeta function]] in the interval $(d/2,d)$

* $(error\;term)$ is some term, estimates of which are the main content of this class of theorems.

## References

The original estimates are due to [[Atle Selberg]], Hejhal, Huber, and [[Peter Sarnak]].

* [[Peter Sarnak]], _Prime Geodesic Theorems_, Stanford University, 1980

Further developments include

* Koyama _Prime geodesic theorem for arithmetic compact surfaces_, Int Math Res Notices (1998) 1998 ([web](http://imrn.oxfordjournals.org/content/1998/8/383.extract))

* Maki Nakasuji, _Prime Geodesic Theorem for Higher-Dimensional Hyperbolic Manifold_, Transactions of the AMS, Vol. 358, No. 8, 2006 ([JSTOR](http://www.jstor.org/stable/3845331), [pdf](http://www.ams.org/journals/tran/2006-358-08/S0002-9947-06-04122-5/S0002-9947-06-04122-5.pdf))

* {#SoundararajanYoung13} K. Soundararajan, Matthew P. Young, _The Prime Geodesic Theorem_,J. Reine Angew. Math. 676 (2013), 105-120 ([arXiv:1011.5486](http://arxiv.org/abs/1011.5486))

Discussion for the case of simple geodesics in hyperbolic surfaces contains

* [[Igor Rivin]], _Simple Curves on Surfaces_, Geometriae Dedicata August 2001, Volume 87, Issue 1-3, pp 345-360 ([web](http://link.springer.com/article/10.1023%2FA%3A1012010721583))

* [[Maryam Mirzakhani]], _Growth of the number of simple closed geodesics on hyperbolic surfaces_, Pages 97-125 from Volume 168 (2008), Issue 1 ([web](http://annals.math.princeton.edu/2008/168-1/p03))


prime ideal

# Prime ideals
* table of contents
{: toc}

## Idea

_Prime ideals_ are supposed to be a generalization of [[prime numbers]] from elements of the [[ring]] of [[integers]] to [[ideals]] in the sense of 'ideal elements' of an arbitrary [[ring]] (usually [[commutative ring|commutative]], but also possibly something more general than a ring).  It\'s not clear that they do so; [[maximal ideals]] may do a better job.  (In particular, [[zero]] is *not* a prime number, and the [[zero ideal]] of $\mathbb{Z}$ is not a maximal ideal either; however, it *is* a prime ideal.)  Nevertheless, they have assumed an importance that dwarfs any question of original motivation; indeed, the general definition of [[prime element]] follows prime ideals rather than prime numbers.  (An element of a ring is prime iff its [[principal ideal]] is prime; $0$ is a prime element of $\mathbb{Z}$ but not a prime number.)

## Definitions

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

Let $R$ be a [[rig]] (assumed unital and associative as usual), and let $P$ be a [[two-sided ideal]] in $R$.  Then $P$ is __prime__ if $P$ is [[proper ideal|proper]] and $x$ or $y$ belongs to $P$ whenever $x a y$ does for all $a$:

$$ \forall\, x \in R,\; \forall\, y \in R,\; (\forall\, a \in R,\; x a y \in P) \;\Rightarrow\; x \in P \;\vee\; y \in P .$$

Also, $P$ is __completely prime__ if $P$ is proper and $x$ or $y$ belongs to $P$ whenever $x y$ does:

$$ \forall\, x \in R,\; \forall\, y \in R,\; x y \in P \;\Rightarrow\; x \in P \;\vee\; y \in P .$$
=--

Note that every completely prime ideal is prime (using that $R$ is unital).  The converse holds if the rig is commutative (using that $P$ is an ideal).  So in [[commutative algebra]] one usually uses the (simpler) definition of completely prime ideal as the definition of prime ideal:

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

Let $R$ be a [[commutative ring|commutative]] rig, and let $P$ be an [[ideal]] in $R$.  Then $P$ is __prime__ if $P$ is [[proper ideal|proper]] and $x$ or $y$ belongs to $P$ whenever $x y$ does:

$$ \forall\, x \in R,\; \forall\, y \in R,\; x y \in P \;\Rightarrow\; x \in P \;\vee\; y \in P .$$
=--

Essentially the same definition applies in [[order theory]], using the analogy (which is more than an analogy in the case of a [[distributive lattice]]) between multiplication in a rig and the [[meet]] in an order:

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

Let $R$ be a [[lattice]], and let $P$ be an [[ideal]] in $R$.  Then $P$ is __prime__ if $P$ is [[proper ideal|proper]] and $x$ or $y$ belongs to $P$ whenever their meet does:

$$ \forall\, x \in R,\; \forall\, y \in R,\; x \wedge y \in P \;\Rightarrow\; x \in P \;\vee\; y \in P .$$
=--

There is an infinitary version of this that is called 'complete' but is *not* equivalent to the notion in Definition \ref{rig} of a completely prime ideal from the noncommutative theory:

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

Let $R$ be a [[complete lattice]], and let $P$ be an [[ideal]] in $R$.  Then $P$ is __completely prime__ if some element belongs to $P$ whenever the meet of a subset of $R$ does:

$$ \forall\, X \subseteq R,\; \bigwedge X \in P \;\Rightarrow\; \exists\, x \in R,\; x \in X \;\wedge\; x \in P .$$
=--

With a little sublety, this makes sense even when [[meets]] (and [[joins]]) don't always exist:

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

Let $R$ be a [[preorder]], and let $P$ be an ideal in $R$.  Then $P$ is __prime__ if $P$ is [[proper ideal|proper]] and $x$ or $y$ belongs to $P$ whenever every $z$ that precedes both $x$ and $y$ does:

$$ \forall\, x \in R,\; \forall\, y \in R,\; (\forall\, z \in R,\; z \leq x \;\Rightarrow\; z \in P) \;\Rightarrow\; (\forall\, z \in R,\; z \leq y \;\Rightarrow\; z \in P) \;\Rightarrow\; x \in P \;\vee\; y \in P .$$

Also, $P$ is __completely prime__ if $P$ if some element belongs to $P$ whenever every $z$ that precedes every element of a subset of $R$ does:

$$ \forall\, X \subseteq R,\; (\forall\, z \in R,\; (\forall\, x \in R,\; x \in X \;\Rightarrow\; z \leq x) \;\Rightarrow\; z \in P) \;\Rightarrow\; \exists\, x \in R,\; x \in X \;\wedge\; x \in P .$$
=--

Note that a [[prime filter]] in a proset $R$ is a prime ideal in the [[opposite order]] $R^op$, and a [[completely prime filter]] in $R$ is a completely prime ideal in $R^op$.  Again, the meaning of 'completely prime' here is unrelated to its meaning in Definition \ref{rig}.

All of these definitions may be justified by looking at the [[quantale]] of ideals.  As discussed at [[ideals in a monoid]], there is for two-sided ideals an operation of ideal multiplication, making the [[lattice of ideals|ideal lattice]] $Idl(R)$ a quantale (cf. [[Day convolution]]).  Namely, if $I, J$ are ideals, then their product $I J$ is the ideal generated by all products $x y$ with $x \in I, y \in J$ in the case of rigs, or generated by all meets $x \wedge y$ in the case of lattices, or generated by all $z$ satisfying $z \leq x$ and $z \leq y$ in the case of general prosets.  (Note that for a lattice or other proset, $I J$ is equal to the [[intersection]] $I \cap J$.)

With $I J$ suitably defined, every definition above (except the 'complete' ones) can be subsumed below:

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

Let $R$ be a rig or a proset, and let $P$ be an ideal in $R$.  Then $P$ is __prime__ if $P$ is [[proper ideal|proper]] and $I$ or $J$ is contained in $P$ whenever $I J$ is:

$$ \forall\, I \in Idl(R),\; \forall\, J \in Idl(R),\; I J \subseteq P \;\Rightarrow\; I \subseteq P \;\vee\; J \subseteq P .$$
=--

Because $I J = I \cap J$ in order theory, prime ideals there are the same as [[strongly irreducible ideals]], so the completely prime ideals of that theory are really just [[completely strongly irreducible ideals]] (generalizing from a pair of ideals to an arbitrary set of ideals).  In contrast, the completely prime ideals of noncommutative ring theory are not of much interest; they are a naïve definition that works in the commutative case but not so well in the noncommutative case.

