VectBund

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
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[[!include bundles - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}

## Idea

This entry is concerned with [[categories]] $VectBund$ of [[vector bundles]] whose [[morphisms]] are allowed to cover non-trivial [[maps]] between their base spaces.

This means that after choosing a [[ground field]] and an ambient category of [[spaces]] (e.g. [[Sets]] [[TopologicalSpaces]], [[SmoothManifolds]], etc.):

* the [[objects]] of $VectBund$ are [[vector bundles]] over any base space $B$ in the ambient category, which we shall denote like this:

  $$
    \left[
      \array{
        E
        \\
        \big\downarrow
        \\
        B
      }
    \right]
  $$

* the [[morphisms]] of $VectBund$ are [[commuting squares]] (in the ambient category of spaces) of the form

  $$
      \array{
        E &\overset{\phi}{\longrightarrow}& E'
        \\
        \big\downarrow && \big\downarrow
        \\
        B &\underset{f}{\longrightarrow}& B'
      }
  $$

  which are [[fiber]]-wise [[linear maps]].

  Notice here that given the base map $f$, such a diagram is equivalent to a homomorphism of vector bundles over $B$ from $E$ into the [[pullback vector bundle]] $f^\ast E'$, which we may denote by the same symbol $\phi$:

  $$
    \array{
      E && \overset{\phi}{\longrightarrow} && f^\ast E'
      \\
      & \searrow && \swarrow
      \\
      && B
    }
  $$

Accordingly, for each base space $B$ there is a [[full subcategory]]-inclusion

$$
  VectBund(B)
    \xhookrightarrow{\phantom{--}}
  VectBund
$$

of the category [[VectBund(X)|VectBund(B)]] of vector bundles over $B$,

## Properties

### Bireflective inclusion of zero-bundles
 {#BireflectiveInclusionOfZeroBundles}

The construction which to any base space $B$ assigns the [[zero-vector bundle]] over $B$ constitutes a [[bireflective subcategory]]-inclusion (see [there](bireflective+subcategory#ZeroVectorBundlesAmongVectorBundles)) of the category of base spaces into that over vector bundles over these base spaces.

### Closed monoidal structures
 {#ClosedMonoidalStructures}

For a fixed base space $B$, the [[monoidal category]]-[[structures]] on [[VectBund(B)|$VectBund_B$]] are well known, given [[fiber]]-wise by the respective structures on [[VectorSpaces]]:

* the [[cartesian monoidal structure]] on $VectBund(B)$ is given by [[direct sum of vector bundles]], which in turn is [[fiber]]-wise the [[direct sum]] of [[vector spaces]];

* the other [[symmetric monoidal structure]] on $VectBund(B)$ is given by [[tensor product of vector bundles]], which in turn is [[fiber]]-wise the [[tensor product of vector spaces]].

But analogous -- if subtly different -- monoidal structures exist on the total category $VectBund$, discussed in the following:

* [Cartesian monoidal structure on VectBund](#TensorMonoidalStructur)

* [Tensor monoidal structure on VectBund](#TensorMonoidalStructur).

In discussing this now, we (have to and want to) assume that the ambient [[category]] of [[TopologicalSpaces]] is itself [[cartesian closed category|cartesian closed]] (a "[[convenient category of topological spaces]]"), such as that of [[compactly generated topological spaces]] or of [[D-topological spaces]].

#### Cartesian monoidal structure
 {#CartesianMonoidalStructure}

The [[cartesian product]] in $VectBund$ is readily checked to be the "external direct sum", namely the result of [[pullback of vector bundles|pulling back]] the two vector bundles to the [[product space]] of their base spaces along the [[projection maps]]

$$
  \array{
    &&
    B \times B'
    \\
    & \mathllap{ {}^{pr_{B}} }\swarrow
    &&
    \searrow \mathrlap{  {}^{ pr_{B'} } }
    \\
    B && && B'
  }
$$

and then forming the [[direct sum of vector bundles]] there:

\[
  \label{CartesianProduct}
  \left[
    \begin{array}{c}
      E
      \\
      \big\downarrow
      \\
      B
    \end{array}
  \right]
  \times
  \left[
    \begin{array}{c}
      E'
      \\
      \big\downarrow
      \\
      B'
    \end{array}
  \right]
  \;\;\simeq\;\;
  \left[
    \begin{array}{c}
      (pr_{B})^\ast E
      \oplus
      (pr_{B'})^\ast {E'}
      \\
      \big\downarrow
      \\
      B \times B'
    \end{array}
  \right]
\]

From this one finds that the cartesian [[mapping space]] ([[internal hom]])

$$
  Maps(-,-)
  \;\colon\;
  VectBund^{op}
  \times
  VectBund
  \longrightarrow
  VectBund
$$

is given by forming the vector bundle whose base is the space of vector bundle homomorphisms $\phi$ covering maps of base spaces $f$, and whose fiber over $(f,\phi)$ is (independent of $\phi$ and) given by the [[vector space|vector]]-[[space of sections]] $\Gamma_B(-)$ of the [[pullback vector bundle]] $f^\ast E'$ of the [[codomain]] bundle $E'$ along $f$:

$$
  Maps
  \left(
    \left[
      \begin{array}{c}
        E
        \\
        \big\downarrow
        \\
        B
      \end{array}
    \right]
    \;,\;\;
    \left[
      \begin{array}{c}
        E'
        \\
        \big\downarrow
        \\
        B'
      \end{array}
    \right]
  \right)
  \;\;\simeq\;\;
  \left[
    \begin{array}{c}
      (f,\phi)
      \mapsto
      \Gamma_B\big(f^\ast E'\big)
      \\
      \big\downarrow
      \\
      \left\{
        \array{
          f \colon B \to B',
          \\
          \phi
          \colon E \underset{lin/B}{\to} f^\ast E'
        }
        \;\;
      \right\}
    \end{array}
  \right]
$$

Namely, one checks the defining [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism)

$$
  Hom
  \big(
    -
    ,\,
    Map(-,-)
  \big)
  \;\;\simeq\;\;
  Hom
  \big(
    - \times -
    ,\,
    -
  \big)
$$

by testing it on vector bundles over the [[point space]] (ignoring the topology for the moment)  and using (eq:CartesianProduct) on the right:

$$
  Hom
  \left(
  \left[
    \begin{array}{c}
      V
      \\
      \big\downarrow
      \\
      \ast
    \end{array}
  \right]
  \;,\;\;
  Maps
  \left(
    \left[
      \begin{array}{c}
        E
        \\
        \big\downarrow
        \\
        B
      \end{array}
    \right]
    \;,\;\;
    \left[
      \begin{array}{c}
        E'
        \\
        \big\downarrow
        \\
        B'
      \end{array}
    \right]
  \right)
  \right)
  \;\;
  \underoverset
  {\sim}
  {
    \Big(
      v
        \mapsto
      \big(
        (f,\phi)
        ,
        \sigma_v
      \big)
    \Big)
    \;\mapsto\;
    \big(f, (\phi + \sigma_{(-)}) \big)
  }{\longrightarrow}
  \;\;
  Hom
  \left(
  \left[
    \begin{array}{c}
      E \oplus V
      \\
      \big\downarrow
      \\
      B
    \end{array}
  \right]
  \;,\;\;
  \left[
    \begin{array}{c}
      E'
      \\
      \big\downarrow
      \\
      B'
    \end{array}
  \right]
  \right)
$$

#### Tensor monoidal structure
 {#TensorMonoidalStructure}

Consider now the [[symmetric monoidal structure]] on $VectBund$ given by the [[external tensor product]] [[external tensor product of vector bundles|of vector bundles]]

$$
  \otimes
  \;\colon\;
  VectBund \times VectBund
  \longrightarrow
  VectBund
$$

which itself is constructed by first [[pullback vector bundle|pulling back]] both bundles to the [[product space]] of their base spaces, and then forming their [[tensor product of vector bundles]] there:

\[
  \label{TensorProduct}
  \left[
    \begin{array}{c}
      E
      \\
      \big\downarrow
      \\
      B
    \end{array}
  \right]
  \otimes
  \left[
    \begin{array}{c}
      E'
      \\
      \big\downarrow
      \\
      B'
    \end{array}
  \right]
  \;\;\coloneqq\;\;
  \left[
    \begin{array}{c}
      (pr_{B})^\ast E
      \otimes
      (pr_{B'})^\ast {E'}
      \\
      \big\downarrow
      \\
      B \times B'
    \end{array}
  \right]
\]

The corresponding [[internal hom]]

$$
  LMaps
  \;\colon\;
  VectBund^{op} \times VectBund
  \longrightarrow
  VectBund
$$

forms the vector bundle whose base space is the [[compact-open topology|mapping space]] of base spaces, and whose [[fiber]] over $f \colon B \to B'$ is the [[vector space]] $[E, f^\ast E']_B$ of vector bundle homomorphisms $E \to f^\ast E'$ over $B$:

$$
  LMaps
  \left(
    \left[
      \begin{array}{c}
        E
        \\
        \big\downarrow
        \\
        B
      \end{array}
    \right]
    \;,\;\;
    \left[
      \begin{array}{c}
        E'
        \\
        \big\downarrow
        \\
        B'
      \end{array}
    \right]
  \right)
  \;\;\coloneqq\;\;
  \left[
    \begin{array}{c}
      f
      \mapsto
      \big[E,  f^\ast E'\big]_B
      \\
      \big\downarrow
      \\
      \left\{
        \array{
          f \colon B \to B'
        }
        \;\;
      \right\}
    \end{array}
  \right]
$$

To see this, one checks the required hom-isomorphism

$$
  Hom
  \big(
    -
    ,\,
    LMap(-,-)
  \big)
  \;\;\simeq\;\;
  Hom
  \big(
    - \otimes -
    ,\,
    -
  \big)
$$

over the point, now using (eq:TensorProduct) on the right:

$$
  Hom
  \left(
  \left[
    \begin{array}{c}
      V
      \\
      \big\downarrow
      \\
      \ast
    \end{array}
  \right]
  \;,\;\;
  LMaps
  \left(
    \left[
      \begin{array}{c}
        E
        \\
        \big\downarrow
        \\
        B
      \end{array}
    \right]
    \;,\;\;
    \left[
      \begin{array}{c}
        E'
        \\
        \big\downarrow
        \\
        B'
      \end{array}
    \right]
  \right)
  \right)
  \;\;
  \underoverset
  {\sim}
  {
    \Big(
      v
        \mapsto
      \big(f, \sigma_v\big)
    \Big)
    \;\mapsto\;
    \big(f, \sigma_{(-)} \big)
  }{\longrightarrow}
  \;\;
  Hom
  \left(
  \left[
    \begin{array}{c}
      E \otimes V
      \\
      \big\downarrow
      \\
      B
    \end{array}
  \right]
  \;,\;\;
  \left[
    \begin{array}{c}
      E'
      \\
      \big\downarrow
      \\
      B'
    \end{array}
  \right]
  \right)
$$

#### Distributive monoidal structure
 {#DistributiveMonoidalStructure}

With respect to the [[coproduct]] and the [[external tensor product]] [[external tensor product of vector bundles|of vector bundles]], $VectBund$ is a [[distributive monoidal category]]. We spell this out in detail for the case of [[discrete topological space|discrete]] base spaces (sets):

\begin{definition}
  For any [[ground field]], write [[VectBund|$Vect_{Set}$]] for the [[category]] of [[indexed sets]] of [[vector spaces]].
\end{definition}
\begin{remark}
We may and will present [[objects]] $\mathcal{V}$ of [[VectBund|$Vect_{Set}$]] as [[pairs]] consisting of a [[set]] $S$ and a [[function]] $\mathcal{V}_{(-)}$ (really a [[functor]] on the [[discrete category]] on $S$) to [[Vect]]:
$$
  \left(
    \array{
      S &\longrightarrow& Vect
      \\
      s &\mapsto& \mathcal{V}_s
    }
  \right)
  \;\;
  \in
  \;\;
  Vect_{Set}
  \mathrlap{\,.}
$$
\end{remark}

\begin{definition}
\label{ExternalTensorProductInVectSet}
The "[[external tensor product|external]]" [[tensor product]]  on [[VectBund|$Vect_{Set}$]] is the [[functor]]
$$
  \boxtimes
  \,\colon\,
  Vect_{Set} \times Vect_{Set}
    \longrightarrow
  Vect_{Set}
$$
given by
$$
  \big(
    \mathcal{V}_{(-)} \,\colon\, S \to \Vect
  \big)
  \,\boxtimes\,
  \big(
    \mathcal{V}'_{(-)} \,\colon\, S' \to \Vect
  \big)
  \;\;
  \coloneqq
  \;\;
  \left(
    \array{
      S \times S' &\longrightarrow& \Vect
      \\
      (s, s')
        &\mapsto&
      \mathcal{V}_s \otimes \mathcal{V}'_{s'}
    }
  \right)
  \mathrlap{\,.}
$$
\end{definition}

\begin{proposition}
\label{CoproductInVectSet}
  The [[coproduct]] in [[VectBund|$Vect_{Set}$]] is given by [[disjoint union]] of [[bundles]]:
$$
  \big(
    \mathcal{V}^{(1)}_{(-)} \,\colon\, S_1 \to \Vect
  \big)
  \;
  \sqcup
  \;
  \big(
    \mathcal{V}^{(1)}_{(-)} \,\colon\, S_2 \to \Vect
  \big)
  \;\;
  \simeq
  \;\;
  \left(
    \array{
      S_1 \sqcup S_2 &\longrightarrow& \Vect
      \\
      s_i &\mapsto& \mathcal{V}^{(i)}_{s_i}
    }
  \right)
$$
\end{proposition}
\begin{proof}
  It is immediate to check the [[universal property]] characterizing the coproduct.
\end{proof}

