principal 3-bundle

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#### Cohomology
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#Contents#
* table of contents
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## Idea

A **principal 3-bundle** is a $G$-[[principal ∞-bundle]] over a [[3-groupoid|3-group]] $G$.

For $G = \mathbf{B}^2 U(1)$ the circle Lie 3-group, the total space of a $\mathbf{B}^2 U(1)$-principal 3-bundle is modeled by a [[bundle 2-gerbe]].

## Related concepts

* [[connection on a 3-bundle]]

* [[nonabelian bundle 2-gerbe]]

[[!redirects principal 3-bundles]]


principal bundle

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#### Bundles
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

For $G$ a [[group]] ([[internalization|internal]] to some [[category]], traditionally that of [[topological spaces]]) and $X$ some other object, a _$G$-principal bundle_  over $X$ -- also called a $G$-[[torsor]] over $X$ -- is a [[bundle]] $P \to X$ equipped with a $G$-[[action]] $\rho : P \times G \to P$ on $P$ over $X$, such that

* the action is _principal_ meaning that

  * the [[shear map]] $(p_1, \rho) \colon  P \times G \to P \times_X P$ is an [[isomorphism]], which in turn means that the action is _[[free action|free]]_ and _[[transitive action|transitive]]_ over $X$, hence that each [[fiber]] of $P \to X$ looks like $G$ once we choose a base point;

  and / or / equivalently (depending on technical details, see below)

  * $P \to X$ is isomorphic to the [[quotient]] map $P \to P/G$.

and usually it is required that

* the bundle is _locally trivial_ in that there is a [[cover]] $\phi : U \to X$ and an isomorphism of $G$-actions between the [[pullback]] $\phi^* P$ and the _trivial $G$-principal bundle_ $U \times G \to U$ over $U$.

A central property of $G$-principal bundles over $X$ is that they are a geometric model of the degree-1 [[nonabelian cohomology]] of $X$ with coefficients in $G$. More precisely (subject to some technical details discussed below) there is a [[natural isomorphism]]

$$
  H^1(X, G) \simeq G Bund(X)/{\sim}
$$

between the degree-1 $G$-[[cohomology]] of $X$ and the [[isomorphism classes]] of $G$-principal bundles over $X$.

The "naturality" of this relation is more pronounced when one refines it from cohomology to [[cocycle]] [[groupoids]]. This is discussed below in [some section](#SomeSection),

## Definition

We discuss first the definition of principal bundles

* [In the category of topological spaces](#InTopologicalSpaces)

This is historically and traditionally the default setup. But the theory exists in and is usefully regarded from a more abstract perspective, which, most naturally, is that of a [[(2,1)-topos]]. This we introduce and discuss in detail in

* [In an (2,1)-topos](#InA2Topos).

Finally in

* [Other internalizations](#OtherInternalizations)

we discuss how the traditional setup and many other contexts are recovered from and illuminated by that abstract perspective.

### In the category of topological spaces
 {#InTopologicalSpaces}

We discuss here principal bundles in the context [[Top]] of [[topological spaces]]. So the group $G$ here is a _[[topological group]]_.

This is the original and oldest branch of the theory. There is a modern established default of the definition, but many slight but crucial variants exists in the literature and are relevant in applications. We start with the modern default notion and then look into its variants.

$\,$

Let $G$ be a [[topological group]].

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

The **trivial $G$-principal bundle** on a [[topological space]] $X$ is the [[product]] space $X \times G$ equipped with

* the [[projection]] map $p_1 : X \times G \to X$ ;

* the [[action]] of $G$ on $X \times G$ by right multiplication of $G$ on itself.

=--

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

A $G$-**principal bundle** over a [[topological space]] $X$ is a topological space $P$ equipped with

* a [[continuous function]] $p : P \to X$;

* an [[action]] $\rho : P \times G \to P$ of $G$ on $P$ over $X$, hence fitting into a [[coequalizer|coequalizing]] [[diagram]]

  $$
    \array{
      P \times G
      \\
      {}^{\mathllap{p_1}} \downarrow \downarrow^{\mathrlap{\rho}}
      \\
      P
      \\
      \downarrow^{\mathrlap{p}}
      \\
      X
    }
  $$

such that this is _locally trivial_ in the sense that

* there exists a [[cover]] $U \to X$  and a [[continuous map]] $P|_U \to U \times G$ from the [[pullback]] of $P$ to the cover to the trivial $G$-pricipal bundle on the cover,
def. \ref{TrivialTopologicalPrincipalBundle}, which is an [[isomorphism]] of $G$-[[actions]] over $U$

$$
  \array{
    U \times G& \stackrel{\simeq_G}{\leftarrow}
     & P|_U &\to& P
    \\
    &{}_{\mathllap{p_1}}\searrow&\downarrow && \downarrow^{\mathrlap{p}}
    \\
    &&U &\to& X
  }
  \,.
$$

=--

In the [references](#References) listed below, this appears for instance as ([Mitchell, section 2](#Mitchell), ...)

A central property of the above definition of principal bundle is

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

For $P \to X$ a $G$-principal bundle, it is naturally [[isomorphism|isomorphic]] to the [[quotient]] projection
$P \to P/G \simeq X$ of the $G$-[[action]].

=--

\begin{remark}\label{CartanPrincipalBundles}
**(Cartan principal bundles)**

Historically this quotient property of a free continuous action was sometimes taken as the very definition of "principal bundle" without requiring [[local triviality]], e. g. in ([Cartan, 1949-1950](#Cartan)), where this perspective is attributed to [[Henri Cartan]]. A standard modern textbook following this tradition is ([Husemöller](#Husemoeller)).

Therefore in order to avoid ambiguous terminology in the following, we will now follow ([Palais 61, Def. 1.1.2](#Palais61)) and refer to this alternative definition of principal bundle as that of _Cartan principal bundle_:

\end{remark}

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

Let $G$ be a [[locally compact topological group]] and $P$ a [[completely regular topological space]] equipped with a [[continuous function]] [[action]] $\rho : P \times G \to P$.
If $G$ acts _[[free action|freely]]_ on $P$, (no element $g \in G$ except the neutral element has any fixed points in $P$ under the action) then the [[coprojection]]

$$
  P \to P/G =: X
$$

to the [[quotient topology|topological quotient]] is called a **$G$-principal bundle in the wide sense**. If furthermore the division map

$$
  P \times_X P \to G
$$

is a [[continuous function]], then this is called a **Cartan principal bundle** ([Palais 61, around theorem 1.1.3](#Palais61)), following ([Cartan](#Cartan)).

(...)

=--

### In a $(2,1)$-topos
 {#InA2Topos}

It is no surprise that there is a good theory of principal bundles internal to every [[topos]]. However, it turns out that the most "natural home" of the theory is the [[higher category theory|higher category theoretic]] context of a _[[(2,1)-topos]]_ $\mathbf{H}$. This we discuss now, and then relate it to the traditional notion and to various other generalizations. More along these lines is at _[[geometry of physics -- principal bundles]]_.

Notably the existence of [[universal principal bundles]] finds its fundamental "explanation" here, where they are seen to be but a presentation of the construction of the [[homotopy fiber]] functor, which establishes the [[equivalence of categories|equivalence of groupoids]]

$$
  \mathbf{H}(X, \mathbf{B}G) \stackrel{\simeq}{\to} G Bund(X)
  \,,
$$

where on the left we have the [[groupoid]] of [[cocycles]] with coefficients in the internal [[delooping]] $\mathbf{B}G$ of the [[group object in an (infinity,1)-category|group object]] $G$: the [[moduli stack]] of $G$-principal bundles.

In this context, all of the non-natural aspects of the traditional theory of principal bundles disappear, for instance

* _every_ $G$-principal bundle is locally trivial in a $(2,1)$-topos $\mathbf{H}$;

* accordingly there is no mismatch between the various definitions anymore as in the context of topological spaces: the condition of principality becomes equivalent to the quotient space condition.

Moreover, all these fact are fairly direct consequences simply of the [[Giraud axioms]] that characterize [[(2,1)-toposes]] in the first place.

Conversely, the traditional theory nicely naturally embeds into a [[(2,1)-topos]] -- for instance that of [[(2,1)-sheaves]] over the [[site]] [[Top]] (or rather some [[small site|small]] [[dense subsite]]   thereof) -- and the higher topos theory helps to study it there.

The failure of various definitions to match in the traditonal context becomes the fact that the [[colimits]] involved get "corrected" to [[homotopy colimits]] after embedding into a [[(2,1)-topos]]. For instance if an $G$-[[action]] on some object $P$ is not suitably free, then the $(2,1)$-topos theory still produces a healthy principal bundle by replacing the base space by a base [[groupoid]]/[[stack]]. In fact, this way _every_ action becomes principal over its [[homotopy colimit|homotopy quotient]]. Notably the trivial $G$-action on the [[terminal object]] $*$ becomes principal over the [[action groupoid]] $*//G$ and the resulting $G$-principal bundle is nothing but the [[universal principal bundle|universal]] one.

As the notation suggests, thus formulating the theory in [[(2,1)-topos]] theory immediately generalizes it to [[(∞,1)-topos theory]]. This is discussed at _[[principal ∞-bundle]]_.

(...)

> The following is old material collected from elsewhere that is going to be rearranged....

#### In terms of fiber sequences

This indicates the more fundamental way to _define_ $G$-principal bundles in the first place:

Recall (from [[fiber sequence]]) that for every [[group]] there is the one-object [[groupoid]] $\mathbf{B}G$. Under the [[Yoneda embedding]] this [[representable functor|represents]] a [[simplicial presheaf|prestack]]. Write $\bar{\mathbf{B}G}$ for the corresponding [[stack]] obtained by [[(infinity,1)-sheafification|stackification]]. This is our $G Bund(-)$

$$
  G Bund(-) = \bar{\mathbf{B} G}(-)
  \,.
$$

This perspective in turn is by [[category theory|general abstract nonsense]] equivalent to the following useful description:

Let $H$ be the suitable [[(∞,1)-topos]] internal to which one looks at $G$-principal bundles. For instance for topological bundles this would be [[Top]]. For smooth bundles it would be the [[(∞,1)-category of (∞,1)-sheaves]] on [[Diff]], etc.

Then every element in $G Bund(X) \simeq \bar{\mathbf{B} G}(X)$ is given by a morphism in $H(X,\mathbf{B}G)$, which may be thought of as an [[anafunctor]] to $\mathbf{B}G$ from the (categorially) [[discrete category]] $X$; the $G$-principal bundle from the beginning of the above definition is just the [[homotopy limit|homotopy pullback]] of the [[point]] along this map, i.e. the [[fibration sequence|homotopy fiber]] of $X \to \mathbf{B}G$:

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

This diagram, incidentally, directly tells us about another important property of $G$-principal bundles: they all canonically trivialize when pulled back to their own total space $P$.

This is what the homotopy commutativity of the above homotopy pullback diagram says: the cocycle $X \to \mathbf{B}G$ pulled back to the bundle $P \to X$ that it classifies becomes $P \to X \to \mathbf{B}G$, which is homotopic to the trivial cocycle (the one that factors through the [[point]]) on $P$.