Finally, note that most of these definitions have an extra clause that a prime ideal must be [[proper ideal|proper]].  This can be justified by removing [[bias]].  We state here the unbiased version of \ref{universal}:

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

Let $R$ be a rig or a proset, and let $P$ be an ideal in $R$.  Then $P$ is __prime__ if some ideal in the list is contained in $P$ whenever a product of a [[finite list]] of ideals is contained in $P$:

$$ \forall\, n \in \mathbb{N},\; \forall\, I \in Idl(R)^n,\; \prod_{k\in[n]} I_k \subseteq P \;\Rightarrow\; \exists\, k \in [n],\; I_k \subseteq P .$$
=--

As is typical, $n = 1$ is trivial, $n \gt 2$ can be proved from $n = 2$ by [[induction]], and $n = 0$ is the mysterious preliminary clause, in this case that $P$ is proper.  Note that completely prime ideals in a proset arise by generalizing from finite $n$ to arbitrary [[cardinality]].

Unbiased versions of the other definitions are fairly straightforward.  (For prime ideals in a noncommutative rig, the unbiased definition involves a product of the form $a_0 x_0 a_1 x_1 \cdots x_{n-2} a_{n-1} x_{n-1} a_n$; $a_0$ and $a_n$ can be ignored when $n \gt 0$, which is why only $a_1$ appears in the biased definition.  The definition of completely prime ideals in order theory is already unbiased, since $X$ could always be the [[empty set]].)

Sometimes it\'s more fruitful to consider the [[complement]] of a prime ideal.  This is known in [[constructive mathematics]] as an [[anti-ideal]], and this becomes a necessary perspective there, as many common examples fail to satisfy the above definitions constructively.  (To support anti-ideals, a rig must be equipped with a [[tight apartness relation]], which is vacuous in [[classical mathematics]].)  However, the concept is useful even classically.

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

Let $R$ be a [[algebra with apartness|rig with apartness]], and let $M$ be a two-sided [[anti-ideal]] in $R$.  Then $M$ is __prime__ if $M$ is [[proper antiideal|proper]] (that is [[inhabited subset|inhabited]]) and $x a y$ belongs to $M$ for some $a$ whenever $x$ and $y$ do:

$$ \forall\, x \in R,\; \forall\, y \in R,\; x \in M \;\Rightarrow\; y \in M \;\Rightarrow\; \exists\, a \in R,\; x a y \in M .$$

Also, $M$ is __completely prime__ if $M$ is proper and $x y$ belongs to $M$ whenever $x$ and $y$ do:

$$ \forall\, x \in R,\; \forall\, y \in R,\; x \in M \;\Rightarrow\; y \in M \;\Rightarrow\; x y \in M .$$
=--

If we ignore the requirement that $M$ be an anti-ideal, then we say that $M$ is an __m-system__ if it is inhabited and satisfies the binary condition of a prime ideal and __multiplicatively closed__ if it owns $1$ and satisfies the binary condition of a completely prime ideal.  (A proper anti-ideal necessarily owns $1$, but an m-system might not, even though by definition it must be inhabited.)  Thus, an ideal in a commutative ring is (classically) prime iff its complement is multiplicatively closed (and analogously for noncommutative rings).

## Examples

In a [[matrix ring]] $M_n(k)$ over a [[field]] $k$, the [[zero ideal]] is prime (really because a matrix ring is a [[simple ring]], where the zero ideal is a [[maximal ideal]]), but (for $n \gt 1$) not *completely* prime.

In the [[ring of integers]] (or the rig of [[natural numbers]]), the prime ideals are precisely the [[principal ideals]] of the [[prime numbers]] together with the [[zero ideal]].  (This is the motivating example, despite not lining up perfectly.)

[[!redirects prime ideal]]
[[!redirects prime ideals]]

[[!redirects completely prime ideal]]
[[!redirects completely prime ideals]]


prime ideal theorem

# Contents
* table of contents
{:toc}

## Idea

A __prime ideal theorem__ is a theorem stating that every [[proper ideal]] is contained in some [[prime ideal]].  A prime ideal theorem is typically equivalent to the [[ultrafilter principle]] (UF), a weak form of the [[axiom of choice]] (AC).

We say 'a' prime ideal theorem (PIT) instead of 'the' prime ideal theorem, since we have not said what the ideals are in. We list some representative examples of prime ideal theorems, all of which are equivalent to (UF) in ZF or even in BZ (bounded Zermelo set theory):

*  The PIT for (commutative) [[rings]].

*  The PIT for (bounded) [[distributive lattices]].

*  The PIT for [[Boolean algebras]].

*  The PIT for [[rigs]], which subsumes all of the above. In fact one can generalize quite a bit further, as we indicate below when we discuss prime element theorems.

The usual way to prove a prime ideal theorem is with the help of [[Zorn's Lemma]], unless one is specifically trying to use something weaker like UF (as we do in this article). In fact [[Zorn's lemma]] can be used to prove a [[maximal ideal theorem]], which is stronger: in all the usual examples maximal ideals are prime so that a [[prime ideal theorem]] becomes a corollary.

There are stronger forms of prime ideal theorems which assert the existence of a prime ideal that does not intersect a given multiplicatively closed subset $\mathfrak{m}$. However, these can often be reduced to an ordinary PIT, by first localizing or inverting the elements in $\mathfrak{m}$, and then applying a PIT to the localization.

## A ladder of prime ideal theorems

Prime ideal theorems may be arranged as a sequence of results and techniques that gradually increase in sophistication; we sketch one possible development here, filling in details in the remainder of the article.

A first easy wave of results concerns the equivalence between the Boolean PIT (or BPIT) and UF. While these are easy, the BPIT is a theoretical underpinning of fundamental techniques such as the [[compactness theorem]] for [[first-order logic]], which leads to the next rung on the ladder.

The next step derives the PIT for distributive lattices as a simple application of the compactness theorem. In effect we write down a simple [[propositional theory]] where a [[model]] is tantamount to a prime ideal in a given distributive lattice. Part of the work involves checking finite satisfiability of the theory, where we may invoke a simple [[Stone duality]] between finite distributive lattices and finite posets to get the job done.

With the PIT for distributive lattices in hand, we prove a key result due to Banaschewski, which may be summarized roughly as saying that a (nontrivial) compact [[frame]] admits a prime element. This is an early result in [[Stone Spaces]], related to the spatiality of compact regular [[locales]].

Banaschewski's lemma is a key of entry into prime ideal theorems of general type. The general idea is that ideals in a monoid (now in suitably nice monoidal categories) tend to form compact [[quantales]], whose prime elements correspond to prime ideals. One then invokes a simple construction which associates to each compact quantale a quotient compact frame, in such a way that the existence of a prime element in the quantale is reduced to existence of a prime element in the compact frame *à la* Banaschewski's lemma. As promised, this gives prime ideal theorems of fairly general type (call them collectively "GPIT").

As suggested earlier, any one of these prime ideal theorems implies the BPIT as a special case, and so we come full circle:

$$UF \Rightarrow BPIT \Rightarrow PIT(DistLat) \Rightarrow Banaschewski \Rightarrow GPIT \Rightarrow BPIT \Rightarrow UF.$$

We now turn to details.

### Boolean PIT and the ultrafilter principle

The Boolean prime ideal theorem or BPIT is equivalent to the [[ultrafilter principle]] UF.

The reasoning may be summarized as follows: the ultrafilter principle implies the [[Tychonoff theorem]] for compact Hausdorff spaces; see this [Remark](/nlab/show/Tychonoff+theorem#TyCH) and the argument immediately preceding it. The Tychonoff theorem for compact Hausdorff spaces in turn implies that every Boolean ring $B$ has a maximal (and therefore prime) ideal; see [here](/nlab/show/Tychonoff+theorem#nonempty). Consequently, for any ideal $I$ of a Boolean algebra $B$, the quotient Boolean ring $B/I$ has a prime ideal $P$, and the pullback $q^{-1}(P) \subseteq B$ of the quotient map $q: B \to B/I$ produces a prime ideal in $B$ which contains a given ideal $I$, thus proving the BPIT from UF.

(In passing, one may note that prime ideals in Boolean algebras (or Boolean rings) $B$ are the same as maximal ideals. For, the corresponding Boolean quotient ring is an [[integral domain]] only if it is the [[field]] $\mathbb{Z}/(2)$, since [[idempotent element|idempotency]] of elements yields $e(1-e) = 0$ and then $e = 0$ or $e = 1$ assuming no [[zero divisors]].)

Finally, maximal ideals are complementary to ultrafilters (see [here](/nlab/show/Boolean+algebra#aremax)). So UF, which says that every [[filter]] in $B = P X$ is contained in an ultrafilter, translates into a special case of the BPIT: every ideal of $P X$ (consisting of negations of elements of a corresponding filter) is contained in a prime/maximal ideal. This brings us full circle: BPIT implies UF implies Tychonoff(CH) implies BPIT.

