transgression in group cohomology

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###Context###
#### Group Theory
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[[!include group theory - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

For $G$ a [[discrete group]] (often taken to be a [[finite group]]) and for $A$ any ([[discrete group|discrete]]) [[abelian group]], there is a [[transgression]] [[homomorphism]] of [[cohomology groups]]

\[
  \label{TransgressionMapOnGroupCohomology}
  \begin{array}{rcl}
  H_{grp}^{\bullet + 1}(G;\,A)
  \;\;=\;\;
  \\
  H^{\bullet + 1}(B G;\, A)
  &
  \xrightarrow{
    \;\;\;\;
    \tau
    \;\;\;\;
  }
  &
  H^n
  \big(
    \Lambda B G,
    \,
    A
  \big)
  \;\simeq
  \\
  &&
  \underset{
    \mathclap{
      \array{
      \phantom{-}
      \\
      [g] \in G^{ad}/G
      }
    }
  }{\oplus}
  \;
  H^n_{grp}\big(C_g;\, A \big)
  \end{array}
\]

from the [[group cohomology]] of $G$ to the [[groupoid cohomology]], in one degree lower, of the [[inertia groupoid]]  $\Lambda B G$ of its [[delooping groupoid]] $B G \simeq (G \rightrightarrows \ast)$.

Since the [[inertia groupoid]] of the [[delooping groupoid]] is [[equivalence of groupoids|equivalent]] to a [[disjoint union]] over [[conjugacy classes]] $[g] \in G^{ad}/G$ of [[delooping groupoids]] of [[centralizer subgroups]] $C_g$, with $C_e = G$,

\[
  \label{DecompositionOfTheInertiaGroupoid}
  \Lambda B G
  \;\simeq\;
  \underset{ [g] \in G_{ad}/G }{\coprod}
  B C_g
  \;\;\;\simeq\;\;\;
  B G \,\sqcup\,
  \underset{ [g] \neq [e] }{\coprod}
  B C_g
  \,,
\]

this induces a [[corestriction|corestricted]] transgression map within the [[group cohomology]] of $G$, and, more generally, to the [[group cohomology]] of any of its [[centralizer subgroups]]:

$$
  H_{grp}^{\bullet + 1}(G;\,A)
  \xrightarrow{
    \;\;\;
    \tau_{[e]}
    \;\;\;
  }
  H_{grp}^{\bullet}(G;\,A)
  \,,
  \;\;\;\;\;\;\;
  H_{grp}^{\bullet + 1}(G,A)
  \xrightarrow{
    \;\;\;
    \tau_{[g]}
    \;\;\;
  }
  H_{grp}^{\bullet}(C_g;\,A)
  \,.
$$

It is a [[folklore]] [[theorem]] that the transgression (eq:TransgressionMapOnGroupCohomology) maps an $(n+1)$-cocycle $c \colon G^{\times_{n+1}} \to A$ to the alternating [[sum]] (of [[functions]] with values in the [[abelian group]] $A$ with group operation denoted "$+$")

\[
  \label{SumFormulaForTransgressedCocycle}
  \begin{array}{l}
  \tau(c)
  \big(
    \gamma, g_{n-1}, \cdots, g_1, g_0
  \big)
  \\
  \;=\;
  \pm
  \underset{
    0 \leq j \leq n
  }{\sum}
  (-1)^j
  \cdot
  c\big(
    g_{n-1},
     \cdots,\,
    g_{n-j},\,
    Ad_j(\gamma),\,
    g_{n-j-1},\,
    \cdots,\,
    g_0
  \big)
  \,,
  \end{array}
\]

where

$$
  \gamma
  \xrightarrow{ g_{n-1} }
  Ad_{1}(\gamma)
  \xrightarrow{ g_{n-2} }
  \cdots
  \xrightarrow{\; g_0 \;}
  Ad_{n}(\gamma)
  \;\;\;\;\;
  \in
  \;\;
  \Lambda B G
$$

is any sequence of $n$ composable [[morphisms]] in the [[inertia groupoid]], and where we use a shorthand for the [[adjoint action]] of $G$ on itself:

$$
  \begin{aligned}
    Ad_{j}(\gamma)
    & \;\coloneqq\;
    Ad_{(g_{n-1}\cdots g_{n-j})}(\gamma)
    \\
    & \;\coloneqq\;
    (g_{n-1} \cdots g_{n-j})^{-1}
      \cdot
    \gamma \cdot (g_{n-1} \cdots g_{n-j})
    \,,
  \end{aligned}
$$

(which [[restriction|restricts]] to $Ad_j(\gamma) = \gamma$ upon [[corestriction]] to the connected component on the right of (eq:DecompositionOfTheInertiaGroupoid) that is indexed by $[\gamma]$).

In historically influential examples, for the case $n = 4$ and $A = \mathbb{Z}$ or, equivalently, $n = 3$ and $A = $ [[U(1)]], this formula (eq:SumFormulaForTransgressedCocycle):

* implicitly underlies [[Dijkgraaf & Witten 1990, p. 14](#DijkgraafWitten90)] the discussion of [[Dijkgraaf-Witten theory]];

* explains [[Willerton 2008](#Willerton08)] the nature of the "[[twisted Drinfeld double]]" of the [[group algebra]] of $G$;

* governs [[Dove 2019, Sec. 6.4](#Dove19)] the expression of 4-[[twisted cohomology|twisted]] [[equivariant elliptic cohomology]] at the [[Tate curve]] in terms of 3-[[twisted equivariant K-theory|twisted equivariant]] [[Tate K-theory]].

Below we mean to spell out a general abstract definition (Def. \ref{Transgression}) of the transgression map (eq:TransgressionMapOnGroupCohomology) and a [full proof](#ProofOfTheComponentFormula) of its component formula (eq:SumFormulaForTransgressedCocycle), amplifying that its form is a direct consequence of -- besides some basic [[homotopy theory]]/[[homological algebra]] which we review [below](#BackgroundAndLemmata)  -- the classical [[Eilenberg-Zilber theorem]], i.e. the [[Eilenberg-Zilber/Alexander-Whitney deformation retraction]] (which was partially re-discovered in [Willerton 2008, Sec. 1](#Willerton08) under the name "Parmesan theorem").

\linebreak

## Background and Lemmata
 {#BackgroundAndLemmata}

### Homotopy and homological algebra

Some relevant basics of [[homotopy theory]] in relation to [[homological algebra]]:

\begin{definition}\label{HomotopyCategories}
**([[homotopy categories]])**

We write:

* $Ho(sSet)$ for the [[classical homotopy category]], realized as the [[homotopy category of a model category|homotopy category of]] the [[classical model structure on simplicial sets]];

* $Ho(sAb)$ for the [[homotopy category of a model category|homotopy category of]] the [[model structure on simplicial abelian groups]];

* $Ho(Ch_+)$ for the [[homotopy category of a model category|homotopy category of]] the [[model structure on connective chain complexes]].

For $A \in $ [[Ab]], and $n \in \mathbb{N}$ we write

$$
  A[n]
    \,\in\,
  Ch_+
    \xrightarrow{Loc_{\mathrm{W}}}
  Ho(Ch_+)
$$

for the [[chain complex]] concentrated on $A$ in degree $n$.

We denote (using the same symbols for [[derived functors]] as for the original [[functors]]):

* the [[derived adjunction]] of the ([[free construction|free]] [[simplicial abelian group]] $\dashv$ [[underlying]] [[simplicial set]])-[[Quillen adjunction]] ([here](model+structure+on+simplicial+abelian+groups#eq:QuillenAdjunctionWithClassicalModelStructureOnSimplicialSets)) by

  \[
    \label{DerivedFreeSimplicialAbelianGroupAdjunction}
    Ho(sAb)
      \underoverset
       {\underset{undrlg}{\longrightarrow}}
       {\overset{\mathbb{Z}(-)}{\longleftarrow}}
       {\;\;\;\;\;\;\bot\;\;\;\;\;\;}
    Ho(sSet)
  \]

* the [[derived adjunction|derived]] [[adjoint equivalence]]  of the [[Dold-Kan correspondence|Dold-Kan]] [[Quillen equivalence]] ([here](Dold-Kan+correspondence#ModelCatVersion)) by

  \[
    \label{DerivedDoldKanAdjunction}
    Ho(Ch_+)
      \underoverset
        {\underset{DK}{\longrightarrow}}
        {\overset{N_\bullet}{\longleftarrow}}
        {\;\;\;\;\;\;\bot_{\mathrlap{\simeq}}\;\;\;\;\;\;}
    Ho(sAb)
  \]

* the [[derived adjunction|derived]] [[internal-hom]]-[[Quillen adjunction]] ([here](monoidal+model+category#InternalHomQuillenAdjunction)), for any $X \in $ [[SimplicialSets]], by

  \[
    \label{DerivedInternalHomAdjunction}
    Ho(sSet)
      \underoverset
        {\underset{[X,-]}{\longrightarrow}}
        {\overset{X \times (-)}{\longleftarrow}}
        {\;\;\;\;\;\;\bot\;\;\;\;\;\;}
    Ho(sSet)
    \,,
  \]

  (whose underived [[right adjoint]] is the [[simplicial mapping complex]]-construction);

* the [[derived adjunction|derived]] [[internal-hom]]-[[Quillen adjunction]] ([here](monoidal+model+category#InternalHomQuillenAdjunction)) for $\mathbb{Z}[1] \,\in\,$ [[ConnectiveChainComplexes]]:

  \[
    \label{DerivedDegreeShiftAdjunction}
    Ho(Ch_+)
      \underoverset
        {\underset{ V_\bullet \mapsto V_{\bullet + 1} }{\longrightarrow}}
        {\overset{ V_\bullet \mapsto V_{\bullet - 1} }{\longleftarrow}}
        {\;\;\;\;\;\;\bot\;\;\;\;\;\;}
    Ho(Ch_+)
  \]

\end{definition}

### Simplicial classifying spaces

\begin{remark}\label{DeloopingGroupoidAndSimplicialClassifyingSpaceOfFiniteGroup}
**([[delooping groupoid]] and [[simplicial classifying space]] of [[finite group]])** \linebreak
The [[nerve]] of the [[delooping groupoid]] of a [[discrete group]] $G$ is [[isomorphism|isomorphic]] to the [[simplicial classifying space]] of $G$ (see [this Example](simplicial+classifying+space#SimplicialClassifyingSpaceOfAnOrdinaryGroup)):
$$
  N
  \big(
    G \rightrightarrows \ast
  \big)
  \;\simeq\;
  \overline{W} G
  \;\;\;
  \in
  \;
  sSet
  \,.
$$
For notational brevity we will be referring to $\overline{W}G$ in the following, but it may be helpful to keep thinking of this isomorphically as the [[nerve]] of the [[delooping groupoid]], $\overline{W} G \,=\, N (G \rightrightarrows \ast)$.
From this perspective, an [[n-simplex]] in $\overline{W}G$, which is an [[n-tuple]] of [[group]] [[elements]], is suggestively denoted as a sequence of [[morphisms]]:

$$
  \big(\overline{W}G\big)_{n}
  \;\;
    =
  \;\;
  \Big\{
    \bullet
    \xrightarrow{g_{n-1}}
    \bullet
    \xrightarrow{\phantom{--}}
    \cdots
    \bullet
    \xrightarrow{\phantom{--}}
    \bullet
    \xrightarrow{\; g_1 \;}
    \bullet
    \xrightarrow{\; g_0 \;}
    \bullet
    \;\big\vert\;
    g_i \in G
  \Big\}
  \,.
$$

\end{remark}

We denote the [[image]] of $\overline{W}G$ in the [[classical homotopy category]] by:

$$
  B G
  \;=\;
  \mathbf{B} G
  \;=\;
  Loc_{\mathrm{W}}
  \big(
    \overline{W}G
  \big)
  \;\;\;
  \in
  \;
  Ho(sSet)
$$

(where the first equality reflects that $G$ is assumed to be [[discrete group|discrete]]).

### Group(oid) cohomology

Some relevant basics of [[cohomology]], for the cases of [[ordinary cohomology]] and [[group cohomology]]:

\begin{definition}\label{EilenbergMacLaneSpaces}
**([[Eilenberg-MacLane spaces]])** \linebreak
  For $A \,\in\,$ [[Ab]], and $n \in \mathbb{N}$, we write
  $$
    B^n A \,=\, K(A,n)
    \,\coloneqq\,
    underlg \circ DK(A[n])
    \;\;\;
    \in
    \;
    sAb
      \xrightarrow{ Loc_{\mathrm{W}}}
    Ho(sAb)
     \xrightarrow{undrlg}
    Ho(sSet)
  $$
  for (the [[homotopy type]] of) the [[Eilenberg-MacLane space]] with $A$ in degree $n$ -- here constructed as the [[underlying]] [[simplicial set]] of the [[simplicial abelian group]] which is the image under the [[Dold-Kan construction]] of the [[chain complex]] that is concentrated on $A$ in degree $n$.
\end{definition}

\begin{definition}\label{OrdinaryCohomology}
**([[ordinary cohomology]])** \linebreak
  For $X \in Ho(sSet)$,  $A \,\in\, Ab$  and $n \in \mathbb{N}$,
  the degree-$n$ *[[ordinary cohomology]]* of $X$ with [[coefficients]] in $A$ is the following [[hom-set]] in the [[classical homotopy category]]:
  $$
    H^n(X;\, A)
    \;=\;
    Ho(sSet)
    \big(
      X, \, B^n A
    \big)
    \,,
  $$
  where on the right we have the [[Eilenberg-MacLane space]] from Def. \ref{EilenbergMacLaneSpaces}.
\end{definition}

The following is an immediate re-casting of the traditional definition of [[group cohomology]]:

\begin{definition}\label{GroupCohomology}
**([[group cohomology]])** \linebreak
 For $G \,\in\, $ [[Groups]] and $A \,\in\,$ [[AbelianGroups]],
the [[group cohomology]] of $G$ with [[coefficients]] in $A$ is, in degree $n \in \mathbb{N}$, the [[hom-group]]

$$
  H^n_{grp}(G;\,A)
  \;=\;
  Ho(Ch_+)
  \big(
    N_\bullet
     \circ
    \mathbb{Z}(\overline{W}G),
    \,
    A[n]
  \big)
  \,.
$$