The homotopy pullback here is conveniently and traditionally computed as an ordinary pullback of a [[fibration|fibrant replacement]] of the pullback diagram. The canonical such fibrant replacement is obtained by replacing ${*} \to \mathbf{B}G$ by $\mathbf{E}G \to \mathbf{B}G$, with $\mathbf{E}G$ an object [[weak equivalence|weakly equivalent]] to the point, called the **$G$-[[universal principal bundle]]**.

With that the above homotopy pullback is computed as the ordinary [[pullback]]

$$
  \array{
     P &\to& \mathbf{E}G
     \\
     \downarrow && \downarrow
     \\
     X &\to& \mathbf{B} G
  }
$$

So every $G$-principal bundle $P \to X$ is the pullback along a classifying map $X \to \mathbf{B}G$ (in the right $(\infty,1)$-categorical context, otherwise a span such as an [[anafunctor]]) of the $G$-[[universal principal bundle]].

#### The $G$-action from the homotopy pullback

Given the definition of the bundle $P$ in terms of a [[homotopy pullback]] of ${*} \to \mathbf{B}G$ we re-obtain the $G$-[[action]] on $P$ as follows
(with an eye towards its generalization to [[principal ∞-bundles]]).

Let

$$
  \cdots G \times G \stackrel{\stackrel{d_2}{\longrightarrow}}{\stackrel{\stackrel{d_1}{\longrightarrow}}{\stackrel{d_0}{\longrightarrow}}}
  G \stackrel{\stackrel{d_1}{\longrightarrow}}{\stackrel{d_0}{\longrightarrow}}
  {*}
  \to
  \mathbf{B}G
$$

be the [[quotient object|effective]] [[groupoid object in an (∞,1)-category]] that exhibits the [[delooping]] $\mathbf{B}G$ of $G$.

Form the [[homotopy pullback]] of the classifying morphism $X \to \mathbf{B}G$ along the $d_0$-[[simplicial object|face maps]] of this diagram. This yields a diagram

$$
  \array{
    \cdots
    &
     P \times G \times G
    &
    \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
    &
     P \times G
    &
    \stackrel{\longrightarrow}{\longrightarrow}
    &
    P
    &
    \stackrel{}{\longrightarrow}
    &
    X
    \\
    & \downarrow
    &&
    \downarrow
    &&
    \downarrow
    &&
    \downarrow
    \\
    \cdots
    &
     G \times G
    &
    \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
    &
     G
    &
    \stackrel{\longrightarrow}{\longrightarrow}
    &
    {*}
    &
    \stackrel{}{\longrightarrow}
    &
    \mathbf{B}G
  }
$$

where all squares formed by the lowest horizontal morphisms are [[homotopy pullback]] squares, by construction, and where the remaining horizontal morphisms in the top row are induced by the universal property of the [[homotopy pullback]] and the morphisms downstairs.

The claim is that

* the top row encodes the [[action]] of $G$ on $P$
  in that the action is the morphism indicated $\rho$ in
  $$
   \cdots
   \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
   P \times G
    \stackrel{\stackrel{\rho}{\longrightarrow}}{\stackrel{p_1}{\longrightarrow}}
   P
   \to X
  $$

* and it exhibits $P \times G^{\times (n-1)}$ as the [[groupoid object in an (∞,1)-category]] being the [[?ech nerve]] of $P \to X$:

  $$
  \array{
    \cdots
    &
     P \times_X P \times_X P
    &
    \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
    &
     P \times_X P
    &
    \stackrel{\longrightarrow}{\longrightarrow}
    &
    P
    &
    \stackrel{}{\longrightarrow}
    &
    X
    \\
    & \uparrow^{\simeq}
    &&
    \uparrow^{\simeq}
    &&
    \uparrow^{\simeq}
    &&
    \uparrow^{\simeq}
    \\
    \cdots
    &
     P \times G \times G
    &
    \stackrel{\to}{\stackrel{\longrightarrow}{\longrightarrow}}
    &
     P \times G
    &
    \stackrel{\stackrel{\rho}{\longrightarrow}}{\longrightarrow}
    &
    P
    &
    \stackrel{}{\longrightarrow}
    &
    X
    \\
    & \downarrow
    &&
    \downarrow
    &&
    \downarrow
    &&
    \downarrow
    \\
    \cdots
    &
     G \times G
    &
    \stackrel{\to}{\stackrel{\longrightarrow}{\longrightarrow}}
    &
     G
    &
    \stackrel{\longrightarrow}{\longrightarrow}
    &
    {*}
    &
    \stackrel{}{\longrightarrow}
    &
    \mathbf{B}G
  }
  $$

Here the second statement in particular encodes the familiar way to formulate **principality of the action** $\rho$, in that it says that

$$
  P \times G \stackrel{p_1 \times \rho}{\to} P \times_X P
$$

is an [[isomorphism]].

We now unwrap the first statement in gory detail to make clear that this abstract nonsense does reproduce the familiar definition of the action of $G$ on $P$.

#### Unwinding the abstract description
 {#UnwindingTheAbstractDefinition}

We now rederive the [[action]] $\rho$ of $G$ on $P$ given just the classifying map $X \to \mathbf{B} G$ by spelling out the details implied by the above abstract description.

Whatever the precise context is (topological, smooth, etc.) we may assume that we are at least in a [[category of fibrant objects]]. Then the classifying morphism $X \to \mathbf{B}G$ is represented by an [[anafunctor]], namely a cocycle

$$
  \array{
    C(U) &\stackrel{g}{\to} & \mathbf{B}G
    \\
    \downarrow^{\mathrlap{\in W \cap F}}
    \\
    X
  }
$$

in [[?ech cohomology]] coming from some [[cover]] $\{U_i \to X\}$ of $X$.

The [[?ech nerve]] $C(U)$ has

* objects = $\{(x,i) | x \in U_i\}$

* morphisms = $\{ (x,i) \stackrel{}{\longrightarrow} (x,j) | x \in U_{i j}\}$ .

The [[functor]] $g : C(U) \to \mathbf{B}G$ sends

$$
  g : ((x,i) \to (x,j)) \mapsto (\bullet \stackrel{g_{i j}(x)}{\longrightarrow}\bullet )
$$

for $g_{i j} \in Functions(U_{i j}, G)$ as described in detail at [[?ech cohomology]].

With $\mathbf{E}G = \{g \stackrel{h}{\to} g h | g,h \in G \}$ the fibrant replacement of the point, which we shall find it helpful to think of as given by

* objects = $\left\{
      \array{
         && \bullet
         \\
         & {}^g\swarrow
         \\
         \bullet
      }
    \right\}
  $

* morphisms = $\left\{
      \array{
         && \bullet
         \\
         & {}^g\swarrow && \searrow^{g' = g h}
         \\
         \bullet &&\stackrel{h}{\longrightarrow}&& \bullet
      }
    \right\}
  $

we compute the [[homotopy pullback]] as the
homotopy fiber product given by the ordinary [[pullback]] (see [[category of fibrant objects]] for details)

$$
  \array{
    P &\to& \mathbf{E}G
    \\
    \downarrow && \downarrow
    \\
    C(U) &\stackrel{g}{\longrightarrow}&
    \mathbf{B}G
  }
  \,.
$$

So we read off that $P$ is the [[groupoid]] with

* objects =
  $$
   \left\{
   \array{
     && \bullet
     \\
     & {}^{g}\swarrow
     \\
     \bullet
     \\
     (x,i)
   }
   \right\}
  $$

* morphisms =
  $$
   \left\{
   \array{
     && \bullet
     \\
     & {}^{g}\swarrow && \searrow^{g'}
     \\
     \bullet &&\stackrel{g_{i j }(x)}{\to}&& \bullet
     \\
     (x,i) &&\stackrel{}{\to}&& (x,j)
   }
   \right\}
  $$

With $P$ determined as an ordinary pullback of a replacement it is convenient for the following to realize it in turn as the pullback-up-to-2-cell in

$$
  \array{
   P &\to& {*}
   \\
   \downarrow &\Downarrow^{\eta_P}& \downarrow
   \\
   C(U) &\to& \mathbf{B}G
  }
 \,.
$$

A moment reflection shows that the component of the [[natural transformation]] $\eta_P$ here is

$$
  \eta_P :
  \left(
     \array{
     && \bullet
     \\
     & {}^{g}\swarrow
     \\
     \bullet
     \\
     (x,i)
   }
  \right)
  \;\;\;
  \mapsto
  \;\;\;
  (\bullet \stackrel{g}{\to}\bullet)
$$

  At the same time recall from the discussion at [[delooping]] that the component of the transformation $\eta_G$ in

$$
  \array{
   G &\to& {*}
  \\
  \downarrow &\Downarrow^{\eta_G}& \downarrow
   \\
   {*} &\to& \mathbf{B}G
  }
$$

is

$$
  \eta_G : g \mapsto (\bullet \stackrel{g}{\to} \bullet)
  \,.
$$

Taken together this shows that  the universal morphism $P \times G \to P$ induced from the commutativity of

$$
 \array{
  &&&& P \times G
  \\
  && & \swarrow && \searrow
  \\
  && P &&&& G
  \\
  & \swarrow && \searrow && \swarrow && \searrow
  \\
  C(U) &&\Downarrow^{\eta_P}&& {*} &&\Downarrow^{\eta_G}&& {*}
  \\
  &&\searrow&& \downarrow && \swarrow
  \\
  &&&& \mathbf{B}G
 }
$$

and from the [[homotopy pullback]] property of

$$
  \array{
   P &\to& {*}
   \\
   \downarrow &\Downarrow^{\eta_P}& \downarrow
   \\
   C(U) &\to& \mathbf{B}G
  }
$$

is simply given by the composition of these two component maps

$$
  P  \times G  \to P :
  \left(
  \left(
     \array{
     && \bullet
     \\
     & {}^{g}\swarrow
     \\
     \bullet
     \\
     (x,i)
   }
  \right)
  \,,
  \array{
    \bullet
    \\
    \downarrow^{g'}
    \\
    \bullet
  }
  \right)
  \; \;\; \;
  \mapsto
  \; \;\; \;
  \left(
     \array{
     && \bullet
     \\
     && \downarrow^{g'}
     \\
     && \bullet
     \\
     & {}^{g}\swarrow
     \\
     \bullet
     \\
     (x,i)
   }
  \right)
  \,.
$$

But this is manifestly the right (being both: from the right and correct :-) [[action]] $\rho : P \times G \to G$ of $G$ on $P$.

### Other internalizations
 {#Other internalizations}

We discuss here aspects of formulating a theory of principal bundles in contexts different from those already discussed above.

(...)

#### Higher generalizations

In [[higher category theory]] the notion of principal bundle has various [[vertical categorification]]s. See

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

* [[principal ∞-bundle]]

## Properties

### Gauge groupoid

For $P \to X$ a $G$-principal bundle,
its **[[Atiyah Lie groupoid]]** is

$$
  ((P \times P)/G
    \stackrel{\stackrel{p\circ p_1}{\to}}{\underset{p \circ p_2}{\to}}
    X
  )
$$

with the evident composition operation.