### BPIT implies prime ideal theorem for distributive lattices

Now let us prove that the Tychonoff theorem for CH spaces implies the PIT for distributive lattices $D$: that any proper ideal $I$ of $D$ is contained in a prime ideal of $D$. By passing to the quotient lattice $D/I$, it suffices to show that $D/I$ has a prime ideal $P$, since the inverse image $\phi^{-1}(P)$ of a prime ideal $P$ along the quotient map $\phi: D \to D/I$ is again prime.

+-- {: .num_theorem}
###### Theorem
(UF)
Any nontrivial distributive lattice $D$ has a prime ideal $P$.
=--

To prove this, we write down a propositional theory of "a prime ideal $P$ of $D$" (and consider its [[Lindenbaum algebra]]). Form a free Boolean algebra $Bool(U D)$ freely generated by the underlying set $U D$ of $D$. So each $x \in U D$ corresponds to a generator $P_x \in Bool(U D)$, which we will think of as standing for a proposition $P_x = $ "$x \in P$". The conditions that enforce "$P$ is a prime ideal of $D$" are the axioms of our theory:

* ($P$ is closed under finite joins) $P_0$, $P_a \wedge P_b \Rightarrow P_{a \vee b}$.

* ($P$ is downward closed) For each $b \in U D$, $P_a \Rightarrow P_{a \wedge b}$.

* ($P$ is proper) $\neg P_1$.

* (primality condition) $P_{a \wedge b} \Rightarrow (P_a \vee P_b)$.

These axioms (certain elements of $Bool(U D)$) then generate a [[filter]] $\mathcal{F}$. As soon as we know $\mathcal{F}$ is proper ("finite satisfiability of the theory"), the BPIT ensures that $F$ is contained in an ultrafilter $\mathcal{U}$, corresponding to a [[model]] or a Boolean ring homomorphism $Bool(U D)/\mathcal{F} \to \mathbf{2}$ out of the Lindenbaum algebra. For that model, the collection $\{a \in D: P_a \in \mathcal{U}\}$ then forms a prime ideal $P$ of $D$, as desired. So:

+-- {: .proof}
###### Proof
(The filter $\mathcal{F}$ is proper.) To see that the filter generated by finitely many axioms does not contain $0 \in Bool(U D)$, consider the sublattice of $D$ generated by the finitely many elements $a$ named in atomic subformulas of such axioms. A finitely generated distributive lattice is finite (indeed, by [[Stone duality]], the free distributive lattice on a finite number $n$ or elements is isomorphic to the poset of downward-closed subsets of the poset $\mathbf{2}^n$). So all we need to do is show that any nontrivial finite distributive lattice $L$ has a prime ideal. This also follows from Stone duality: a distributive lattice homomorphism $L \to \mathbf{2}$ corresponds to a point of the Stone dual $S$, and if $L$ is nontrivial then $S$ must be inhabited by a point $x$, so lattice homomorphisms $L \to \mathbf{2}$ exist and the kernel of such is a maximal and therefore prime ideal.
=--

### Banaschewski's lemma

During the 1980's, [Banaschewski](#Ban) discovered the following result (a breakthrough in the study of prime ideal theorems, according to [Marcel Erné](http://www.researchgate.net/publication/242573716_Prime_Ideal_Theorems_and_systems_of_finite_character)). We state and prove the lemma for now, and show how it is used in later sections.

+-- {: .num_theorem}
###### Theorem
**(Banaschewski's lemma)**
Every nontrivial distributive complete lattice $L$ whose top element $1$ is [[compact element|compact]], e.g., a compact [[frame]], has a prime element.
=--

+-- {: .num_remark}
###### Remarks
Here "nontrivial" means $0 \neq 1$: distinct top and bottom elements. This is needed because prime ideals are required to be proper (see [[too simple to be simple]]). An element $p$ is *prime* if the principal ideal it generates is prime, i.e., if $a \wedge b \leq p$ implies $a \leq p$ or $b \leq p$. In a distributive lattice, this is the same as saying $p = a \wedge b$ implies $p = a$ or $p = b$ (meet-irreducibility).
=--

+-- {: .proof}
###### Proof
First we observe that a coproduct of nontrivial bounded distributive lattices $\{X_i\}_{i \in I}$ (in the category of bounded distributive lattices) is also nontrivial. Again this is a kind of finite satisfiability result: if such a coproduct were trivial, i.e., if $0 = 1$ were satisfied in the coproduct (constructed syntactically as the free distributive lattice $F = F(\sum_i X_i)$ modulo equations that identify finite meets and joins in each $X_i$ with formal meets and joins in $F$), then $0 = 1$ would be derivable from finitely many equations involving finitely many elements $x_i$ in the $X_i$, and hence $0 = 1$ would be derivable in a finite coproduct of finite bounded nontrivial distributive lattices generated by these elements. This cannot happen, by considering their [[Stone duality]] with finite [[posets]] and products thereof: finite products of [[inhabited set|inhabited]] finite posets are also inhabited.

With that in hand, consider the [[coproduct]] $M = \sum_{a \in L \setminus \{1\}} \uparrow a$ of all the nontrivial principal filters of $L$, taken in the category of bounded distributive lattices; let $i_a: \uparrow a \to M$ denote the coproduct inclusion. $M$ has a prime ideal $P$ by the PIT for distributive lattices; the pullback $P_a \coloneqq i_a^{-1}(P)$ is a prime ideal of $\uparrow a$. Of course $P_a \subseteq L \setminus \{1\}$ by properness of prime ideals.

Put $S = L \setminus \{1\}$. This is a [[dcpo]] by completeness of $L$ and compactness of $1$; since proper ideals $I \subseteq S$ are directed subsets, we see that $S$ admits a sup $\sigma(a) \coloneqq \bigvee P_a$ for each $a \in S$. We have $a \leq \sigma(a)$ for all $a \in S$. By the [[Zorn's lemma|Bourbaki-Witt fixed point theorem]], the inflationary operator $\sigma: S \to S$ has a fixed point, say $c: c = \sigma(c)$. For that $c$, note that $P_c \subseteq \uparrow c$ is just $\{c\}$.

So $\{c\}$ is a prime ideal in $\uparrow c$. By meet-irreducibility, this means $c$ is a prime element in $L$.
=--

+-- {: .num_remark}
###### Remark
When the statement is about compact frames, it can be rephrased as saying that every nontrivial compact locale has at least one point; compare [[Stone Spaces]], Lemma 1.9.
=--

### Prime elements in quantales of ideals

To show how Banaschewski's lemma is applied, we consider for example the prime ideal theorem for (possibly noncommutative) rings. We follow the common convention that rings have units.

Let $R$ be a ring, and let $Idl(R)$ be the sup-lattice of two-sided ideals of $R$. Under multiplication of ideals $A \cdot B$, we obtain a [[quantale]] structure on $Idl(R)$ whose multiplicative unit is the top element $R$. In general we will call a quantale *affine* (or [[semicartesian monoidal category|semicartesian]]) if its unit is the top element.

The top element $1$ of $Idl(R)$ is also a *[[compact element]]*. In general, and generalizing a definition from the theory of [[frames]]/[[locales]], we will say that a quantale is *compact* if its top element is a compact element.

Finally, in a quantale $Q$, we say an element $p \in Q$ is *prime* if for all $a, b \in Q$ we have $a b \leq p$ implies $a \leq p$ or $b \leq p$. Prime elements in $Idl(R)$ are precisely prime ideals of $R$.

+-- {: .num_theorem #quantale}
###### Theorem
(ZF + UF) Every nontrivial compact affine quantale has a prime element.
=--

The strategy (following [Paseka](#Pas)) is to reduce the claim to an application of Banaschewski's lemma.

We begin by introducing an equivalence relation on $Q$: say $a \equiv b$ if $(\forall_c)\; 1 = a \vee c \Leftrightarrow 1 = b \vee c$.

+-- {: .num_lemma}
###### Lemma
The relation $\equiv$ is a quantale congruence.
=--

+-- {: .proof}
###### Proof
First we show that $\equiv$ respects the quantale multiplication $\cdot$. Suppose $a \equiv b$ and $x \equiv y$. So for any $c$, if $1 \leq a x \vee c$, it follows that $1 \leq a \vee c$ and $1 \leq x \vee c$, whence $1 \leq b \vee c$ and $1 \leq y \vee c$. Then

$$1 \leq b \vee c = (b \cdot 1) \vee c = b(y \vee c) \vee c = b y \vee b c \vee c = b y \vee c$$

and similarly $1 = b y \vee c$ implies $1 = a x \vee c$. So $a x \equiv b y$.