\end{definition}

\begin{proposition}
\label{GroupCohomologyIsOrdinaryCohomologyOfClassifyingSpace}
**([[group cohomology]] is [[ordinary cohomology]] of [[classifying space]])** \linebreak
 For $G \,\in\, $ [[Groups]] and $A \,\in\,$ [[AbelianGroups]],
 the [[group cohomology]] (Def. \ref{GroupCohomology}) of $G$ with coefficient in $A$ is [[natural isomorphism|naturally isomorphic]] to the [[ordinary cohomology]] of the [[simplicial classifying space]] of $G$ with [[coefficients]] in $B^n A$:

$$
  H^n_{grp}(G;\, A)
  \;\simeq\;
  H^n(B G;\, A)
  \,.
$$

\end{proposition}

\begin{proof}

By the [hom-isomorphisms](adjoint+functor#InTermsOfHomIsomorphism) of the [above](#HomotopyCategories) [[derived adjunctions]]:

$$
  \begin{aligned}
    H^n_{grp}(G;\, A)
    & \;0\;
    Ho(Ch_+)
    \big(
      N_\bullet \circ \mathbb{Z}(\overline{W}G),
      \,
      A[n]
    \big)
    \\
    & \;\simeq\;
    Ho(sAb)
    \big(
      \mathbb{Z}(\overline{W}G)
      ,\,
      DK(A[n])
    \big)
    \\
    & \;\simeq\;
    Ho(sSet)
    \big(
      \overline{W}G
      ,\,
      undrlng \circ DK(A[n])
    \big)
    \\
    & \;=\;
    Ho(sSet)
    \big(
      B G
      ,\,
      B^n A
    \big)
    \\
    & \;=\;
    H^n(B G;\, A)
    \,.
  \end{aligned}
$$

\end{proof}

This Prop. \ref{GroupCohomologyIsOrdinaryCohomologyOfClassifyingSpace}, in view of Rem. \ref{DeloopingGroupoidAndSimplicialClassifyingSpaceOfFiniteGroup}  justifies the following definition:

\begin{definition}\label{GroupoidCohomology}
**([[groupoid cohomology]])** \linebreak
For $\mathcal{G} \,=\, (\mathcal{G}_1 \rightrightarrows \mathcal{G}_0) \,\in\,$ [[Groupoids]] and $A \,\in\,$ [[Ab]], $n \,\in\, \mathbb{N}$, the degree-$n$ *[[groupoid cohomology]]* of $\mathcal{G}$ with [[coefficients]] in $A$ is the [[ordinary cohomology]] (Def. \ref{OrdinaryCohomology}) of the [[homotopy type]] of the [[nerve]] of $\mathcal{G}$, regarded in the [[classical homotopy category]]:

$$
  H^n
  \big(
    \mathcal{G};\,
    A
  \big)
  \;\coloneqq\;
  Ho(sSet)
  \big(
    N(\mathcal{G}),
    \,
    B^n A
  \big)
  \,.
$$
\end{definition}

### Products of simplices

Some basic facts about [[products of simplicial sets]]:

\begin{proposition}\label{NonDegenerateSimplicesInProductOfSimplices}
**(non-degenerate $(p+q)$-simplices in $\Delta[p] \times \Delta[q]$)**
\linebreak
For $p,q \,\in\, \mathbb{N}$
the non-degenerate simplices in the [[Cartesian product]] (Prop. \ref{CartesianProductOfSimplicialSetsIsComponentwise})

$$
  \Delta[p] \times \Delta[q]
$$

of standard [[simplices]] in [[sSet]] correspond, under the [[Yoneda lemma]], to precisely those morphisms of [[simplicial sets]]

\[
  \label{GenericSimplexInProductOfSimplices}
  \Delta[p+q]
  \xrightarrow{\;\; \sigma \;\;}
  \Delta[p] \times \Delta[q]
\]

which satisfy the following equivalent conditions:

* as [[morphisms]] of [[posets]], they are [[strictly monotone]];

* as [[permutations]] of $(p+q)$ [[elements]] they are $(p,q)$-[[shuffles]];

* as [[morphisms]] of [[finitely generated object|finitely generated]] [[categories]] they take generating morphisms to generating morphisms;

\end{proposition}

Such morphisms may hence be represented by [[paths]]:

* on a $(p+1)\times(q+1)$-lattice,

* from one corner to its opposite corner,

* consisting of $p+q$ unit steps,

* each either horizontally or vertially:

\begin{tikzcd}
  [
    row sep=large
  ]
    {(0,0)}
    \ar[r]
    \ar[d]
    \ar[
      r,-,
      color=blue,
      opacity=.25,
      line width=6pt
    ]
    &
    %
    {(1,0)}
    \ar[r]
    \ar[d]
    \ar[
      d,-,
      color=blue,
      opacity=.25,
      line width=6pt
    ]
    &
    {(2,0)}
    \ar[rr,dotted]
    \ar[d]
    &
    &
    {(p,0)}
    \ar[d]
    \\
    {(0,1)}
    \ar[r]
    \ar[d]
    &
    {(1,1)}
    \ar[r]
    \ar[d]
    \ar[
      d,-,
      color=blue,
      opacity=.25,
      line width=6pt
    ]
    &
    {(2,1)}
    \ar[rr, dotted]
    \ar[d]
    &
    \cdots
    &
    {(p,1)}
    \ar[d]
    \\
    {(0,2)}
    \ar[r]
    \ar[dd, dotted]
    &
    {(1,2)}
    \ar[r]
    \ar[dd, dotted]
    \ar[
      r,-,
      color=blue,
      opacity=.25,
      line width=6pt
    ]
    &
    {(2,2)}
    \ar[rr, dotted]
    \ar[dd, dotted]
    \ar[ddrr, dotted]
    \ar[
      ddrr,-,
      color=blue,
      opacity=.25,
      line width=6pt,
      dotted
    ]
    &
    &
    {(p,2)}
    \ar[dd, dotted]
    \\
    &
    &
    &&
    \\
    {(0,q)}
    \ar[r]
    &
    {(1,q)}
    \ar[r]
    &
    {(2,q)}
    \ar[rr, dotted]
    &
    &
    {(p,q)}
\end{tikzcd}

\begin{proof}
From [this Prop.](product+of+simplices#CartesianProductOfSimplicialSetsIsComponentwise)
it is clear (see [this Remark](product+of+simplices#DegenerateSimplicesInProductOfSimplicialSets)) that a
simplex $\sigma \,\colon\, \Delta[p+q] \xrightarrow{\;} \Delta[p] \times \Delta[q]$  is degenerate
precisely if, when regarded as a path as above, it contains a [[constant function|constant]] step, i.e. one which moves neither horizontally nor vertically. But then --- by degree reasons, since we are looking at paths of $p + q$ steps in a lattice of side length $p$ and $q$ --- it must be that the path proceeds by $p + q$ unit steps.
\end{proof}

### Nerve of the inertia groupoid

Some basic facts about the [[nerve]] of an [[inertia groupoid]]:

\begin{proposition}
\label{InertiaGroupoidOfDeloopingGroupoidIsAdjointActionGroupoid}
**([[inertia groupoid]] of [[delooping groupoid]] is [[adjoint action|adjoint]] [[action groupoid]])** \linebreak
The inertia groupoid $\Lambda \mathbf{B} G$ is [[isomorphism|isomorphic]] to the [[action groupoid]] of the [[adjoint action]] of $G$ on itself:
$$
  \Lambda \mathbf{B}G
  \;\simeq\;
  G_{ad} \sslash G
  \;=\;
  \left(
    G \times G
    \underoverset
      {Ad_{(-)}(-)}
      {pr_2}
      {\rightrightarrows}
      G
  \right)
$$
\end{proposition}
This follows by immediate inspection. For more discussion see at *[[free loop space of a classifying space]]* the section *[Examples -- For finite groups](free+loop+space+of+classifying+space#ExampleDiscreteGroups)*.

\begin{remark}
The [[groupoid convolution algebra]] of the [[inertia groupoid]] of the [[delooping groupoid]] $\mathbf{B}G$ is the [[Drinfeld double]] (see there for more) of the [[group algebra]] of $G$.
\end{remark}

\begin{definition}\label{MinimalSimplicialCircle}
**([[minimal simplicial circle]])** \linebreak
  Write
  $$
    S
    \;\coloneqq\;
    \Delta[1]/\partial\Delta[1]
    \;\;\;
    \in
    \;
    sSet
  $$
  for the [[simplicial set]] with exactly two non-degenerate cells,

* one of which in degree 0, which we denote by $[0] = [1]$,

* and one in degree 1, which we denote by $\big[ [0], [1] \big]$.

\end{definition}

\begin{proposition}\label{NormalizedChainComplexOfMinimalSimplicialCircle}
**([[normalized chain complex]] of [[minimal simplicial circle]])**
\linebreak
  The [[normalized chain complex]] of the [[free simplicial abelian group]] of the [[minimal simplicial circle]] $S$ (Def. \ref{MinimalSimplicialCircle}) has the group of [[integers]] in degrees 0 and 1, and all [[differentials]] are [[zero morphism|zero]]:
$$
  N_\bullet \circ \mathbb{Z}(S)
  \;\simeq\;
  \left[
  \array{
    \vdots
    \\
    \big\downarrow
    \\
    0
    \\
    \big\downarrow
    \\
    \mathbb{Z}
    \\
\big\downarrow\mathrlap{ \scriptsize{0} }
    \\
    \mathbb{Z}
  }
  \;\;
  \right]
  \;\simeq\;
  \mathbb{Z}[1] \,\oplus\, \mathbb{Z}
  \,.
$$
\end{proposition}

The following proposition holds on general grounds (duscussed at *[[free loop space of classifying space]]*), but the explicit component-based proof we give now is necessary, further below, in order to understand the [[transgression]]-formula for [[cocycles]] in the [[group cohomology]] of $G$ to cocycles on the inertia groupoid.
\begin{proposition}\label{RelationToSimplicialHomComplexIntoClassifyingSpace}
The [[nerve]] of the [[inertia groupoid]] of a [[delooping groupoid]] of a [[finite group]] $G$ is [[isomorphism|isomorphic]] to the [[simplicial hom complex]] out of the [[minimal simplicial circle]] $S$ (Def. \ref{MinimalSimplicialCircle}) into the [[simplicial classifying space]] $\overline{W}G$ (Rem. \ref{DeloopingGroupoidAndSimplicialClassifyingSpaceOfFiniteGroup}):

$$
  N\big( \Lambda \mathbf{B}G\big)_\bullet
  \;\simeq\;
  [S,\overline{W}G]_\bullet
  \,.
$$
\end{proposition}
\begin{proof}
This is a specialization of the general fact that the simplicial nerve repects [[mapping objects]] (see [this Prop.](nerve#NerveOfCategoriesPreservesMappingObjects) at *[[nerve]]*). We spell it out for the present case:

We claim that the isomorphism is given by sending, for each $n \in \mathbb{N}$, any [[n-simplex]] $(\gamma;\, g_{n-1}, \cdots, g_1, g_0)$ of $N\big( Func( \mathbf{B}\mathbb{Z}, \, \mathbf{B}G )\big)$, being a sequence of [[natural transformations]] of the form
$$
  \array{
    \bullet
    &\xrightarrow{g_{n-1}}&
    \bullet
    &\xrightarrow{g_{n-2}}&
    \cdots
    &\xrightarrow{g_{n-j-1}}&
    \bullet
    &\xrightarrow{g_{n-j-2}}&
    \bullet
    &\xrightarrow{g_{n-j-3}}&
    \cdots
    &\xrightarrow{g_{0}}&
    \bullet
    \\
    \big\downarrow
    {}^{\mathrlap{
      \gamma
    }}
    &&
    \big\downarrow
    {}^{\mathrlap{
      g_{n-1}^{-1} \cdot \gamma \cdot g_{n-1}
    }}
    &&
    &&
    \big\downarrow
    &&
    \big\downarrow
    && &&
    \big\downarrow\mathrlap{^{
      ( g_{n-1} \cdots g_{0} )^{-1}
      \cdot
      \gamma
      \cdot
      ( g_{n-1} \cdots g_{0} )
    }}
    \\
    \bullet
    &\xrightarrow{g_{n-1}}&
    \bullet
    &\xrightarrow{g_{n-2}}&
    \cdots
    &\xrightarrow{g_{n-j-1}}&
    \bullet
    &\xrightarrow{g_{n-j-2}}&
    \bullet
    &\xrightarrow{g_{n-j-3}}&
    \cdots
    &\xrightarrow{g_{0}}&
    \bullet
    \mathrlap{\,,}
  }
$$

to the homomorphism of simplicial sets

$$
  \Delta[n] \times S \xrightarrow{\;\;} \overline{W}G
  \,,
$$

which, in turn, sends a non-degenerate $(n+1)$-simplex in $\Delta[n] \times S$ of the form (in the path notation discussed at *[[product of simplices]]*)

$$
  \array{
    (0,[0])
    &\to&
    (1,[0])
    &\to&
    \cdots
    &\to&
    (j,[0])
    \\
    && && && \big\downarrow
    \\
    && && &&
    (j,[1])
    &\to&
    (j+1,[1])
    &\to&
    \cdots
    &\to&
    (n,[1])
  }
$$

to the $n+1$-simplex in $\overline{W}G$ (Rem. \ref{DeloopingGroupoidAndSimplicialClassifyingSpaceOfFiniteGroup}) of the form

$$
  \array{
    \bullet
    &\xrightarrow{g_{n-1}}&
    \bullet
    &\xrightarrow{g_{n-2}}&
    \cdots
    &\xrightarrow{g_{n-j}}&
    \bullet
    \\
    && && &&
    \big\downarrow
    {}^{\mathrlap{
      ( g_{n-1} \cdots g_{n-j} )^{-1}
      \cdot
      \gamma
      \cdot
      ( g_{n-1} \cdots g_{n-j} )
    }}
    \\
    && && &&
    \bullet
    &\xrightarrow{g_{n-j}}&
    \bullet
    &\xrightarrow{g_{n-j-1}}&
    \cdots
    &\xrightarrow{g_{0}}&
    \bullet
  }
$$