\begin{proposition}
The [[external tensor product]] $\boxtimes$
(Def. \ref{ExternalTensorProductInVectSet})
[[distributive monoidal category|distributes]] over the coproduct (Prop. \ref{CoproductInVectSet}):
$$
  \mathcal{E} \boxtimes ( \mathcal{V}^{(1)} \sqcup  \mathcal{V}^{(2)})
  \;\simeq\;
  \big(
    \mathcal{E} \boxtimes \mathcal{V}^{(1)}
  \big)
  \,\sqcup\,
  \big(
    \mathcal{E} \boxtimes \mathcal{V}^{(2)}
  \big)
$$
and hence gives a [[distributive monoidal category]]:
$$
  \big(
    Vect_{Set}
    ,
    \sqcup
    ,
    \boxtimes
  \big)
  \;\in\;
  DistMonCat
  \mathrlap{\,.}
$$
\end{proposition}
\begin{proof}
Unwinding the above definitions and using that [[Set]] is a [[distributive category]], we have the following sequence of [[natural isomorphisms]]:
$$
  \begin{array}{l}
    \mathcal{E}
    \,\boxtimes\,
    \big(
      \mathcal{V}^{(1)}
      \,\sqcup\,
      \mathcal{V}^{(2)}
    \big)
    \\
    \;\equiv\;
    \big(
      \mathcal{E}_{(-)} \,\colon\, S_E \to Vect
    \big)
    \,\boxtimes\,
    \Big(
      \big(
        \mathcal{V}^{(1)}_{(-)} \,\colon\, S_1 \to Vect
      \big)
      \,\sqcup\,
      \big(
        \mathcal{V}^{(2)}_{(-)} \,\colon\, S_2 \to Vect
      \big)
    \Big)
    \\
    \;\simeq\;
    \big(
      \mathcal{E}_{(-)} \,\colon\, S_E \to Vect
    \big)
    \,\boxtimes\,
    \left(
      \array{
        S_1 \sqcup S_2 &\longrightarrow& Vect
        \\
        s_i &\mapsto& \mathcal{V}^{(i)}_{s_i}
      }
    \right)
    \\
    \;\simeq\;
    \left(
      \array{
        S_E \times (S_1 \sqcup S_2)
        &\longrightarrow& Vect
        \\
        (s_E , s_i) &\mapsto&
        \mathcal{E}_{s_E} \otimes \mathcal{V}^{(i)}_{s_i}
      }
    \right)
    \\
    \;\simeq\;
    \left(
      \array{
        (S_E \times S_1) \,\sqcup\, (S_2 \times S_2)
        &\longrightarrow& Vect
        \\
        (s_E , s_i) &\mapsto&
        \mathcal{E}_{s_E} \otimes \mathcal{V}^{(i)}_{s_i}
      }
    \right)
    \\
    \;\simeq\;
    \left(
      \array{
        S_E \times S_1
        &\longrightarrow& Vect
        \\
        (s_E , s_1)
         &\mapsto&
        \mathcal{E}_{s_E} \otimes \mathcal{V}^{(1)}_{s_1}
      }
    \right)
    \,\sqcup\,
    \left(
      \array{
        S_E \times S_2
        &\longrightarrow& Vect
        \\
        (s_E , s_2)
         &\mapsto&
        \mathcal{E}_{s_E} \otimes \mathcal{V}^{(2)}_{s_2}
      }
    \right)
    \\
    \;\equiv\;
    \mathcal{E} \boxtimes \mathcal{V}^{(1)}
    \,\sqcup\,
    \mathcal{E} \boxtimes \mathcal{V}^{(2)}
  \end{array}
$$
\end{proof}

#### Amalgamation of monoidal and parameter structures
 {#AmalgamationOfMonoidalAndParameterStructures}

> under construction

We want to point out a way in which $VectBund$ with its [[external tensor product]]-structure is the canonical "amalgamation" of

1. vector spaces equipped with their tensor product,

1. vector spaces equipped with parametrization, namely vector bundles,

at least in the case that the base space is discrete.

The argument generalizes immediately to [[flat vector bundles]] over arbitrary base spaces (by enhancing the following parameter [[sets]] to [[fundamental groupoids]]), and in fact to [[(infinity,1)-vector bundle|$(\infty,1)$-vector bundles]] with [[flat infinity-connection|flat $\infty$-connections]] (by enhancing the [[fundamental groupoids]] further to [[fundamental infinity-groupoids|fundamental $\infty$-groupoids]] and the [[category of vector spaces]] to the [[(infinity,1)-category of chain complexes]]).

In order to formalize this idea, we need to work inside a [[2-category]] which suitably subsumes all of the following
[[very large category|very large]] [[(2,1)-categories]]:

* $Cat$ of [[categories]]

* $MonCat$ of [[monoidal categories]];

* $CoCartCart$ of [[cocartesian monoidal categories]];

* $DistMonCat$ of [[distributive monoidal categories]] (with [[underlying]] [[cocartesian monoidal categories]]).

Here we understand that

* the [[1-morphisms]] in these [[2-categories]] are [[functors]] which respect all given [[tensor products]] up to [[isomorphism]], i.e. [[strong monoidal functors]]

* the [[2-morphisms]] are [[natural isomorphisms]] between these functors.

The system of [[forgetful functor|forgetful]] [[2-functors]] between these [[(2,1)-categories]] forms a square [[diagram]] which we may regard as the [[image]] of a corresponding [[contravariant functor|contravariant]] [[pseudofunctor]] from the [[commuting square]]-[[diagram category]] to [[2Cat]]:

$$
  \mathbf{C}
  \;\;
    \colon
  \;\;
  \array{
    (1,0) &\longrightarrow& (1,1)
    \\
    \big\uparrow && \big\uparrow
    \\
    (0,0) &\longrightarrow& (1,0)
  }
  \;\;\;\;
    \mapsto
  \;\;\;\;
  \array{
     MonCat &\longleftarrow& DistMonCat
     \\
     \big\downarrow && \big\downarrow
     \\
     Cat &\longleftarrow& CoCartCat
  }
$$

The 2-categorical context which we are after is then the [[(infinity,1)-Grothendieck construction|(2,1)-]][[Grothendieck construction]] $\int \mathbf{C}$ on this [[pseudofunctor]]: Here [[1-morphisms]] are functors which strictly preserve the [[tensor products]] present on their [[domain]] category, but whose [[codomain]] may be equipped with further [[tensor products]] (though not with fewer tensor products).

**Claim.** In $\int \mathbf{C}$ the following [[diagram]] is a ([[homotopy pushout|homotopy]]) [[pushout]]

\begin{tikzcd}
  \big(
    \mathrm{Vect}
    ,
    \otimes
  \big)
  \ar[rr]
  &&
  \big(
    \mathrm{Vect}_{\mathrm{Set}}
    ,
    \sqcup
    ,
    \boxtimes
  \big)
  \\
  \mathrm{Vect}
  \ar[u]
  \ar[
    rr
  ]
  &&
  \big(
    \mathrm{Vect}_{\mathrm{Set}}
    ,
    \sqcup
  \big)
  \ar[u]
\end{tikzcd}

\begin{proof}
We check the [[universal property]]. To this end, observe that any [[cocone]] under the diagram must have as tip a [[distributive monoidal category]], since only these can receive morphisms in $\int \mathbf{C}$ both from a monoidal and from a cocartesian category.

This means that a general [[cocone]] looks like the following solid diagram:

\begin{tikzcd}
  &&
  &[-10pt]
  \big(
    \mathcal{C}
    ,
    \sqcup
    ,
    \otimes
  \big)
  \\[-10pt]
  \big(
    \mathrm{Vect}
    ,
    \otimes
  \big)
  \ar[rr]
  \ar[urrr]
  &&
  \big(
    \mathrm{Vect}_{\mathrm{Set}}
    ,
    \sqcup
    ,
    \boxtimes
  \big)
  \ar[
    ur,
    dashed
  ]
  \\
  \mathrm{Vect}
  \ar[u]
  \ar[
    rr
  ]
  &&
  \big(
    \mathrm{Vect}_{\mathrm{Set}}
    ,
    \sqcup
  \big)
  \ar[u]
  \ar[uur]
\end{tikzcd}

We need to see that this uniquely admits the dashed arrow. Unwinding the definitions, the existence of the dashed arrow means that any functor $F \,\colon\, Vect_{Set} \to \mathcal{C}$ which preserves both the coproduct of vector bundles as well as the tensor product on fibers already respects the [[external tensor product]]. But this follows by the [[distributive monoidal category|distributivity]] and using that any [[set]] is the [[coproduct]] of its elements:

$$
  \begin{array}{l}
    F\big( \mathcal{E} \boxtimes \mathcal{E}' \big)
    \\
    \;\simeq\;
    F
    \Bigg(
      \underset{s \in S}{\coprod}
      \left(
        \array{
          \{s\} &\longrightarrow& Vect
          \\
          s &\mapsto& \mathcal{E}_s
        }
      \right)
      \;\;
      \boxtimes
      \;\;
      \underset{s' \in S'}{\coprod}
      \left(
        \array{
          \{s'\} &\longrightarrow& Vect
          \\
          s' &\mapsto& \mathcal{E}'_{s'}
        }
      \right)
    \Bigg)
    \\
    \;\simeq\;
    F
    \Bigg(
      \underset{(s, s') \in S \times S'}{\coprod}
      \Big(
        \big(
          (s \colon \{s\}) \mapsto \mathcal{E}_s
        \big)
        \boxtimes
        \big(
          (s' \colon \{s'\}) \mapsto \mathcal{E}_{s'}
        \big)
      \Big)
    \Bigg)
    \\
    \;\simeq\;
    \underset{(s, s') \in S \times S'}{\coprod}
    \bigg(
      \Big(
        F
        \big(
          (s \colon \{s\}) \mapsto \mathcal{E}_s
        \big)
       \Big)
        \otimes
       \Big(
        F
        \big(
          (s' \colon \{s'\}) \mapsto \mathcal{E}_{s'}
        \big)
      \Big)
    \bigg)
    \\
    \;\simeq\;
      \Big(
        F
        \big(
          \underset{s \in S}{\coprod}
          (s \colon \{s\}) \mapsto \mathcal{E}_s
        \big)
       \Big)
        \otimes
       \Big(
        F
        \big(
          \underset{s' \in S'}{\coprod}
          (s' \colon \{s'\}) \mapsto \mathcal{E}_{s'}
        \big)
      \Big)
    \\
    \;\simeq\;
    F(\mathcal{E}) \otimes F(\mathcal{E}')
    \,.
  \end{array}
$$

\end{proof}

## Related concepts

* [[Mod]]

* [[VectBund(B)]]

* [[tangent (infinity,1)-category|tangent $\infty$-category]]

* [[parameterized spectra]]

* [[doubly closed monoidal category]]

[[!redirects VectorBundles]]


vector

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

#Contents#
* table of contents
{:toc}

## Idea

A **vector** is an element in a [[vector space]].

The archetypical examples are [[tangent vectors]].

## Related concepts

* [[wave vector]]
* [[direction of a vector]]

[[!redirects vectors]]


vector boson

#Contents#
* table of contents
{:toc}

## Idea

A [[boson]] with [[spin]] 1.

## Examples

In the [[standard model of particle physics]]: the [[photon]], the [[W-boson]], the [[Z-boson]] and the [[gluon]].

## References

* Wikipedia, _[Vecor boson](http://en.wikipedia.org/wiki/Vector_boson)_

[[!redirects vector bosons]]


vector bundle

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

# Vector bundles
* table of contents
{: toc}

## Idea

Given some context of [[geometry]], then a _vector bundle_ is a collection of [[vector spaces]] that varies in a geometric way over a given base [[space]] $X$: over each [[generalized element|element]] $x \in X$ there is a [[vector space]] $V_x$, called the _[[fiber]]_ over $x$, and as $x$ varies in $X$, the fibers vary along in a geometric way. One also says that vector bundles are _[[fiber bundles]]_ whose fiber carries [[vector space]]-structure. Hence the theory of vector bundle is _parameterized_ [[linear algebra]]. The vector bundles over a fixed base $X$ form a category [[Vect(X)|$VectBund_X$]], and as the base space is allowed to vary these fit into a global category [[VectBund]].

For example

* in [[topology]] a _[[topological vector bundle]]_ is a collection of [[vector spaces]] which "vary continuously" over a [[topological space]] $X$,

* in [[differential geometry]] a _[[differentiable vector bundle]]_ is a collection of vector space which "varies differentiably" over a [[differentiable manifold]],

* in [[algebraic geometry]] an _[[algebraic vector bundle]]_ is a collection of vector spaces which vary algebraically over a [[scheme]].

and so on.

One requires that "locally", on small enough patches of the base space $X$, the variation of the fibers is constant up to isomorphism (one says the vector bundle is "locally trivial"), but the key point of vector bundles is that there may be non-trivial structure in how the collection of vector spaces "globally glues together".

For example if $X = S^1$ is the [[circle]] regarded as a [[topological space]] in the standard way, and if we consider [[real vector spaces]], then there are up to [[isomorphism]] two different $\mathbb{R}$-vector bundles over $S^1$ whose [[fibers]] look like the 1-dimensional real vector space $\mathbb{R}$ itself, namely

1. the [[cylinder]]

1. the [[Möbius strip]]:

(In these pictures each vertical interval is to be thought of as a stand-in for a copy of the [[real line]] $\mathbb{R}$.)

Clearly for the cylinder nothing special happens to the fibers as one moves around the circle (one says this is a _trivial vector bundle_) while the Möbius strip is "locally trivial" but globally has a twist: as one moves once around the circle the original fiber comes back identified with its reflection at the origin.

  graphics grabbed from Hatcher

An important class of examples of vector bundles are [[tangent bundles]] of [[differentiable manifolds]] $X$. Here the [[vector space]] at each point of $X$ is the [[tangent space]] of that point, the space of all [[tangent vectors]] based at that point. The graphics on the right shows one of the tangent space of the [[2-sphere]].