The principal bundle $P \to X$ is recovered from its
Atiyah Lie groupoid, up to isomorphism, as the source fiber over
any point.

This is a classical statement due to Ehresmann ... .
See for instance ([Androulidakis](#Androulidakis)).

## Examples

### Hopf fibration

A standard example of a nontrivial [[circle group]]-principal bundle -- a [[circle bundle]] -- is the [[Hopf fibration]] $S^3 \to S^2$, which has the structure of an $S^1$-principal bundle in [[topological space]]s.

### Pullbacks of universal bundles

Generally, if we accept that we have a large supply of continuous maps between [[topological spaces]], we obtain a $G$-principal bundle $f^* \mathcal{E}G \to X$ on a space $X$ for each continuous map $f : X \to \mathcal{B}G$ to the [[classifying space]] of $G$, by [[pullback]] of the [[generalized universal bundle|universal bundle]] $\mathcal{E}G \to \mathcal{B}G$ along $f$.

### Quotients by Lie group actions
 {#QuotientsByLieGroupActions}

We consider [[actions]] by [[topological groups]] and [[Lie groups]].

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

For $X$ a [[smooth manifold]] and $G$ a [[compact Lie group]] equipped with a [[free action|free]] smooth [[action]] on $X$, then the [[quotient]] [[projection]]

$$
  X \longrightarrow X/G
$$

is a $G$-[[principal bundle]] (hence in particular a [[Serre fibration]]).

=--

This is originally due to ([Gleason 50](#Gleason50)). See e.g. ([Cohen, theorem 1.3](#Cohen))

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

Let $P$ be a [[completely regular topological space]] and let $G$ be a [[Lie group]] equipped with a free [[action]] on $P$. Then the [[quotient]] map $P \to P/G$
is a $G$-principal bundle -- in that it is locally trivial  -- precisely if the division map

$$
 P \times_{P/G} P \to G
$$

is a [[continuous function]].

=--

This is ([Palais, theorem 4.1](#Palais)).

### Coset projections
 {#CosetSpaces}

We discuss principal bundles of the form $P \to P/G$ for $G \hookrightarrow P$ a [[subgroup]] of a [[topological group]], hence with base space a [[coset]] space.

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

For $G$ a [[Lie group]] and $H \subset G$ a [[compact Lie group|compact]] [[subgroup]], then the [[coset]] [[quotient]] [[projection]]

$$
  G \longrightarrow G/H
$$

is an $H$-[[principal bundle]] (hence in particular a [[Serre fibration]]).

=--

This is a direct corollary of prop. \ref{QuotientProjectionForCompactLieGroupActingFreelyOnManifoldIsPrincipa}.
Originally this statement is due to ([Samelson 41](#Samelson41)).

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

For $G$ a [[compact Lie group]] and $K \subset H \subset G$ [[closed subspace|closed]] [[subgroups]], then the [[projection]] map

$$
  p \;\colon\; G/K \longrightarrow G/H
$$

is a locally trivial $H/K$-[[fiber bundle]] (hence in particular a [[Serre fibration]]).

=--

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

Observe that the projection map in question is equivalently

$$
  G \times_H (H/K) \longrightarrow G/H
  \,,
$$

(where on the left we form the [[Cartesian product]] and then divide out the [[diagonal action]] by $H$). This exhibits it as the $H/K$-[[fiber bundle]] [[associated bundle|associated]] to the $H$-[[principal bundle]] of corollary \ref{QuotientProjectionForCompactLieSubgroupIsPrincipal}.

=--

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

If $P$ is a [[topological group]] and $G \hookrightarrow P$
a closed [[Lie group|Lie]] subgroup, then the quotient map
$P \to P/G$ is a locally trivial $G$-principal bundle.

=--

This is a corollary of theorem \ref{PalaisTheorem} ([Palais 61](#Palais61)).

Examples where $P \to P/G$ is **not** locally trivial are in
([Karube](#Karube)), see also ([Mostert](#Mostert)):

+-- {: .num_example }
###### Example (counter example)

Let $P$ be the [[product]] of infinitely many [[circles]], and let $G$ be the product of their [[order]] 2 [[subgroups]]. This cannot have local  [[section]] because $P$ is [[locally connected topological space|locally connected]] and $G$ is not. Therefore $P$ is not even locally [[homeomorphism|homeomorphic]] to $(P / G) \times G$.

=--

## Gauge theory

In [[physics]], principal bundles [[connection on a bundle|with connection]] and their higher categorical analogs model [[gauge fields]]. See at _[[fiber bundles in physics]]_.

In fact, the history of the development of the theory of principal bundles and [[gauge theory]] is closely related. In the early 1930s Dirac and Hopf independently introduced $U(1)$-principal bundles: Dirac, somewhat implicitly, in his study of the [[electromagnetic field]] as a background for [[quantum mechanics]], Hopf in terms of the fibration named after him. However, from there it took apparently many years for the first publication to appear that explicitly states that these two considerations are aspects of the same phenomenon.

## Related concepts

* [[slice theorem]]

* **([[formally principal bundle|formally]]) principal bundle** / ([[pseudo-torsor|pseudo]]-)[[torsor]] / [[groupoid principal bundle]] / [[associated bundle]]

  * [[universal principal bundle]]

  * [[caloron correspondence]]

  * [[Galois cover]]

  * [[bibundle]]

  * [[groupoid principal bundle]]

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

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

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

  [[universal principal ∞-bundle]]
* [[noncommutative principal bundle]]

* [[n-vector bundle]]

[[!include gauge field - table]]

* [[horizontal differential form]]

* [[basic differential form]]

* [[equivariant de Rham cohomology]]

## References
 {#References}

### General

An original reference on the notion of a principal bundle as a quotient map by a free continuous action of a topological group is

* {#Cartan} _Séminaire [[Henri Cartan]]_ 1949-1950 *Topologie algébrique* ([numdam:SHC_1948-1949__1](http://www.numdam.org/volume/SHC_1948-1949__1))

some of which is recollected in ([Palais 61](#Palais61)).

See also the references at *[[classifying space]]*.

Textbook accounts:

* {#Steenrod51} [[Norman Steenrod]], section I.7 of _The topology of fibre bundles_, Princeton Mathematical Series 14, Princeton Univ. Press, 1951 ([jstor:j.ctt1bpm9t5](https://www.jstor.org/stable/j.ctt1bpm9t5))

* {#Husemoeller} [[Dale Husemöller]], _Fiber bundles_, Springer (1994) ([doi:10.1007/978-1-4757-2261-1](https://doi.org/10.1007/978-1-4757-2261-1))

* [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurco]], [[Martin Schottenloher]], _[[Basic Bundle Theory and K-Cohomology Invariants]]_, Lecture Notes in Physics, Springer 2008 ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf))

* [[Ralph Cohen]], _The topology of fiber bundles_, Stanford University (2017) ([pdf](http://math.stanford.edu/~ralph/fiber.pdf), [OMN:201707.110706](https://www.ams.org/open-math-notes/omn-view-listing?listingId=110706))

* [[Loring Tu]], Parts I-II in: *Introductory Lectures on Equivariant Cohomology*, Annals of Mathematics Studies **204**, AMS 2020 ([ISBN:9780691191744](https://press.princeton.edu/books/hardcover/9780691191744/introductory-lectures-on-equivariant-cohomology))

With an eye towards application in [[mathematical physics]]:

* [[Mikio Nakahara]], Chapter 9 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), <a href="http://alpha.sinp.msu.ru/~panov/LibBooks/GRAV/(Graduate_Student_Series_in_Physics)Mikio_Nakahara-Geometry,_Topology_and_Physics,_Second_Edition_(Graduate_Student_Series_in_Physics)-Institute_of_Physics_Publishing(2003).pdf">pdf</a>)

* [[Gerd Rudolph]], [[Matthias Schmidt]], Section 1.1: *Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields*, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

For principal bundles in the smooth context see most textbooks on _[[differential geometry]]_, for instance

* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], _[[Connections, Curvature, and Cohomology]]_ Academic Press (1973)

also around section 3.1 of

* {#Moerdijk} [[Ieke Moerdijk]], _On the classification of regular Lie groupoids_ ([pdf](http://igitur-archive.library.uu.nl/math/2007-0201-202453/moerdijk_02_on_the_classification.pdf))

Questions related to the existence [[slice theorem|slices]] of [[topological G-spaces]], of sections of $G$-bundles and conditions for [[proper map|properness]] of some related maps are treated in

* {#Palais61} [[Richard Palais]], _On the Existence of Slices for Actions of Non-Compact Lie Groups_, Annals of Mathematics
Second Series, Vol. 73, No. 2 (Mar., 1961), pp. 295-323 ([jstor:1970335](https://www.jstor.org/stable/1970335), [doi:10.2307/1970335](https://doi.org/10.2307/1970335), [pdf](http://vmm.math.uci.edu/ExistenceOfSlices.pdf))

Lecture notes on principal bundles include

* {#Mitchell}  [[Stephen A. Mitchell]], *Notes on principal bundles and classifying spaces* (2011) [[pdf](https://sites.math.washington.edu/~mitchell/Notes/prin.pdf), [[MitchellPrincipalBundles.pdf:file]]]

### Internal to more general categories
 {#ReferencesInternalToMoreGeneralCategories}

Discussion of [[fiber bundles]] [[internalization|internal]] to [[finitely complete categories]]:

* [[Anders Kock]], *Fibre bundles in general categories*, Journal of Pure and Applied Algebra **56** 3 (1989) 233-245 [<a href="https://doi.org/10.1016/0022-4049(89)90059-5">doi:10.1016/0022-4049(89)90059-5</a>]

* [[Anders Kock]], *Generalized fibre bundles*, in: Categorical Algebra and its Applications, Lecture Notes in Mathematics **1348** (2006) 194-207 [[doi:10.1007/BFb0081359](https://doi.org/10.1007/BFb0081359)]

* {#SatiSchreiber21} [[Hisham Sati]], [[Urs Schreiber]], Part 2 of: *[[schreiber:Equivariant principal infinity-bundles|Equivariant principal $\infty$-bundles]]* [[arXiv:2112.13654](https://arxiv.org/abs/2112.13654)]

  > (in the generality of [[equivariant principal bundles]])

See also:

* C. Townsend, _Principal bundles as Frobenius adjunctions with application to geometric morphisms_, Math. Proc. Camb. Phil. Soc. 159(03) (2015), 433-444 [pdf](http://www.christophertownsend.org/Documents/Principal.pdf)

* [[Tomasz Brzezinski]], _On synthetic interpretation of quantum principal bundles_, AJSE D - Mathematics 35(1D): 13-27, 2010 [arxiv:0912.0213](http://uk.arxiv.org/abs/0912.0213)

### In relation to Lie groupoids
 {#ReferencesInRelationToLieGroupoids}

Discussion of [[Atiyah Lie groupoids]] associated to principal bundles and the reconstruction of principal bundles from their Atiyah Lie groupoids is due to

* Ehresmann, ...