\end{proof}

As a consequence:

\begin{proposition}\label{FormOfTheSimplicialEvaluationMap}
  The [[evaluation map]]
  $$
    [S,\overline{W}G] \times S
    \xrightarrow{\;\;}
    \overline{W}G
  $$
  (out of the [[product of simplicial sets|product]] of the [[simplicial hom complex]] out of $S$ with $S$)
  maps non-degenerate $n+1$-simpleces of $[S,\overline{W}G] \times S$ as follows:

\begin{tikzcd}[
    column sep=30pt,
    row sep=4pt
  ]
  S_{n+1}
  \times
  \mathrm{Hom}
  \big(
    S \times \Delta[n+1]
    ,\,
    \overline{W}G
  \big)
  \ar[
    rr,
    "{ \mathrm{ev} }"
  ]
  &&
  \big(
    \overline{W}G
  \big)_{n+1}
  \\
  \Big(
  s_{(1,\cdots, \widehat{k}, \cdots, n)} \ell
  ,\,
  s_k
  \big(
    \gamma
    ;\,
    g_{n-1}
    ,\,
    \cdots
    ,\,
    g_{0}
  \big)
  \Big)
  &\mapsto&
  \big(
    g_{n-1}
    ,\,
    \cdots,
    g_{n-k}
    ,\,
    \mathrm{Ad}_k(\gamma)
    ,\,
    g_{n - (k+1)}
    ,\,
    \cdots
    ,\,
    g_0
  \big)
\end{tikzcd}

\end{proposition}
\begin{proof}
This follows by unwinding the component formula for the [[evaluation map]] on [[simplicial mapping complexes]] ([this Prop.](function+complex#ComponentFormulaForEvaluationMap)):

Recall that we denote by
\[
  \label{CellInNerveOfInertiaGroupoid}
 \big(
  \gamma
  ;\,
  g_{n-1}
  ,\,
  g_{n-2}
  ,\,
  \cdots
  ,\,
  g_{0}
 \big)
 \;\; \in \;\;
 \mathrm{Hom}
 \big(
   S \times \Delta[n]
   ,\,
   \overline{W}G
 \big)
 \;\simeq\;
 \Big(
 N
 \mathrm{Map}
  \big(
   { \mathbf{B}\mathbb{Z} }
   ,\,
   { \mathbf{B}G }
  \big)
 \Big)_{n}
\]
the $n$-cell in the [[nerve]] of the [[inertia groupoid]] that corresponds to the sequence of natural transformation which start at the functor
$$
  \gamma
    \,\in\,
   G
    \,\simeq\,
  \mathrm{Hom}_{\mathrm{Grp}}(\mathbb{Z}, G)
    \,\simeq\,
  \mathrm{Hom}\big(
    \mathbf{B}\mathbb{Z}
    ,\,
    \mathbf{B}G
  \big)
$$
and successively have components $g_{n-\bullet} \in G$.

> Throughout we are writing "$Hom$" for [[hom-sets]] and "$Map$" for [[hom-objects]], i.e. for [[internal homs]] and we keep tacitly going back and forth through the [[bijections]] in (eq:CellInNerveOfInertiaGroupoid).

If we abbreviate (this follows conventions familiar in discussion of *[[transgression in group cohomology]]*)
$$
  \begin{aligned}
    Ad_{k}(\gamma)
    & \;\coloneqq\;
    Ad_{(g_{n-1}\cdots g_{n-k})}(\gamma)
    \\
    & \;\coloneqq\;
    (g_{n-1} \cdots g_{n-k})^{-1}
      \cdot
    \gamma \cdot (g_{n-1} \cdots g_{n-k})
    \,,
  \end{aligned}
$$
then this corresponds to a sequence of composable morphisms in the inertia groupoid of this form:
$$
  \gamma
  \xrightarrow{ g_{n-1} }
  Ad_{1}(\gamma)
  \xrightarrow{ g_{n-2} }
  \cdots
  \xrightarrow{\; g_0 \;}
  Ad_{n}(\gamma)
  \;\;\;\;\;
  \in
  \;\;
  \Lambda B G
$$

The simplicial map (eq:CellInNerveOfInertiaGroupoid) maps the non-degenerate $(n+1)$-cells in the [[product of simplicial sets]] (see the discussion [there](product+of+simplices#NonDegenerateCellsInAProductOfSimplices)) of $S$ (eq:MinimalSimplicialCircle) with the [[simplicial simplex|simplicial $n$-simplex]] as follows:

\begin{imagefromfile}
    "file_name": "ActionOfCellInNerveOfIniertiaGroupoid-221205.jpg",
    "width": "800",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

This is not exactly what we need unwinding the [[evaluation map]], but it is close: the image of (eq:CellInNerveOfInertiaGroupoid) under the $k$th [[degeneracy map]] evidently gives the following mapping:

\begin{imagefromfile}
    "file_name": "ActOfDegOfCellInNerveOfInertiaGrpd-221205b.jpg",
    "width": "850",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

This is the type of mapping that appears in the component formula for the [[evaluation map]] on [[function complexes]] (from [that Prop.](function+complex#ComponentFormulaForEvaluationMap)), and so the claim follows.
\end{proof}

### Transgression in cohomology

\begin{definition}\label{Transgression}
  **([[transgression]])**
  For $\mathcal{A} \,\in\, sAb$ a [[simplicial abelian group]], hence with [[free loop space of classifying space|free loop space of its classifying space]] given
  (via [this Prop.](free+loop+space+of+classifying+space#FreeLoopSpaceOfClassifyingSpaceOfSimplicialAbelianGroup)) by
  \[
    \label{FreeLoopSpaceOfClassifyingSpaceOfAbelianSimplicialGroup}
    \Lambda \mathbf{B}\mathcal{A}
    \;\simeq\;
    \mathcal{A}
    \times
    \mathbf{B}\mathcal{A}
  \]
  we say that *[[transgression]]* in $\mathcal{A}$-cohomology
  is
  $$
    H
    \big(
      -;\,
      \mathbf{B}\mathcal{A}
    \big)
    \xrightarrow{ [S, -] }
    H
    \big(
      \Lambda(-);\,
      \mathcal{A} \times \mathbf{B}\mathcal{A}
    \big)
    \xrightarrow{\;\; pr_2 \;\; }
    H
    \big(
      \Lambda(-);\,
      \mathcal{A}
    \big)
    \,.
  $$
\end{definition}

\begin{proposition}\label{FreeLoopSpaceOfEMSpaceIdentifiedViaEilenbergZilber}
**([[free loop space of classifying space]] identified via [[Eilenberg-Zilber map]])** \linebreak
For $n \in \mathbb{N}_+$ and
for $A \,\in\, Ab$ the [[integers]] or the [[circle group]],
the following [[composition|composite]] of is a [[simplicial weak equivalence]]
\[
  \label{ComparisonMorphismFromFreeLoopSpaceViaEilenbergZilber}
  \begin{aligned}
    &
    \big[
      S,
      \,
      B^n A
    \big]_\bullet
    \\
    & \;=\;
    \big[
      S,
      \,
      undrlng \circ DK(A[n])
    \big]_\bullet
    \\
    &
    \;\simeq\;
    sSet
    \big(
      S \times \Delta[\bullet],
      \,
      undrlng \circ DK(A[n])
    \big)
    \\
    & \;\simeq\;
    Ch_+
    \big(
      N \circ \mathbb{Z}(S \times \Delta[\bullet]),
      \,
      A[n]
    \big)
    \\
    & \;\xrightarrow{EZ_S}\;
    Ch_+
    \big(
      N \circ \mathbb{Z}(S)
      \,\otimes\,
      N \circ \mathbb{Z}(\Delta[\bullet]),
      \,
      A[n]
    \big)
    \\
    & \;\simeq\;
    Ch_+
    \big(
      N \circ \mathbb{Z}(\Delta[\bullet]),
      \,
      [
        N \circ \mathbb{Z}(S),
        \,
        A[n]
      ]
    \big)
    \\
    & \;\simeq\;
    \Big(
    undrlng \circ DK
    \big(
      [
        \underset{
          \mathbb{Z} \oplus \mathbb{Z}[1]
        }{
        \underbrace{
          N \circ \mathbb{Z}(S)
        }
        },
        \,
        A[n]
      ]
    \big)
    \Big)_\bullet
    \\
    & \;\simeq\;
    \Big(
    undrlng \circ DK
    \big(
      A[n] \oplus A[n-1]
    \big)
    \Big)_\bullet
    \;=\;
    B^n A \times B^{n-1} A
  \end{aligned}
\]

\end{proposition}
Here all [[isomorphisms]] are [hom-isomorphisms](adjoint+functor#InTermsOfHomIsomorphism) of the [above](#HomotopyCategories) [[adjoint functor|adjunctions]], the step denoted $EZ_S$ is pre-composition with the [[Eilenberg-Zilber map]], and under the brace we are using Prop. \ref{NormalizedChainComplexOfMinimalSimplicialCircle}.

\begin{proof}
  By the fact that the [[Eilenberg-Zilber map]] has a [[left inverse]] given by the [[Alexander-Whitney map]] $AW$ (see at *[[Eilenberg-Zilber/Alexander-Whitney deformation retraction]]*), the analogous composite with $AW_S$ instead of $EZ_S$ yields a left inverse morphism, which hence [[retraction|retracts]] the [[homotopy groups]] of $B^n A \times B^{n-1}A$ onto those of $\big[S, B^n A \big]$.
But, by (eq:FreeLoopSpaceOfClassifyingSpaceOfAbelianSimplicialGroup), the latter is a product of [[Eilenberg-MacLane spaces]]
with homotopy groups $A$ concentrated in degrees $n$ and $n -1 $. By assumption on $A$ the only retractions of $A$ onto itself is the identity, so that $EZ_S$ must induce the identity morphism of homotopy groups.
\end{proof}

## Proof of the component formula
 {#ProofOfTheComponentFormula}

We prove that the formula (eq:SumFormulaForTransgressedCocycle) indeed expresses transgression in group cohomology.

\begin{proof}

Consider the following sequence of [[natural isomorphisms]] of [[hom-sets]] of the above [[homotopy categories]] (Def. \ref{HomotopyCategories}):

\[
  \label{TransgressionDecomposed}
  \begin{aligned}
    H^n_{grp}
    \big(
      G;\, A
    \big)
    &
    \;=\;
    Ho(sSet)
    \big(
      N_\bullet \circ \mathbb{Z}(\overline{W}G),
      \,
      A[n]
    \big)
    \\
    &
    \;\simeq\;
    Ho(sSet)
    \big(
      \overline{W}G, \, undrlng \circ DK\big( A[n] \big)
    \big)
    \\
    =
    H^n
    \big(
      B G;\, A
    \big)
    &
    \;\overset{[S,-]}{\to}\;
    Ho(sSet)
    \Big(
      [S,\overline{W}G],\, \, \big[S, undrlng \circ DK\big( A[n] \big)\big]
    \Big)
    \\
    &
    \;\simeq\;
    Ho(sSet)
    \Big(
      [S,\overline{W}G] \times S,
      \, undrlng \circ DK\big( A[n] \big)
    \Big)
    \\
    &
    \;\simeq\;
    Ho(sAbGrp)
    \Big(
      \mathbb{Z}\big( [S, \overline{W}G] \times S \big),
      \,
      DK\big(A[n]\big)
    \Big)
    \\
    &
    \;\simeq\;
    Ho(Ch_+)
    \Big(
      N_\bullet \circ \mathbb{Z}\big( [S, \overline{W}G] \times S \big),
      \,
      A[n]
    \Big)
    \\
    &
    \;\overset{EZ}{\simeq}\;
    Ho(Ch_+)
    \Big(
      N_\bullet \circ \mathbb{Z}\big([S, \overline{W}G]\big)
        \otimes
      \underset{
        \mathbb{Z}[1]
        \oplus
        \mathbb{Z}
      }{
        \underbrace{N_\bullet \circ \mathbb{Z}(S)}
      },
      \,
      A[n]
    \Big)
    \\
    &
    \;\simeq\;
    Ho(Ch_+)
    \Big(
      N_\bullet \circ \mathbb{Z}\big([S, \overline{W}G]\big),
      \,
      A[n] \oplus A[n-1]
    \Big)
    \\
    &
    \;\overset{pr_2}{\to}\;
    Ho(Ch_+)
    \Big(
      N_\bullet \circ \mathbb{Z}\big([S, \overline{W}G]\big),
      \,
      A[n-1]
    \Big)
    \\
    & \;=\;
    H^{n-1}
    \big(
      \Lambda B G;
      \,
      A
    \big)
    \,.
  \end{aligned}
\]

Here

* the first three lines recall the identification from Prop. \ref{GroupCohomologyIsOrdinaryCohomologyOfClassifyingSpace};

* the fourth line is the [component function](functor#eq:ComponentFunctionOnHomSets) on [[hom-sets]] of the [[derived functor|derived]] [[internal hom]]/[[simplicial mapping complex]]-[[functor]] out of the [[minimal simplicial circle]] $S$ (Def. \ref{MinimalSimplicialCircle});

* the fifth line is the [hom-isomorphisms](adjoint+functor#InTermsOfHomIsomorphism) of the derived [[free abelian group|free]] [[simplicial abelian group]]-adjunction (eq:DerivedFreeSimplicialAbelianGroupAdjunction);

* the sixth line is the [hom-isomorphisms](adjoint+functor#InTermsOfHomIsomorphism) of the derived [[Dold-Kan equivalence]] (eq:DerivedDoldKanAdjunction);

* the seventh line is pre-composition with the [[Eilenberg-Zilber map]], using that this is a [[quasi-isomorphism]] (and hence an [[isomorphism]] in the [[homotopy category]] [[model structure on connective chain complexes|on chain complexes]]) by the [[Eilenberg-Zilber theorem]];

  under the brace we are using Prop. \ref{NormalizedChainComplexOfMinimalSimplicialCircle};

* the eighth line is the [hom-isomorphisms](adjoint+functor#InTermsOfHomIsomorphism) of the derived degree-shift adjunction (eq:DerivedDegreeShiftAdjunction)

  (observing that the [[tensor product of chain complexes|tensor product]] of [[normalized chain complexes]] of [[free simplicial abelian groups]] is a [[cofibrant object|cofibrant]] representative of the correct [[derived functor]]-image -- using that all simplicial sets are [[cofibrant object|cofibrant]], that $\mathbb{Z}(-)$ and $N_\bullet$ are [[left Quillen functors]], and that $(Ch_+, \otimes)$ is a [[monoidal model category]] ([this Prop.](model+structure+on+chain+complexes#ProjectiveModelStructureOnConnectiveChainComplexesIsMonoidal)), so that the [[tensor product of chain complexes|tensor product]] with $\mathbb{Z} \oplus \mathbb{Z}[1]$ is the correct [[left derived functor]]);

* the ninth line is the [[projection]] onto the second [[direct sum|direct summand]];

* the last line is Def. \ref{GroupoidCohomology} of [[groupoid cohomology]].