Dually, given an [[embedding of differentiable manifolds]] into a [[Euclidean space]], then the [[normal vectors]] to the tangent bundle span a vector bundle called the _[[normal bundle]]_ of the embedding.

All the usual operations on [[finite dimensional vector spaces]] in [[linear algebra]] generalize to vector bundles by applying them [[fiber]]-wise. For instance there is [[direct sum of vector bundles]] and the [[tensor product of vector bundles]] over the same base space.

To the extent that the base [[space]] $X$ is encoded in its [[algebra of functions]] (tautologically in [[algebraic geometry]] or via [[Gelfand duality]] in [[topology]]), the [[Serre-Swan theorem]] asserts that vector bundles over $X$ are equivalently encoded in the [[projective modules]] over these algebras constituted by their [[sections]].

Vector bundles have various applications and uses:

1. their [[Grothendieck group]] under [[direct sum of vector bundles]] yields [[topological K-theory]], an interesting [[generalized (Eilenberg-Steenrod) cohomology theory]];

1. a [[reduction of the structure group]] of vector bundles encodes actual [[geometry]] on the base space; when applied to [[tangent bundles]] such _[[G-structures]]_ on vector bundles encode for instance [[orthogonal structure]], [[Riemannian geometry]], [[complex geometry]], [[symplectic geometry]], [[conformal geometry]] etc. (in general: [[Cartan geometry]]); when applied to [[normal bundles]] these [[G-structures]] give rise, via [[Thom's theorem]], to [[Thom spectra]] and [[cobordism theory]];

1. equipping differentiable vector bundles with [[connection on a vector bundle]] is the basis for [[Chern-Weil theory]] and for the application of vector bundles in [[physics]], where they model [[gauge fields]] and [[instanton sectors]]; see also at _[[fiber bundles in physics]]_.

## Definition

### Standard

See at _[[topological vector bundle]]_

### Sheaf-theoretic version

Vector bundles can also be defined via [[sheaf and topos theory|sheaf theory]], which permits easy transport to general [[Grothendieck toposes]]. Let $Sh(X)$ be the [[category]] of ([[set]]-valued) [[sheaf|sheaves]] on $X$. The sheaf of continuous local sections of the product projection

$$X \times \mathbb{R} \to X$$

forms a [[local ring]] object $R$; when interpreted in the [[internal logic]] of $Sh(X)$, it is the Dedekind [[real numbers object]]. Then, according to a [[Serre-Swan theorem|theorem of Richard Swan]], in its sheaf-theoretic incarnation a vector bundle is the same thing as a [[projective module|projective R-module]].

* A theorem of Kaplansky states "every [[projective module]] over a [[local ring]] is [[free module|free]]". When interpreted in [[sheaf semantics]] ([[Kripke-Joyal semantics]]), the [[existential quantifier]] implicit in "free" is interpreted _locally_, so we can consider a vector bundle as a locally free module over the Dedekind reals.

### Virtual vector bundles

In one class of models for [[K-theory]] -- [[generalized (Eilenberg-Steenrod) cohomology]] theory -- cocycles are represented by $\mathbb{Z}_2$-graded vector bundles (pairs of vector bundles, essentially) modulo a certain equivalence relation. In that context it is sometimes useful to consider a certain variant of infinite-dimensional $\mathbb{Z}_2$-graded vector bundles called [[vectorial bundle | vectorial bundles]].

Much else to be discussed...

## Examples

* [[canonical bundle]]

* [[valence bundle]]

* ...

## Related concepts

* [[principal bundle]], [[associated bundle]]

* **vector bundle**:

  * [[VectBund(X)]], [[VectBund]]

  * [[real vector bundle]]

  * [[complex vector bundle]]

    * [[holomorphic vector bundle]], [[pseudoholomorphic vector bundle]]

  * [[universal vector bundle]]

  * [[rank]] of a vector bundle

  * [[dual vector bundle]]

  * [[module bundle]]

  * [[direct sum of vector bundles]]

  * [[short exact sequence of vector bundles]]

  * [[connection on a vector bundle]]

  * [[flat vector bundle]]

  * [[real vector bundle]], [[complex vector bundle]]

  * [[super vector bundle]]

* [[measurable field of Hilbert spaces]]

* [[2-vector bundle]]

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

## Literature
 {#Literature}

* [[Glenys Luke]], [[Alexandr S. Mishchenko]], *Vector bundles and their applications*, Math. and its Appl. **447** Kluwer (1998) [[doi:10.1007/978-1-4757-6923-4](https://doi.org/10.1007/978-1-4757-6923-4), [MR99m:55019](http://www.ams.org/mathscinet-getitem?mr=99m:55019)]

* А. С. Мищенко, _Векторные расслоения и их применения_ (Russian;  A. S. Mishchenko, Vector bundles and their applications) Nauka, Moscow, 1984. 208 pp.

* Howard Osborn, _Vector bundles. Vol. 1. Foundations and Stiefel-Whitney classes_, Pure and Appl. Math. __101__, Academic Press 1982. xii+371 pp. [MR85e:55001](http://www.ams.org/mathscinet-getitem?mr=85e:55001)

* [[Dale Husemöller]], _Fibre bundles_, McGraw-Hill 1966 (300 p.); Springer GTM 1975 (327 p.), 1994 (353 p.).

* [[Dale Husemöller]], [[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))

An exposition with an eye towards [[gauge theory]] is in section 16.1 of

* [[Theodore Frankel]], _[[The Geometry of Physics - An Introduction]]_

* [[Raoul Bott]], [[Loring Tu]], _Differential forms in algebraic topology_, Graduate Texts in Mathematics __82__, Springer 1982. xiv+331 pp.

Discussion with an eye towards [[K-theory]] is in

* [[Max Karoubi]], _K-theory. An introduction_, Grundlehren der Mathematischen Wissenschaften __226__, Springer 1978. xviii+308 pp.

* {#Hatcher} [[Allen Hatcher]], _Vector bundles and K-Theory_, (partly finished book) [web](https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html)

[[!redirects vector bundles]]


vector field

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###Context###
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A __vector field__ is a [[section]] of a [[vector bundle]].

More specifically, a vector field is a _[[tangent vector field]]_ which is a [[section]] of a [[tangent bundle]]. Hence this is a function which picks a [[tangent vector]] at each point of a manifold, such that this assignment is suitably [[differentiable function|differentiable]].

Equivalently this is a rank $(1,0)$-[[tensor field]] on $X$

## Properties

### As derivations of smooth functions

Vector fields may be identified with [[derivations]]
on the [[algebra of functions|algebra of]] [[smooth functions]].
See the article *[[derivations of smooth functions are vector fields]]*.

## Related concepts

* [[flow of a vector field]]

* [[differential form]]

* [[derivations of smooth functions are vector fields]]

* [[frame field]]

* [[Hamiltonian vector field]]

* [[Reeb vector field]]

* [[evolutionary vector field]]

* [[field (physics)]]

## References

For instance section 1.3 and 1.4 of

* [[Theodore Frankel]], _[[The Geometry of Physics - An Introduction]]_

[[!redirects vector fields]]


vector G-space

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### Context
#### Representation theory
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[[!include representation theory - contents]]
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#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
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#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $G$ be a [[finite group]]. We write

\[
  \label{OrbitCategory}
  G Orbits \;\subset\; Categories
\]

for the [[orbit category]] of $G$.

+-- {: .num_defn #FiniteDimensionalRationalVectorGSpaces}
###### Definition
**([[rational vector space|rational]] [[vector G-spaces]])**

We say that the _category of finite-dimensional vector G-spaces_ is
the [[functor category|category of functors]] from the [[opposite category|opposite]] of the [[orbit category]] to the [[category]] of [[finite-dimensional vector space|finite-dimensional]] [[rational vector spaces]]:

\[
  \begin{aligned}
  Vector G Spaces^{fin}_{\mathbb{Q}}
  &
    \coloneqq
  \;
  PSh
  \Big(
    G Orbits
    \,,\,
    VectorSpaces_{\mathbb{Q}}^{\mathrm{fin}}
  \Big)
  \\
  &
  \coloneqq
  \;
  Functors
  \Big(
    G Orbits^{op}
    \,,\,
    VectorSpaces_{\mathbb{Q}}^{\mathrm{fin}}
  \Big)
  \end{aligned}
\]

Its [[opposite category]] we call the
_category of finite-dimensional dual vector G-spaces_:

\[
  \label{FunctorCategoryFromOrbitCategoryToOppositeRationalVectorSpaces}
  \begin{aligned}
  \big(
    Vector G Spaces^{fin}_{\mathbb{Q}}
  \big)^{op}
  &
    \simeq
  \;
  PSh
  \Big(
    G Orbits^{op}
    \,,\,
    VectorSpaces_{\mathbb{Q}}^{\mathrm{fin}}
  \Big)
  \\
  &
  =
  \;
  Functors
  \Big(
    G Orbits
    \,,\,
    VectorSpaces_{\mathbb{Q}}^{\mathrm{fin}}
  \Big)
  \end{aligned}
\]

(using in the first line that forming [[dual linear maps]] is an [[equivalence of categories]] from [[finite dimensional vector spaces]] to their [[opposite category]].)

In generalization of (eq:FunctorCategoryFromOrbitCategoryToOppositeRationalVectorSpaces), dropping the finiteness condition, we write

\[
  \label{DualVectorGSpaces}
  DualVector G Spaces
  \;\coloneqq\;
  Functors
  \Big(
    G Orbits
    \,,\,
    VectorSpaces_{\mathbb{Q}}
  \Big)
\]

=--

The category (eq:DualVectorGSpaces) is denoted $Vec_G^\ast$ in ([Triantafillou 82](#Triantafillou82)).

## Properties

### Projective objects
 {#ProjectiveObjects}

> Beware that this section uses different notational conventions than the rest of the entry. None of the rest of the entry is necessary for reading this section here.

Notation:

* $G$ a [[finite group]],

* $N(H)/H$ the "[Weyl group](Weyl+group#InEquivariantHomotopyTheory)" of a [[subgroup]] $H \subset G$,

* $Orb_G$ the [[orbit category]] of $G$, equivalently thought of as the [[full subcategory]] of [[G-sets]] $G/H$ of [[cosets]] for [[subgroups]] $H \subset G$,

* $\mathbb{Q}$ the [[ground field]] of [[rational numbers]],

* $Mod_{\mathbb{Q}}^{Orb_G^{op}}$ the [[presheaf category]] on $Orb_G$ with [[coefficients]] in [[rational vector spaces]],

* $\mathbb{Q}[K]$ the [[group algebra]] over the [[rational numbers]], of a [[finite group]] $K$.

\linebreak

\begin{definition}
\label{TheProjectiveGenerators}
([T82, Def. 3.1](#Triantafillou82))
\linebreak

  For

  * $H \subset G$,

  * $V_H \,\in\, Mod_{ \mathbb{Q}[N(H)/H] }$

define

$$
  \begin{array}{l}
    \underline{V}_H
    \;\in\;
    Mod_{\mathbb{Q}}^{Orb_G^{op}}
    \\
    \underline{V}_H
    \;\equiv\;
    \mathbb{Q}\big[
      Orb_G(-,G/H)
    \big]
    \underset{
      \mathbb{Q}[N(H)/H]
    }{\otimes}
    V_H
    \mathrlap{\,.}
  \end{array}
$$
\end{definition}

\begin{remark}\label{UnderlineVHatStageH}
  Since the [[Weyl group]] of $H$ is (see [there](Weyl+group#eq:AsAutomorphismGroupInOrbitCategory)) the [[endomorphism monoid]] of $G/H$ in the $G$-[[orbit category]]
  $$
    N(H)/H
    \;\simeq\;
    {Orb_G}(G/H,G/H)
    \;\;
  $$
  we have with Def. \ref{TheProjectiveGenerators} canonical isomorphism:
\[
  \begin{array}{ccl}
    \underline{V}_H(G/H)
    &\equiv&
    \mathbb{Q}\big[
      Orb_G(G/H,G/H)
    \big]
    \underset{
      \mathbb{Q}[N(H)/H]
    }{\otimes}
    V_H
    \\
    &\simeq&
    V_H
    \mathrlap{\,.}
  \end{array}
\]
\end{remark}

\begin{proposition}
  \label{ProjectiveGeneratorsAreIndeedProjective}
  ([T82, Prop. 3.2](#Triantafillou82))
  \linebreak
  The objects $\underline{V}_H$ from Def. \ref{TheProjectiveGenerators} are [[projective object|projective]] in $Mod_{\mathbb{Q}}^{Orb_G^{op}}$.
\end{proposition}
\begin{proof}
  We need to show that dashed [[lifts]] in the following [[diagrams]] exist, where $p$ is an [[epimorphism]]:
\begin{tikzcd}[sep=20pt]
  && M
  \ar[
    dd,
    ->>,
    "{ p }"
  ]
  \\
  \\
  \underline{V}_H
  \ar[
     rr,
     "{ f }"
  ]
  \ar[
    uurr,
    dashed,
    "{ \widehat{f} }"
  ]
  &&
  N
\end{tikzcd}
Since plain [[rational vector spaces]] are [[free modules]] (by the [[basis theorem]]), hence [[projective modules]], hence [[projective objects]] in [[Vect]], each $Orb_G$-component of this diagram separately has such a lift,  in particular we may choose a lift $\widehat{f_{G/H}}$ at stage $G/H$:

\begin{tikzcd}[sep=20pt]
  &&
  M(G/H)
  \ar[
    dd,
    ->>,
    "{ p_{G/H} }"
  ]
  \\
  \\
  V_H
  \ar[
     rr,
     "{ f_{G/H} }"{swap}
  ]
  \ar[
    uurr,
    dashed,
    "{ \widehat{f_{G/H}} }"
  ]
  &&
  N(G/H)
  \mathrlap{\,,}
\end{tikzcd}

where we have identified $V_H$ in the bottom left via Rem. \ref{UnderlineVHatStageH}.