Further discussion along these lines is for instance in

* [[Iakovos Androulidakis]], _Classification of extensions of principal bundles and transitive Lie groupoids with prescribed kernel and cokernel_ ([arXiv:math/0402007](http://arxiv.org/abs/math/0402007))
 {#Androulidakis}

### Examples

Discussion of topological quotients of groups $G \to G/H$ as principal $H$-bundles is in

* {#Gleason50} [[Andrew Gleason]], _Spaces with a compact Lie group of transformations_, Proc. of A.M.S 1, (1950), 35 - 43.

* [Palais 61](#Palais61)

Explicit examples and counter examples of coset principal bundles are discussed in

* Takashi Karube, _On the local cross-sections in locally compact groups_, J. Math. Soc. Japan 10 (1958) 343–347 ([Euclid](http://projecteuclid.org/euclid.jmsj/1261148669))
 {#Karube}

* Paul Mostert, _Local cross sections  in locally compact groups_ ([pdf](http://www.pages.drexel.edu/~gln22/Local%20Cross%20Sections%20in%20Locally%20Compact%20Groups.pdf))
 {#Mostert}

Relations between classes of continuous and of smooth principal bundles are discussed in

* Christoph Müller, [[Christoph Wockel]], _Equivalences of smooth and continuous principal bundles with infinite-dimensional structure groups_, Advances in Geometry. Volume 9, Issue 4, Pages 605–626 (2009)

### Extensions
 {#ReferencesExtensions}

Extensions of principal bundles are discussed for instance in

* [[Kirill Mackenzie]], _On extensions of principal bundles_, Annals of Global Analysis and Geometry Volume 6, Number 2 (1988),

* I. Androulidakis, _Classification of extensions of principal bundles and transitive Lie groupoids with prescribed kernel and cokernel_, J. Math. Phys. 45, 3995 (2004); ([pdf](http://users.uoa.gr/~iandroul/A-gpdextnclass.pdf))

### Automorphism groups

The [[automorphism groups]] of principal bundles are discussed for instance in

* M.C. Abbati, R. Cirelli, A. Mania, P. Michor _The Lie group of automorphisms of a principal bundle_ 1989 JGP **6** 215 ([pdf](http://www.mat.univie.ac.at/~michor/AutomorphismPrincipleBundle.pdf), [doi](http://dx.doi.org/10.1016/0393-0440%2889%2990015-6))

[[!redirects principal bundles]]
[[!redirects principal fiber bundle]]
[[!redirects principal fiber bundles]]
[[!redirects principal fibre bundle]]
[[!redirects principal fibre bundles]]


#Contents#
* tic
{:toc}

+--{: .query}
This entry is about ( a fragment of) a paper by [[Anders Kock]]
=--

##Pregroupoids derived from a groupoid##
Let $G:=\{G_0\to G_1\}$ a groupoid, $A,B\subset G_0$ subsets.

Then the object $G(A,B)$ defined to be the set of arrows in $G_1$ whose source is in $A$ and whose target is in $B$ equipped with the two evident structure maps
$$s:G(A,B)\to A$$
$$t:G(A,B)\to B$$
is a [[heap|pregroupoid]]. If $A=B$ this is a full subgroupoid of $G$.

##Pregroupoid associated to a principal bundle##
Let $p:P\to A$ be a principal right $G$-bundle for a group $G$. Then the span $A\xleftarrow{p}P\xrightarrow{!} * $ is a pregroupoid.

##The category of pregroupoids##
There is a category $\mathbf{PGrpd}$ with pregroupoids as objects and triples $(c_0,c,c_1)$ of morphisms making
$$\array{A&\xleftarrow{a}&^X&\xrightarrow{b}&B\\ {}_{c_0}\downarrow&&{}_c\downarrow&&{}_{c_1}\downarrow\\A^\prime&\xleftarrow{a^\prime}&X^\prime&\xrightarrow{b^\prime}&B^\prime}$$
commute.

There is a forgetful functor
$$V:\begin{cases}Grpd\to PGrpd\\G\to G(G_0,G_0)\end{cases}$$

##Groupoids associated to a pregroupoid
There are three candidates of a groupoid associated to a pregroupoid: $PP^{-1}$, $P^{-1}P$ and $E(P)$. $PP^{-1}$ and $P^{-1}P$ are called edges of the pregroupoid $P$, since they appear as edges of some bisimplicial set. $E(P)$ is called enveloping groupoid for the pregroupoid $P$. $E(P)$ contains the edges of $P$ as subgroupoids.

###The Ehresmann groupoid $PP^{-1}$
For a pregroupoid $A\xleftarrow{\alpha} P\xrightarrow{\beta} B$ the **Ehresmann groupoid** $PP^{-1}$ is defined by $PP^{-1}_0:=A$ and
$$PP^{-1}(a,a^\prime):=xy^{-1}:=\{(x,y)|\alpha(x)=a,\,\alpha(y)=a^\prime\}/\sim$$
where $(x,y)\sim(u,z)$ iff $u=xy^{-1}z$

###The ''inverse Ehresmann'' groupoid $PP^{-1}$
For a pregroupoid $A\xleftarrow{\alpha} P\xrightarrow{\beta} B$ the **''inverse Ehresmann'' groupoid** $P^{-1}P$ is defined by $P^{-1}P_0:=B$ and
$$P^{-1}P(b,b^\prime):=y^{-1}z:=\{(y,z)|\beta(y)=b,\,\alpha(z)=b^\prime\}/\sim$$
where $(y,z)\sim(x,u)$ iff $u=xy^{-1}z$

If $P$ is a principal $G$-bundle, the groupoid $P^{-1}P$ is canonically isomorphic to the one object groupoid $G$ by
$$\begin{cases}P^{-1}P\to G\\(y^{-1}z:y\to z)\to g,&yg=z\end{cases}$$

##The enveloping groupoid $E(P)$

Let $A\xleftarrow{\alpha} P\xrightarrow{\beta} B$ be a pregroupoid.

The **enveloping groupoid $E(P)$ for $P$** is defined by $E(P)_0:=A\coprod B$ and $E(P)_1:=PP^{-1}\coprod P^{-1}P\coprod P\coprod P^{-1}$, where $P^{-1}$ denotes the pregroupoid $P$ with $\alpha$ and $\beta$ interchanged. And calculations (see [arxiv](http://home.imf.au.dk/kock/pbgc4.pdf) p.9) show that this is a groupoid.

##Properties

* The edges of $P$ act on $P$ principally on the left and the right, respectively and the actions commute with each other.

* Let $p:P\to A$ be a principal $G$ bundle. Then there is a groupoid $E$ with $E_0=A\coprod *$ and $E_1=P$. In this cases $G=E(*,*)$

* The functor $E(-):PGrdp\to Grpd$ is a faithful left adjoint to the forgetful functor $V$, the unit for the adjunction is injective.

##References
Anders Kock:

* Principal bundles, groupoids and connections ([arxiv](http://home.imf.au.dk/kock/pbgc4.pdf)),


principal connection

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The term _principal connection_ is synonymous to _[[connection on a bundle|connection]] (see there for more) on a [[principal bundle]]_.

## Related concepts

* [[connection on a bundle]]

  * [[parallel transport]], [[holonomy]]

* [[principal connection]]

  * [[affine connection]], [[Levi-Civita connection]] [[Cartan connection]]

* [[connection on a 2-bundle]]

* [[connection on an infinity-bundle]]

  * [[higher parallel transport]]

[[!include higher Atiyah groupoid - table]]

[[!redirects principal connections]]

[[!redirects connection on a principal bundle]]

[[!redirects connections on principal bundles]]


principal ideal

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

#Contents#
* table of contents
{:toc}

## Definition

A **(left/right/2-sided) principal ideal** in a [[ring]] $R$ is a left/right/2-sided [[ideal]] $I$ generated by an element $x \in R$, or equivalently a [[left module|left sub-$R$-module]]/[[right module|right sub-$R$-module]]/[[bimodule|sub-$R$-$R$-bimodule]] [[generators and relations|generated by]] $x$.

This means there exists an element $x \in I$ such that $y$ is a multiple of $x$ whenever $y \in I$; we say that $I$ is __generated__ by $x$. Thus every element $x$ generates a unique principal ideal, the set of all left/right/two-sided multiples of $x$: $a x$, $x b$, or $a x b$ if we are talking about left/right/two-sided ideals in a ring. Clearly, every ideal $I$ is a join over all the principal ideals $P_x$ generated by the elements $x$ of $I$.

### In commutative rings

In commutative rings, since the set of all principal ideals is isomorphic to the quotient of (the multiplicative monoid structure on) $R$ by the group of units, a **principal ideal** $I$ is equivalently an element of the quotient monoid $I \in R/R^\times$.

## See also

* [[Bézout ring]]

* [[principal ideal domain]]

[[!redirects principal ideals]]
[[!redirects principal-ideal]]


principal ideal domain

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

# Contents
* table of contents
{: toc}

## Definition

A [[ring]] $R$ is a **principal ideal domain** if

1. it is an [[integral domain]] (hence in particular a [[commutative ring]])

2. every [[ideal]] in $R$ is a [[principal ideal]].

Often _pid_ is used as an abbreviation of "principal ideal domain".

A _pid_ can also be characterized as an [[integral domain]] wich possesses a [[Dedekind-Hasse norm]], making very clear the fact that it is a generalization of an [[Euclidean domain]].

## Examples

* any [[field]]

* the ring $\mathbb{Z}$ of [[integers]]

* a [[polynomial ring]] with [[coefficients]] in a field (in fact, for any commutative ring $R$, the ring $R[x]$ is a pid if and _only if_ $R$ is a field)

* a [[valuation ring|discrete valuation ring]] (for example, a ring of formal power series over a field)

* any [[Euclidean domain]]

* in the ring of [[entire function|entire]] [[holomorphic functions]] on $\mathbb{C}$ every finitely generated ideal is principal  ([Helmer 40](#Helmer40)), but the ring is only a [[Bézout domain]].

That both the [[integers]] and the [[polynomial rings]] $\mathbb{F}_q[x]$ over [[finite fields]] are principal integral domains with [[finite group|finite]] [[group of units]] is one aspect of the close similarity between the two that is the topic of the [[function field analogy]]. That also the [[holomorphic functions]] on the [[complex plane]] form a [[Bézout domain]] may then be viewed as part of the further similarity that relates the previous two to topics such as [[geometric Langlands duality]]. See at _[[function field analogy -- table]]_ for more on this.

## Properties

+-- {: .num_lemma}
###### Lemma
In a pid $R$, an element $x$ is irreducible iff $(x)$ is a [[maximal ideal]].
=--

+-- {: .proof}
###### Proof
Suppose $x$ is irreducible and $a \notin (x)$. Then the ideal generated by $a, x$ is principal, say $(b)$. Then $x = b y$ and since $x$ is irreducible, one of $b, y$ is a unit; if $y$ is a unit then $(x) = (b) = (a, x)$ and thus $a \in (x)$, contradiction. So then $b$ must be a unit, i.e., $(b)$ is the improper ideal. Thus $(x)$ has no proper extension: $(x)$ is maximal.