[[diagram chase|Chasing]] a group cocycle (Def. \ref{GroupCohomology}) through this sequence, and using Prop. \ref{FormOfTheSimplicialEvaluationMap} when it gets sent through the [[Eilenberg-Zilber map]], manifestly outputs the sum formula (eq:SumFormulaForTransgressedCocycle) to be proven.

Hence it only remains to see that this concrete composite (eq:TransgressionDecomposed) is equal to the abstractly defined transgression map (Def. \ref{Transgression}).

{#VerifyingTheMapIsCorrect} This follows by Prop. \ref{FreeLoopSpaceOfEMSpaceIdentifiedViaEilenbergZilber}. In detail, since:

1. $A[n] \in Ch_+$ is a [[fibrant object]] (like every object in the projective [[model structure on connective chain complexes]]);

1. $[S,\overline{W}G] \in sSet$ is a [[cofibrant objects]] (like every object in the [[classical model structure on simplicial sets]]);

the chain of hom-isomorphisms of [[derived adjunctions]] in (eq:TransgressionDecomposed) is covered (through the [[quotient]] by [[left homotopy]] that defines the [[homotopy category of a model category]] according to [this Lemma](homotopy+category+of+a+model+category#HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory)) by "the same" chain of hom-isomorphisms of plain adjunctions. Using here that every simplicial set $X$ is the [[colimit]] over its [[category of elements|elements]] $\Delta[k] \in el(X)$ ([this Prop.](simplicial+set#SimplicialSetIsColimitOfItsSimplices)) and that the [[1-category]]-theoretic [[hom functors]] turn this colimit, in their first argument, into a [[limit]] ([this Prop.](hom-functor+preserves+limits#HomFunctorPreservesLimits)), we find that the composite in (eq:TransgressionDecomposed) is covered by

$$
  \begin{aligned}
    &
    sSet
    \Big(
      \big[S, \overline{W}G \big],
      \,
      \big[S, undrlng \circ DK(A[n]) \big]
    \Big)
    \\
    & \;\simeq\;
    sSet
    \Big(
      \underset{
        \underset{
          \mathclap{
            {\Delta[k] \in}
            \atop
            {el\big([S,\overline{W}G]\big)}
          }
        }
        {\longrightarrow}
      }{\lim}
      \Delta[k],
      \,
      \big[S, undrlng \circ DK(A[n]) \big]
    \Big)
    \\
    & \;\simeq\;
    \underset{
      \underset{
        \mathclap{
          {\Delta[k] \in}
          \atop
          {el\big([S,\overline{W}G]\big)}
        }
      }
      {\longleftarrow}
    }{\lim}
    \,
    sSet
    \Big(
      \Delta[k],
      \,
      \big[S, undrlng \circ DK(A[n]) \big]
    \Big)
    \\
    & \;\simeq\;
    \underset{
      \underset{
        \mathclap{
          {\Delta[k] \in}
          \atop
          {el\big([S,\overline{W}G]\big)}
        }
      }
      {\longleftarrow}
    }{\lim}
    \,
    Ch_+
    \Big(
      N_\bullet \circ \mathbb{Z}
      \big(
        S \times \Delta[k]
      \big),
      \,
      A[n]
    \Big)
    \\
    & \;\xrightarrow{EZ_S}\;
    \underset{
      \underset{
        \mathclap{
          {\Delta[k] \in}
          \atop
          {el\big([S,\overline{W}G]\big)}
        }
      }
      {\longleftarrow}
    }{\lim}
    \,
    Ch_+
    \Big(
      N_\bullet \circ \mathbb{Z}(S)
      \,\otimes\,
      N_\bullet \circ \mathbb{Z}(\Delta[k]),
      \,
      A[n]
    \Big)
    \\
    & \;\simeq\;
    \underset{
      \underset{
        \mathclap{
          {\Delta[k] \in}
          \atop
          {el\big([S,\overline{W}G]\big)}
        }
      }
      {\longleftarrow}
    }{\lim}
    \,
    sSet
    \Big(
      \Delta[k],
      \,
      undrlng \circ DK
      \big(
        A[n] \oplus A[n-1]
      \big)
    \Big)
    \\
    & \;\simeq\;
    sSet
    \Big(
    \underset{
      \underset{
        \mathclap{
          {\Delta[k] \in}
          \atop
          {el\big([S,\overline{W}G]\big)}
        }
      }
      {\longrightarrow}
    }{\lim}
      \Delta[k],
      \,
      undrlng \circ DK
      \big(
        A[n] \oplus A[n-1]
      \big)
    \Big)
    \\
    & \;\simeq\;
    sSet
    \Big(
      [S,\overline{W}G],
      \,
      undrlng \circ DK
      \big(
        A[n] \oplus A[n-1]
      \big)
    \Big)
    \,.
  \end{aligned}
$$

This is manifestly the image under $sSet\big( [S,\overline{W}G], - \big)$ of the correct morphism (eq:ComparisonMorphismFromFreeLoopSpaceViaEilenbergZilber) from Prop. \ref{FreeLoopSpaceOfEMSpaceIdentifiedViaEilenbergZilber}.
\end{proof}

## References

The transgression map is alluded to in

* {#DijkgraafWitten90} [[Robbert Dijkgraaf]], [[Edward Witten]], p. 14 of: _[[DW.pdf:file]]_, Commun. Math. Phys. __129__ (1990) 393 ([euclid:cmp/1104180750](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-129/issue-2/Topological-gauge-theories-and-group-cohomology/cmp/1104180750.full))

An indication of a proof, implicitly using ingredients of the [[Eilenberg-Zilber map]] (re-discovered under the name "Parmesan map"):

* {#Willerton08} [[Simon Willerton]], Section 1 of: *The twisted Drinfeld double of a finite group via gerbes and finite groupoids*, Algebr. Geom. Topol. 8 (2008) 1419-1457 [[arXiv:math/0503266](https://arxiv.org/abs/math/0503266), [doi:10.2140/agt.2008.8.1419](https://doi.org/10.2140/agt.2008.8.1419)]

The transgression formula itself (without derivation) is also considered, in a context of [[twisted equivariant K-theory|twisted]] [[orbifold K-theory]], in:

* [[Alejandro Adem]], [[Yongbin Ruan]], [[Bin Zhang]], Section 4 of: _A Stringy Product on Twisted Orbifold K-theory_, Morfismos (10th Anniversary Issue), Vol. 11, No 2 (2007), 33-64.  ([arXiv:math/0605534](https://arxiv.org/abs/math/0605534), [Morfismos pdf](www.morfismos.cinvestav.mx/Portals/morfismos/SiteDocs/Articulos/Volumen11/No2/Zhang/arz.pdf))

* [[Jean-Louis Tu]], [[Ping Xu]], Section 3 of: *The ring structure for equivariant twisted K-theory*, J. Reine Angew. Math. 635 (2009), 97–148 ([arXiv:math/0604160](https://arxiv.org/abs/math/0604160), [doi:10.1515/CRELLE.2009.077](https://doi.org/10.1515/CRELLE.2009.077))

and specifically in the context of equivariant [[Tate K-theory]]:

* {#Dove19} [[Thomas Dove]], _Twisted Equivariant Tate K-Theory_ ([arXiv:1912.02374](https://arxiv.org/abs/1912.02374))

Generalization to Real $\mathbb{Z}/2$-equivariant cohomology (as appropriate for [[twisted cohomology|twists]] of [[KR-theory]]):

* [[Behrang Noohi]], [[Matthew B. Young]], *Twisted loop transgression and higher Jandl gerbes over finite groupoids*, Algebr. Geom. Topol. **22** (2022) 1663-1712 [[arXiv:1910.01422](https://arxiv.org/abs/1910.01422), [doi:10.2140/agt.2022.22.1663](https://doi.org/10.2140/agt.2022.22.1663)]

The above discussion and proof that transgression is essentially the [[internal hom]] out of the circle into the cocycle-as-a-functor is taken from:

* {#SatiSchreiber22} [[Hisham Sati]], [[Urs Schreiber]], Thm. 3.4 of: *[[schreiber:Cyclification of Orbifolds|Cyclification of Orbifolds]]*, Comm. Math. Phys. (2023)

following earlier observations in:

* [[Urs Schreiber]], [[Zoran Škoda]], [§8.2](https://arxiv.org/pdf/1004.2472.pdf#page=37) & [§9.6.2](https://arxiv.org/pdf/1004.2472.pdf#page=47) of: *Categorified symmetries* [[arXiv:1004.2472](https://arxiv.org/abs/1004.2472)]

[[!redirects transgressions in group cohomology]]

[[!redirects transgression of group cocycles]]
[[!redirects transgressions of group cocycles]]


transgression of bundle gerbes

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

## Motivation

Let $X$ be a [[smooth manifold]], Write $L X = C^\infty(S^1,X)$
for the [[free loop space object|free loop space]].

then [[transgression]] gives a map on [[cohomology]]

$$
  \tau : H^k(X) \to H^{k-1}(L X)
$$

**Example**

$\mathbb{Z}_2$-coefficients, $k = 2$

$$
  \array{
     H^2(X, \mathbb{Z}_2) &\stackrel{\tau}{\to}& H^1(L X, \mathbb{Z}_2)
     \\
     \xi &\mapsto& \tau(\xi)
  }
$$

where $\xi$ is the second [[Stiefel-Whitney class]] we have that $X$ has  [[spin structure]] precisely if $\xi = 0$ is the trivial class.
This implies of course that also $\tau(\xi)$ vanishes. Atiyah showed that if
the [[fundamental group]] $\pi_1(X) = 1$ of $X$ vanishes, i.e. if $X$ is a
[[simply connected space]], that the also the converse holds: $X$ is spin if
$\tau(\xi)$ vanishes in the cohomology of the loop space.

**Questions**

1. What is the relation between $\xi$ and $\tau(\xi)$ in general, that would make $\tau$
   a bijection.

1. What are relation between trivializations of $\xi$ and those of $\tau(\xi)$
   that would make $\tau$ a [[functor]] -- such that this makes transgression
   an [[equivalence of categories]].

## Transgression as a functor

Let $A$ be an [[abelian group|abelian]] [[Lie group]]. Write $H^2(X,A)$ for the
[[abelian sheaf cohomology]].

we want to realize this as the connected components of a [[2-groupoid]]
$Grb_A^\nabla(X)$ of [[bundle gerbe]]s with connection on $X$.

Similarly we want to refine $H^1(L X, A)$ to a groupoid $Bun_A^\nabla(L X)$
of [[connection on a bundle|connections]] on smooth $A$-[[principal bundle]]s.

[[Jean-Luc Brylinski]] and MacLaughlin define a [[functor]]

$$
   L : Grb_{A}^\nabla(X) \to Bun_A^\nabla(L X)
   \,.
$$

by

$$
  \mathcal{G} \mapsto L \mathcal{G}|_{\beta}
   :=
   Hom_{Grb_A^\nabla(S^1)}(\beta^* \mathcal{G}, I_0)
$$

for $\beta \in L X$ and where $I_0$ denotes the trivial gerbe on the circle.

We want to understand the image of this transgression map, i.e. to characterize those bundles over $L X$ that can be obtained by transgression of a gerbe on $X$.

**Definition** Let $P$ be an $A$-[[principal bundle]] over $L X$, then
a **fusion product** on $P$ is a bundle [[isomorphism]] $\lambda$
that is [[fiber]]wise given for a triple of paths

$$
  \gamma_i : x \to y \,,\;\;\;\;\; i \in \{1,2,3\}
$$

$$
  \lambda_{\gamma_1, \gamma_2,\gamma_3}
  :
  P_{\bar\gamma_2 \star \gamma_1}
  \otimes
  P_{\bar \gamma_3 \star \gamma_2}
  \to
  P_{\bar \gamma_3 \star \gamma_1}
$$

Brylinske-MacLaughlin have a similar fusion product but over figur-8s of paths.
This however gives associativity only up to homotopy. Here we are aiming for
a product that is strictly associative.

**Definition** A connection on the fusion  bundle $(P,\lambda)$ is called

1. **compactible** if $\lambda$ is connection-preserving;

1. **symmetrizing**, if

   $$
     R_\pi(\lambda(q_1 \otimes q_2))
     =
     \lambda(R_\pi(q_2) \otimes R_\pi(q_1))
     \,,
   $$

   where $R_\pi$ is a lift of the

   $$
     \array{
       P &\stackrel{}{\to}& P
       \\
       \downarrow && \downarrow
       \\
       L X &\stackrel{r_\pi}{\to}& L X
     }
   $$

   lifts the loop rotation operation by an angle $\pi$ from loop space to the bundle
   over loop space.

   We can take $R$ to be the [[parallel transport]] of the connection on loop space
   along the canonical path in loop space that connects a loop to its rotated loop.