With this local lift in hand, we obtain a global lift by setting:

\begin{tikzcd}[row sep=0pt]
  &
  \underline{V}_H
  \ar[rr, "{ \widehat{f} }"]
  &&
  M
  \\
  G/K
  &
  \mathbb{Q}\big[
    \mathrm{Orb}_G(G/K,G/H)
  \big]
  \underset{
    \mathclap{
      \raisebox{-2pt}{
        \scalebox{.6}{$
          \mathbb{Q}[N(H)/H]
        $}
      }
    }
  }{\otimes}
  V_H
  \ar[
    rr,
    "{ \widehat{f}_{G/K} }"
  ]
  &&
  M(G/H)
  \\
  &
  \big(
    G/K \xrightarrow{\phi} G/H
  \big)
  \otimes
  v_H
  \ar[
    rr,
    phantom,
    "{ \longmapsto }"
  ]
  &&
  M(\phi)\big(
    \widehat{f_{G/H}}(v_H)
  \big)
  \mathrlap{\,.}
\end{tikzcd}

This is clearly a [[natural transformation]] (by the [[contravariant functor|contravariant functoriality]]) of $M$), and it is a lift by [[natural transformation|naturality]] of $p$ and $f$:

$$
  \begin{array}{l}
    p_{G/K}
    \circ
    \widehat{f}_{G/K}
    \big(
      (G/K \xrightarrow{\phi} G/H)
      \otimes
      v_H
    \big)
    \\
    \;\equiv\;
    p_{G/K}
      \circ
    M(\phi)
      \circ
    \widehat{f_{G/H}}
    \big(
      (G/H \xrightarrow{id} G/H)
      \otimes
      v_H
    \big)
    \\
    \;=\;
    N(\phi)
      \circ
    p_{G/H}
      \circ
    \widehat{f_{G/H}}
    \big(
      (G/H \xrightarrow{id} G/H)
      \otimes
      v_H
    \big)
   \\
   \;=\;
    N(\phi)
      \circ
    f_{G/H}
    \big(
      (G/H \xrightarrow{id} G/H)
      \otimes
      v_H
    \big)
    \\
    \;=\;
    f_{G/K}
      \circ
    \underline{V}_H(\phi)
    \big(
      (G/H \xrightarrow{id} G/H)
      \otimes
      v_H
    \big)
    \\
    \;=\;
    f_{G/K}
    \big(
      (G/K \xrightarrow{\phi} G/H)
      \otimes
      v_H
    \big)
    \mathrlap{\,.}
  \end{array}
$$

\end{proof}

\begin{proposition}\label{CharacterizationOfProjectives}
([T82, Prop. 3.4](#Triantafillou82))
\linebreak
 Every [[projective object]] $P \,\in\, Mod_{\mathbb{Q}}^{Orb_G^{op}}$ is a [[direct sum]] of projective generators as in Prop. \ref{TheProjectiveGenerators}.
\end{proposition}

\begin{proof}
We make a bunch of choices:

First, in each [[conjugacy class]] $[H]$ of [[subgroups]] $G$ choose one representative $H \subset G$.

For that $H \hookrightarrow G$, consider the joint [[span]] of the [[images]] of

$$
  P(G/H \twoheadrightarrow G/H')
  \;\colon\;
  P(G/H') \to P(G/H)
$$

for all intermediate [[subgroup]]-inclusions $H \hookrightarrow H' \hookrightarrow G$:

$$
  0
  \to
  \underset{ H' \supset H }{
    \textstyle{\sum}
  }
  im\big(
    P(G/H \twoheadrightarrow G/H')
  \big)
  \hookrightarrow
  P(G/H)
  \twoheadrightarrow
  P(G/H)
    \big/
  \underset{ H' \supset H }{
    \textstyle{\sum}
  }
  im\big(
    P(G/H \twoheadrightarrow G/H')
  \big)
  \to
  0
  \,.
$$

On the right we have exhibited the [[quotient vector space]] of the inclusion of the joint images on the left (hence the joint [[cokernel]]) making a [[short exact sequence]] of [[rational vector spaces]].

Since [every short exact sequence of vector spaces splits](split+exact+sequence#OfVectorSpaces), we may next *choose* a [[split exact sequence|splitting]]:

\[
  \label{TheSplitting}
  \sigma_H
    \;\;\colon\;\;
  P(G/H)
    \big/
  \underset{ H' \supset H }{
    \textstyle{\sum}
  }
  im\big(
    P(G/H \twoheadrightarrow G/H')
  \big)
    \xhookrightarrow{\phantom{---}}
  P(G/H)
  \,,
\]

whose [[image]] we denote by

\[
  \label{VHInProjectiveCover}
  V_H
  \;\;\;\coloneqq\;\;\;
  \sigma
  \Big(
    P(G/H)
      \big/
    \underset{ H' \supset H }{
      \textstyle{\sum}
    }
    im\big(
      P(G/H \twoheadrightarrow G/H')
    \big)
  \Big)
  \xhookrightarrow{\phantom{---}}
  P(G/H)
  \,.
\]

(In fact, we need this splitting $N(H)/H$-equivariantly: Since we are in [[characteristic zero]] this follows by the fact that every $N(H)/H$-[[linear representation|representation]] splits as a [[direct sum]] of [[irreducible representations]], and by the first part of [[Schur's lemma]], which says that there are no non-zero maps between distinct such direct summands.)

Via these (images of) chosen splittings (eq:VHInProjectiveCover), we may define a morphism in $Mod_{\mathbb{Q}}^{Orb_G^{op}}$ as follows, out of the direct sum of their underlined versions from \ref{TheProjectiveGenerators}:

\[
  \label{ComparisonMorphismForProjectiveDecomposition}
  \array{
    p
    &\colon&
    \underset{
      [H]
    }{\oplus}
    \underline{V}_H
    &\longrightarrow&
    P
    \\
    p_{G/K}
    &\colon&
    \underset{
      [H]
    }{\oplus}
    \mathbb{Q}
    \big[
      Orb_G(G/K, G/H)
      \underset{
        \mathbb{Q}[N(H)/H]
      }{\otimes}
      V_H
    \big]
    &\longrightarrow&
    P(G/K)
    \\
    &&
    \big(
      G/K \xrightarrow{ f } G/H
    \big)
    \,\otimes\,
    v_H
    &\mapsto&
    P(f)(v_H)
    \,,
  }
\]

which is manifestly [[natural transformation|natural]] in $G/K$ (via [[contravariant functor|contravariant functoriality]] of $P$) and hence well-defined:

\begin{tikzcd}[
  row sep=0pt,
  column sep=10pt
]
  &
  \bigoplus_{[H]}
  \underline{V}_H
  \ar[
    rrrr,
    "{ p }"
  ]
  &[-30pt]
  &&
  &[-20pt]
  P
  \\
  G/K
  &
  \underset{[H]}{\bigoplus}
  \,
  \mathbb{Q}
  \big[
    \mathrm{Orb}_G\big(
      G/K
      ,\,
      G/H
    \big)
    \underset{
      \mathclap{
        \raisebox{-2pt}{
          \scalebox{.6}{$
            \mathbb{Q}[N(H)/H]
          $}
        }
      }
    }{\otimes}
    V_H
  \big]
  \ar[
    rrrr,
    "{ p_{G/K} }"
  ]
  \ar[
    dddd,
    "{
  \underset{[H]}{\bigoplus}
  \,
  \mathbb{Q}
  [
    \mathrm{Orb}_G(
      \phi
      ,\,
      G/H
    )
    \underset{
      \mathclap{
        \raisebox{-2pt}{
          \scalebox{.6}{$
            \mathbb{Q}[N(H)/H]
          $}
        }
      }
    }{\otimes}
    V_H
  ]
    }"{description}
  ]
  &&&&
  P(G/K)
  \ar[
    dddd,
    "{ P(\phi) }"
  ]
  \\[-10pt]
  &
  &
  \big(
    G/K \xrightarrow{f} G/H
  \big)
  \otimes
  v_H
  \ar[
    rr,
    phantom,
    "{ \longmapsto }"
  ]
  \ar[
    dd,
    phantom,
    "{ \longmapsto }"{rotate=-90}
  ]
  &&
  P(f)(v_H)
  \ar[
    dd,
    phantom,
    "{ \longmapsto }"{rotate=-90}
  ]
  \\
  \phantom{A}
  \\
  &
  &
  \big(
    G/K' \xrightarrow{ \phi }
    G/K \xrightarrow{f} G/H
  \big)
  \otimes
  v_H
  \ar[
    rr,
    phantom,
    "{ \longmapsto }"
  ]
  &&
  P(\phi)\big(P(f)(v_H)\big)
  \\
  G/K'
  \ar[
    uuuu,
    "{ \phi }"
  ]
  &
  \underset{[H]}{\bigoplus}
  \,
  \mathbb{Q}
  \big[
    \mathrm{Orb}_G\big(
      G/K'
      ,\,
      G/H
    \big)
    \underset{
      \mathclap{
        \raisebox{-2pt}{
          \scalebox{.6}{$
            \mathbb{Q}[N(H)/H]
          $}
        }
      }
    }{\otimes}
    V_H
  \big]
  \ar[
    rrrr,
    "{ p_{G/K'} }"{swap}
  ]
  &&&&
  P(G/K')
\end{tikzcd}

Since all the direct summands on the left are projective by Prop. \ref{ProjectiveGeneratorsAreIndeedProjective}, it is now sufficient to prove that (eq:ComparisonMorphismForProjectiveDecomposition) is an [[isomorphism]]. Since isomorphisms in [[functor categories]] are detected objectwise and  since [[rational vector spaces]] form a [[balanced category]] (see [there](balanced+category#AbelianCategoriesAreBalanced)) for this it is sufficient to show that for all $K \subset G$ the morphism $p_{G/K}$ (eq:ComparisonMorphismForProjectiveDecomposition) is both an [[epimorphism]] and a [[monomorphism]].

First to see that that $p_{G/K}$ is an epimorphism: To start with, it is clearly surjective onto the summand $V_K$. Hence it is next sufficient to show that given $v_K \in P(G/K)$ which is in the image under $P(G/K \twoheadrightarrow G/H)$ of some $\widehat{v}_H \in P(G/H)$ then it is also in the image of $p_{G/K}$. As before, this is clear for those $\widehat{v}_H \in V_H$. Hence next, as before, it is sufficient to show this for those $\widehat{v}_H$ which are in the image under some $P(G/H \twoheadrightarrow G/H')$ of some $v_{H'} \in P(G/H')$... And so on. Since $G$ is a [[finite group]], this recursive argument eventually terminates with $V_G = P(G/G)$.

Finally, to see that $p_{G/K}$ (eq:ComparisonMorphismForProjectiveDecomposition) is a monomorphism. It is here (only) that we use the assumption that $P$ is [[projective object|projective]]. With the previous point, this implies a [[lift]] $p'$ in the following diagram in $Mod_{\mathbb{Q}}^{Orb_G^{op}}$:

\begin{tikzcd}[sep=25pt]
  &
  \underset{[H]}{\oplus} \underline{V}_H
  \ar[
    d,
    ->>,
    "{ p }"
  ]
  \\
  P
  \ar[r, equals]
  \ar[ur, dashed, "{ p' }"]
  &
  P
\end{tikzcd}

Hence if $v,w \,\in\, P(G/K)$ such that $p'_{G/K}(v) = p'_{G/K}(w)$ then $p_{G/K} \circ  p'_{G/K}(v) = p_{G/k} \circ p'_{G/K}(w)$ hence $v = w$, whence each $p'_{G/K}$ is injective.
\end{proof}

\begin{corollary}([T82, Prop. 3.6](#Triantafillou82))
 Every object $N \,\in\, Mod_{\mathbb{Q}}^{Orb_G}$ admits a [[projective cover]] in the sense of a [[projective object]] $\underset{[H]}{\oplus} \underline{V}_H$ and an [[epimorphism]] $p \,\colon\,\underset{[H]}{\oplus} \underline{V}_H \twoheadrightarrow N$.
\end{corollary}
\begin{proof}
  The construction and verification is verbatim as in the proof of Prop. \ref{CharacterizationOfProjectives}, omitting only the proof of injectivity in the last step.
\end{proof}

\linebreak

### Injective objects (1)
 {#InectiveObjects}

We spell out aspects of the discussion of injective objects in the copresheaf category dual to the [above](#ProjectiveObjects) discussion of projective objects in the presheaf category but left implicit in [Triantafillou 1982, p. 517](#Triantafillou82).

Notation:

* $G$ a [[finite group]],

* $N(H)/H$ the "[Weyl group](Weyl+group#InEquivariantHomotopyTheory)" of a [[subgroup]] $H \subset G$,

* $Orb_G$ the [[orbit category]] of $G$, equivalently thought of as the [[full subcategory]] of [[G-sets]] $G/H$ of [[cosets]] for [[subgroups]] $H \subset G$,

* $\mathbb{Q}$ the [[ground field]] of [[rational numbers]],

* $Mod_{\mathbb{Q}}^{Orb_G}$ the [[copresheaf]] [[functor category]] on $Orb_G$ with [[coefficients]] in [[rational vector spaces]],

* $\mathbb{Q}[K]$ the [[group algebra]] over the [[rational numbers]], of a [[finite group]] $K$.

\linebreak

\begin{definition}
\label{TheInjectiveGenerators}
(dual to [T82, Def. 3.1](#Triantafillou82))
\linebreak

  For

  * $H \subset G$,

  * $V^H \,\in\, Mod_{ \mathbb{Q}[N(H)/H] }$

define

$$
  \begin{array}{l}
    \underline{V}^H
    \;\in\;
    Mod_{\mathbb{Q}}^{Orb_G}
    \\
    \underline{V}^H
    \;\equiv\;
    Mod_{\mathbb{Q}[N(H)/H]}
    \Big(
    \mathbb{Q}\big[
      Orb_G(-,G/H)
    \big]
    ,\,
    V^H
    \Big)
    \mathrlap{\,.}
  \end{array}
$$
\end{definition}

\begin{proposition}\label{CharacterizationOfInjectives}
(dual to [T82, Prop. 3.4](#Triantafillou82))
\linebreak
 Every [[injective object]] $I \,\in\, Mod_{\mathbb{Q}}^{Orb_G}$ is a [[direct sum]] of injective generators as in Prop. \ref{TheInjectiveGenerators}.
\end{proposition}

\begin{proof}
We make a bunch of choices:

First, in each [[conjugacy class]] $[H]$ of [[subgroups]] $G$ choose one representative $H \subset G$.