Now we show that $\equiv$ respects joins, i.e., if $x_i \equiv y_i$ for all $i \in I$, then $\bigvee_i x_i \equiv \bigvee_i y_i$. Suppose $1 = c \vee \bigvee_i x_i$. Since $Q$ is compact, there is a finite set of the $x_i$, say $x_1, \ldots, x_n$, such that $1 = c \vee x_1 \vee \ldots \vee x_n$. From $x_n \equiv y_n$ and $1 = (c \vee x_1 \vee \ldots \vee x_{n-1}) \vee x_n$, we derive

$$1 = (c \vee x_1 \vee \ldots \vee x_{n-1}) \vee y_n = (y_n \vee c) \vee x_1 \vee \ldots \vee x_{n-1}$$

and we proceed by induction to show $1 = c \vee y_1 \vee \ldots \vee y_n$, which implies $1 \leq c \vee \bigvee_{i \in I} y_i$. Similarly, $1 = c \vee \bigvee_i y_i$ implies $1 = c \vee \bigvee_i x_i$, and so we are done.
=--

+-- {: .num_prop #quot}
###### Proposition
The quantale $\tilde{Q}$ formed as the quotient $Q/\equiv$ is a nontrivial compact frame.
=--

+-- {: .proof}
###### Proof
Clearly $1 \leq x x \vee c$ implies $1 \leq x x \vee c \leq x \vee c$. But also if $1 \leq x \vee c$, then

$$1 \leq x \vee c = x \cdot 1 \vee c = x\cdot (x \vee c) \vee c = x x \vee x c \vee c = x x \vee c$$

and thus $x x \equiv x$ for all $x$. Being an idempotent affine quantale, $\tilde{Q}$ is a frame.

We have $\neg (0 \equiv 1)$ since ($0 \vee c = 1$ iff $1 \vee c = 1$) fails for $c = 0$, under the assumption $Q$ is nontrivial. So $\tilde{Q}$ is nontrivial. The same proof shows that for *any* $i \in Q$, we have $\neg (i \equiv 1)$ if $\neg (i = 1)$.

To show $\tilde{Q}$ is compact, suppose $\{x_i\}_{i \in I}$ is a family in $Q$ such that $\bigvee_i x_i \equiv 1$, in other words such that $c \vee \bigvee_i x_i = 1$ for all $c \in Q$. Setting $c = 0$ and applying compactness of $Q$, there is a finite subfamily $F$ such that $\bigvee_{i \in F} x_i = 1$, and then $\bigvee_{i \in F} x_i \equiv 1$ which proves compactness of $\tilde{Q}$.
=--

The quantale quotient map $Q \to \tilde{Q}$ preserves arbitrary sups and thus has a right adjoint $\pi: \tilde{Q} \to Q$, by the poset form of the [[adjoint functor theorem]]. If we denote the $\equiv$-class of an element $a \in Q$ by $[a]$, then for $c \in \tilde{Q}$ the element $\pi(c)$ is given by the explicit formula

$$\pi(c) = \bigvee \{y \in Q: [y] \leq c\}.$$

+-- {: .num_lemma #inverse}
###### Lemma
Given a regular epi $[-]: Q \to \tilde{Q}$ in the category of quantales, with right adjoint $\pi: \tilde{Q} \to Q$, if $c$ is prime in $\tilde{Q}$, then $\pi(c)$ is prime in $Q$.
=--

+-- {: .proof}
###### Proof
For elements $a, b \in Q$ we have

$$\array{
a b \leq \pi(c) & iff & [a b] \leq c & since \; [-] \dashv \pi \\
 & iff & [a] \cdot [b] \leq c & since \; [-]\; is\; \mathrm{a}\; quantale\; map \\
 & iff & ([a] \leq c) \vee ([b] \leq c) & since \; c \; is\; prime \\
 & iff & (a \leq \pi(c)) \vee (b \leq \pi(c)) & since \; [-] \dashv \pi
}$$

which completes the proof.
=--

The proof of Theorem \ref{quantale} is immediate:

+-- {: .proof}
###### Proof
Given that $Q$ is a nontrivial compact affine quantale, $\tilde{Q}$ is a nontrivial compact frame by Proposition \ref{quot}. Then $\tilde{Q}$ has a prime element $c$ by Banaschewski's lemma, whence $Q$ has a prime element by Lemma \ref{inverse}.
=--

+-- {: .num_cor}
###### Corollary
**(PIT for rings)**
Every unital ring $R$ has a prime ideal under ZF + (UF).
=--

+-- {: .num_remark}
###### Remark
See also [Banaschewski-Harting](#BH). We remark that the nucleus $k$ on $Idl(R)$ formed by the composition $k = \pi \circ [-]: Idl(R) \to Idl(R)$ has the following properties:

* $k(I \cap J) = k(I J) = k(I) \cap k(J)$,

* $k(J) = 1$ implies $J = 1$.

These properties are all that is needed to verify that the fixed points of $k$ (corresponding to the quotient $\widetilde{Idl(R)}$) forms a compact frame to which we can apply Banaschewski's lemma. For example, the nucleus obtained by taking the Jacobson [[radical]] of an ideal has these properties, as do other radicals considered in noncommutative ring theory (Levitski radical, Brown-McCoy radical).
=--

In fact the same methods show that we can strengthen Theorem \ref{quantale} to get what we really mean by a PIT:

+-- {: .num_cor #pit}
###### Corollary
Let $Q$ be a nontrivial compact affine quantale, and let $i \in Q$ be any *proper* element, i.e., assume $\neg (i = 1)$. Then there exists a prime element $p \in Q$ such that $i \leq p$.
=--

+-- {: .proof}
###### Proof
Let $k = \pi \circ [-]: Q \to Q$ be the [nucleus](/nlab/show/quantale#quantale_congruences_and_nuclei) corresponding to the congruence $\equiv$ (i.e., the regular epi $[-]: Q \to \tilde{Q}$). Since $i \neq 1$, we also have $k(i) \neq 1$ (i.e., $\neg (i \equiv 1)$), as shown in the proof of Proposition \ref{quot}. So $\tilde{Q} = Fix(k)$ and the upwards-closed set $k(i) \downarrow Fix(k)$ are both nontrivial compact affine frames, and there is a prime element $p$ of $k(i) \downarrow Fix(k)$. We have a regular epi of quantales $r$ defined as the composite

$$Q \stackrel{[-]}{\to} Fix(k) \stackrel{k(i) \vee -}{\to} k(i) \downarrow Fix(k)$$

(here using the fact that $k(i) \vee -$ preserves finite meets, since $Fix(k)$ is a distributive lattice). The right adjoint of $r$ is the inclusion $\rho: k(i) \downarrow Fix(k) \hookrightarrow Q$, and it follows from Lemma \ref{inverse} that $p = \rho(p)$ is a prime of $Q$ such that $k(i) \leq p$. Since $i \leq k(i)$, we conclude $i \leq p$.
=--

### Generalized prime ideal theorems

Now we consider more general settings in which we can speak of (two-sided) ideals and prime ideals, and to which we can apply Theorem \ref{quantale}.

For example, let $\mathbf{C}$ be a [[complete category|complete]], [[cocomplete category|cocomplete]] [[biclosed monoidal category]] (e.g., a [[cosmos]]). If $A$ is a [[monoid]] object in $\mathbf{C}$, then we can speak of [[ideal in a monoid|ideals]] in $A$, and typically $Idl(A)$ will inherit good properties from $\mathbf{C}$. If for example $\mathbf{C}$ is regular and well-powered, then $Idl(A)$ forms a quantale under multiplication of ideals, as discussed [here](/nlab/show/ideal+in+a+monoid#ideals_forming_a_quantale).

If we further assume that the monoidal unit $I$ of $\mathbf{C}$ is [[compact object|compact]] (i.e., finitely presented), then the top element of $Idl(A)$ generated by the monoid unit $e: I \to A$ will be a compact element. Then, defining a prime ideal in $A$ as usual to be a prime element in $Idl(A)$, we deduce existence of prime ideals from Theorem \ref{quantale}.