1. **superficial** (German: _oberflächlich_ -- this is a joke with translations)
   if it behaves like a surface holonomy in that

   1. if $\phi \in L L X$, $\tilde \phi : S^1 \times S^1 \to X$ has rank one, then
      $Hol_P(\phi) = 1$;

   1. if $\phi_1, \phi_2  \in L L X$ such that $\tilde \phi_1, \tilde \phi_2$
      are rank-2-homotopic (i.e. think homotopic) then $Hol_p(\phi_1) = Hol_p(\phi_2)$.

**Definition** An $A$-**fusion bundle** with connection over $L X$ is an
$A$-[[nLab:principal bundle]] over $L X$ with fusion product and compatible,
symmetrizing and superficial connection.

**Lemma** Transgression lifts

$$
  \array{
    Grb_A^\nabla(X) &&\stackrel{\tilde K}{\to}&&
    FusBund_A^\nabla(L X)
    \\
    & {}_{\mathllap{L}}\searrow && \swarrow_{\mathrlap{forget}}
    \\
    && Bun_A^\nabla(L X)
  }
$$

**Theorem** Lifted transgression $\tilde L$ is an equivalence of categories

## Application: Spin structures and loop space orientation

Assume $\mathcal{G}$ is the $\mathbb{Z}_2$-[[lifting gerbe]] for [[spin structure]] on $X$ whose characteristic class is

$$
  [\mathcal{G}] = \xi \in H^3(X, \mathbb{Z}_2)
$$

the [[Stiefel-Whitney class]] of $X$. So [[spin structure]]s on $X$ are in corresppndence with trivializations of $\mathcal{G}$.

On the other hand we have that [[orientation]]s of $L X$ correspond to [[section]]s of $L \mathcal{G}$. Inside there are the fusion preserving sections, which by the above are equivalent to trivializations of $\mathcal{G}$.

So we find that in general [[spin structure]]s on $X$ are not in bijection to just all orientations of $L X$, but precisely ot the fusion-compatible ones.

##References

* [[Konrad Waldorf]], _Transgression to Loop Spaces and its Inverse_

  _I: Diffeological Bundles and Fusion Maps _ ([arXiv:0911.3212](http://arxiv.org/abs/0911.3212))

  _II: Gerbes and Fusion Bundles with Connection_ ([arXiv:1004.0031](http://arxiv.org/abs/1004.0031))

For an extension of this formalism to include D-branes, see

* [[Severin Bunk]], [[Konrad Waldorf]], _Transgression of D-branes_, Adv. Theor. Math. Phys. 25(5):1095-1198 (2021), ([arXiv:1808.04894] (https://arxiv.org/abs/arXiv:1808.04894))


transgression of differential forms

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[differential form]] $\omega$ of degree $n$ on some [[smooth space]] $X$ and given a [[closed manifold|closed]] [[smooth manifold]] $\Sigma$ of [[dimension]] $k \leq n$, then there is canonically induced a differential form $\tau_\Sigma \omega$ of degree $n-k$ on the [[mapping space]] $[\Sigma,X]$: its restriction to any smooth family $\Phi_{(-)}$ of smooth functions $\Phi_u \colon \Sigma \to X$ is the result of first forming the [[pullback of differential forms]] of $\omega$ along $\Phi_{(-)}$ and then forming the [[integration of differential forms]] of the result over $\Sigma$:

$$
  \tau_{\Sigma} \omega\vert_{\Phi_{(-)}}
    \coloneqq
  \int_\Sigma (\Phi_{(-)})^\ast \omega
  \,.
$$

This differential form $\tau_\Sigma \omega$ on the mapping space is called the _[[transgression]]_ of $\omega$ with respect to $\Sigma$

This construction has a variety of immediate generalizations, for instance $\Sigma$ may have [[manifold with boundary|boundary]] and [[manifold with corners|corners]], and it may be a [[supermanifold]] and/or a [[formal manifold]]; and the [[mapping space]] may be generalized to a [[space of sections]] of a given [[fiber bundle]]. Finally, the construction also generalizes to coefficients richer than differential forms, such as [[cocycles]] in [[differential cohomology]], but this is no longer the topic of the present entry.

Important examples of transgression of differential forms appear in [[Lagrangian field theory]] (in the sense of [[physics]]) defined by a [[Lagrangian density|Lagrangian form]] on the [[jet bundle]] of a [[field bundle]]. Here the transgression of the Lagrangian itself (along [[jet prolongations]] of [[field (physics)|fields]]) is the corresponding _[[action functional]]_, the transgression of its [[Euler-Lagrange operator|Euler-Lagrange variational derivative]] is the 1-form whose vanishing is the _[[equations of motion]]_ and the transgression of the induced [[pre-symplectic current]] is the _[[pre-symplectic form]] on the [[covariant phase space]]_ of the field theory.  These examples are discussed below at _[Transgression of variational differential forms](#TransgressionOfVariationalDifferentialForms)_.

## Definition

There are two definitions of transgression of differential forms: A traditional formulation is def. \ref{TransgressionOfDifferentialFormsToMappingSpacesViaEvaluationMap} below, which transgresses by [[pullback of differential forms]] along the [[evaluation map]], followed by [[integration of differential forms]].

Another definition is useful, which makes more use of the existence of smooth classifying spaces for differential forms in [[smooth sets]], this we consider as def. \ref{ParameterizedIntegrationOfDifferentialForms} below.

That these two definitions are indeed equivalent is the content of prop. \ref{EquivalenceOfTransgressionOfDifferentialFormsToMappingSpaces} below

### Preliminaries on smooth sets

Since the concept of transgression of differential forms involves [[mapping spaces]] between, in particular, [[smooth manifolds]], it is most conveniently formulated in terms of the concept of [[generalized smooth spaces]] called _[[smooth sets]]_. For the following discussion we assume background on [[smooth sets]] as introduced in

* _[[geometry of physics -- smooth sets]]_

* _[[geometry of physics -- differential forms]]_.

(This entry itself here overlaps with _[[geometry of physics -- integration]]_, where more background may be found.)

Recall form the discussion there that a [[smooth set]] $X$ is _defined_ by specifying, in a consistent way, what counts as a smooth functions $U \to X$  from a [[Cartesian space]] $U$ (a "plot" of $X$). Given two smooth sets $X$ and $Y$ then a [[smooth function]] $f \;\colon\; X \longrightarrow Y$ is a function that takes plots $U \overset{\phi}{\to} X$ of $X$ to plots $f \circ \phi \colon U \to Y$ of $Y$.

A key example of a smooth set which is in general not a [[smooth manifold]] is the [[mapping space]] $[X,Y]$ between two smooth sets $X$ and $Y$, hence the set of all smooth functions $X \to Y$ equipped with a smooth structure itself. Namely a plot $\phi_{(-)} \colon U \to [X,Y]$ is defined to be a smooth function $\phi_{(-)}(-) \colon U \times X \to Y$ out of the [[Cartesian product]] of $U$ with $X$ to $Y$,  hence a "U-parameterized smooth family of smooth functions".

An example of a smooth set which is far from being a smooth manifold is for $n \in \mathbb{N}$ the smooth set $\mathbf{\Omega}^n$ which is the "smooth [[classifying space]]" for [[differential n-forms]], defined by the rule that a smooth function $\phi \colon U \to \mathbf{\Omega}^n$ is equivalently a smooth differential $n$-form on $U$ (to be thought of as the [[pullback of differential forms|pullback]] of a "universal $n$-form" on $\mathbf{\Omega}^n$ along $\phi$).
It follows from this in particular that for $X$ any [[smooth manifold]] then smooth functions $X \to \mathbf{\Omega}^n$ are equivalent to smooth $n$-forms on $X$. Accordingly we may say that for $X$ any [[smooth set]] (which may be far from being a smooth manifold) then a differential $n$-form on $X$ is equivalently a smooth function $X \to \mathbf{\Omega}^n$. Under this identification the operation of [[pullback of differential forms]] along some smooth function $f \colon Y \to X$ is just [[composition]] of smooth functions $f^\ast \omega \colon Y \overset{f}{\to} X \overset{\omega}{\to} \mathbf{\Omega}^n$.

These examples may be combined: the [[mapping space]] $[\Sigma, \mathbf{\Omega}^n]$ is a kind of smooth classifying space for differential forms _on_ $\Sigma$: a smooth function $\omega_{(-)} \colon U \to [\Sigma,\mathbf{\Omega}^n]$ into this space is, by the above, a [[differential n-form]] on the [[Cartesian product]] $U \times \Sigma$.

(There is a smooth space that has more right to be called "the" classifying space of differential $n$-foms on $\Sigma$, namely the [[concretification]] $\sharp_1 [\Sigma, \mathbf{\Omega}^n]$, but for the discussion of trangression actually the unconcretified space is the right one to use.)

### Via parameterized integration of differential forms

+-- {: .num_defn #ParameterizedIntegrationOfDifferentialForms}
###### Definition
**(parameterized [[integration of differential forms]])

Let

1. $X$ be a [[smooth set]];

1. $n \geq k \in \mathbb{N}$;

1. $\Sigma_k$ be a [[compact topological space|compact]] [[smooth manifold]] of [[dimension]] $k$.

Then we write

$$
  \int_{\Sigma}
    \;\colon\;
  [\Sigma_k, \mathbf{\Omega}^n]
    \longrightarrow
  \mathbf{\Omega}^{n-k}
$$

for the [[smooth function]] which takes a plot $\omega_{(-)} \colon U \to [\Sigma, \mathbf{\Omega}^k]$, hence equivalently a differential $n$-form $\omega_{(-)}(-)$ on $U \times \Sigma$ to the result of [[integration of differential forms]] over $\Sigma$:

$$
  \int_{\Sigma} \omega_{(-)}(-) \coloneqq \int_\Sigma \omega_{(-)}
  \,.
$$

=--

+-- {: .num_defn #TransgressionOfDifferentialFormsToMappingSpaces}
###### Definition
**(transgression of differential forms to [[mapping spaces]])

Let

1. $X$ be a [[smooth set]];

1. $n \geq k \in \mathbb{N}$;

1. $\Sigma_k$ be a [[compact topological space|compact]] [[smooth manifold]] of [[dimension]] $k$.

Then the operation of _transgression of differential $n$-forms_ on $X$ with respect to $\Sigma$ is the [[function]]

$$
  \tau_\Sigma
   \coloneqq
  \int_\Sigma [\Sigma,-]
   \;\colon\;
  \Omega^n(X) \to \Omega^{n-k}([\Sigma,X])
$$

from differential $n$-forms on $X$ to differential $n-k$-forms on the [[mapping space]] $[\Sigma,X]$ which takes the differential form corresponding to the smooth function

$$
  (X \stackrel{\omega}{\to} \Omega^n) \in \Omega^n(X)
$$

to the differential form corresponding to the following composite smooth function:

$$
  \tau_\Sigma \omega
   \coloneqq
  \int_{\Sigma} [\Sigma,\omega]
   \;\colon\;
  [\Sigma, X]
    \stackrel{[\Sigma, \omega]}{\to}
  [\Sigma, \Omega^n]
    \stackrel{\int_{\Sigma}}{\to}
  \Omega^{n-k}
  \,,
$$

where $[\Sigma,\omega]$ is the [[mapping space]] [[functor]] on [[morphisms]] and $\int_{\Sigma}$ is the parameterized integration of differential forms from def. \ref{ParameterizedIntegrationOfDifferentialForms}.

More explicitly in terms of plots this means equivalently the following

A plot of the [[mapping space]]

$$
  \phi_{(-)} \;\colon\; U \to [\Sigma, X]
$$

is equivalently a [[smooth function]] of the form

$$
  \phi_{(-)}(-) \;\colon\; U \times \Sigma \to X
  \,.
$$

The smooth function $[\Sigma,\omega]$ takes this smooth function to the plot

$$
  U \times \Sigma \to X
   \overset{\phi_{(-)}(-)}{\longrightarrow}
  X
   \overset{\omega}{\longrightarrow}
  \mathbf{\Omega}^{n}
$$

which is equivalently a differential form

$$
  (\phi_{(-)}(-))^\ast \omega \in \Omega^n(U \times \Sigma)
  \,.
$$

Finally the smooth function $\int_\Sigma$ takes this to the result of [[integration of differential forms]] over $\Sigma$:

$$
  \tau_{\Sigma}\omega\vert_{\phi_{(-)}}
  \;=\;
  \int_\Sigma (\phi_{(-)}(-))^\ast \omega
  \;\in\;
  \Omega^{n-k}(U)
  \,.
$$

=--

### Via pullback along the evaluation map

+-- {: .num_defn #TransgressionOfDifferentialFormsToMappingSpacesViaEvaluationMap}
###### Definition
**(transgression of differential forms to mapping space via evaluation map)**

Let

1. $X$ be a [[smooth set]];

1. $n \geq k \in \mathbb{N}$;

1. $\Sigma_k$ be a [[compact topological space|compact]] [[smooth manifold]] of [[dimension]] $k$.

Then the operation of _transgression of differential $n$-forms_ on $X$ with respect to $\Sigma$ is the [[function]]

$$
  \tau_\Sigma
   \coloneqq
  \int_\Sigma ev^\ast
   \;\colon\;
  \Omega^n(X)
    \overset{ev^\ast}{\longrightarrow}
  \Omega^n(\Sigma \times [\Sigma, X])
    \overset{\int_\Sigma}{\longrightarrow}
  \Omega^{n-k}([\Sigma,X])
$$

from differential $n$-forms on $X$ to differential $n-k$-forms on the [[mapping space]] $[\Sigma,X]$ which is the [[composition|composite]] of forming the [[pullback of differential forms]] along the [[evaluation map]] $ev \colon  [\Sigma, X] \times \Sigma \to X$ with [[integration of differential forms]] over $\Sigma$.

=--

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

The two definitions of transgression of differential forms to mapping spaces from def. \ref{TransgressionOfDifferentialFormsToMappingSpaces} and def. \ref{TransgressionOfDifferentialFormsToMappingSpacesViaEvaluationMap} are equivalent.

=--

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

We need to check that for all plots $\gamma \colon U \to [\Sigma, X]$
the pullbacks of the two forms to $U$ coincide.