For that $H \hookrightarrow G$, consider the [[intersection]] of the [[kernels]] of

$$
  I(G/H \twoheadrightarrow G/H')
  \;\colon\;
  I(G/H) \to I(G/H')
$$

for all intermediate [[subgroup]]-inclusions $H \hookrightarrow H' \hookrightarrow G$,

and an $N(H)/H$-equivariant splitting

\[
  \label{TheSplittingForInjectiveEnvelopes}
  \tau_H
    \;\;\colon\;\;
  I(G/H)
  \twoheadrightarrow
  \underset{ H' \supset H }{
    \textstyle{\bigcap}
  }
  ker\big(
    I(G/H \twoheadrightarrow G/H')
  \big)
  \,\equiv\,
  V^H
  \,.
\]

With this, we may define a morphism in $Mod_{\mathbb{Q}}^{Orb_G}$ as follows,

\[
  \label{ComparisonMorphismForInjectiveDecomposition}
  \array{
    i
    &\colon&
    I
    &\longrightarrow&
    \underset{
      [H]
    }{\oplus}
    \underline{V}^H
    \\
    i_{G/K}
    &\colon&
    I(G/K)
    &\longrightarrow&
    \underset{
      [H]
    }{\oplus}
    Mod_{\mathbb{Q}[N(H)/H]}
    \Big(
      \mathbb{Q}
      \big[
        Orb_G(G/K, G/H)
        ,\,
        V^H
      \big]
    \Big)
    \\
    &&
    i_K
    &\mapsto&
    \underset{[H]}{\oplus}
    \Big(
      \big(
        G/K \xrightarrow{f} G/H
      \big)
      \mapsto
      \tau_{H} \circ I(f)(i_K)
    \Big)
    \,,
  }
\]

where it is the [[functor|functoriality]] of $I$ which implies both that the maps on the right are $N(H)/H$-equivariant and that this transformation is [[natural transformation|natural]] in $G/K$:

\begin{tikzcd}[
  row sep=0pt,
  column sep=10pt
]
  &
  I
  \ar[rrrr, "{ i }"]
    &[-10pt]
    &&
    &[-50pt]
  \bigoplus_{[H]}
  \underline{V}^H
  \\
  G/K
  \ar[
    dddd,
    "{ \phi }"
  ]
  &
  I(G/K)
  \ar[
    rrrr,
    "{ i_{G/K} }"
  ]
  \ar[
    dddd,
    "{ I(\phi) }"
  ]
  &&&&
  \underset{[H]}{\bigoplus}
  \,
  \mathrm{Mod}_{N(H)/H}
  \Big(
    \mathrm{Orb}_G
      \big(G/K ,\, G/H \big)
    ,\,
    V^H
  \Big)
  \ar[
    dddd,
    "{
  \mathrm{Mod}_{N(H)/H}
  \Big(
    \mathrm{Orb}_G
      \big(\phi ,\, G/H \big)
    ,\,
    V^H
  \Big)
    }"{description}
  ]
  \\
  & &
  i_K
  \ar[
    rr,
    phantom,
    "{ \longmapsto }"
  ]
  \ar[dd, phantom, "{ \longmapsto }"{rotate=-90}]
    &&
  \oplus_{[H]}
  \Big(
    \big( G/K \xrightarrow{f} G/H \big)
    \mapsto
    \tau_H \circ I(f)(i_K)
  \Big)
  \ar[dd, phantom, "{ \longmapsto }"{rotate=-90}]
  \\
  \phantom{A}
  \\
  &&
  I(\phi)(i_K)
  \ar[
    rr,
    phantom,
    "{ \longmapsto }"
  ]
    &&
  \oplus_{[H]}
  \Big(
    \big( G/K' \xrightarrow{f'} G/H \big)
    \mapsto
    \tau_H \circ I(f')\circ I(\phi) (i_K)
  \Big)
  \\
  G/K'
  &
  I(G/K')
  \ar[
    rrrr,
    "{ i_{G/K'} }"{swap}
  ]
  &&&&
  \underset{[H]}{\bigoplus}
  \,
  \mathrm{Mod}_{N(H)/H}
  \Big(
    \mathrm{Orb}_G
      \big(G/K' ,\, G/H \big)
    ,\,
    V^H
  \Big)
\end{tikzcd}

Now to check that this map $i$ is in fact an isomorphism if $I$ is injective (...)
\end{proof}

(...)

### Injective objects (2)

> The following is another survey of aspects of the injective objects from [Triantafillou 1982](#Triantafillou82). This is from a different edit using different notational conventions than the previous subsection. Eventually both subsections should be harmonized and merged.

+-- {: .num_example #RestrictionOfVectorGSpacesToWeyGroupRepresentations}
###### Example
(restriction of vector $G$-spaces to Weyl group representations)

Let $H \subset G$ any [[subgroup]]. Notice that its [[Weyl group]] is the [[automorphism group]] of its [[coset space]] in the [[orbit category]]:

\[
  \label{WeylGroup}
  G Orbits
  \big(
    G/H
    \,,\,
    G/H
  \big)
  \;\;
  \simeq
  \;\;
  Aut_{G Orbits}
  \big(
    G/H
  \big)
  \;\;
    \simeq
  \;\;
  W_G(H)
  \;\;
    \coloneqq
  \;\;
  N_G(H)/H
\]

This gives a [[full subcategory]]-inclusion

$$
  \mathbf{B}
  W_G(H)
  \;
  \overset{\;\;i_H\;\;}{\hookrightarrow}
  \;
  G Orbits
$$

of the [[delooping]] category of they Weyl group into the [[orbit category]] of $G$ (eq:OrbitCategory), and hence a restriction functor

\[
  \label{RestrictionFromOrbitCategoryRepresentations}
  W_G(H) Representations^{fin}_{l,\mathbb{Q}}
  \overset{
     \;\;\;
     (-)^\ast
     \,\circ\,
     i_H^\ast
     \;\;\;
  }{\longleftarrow}
  \big(
    Vector G Spaces_{\mathbb{Q}}^{fin}
  \big)^{op}
\]

or more generally

\[
  \label{NonFiniteRestrictionFromOrbitCategoryRepresentations}
  W_G(H) Representations^{fin}_{r,\mathbb{Q}}
  \overset{
     \;\;\;
     i_H^\ast
     \;\;\;
  }{\longleftarrow}
  DualVector G Spaces
\]

=--

By the general [[end]]-formula for [[right Kan extension]] ([here](Kan+extension#eq:RightKanExtensionViaEndFormula)), this restriction functor has a [[right adjoint]], given as follows:

+-- {: .num_defn #InjectiveAtomsOfGEquivariantDualVectorSpaces}
###### Definition
**(injective atoms of dual vector $G$-spaces)**

For $H \subset G$ a [[subgroup]] and

$$
  V \;\in\; W_G(H) Representations^{fin}_{l,\mathbb{Q}}
$$

a [[rational vector space|rational]] [[finite number|finite]] [[dimension|dimensional]] left [[representation]] of the [[Weyl group]] of $H$ in $G$, write

$$
  \array{
    G Orbits
    &
      \overset{
        I_H(V)
      }{\longrightarrow}
    &
    \mathbb{Q}VectorSpaces
    \\
    G/K
    &\mapsto&
    W_G(H) Representations
    \Big(
      \mathbb{Q}
      \big[
        G Orbits
        (
          G/K, G/H
        )
      \big]
      \,,\,
      V^\ast
    \Big)
  }
$$

for the [[functor]] from the $G$-[[orbit category]] to [[rational vector spaces]] which assigns to a [[coset space]] $G/K$ the vector space of [[homomorphisms]] of right [[actions]] by the [[Weyl group]] (eq:WeylGroup) from the [[hom-set]] $G Orbits\big(G/K, G/H \big)$ to the [[dual vector space]] equipped with its [[dual linear map|dual action]].

More generally, for

$$
  V^\ast \;\in\; W_G(H) Representations_{r,\mathbb{Q}}
$$

set

$$
  \array{
    G Orbits
    &
      \overset{
        I_H(V^\ast)
      }{\longrightarrow}
    &
    \mathbb{Q}VectorSpaces
    \\
    G/K
    &\mapsto&
    W_G(H) Representations
    \Big(
      \mathbb{Q}
      \big[
        G Orbits
        (
          G/K, G/H
        )
      \big]
      \,,\,
      V^\ast
    \Big)
  }
$$

This construction extends to a [[functor]] [[right adjoint]] to the restriction (eq:RestrictionFromOrbitCategoryRepresentations):

$$
  W_G(H) Representations_{\mathbb{Q}}
  \underoverset{
    \underset{
      \;\;\;
        I_H
      \;\;\;
    }{
      \longrightarrow
    }
  }{
    \overset{
       \;\;\;
       (-)^\ast
       \,\circ\,
       i_H^\ast
       \;\;\;
    }{\longleftarrow}
  }
  {\bot}
  DualVector G Spaces
$$

=--

([Triantafillou 82, (4.1)](#Triantafillou82), [Golasinski 97a, Lemma 1.1](#Golasinski97a), [Scull 08, Def. 2.2, Lemma 2.3](#Scull08))

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

$\,$

**(i)**
The objects of the form $I_H(V^\ast)$ (Def. \ref{InjectiveAtomsOfGEquivariantDualVectorSpaces}) are [[injective objects]] in dual [[vector G-spaces]] (Def. \ref{FiniteDimensionalRationalVectorGSpaces}).

**(ii)** Every [[injective]] dual vector $G$-space is a [[direct sum]] of objects of this form, specifically (see Def. \ref{InjectiveEnvelope} below):

$$
  \underline{V}
  \;\in\;
  DualVector G Spaces
  \;\;\;
  \text{is injective}
  \;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;
  \underline{V}
  \;\simeq\;
  \underset{
   \big[
     H \subsetneqq G
   \big]_{conj}
  }{\bigoplus}
  \,
  I_H
  \left(
    \underset{
      K \supset H
    }{\bigcap}
    ker
    \big(
      \underline{V}(G/H)
        \overset{
          \underline{V}(G/H \to G/K)
        }{\longrightarrow}
      \underline{V}(G/K)
    \big)
  \right)
$$

=--

([Triantafillou 82, Section 3 and p. 10](#Triantafillou82), [Scull 08, Lemma 2.4, Prop. 2.5](#Scull08))

+-- {: .num_example #EquivariantPLDeRhamComplex}
###### Example
**(equivariant PL de Rham complex  in injective dual vector $G$-space)**

Let $S \in G SimplicialSets$ a [[simplicial set]] equipped with $G$-[[action]], say that the _[[equivariant PL de Rham complex]]_ is the [[functor]] on the [[orbit category]]

$$
  \array{
   G Orbits
    &
    \overset{
      \Omega^\bullet_{PLdR}
      \big(
        Maps(-,X)^G
      \big)
    }{\longrightarrow}
    &
    dgcAlgebras
    \\
    G/H
    &\mapsto&
    \Omega^\bullet_{PLdR}
    \big(
      X^H
    \big)
  }
$$

which to a [[coset space]] $G/H$ assigns the [[PL de Rham complex]] of the $H$-[[fixed locus]] $X^H \subset X$.

Then the underlying dual [[vector G-space]]

$$
  \array{
   G Orbits
    &
    \overset{
      \Omega^\bullet_{PLdR}
      \big(
        Maps(-,X)^G
      \big)
    }{\longrightarrow}
    &
    dgcAlgebras
    &\overset{}{\longrightarrow}&
    VectorSpaces_{\mathbb{Q}}
  }
$$

is an [[injective object]] (degreewise, in fact).

=--

([Triantafillou 82, Prop. 4.3](#Triantafillou82))

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

Any [[equivariant PL de Rham complex]] (Def. \ref{EquivariantPLDeRhamComplex}) is a [[fibrant object]] in the [[model structure on equivariant connective dgc-algebras]].

=--

(also [Scull 08, Lemma 5.2](#Scull08))

+-- {: .num_defn #InjectiveEnvelope}
###### Definition
**([[injective envelope]] of [[dual vector G-spaces]])**

For $\underline{V} \in DualVector G Spaces$ (eq:DualVectorGSpaces),
its _[[injective envelope]]_ is

$$
  \underset{
   \big[
     H \subset G
   \big]_{conj}
  }{\bigoplus}
  \,
  I_H
  \left(
    \underset{
      K \supsetneqq H
    }{\bigcap}
    ker
    \big(
      \underline{V}(G/H)
        \overset{
          \underline{V}(G/H \to G/K)
        }{\longrightarrow}
      \underline{V}(G/K)
    \big)
  \right)
  \,,
$$

where

1. the [[direct sum]] is over [[conjugacy classes]] of [[subgroups]], with $H \subset G$ on the right any one representative of its conjugacy class,

1. for $H = G$ the argument of $I_H$ is taken to be all of $\underline{V}(G/G)$,

1. $I_H(-)$ is the injective atom construction from Def. \ref{InjectiveAtomsOfGEquivariantDualVectorSpaces}.

=--

([Triantafillou 82, p. 10](#Triantafillou82), [Scull 01, Prop. 7.34](#Scull01),  [Scull 08, Def. 2.6](#Scull08))

+-- {: .num_defn #TensorProductOfInjectiveDualVectorSpaces}
###### Lemma
**(tensor product of injective dual vector $G$-spaces)**

The object-wise [[tensor product]] of two finite-dimensional [[injective object|injective]] [[dual vector G-spaces]] (Def. \ref{FiniteDimensionalRationalVectorGSpaces}) is again injective.