Here is a sample result along these lines:

+-- {: .num_theorem #gpit}
###### Theorem
Let $T$ be a [[commutative theory|commutative Lawvere theory]]. Then the category of algebras $Set^T$ is a well-powered regular cosmos in which the monoidal unit is compact. Consequently, for every monoid $A$ of $Set^T$ and every proper ideal $I \hookrightarrow A$, there is a prime ideal $P$ containing $I$ (by applying Corollary \ref{pit} to $Idl(A)$).
=--

+-- {: .num_cor}
###### Corollary
**(PIT for rigs)**
Every proper ideal in a [[rig]] is contained in a prime ideal.
=--

+-- {: .proof}
###### Proof
Take $T$ to be the theory of [[commutative monoids]], and apply Theorem \ref{gpit}.
=--

## References

* [[Bernhard Banaschewski]] and Roswitha Harting, _Lattice Aspects of Radical Ideals and Choice Principles_, Proc. London Math. Soc. s3-50 (3) (1985), 385-404. ([web](http://plms.oxfordjournals.org/content/s3-50/3/385.extract))
 {#BH}

* Bernhard Banaschewski, _Prime Elements from Prime Ideals_, Order 2 (1985), 211-213. ([Springer link](http://link.springer.com/article/10.1007%2FBF00334858))
 {#Ban}

* Jan Paseka, _A note on the prime ideal theorem_, Acta Universitatis Carolinae, Mathematica et Physica, Vol. 30 (1989), No. 2, 131-136. ([pdf](http://dml.cz/bitstream/handle/10338.dmlcz/701805/ActaCarolinae_030-1989-2_20.pdf))
 {#Pas}


prime knot

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## Idea

A _prime knot_ is a [[knot]] that cannot be decomposed, that is to say, a knot that cannot be constructed from the [[knot sum]] of two non-trivial knots. In the context of [[knot complexity]], whether or not a knot is prime is nontrivial. Any knot which is not _prime_ is called a [[composite knot]].

## Examples
* The [[unknot]]

* The fundamental example of a _prime knot_ is the [[trefoil]] with three [[crossings]]

* The family of [[torus knots]]

## Related concepts

* [[prime number]]

* [[prime geodesic]]

## References

* Adams, Colin, C., 1994, _The Knot Book—An elementary Introduction to the Mathematical Theory of Knots_,  W. H. Freedman and Co.

[[!redirects prime knots]]


prime number

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\linebreak

> A prime number is that which is measured [=divided by smaller number] by a monad [=unit] alone.

> [[[Euclid]], Def. 11 of *[[Elements]]* Book VII (~ 400-300 BC), see [here](monad+disambiguation#ElementsVIIDef11)]

## Definition

A _prime number_ is a nonzero [[natural number]] that cannot be written as a [[multiplication|product]] of finitely many natural numbers (all) other than itself, hence a natural number greater than 1 that is [[division|divisible]] only by 1 and itself.

This means that every [[natural number]] $n \in \mathbb{N}$ is, up to re-ordering of factors, _uniquely_ expressed as a product of a tuple of prime numbers:

$$
  n \;=\; 2^{n_1}  \cdot 3^{n_2} \cdot  5^{n_3} \cdot 7^{n_4} \cdot  11^{ n_5 } \cdots
$$

This is called the _prime factorization_ of $n$.

Notice that while the number $1 \in \mathbb{N}$ is, clearly, only divisible by one and by itself, hence might look like it deserves to be counted as a prime number, too, this would break the uniqueness of this prime factorization. In view of the general phenomenon in classifications in [[mathematics]] of some objects being _[[too simple to be simple]]_ one could say that 1 is "too prime to be prime".

However, historically, some authors did count 1 as a prime number, see e.g. [Roegel 11](#Roegel11).

## Relation to ideals and arithmetic geometry
  {#RelationToIdeals}

A number is prime if and only if it generates a [[maximal ideal]] in the [[rig]] $\mathbb{N}$ of [[natural numbers]].

Prime numbers do *not* quite match the [[prime elements]] of $\mathbb{N}$, since $0$ generates a [[prime ideal]] but not a maximal ideal; instead they match the [[irreducible elements]] ([Wikipedia](http://secure.wikimedia.org/wikipedia/en/wiki/Irreducible_element)).

From the [[Isbell duality|Isbell-dual]] point of view, where a [[commutative ring]] such as the [[integers]] $\mathbb{Z}$ is regarded as the [[ring of functions]] on some [[variety]], namely on [[Spec(Z)]], the fact that prime numbers $p$ correspond to [[maximal ideals]] means that they correspond to the _[[closed points]]_ in this variety (see [this Example](Zariski+topology#SpecZ)), one also writes

$$
  (p) \in Spec(\mathbb{Z})
  \,.
$$

This [[Isbell duality|dual]] perspective on [[number theory]] as being the [[geometry]] ([[algebraic geometry]]) over [[Spec(Z)]] is called _[[arithmetic geometry]]_.

[[!include function field analogy -- table]]

## Properties

### Riemann hypothesis

see at _[[Riemann hypothesis]]_

### Goldbach conjecture

see at _[[Goldbach conjecture]]_

### Asymptotic distribution

see at _[[prime number theorem]]_

## Specific classes of prime numbers

* [[Fermat prime]]
* [[Mersenne prime]]
* [[twin prime]]

## Related concepts

* [[prime factor]]

* [[coprime integers]]

* [[Bertrand's postulate]]

* [[p-localization]]

* [[prime geodesic]]

* [[prime knot]]

* [[prime power]]

## References

For historical discussion see

* {#Roegel11} Denis Roegel, _A reconstruction of Lehmer's table of primes (1914)_, 2011 ([pdf](http://locomat.loria.fr/lehmer1914/lehmer1914doc.pdf))

See also

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

[[!redirects prime number]]
[[!redirects prime numbers]]
[[!redirects theory of primes]]

[[!redirects prime]]
[[!redirects primes]]


prime number theorem

## Idea

What is known as the _prime number theorem_ is a description of the [[asymptotic]] distribution of [[prime numbers]]. It says that the number of primes less than a positive real number $x$ is asymptotic to $x/\log(x)$, where $\log x$ is the natural [[logarithm]].

## Proof

See [[proof of the prime number theorem]], which is based on a well-known account by Don Zagier (who in turn was explicating a short proof due to Donald J. Newman) -- see the references there.

## Related concepts

* An [[analogy|analog]] of the theorem exists for [[prime geodesics]] in [[hyperbolic manifolds]], accordingly called the _[[prime geodesic theorem]]_. Just as the zeros of the [[Riemann zeta function]] control the error term in the prime number theorem, so the zeros of the [[Selberg zeta function]] control the error term in the prime geodesic theorem.

## References

* Wikipedia, _[Prime number theorem](http://en.wikipedia.org/wiki/Prime_number_theorem)_


prime power

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## Definition

In the natural numbers, a **prime power** $p^n$ is a positive power $n \geq 1$ of a [[prime number]] $p$.

## Properties

The [[fundamental theorem of arithmetic]] states that every positive natural number is a finite product of prime powers.

## Uses in other parts of mathematics

In [[group theory]], a group is a [[p-primary group]] if its order is a [[prime power]] of the given [[prime number]] $p$.

The [[fundamental theorem of finite abelian groups]] states that every [[finite abelian group]] is the [[direct sum]] of [[cyclic groups]] of prime power [[order of a group|order]].

In ring theory, the [[integers modulo n]] which are [[local rings]] are precisely the [[integers modulo]] a [[prime power]].

## See also

* [[prime number]]

## External links

* Wikipedia, *[Prime power](https://en.wikipedia.org/wiki/Prime_power)*.


prime power local ring

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## Idea

Prime power local rings are the integers modulo $n$ which are local rings, in the same way that [[prime fields]] are the integers modulo $n$ which are fields.

## Definition

For $p \geq 2$ a [[prime number]] and for $n \geq 1$ a positive [[natural number]], the [[quotient]] $\mathbb{Z}/p^n\mathbb{Z}$ of the [[ring]] of [[integers]] by the [[prime power]] $p^n$ is a [[local ring]], a **prime power local ring**.

## Properties

The [[ideal]] of non-invertible elements is the ideal $p(\mathbb{Z}/p^n\mathbb{Z})$, and the quotient of $\mathbb{Z}/p^n\mathbb{Z}$ by the ideal of non-invertible elements is the [[prime field]] $\mathbb{Z}/p\mathbb{Z}$. Since the ideal of non-invertible elements is a [[nilradical]], every prime power local ring is an [[Artinian ring]], which implies that it is a [[local Artinian ring]].

The underlying [[abelian group]] of a prime power local ring is a [[p-primary group]].

The cardinality of the [[group of units]] of $\mathbb{Z}/p^n\mathbb{Z}$ (the [[Euler totient function]] $\phi(p^n)$) is given by $\phi(p^n) = p^{n - 1} (p - 1)$.

In addition, every prime power local ring with [[trivial nilradical]] is a [[prime field]].