For def. \ref{TransgressionOfDifferentialFormsToMappingSpacesViaEvaluationMap} we get

$$
  \gamma^\ast \int_\Sigma \mathrm{ev}^\ast A
  =
  \int_\Sigma (\gamma,\mathrm{id}_\Sigma)^\ast \mathrm{ev}^\ast A
  \;
  \in \Omega^n(U)
$$

Here we recognize in the integrand the pullback along
the $( (-)\times \Sigma \dashv [\Sigma,-])$-[[adjunct]] $\tilde \gamma : U \times \Sigma \to \Sigma$ of $\gamma$,
which is given by applying the [[left adjoint]] $(-)\times \Sigma$ and then postcomposing with the adjunction counit $\mathrm{ev}$:

$$
  \array{
    U \times \Sigma
    &
     \overset{(\gamma, \mathrm{id}_\Sigma)}{\longrightarrow}
    &
    [\Sigma,X] \times \Sigma
    &
      \overset{\mathrm{ev}}{\longrightarrow}
    &
    X
  }
  \,.
$$

Hence the integral is now

$$
  \cdots = \int_{\Sigma} \tilde \gamma^\ast A
  \,.
$$

This is the operation of the top horizontal composite in the following
[[natural transformation|naturality square]] for [[adjuncts]], and so the claim follows by its [[commuting diagram|commutativity]]:

$$
  \array{
    \tilde \gamma \in
    &
    \mathbf{H}(U \times\Sigma, X)
    &
      \overset{\mathbf{H}(U \times \Sigma,A)}{\longrightarrow}
    &
    \mathbf{H}(U \times \Sigma, \mathbf{\Omega}^{n+k})
    &
      \overset{\int_\Sigma(U)}{\longrightarrow}
    &
    \Omega^n(U)
    \\
    &
    {}^{\mathllap{\simeq}}\downarrow
      &&
    {}^{\mathllap{\simeq}}\downarrow
      &&
    {}^{\mathllap{\simeq}}\downarrow
    \\
    \gamma \in
    &
    \mathbf{H}(U,[\Sigma,X])
    &
      \overset{\mathbf{H}(U,[\Sigma,A])}{\longrightarrow}
    &
    \mathbf{H}(U,[\Sigma,\mathbf{\Omega}^{n+k}])
    &
      \overset{\mathbf{H}(U,\int_\Sigma)}{\longrightarrow}
    &
    \mathbf{H}(U,\mathbf{\Omega}^n)
  }
$$

(here we write $\mathbf{H}(-,-)$ for the [[hom functor]] of [[smooth sets]]).

=--

### Transgression of variational differential forms
 {#TransgressionOfVariationalDifferentialForms}

An important variant of transgression of differential forms is the transgression of [[variational bicomplex|variational differential forms]] along [[jet prolongation]].

In the following let $\Sigma$ be a fixed [[smooth manifold]]. We will refer to this as "[[spacetime]]", but for the present purpose it may be an smooth manifold without further structure.

+-- {: .num_defn #Fields}
###### Definition
**([[field (physics)|fields]] and their [[space of histories]])**

Given a [[spacetime]] $\Sigma$, then a _[[type]] of fields_ on $\Sigma$
is a [[smooth set|smooth]] [[fiber bundle]]

$$
  \array{E \\ \downarrow^{\mathrlap{fb}} \\ \Sigma }
$$

called the _[[field bundle]]_,

Given a type of fields on $\Sigma$ this way, then a  _[[field (physics)|field]] [[trajectory]]_ (or _field history_) of that type on $\Sigma$
is a smooth  [[section]] of this [[bundle]], namely a [[smooth function]] of the form

$$
  \Phi \colon \Sigma \longrightarrow E
$$

such that composed with the [[projection]] map it is the [[identity function]], i.e. such that

$$
  fb \circ \Phi = id
  \phantom{AAAAAAA}
  \array{
    && E
    \\
    & {}^{\mathllap{\Phi}}\nearrow & \downarrow^{\mathrlap{fb}}
    \\
    \Sigma & = & \Sigma
  }
  \,.
$$

The corresponding _field [[space of histories]]_ is the [[smooth set|smooth]] [[space of sections|space of all these]], to be denoted

$$
  \Gamma_\Sigma(E) \in \mathbf{H}
  \,.
$$

This is a [[smooth set]] by declaring that a smooth family $\Phi_{(-)}$ of field configurations,
parameterized over any [[Cartesian space]] $U$ is a smooth function

$$
  \array{
    U \times \Sigma &\overset{\Phi_{(-)}(-)}{\longrightarrow}& E
    \\
    (u,x) &\mapsto& \Phi_u(x)
  }
$$

such that for each $u \in U$ we have $p \circ \Phi_{u}(-) = id_\Sigma$, i.e.

$$
  \array{
    && E
    \\
    & {}^{\mathllap{\Phi_{(-)}(-)}}\nearrow & \downarrow^{\mathrlap{fb}}
    \\
    U \times \Sigma &\underset{pr_2}{\longrightarrow}& \Sigma
  }
  \,.
$$

More generally, let $S \hookrightarrow \Sigma$ be a [[submanifold]] of spacetime. We write
$N_\Sigma(S) \hookrightarrow \Sigma$ for its [[infinitesimal neighbourhood]] in $\Sigma$.

If $E \overset{fb}{\to} \Sigma$ is a [[field bundle]]
then the _[[space of histories]] of fields restricted to $S$_, to be denoted

$$
  \Gamma_{S}(E) \coloneqq \Gamma_{N_\Sigma(S)}( E\vert_{N_\Sigma S} ) \in \mathbf{H}
$$

is the [[space of sections]] restricted to the [[infinitesimal neighbourhood]] $N_\Sigma(S)$.

There is a canonical [[evaluation]] [[smooth function]]

$$
  \label{FieldEvaluation}
  ev_S
   \;\colon\;
  N_\Sigma S \times \Gamma_{S}(E)
    \longrightarrow
  E
$$

which takes a [[pair]] consisting of an [[generalized element|element]] in $N_\Sigma S$
and a field configuration to the value of the field configuration at that point.

=--

+-- {: .num_defn #SpacetimeSupport}
###### Definition
**(spacetime support)**

Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] over a [[spacetime]] $\Sigma$,
with induced [[jet bundle]] $J^\infty_\Sigma(E)$

For every [[subset]] $S \subset \Sigma$ let

$$
  \array{
    J^\infty_\Sigma(E)\vert_S
      &\overset{\iota_S}{\hookrightarrow}&
    J^\infty_\Sigma(E)
    \\
    \downarrow &(pb)& \downarrow
    \\
    S &\hookrightarrow& \Sigma
  }
$$

be the corresponding restriction of the [[jet bundle]] of $E$.

The _spacetime support  _ $supp_\Sigma(A)$ of a [[differential form]] $A \in \Omega^\bullet(J^\infty_\Sigma(E))$
on the [[jet bundle]] of $E$ is the [[topological closure]] of the maximal subset $S \subset \Sigma$
such that the restriction of $A$ to the jet bundle restrited to this subset vanishes:

$$
  supp_\Sigma(A) \coloneqq Cl( \{ x \in \Sigma |  \iota_{\{x\}^\ast A = 0} \} )
$$

We write

$$
  \Omega^{r,s}_{\Sigma,cp}(E)
  \coloneqq
  \left\{
    A \in \Omega^{r,s}_\Sigma(E)
    \;\vert\;
    supp_\Sigma(A) \, \text{is compact}
  \right\}
    \;\hookrightarrow\;
  \Omega^{r,s}_\Sigma(E)
$$

for the subspace of differential forms on the jet bundle whose spacetime support is a [[compact subspace]].

=--

+-- {: .num_defn #TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}
###### Definition
**(transgression of variational differential forms to [[field (physics)|field]] [[space of histories]])

Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] over a [[spacetime]] $\Sigma$ (def. \ref{Fields}), with induced [[jet bundle]] $J^\infty_\Sigma(E)$

For
$\Sigma_r \hookrightarrow \Sigma$ be a submanifold of [[spacetime]] of dimension $r \in \mathbb{N}$,
then _transgression of variational differential forms to $\Sigma_r$_ is the function

$$
  \tau_{\Sigma_r}
   \;\colon\;
  \Omega^{r,\bullet}_{\Sigma,cp}(E)
    \overset{  }{\longrightarrow}
  \Omega^\bullet\left(
    \Gamma_{\Sigma_r}(E)
   \right)
$$

which sends a differential form $A \in \Omega^{r,\bullet}_{\Sigma,cp}(E)$
to the differential form $\tau_{\Sigma_r} \in \Omega^\bullet(\Gamma_{\Sigma_r}(E))$
which to a smooth family on field configurations

$$
  \Phi_{(-)} \;\colon\; U \times N_\Sigma \Sigma_r \longrightarrow E
$$

assigns the differential form given by first forming the [[pullback of differential forms]] along the family of [[jet prolongation]] $j^\infty_\Sigma(\Phi_{(-)})$ followed by the [[integration of differential forms]] over $\Sigma_r$:

$$
  (\tau_{\Sigma}A)_{\Phi_{(-)}}
    \;\coloneqq\;
  \int_{\Sigma_r}  (j^\infty_\Sigma(\Phi_{(-)}))^\ast
  \;\in\;
  \Omega^\bullet(U)
  \,.
$$

=--

## Properties

### Relative transgression over manifolds with boundary
 {#RelativeTransgressionOverManifoldsWithBoundary}

+-- {: .num_prop }
###### Proposition
**(relative transgression over [[manifolds with boundary]])**

1. $X$ be a [[smooth set]];

1. $\Sigma_k$ be a [[compact topological space|compact]] [[smooth manifold]] of [[dimension]] $k$ with [[manifold with boundary|boundary]] $\partial \Sigma$

1. $n \geq k \in \mathbb{N}$;

1. $\omega \in \Omega^n_{X}$ a [[closed differential form]].

Write

$$
  (-)\vert_{\partial \Sigma}
    \;\coloneqq\;
  [\partial \Sigma \hookrightarrow \Sigma, X]
  \;\colon\;
  [\Sigma, X]
   \longrightarrow
  [\partial \Sigma, X]
$$

for the smooth function that restricts smooth functions on $\Sigma$ to smooth functions on the [[boundary]] $\partial \Sigma$.

Then the operations of transgression of differential forms (def. \ref{TransgressionOfDifferentialFormsToMappingSpaces}) to $\Sigma$ and to $\partial \Sigma$, respectively, are related by

$$
  d \left(
    \tau_{\Sigma}(\omega)
  \right)
  =
  (-1)^{k+1}
  ((-)\vert_{\partial \Sigma})^\ast \tau_{\partial \Sigma}(\omega)
    \phantom{AAAAAAAA}
  \array{
    [\Sigma, X]
      &\overset{ \tau_{\Sigma}(\omega) }{\longrightarrow}&
    \mathbf{\Omega}^{n-k}
    \\
    {}^{\mathllap{(-)\vert_{\partial \Sigma} }}\downarrow
      &&
    \downarrow^{\mathrlap{ (-1)^{k+1} d}}
    \\
    [\partial \Sigma, X]
      &\underset{ \tau_{\partial\Sigma}(\omega) }{\longrightarrow}&
    \mathbf{\Omega}^{n-k+1}
  }
  \,.
$$

In particular this means that if the compact manifold $\Sigma$ happens to have no boundary (is a [[closed manifold]]) then transgression over $\Sigma$ takes closed differential forms to closed differential forms.

=--

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

Let $\phi_{(-)}(-) \colon U \times \Sigma \to X$ be a plot of the mapping space $[\Sigma, X]$. Notice that the [[de Rham differential]] on the [[Cartesian product]] $U \times \Sigma$ decomposes as

$$
  d = d_U + d_\Sigma
  \,.
$$

Now we compute as follows:

$$
  \begin{aligned}
    d \tau_{\Sigma}\omega\vert_{\phi_(-)}
    & =
    d_U \int_\Sigma (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    (-1)^k \int_\Sigma d_U (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    (-1)^k \int_\Sigma (d - d \Sigma) (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    (-1)^k \int_\Sigma d (\phi_{(-)}(-))^\ast \omega
    -
    (-1)^k \int_\Sigma d_\Sigma (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    (-1)^k \int_\Sigma (\phi_{(-)}(-))^\ast \underset{= 0}{\underbrace{d \omega}}
    -
    (-1)^k \int_\Sigma d_\Sigma (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    -
    (-1)^k \int_\Sigma d_\Sigma (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    -(-1)^k \int_{\partial \Sigma} (\phi_{(-)}(-))^\ast \omega
    \\
    & =
    -(-1)^k \tau_{\partial \Sigma} \omega \vert_{\phi_{(-)}}
  \end{aligned}
$$

where in the second but last step we used [[Stokes' theorem]].

=--

### Variational transgression picks out the vertical differential forms

+-- {: .num_example }
###### Example
**(some transgressions of variational differential forms)**

We spell out the result transgression of variational differential forms (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces})
of some variational differential forms on the [[jet bundle]] of a trivial vector field bundle to the [[space of histories]] $\Gamma_\Sigma(E)$
of fields (def. \ref{Fields}).

We describe the resulting transgressed differential forms restricted to any smooth family of field configurations

$$
  \Phi_{(-)} \;\colon\; U \times \Sigma \longrightarrow E
  \,.
$$

Let $b \in C^\infty_{cp}(\Sigma)$ be any [[bump function]] on spacetime. Its product with the [[volume form]]
(as in example \ref{BasicFactsAboutVarationalCalculusOnJetBundleOfTrivialVectorBundle}) is then a
horizontal $p+1$-form on the jet bundle with compact spacetime support.

$$
  b dvol_\Sigma \in \Omega^{0,0}_{\Sigma,cp}(E)
$$

The transgression of this 0-form to the [[space of histories]] of fields

$$
  \tau_\Sigma (b dvol_\Sigma) \in \Omega^0( \Gamma_\Sigma(E) )
$$

is the differential form on $\Gamma_\Sigma(E)$ which restricted to the given family of
field configurations $\Phi_{(-)} \colon u \mapsto \Phi_u$ yields the function

$$
  \tau_\Sigma (b dvol_\Sigma)\vert_\Phi \colon u \mapsto \int_\Sigma b dvol_\Sigma
$$

which is simply the constant function with value the integral of $b$ against the given volume form.