=--

This is proven as [Golasinski 97b, Lemma 3.6](#Golasinski97b) (use [Golasinski 97b, Remark 1.2](#Golasinski97b) to see that the Lemma does apply to the ordinary tensor product of finite-dimensional vector spaces).

Beware that incorrect versions of this statement had been circulating; for discussion of the literature see [Golasinski 97b, p. 3](#Golasinski97b) and [Scull 01, Prop. 7.36](#Scull01)

## Examples

### Over $G = \mathbb{Z}_2$
  {#ExamplesOverCyclicGroupOfOrder2}

+-- {: .num_example #OrbitCategoryOfCyclicGroupOfOrder2}
###### Example
**([[orbit category]] of [[cyclic group of order 2|Z/2Z]])**

For equivariance group the [[cyclic group of order 2]]:

$$
  G \;\coloneqq\;  \mathbb{Z}_2 \;\coloneqq\; \mathbb{Z}/2\mathbb{Z}
  \,.
$$

the [[orbit category]] looks like this:

\[
  \label{OrbitCategoryOfZMod2}
  \mathbb{Z}_2
  Orbits
  \;=\;
  \left\{
  \array{
    \mathbb{Z}_2/1
    &
      \overset{
        \phantom{AAAAA}
      }{
        \longrightarrow
      }
    &
    \mathbb{Z}_2/\mathbb{Z}_2
    \\
    Aut = \mathbb{Z}_2
    &&
    Aut = 1
  }
  \right\}
\]

i.e.:

$$
  \begin{aligned}
    \mathbb{Z}_2 Orbits
    \big(
      \mathbb{Z}_2/\mathbb{Z}_2
      \,,\,
      \mathbb{Z}_2/\mathbb{Z}_2
    \big)
    \;\simeq\;
    1
    \\
    \mathbb{Z}_2 Orbits
    \big(
      \mathbb{Z}_2/1
      \,,\,
      \mathbb{Z}_2/\mathbb{Z}_2
    \big)
    \;\simeq\;
    \ast
    \\
    \mathbb{Z}_2 Orbits
    \big(
      \mathbb{Z}_2/\mathbb{Z}_2
      \,,\,
      \mathbb{Z}_2/1
    \big)
    \;\simeq\;
    \varnothing
    \\
    \mathbb{Z}_2 Orbits
    \big(
      \mathbb{Z}_2/1
      \,,\,
      \mathbb{Z}_2/1
    \big)
    \;\simeq\;
    \mathbb{Z}_2
  \end{aligned}
$$

=--

Write

$$
  \mathbf{1}, \mathbf{1}_{sgn}
  \;\in\;
  \mathbb{Z}_2 Representations
$$

for the two [[irreducible representations]] (the [[trivial representation]] and the [[sign representation]], respectively) of the [[Weyl group]] $W_{\mathbb{Z}_2}(1) = \mathbb{Z}_2$.

Their induced injective dual vector $\mathbb{Z}_2$-spaces, according to Def. \ref{InjectiveAtomsOfGEquivariantDualVectorSpaces}, are:

$$
  I_1(\mathbf{1})
  \;\;
  \colon
  \;\;
  \;\;\;\;\;
  \array{
    \mathbb{Z}_2/1
    &\mapsto&
    \mathbb{Z}_2 Reps
    \Big(
      \underset{
        \simeq \, \mathbf{1} \oplus \mathbf{1}_{sgn}
      }{
      \underbrace{
        \mathbb{Q}
        \big[
          \mathbb{Z}_2 Orbits( \mathbb{Z}_2/1, \mathbb{Z}_2/1 )
        \big]
      }
      }
      \,,\,
      \mathbf{1}
    \Big)
    &
    \simeq
    &
    \mathbf{1}
    \\
    \big\downarrow
    &&
    \\
    \mathbb{Z}_2/\mathbb{Z}_2
    &\mapsto&
    \mathbb{Z}_2 Reps
    \Big(
      \underset{
        \simeq \, 0
      }{
      \underbrace{
        \mathbb{Q}
        \big[
          \mathbb{Z}_2 Orbits( \mathbb{Z}_2/\mathbb{Z}_2, \mathbb{Z}_2/1 )
        \big]
      }
      }
      \,,\,
      \mathbf{1}
    \Big)
    & \simeq
    &
    0
  }
$$

and

$$
  I_1(\mathbf{1}_{sgn})
  \;\;
  \colon
  \;\;
  \;\;\;\;\;
  \array{
    \mathbb{Z}_2/1
    &\mapsto&
    \mathbb{Z}_2 Reps
    \Big(
      \underset{
        \simeq \, \mathbf{1} \oplus \mathbf{1}_{sgn}
      }{
      \underbrace{
        \mathbb{Q}
        \big[
          \mathbb{Z}_2 Orbits( \mathbb{Z}_2/1, \mathbb{Z}_2/1 )
        \big]
      }
      }
      \,,\,
      \mathbf{1}_{sgn}
    \Big)
    &
    \simeq
    &
    \mathbf{1}_{sgn}
    \\
    \big\downarrow
    &&
    \\
    \mathbb{Z}_2/\mathbb{Z}_2
    &\mapsto&
    \mathbb{Z}_2 Reps
    \Big(
      \underset{
        \simeq \, 0
      }{
      \underbrace{
        \mathbb{Q}
        \big[
          \mathbb{Z}_2 Orbits( \mathbb{Z}_2/\mathbb{Z}_2, \mathbb{Z}_2/1 )
        \big]
      }
      }
      \,,\,
      \mathbf{1}_{sgn}
    \Big)
    & \simeq
    &
    0
  }
$$

Similarly, write

$$
  \mathbf{1} \;\in\; 1 Representations
$$

for the unique [[irrep]] of the [[Weyl group]] $W_{\mathbb{Z}_2}(\mathbb{Z}_2) = 1$.

Its induced injective dual vector $\mathbb{Z}_2$-spaces, according to Def. \ref{InjectiveAtomsOfGEquivariantDualVectorSpaces}, is:

$$
  I_{\mathbb{Z}_2}(\mathbf{1})
  \;\;
  \colon
  \;\;
  \;\;\;\;\;
  \array{
    \mathbb{Z}_2/1
    &\mapsto&
    1 Reps
    \Big(
      \underset{
        \simeq \, \mathbf{1}
      }{
      \underbrace{
        \mathbb{Q}
        \big[
          \mathbb{Z}_2 Orbits( \mathbb{Z}_2/1, \mathbb{Z}_2/\mathbb{Z}_2 )
        \big]
      }
      }
      \,,\,
      \mathbf{1}
    \Big)
    &
    \simeq
    &
    \mathbf{1}
    \\
    \big\downarrow
    &&
    &&
    \big\downarrow{}^{\mathrlap{\mathrm{id}}}
    \\
    \mathbb{Z}_2/\mathbb{Z}_2
    &\mapsto&
    1 Reps
    \Big(
      \underset{
        \simeq \, \mathbf{1}
      }{
      \underbrace{
        \mathbb{Q}
        \big[
          \mathbb{Z}_2 Orbits( \mathbb{Z}_2/\mathbb{Z}_2, \mathbb{Z}_2/\mathbb{Z}_2 )
        \big]
      }
      }
      \,,\,
      \mathbf{1}
    \Big)
    & \simeq
    &
    \mathbf{1}
  }
$$

## Related concepts

* [[equivariant chain complex]]

* [[equivariant dgc-algebra]], [[model structure on equivariant dgc-algebras]]

* [[equivariant rational homotopy theory]]

## References

* {#Triantafillou82} [[Georgia Triantafillou]], _Equivariant minimal models_, Trans. Amer. Math. Soc. **274** (1982) 509-532  [[jstor:1999119](http://www.jstor.org/stable/1999119)]

* {#Golasinski97a} [[Marek Golasiński]], _Componentwise injective models of functors to DGAs_, Colloquium Mathematicum, Vol. 73, No. 1 (1997)  ([dml:21048](https://eudml.org/doc/210480), [[GolasinskiInjectiveModels.pdf:file]])

* {#Golasinski97b} [[Marek Golasiński]], _Injective models of $G$-disconnected simplicial sets_,  Annales de l'Institut Fourier, Volume 47 (1997) no. 5, p. 1491-1522 ([numdam:AIF_1997__47_5_1491_0](http://www.numdam.org/item/?id=AIF_1997__47_5_1491_0))

* {#Scull01} [[Laura Scull]], _Rational $S^1$-equivariant homotopy theory_, Transactions of the AMS, Volume 354, Number 1, Pages 1-45 2001 ([pdf](http://www.ams.org/journals/tran/2002-354-01/S0002-9947-01-02790-8/S0002-9947-01-02790-8.pdf), [doi:10.1090/S0002-9947-01-02790-8](https://doi.org/10.1090/S0002-9947-01-02790-8))

* {#Scull08} [[Laura Scull]], _A model category structure for equivariant algebraic models_, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 ([doi:10.1090/S0002-9947-07-04421-2](https://doi.org/10.1090/S0002-9947-07-04421-2))

[[!redirects vector G-spaces]]

[[!redirects dual vector G-space]]
[[!redirects dual vector G-spaces]]

[[!redirects equivariant vector space]]
[[!redirects equivariant vector spaces]]


vector meson

+-- {: .rightHandSide}
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _vector meson_ is a [[meson]] which (in [[Wigner classification]]) is in the [[vector representation]] or [[pseudo-vector representation]] of the [[Lorentz group]].

This is in contrast to [[scalar mesons]].

## Examples

Examples of [[vector mesons]]:

* [[omega meson]]

* [[rho meson]]

* [[K* meson]]

Examples of [[pseudo-vector mesons]]:

* [[a1-meson]]

* [[f1-meson]]

\begin{imagefromfile}
    "file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": 0,
        "right": 10,
        "bottom": 0,
        "left": 20
    }
\end{imagefromfile}

## Properties

* [[hidden local symmetry]]

* [[vector meson dominance]]

## Related concepts

* [[scalar meson]], [[tensor meson]]

## References

(See also the references at _[[meson]]_.)

* F. Nichitiu, _An Introduction to the vector meson_, Swansea Hadron Spect. 1995:0219-240 ([spire:405666](https://inspirehep.net/literature/405666))

* Eef van Beveren, George Rupp, _Scalar and axial-vector mesons_, Eur. Phys. J. A31:468-473, 2007 ([arXiv:hep-ph/0610199](https://arxiv.org/abs/hep-ph/0610199))

See also:

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

[[!redirects vector mesons]]

[[!redirects pseudovector meson]]
[[!redirects pseudovector mesons]]
[[!redirects pseudo-vector meson]]
[[!redirects pseudo-vector mesons]]

[[!redirects vector meson field]]
[[!redirects vector meson fields]]


vector meson dominance

+-- {: .rightHandSide}
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#### Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The phenomenon or principle called _vector meson dominance_ (VMD) is a tight relation between [[quantum hadrodynamics]] and [[quantum electrodynamics]], where the electrically neutral [[light meson|light]] [[vector meson]] [[field (physics)|fields]] $V_\mu$ (the neutral [[rho-meson]] $\rho^0_\mu$, the [[omega-meson]] $\omega_\mu$ and the [[phi-meson]]) are seen to be on par with, or even identified with, the [[electromagnetism|electromagnetic]] [[hadronic current]] $J^{hadr}$, at least to some approximation.

### Phenomenology
 {#Phenomenology}

In [[particle physics]] [[phenomenology]], _vector meson dominance_ is the observation that the [[interaction]] of [[hadrons]] with [[photons]] is dominated by [[interactions]] that procceed via excange of [[vector mesons]].

Specifically, the [[decay]] of [[electron]]/[[positron]]-[[pairs]] into [[hadrons]]

$$
  e^+ + e^- \to H
$$

(reverse to the [[purely leptonic decay]] $H \to e^+ e^-$ of [[hadrons]]) is dominated by the [[light meson|light]] [[vector mesons]], in that the [[scattering cross section]] $\sigma(e^+ e^- \to H)$ is peaked at those [[energies]] corresponding to the [[rest masses]] of the vector mesons:

\begin{imagefromfile}
  "file_name": "VectorMesonDominanceInDileptonCrossSectionI.jpg",
  "width": 420,
  "caption": "[Piller-Weise 90, Fig. 1](#PillerWeise90)",
  "float": "right",
  "margin": {
    "top": 0,
    "right": 10,
    "bottom": 10,
    "left": 0,
    "unit": "px"
  }
\end{imagefromfile}

The first graphics shows the measured cross section (in [[physical unit|units]] of that of the decay $e^+ e^- \to \mu^+ \mu_-$ of [[electron]] [[dileptons]] into [[muon]] [[dileptons]]): The [[light vector mesons]] [[rho-meson|ρ]]$^0$, [[omega-meson|ω]] and [[phi-meson|ϕ]] correspond to the the spikes below 1 [[GeV]], the further spikes correspond to [[heavy vector mesons]], with [[charmonium]] $J/\psi$ around 4 [[GeV]].

\begin{imagefromfile}
  "file_name": "VectorMesonDominanceInDileptonCrossSectionII.jpg",
  "width": 420,
  "caption": "[Piller-Weise 90, Fig. 2](#PillerWeise90)",
  "float": "left",
  "margin": {
    "top": 0,
    "right": 10,
    "bottom": 10,
    "left": 0,
    "unit": "px"
  }
\end{imagefromfile}

The second graphics shows the same data, now zoomed into the region of the [[light vector mesons]]. One sees clearly that in this region the [[graph of a function|graph]] of the [[scattering cross section]] is completely dominated first of all by the broad [[rho-meson|ρ]]$^0$ peak at a [[mass]] of $\sim 770$ [[MeV]] (see also [Murphy-Yount 71, Fig. 4](#MurphyYount71)), accompanied just by two sharp spikes, corresponding to the [[omega-meson|ω]] at $\sim 783$ [[MeV]] and the [[phi-meson|ϕ]] at $\sim 1020$ [[MeV]] (see also [Schildknecht 72, Table 1](#Schildknecht72)).