For any ring $R$, if $(x)$ is maximal and $x = a b$ for a non-unit $a$, then the inclusion $(x) \subseteq (a)$ is an equality by maximality, so $a = x v$ for some $v$. Then $x = x v b$. In an integral domain we conclude $1 = v b$; thus $b$ is a unit.
=--

+-- {: .num_prop #noeth}
###### Proposition
A pid is a [[Noetherian ring]].
=--

+-- {: .proof}
###### Proof
The union of an ascending chain of ideals $I_1 \subseteq I_2 \subseteq \ldots$ is an ideal $I$; if $I = (x)$, then $x$ belongs to one of the $I_n$, whereupon $I = I_n$.
=--

+-- {: .num_prop}
###### Proposition
A PID is a [[unique factorization domain]].
=--

+-- {: .proof}
###### Proof
Working in classical logic, Proposition \ref{noeth} says that the reverse inclusion relation on the set of nonzero ideals is [[well-founded relation|well-founded]]. Let $A$ be the subset of ideals $(a)$ that are products of finitely many (maybe zero!) maximal principal ideals. For any proper ideal $(x) \neq (0)$, if every $(t)$ properly containing $(x)$ can be factored into maximals, then so can $(x)$. (Spelling this out: either $(x)$ is maximal/irreducible, or factors as $(s)(t)$ where both $s$ and $t$ are non-units; $(s)$ and $(t)$ factor into maximals by hypothesis, and therefore so does $(x)$.) Thus $A$ is an inductive set, so by well-foundedness it contains every ideal $(x) \neq (0)$, i.e., $x$ can be factored into irreducibles.

For uniqueness of the factorization, we first remark that if $p$ is irreducible and $p|a b$, then $p|a$ or $p|b$. (For $R/(p)$ is a [[field]] and thus *a fortiori* an [[integral domain]], so that if $a b \equiv 0 \; mod p$, then $a \equiv 0 \; mod p$ or $b \equiv 0 \; mod p$.) Thus if $p_1 p_2 \ldots p_m = q_1 q_2 \ldots q_n$ are two factorizations into irreducibles of the same element, then $p_1$ divides one of the irreducibles $q_i$, in which case $(p_1) = (q_i)$ and each is a unit times the other, meaning we can cancel $p_1$ on both sides and argue by induction.
=--

## Structure theory of modules
 {#StructureTheoryOfModules}

### Free and projective modules

+-- {: .num_theorem #free}
###### Theorem
If $R$ is a pid, then any [[submodule]] $M$ of a [[free module]] $F$ over $R$ is also free. (For the converse statement, see [here](http://ncatlab.org/nlab/show/free+module#submod).)
=--

+-- {: .proof}
###### Proof
By freeness of $F$, there exists an isomorphism $F \cong \sum_{j \in J} R_j$, a [[coproduct]] of copies $R_j$ of $R$ (as a module over the ring $R$) indexed over a set $J$, which we assume [[well-ordering|well-ordered]] using the [[axiom of choice]]. Define submodules of $F$:

$$F_{\leq j} \coloneqq \sum_{i \leq j} R_i, \qquad F_{\lt j} \coloneqq \sum_{i \lt j} R_i\,.$$

Any element of $M \cap F_{\leq j}$ can be written uniquely as $(x, r)$ where $x \in F_{\lt j}$ and $r \in R_j$. Define a [[homomorphism]]

$$p_j \colon M \cap F_{\leq j} \to R$$

by $p_j((x, r)) = r$. The [[kernel]] of $p_j$ is $M \cap F_{\lt j}$, and we have an [[exact sequence]]

$$0 \to M \cap F_{\lt j} \to M \cap F_{\leq j} \to \im p_j \to 0$$

where $\im p_j$ is a submodule (i.e., an [[ideal]]) of $R$, hence generated by a single element $r_j$. Let $K \subseteq J$ consist of those $j$ such that $r_j \neq 0$, and for $k \in K$, choose $m_k$ such that $p_k(m_k) = r_k$. We claim that $\{m_k: k \in K\}$ forms a [[basis]] for $M$.

First we prove [[linear independence]] of $\{m_k\}$. Suppose $\sum_{i=1}^{n} a_i m_{k_i} = 0$, with $k_1 \lt k_2 \lt \ldots \lt k_n$. Applying $p_{k_n}$, we get $a_n p_{k_n}(m_{k_n}) = a_n r_{k_n} = 0$. Since $r_{k_n} \neq 0$, we have $a_n = 0$ (since we are working over a domain). The assertion now follows by [[induction]].

Now we prove that the $m_k$ generate $M$. Assume otherwise, and let $j \in J$ be the least $j$ such that some $m \in M \cap F_{\leq j}$ cannot be written as a linear combination of the $m_k$, for $k \in K$. If $j \notin K$, then $m \in M \cap F_{\lt j}$, so that $m \in M \cap F_{\leq i}$ for some $i \lt j$, but this contradicts minimality of $j$. Therefore $j \in K$. Now, we have $p_j(m) = r \cdot r_j$ for some $r$; put $m' = m - r m_j$. Clearly

$$p_j(m') = p_j(m) - r \cdot p_j(m_j) = 0$$

and so $m' \in M \cap F_{\lt j}$. Thus $m' \in M \cap F_{\leq i}$ for some $i \lt j$. At the same time, $m'$ cannot be written as a linear combination of the $m_k$; again, this contradicts minimality of $j$. Thus the $m_k$ generate $M$, as claimed.
=--

Since the [[integers]] $\mathbb{Z}$ form a pid, and [[abelian groups]] are the same as $\mathbb{Z}$-[[modules]], we have

+-- {: .num_cor}
###### Corollary
**(Dedekind)**
A [[subgroup]] of a [[free abelian group]] is also free abelian.
=--

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

The analog of this statement for possibly non-abelian [[groups]] is the _[[Nielsen-Schreier theorem]]_.

=--

Also, since [[projective modules]] are [[retracts]] of free modules, we have

+-- {: .num_cor}
###### Corollary
Projective modules over a pid are free. In particular, submodules of projective modules are projective.
=--

### Normal forms
 {#SmithNormalForm}

+-- {: .num_prop #MatricesOverPrincipalIdealDomainsHaveSmithNormalForm}
###### Proposition
**([[matrices]] over [[principal ideal domains]] [[matrix equivalence|equivalent]] to [[Smith normal form]])**

For $R$ a [[commutative ring]] which is a [[principal ideal domain]], every [[matrix]] $A \in Mat_{n \times m}(R)$ with entries in $R$ is [[matrix equivalence|matrix equivalent]] to a [[diagonal matrix]] filled up with zeros:

There exist [[invertible matrices]] $P \in Mat_{n \times n}(R)$ and $Q \in Mat_{m \times m}(R)$ such that the [[matrix product|product matrix]] $P A Q$ is a [[diagonal matrix]] filled up with zeros:

$$
  P A Q
  \;=\;
$$

such that, moreover, each $a_i$ [[divisor|divides]] $a_{i+1}$.

=--

For [[matrices]] with coefficients in the pid of [[integers]] see also

* _[[Hermite normal form]]_.

### Torsion-free modules

+-- {: .num_prop #torsionfree}
###### Proposition
A finitely generated [[torsion subgroup|torsionfree]] module $M$ over a pid $R$ is free.
=--

+-- {: .proof}
###### Proof
Let $S$ be a finite set of generators of $M$, and let $T \subseteq S$ be a maximal subset of linearly independent elements. (Unless $M = 0$, then $T$ has at least one element, because $M$ is torsionfree.) We claim that $M$ can be embedded as a submodule of the free module $F$ generated by $T$ (which in turn is the span of $T$ as a submodule $F \subseteq M$). By Theorem \ref{free}, it follows that $M$ is free.

Let $x_1, \ldots, x_n$ be the elements of $T$. It follows from maximality of $T$ that for any $m \in S - T$, there is a linear relation

$$r_m m + r_1 x_1 + \ldots + r_n x_n = 0$$

with $r_m \neq 0$. For each $m$ in the complement $S - T$, pick such an $r_m$, and form $r = \prod_{m \in S - T} r_m$. Then the image of the scalar multiplication $\lambda_r \colon M \to M$ factors through $F \subseteq M$, and $M \to \lambda_r(M)$ is monic because $M$ is torsionfree. This completes the proof.
=--

+-- {: .num_prop #flat}
###### Proposition
Let $R$ be a pid. Then an $R$-module $M$ is torsionfree if and only if it is [[flat module|flat]].
=--

+-- {: .proof}
###### Proof
Suppose $M$ is flat. Let $K$ be the field of fractions of $R$; since $R$ is a domain, we have a monic $R$-module map $R \to K$. By flatness, we have an induced monomorphism $M \cong R \otimes_R M \to K \otimes_R M$. For any nonzero $r \in R$, the naturality square

$$\array{
M & \to & K \otimes_R M \\
 \mathllap{\lambda_r} \downarrow & & \downarrow \mathrlap{1 \otimes \lambda_r} \\
M & \to & K \otimes_R M
}$$

commutes, and since the map $1 \otimes \lambda_r$ is multiplication by a non-zero scalar on a vector space, it follows that this map and therefore also $\lambda_r$ is monic, i.e., $M$ is torsionfree.

In the other direction, suppose $M$ is torsionfree. Any module is the filtered colimit over the system of finitely generated submodules and inclusions between them; in this case all the finitely generated submodules of $M$ are torsion-free and hence free, by Proposition \ref{torsionfree}. Thus $M$ is a filtered colimit of free modules; it is therefore flat by a standard result proved [here](http://ncatlab.org/nlab/show/flat+module#filtfree).
=--

### Structure theory of finitely generated modules

_[[structure theorem for finitely generated modules over a principal ideal domain]]_

## In constructive mathematics

In [[constructive mathematics]], many important rings may fail to be principal ideal domains in the naïve sense; the notion of _[[Bézout domain]]_, in which only the finitely generated ideals are required to be principal, is better behaved.

For instance, the ring of integers is a principal ideal domain if and only if the [[excluded middle|law of excluded middle]] holds: In one direction, the usual proofs rely on being able to [[decidable proposition|decide]] whether any particular integer belongs to the ideal or not. For the converse, let $\varphi$ be an arbitrary proposition. Consider the ideal $\{ x \in \mathbb{Z} | (x = 0) \vee \varphi \}$. By assumption, it is generated by some number $n$. Since the integers are [[decidable equality|discrete]], it holds that $n = 0$ or $n \neq 0$. In the first case $\neg\varphi$ holds, in the second $\varphi$.

However, this ideal cannot be proved to be finitely generated either.  If an ideal is generated by $n_1, \ldots, n_k$, then we may form their [[gcd]] one step at a time, which we can do algorithmically.  Therefore, $\mathbb{Z}$ remains a Bézout domain.