## See also

* [[prime field]]

* [[integers modulo n]]

[[!redirects prime power local rings]]


prime ring

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## Related concepts

* [[ring]]

* [[Martindale ring of quotients]]

## References

See also

* [[eom]], _[Prime ring](https://www.encyclopediaofmath.org/index.php/Prime_ring)_

[[!redirects prime rings]]


prime spectrum

## Idea

The __prime [[spectrum of a ring]]__  (associative, but not necessarily commutative or unital) is the set of [[prime ideal]]s of the ring. For noncommutative rings however sometimes spectra of [[primitive ideal]]s are more interesting.

The prime spectrum of a commutative unital ring extends to a contravariant functor $Spec : CRing\to Set$.

The prime spectrum $Spec R$ of a commutative unital ring $R$ has a natural Zariski topology $\tau_{Zar}$,  given by the basis of open sets $D_f = Spec R[f^{-1}]\subset Spec R$ where $R\ni f\neq 0$. The prime spectrum considered as a topological space is  also called __Zariski spectrum__. A ring map $h:R\to S$ induces a continuous map $Spec h: Spec S \to Spec R$, so there is a  contravariant functor $Spec : CRing\to Top$.

The correspondence $D_f\mapsto R[f^{-1}]$ for all $f\neq 0$ extends to a unique sheaf $\mathcal{O}=\mathcal{O}_{Spec R}$ of commutative local rings on the Zariski topology on $Spec R$. The ringed space $(Spec R,\tau_{Zar},\mathcal{O})$ so constructed is also called the prime spectrum of the commutative ring $R$. An __affine scheme__ as a locally ringed space is any ringed space which is isomorphic to the prime spectrum of a commutative ring.

Any morphism of commutative rings $h:R\to S$ also induces the comorphism of structure sheaves on spectra, hence a morphism in locally ringed spaces. This way one obtains a contravariant functor $Spec : CRing\to lRingedSpace$ which is fully faithful (and its essential image is the strictly full subcategory whose objects are affine schemes).

## Related concepts

* [[prime spectrum of a monoidal stable (∞,1)-category]]

* [[Stone spectrum]]

* [[Gelfand spectrum]]

* [[Gelfand duality]]

[[!redirects Zariski spectrum]]


prime spectrum of a symmetric monoidal stable (∞,1)-category

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* table of contents
{:toc}

## Idea

The [[categorification]] of the concept of ([[commutative ring|commutative]]) [[ring]], hence the concept of (commutative) _[[2-ring]]_ in [[homotopy theory]]/[[(∞,1)-category theory]] may be taken to be that of ([[symmetric monoidal (infinity,1)-category|symmetric]]) [[monoidal (∞,1)-category|monoidal]] [[stable (∞,1)-category]].

In analogy to the [[prime spectrum]] of a [[commutative ring]], the prime spectrum of a symmetric monoidal stable $(\infty,1)$-category is the collection of [[thick subcategories]] which are [[prime ideals]], in the evident higher sense, under the given [[tensor product]]. On the level of [[homotopy category of an (infinity,1)-category|homotopy categories]] this concept was introduced in ([Balmer 04](#Balmer04)) as the _[[spectrum of a tensor triangulated category]]_.

## Examples

### The prime spectrum of the category of spectra
 {#OfCategoryOfSpectra}

The prime spectrum of the [[(∞,1)-category of spectra]] ([[p-localization|p-local]] and [[finite spectrum|finite]]) is labeled by the [[Morava K-theories]] ([Balmer 10, prop. 9.4](#Balmer10)). This follows from the the [[thick subcategory theorem]].

This is the topic of [[chromatic homotopy theory]], see at _[[Spec(S)]]_.

Generally, the [[kernel]] of any [[generalized homology theory]] here is a thick ideal, but not generally a prime ideal.

## Related concepts

* [[spectrum of a tensor triangulated category]]

* [[2-algebraic geometry]]

## References

The concept was introduced at the level of [[triangulated categories]] in

* {#Balmer04} [[Paul Balmer]], _The spectrum of prime ideals in tensor triangulated categories_. J. Reine Angew. Math., 588:149–168, 2005 ([arXiv:math/0409360](http://arxiv.org/abs/math/0409360))

* {#Balmer10} [[Paul Balmer]], _Spectra, spectra, spectra—tensor triangular spectra versus Zariski spectra of endomorphism rings_, Algebr. Geom. Topol., 10(3):1521–1563, 2010 ([pdf](http://www.math.ucla.edu/~balmer/research/Pubfile/SSS.pdf))

Review includes

* [[Greg Stevenson]], _Tensor actions and locally complete intersection_ PhD thesis 2011 ([pdf](http://www.math.uni-bielefeld.de/~gstevens/Stevenson_thesis.pdf))

Exposition with an eye towards application in [[chromatic homotopy theory]] in general and [[tmf]] in particular is in

* {#MazelGee13} [[Aaron Mazel-Gee]], _You could've invented $tmf$_, April 2013 ([pdf slides](http://math.berkeley.edu/~aaron/writing/ustars-tmf-beamer.pdf))

Further discussion is in

* [[Beren Sanders]], _Higher comparison maps for the spectrum of a tensor triangulated category_ ([arXiv:1302.4521](http://arxiv.org/abs/1302.4521))

See also

* [[Jack Morava]], _Toward a fundamental groupoid for the stable homotopy category_ ([arXiv:math/0509001](http://arxiv.org/abs/math/0509001))

[[!redirects prime spectrum of a monoidal stable (∞,1)-category]]

[[!redirects prime spectrum of a monoidal stable ∞-category]]
[[!redirects prime spectrum of a monoidal stable infinity-category]]

[[!redirects prime spectrum of a monoidal stable (infinity,1)-category]]

[[!redirects prime spectrum of a tensor triangulated category]]

[[!redirects spectrum of a symmetric monoidal (∞,1)-category]]


PrimeDeGold

RH is true, and go forever Maize and Blue! :-)

+-- {: .standout}
Hello, and welcome to the nLab!

Your recent edit on the theory of primes seems rather out of place here; see the [[HomePage]] and the [[About]] page to see what we do.

We do have an entry on [[prime numbers]] that you might contribute to, but if what you have to say is not related to our interests, then it would probably fit better on another site.

—[[Toby Bartels]]
=--

category: people

[[!redirects spam: PrimeDeGold]]


primitive element

> (This article is about primitive elements in [[coalgebra]] theory, not about primitive elements for finite [[field extensions]].)

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

#Contents#
* table of contents
{:toc}

## Definition

Working over a [[commutative ring]] $R$, recall that a _unital_ or _coaugmented_ [[coalgebra]] is an $R$-[[coalgebra]] $(C, \Delta: C \to C \otimes_R C, \epsilon: C \to R)$ equipped with a coalgebra map $u: R \to C$. Abusing notation, denote $u(1) \in C$ by $1$.

### Primitive element in a coalgebra

An element  $x$ in a [[coalgebra]] $C$ is __primitive__ if $\Delta(x) = 1\otimes x + x\otimes 1$. This condition implies $\epsilon(x) = 0$.

This notion generalizes straightforwardly to unital [[corings]] over $R$ and even more generally to any [[comonoid object|unital comonoid object]] in a [[CMon-enriched monoidal category]] (see below).

### Primitive element in a comodule

Let $N$ be a [[comodule]] over $C$, with co-action map $\Psi \colon N \longrightarrow C \otimes_R N$. Then an element $n \in N$ is called a **primitive element** or **[[coinvariant]]** if $\Psi(n) = 1 \otimes n$.

The $R$-module $prim(N)$ of primitive elements of $N$ is [[natural isomorphism|naturally]] to

1. the module of comodule homomorphisms out of $R$ into $N$

   $$
     prim(N) \simeq Hom_{C}(R,N)
     \,.
   $$

1. the [[cotensor product]] of $N$ with $A$

   $$
     prim(N) \simeq A \Box_R N
     \,.
   $$

### Object of primitive elements of a unital comonoid in a $CMon$-enriched monoidal category

Let $(C,\Delta,\epsilon)$ be a [[comonoid object|unital comonoid object]] in a [[CMon-enriched monoidal category]] $(\mathcal{C},\otimes,I)$. The object $P$ of primitive elements of $C$ is defined if it exists, as the [[equalizer]] in this diagram:

\begin{tikzcd}
P \arrow[r, "i"] & C \arrow[rr, "\Delta", shift left] \arrow[rr, "1 \otimes \eta + \eta \otimes 1"', shift right] &  & C \otimes C
\end{tikzcd}

\begin{proposition}
If the unital comonoid $C$ admits an object $P$ of primitive elements, the [[commutative rig]] $\mathcal{C}[I,I]$ verifies $2 \neq 1$, and the homsets $\mathcal{C}$[A,B] are [[multiplicatively cancellative modules]] over $\mathcal{C}[I,I]$, we then have:

\begin{tikzcd}
P \arrow[r, "i"] \arrow[rd, "0"'] & C \arrow[d, "\epsilon"] \\
                                  & I
\end{tikzcd}