The constancy of this function is due to the fact that $b dvol_\Sigma$ does not depend on the field variables. So consider next the
horizontal $(p+1)$-form

$$
  \phi^a \, b dvol_\Sigma \; \in \Omega^{p+1,0}( E )
  \,.
$$

Its transgression is the function

$$
  \tau_\Sigma( \phi^a \, b dvol_\Sigma  )_\Phi
   \;=\;
   \left(
     u
       \mapsto
     \int_\Sigma \Phi^a_{(u)}(x)  b(x) dvol_\Sigma(x)
   \right)
$$

which assigns to a given field configuration $\Phi_{u}$ in the family the value its $a$-component
integrated against $b dvol_\Sigma$.

Similarly the transgression of $\phi^a_{,\mu}$ is the function

$$
  \tau_\Sigma( \phi^a_{,\mu} \, b dvol_\Sigma  )_\Phi
   \;=\;
   \left(
     u
       \mapsto
     \int_\Sigma \frac{\partial \Phi^a_{u}}{\partial x^\mu}  b(x) dvol_\Sigma(x)
   \right)
$$

which assigns to a field configuration the integral of the value of the $\mu$th derivative of its $a$th
component against $b dvol_\Sigma$.

Next consider a horizontally exact variational form

$$
  d \alpha \in \Omega^{p+1,s}_{\Sigma,cp}(E)
  \,.
$$

By prop. \ref{PullbackAlongJetProlongationIntertwinesHorizontalDerivative} the pullback of this form
along the jet prolongation of fields is exact in the $\Sigma$-direction:

$$
  (j^\infty_\Sigma\Phi_{(-)})^\ast(d \alpha  \wedge b dvol_\Sigma) = d_\Sigma (j^\infty_\Sigma\Phi_{(-)})^\ast\alpha \wedge b dvol_\Sigma
  \,,
$$

(where we write $d = d_U + d_\Sigma$ for the de Rham differential on $U \times \Sigma$). It follows that the
integral over $\Sigma$ vanishes.

Now let

$$
  \delta \alpha \phi^a_{,\mu_1 \cdots \mu_k} \, b dvol_\Sigma
  \in
  \Omega^{p+1,1}_\Sigma(E)
$$

be a variational (vertical) differential 1-form.
Its [[pullback of differential forms]]
along $j^\infty_\Sigma(\Phi_{(-)}) \colon U \times \Sigma \to J^\infty_\Sigma(E)$ has two contributions:
one from the variation along $\Sigma$, the other from variation along $U$.

By prop. \ref{PullbackAlongJetProlongationIntertwinesHorizontalDerivative},
for _fixed_ $u \in U$ the pullback along the jet prolongation vanishes.

On the other hand, for fixed $s \in \Sigma$, the pullback of $\mathbf{d} \phi^a_{\mu_1\cdots \mu_k}$
is
$$
  d_U \frac{ \partial^k \Phi_{(-)}}{\partial x^{\mu^1} \cdots \partial x^{\mu_k}}
$$
while the pullback of $d \phi^a_{\mu_1\cdots \mu_k}$ vanishes at fixed $\Sigma$.

This means that

$$
  \tau_\Sigma( \delta \phi^a_{,\mu_1 \cdots \mu_k} )
  =
  d \tau_{\Sigma}( \phi^a_{_\mu_1 \cdots \mu_k} )
$$

is the de Rham differential (on $U$) of the corresponding function discussed before.

In conclusion:

Under transgression the variational (vertical) derivative on the jet bundle turns into the ordinary de Rham derivative on the [[space of histories]] of fields.

=--

## Examples

We discuss some examples and applications:

* [Gauge coupling action functional of charged particle](#GaugeCouplingActionFunctionalOfChargedParticle)

* [Transgression of Killing form to symplectic form of Chern-Simons theory](#TransgressionOfKillingFormToSymplecticFormOfChernSimons)

### Gauge coupling action functional of charged particle
 {#GaugeCouplingActionFunctionalOfChargedParticle}

Let $X \in \mathbf{H}$ and consider a [[circle group]]-[[principal connection]] $\nabla \colon X \to \mathbf{B}U(1)_{conn}$ over $X$. By the discussion in _[Dirac charge quantization and the electromagnetic field](#DiracChargeQuantizationAndElectromagneticField)_ above this encodes an [[electromagnetic field]] on $X$. Assume for simplicity here that the underlying [[circle principal bundle]] is trivialized, so that then the connection is equivalently given by a differential 1-form

$$
  \nabla = A \;\colon\; X \to \mathbf{\Omega}^1
  \,,
$$

the _[[electromagnetic potential]]_.

Let then $\Sigma = S^1$ be the [[circle]].  The [[transgression]] of the electromagnetic potential to the [[loop space]] of $X$

$$
  \int_{S^1} [S^1, A]
   \;\colon\;
  [S^1, X]
    \stackrel{[S^1, A]}{\to}
  [S^1 , \Omega^1]
    \stackrel{\int_{S^1}}{\to}
  \Omega^0
  \simeq
  \mathbb{R}
$$

is the [[action functional]] for an [[electron]] or other electrically charged [[particle]] in the [[background gauge field]] $A$ is $S_{em} = \int_{S^1} [S^1, A]$.

The [[variational calculus|variation]] of this contribution in addition to that of the [[kinetic action]]  of the electron gives the _[[Lorentz force]]_ law describing the [[force]] exerted by the [[background gauge field]] on the electron.

### Transgression of Killing form to symplectic form of Chern-Simons theory
 {#TransgressionOfKillingFormToSymplecticFormOfChernSimons}

Let $\mathfrak{g}$ be a [[Lie algebra]] with
binary [[invariant polynomial]]
$\langle -,-\rangle \colon \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R}$.

For instance $\mathfrak{g}$ could be a [[semisimple Lie algebra]] and $\langle -,-\rangle$ its [[Killing form]]. In particular if $\mathfrak{g} = \mathfrak{su}(n)$ is a [[matrix Lie algebra]] such as the [[special unitary Lie algebra]], then the Killing form is given by the [[trace]] of the product of two matrices.

This pairing $\langle -,-\rangle$ defines a differential 4-form on the [[smooth space]] of [[Lie algebra valued 1-forms]]

$$
  \langle F_{(-)} \wedge F_{(-)} \rangle
  \colon
  \Omega^1(-,\mathfrak{g})
   \stackrel{F_{(-)}}{\to}
  \Omega^2(-, \mathfrak{g})
   \stackrel{(-)\wedge (-)}{\to}
  \Omega^4(-, \mathfrak{g}\otimes \mathfrak{g})
   \stackrel{\langle-,-\rangle}{\to}
  \Omega^4
$$

Over a [[coordinate patch]] $U \in $ [[CartSp]] this sends a differential 1-form $A \in \Omega^1(U)$ to the differential 4-form

$$
  \langle F_A \wedge F_A \rangle \in \Omega^4(U)
  \,.
$$

The fact that $\langle -, - \rangle$ is indeed an _[[invariant polynomial]]_ means that this indeed extends to a 4-form on the smooth [[groupoid of Lie algebra valued forms]]

$$
  \langle F_{(-)} \wedge F_{(-)}\rangle
  \colon
  \mathbf{B}G_{conn}
  \to
  \Omega^4
  \,.
$$

Now let $\Sigma$ be an [[orientation|oriented]] [[closed manifold|closed]] [[smooth manifold]].  The [[transgression]] of the above 4-form to the [[mapping space]] out of $\Sigma$ yields the 2-form

$$
  \omega
    \coloneqq
  \int_{\Sigma} \langle F_{(-)}\wedge F_{(-)}\rangle
   \colon
  \mathbf{\Omega}^1(\Sigma,\mathfrak{g})
   \hookrightarrow
  [\Sigma, \mathbf{B}G_{conn}]
    \stackrel{[\Sigma, \langle F_{(-)}\wedge F_{(-)}\rangle]}{\to}
  [\Sigma, \Omega^4]
    \stackrel{\int_{\Sigma}}{\to}
  \Omega^2
$$

to the [[moduli stack]] of [[Lie algebra valued 1-forms]] on $\Sigma$.

Over a [[coordinate chart]] $U = \mathbb{R}^n \in $ [[CartSp]] an element $A \in \mathbf{\Omega}^1(\Sigma,\mathfrak{g})(\mathbb{R}^n)$ is a $\mathfrak{g}$-valued 1-form $A$ on $\Sigma \times U$ with no leg along $U$. Its [[curvature]] 2-form therefore decomposes as

$$
  F_A = F_A^{\Sigma} + \delta A
  \,,
$$

where $F_A^{\Sigma} $ is the curvature component with all legs along $\Sigma$ and where

$$
  \delta A
  \coloneqq
  - \sum_{i = 1}^n \frac{\partial}{\partial x^i} A \wedge \mathbf{d}x^i
$$

is the [[variational calculus|variational]] derivative of $A$.

This means that in the 4-form

$$
  \langle F_A \wedge F_A\rangle
  =
  \langle F_A^\Sigma \wedge F_A^\Sigma \rangle
  +
  2 \langle F_A^\Sigma \wedge \delta A\rangle
  +
  \langle \delta A \wedge \delta A\rangle
  \in
  \Omega^4(\Sigma \times U)
$$

only the last term gives a 2-form contribution on $U$. Hence we find that the transgressed 2-form is

$$
  \omega = \int_\Sigma \langle \delta A \wedge \delta A\rangle
   \colon
  \mathbf{\Omega}^1(\Sigma, \mathfrak{g})
   \to
  \Omega^2
  \,.
$$

When restricted further to flat forms

$$
  \mathbf{\Omega^1}_{flat}(\Sigma,\mathfrak{g})
  \hookrightarrow
  \mathbf{\Omega^1}(\Sigma,\mathfrak{g})
$$

which is the [[phase space]] of $\mathfrak{g}$-[[Chern-Simons theory]], then this is the corresponding [[symplectic form]] (by the discussion at _[Chern-Simons theory -- covariant phase space](Chern-Simons+theory#CovariantPhaseSpace)_).

[[!redirects transgressions of differential forms]]

[[!redirects transgression of differential n-forms]]
[[!redirects transgressions of differential n-forms]]

[[!redirects transgression of variational differential forms]]
[[!redirects transgressions of variational differential forms]]


transition system

[[!redirects transition systems]]

# Contents
* table of contents
{:toc}

##Transition systems.##

Transition systems are a well established [[semantics|semantic model]] for both sequential and concurrent systems.

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

A _transition system_ is a structure $T = (S,i,L,Tran)$, where

* $S$ is a set of _states_ with initial state, $i$;

* $L$ is a set of _labels_, sometimes referred to also as _events_;

* $Tran\subseteq S\times L\times S$ is the _transition relation_.

=--

Let $(S, i, L, Tran )$ be a transition system. We write

$$s\stackrel{a}{\to} s'$$

to indicate that $(s, a, s')\in Tran$.

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

 A _morphism_ from one transition system, $T$, to another
$T'$  will be a pair $(\sigma, \lambda)$, in which

* $\sigma$ is a function from the states of $T$ to those of $T'$

* $\lambda$ is a [[partial function]] from the labels of $T$ to those of $T'$ such that for any transition $s\stackrel{a}{\to} s'$ of $T$ if $\lambda(a)$ is defined, then $\sigma(s)\stackrel{\lambda(a)}{\to} \sigma(s')$ is a
transition of $T'$; otherwise, if $\lambda(a)$ is undefined, then $\sigma(s) = \sigma(s')$.
=--

It is useful to rework this definition of morphism using a variant of the idea, discussed at [[partial function]], that replaces a partial function by a total function using the neat trick of adding an additional element $\bot$ to the codomain. (We will use the notation from [[partial function]] freely in what follows.)   This is done here simply by introducing **idle transitions** $(s,\bot,s)$ thought of as going from $s$ to itself, and working with $L_\bot$ as a set of labels instead of  $L$.  (This is very neat here as it corresponds the label $\bot$ to 'do nothing' to the states.)  After completing everything in this way we get new transition systems $T_\bot = (S,i,L_\bot, Tran_\bot)$, etc. and will work with these. (Of course, $Tran_\bot = Tran \cup \{s,\bot,s)\mid s\in S\}$.)  Now a morphism $f$ is the same as given by a pair $\sigma: S\to S'$, as before, and $\lambda: L_\bot\to L'_\bot$, satisfying the compatibility condition that if $(s,a,s')\in Tran_\bot$, then $(\sigma(s),\lambda(a),\sigma(s'))\in Tran'_\bot$, (and that $\lambda_\bot(\bot) = \bot'$).

This way we get a category, $TS$, of transition systems.

The notions of transition system and of morphisms between them is clearly related to (low dimensional) [[cubical sets]] / labelled [[directed graphs]]/labelled [[transition systems]], but we will need to consider labelled  cubical sets.

##Labeled transition systems.##

In the above, we have used the notation $L$ to stand for the set of _events_ and the set of _labels_ for those events. It is sometimes useful to make a distinction between the events themselves and their labels and to explicitly give a labelling as a function.  This is important, for instance, in treating 'relabelling' which leads to [[fibration|categorical fibrational situations]] (see the paper by Winskel and Nielsen, cited below.) In order to make the distinction clearer, we will replace $L$ by $E$ and refer to its elements as 'events' in what follows.

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

A _labelled transition system_ consists of a transition
system $T = (S, i, E, Tran)$ together with a set $L$ of _labels_, and a function $l : E \to L$.  We denote it by $(T,L,l)$.