On the other hand, we could try to modify the concept of principal ideal domain to recover a concept that is identical to the usual one in [[classical mathematics]] but also includes $\mathbb{Z}$.  For instance, we could demand that every [[decidable subset|decidable]] ideal is principal, or (potentially more strongly) that any ideal generated by a decidable subset is principal.  While these seem to work at first, they are too weak to prove that every PID is a Bézout domain, so we should try to think of something better.

Some authors have tried to define a principal ideal domain as a [[Noetherian ring|Noetherian]] [[Bézout ring|Bézout]] [[integral domain|domain]], but it is unknown if this still coincides in constructive mathematics with the definition of principal ideal domain above.

## See also

* [[unique factorization domain]]

* [[principal ideal domain]]

## References

* {#Helmer40} O. Helmer, _Divisibility properties of integral functions_, Duke Math. J.
6 (1940), 345-356.

* Wikipedia, _[Principal ideal domain](http://en.wikipedia.org/wiki/Principal_ideal_domain)_

* Eric Wofsey, _Principal Ideal Domains_, (written for Mathcamp 2009) [pdf](http://www.math.harvard.edu/~waffle/pids.pdf)

* [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))

[[!redirects principal ideal domain]]
[[!redirects principal ideal domains]]
[[!redirects pid]]
[[!redirects pids]]
[[!redirects PID]]
[[!redirects PIDs]]


principal ideal monoid

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

# Contents
* table of contents
{: toc}

## Definition

A [[monoid]] (or [[semigroup]]) $M$ is a __principal left ideal monoid__ if all left [[ideal of a monoid|ideals]] of $M$ are left [[principal ideal of a monoid|principal ideals]] in $M$. Similarily, $M$ is a __principal right ideal monoid__ if all right [[ideal of a monoid|ideals]] of $M$ are right [[principal ideal of a monoid|principal ideals]]. Finally, $M$ is a __principal ideal monoid__ if it is both a principal left ideal monoid and a principal right ideal monoid.

## Examples

A [[group]] is a principal ideal monoid $G$ whose only ideal is isomorphic to $G$ itself.

## Related concepts

* [[ideal of a monoid]]

* [[principal ideal of a monoid]]

* [[GCD monoid]]

* [[unique factorization monoid]]

* [[Bézout monoid]]

* [[group]]


principal ideal of a monoid

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

# Contents
* table of contents
{: toc}

## Definition

Given a [[monoid]] (or [[semigroup]]) $M$ and an element $a \in M$, a __left principal ideal__ in $M$ is a [[subset]] $M a$ of $M$ such that for all $m \in M$, $m a \in M a$.  Similarly, a __right principal ideal__ in $M$ is a [[subset]] $a M$ of $M$ such that for all $m \in M$, $a m \in a M$.  Finally, a __two-sided principal ideal__, or simply __principal ideal__, in $M$ is a subset $\langle a \rangle$ that is both a left ideal and a right ideal.

## Related concepts

* [[ideal of a monoid]]

* [[principal ideal monoid]]

* [[Bézout monoid]]

* [[GCD monoid]]

* [[unique factorization monoid]]

* [[group]]

[[!redirects principal ideal of a monoid]]
[[!redirects principal ideals of a monoid]]
[[!redirects principal ideals of monoids]]
[[!redirects principal ideal in a monoid]]
[[!redirects principal ideals in a monoid]]
[[!redirects principal ideals in monoids]]


principal infinity-bundle

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _principal $\infty$-bundle_ is a [[vertical categorification|categorification]] of [[principal bundle]] from [[groups]] and [[groupoids]] to [[∞-groupoids]], or rather from parameterized groupoids (generalized spaces called [[stacks]]) to parameterized $\infty$-groupoids (generalized spaces called [[∞-stacks]]).

For motivation, background and further details see

* [[motivation for sheaves, cohomology and higher stacks]]
* [[principal bundle]]
* [[gerbe]]
* [[principal 2-bundle]].

A model for principal $\infty$-bundles is given by

* [[simplicial principal bundle]]s.

See also

* [[universal principal ∞-bundle]]

* [[groupal model for universal principal ∞-bundles]].

## Definition

We define $G$-principal $\infty$-bundles in the general context of an [[∞-stack]] [[(∞,1)-topos]] $\mathbf{H}$, with $G$ a [[groupoid object in an (∞,1)-category|group object in the (∞,1)-topos]].

Recall that for $A \in \mathbf{H}$ an object equipped with a point $pt_A : {*} \to A $, its corresponding [[loop space object]] $\Omega A$ is the [[homotopy pullback]]

$$
  \array{
    \Omega A &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    {*} &\to& A
  }
  \,.
$$

Conversely, for $G \in \mathbf{H}$ we say an object $\mathbf{B}G$ is a [[delooping]] of $G$ if it has an essentially unique point and if $G \simeq \Omega \mathbf{B}G$. We call $G$ an **[[∞-group]]**. More in detail, its structure as a [[groupoid object in an (∞,1)-category|group object in an (∞,1)-category]] is exhibited by the [[?ech nerve]]

$$
 \left(
 \array{
   &\cdots&
   {*} \times_{\mathbf{B}G} {*}
   \times_{\mathbf{B}G} {*}
   &\stackrel{\to}{\stackrel{\to}{\to}}&
   {*} \times_{\mathbf{B}G} {*}
   &\stackrel{\to}{\to}&
   {*}
 }
 \right)
 \simeq
 \left(
  \array{
   &\cdots&
   G \times G
   &\stackrel{\to}{\stackrel{\to}{\to}}&
   G
   &\stackrel{\to}{\to}&
   {*}
  }
 \right)
$$

of ${*} \to \mathbf{B}G$.

### $G$-principal $\infty$-bundles {#GeneralPrincBund}

To every cocycle $g : X \to \mathbf{B}G$ is canonically associated its [[homotopy fiber]] $P \to X$, the [[(∞,1)-pullback]]

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

We discuss now that $P$ canonically has the structure of a $G$-[[principal ∞-bundle]] and that $\mathbf{B}G$ is the [[fine moduli space]] for $G$-principal $\infty$-bundles.

+-- {: .num_defn}
###### Definition
**(principal $G$-action)**

Let $G$ be a [[groupoid object in an (∞,1)-category|group object in the (∞,1)-topos]] $\mathbf{H}$. A **principal [[action]]** of $G$ on a morphism $(P \to X) \in \mathbf{H}$ is
a [[groupoid object in an (∞,1)-category|groupoid object]] $P//G$ that sits over $*//G$ in that we have a
  morphism of [[simplicial object|simplicial diagram]]s $\Delta^{op} \to \mathbf{H}$

$$
  \array{
    \vdots && \vdots
    \\
    P \times G \times G
    &\stackrel{(p_2, p_3)}{\to}& G \times G
    \\
   \downarrow\downarrow\downarrow
    && \downarrow\downarrow\downarrow
    \\
    P \times G
    &\stackrel{p_2}{\to}& G
    \\
    \downarrow\downarrow && \downarrow\downarrow
    \\
    P &\stackrel{}{\to}& {*}
  }
$$

in $\mathbf{H}$;

and such that $P \to X$ exhibits the [[(∞,1)-colimit]]

$$
  X \simeq \lim_\to (P//G : \Delta^{op} \to \mathbf{H})
$$

called the **base space** over which the action takes place.

=--

We may think of $P//G$ as the **[[action groupoid]]** of the $G$-action on $P$. For us it _defines_ this $G$-action.

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

The $G$-principal action as defined above satisfies the
**principality condition** in that we have an equivalence of
groupoid objects

 $$
   \array{
     \vdots && \vdots
     \\
     P \times_X P \times_X P
     &\stackrel{\simeq}{\to}&
     P \times G \times G
     \\
    \downarrow\downarrow\downarrow
     && \downarrow\downarrow\downarrow
     \\
     P \times_X P
     &\stackrel{\simeq}{\to}&
     P \times G
     \\
     \downarrow\downarrow && \downarrow\downarrow
     \\
     P &\stackrel{\simeq}{\to}& P
   }
   \,.
 $$

=--

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

This principality condition asserts that the groupoid object $P//G$ is [[groupoid object in an (infinity,1)-category|effective]]. By [[(∞,1)-topos|Giraud's axioms characterizing (∞,1)-toposes]], every groupoid object in $\mathbf{H}$ is effective.

=--

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

For $X \to \mathbf{B}G$ any morphism, its [[homotopy fiber]] $P \to X$
is canonically equipped with a principal $G$-action with base space $X$.

=--

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

First we show that we have a morphism of simplicial diagrams

$$
  \array{
    \vdots && \vdots && \vdots
    \\
    P \times_X P \times_X P
    &\stackrel{\simeq}{\to}& P \times G \times G
    &\to& G \times G
    \\
    \downarrow\downarrow\downarrow
    &&
   \downarrow\downarrow\downarrow
    && \downarrow\downarrow\downarrow
    \\
    P \times_X P &\stackrel{\simeq}{\to}& P \times G
    &\stackrel{p_2}{\to}& G
    \\
    \downarrow\downarrow
    && \downarrow\downarrow && \downarrow\downarrow
    \\
    P &\stackrel{=}{\to}& P &\stackrel{}{\to}& {*}
    \\
    \downarrow && \downarrow && \downarrow
    \\
    X &\stackrel{=}{\to}& X &\stackrel{g}{\to}& \mathbf{B}G
  }
  \,,
$$

with the right square swhere the left horizontal morphisms are equivalences, as indicated. We proceed by induction through the height of this diagram.

The defining [[(∞,1)-pullback]] square for $P \times_X P$ is

$$
  \array{
    P \times_X P &\to& P
    \\
    \downarrow && \downarrow
    \\
    P &\to& X
  }
$$

To compute this, we may attach the defining $(\infty,1)$-pullback square of $P$ to obtain the [[pasting]] diagram

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

and use the [pasting law for pullbacks](http://ncatlab.org/nlab/show/pullback#Pasting), to conclude that $P \times_X P$ is the pullback

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

By definition of $P$ as the homotopy fiber of $X \to \mathbf{B}G$, the lower horizontal morphism is equivbalent to $P \to {*} \to \mathbf{B}G$, so that $P \times_X P$ is also equivalent to the pullback

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

This finally may be computed as the pasting of two pullbacks

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

of which the one on the right is the defining one for $G$ and the remaining one on the left is just an [[(∞,1)-product]].

Proceeding by induction from this case we find analogously that $P^{\times_X^{n+1}} \simeq P \times G^{\times_n}$: suppose this has been shown for $(n-1)$, then the defining pullback square for $P^{\times_X^{n+1}}$ is

$$
  \array{
    P \times_X P^{\times_X^n}
    &\to&
    P
    \\
    \downarrow && \downarrow
    \\
    P^{\times_X^n}&\to& X
  }
  \,.
$$

We may again paste this to obtain

$$
  \array{
    P \times_X P^{\times_X^n}
    &\to&
    P
    &\to&
    *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    P^{\times_X^n}&\to& X &\to& \mathbf{B}G
  }
$$

and use from the previous induction step that

$$
  (P^{\times_X^n} \to X \to \mathbf{B}G)
  \simeq
  (P^{\times_X^n} \to * \to \mathbf{B}G)
$$

to conclude the induction step with the same arguments as before.