### Theory

In the [[effective field theory]] of [[quantum hadrodynamics]], _vector meson dominance_ refers to a proposal for how to formulate the [[theory (physics)|theory]] guided by the above [[phenomenology]]:

Here the dominance is encoded by the _field-current identity_  ([Gell-Mann & Zachariasen 61](#GellMannZachariasen61), [Kroll, Lee & Zumino 76, (1.3)](#KrollLeeZumino76), [Sakurai 69, p. 54 onwards](#Sakurai69), review in [Piller-Weise 90, (4)](#PillerWeise90)):

$$
  J^{hadr}_\mu
  \;\sim\;
  V_\mu
  \,,
$$

essentially identifying the electromagnetic [[hadron current]] with the joint neutral [[light meson|light]] [[vector meson]] [[field (physics)|field]].

This implies in particular that all [[coupling constants]] of [[interactions]] with an [[omega-meson]]/neutral [[rho-meson]] are proportional, by the same factor, to the corresponding electromagnetic coupling (reviewed in [Schildknecht 05, p. 3](#Schildknecht05)).

In terms of a [[Lagrangian density]], this is encoded by meson/photon _mixed terms_ of the following form ([Kroll, Lee & Zumino 76, (2.7)](#KrollLeeZumino76),[Sakurai 69, p. 67](#Sakurai69)):

\[
  \label{VMD1Lagrangian}
  \mathbf{L}_{VMD1}
  \;\sim\;
  d V \wedge \star_4 d A
  \;+\;
  V \wedge \star_4 J^{hadr}
\]

obtained from the Lagrangian density $\mathbf{L}_{EM} \;\sim\; d A \wedge \star_{4} d A + A \wedge \star_4 J_{hadr}$ of [[Maxwell theory]] by exchanging a [[photon]] field variable $A$ with a [[vector meson]] [[field (physics)|field]] $V$ (reviewed in [OCPTW 95, p. 10](#OCPTW95), [Schildknecht 05, p. 4](#Schildknecht05)).

There is also an alternative, supposedly equivalent, Lagrangian formulation, less elegant but now in more widespread use. To distinguish the two one speaks of "VMD1" for the formulation (eq:VMD1Lagrangian) and of "VMD2" for the alternative formulation (reviewed in [OCPTW 95. (26)](#OCPTW95), [Schildknecht 05, (8)](#Schildknecht05)).

## Properties

### Derivation from holography

From [OCPTW 95](#OCPTW95):

> No direct translation between the [[standard model of particle physics|Standard Model]] and [[vector meson dominance|VMD]] has yet been made.

\linebreak

From [Rho et al. 16](AdS-QCD#RhoEtAl16):

> One can make $[$[[chiral perturbation theory]]$]$ consistent with [[QCD]] by suitably matching the [[correlators]] of the [[effective field theory|effective theory]] to those of [[QCD]] at a [[scale]] near $\Lambda$. Clearly this procedure is not limited to only one set of [[vector mesons]]; in fact, one can readily generalize it to an infinite number of hidden [[gauge fields]] in an [[effective field theory|effective]] [[Lagrangian density|Lagrangian]]. In so doing, it turns out that a fifth dimension is "deconstructed" in a (4+1)-dimensional (or 5D) [[Yang-Mills theory|Yang–Mills]] type form. We will see in Part III that such a structure arises, [[top-down model building|top-down]], in [[string theory]].

> $[...]$

> $[$this [[holographic QCD]]$]$ model comes out to describe — unexpectedly well — low-energy properties of both [[mesons]] and [[baryons]], in particular those properties reliably described in quenched [[lattice QCD]] simulations.

> $[...]$

> One of the most noticeable results of this [[AdS/QCD|holographic model]] is the first derivation of vector dominance (VD) that holds both for [[mesons]] and for [[baryons]]. It has been
somewhat of an oddity and a puzzle that [[Jun John Sakurai|Sakurai's]] [[vector meson dominance|vector dominance]] — with the lowest [[vector mesons]] [[rho meson|ρ]] and [[omega meson|ω]] — which held very well for [[pion|pionic]] [[form factors]] at low momentum transfers famously failed for [[nucleon]] [[form factors]]. In this [[AdS/QCD|holographic model]], the [[vector meson dominance|VD]] comes out automatically for both the [[pion]] and the [[nucleon]] provided that the infinite $[$[[Kaluza-Klein mechanism|KK-]]$]$tower is included. While the [[vector meson dominance|VD]] for the [[pion]] with the infinite tower is not surprising given the successful Sakurai VD, that the [[vector meson dominance|VD]] holds also for the [[nucleons]] is highly nontrivial. $[...]$ It turns out to be a consequence of a [[AdS/QCD|holographic]] [[Cheshire cat principle|Cheshire Cat phenomenon]]

## Related concepts

* [[vector meson]]

* [[form factor]]

* [[hadron current]]

[[!include effective field theories of nuclear physics -- contents]]

## References

### General

The original articles:

* {#GellMannZachariasen61} [[Murray Gell-Mann]], Fredrik Zachariasen, _Form Factors and Vector Mesons_, Phys. Rev. 124, 953 (1961) ([doi:10.1103/PhysRev.124.953](https://doi.org/10.1103/PhysRev.124.953))

* {#KrollLeeZumino76} Norman M. Kroll, T. D. Lee, [[Bruno Zumino]], _Neutral Vector Mesons and the Hadronic Electromagnetic Current_, Phys. Rev. 157, 1376 (1967) ([arXiv:10.1103/PhysRev.157.1376](https://doi.org/10.1103/PhysRev.157.1376))

* {#Sakurai69} [[Jun John Sakurai]], Chapter III of: _Currents and Mesons_, Chicago Lectures in Physics, based on notes by George Barry, University of Chicago Press (1969) ([ISBN: 9780226733838](https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3622598.html))

Review:

* {#MurphyYount71} Fred V. Murphy, David E. Yount, _Photons as hadrons_, Sci.Am. 225N1 (1971) 1, 94-104 ([spire:41546](https://inspirehep.net/literature/41546), [doi:10.1038/scientificamerican0771-94](https://doi.org/10.1038/scientificamerican0771-94))

* {#Schildknecht72} [[Dieter Schildknecht]], _Vector meson dominance, photo- and electroproduction from nucleons_,  in: _Photon-Hadron Interactions II_ Springer Tracts in Modern Physics, vol 63. Springer 1972 ([doi:10.1007/BFb0041507](https://doi.org/10.1007/BFb0041507))

* {#PillerWeise90} G. Piller, [[Wolfram Weise]], _Vector meson dominance: Selected topics_ 1990 ([spire310958](https://inspirehep.net/literature/310958), [[PillerWeiseVMD.pdf:file]])

* {#OCPTW95} H. B. O'Connell, B. C. Pearce, A. W. Thomas, A. G. Williams, _Rho-omega mixing, vector meson dominance and the pion form-factor_, Prog. Part. Nucl. Phys. 39:201-252, 1997 ([arXiv:hep-ph/9501251](https://arxiv.org/abs/hep-ph/9501251))

* {#Schildknecht05} [[Dieter Schildknecht]], _Vector Meson Dominance_, Acta Phys. Polon. B37:595-608, 2006 ([arXiv:hep-ph/0511090](https://arxiv.org/abs/hep-ph/0511090))

See also

* Wikipedia, _[Vector meson dominance](https://en.wikipedia.org/wiki/Vector_meson_dominance)_

More:

* H. B. O'Connell, A. G. Williams, M. Bracco, G. Krein, _Vector Meson Mixing and Charge Symmetry Violation_, Phys. Lett. B370:12-16, 1996 ([arXiv:hep-ph/9510425](https://arxiv.org/abs/hep-ph/9510425))

* A. G. Williams, _New results in vector meson dominance and rho meson physics_, Proceedings of the APCTP Workshop on Astro-Hadron Physics, "Properties of Hadrons in Matter", Seoul, 25-31 October, 1997 ([arXiv:hep-ph/9712405](https://arxiv.org/abs/hep-ph/9712405))

* {#BDDL09a}  M. Benayoun, P. David, L. DelBuono, O. Leitner, _A Global Treatment Of VMD Physics Up To The $\phi$: I. $e^+ e^-$ Annihilations, Anomalies And Vector Meson Partial Widths_, Eur. Phys. J. C65:211-245, 2010 ([arXiv:0907.4047](https://arxiv.org/abs/0907.4047))

* {#BDDL09a} M. Benayoun, P. David, L. DelBuono, O. Leitner, _A Global Treatment Of VMD Physics Up To The $\phi$: II. $\tau$ Decay and Hadronic Contributions To $g-2$_, Eur. Phys. J. C68:355-379, 2010 ([arXiv:0907.5603](https://arxiv.org/abs/0907.5603))

* Avner Karasik, _Vector dominance, one flavored baryons, and QCD domain walls from the "hidden" Wess-Zumino term_ ([arXiv:2010.10544](https://arxiv.org/abs/2010.10544))

### Via holographic QCD
 {#ReferencesViaHolographicQCD}

Derivation of vector meson dominance via [[holographic QCD]]:

* D.T. Son, M.A. Stephanov, _QCD and dimensional deconstruction_, Phys. Rev. D69 (2004) 065020 ([arXiv:hep-ph/0304182](https://arxiv.org/abs/hep-ph/0304182))

  (from the point of view of [[hidden local symmetry]])

* Sungho Hong, Sukjin Yoon, [[Matthew Strassler]], _On the Couplings of Vector Mesons in AdS/QCD_, JHEP 0604 (2006) 003 ([arXiv:hep-th/0409118](https://arxiv.org/abs/hep-th/0409118))

* Sungho Hong, Sukjin Yoon, [[Matthew Strassler]], _On the Couplings of the Rho Meson in AdS/QCD_ ([cds:816440](https://cds.cern.ch/record/816440), [arXiv:hep-ph/0501197](https://arxiv.org/abs/hep-ph/0501197))

* Leandro Da Rold, Alex Pomarol, _Chiral symmetry breaking from five dimensional spaces_, Nucl. Phys. B721:79-97, 2005 ([arXiv:hep-ph/0501218](https://arxiv.org/abs/hep-ph/0501218))

and specifically in the [[Witten-Sakai-Sugimoto model]]:

* {#SakaiSugimoto05} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], p. 18 and Section 5 of: _More on a holographic dual of QCD_, Progr. Theor. Phys. 114: 1083-1118, 2005 ([arXiv:hep-th/0507073](https://arxiv.org/abs/hep-th/0507073))

[[!redirects current-field identity]]
[[!redirects current-field identities]]

[[!redirects field-current identity]]
[[!redirects field-current identities]]

[[!redirects VMD]]
[[!redirects VMD1]]
[[!redirects VMD2]]


vector representation

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#### Spin geometry
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=--
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#Contents#
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{:toc}

## Definition

Let $G \coloneqq Spin(V) \overset{\pi}{\to} SO(V)$ be a [[spin group]] [[group extension|extension]] of a [[special orthogonal group]], or more generally a [[Pin group]]-extension of an [[orthogonal group]] (or [[Lorentz group]], ...).

Then a [[spin representation]] of $Spin(V)$ is called the _vector representation_ if it comes via $\pi$ from the defining [[linear representation]] of $SO(V)$ on the [[vector space]] underlying the given [[inner product space]] $V$.

## Related concepts

* [[real spin representation]]

* [[pseudovector representation]]

* [[vector meson]]

[[!redirects vector representations]]


vector space

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# Contents
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{:toc}

## Definition

For $k$ a [[field]] or a [[division ring]], a __vector space__ over $k$
(or a $k$-vector space) is a [[module]] over the [[ring]] $k$.  When the vector space is fixed, its elements are called _vectors_, the field $k$ is referred to as the base field of the ground field of the vector space, and the elements of $k$ are called _scalars_.

Sometimes a vector space over $k$ is called a __$k$-linear space__.  (Compare '$k$-[[linear map]]'.) If $k$ is only a division ring then we
carefully distinguish the left $k$-vector spaces and right $k$-vector spaces.

The [[category]] of vector spaces is typically denoted [[Vect]], or
$Vect_k$ if we wish to make the field $k$ (the *ground field*) explicit. So

$$
  Vect_k \coloneqq k Mod
  \,.
$$

This category has vector spaces over $k$ as objects, and $k$-linear maps between these as morphisms.

### Multisorted notion

Alternatively, one sometimes defines "vector space" as a two-sorted notion; taking the field $k$ as one of the sorts and a module over $k$ as the other. More generally, the notion of "module" can also be considered as two-sorted, involving a ring and a module over that ring.

This is occasionally convenient; for example, one may define the notion of [[topological vector space]] or topological module as an [[internalization]] in $Top$ of the multisorted notion. This procedure is entirely straightforward for topological modules, as the notion of module can be given by a two-sorted Lawvere theory $T$, whence a topological module (for instance) is just a product-preserving functor $T \to Top$. One may then define a topological vector space as a topological module whose underlying (discretized) ring sort is a field.

## Properties

Every [[free object|free]] vector space admits a [[basis of a vector space|basis]].

The _[[basis theorem]]_, which is equivalent to the [[axiom of choice]], states that every vector space is a free vector space.