\end{proposition}
\begin{proof}
In this diagram, the path from $C$ to $C$ by the upper square (with below side equal to $\Delta$) is equal to the identity and the two paths from $P$ to $C \otimes C$ are equal:

\begin{tikzcd}
                 & {} \arrow[rr, "1 \otimes \epsilon", no head] \arrow[d]                                                                       &  & {}                             \\
P \arrow[r, "i"] & C \arrow[rr, "\Delta", shift left] \arrow[rr, "1 \otimes \eta + \eta \otimes 1"', shift right] \arrow[d, "\epsilon"] &  & C \otimes C \arrow[u, no head] \\
                 & I                                                                                &  &
\end{tikzcd}

We thus have that:
$$i = i;\Delta;1 \otimes \epsilon = i;(1 \otimes \eta + \eta \otimes 1);(1 \otimes \epsilon)$$

and:
$$
i;\epsilon = i;\Delta;1 \otimes \epsilon;\epsilon = i;(1 \otimes \eta + \eta \otimes 1);(1 \otimes \epsilon);\epsilon$$

$$
 = i;(1 \otimes \eta);(1 \otimes \epsilon);\epsilon + i;(\eta \otimes 1);(1 \otimes \epsilon);\epsilon
$$

We then have:
<img src="/nlab/files/primitive.png"/>

$\mathcal{C}[P,I]$ being a multiplicatively cancellative $\mathcal{C}[I,I]$-module, if $i;\epsilon \neq 0$, we would have that $1 = 2$ in $\mathcal{C}[I,I]$ which is false, therefore $i;\epsilon = 0$.
\end{proof}

## Properties

### Primitives in a Hopf algebra

A straightforward calculation shows that the module of primitive elements in a [[Hopf algebra]] $H$ (or even in a [[bialgebra]] $H$) is a Lie subalgebra of the underlying [[Lie algebra]] of $H$ (whose bracket is the algebra [[commutator]]).
Thus, taking primitive elements yields a functor

$$P: HopfAlg \to LieAlg$$

(and of course we have more generally a functor $P: BiAlg \to LieAlg$ which is an extension along the full inclusion $HopfAlg \to BiAlg$).

For a Lie algebra $L$, let $U(L)$ be its [[universal enveloping algebra]]:

$$U(L) = T(L)/I$$

where $I$ is the two-sided ideal generated by elements of the form $x y - y x - [x, y]$ where $x, y \in L$. This carries a bialgebra structure whose comultiplication $\delta: U(L) \to U(L) \otimes U(L)$ is uniquely determined by the rule $\delta(x) = 1 \otimes x + x \otimes 1$ for $x \in L$. Since this says $x \in L$ is primitive, the counit $\epsilon: U(L) \to R$ is forced to be the algebra map such $\epsilon(x) = 0$ for all $x \in L$, and also the Hopf antipode is uniquely determined: $\sigma(x) = -x$ for $x \in L$.

The following proposition is entirely straightfoward:

+-- {: .num_prop}
###### Proposition
The functor $U: LieAlg \to BiAlg$ is [[left adjoint]] to the functor $P: BiAlg \to LieAlg$.
=--

This result is essentially tautologous and holds for any commutative ring of arbitrary characteristic. (This despite the fact that the $U(L)$ as defined above is not as well-behaved in nonzero characteristic as one might like; e.g. the [[PBW theorem]] fails.) More information on this adjunction may require more restrictive hypotheses:

+-- {: .num_prop}
###### Proposition
For $R = k$ a [[field]] of [[characteristic zero]], the [[adjunction unit|unit]] $L \to P U(L)$ is an [[isomorphism]].
=--

An immediate consequence is that for such ground fields $k$, the functor $U: LieAlg \to BiAlg$ is [[fully faithful functor|fully faithful]]. Of course, $U$ lands in the full subcategory of cocommutative Hopf algebras, which is exactly the category of [[group objects]] in the [[cartesian monoidal category]] of [[cocommutative coalgebras]].

The [[Milnor-Moore theorem]] gives further information: for Hopf algebras over a field of characteristic zero, the counit $U P(H) \to H$ is a monomorphism, and an isomorphism in case $H$ satisfies a suitable conilpotency condition.

(More needs to be added.)

[[!redirects primitive element in a coalgebra]]

[[!redirects primitive elements]]


primitive recursive arithmetic

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### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In [[proof theory]], **primitive recursive arithmetic**, or **PRA**, is a [[finitist mathematics|finitist]], [[quantifier]]-free formalization of the [[natural numbers]].

**PRA** can express arithmetic propositions involving natural numbers and any [[primitive recursive function]], including the operations of addition, multiplication, and exponentiation.

## As a Skolem theory

Let $T$ be a [[Lawvere theory]], i.e., a category with finite products $T$ equipped with a bijective-on-objects product-preserving functor $X: Fin^{op} \to T$ where $Fin$ is the category of finite [[cardinals]] and functions. Recall that a [[Skolem theory]] is a Lawvere theory $(T, X: Fin^{op} \to T)$ in which $N = X(1)$ carries a structure of parametrized [[natural numbers object]] in $T$.

Abstractly, PRA can be described as the initial Skolem theory. Precise statements to this effect are difficult to pin down in the literature. (More later)

##Related pages

* [[ordinal analysis]]


principal 2-bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _principal 2-bundle_ is the generalization of a $G$-[[principal bundle]] over a [[group]] $G$ to a principal structure over a [[2-group]]. It is a special case of a [[principal ∞-bundle]].

For $G = AUT(H)$ the [[automorphism 2-group]] of a group $H$, $G$-principal bundles are equivalent to $H$-[[gerbe]] (see [[gerbe (general idea)]] for more background.). An $H$-[[nonabelian bundle gerbe]] is a model for the total space of an $AUT(H)$-principal 2-bundle.

An expository introduction to the concepts is at [[infinity-Chern-Weil theory introduction]].

## Definition

For $G$ a topological [[Lie 2-group]], a topological or smooth **$G$-principal 2-bundle** $P \to X$ is a topological or [[Lie groupoid]] that arises as the [[homotopy fiber]] of a [[cocycle]] $X \to \mathbf{B}G$ in [[ETop∞Grpd]] or [[Smooth∞Grpd]], respectively, i.e. as an [[(∞,1)-pullback]] of the form

$$
  \array{
    P &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    X &\to& \mathbf{B}G
  }
  \,.
$$

By the general rules of [[homotopy pullbacks]], this may be modeled by an ordinary pullback of topological or [[Lie 2-groupoid]]s of the form

$$
  \array{
    P &\to& \mathbf{E}G
    \\
    \downarrow && \downarrow
    \\
    C(U) &\to& \mathbf{B}G
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    X
  }
  \,,
$$

where $C(U)$ is the [[Cech nerve]] of a [[good open cover]] $U \to X$
and where $\mathbf{E}G$ is the [[universal principal infinity-bundle|universal principal 2-bundle]] ([RS](#RS)). This says that principal 2-bundles are classified by [[Cech cohomology]] with coefficients in [[delooping]]s of (sheaves of) [[2-group]]s.

## Properties

### Classification
 {#Classification}

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

Let $G$ be a [[well-pointed simplicial topological group|well pointed topological 2-group]]. Write $\mathbf{B}G := \bar W G$ for its [[delooping]] [[simplicial topological space]] and

$$
  B G := \vert \mathbf{B}G\vert \in Top
$$

for the corresponding [[geometric realization of simplicial topological spaces]]. Then $B G$ is a classifying space for topological $G$-principal 2-bundles: for $X$ a (sufficiently nice...) [[topological space]] we have that the [[nonabelian cohomology|nonabelian]] [[Cech cohomology]] on $X$ with coefficients in $\mathbf{B}G$ is naturally in bijections with the set of [[homotopy]] classes of [[continuous function]]s $X \to B G$

$$
  H_{Top}(X, \mathbf{B}G) \simeq [X, B G]
  \,.
$$

=--

This appears as ([BaezStevenson, theorem 1](#BaezStevenson)). It is also a special case of the analogous theorem for topological [[principal infinity-bundles]] in ([RobertsStevenson](#RobertsStevenson)).

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

Let $G$ be a [[Lie 2-group]] with the property that $\pi_0 G$ is a [[smooth manifold]] and the projection $G_0 \to \pi_0 G$ is a [[submersion]]. Then equivalence classes of smooth $G$-principal bundles on a [[smooth manifold]] $X$ are in natural bijection with equivalence classes of topological $G$-principal 2-bundles (regarding $G$ as a topological 2-group)

$$
  H_{Top}(X, \mathbf{B}G) \simeq H_{smooth}(X, \mathbf{B}G)
$$

induced by the natural [[forgetful functor]] [[SmoothMfd]] $\to$ [[Top]].