This shows that $P//G$ is the [[Cech nerve]] of $P \to X$. It remains to show that
indeed $X = {\lim_\to}_n P \times G^{\times^n}$.
For this notice that $* \to \mathbf{B}G$ is an [[effective epimorphism in an (infinity,1)-category]]. Hence so is $P \to X$. This proves the claim, by definition of effective epimorphism.

using this we have
$$
  \begin{aligned}
    X & \simeq \mathbf{B}G \prod_{\mathbf{B}G} X
    \\
      & \simeq \left({\lim_{\to}}_n G^{\times^n}\right) \prod_{\mathbf{B}G} X
      \\
      & \simeq {\lim_{\to}}_n ( G^{\times^n} \prod_{\mathbf{B}G} X )
      \\
      & \simeq {\lim_\to}_n ( P\times G^{\times^n} )
      \\
      & \simeq {\lim_\to} P//G
  \end{aligned}
  \,.
$$

=--

We have established that every [[cocycle]] $X \to \mathbf{B}G$ canonically induced a $G$-principal action on the homotopy fiber $P \to X$. The following definition declares the _$G$-principal $\infty$-bundles_ to be those $G$-principal actions that do arise this way.

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

We say a $G$-principal action of $G$ on $P$ over $X$ is a
$G$-[[principal ∞-bundle]] if the colimit over $P//G \to *//G$
produces a pullback square: the bottom square in

$$
  \array{
    \vdots && \vdots
    \\
    P \times G \times G
    &\to& G \times G
    \\
   \downarrow\downarrow\downarrow
    && \downarrow\downarrow\downarrow
    \\
    P \times G
    &\stackrel{p_2}{\to}& G
    \\
    \downarrow\downarrow && \downarrow\downarrow
    \\
    P &\stackrel{}{\to}& {*}
    \\
    \downarrow && \downarrow
    \\
    X = \lim_\to (P \times G^\bullet) &\stackrel{g}{\to}&
   \mathbf{B}G = \lim_\to( G^\bullet)
  }
  \,.
$$

=--

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

For $G$ an [[infinity-group]] in $\mathbf{H}$ and $X \in \mathbf{H}$ any object, write

$$
  G Bund(X) \subset Grpd(\mathbf{H})/{*//G}
$$

for the [[sub-(infinity,1)-category]] on the [[over-(infinity,1)-category]] of the groupoid objects over $*//G$ on the $G$-principal $\infty$-bundles as above.

=--

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

We have an [[equivalence of (∞,1)-categories]]

$$
  G Bund(X) \simeq \mathbf{H}(X, \mathbf{B}G)
$$

of $G$-principal $\infty$-bundles over $X$ with [[cocycles]] $X \to \mathbf{B}G$.

=--

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

The [[arrow category]] $\mathbf{H}^I$ is still an [[(infinity,1)-topos]] and hence the Giraud-Lurie axioms still hold. This means that by the discussion at [[groupoid object in an (infinity,1)-category]] (using the statement below [[Higher Topos Theory|HTT, cor. 6.2.3.5]]) we have an equivalence

$$
  Grpd(\mathbf{H}^I) \simeq
  (\mathbf{H}^{I})^{I}_{eff}
$$

between groupoid objects in $\mathbf{H}^I$ and [[effective epimorphism in an (infinity,1)-category|effective epimorphisms]] in the arrow category.

Notice that groupoid objects and effective epis in $\mathbf{H}^I$ are given objectwise over the two objects of the interval $I = \Delta[1]$.

Restricting this equivalence along the inclusion

$$
  \mathbf{H}(X, \mathbf{B}G)
    \hookrightarrow
  (\mathbf{H}^I)^I
$$

given by sending a cocycle to its homotopy fiber diagram

$$
 (X \to \mathbf{B}G)
  \mapsto
  \left(
     \array{
        P &\to& *
        \\
        \downarrow && \downarrow
        \\
        X &\to& \mathbf{B}G
     }
  \right)
$$

therefore yields precisely the equivalence in question

$$
  \array{
     G Bund(X) &\hookrightarrow& Grpd(\mathbf{H}^I)
     \\
     \downarrow^\simeq && \downarrow^\simeq
     \\
     \mathbf{H}(X, \mathbf{B}G)
     &\stackrel{hofib}{\hookrightarrow}&
     (\mathbf{H}^I)^I
  }
  \,.
$$

=--

In words this says that the [[cohomology]] on $X$ with coefficients in $G$ classified $G$-principal $\infty$-bundles, and in fact does so on the level of [[cocycle]]s.

### Connections on $G$-principal $\infty$-bundles

For some comments on the generalization of the notion of [[connection on a bundle]] to principal $\infty$-bundles see [[schreiber:differential cohomology in an (∞,1)-topos -- survey]].

## Concrete realizations

We discuss realizations of the general definition in various
[[(∞,1)-toposes]] $\mathbf{H}$.

### In topological spaces

The following general construction was originally due to
Quillen and defines principal _groupoid_ $\infty$-bundles in the [[(∞,1)-topos]] [[Top]]
in its [[presentable (infinity,1)-category|presentation]]
by the [[model structure on simplicial sets]].

Let $C$ be a small [[category]] and let

$$
  \rho_P : C \to SSet
$$

be a functor with values in [[SSet]] such that
it sends all morphisms in $C$ to weak equivalences in
[[SSet]] ([[simplicial homotopy group|weak homotopy equivalences]] of simplicial sets).

Consider first the case that $C$ has a single object, so that it is the [[delooping]] $\mathbf{B}G$ of a [[monoid]]
or [[group]] $G$. Then

Let

$$
  P := \rho_P(\bullet)
$$

be the simplicial set assigned to this single object and let

$$
  X := P//G := hocolim \rho_P
$$

be the corresponding [[action groupoid]] (see there for the description as a weak colimit).

Notice that, as every [[action group]], this comes with
a canonical map $P//G \to \mathbf{B}G$.

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

Given the above, the diagram

$$
  \array{
     P &\to& {*}
     \\
     \downarrow && \downarrow
     \\
     X &\stackrel{g}{\to}& \mathbf{B}G
  }
$$

is a homotopy pullback
(i.e. defines a [[fibration sequence]]).

=--

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

This is originally due to

* D. Quillen, _Higher algebraic K-theory I_,
Springer Lecture notes in Math. 341 (1973)
85--147.

The statement is reproduced in section IV of

* P. G. Goerss and J. F. Jardine, 1999, _Simplicial Homotopy Theory_, number 174 in Progress in Mathematics, Birkhauser. ([ps](http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html))

=--

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

For the simple case that
$G$ is [[group]],
in which case $\rho_C$ necessarily takes values not
just in weak equivalences but is isomorphisms of
simplicial sets, this says that
$P \to X$ is a  $G$-principal $\infty$-bundle.
In particular the _principality_
of the action is manifestly exhibited by the fact that
the base space $X$ is the (weak) quotient of $P$ by the
action of $G$.

The above reproduces manifest the description of ordinary
$G$-principal topological bundles in the incarnation
as groupoids as described in detail at
[[generalized universal bundle]].

More generally, when $G$ is just a [[monoid]]
the above descibes something a bit more general than
an ordinary $G$-[[principal bundle]]
(as then the action of $G$ on the total space may
be by weak equivalences that are not isomorphisms).
=--

Quillen's original construction is more general than
this, concerning in fact 1-[[groupoid]]-principal
$\infty$-bundles:

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

Let now $C$ be a [[category]] and for

$$
  \rho_P : C \to SSet
$$

a functor that sends all morphisms to weak
equivalences of simplicial sets.

Let now for each object $c \in C$

$$
  P_c := \rho_C(c)
$$

be the "bundle of $c$-fibers".

Then for each $c$ the diagram

$$
  \array{
     P_c &\to& {*}
     \\
     \downarrow && \downarrow^{{*} \mapsto c}
     \\
     X &\stackrel{g}{\to}& C
  }
$$

is a homotopy pullback
(i.e. defines a [[fibration sequence]]).

=--

This classical construction is recalled in the introduction of

* Jardine, _Diagrams and torsors_ ([pdf](http://www.math.uiuc.edu/K-theory/0723/diagrams3.pdf))

### In simplicial sets / Kan complexes

See [[simplicial principal bundle]].

### In a petit $(\infty,1)$-topos

For $X$ a [[topological space]] $C = Op(X)$ the [[category of open subsets]] of $X$, let $\mathbf{H} = Sh_{(\infty,1)}(X)$
be the [[(∞,1)-topos]]  of [[∞-stacks]]
on $C$. This is the [[petit topos]] incarnation of $X$.

In its [[presentable (infinity,1)-category|presentation]] by the
[[model structure on simplicial presheaves]] this is the context in which princpal $\infty$-bundles are discussed in

* Jardine, _Diagrams and torsors_ ([pdf](http://www.math.uiuc.edu/K-theory/0723/diagrams3.pdf))

### In a gros $(\infty,1)$-topos

For $C$ a [[site]] of test [[space]], -- for instance duals of [[algebras over a Lawvere theory]] as described at [[function algebras on infinity-stacks]] -- let $\mathbf{H} = Sh_{(\infty,1)}(C)$ be the [[(∞,1)-topos]]  of [[∞-stacks]] on $C$. This is a [[gros topos]].

#### Smooth principal $\infty$-bundles

Smooth principal $\infty$-bundles are realized in the $\infty$-[[Cahiers topos]] as described in some detail at [[∞-Lie groupoid]].

In this context there is a notion of [[connection on a principal ∞-bundle]].

## Examples

### Ordinary principal bundles

For $G$ an ordinary [[Lie group]], a $G$-principal bundle in the $(\infty,1)$-topos $\mathbf{H} = $ [[?LieGrpd]] is an ordinary $G$-[[principal bundle]].

### Circle $n$-bundles

For $G = \mathbf{B}^{n-1} U(1) \in $ [[?LieGrpd]], the circle Lie n-group, a $G$-principal $\infty$-bundle is a **circle $n$-bundle**.

See [[circle n-bundle with connection]].

Classes of examples include

* the [[Chern-Simons circle 3-bundle]].

### Bundle gerbes

* A [[bundle gerbe]]  is a concrete model for the total space [[groupoid]] of the total space of a $\mathbf{B}U(1)$-[[principal 2-bundle]].

  More generally, a [[nonabelian bundle gerbe]] is a concrete model for the [[groupoid]] of the total space of a general [[principal 2-bundle]].

* A [[bundle 2-gerbe]]  is a concrete model for the total space [[2-groupoid]] of the total space of a $\mathbf{B}^2 U(1)$-principal 3-bundle.

  More generally, a [[nonabelian bundle 2-gerbe]] is a concrete model for the [[2-groupoid]] of the total space of a general principal 3-bundle.

  Classes of examples include

  * the [[Chern-Simons bundle 2-gerbe]].