## Related concepts

* **vector space**, [[dual vector space]],

  * [[finite-dimensional vector space]]

  * [[real vector space]], [[complex vector space]]

  * [[topological vector space]], [[convenient vector space]]

  * [[virtual vector space]]

* [[tensor product of vector spaces]]

* [[real structure]], [[complex structure]], [[quaternionic structure]]

* [[vector bundle]],

* [[lattice in a vector space]]

* [[vector G-space]]

* [[2-vector space]], [[n-vector space]]

* [[inner product space]]

* [[linear operator]], [[matrix]], [[determinant]], [[eigenvalue]],  [[eigenvector]]

## References

The concept of vector spaces seem to have been first introduced in:

* [[Giuseppe Peano]], *Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva*, Fratelli Bocca Editori, Torino, 1888, pp. XI, 171 $[$[Mathematica Italiana opere 138](http://mathematica.sns.it/opere/138), [pdf](http://mathematica.sns.it/media/volumi/138/Calcolo%20geometrico%20secondo%20l'Ausdehnungslhere%20di%20H.%20Grassmann_bw.pdf)$]$

  > (referring to [[Hermann Grassmann]]'s *[[Ausdehnungslehre]]*)

An early expository account is in §14 of

* [[Bartel Leendert van der Waerden]], _[[Moderne Algebra]]_, Volume I, Springer, 1930.

The literature on vector spaces is now extremely large, including lots of elementary linear algebra textbooks.

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

Classics include:

* [[Michael Artin]], Algebra
* [[Israel M. Gelfand]], Lectures on linear algebra
* P. R. Halmos, Finite dimensional vector spaces
* [[M M Postnikov]], Lectures on geometry, semester 2: Linear algebra

Affine spaces are sets which are torsors over the abelian group of vectors of a vector space. Thus vector spaces may serve as a basis for the affine and for the Euclidean geometry. This approach has been invented by [[Hermann Weyl]] in 1918. Dieudonné wrote an influential book on such an approach to 2d and 3d Euclidian geometry, in which the basics of vector spaces in low dimension is introduced along the way (the book is intended for high school teachers):

* Jean Alexandre Dieudonné, Linear algebra and geometry

Discussion of vector space in [[univalent foundations of mathematics]] ([[homotopy type theory]]) is announced to appear in:

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: Chapter 8 of: *[[Symmetry]]* (2021) $[$[pdf](https://unimath.github.io/SymmetryBook/)]

[[!redirects vector space]]
[[!redirects vector spaces]]
[[!redirects linear space]]
[[!redirects linear spaces]]

[[!redirects Grassmannian]]


vectorial bundle

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* table of contents
{:toc}

## Idea

A _vectorial bundle_ ([Gomi 08](#Gomi08)) is a $\mathbb{Z}_2$-[[graded vector space|graded]] [[vector bundle]] $E$ of [[finite number|finite]] [[rank of a vector bundle|rank]], equipped with an odd [[endomorphism]] $h \;\colon\; E \to E$. The [[homomorphisms]] of vectorial bundles are such that the [[endomorphism]] $h$ acts like canceling parts of the even and odd degree of $E$ against each other.

This way vectorial bundles lend themselves to the description of [[topological K-theory]]. In particular, they allow a geometric model for [[twisted K-theory]].

## Definition

For $X$ a [[topological space]], the [[category]] $VectrBund(X)$ of **vectorial bundles** on $X$ has

* as [[objects]] $(E \stackrel{h}{\to} E)$ [[finite number|finite]] [[rank of a vector bundle|rank]] Hermitean $\mathbb{Z}_2$-graded [[vector bundle | vector bundles]] $E\to X$ equipped with a self-adjoint [[endomorphism]] $h$ of odd degree. In [[matrix calculus]]

  $$
    E = \left(
      \array{
        E_{0}
        \\
        E_{1}
      }
    \right)
  $$

  $$
     h =
     \left(
       \array{
        0 & h_{10}
        \\
        h_{01} & 0
      }
     \right)
  $$

* as [[morphisms]] $\phi : (E,h) \to (E',h)$ [[equivalence classes]] of morphisms $\phi \colon E \to E'$ of vector bundles such that

  $$
    \array{
      E &\stackrel{\phi}{\longrightarrow}& E
      \\
      {}^{\mathllap{h}}\big\downarrow && \big\downarrow^{\mathrlap{h'}}
      \\
      E &\stackrel{\phi}{\longrightarrow}& E
    }
    \,,
  $$

  where two such maps are regarded as equivalent, $\phi \sim \phi'$,  already if they coincide on  the kernel of $h^2_x$ for each point $x$.

In particular, we have the following two important special cases:

* the case that $h = 0$ -- in this case all eigenvalues of all $h_x^2$ are zero. and hence maps $\phi, \phi' : (E,0) \to (E',0)$ represent the same morphism precisely if they are actually equal as morphisms $\phi, \phi' : E \to E'$ of vector bundles.

  (Notice that there is only the 0-morphism $(E,0) \to (E',h')$ for $h' \neq 0$.)

  This yields a canonical inclusion

  $$
    VectBund(X) \hookrightarrow VectrBund(X)
  $$

  by sending $E \mapsto (E \stackrel{0}{\to} E)$.

* the case that $E = \left( \array{V \\ V}\right)$
  and $h = \left(
       \array{
         0 & Id \\
         Id & 0
       }
    \right)
  $

  Here $E_x|_{\lt \mu \lt 1} = 0$ and hence two morphisms
  $\phi, \phi' : (E,h) \to (E',h')$ are identified already if they agree on the 0-vector. In other words, _all_ morphisms out of such $(E,h)$ are identified. In particular they are all equal to the 0-morphism to $(0,0)$. Therefore the bundles of this form represent the 0-element.

**Definition**

Say two vectorial bundles $(E,h)$, $(E',h')$ on $X$ are **[[concordance|concordant]]** if there is a vectorial bundle on $X \times [0,1]$ which restricts to them at either end, respectively.

Let
$
  (E,h)^{\vee}
  =
$  be the degree-reversed bundle to $(E,h)$.

**Lemma**

There is a concordance

$$
  E \oplus E^\vee \to 0 .
$$

## References

The definition of vectorial bundles is due to Furuta. It is recalled and applied to the study of [[K-theory]] and [[twisted K-theory]] in

* {#Gomi08} [[Kiyonori Gomi]], _Twisted K-theory and finite-dimensional approximation_ ([arXiv:0803.2327](http://arxiv.org/abs/0803.2327))

[[!redirects vectorial bundles]]


Veeravalli Varadarajan

* [webpage](http://www.math.ucla.edu/~vsv/)

## selected writings

* {#Varadarajan04} [[Veeravalli Varadarajan]], _[[Supersymmetry for mathematicians]]: An introduction_, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I 2004

  (on [[supersymmetry]])

* [[Riccardo D'Auria]], [[Sergio Ferrara]], M. A. Lledó, [[Veeravalli Varadarajan]], _Spinor Algebras_, J.Geom.Phys. 40 (2001) 101-128 ([arXiv:hep-th/0010124](http://arxiv.org/abs/hep-th/0010124))

  (on [[spin groups]] and [[spin representations]])

* {#Varadarajan01} [[Veeravalli Varadarajan]], _Spin(7)-subgroups of SO(8) and Spin(8)_, Expositiones Mathematicae, 19 (2001): 163-177 (doi:10.1016/S0723-0869(01)80027-X, [pdf](https://core.ac.uk/download/pdf/81114499.pdf))

  (on [[Spin(7)]]-[[subgroups]] of [[Spin(8)]])

## related $n$Lab entries

* [[spin representation]]

* [[Spin(7)]]

* [[supersymmetry]], [[geometry of physics -- supersymmetry]]

* [[conformal compactification]]

category: people


Velayudhan Parameswaran Nair

* [spire page](https://inspirehep.net/authors/996257)

## Selected writings

Precursor to [[twistor string theory]]:

* [[Velayudhan Parameswaran Nair]], *A current algebra for some gauge theory amplitudes*, Physics Letters B Volume 214, Issue 2, 17 November 1988, Pages 215-218 (doi:10.1016/0370-2693(88)91471-2, [spire:24212](https://inspirehep.net/literature/24212))

category: people

[[!redirects V. Parameswaran Nair]]
[[!redirects V. P. Nair]]


Velo-Zwanziger problem

+-- {: .rightHandSide}
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#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Velo-Zwanziger problem_ is is a phenomenon in theoretical [[physics]]: the "evident" [[action functionals]] for [[field (physics)|fields]] which

1. have [[spin]] higher than 1

1. are in an irreducible [[mass|massive]] representation of the Lorentz group

1. have [[minimal coupling]] to [[electromagnetism]]

tend to have [[equations of motion]] which are not [[hyperbolic differential operator|hyperbolic]] and hence may violate the usual [[causal locality|causality]] relation of [[general relativity]].

## References

The original references are

* G. Velo, D. Zwanziger, Phys. Rev. 186 (1969) 1337,

* G. Velo, D. Zwanziger, Phys. Rev. 188 (1969) 2218;

* G. Velo, Nucl. Phys. B 43 (1972) 389

* M. Hortacsu, Phys. Rev. D 9, 928 (1974).

Adjustments of the [[action functional]] for [[spin]] [[mass|massive]] $3/2$-particles ([[Rarita-Schwinger field]]) by non-[[minimal coupling]] that evades the Velo-Zwanziger problem is discussed in

* [[Massimo Porrati]], Rakibur Rahman, _Causal Propagation of a Charged Spin 3/2 Field in an External Electromagnetic Background_,  	Phys. Rev. D80:025009, 2009 ([arXiv:0906.1432](http://arxiv.org/abs/0906.1432))

Discussion for the higher [[string]] excitations that appear in [[string theory]] is for instance in

* [[Massimo Porrati]], Rakibur Rahman, [[Augusto Sagnotti]], _String Theory and The Velo-Zwanziger Problem_, Nucl. Phys. B846:250-282, 2011 ([arXiv:1011.6411](http://arxiv.org/abs/1011.6411))


velocity

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# Contents
* table of contents
{: toc}

## Definition

Given a [[Cartesian space]] $V$ and a [[smooth curve]] $r:\mathbb{R} \to V$, the **velocity** of $r$ is given by the [[derivative]] of $r$: $v \coloneqq \partial_t r$.

## Related concepts

* [[curve]]

* [[acceleration]]

* [[jerk]]

In Euclidean spaces:

* [[speed]]

* [[rapidity]]

* [[areal velocity]]

* [[torque]]

## References

See the references at *[[differential geometry of curves and surfaces]]*.

See also

+ Wikipedia, _[Velocity](https://en.wikipedia.org/wiki/Velocity)_


Veneziano amplitude

+-- {: .rightHandSide}
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#### Quantum field theory
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[[!include functorial quantum field theory - contents]]
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=--

#Contents#
* table of contents
{:toc}

## Idea

The _Veneziano amplitude_ is the tree-level [[string scattering amplitude]] for the scattering of four [[open string|open]] [[bosonic strings]] in their [[tachyon]] [[quantum state|states]]. Hence equivalently it is a certain [[n-point function|4-point function]] in a certain [[boundary field theory|boundary]] [[2d CFT]].

Historically, this amplitude first arose in [[dual resonanace models]] and the realization that it has a natural interpretation as the scattering of [[strings]] was one of the main original motivations for the development of [[string theory]].

## Related concepts

* [[Virasoro amplitude]]

* In [[p-adic string theory]] (see there) one considers [[adelic integral]]-variants of the Veneziano amplitude.

* [[vacuum amplitude]]

## References

* [[Michael Green]], [[John Schwarz]], [[Edward Witten]], section 1.1 of _Superstring theory_ Volume 1, Cambridge Monographs on Mathematical Physics, 1987

* [[Gerard 't Hooft]], section 1.2 and section 6 of _Introduction to string theory_ 2004 ([pdf](http://www.phys.uu.nl/~thooft/lectures/stringnotes.pdf))

* _The Veneziano amplitude_ [pdf](http://www.thphys.uni-heidelberg.de/~palti/Stringcourse/problemset8.pdf)

Discussion in the context of [[Riemann hypothesis and physics]] is in

* {#HJM15} [[Yang-Hui He]], Vishnu Jejjala, Djordje Minic, _From Veneziano to Riemann: A String Theory Statement of the Riemann Hypothesis_ ([arXiv:1501.01975](http://arxiv.org/abs/1501.01975))

Textbook review includes (16.49) in

* [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], _String Scattering Amplitudes and Low Energy Effective Field Theory_, chapter 16 in _Basic Concepts of String Theory_ Part of the series Theoretical and Mathematical Physics pp 585-639 Springer 2013

See also:

* Clifford Cheung, Grant N. Remmen, *Veneziano Variations: How Unique are String Amplitudes?* [[arXiv:2210.12163](https://arxiv.org/abs/2210.12163)]

* Nicholas Geiser, Lukas W. Lindwasser, *Generalized Veneziano and Virasoro amplitudes* [[arXiv:2210.14920](https://arxiv.org/abs/2210.14920)]

[[!redirects Veneziano amplitudes]]


Venkata S. R. Redrouthu

* [institute page](https://www.asifequbal.com/team/venkata-subbarao-redrouthu)

* [GoogleScholar page](https://scholar.google.com/citations?user=vFG-qJYAAAAJ)

category: people

[[!redirects Venkata Redrouthu]]


Venkatesa Chandrasekaran

* [InSpire page](https://inspirehep.net/authors/1812047)

## Selected writings

On [[von Neumann algebra factors]] [[algebra of observables|of observables]] for [[quantum field theory on curved spacetime|quantum field theory on]] [[de Sitter spacetime]]:

* [[Venkatesa Chandrasekaran]], [[Roberto Longo]], [[Geoff Penington]], [[Edward Witten]], *An Algebra of Observables for de Sitter Space*, Journal of High Energy Physics **2023**  82 (2023) [[arXiv:2206.10780](https://arxiv.org/abs/2206.10780), doi:10.1007/JHEP02(2023)082]

category: people


Vera Serganova

* [personal page](https://math.berkeley.edu/people/faculty/vera-serganova)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Vera_Serganova)

## Selected writings

On [[Lie groups]] and [[quantum groups]]:

* [[Richard Borcherds]], [[Mark Haiman]], [[Theo Johnson-Freyd]], [[Nicolai Reshetikhin]], [[Vera Serganova]], *Berkeley Lectures on Lie Groups and Quantum Groups*, 2020 ([pdf](http://categorified.net/LieQuantumGroups.pdf))

category: people