=--

This appears as ([Nikolaus-Waldorf 11, prop. 4.1](#NikolausWaldorf11)).

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

For [[Lie 2-group]]s with the above property, also smooth $G$-principal 2-bundles have [[classifying space]] $B G = \vert \mathbf{B}G\vert$.

=--

## Connections

An ordinary [[principal bundle]] may be equipped with a [[connection on a bundle|connection]] by refining the cocycle

$$
  X \to \mathbf{B} G
$$

to a cocycle

$$
  P_1(X) \to \mathbf{B} G
$$

where $P_1(X)$ is the [[path groupoid]] of $X$.

Similarly, 2-bundles may be equipped with connections by refining their cocycles $X \to \mathbf{B}H$ to cocycles out of a higher path groupoid. Details on this are at [[schreiber:differential cohomology in a cohesive topos]].

## Examples

### T-Folds

Principal 2-bundles for the [[T-duality 2-group]] serve to model [[T-folds]] in [[string theory]] ([Nikolaus-Waldorf 18](#NikolausWaldorf18))

## Related concepts

* [[principal bundle]] / [[torsor]] / [[associated bundle]]

* **principal 2-bundle** / [[gerbe]] / [[bundle gerbe]]

  * [[principal 2-connection]]

  * [[central extension of groupoids]]

  * [[line 2-bundle]]

* [[principal 3-bundle]] / [[2-gerbe]] / [[bundle 2-gerbe]]

* [[principal ∞-bundle]] / [[associated ∞-bundle]] / [[∞-gerbe]]

  * [[circle n-bundle with connection]]

## References

The general description of higher bundles internal to generalized spaces modeled as [[∞-stacks]] is discussed in

* [[J. F. Jardine]], _Cocycle categories_ ([pdf](http://arxiv.org/abs/math.AT/0605198)) .

The above situation of ordinary $G$-[[principal bundles]] is section 2.1 _[[torsor|Torsors]] for [[sheaves]] of groups_ in that article. The generalization to principal 2-bundles and [[principal ∞-bundles]] is then briefly indicated in section 2.2, _Diagrams and torsors_ .

The point is that in the [[(∞,1)-topos]] of topological or smooth or whatever [[∞-groupoids]] (i.e. in the [[(∞,1)-category]] of [[∞-stacks]] on our category of [[space and quantity|test spaces]]) the above situation generalizes straightforwardly:

For $G$ a [[2-group]], a $G$-principal $2$-bundle is a fibration of groupoids $P \to X$ that arises as the [[fibration sequence|homotopy fiber]] of a classifying morphism $X \to \mathbf{B}G$ (a $2$-[[2-anafunctor|anafunctor]])

$$
 \array{
  P &\to& {*}
  \\
  \downarrow && \downarrow
  \\
  X &\to& \mathbf{B}G
 }
  \,.
$$

This may be modeled by the pullback of the [[universal principal infinity-bundle|universal principal 2-bundle]] as described in

* [[Urs Schreiber]], [[David Roberts]], _The inner automorphism 3-group of a strict 2-group_ ([arXiv](http://arxiv.org/abs/0708.1741))
{#RS}

As ordinary [[principal bundles]], the gadgets obtained this way may be described from various points of view, using [[anafunctor]] cocycles $g : X \stackrel{\simeq}{\to}\leftarrow Y \to \mathbf{B}H$ in [[nonabelian cohomology]], or the corresponding total spaces being 2-[[torsors]] equipped with 2-group [[action]], or certain variants of this.

Maybe the earliest explicit description of a principal $\infty$-bundle using a [[geometric definition of higher category]] is

* [[Paul Glenn]], _Realization of cohomology classes in arbitrary exact categories_, J. Pure Appl. Algebra 25, 1982, no. 1, 33- 105 (doi:10.1016/0022-4049(82)90094-9)

This describes [[torsors]] over [[∞-groupoids]] in terms of the corresponding $\infty$-[[action groupoids]].

This theory of higher bundles and [[gerbes]] was made to look manifestly like a systematic [[vertical categorification|categorification]] of the familiar description of ordinary [[principal bundles]] in terms of cocycles and local trivializations in

*  Luca Mauri, PhD thesis, 1998 ([pdf](http://home.aubg.bg/faculty/aganchev/Category%20Theory/Mauri%20%28papers%29/mauri-2.pdf));

An abridged version is

*  L. Mauri, M. Tierney, _Two-descent, two-torsors and local equivalence_ , J. Pure Appl. Algebra 143 (1999), 313--327.

The first article in the differential-geometric context was

* [[Toby Bartels]], _2-Bundles_ ([arXiv](http://arxiv.org/abs/math.CT/0410328), [web](http://toby.bartels.name/2bundles/))

One should notice that if one uses categories internal to [[diffeological spaces]], then these are (under their [[nerve]]) in particular [[simplicial presheaf|simplicial presheaves]], and that the [[anafunctors]] used as morphisms between these simplicial presheaves represent precisely the morphisms the corresponding [[(∞,1)-category of (∞,1)-sheaves]] using the [[model structure on simplicial presheaves]] or, more lightweight, the structure of a Brown [[category of fibrant objects]] on $\infty$-groupoid valued sheaves.

A description of higher principal bundles (see also [[principal ∞-bundle]]) in terms of the [[model structure on simplicial presheaves]] appears in

* [[J. F. Jardine]], Z. Luo, _Higher principal bundles_, Mathematical Proceedings of the Cambridge Philosophical Society,  Volume 140, Issue 2 March 2006 , pp. 221-243 ([pdf](http://www.math.uiuc.edu/K-theory/0681/cocycles6.pdf), [doi:10.1017/S0305004105008911](https://doi.org/10.1017/S0305004105008911))

The relation of such 2-categorical constructions of 2-bundles to the one of simplicially modeled $\infty$-bundles by Glenn was established in

* [[Igor Bakovic]], _Bigroupoid 2-torsors_ ([pdf](http://edoc.ub.uni-muenchen.de/9209/1/Bakovic_Igor.pdf)).

Still more explicit descriptions of these constructions are given in

* Christoph Wockel, _A global perspective to gerbes and their gauge stacks_ ([arXiv](http://arxiv.org/abs/0803.3692)) .

These constructions either work internal to [[Diff]] or internal to some [[topos]].

More generally, a principal 2-bundle is a ([[n-truncated object of an (infinity,1)-category|2-truncated]] [[principal ∞-bundle]]) in a [[(∞,1)-topos]] of [[∞-stack]]s over some [[site]].

This is for instance in

* [[Behrang Noohi]], E. Aldrovandi, _Butterflies II: Torsors for 2-group stacks_,[arXiv](http://arxiv.org/abs/0910.1818)

Notice that [[torsor]] is just another word for (internal) [[principal bundle]].

Classification results of principal 2-bundles are in

* [[John Baez]], [[Danny Stevenson]], _The classifying space of a topological 2-group_ Algebraic Topology
Abel Symposia, 2009, Volume 4, 1-31 ([arXiv:0801.3843](http://arxiv.org/abs/0801.3843))
 {#BaezStevenson}

* [[David Roberts]], [[Danny Stevenson]], _Simplicial principal bundles in parametrized spaces_ (web)
 {#RobertsStevenson}

An extensive discussion of various models of principal 2-bundles is in

* {#NikolausWaldorf11} [[Thomas Nikolaus]], [[Konrad Waldorf]], _Four Equivalent Versions of Non-Abelian Gerbes_, Pacific J. Math., 264-2 (2013), 355-420 ([arXiv:1103.4815](http://arxiv.org/abs/1103.4815), [DOI: 10.2140/pjm.2013.264.355](https://msp.org/pjm/2013/264-2/p04.xhtml))

For a comprehensive account in the general context of [[principal infinity-bundles]] see

* [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal ∞-bundles -- theory, presentations and applications]]_ ([arXiv:1207.0248](http://arxiv.org/abs/1207.0248), [arXiv:1207.0249](http://arxiv.org/abs/1207.0249))

For more references see at _[[principal 2-connection]]_.

The example for structure group the [[T-duality 2-group]] is discussed, as a formalization of [[T-folds]], in

* {#NikolausWaldorf18} [[Thomas Nikolaus]], [[Konrad Waldorf]], _Higher geometry for non-geometric T-duals_ ([arXiv:1804.00677](https://arxiv.org/abs/1804.00677)

[[!redirects principal 2-bundles]]