### Normal morphisms of $\infty$-groups

A principal $\infty$-bundle over a [[0-connected]] object / [[delooping]] object $\mathf{B}K$ is a _[[normal morphism of ∞-groups]]_. See there for more details.

## Related concepts

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

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

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

* **principal $\infty$-bundle** / [[associated ∞-bundle]] / [[∞-gerbe]], [[twisted ∞-bundle]], [[groupoid-principal ∞-bundle]]

* [[vector bundle]]

* [[(∞,1)-vector bundle]] / [[(∞,n)-vector bundle]]

## References
 {#References}

The notion of principal $\infty$-bundle (often addressed in the relevant literature in the language of [[torsors]]) appears in the context of the [[simplicial presheaf]] [[model structure on simplicial presheaves|model]] for generalized spaces in

* [[Rick Jardine]], Z. Luo, _Higher order principal bundles_, Mathematical Proceedings of the Cambridge Philosophical Society **140** 2 (2006) 221-243 [[doi:10.1017/S0305004105008911](https://doi.org/10.1017/S0305004105008911)]

* [[Rick Jardine]], _Cocycle categories_, in *Algebraic Topology* Abel Symposia **4**  (2009) [[arXiv:math.AT/0605198](http://arxiv.org/abs/math.AT/0605198), [doi:10.1007/978-3-642-01200-6_8](https://doi.org/10.1007/978-3-642-01200-6_8)]

An earlier description in terms of simplicial objects is

* P. Glenn, _Realization of cohomology classes in arbitrary exact categories_, J. Pure Appl. Algebra **25** 1 (1982)  33-105 [doi:10.1016/0022-4049(82)90094-9]

In that article not the total space of the bundle $P \to X$ is axiomatized, but the $\infty$-[[action groupoid]] of the action of $G$ on it.

See the remarks at [[principal 2-bundle]].

See also

* {#Wendt10} [[Matthias Wendt]], _Classifying spaces and fibrations of simplicial sheaves_, J. Homotopy Relat. Struct. **6** 1 (2011) 1-38 [[arXiv:1009.2930](http://arxiv.org/abs/1009.2930)]

on [[associated ∞-bundles]].

The fully general abstract formalization in [[(∞,1)-topos theory]] as discussed here was first indicated in

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], *[[schreiber:Twisted Differential String and Fivebrane Structures]]* (2009)

* [[Urs Schreiber]], [[Zoran Škoda]], [§7.1](https://arxiv.org/pdf/1004.2472.pdf#page=20) of: *Categorified symmetries* [[arXiv:1004.2472](https://arxiv.org/abs/1004.2472)]

with more comprehensive accounts in:

* {#NSS12} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal ∞-bundles -- models and general theory|Principal $\infty$-bundles -- General theory]]_, J. Hom. Rel. Struc. **10** 4 (2015) 749-801 [[arXiv:1207.0248](https://arxiv.org/abs/1207.0248), [doi:10.1007/s40062-014-0083-6](http://link.springer.com/article/10.1007/s40062-014-0083-6)]

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ (2013)

* [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Equivariant principal infinity-bundles|Equivariant Principal $\infty$-bundles]]* [[arXiv:2112.13654](https://arxiv.org/abs/2112.13654)]

For some additional developments and applications to ∞-group extensions, see also

* [[Severin Bunk]], _Principal ∞-bundles and smooth String group models_, Mathematische Annalen **387** 689–743  [[doi:10.1007/s00208-022-02462-0](https://doi.org/10.1007/s00208-022-02462-0), [arXiv:2008.12263] (http://arxiv.org/abs/2008.12263)]

A comparison of smooth principal ∞-bundles and diffeological principal bundles for diffeological groups is in

* [[Emilio Minichiello]], _Diffeological Principal Bundles and Principal Infinity Bundles_, [[arXiv:2202.11023] (http://arxiv.org/abs/2202.11023)]

The [[classifying spaces]] for a large class of principal $\infty$-bundles are discussed in

* {#RobertsStevenson} [[David Roberts]], [[Danny Stevenson]], _Simplicial principal bundle in parameterized spaces_ [[arXiv:1203.2460](http://arxiv.org/abs/1203.2460)]

* {#Stevenson} [[Danny Stevenson]], _Classifying theory for simplicial parametrized groups_ [[arXiv:1203.2461](http://arxiv.org/abs/1203.2461)]

A fairly comprehensive account of the literature is also in the introduction of [NSS 12, "Presentations"](#NSS12).

For $\mathbf{H}= \infty Grpd$ the statement that homotopy types over $B G$ are equivalently $G$-[[infinity-actions]] is maybe due to

* [[Emmanuel Dror Farjoun]], [[William Dwyer]], [[Daniel Kan]], _Equivariant maps which are self homotopy equivalences_, Proc. Amer. Math. Soc. **80** 4 (1980) 670672 [[jstor:2043448](http://www.jstor.org/stable/2043448)]

This is mentioned for instance as exercise 4.2in

* {#Dwyer2008} [[William Dwyer]], _Homotopy theory of classifying spaces_, Lecture notes Copenhagen (June, 2008) [pdf](http://www.math.ku.dk/~jg/homotopical2008/Dwyer.CopenhagenNotes.pdf)

Closely related discussion of homotopy fiber sequences and homotopy action but in terms of [[Segal spaces]] is in section 5 of

* [[Matan Prezma]], _Homotopy normal maps_ ([arXiv](http://arxiv.org/pdf/1011.4708v7.pdf))

There, conditions are given for a morphism $A_\bullet \to B_\bullet$ to a [[reduced Segal space]] to have a fixed homotopy fiber, and hence encode an action of the loop group of $B$ on that fiber.

Discussion in [[higher differential geometry]] of [[Kaluza-Klein compactification]] along [[principal ∞-bundles]], relating to [[double field theory]], [[T-folds]], [[non-abelian T-duality]], [[type II geometry]], [[exceptional geometry]]:

* [[Luigi Alfonsi]], _Global Double Field Theory is Higher Kaluza-Klein Theory_ ([arXiv:1912.07089](https://arxiv.org/abs/1912.07089))

* [[Luigi Alfonsi]], _The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective_ ([arXiv:2007.04969](https://arxiv.org/abs/2007.04969))

Review:

* [[Severin Bunk]], *$\infty$-Bundles*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]* [[arXiv:2308.04196](https://arxiv.org/abs/2308.04196)]

[[!redirects principal infinity-bundles]]
[[!redirects principal ∞-bundle]]
[[!redirects principal ∞-bundles]]


principal minor

### Principal minors of matrices with commutative entries

Given a matrix $A = (a^i_j)_{i,j = 1,\ldots n}$ with entries in a commutative ring $R$, the principal minors are the [[minor]]s of the upper left corner square matrices, i.e.

$$
det (a^i_j)_{i,j = 1,\ldots k}
$$

where $k = 1,\ldots, n$ (i.e there are $n$ principal minors).

Sometimes it is convenient to consider the principal minors of the matrix in the oppositely ordered basis, i.e. the minors of the lower right corner

$$
det (a^i_j)_{i,j = n-k+1,\ldots n}
$$

Of course, as it is true for the expression minor in general, sometimes it denotes the determinant of the submatrix and for some authors it is the submatrix itself called that way.

The principal minors appear as the denominators in the [[Gauss decomposition]] of matrices, and consequently in the description of relations between matrix elements in a linear group $GL(n)$ (or $SL(n)$) and canonical coordinates on the big Bruhat cell on its [[flag variety]].

### Principal quasiminors

Quasideterminant of a matrix $|A|_{ij}$ with entries in a noncommutative [[ring]], generalizes (up to sign), from the commutative case, the ratio of $n\times n$-determinant and the determinant of the $(n-1)\times(n-1)$ submatrix obtained by crossing out the $i$-th row and $j$-th column. In particular the determinant of $A$ in the commutative case can be obtained as a product of all its principal quasiminors, provided the latter are defined (quasideterminants are rational expressions not necesarilly always defined).

Principal quasiminors play role in the study of noncommutative flag varieties.

### Quantum group case

Similar role play quantum principal minors in the study of quantum flag variety, cf. sec, 14 in

* [[Zoran Škoda]], _Localizations for construction of quantum coset spaces_, in "Noncommutative geometry and Quantum groups", W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265–298, Warszawa 2003, [math.QA/0301090](http://arxiv.org/abs/math.QA/0301090)

[[!redirects principal minors]]


principal submatrix

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## Definition

Given an index [[set]] *I* and a [[matrix]] $A = (A_{i j})_{i,j \in I}$, a *principal submatrix* of $A$ is a matrix of the form $(A_{i j})_{i,j \in I'}$ for $I' \subset I$ some [[subset]].

## Related entries

* [[determinant]]

* [[Cauchy's interlace theorem]]

[[!redirects principal submatrices]]


principal symbol

## Idea

A *principal symbol* is the highest homogeneous degree part of a [[symbol of a differential operator|symbol]] of a [[pseudodifferential operator]]. Some authors call a principal symbol a "symbol", and [[symbol of a differential operator|symbol]] the "total symbol".

## References

* [wikipedia](https://en.wikipedia.org/wiki/Differential_operator)
* Y. Kordyukov, F. Sukochev, D. Zanin, _$C^\ast$-algebraic approach to the principal symbol. III_, [arXiv:2309.04500](https://arxiv.org/abs/2309.04500)


Principia Mathematica

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> For [[Isaac Newton|Newton]]'s *[[Philosophiæ naturalis principia mathematica]]* see there.

***

This page collects material related to

* [[Bertrand Russell]], [[Alfred Whitehead]]

  \linebreak

  **Principia Mathematica**

  \linebreak

  (1910, 1927)

  [ISBN:9780521067911](https://www.cambridge.org/ae/academic/subjects/mathematics/logic-categories-and-sets/principia-mathematica?format=WX&isbn=9780521067911)

  [pdf](https://lesharmoniesdelesprit.files.wordpress.com/2015/11/whiteheadrussell-principiamathematicavolumei.pdf)

an early account aiming to lay [[foundations of mathematics]] via [[formal logic]].

Among other things, this book, inspired by work of [[Gottlob Frege]] (such as in following his notion of *[[judgement]]*, see [there](judgment#History)), introduces concepts of [[type theory|typing]], called the "extensional hierarchy of type" to prevent [[paradoxes]] of [[set theory]] such as [[Russell's paradox]], and a
"[ramified hierarchy of types](http://en.wikipedia.org/wiki/Principia_Mathematica#Ramified_types_and_the_axiom_of_reducibility)", in order to avoid [[paradoxes]] such as the [[liar's paradox]].
(See also [SEP -- Type theory](http://plato.stanford.edu/entries/type-theory/)).

## References

* Wikipedia, _[Principia Mathematica](http://en.wikipedia.org/wiki/Principia_Mathematica)_

* Stanford Encyclopedia, *[Principia Mathematica](https://plato.stanford.edu/entries/principia-mathematica/)*

category: reference

[[!redirects ramified type]]
[[!redirects ramified types]]