A _morphism_, $(\sigma, \tau , \lambda) : (T_1 , L_1 , l_1) \to (T_2 , L_2 , l_2 )$ between labeled transition
systems consists of a morphism $(\sigma, \tau) : T_1 \to T_2$ between the underlying transition
systems together with a [[partial function]] $\lambda : L_1 \to L_2$ such that $l_2 \circ \tau = \lambda \circ l_1$.
=--
We
write $LTS$ for the category of labeled transition systems.

\begin{remark}
The term *labelled transition system* can also refer to the data of a domain, a set of labels and a transition relation, without any distinguished initial or final sets.
\end{remark}

##TSs as relational structures##

We can view a transition system as a [[relational structure]]. The set of states is the 'set of worlds' and for each event, $e \in E$ we define a relation $R_e \subseteq S\times S$ by $(s,s')\in R_e$ if and only if, $(s,e,s')\in Trans$.  We thus derive a relation for each event and, conversely, if we know the family $\{R_e \mid e\in E\}$, then we can rebuild $Trans$ in the obvious way.

##Higher dimensional analogues

There are tentative definitions of

[[higher dimensional transition system]],

which take a more [[nPOV]] of this theory.

## See also ##

* [[bisimulation]]

* [[models for concurrency]]

## References

* [[Mogens Nielsen]], [[Glynn Winskel]]: *Models for concurrency*, in: *Handbook of Logic in Computer Science* vol 4, Oxford Univ. Press (1994) 1-148 [[pdf](https://www.cl.cam.ac.uk/~gw104/winskel-nielsen-models-for-concurrency.pdf), [[WinskelNielsen-Concurrency.pdf:file]], [doi:10.5555/218623.218630](https://dl.acm.org/doi/10.5555/218623.218630), [ISBN:9780198537809](https://global.oup.com/academic/product/handbook-of-logic-in-computer-science-9780198537809?lang=en&cc=at)]

* [[Eric Goubault]] and [[Samuel Mimram]], *[Formal Relationships Between Geometrical and Classical Models for Concurrency](http://arxiv.org/abs/1004.2818)*

Discussion in relation to [[bisimulations]] and [[open morphisms]]:

* {#JoyalNielsenWisnkel94} [[André Joyal]], [[Mogens Nielsen]], [[Glynn Winskel]]: *Bisimulation from open maps*, BRICS report series RS-94-7 (1994) [[doi:10.7146/brics.v1i7.21663](https://doi.org/10.7146/brics.v1i7.21663), [pdf](http://www.brics.dk/RS/94/7/BRICS-RS-94-7.pdf)], Information and Computation **127** 2 (1996) 164-185 [[doi:10.1006/inco.1996.0057](https://doi.org/10.1006/inco.1996.0057)]

[[!redirects labeled transition system]]
[[!redirects labelled transition system]]
[[!redirects labeled transition systems]]
[[!redirects labelled transition systems]]

category:computer science


transitive action

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

#Contents#
* table of contents
{:toc}

## Definition

An [[action]]

$$
  (-)\cdot(-)
  \;\colon\;
  G \times X \to X
$$

of a [[group]] $G$ on a [[set]] $X$ is called _transitive_ if it has a single [[orbit]], i.e. for any two elements, $x, y \in X$, there exists $g\in G$ such that $y = g \cdot x$.

This is equivalent to saying that the [[shear map]]

$$
  \array{
    G \times X
    &
    \overset
      { (pr_2, \cdot) }
      {\longrightarrow}&
    X \times X
    \\
    (g, x)
    &
    \mapsto
    &
    \big( x, g \cdot x \big)
    \,.
  }
$$

is in [[epimorphism]]. In this form the definition makes sense for [[action objects]] [[internalization|internal]] to any ambient [[category]] with [[finite products]].

Beware that often it is assumed that the underlying [[object]] $X$ of a transitive action is [[inhabited]] (but not always, see at *[[pseudo-torsor]]).

For $k\ge 0$, an action $G \times X \to X$ is said to be **$k$-transitive** if the componentwise-action $G \times X^{\underline{k}} \to X^{\underline{k}}$ is transitive, where $X^{\underline{k}}$ denotes the set of tuples of $k$ distinct points (i.e., [[injective functions]] from $\{1,\dots,k\}$ to $X$).  For instance, an action of $G$ on $X$ is 3-transitive if any pair of triples $(a_1,a_2,a_3)$ and $(b_1,b_2,b_3)$ of points in $X$, where $a_i \ne a_j$ and $b_i \ne b_j$ for $i\ne j$, there exists $g \in G$ such that $(b_1,b_2,b_3) = (g a_1,g a_2,g a_3)$.

A transitive action that is also [[free action|free]] is called _[[regular action|regular]]_ action. See also at _[[torsor]]_.

## Properties

A set equipped with a transitive action of $G$ (and which is [[inhabited]]) is the same thing as a [[connected object]] in the category of [[G-sets]]. A $G$-set may be decomposed uniquely as a coproduct of transitive $G$-sets.

## Examples

* Any group $G$ acts transitively on itself by multiplication $\cdot : G \times G \to G$, which is called the (left) [[regular representation]] of $G$.

* The [[alternating group]] $A_n$ acts transitively on $\{1,\dots,n\}$ for $n \gt 2$, and in fact it acts $(n-2)$-transitively for all $n \ge 2$.

* The [[modular group]] $PSL(2,\mathbb{Z})$ acts transitively on the [[rational number|rational]] [[projective line]] $\mathbb{P}^1(\mathbb{Q}) = \mathbb{Q} \cup \{\infty\}$. The [[projective general linear group]] $PGL(2,\mathbb{C})$ acts 3-transitively on the [[Riemann sphere]] $\mathbb{P}^1(\mathbb{C})$.

* An action of $\mathbb{Z}$ (viewed as the [[free group]] on one generator) on a set $X$ corresponds to an arbitrary [[permutation]] $\pi : X \to X$, but the action is transitive just in case $\pi$ is a _cyclic_ permutation.

## As an action on cosets

Let $* : G \times X \to X$ be a transitive action and suppose that $X$ is [[inhabited]].  Then $*$ is equivalent to the action of $G$ by multiplication on a [[coset]] space $G/H$, where the subgroup $H$ is taken as the [[stabilizer subgroup]]
$$H = G_x = \{ g \in G \mid g * x = x \}$$
of some arbitrary element $x \in X$.  In particular, the transitivity of $*$ guarantees that the $G$-equivariant map $G/H \to X$ defined by $g H \mapsto g * x$ is a bijection.  (Note that although the subgroup $H = G_x$ depends on the choice of $x$, it is determined up to conjugacy, and so the coset space $G/H$ is independent of the choice of element.)

## Related concepts

* [[combinatorial map]]

* [[homogeneous space]]

* [[torsor]]

* [[transitive action]], [[free action]], [[semi-free action]], [[regular action]]

* [[proper action]], [[properly discontinuous action]]

## References

* Helmut Wielandt.  _Finite Permutation Groups_.  Academic Press, 1964.

* [Fundamental theorem of group actions](http://groupprops.subwiki.org/wiki/Fundamental_theorem_of_group_actions) on the Group Properties Wiki.

[[!redirects transitive actions]]

[[!redirects transitive]]


transitive closure

# (Reflexive)-transitive closures
* table of contents
{: toc}

## Of a relation

The __transitive closure__ of a binary [[relation]] $\sim$ on a [[set]] $S$ is the smallest [[transitive relation]] that contains $\sim$.

Often one wants the __reflexive-transitive closure__ of $\sim$, which is the smallest transitive relation that contains $\sim$ and is also [[reflexive relation|reflexive]].

These can be defined explicitly as follows: $ x \sim^* y$ if and only if, for some [[natural number]] $n$, there exists a sequence $(r_0, \ldots, r_n)$ such that
$$ x = r_0 \sim \cdots \sim r_n = y .$$
If you accept $0$ as a natural number, then this defines the reflexive-transitive closure; if not, then this defines the transitive closure.

For the transitive closure, it\'s also possible to rephrase the above slightly (using only $r_1$ through $r_{n-1}$) to avoid any reference to [[equality]].  Thus [[prerelations]] have transitive closures but not necessarily reflexive-transitive closures.

## Of a material set

In material [[set theory]], the __transitive closure__ of a [[pure set]] $X$ is the [[transitive set]] $X^*$ whose members are the elements of the [[downset]] of $X$ under the transitive closure (in the previous sense) of $\in$.  That is, it consists of the members of $X$, their members, their members, and so on.  The result is a [[transitive set]], the smallest transitive set that contains $X$ as a [[subset]].

Analogously, the __reflexive-transitive closure__ of $X$ may be defined as the transitive closure of the [[successor]] $X \cup \{X\}$ of $X$, or equivalently the transitive closure of $\{X\}$ itself.  Either way, this is the same as $X^* \cup \{X\}$.  This is the smallest transitive set that contains $X$ as a [[member]].  (It is not, however, normally a [[reflexive set]]; that is, it does not belong to itself.)

[[!redirects transitive closure]]
[[!redirects transitive closures]]

[[!redirects reflexive-transitive closure]]
[[!redirects reflexive-transitive closures]]


transitive relation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### (0,1)-categoty theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

A __transitive relation__ is a [[semicategory]] or [[magmoid]] $A$ such that for all objects $a$ and $b$ in $A$, there are no more than one morphism with domain $a$ and codomain $b$. Equivalently, a transitive relation is a semicategory or magmoid [[enriched category|enriched]] on [[truth values]].

[[set theory|Set theoretically]], a (binary) [[relation]] $\sim$ on a [[set]] $A$ is __transitive__ if in every chain of $3$ pairwise related elements, the first and last elements are also related:
$$\forall (x, y, z: A),\; x \sim y \;\wedge\; y \sim z \;\Rightarrow\; x \sim z$$
which generalises from $3$ to any finite, positive number of elements.

In the language of the $2$-poset [[Rel]] of sets and relations, a relation $R: A \to A$ is __transitive__ if it contains its composite with itself:
$$R^2 \subseteq R$$
from which it follows that $R^n \subseteq R$ for any positive [[natural number]] $n$.  To include the case where $n = 0$, we must explicitly state that the relation is [[reflexive relation|reflexive]].

Transitive relations are often understood as [[orders]].

[[!redirects transitive relation]]
[[!redirects transitive relations]]

[[!redirects transitivity]]


transitive set

# Transitive sets
* table of contents
{: toc}

## Definition

In [[unsorted set theory|unsorted]] [[material set theory]], a [[pure set]] $S$ is called __transitive__ if $a\in b\in S$ implies that $a\in S$.  Note that this does *not* mean that $\in$ is a [[transitive relation|transitive]] binary relation on $S$ itself.  (In fact, assuming the [[axiom of foundation]], $\in$ is a transitive relation on $S$ precisely when $S$ is a von Neumann [[ordinal number]].)

Transitive sets are a natural place to look for [[inner model]]s of a material set theory: that is, sets $S$ such that $S$ together with the restriction of $\in$ to $S$ satisfies some or all of the axioms of the set theory.  This is especially so because of [[Mostowski's collapsing lemma]] that any [[extensional relation|extensional]] [[well-founded relation]] is isomorphic to a transitive set.

## Transitive closures

In [[ZFC]], one can prove that every pure set $x$ is contained in a least transitive pure set, called its __transitive closure__.  This can be defined as the set of all $y$ such that there is a chain $y = x_n \in \cdots \in x_0 = x$ for some [[natural number]] $n$.  The proof that this set exists requires the [[axiom of replacement]] and the property of [[induction]] for unbounded formulas (which follows from the [[axiom of separation]]).  In set theories that are too weak to prove the existence of transitive closures in this way, their existence is sometimes assumed explicitly, as the __axiom of transitive closure__.

## Modeling pure sets

Given the existence of transitive closures, [[pure sets]] can be identified with subsets of transitive sets, and hence (given Mostowski's lemma) with subsets of extensional well-founded relations.  This latter characterization is completely structural, and thus can be used to model pure sets in [[structural set theory]].

category: foundational axiom

[[!redirects transitive set]]
[[!redirects transitive sets]]

[[!redirects transitive closure]]
[[!redirects transitive closures]]
[[!redirects axiom of transitive closure]]


translation functor

## Definition

Let $\mathfrak{g}$ be a semisimple complex Lie algebra and $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra. Given $\lambda,\mu\in\mathfrak{h}^*$ such that the difference is an integral weight $\mu-\lambda\in\Lambda$. The orbit of the difference $\mu-\lambda$ under the Weyl group contains a unique positive integral weight $\bar{nu}$. Given any weight $\tau$ denote by $\chi_\tau$ the central character corresponding to the weight $\tau$. The Bernstein-Gelfand-Gelfand category $\mathcal{O}$ is the internal direct sum of the subcategories $\mathcal{O}_{\chi}$
where $\chi$ runs through [[central character]]s of the form $\chi_\tau$ and where the full subcategory $\mathcal{O}_\chi\subset\mathcal{O}$ for a central character $\chi$ by the definition consists of all modules $M$ in $\mathcal{O}$ such that for each $z$, the action $(z-\chi(z))^n \cdot v = 0$ for some $n = n(z)\gt 0$. There are canonical functors of projection  $pr_\lambda:\mathcal{O}\to\mathcal{O}_{\chi_\lambda}$. The functors

$$
T^\mu_\lambda : M\to pr_\mu (L_{\bar{nu}}\otimes pr_\lambda M),
$$

$T^\mu_\lambda : \mathcal{O}\to\mathcal{O}_{\chi_\mu}$ and its restriction to $\mathcal{O}_{\chi_\lambda}$ is called the translation functor (because it changes the subcategories corresponding to different central characters).

## Properties and applications

Translation functors are exact and preserve projective objects.

Used also in categorification in Lie theory...

## Literature

Translation

* J. C. Jantzen, _Moduln mit einem höchsten Gewicht_, (Modules with a highest weight)
Lecture Notes in Math. __750__, Springer 1979. ii+195 pp.

The translation functors are explained in detail in chapter 7 from

* James E. Humphreys, _Representations of semisimple Lie algebras in the BGG categry $\mathcal{O}$_, Grad. Studied in Math. __94__, AMS 2008

Zuckerman came independently from Jantzen to the idea of using the translation functors

* Greg Zuckerman, _Tensor products of finite and infinite dimensional representations of semisimple Lie groups_, Ann. of Math. __106__ (1977) 295-308

[[!redirects translation functor]]


translation group

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* table of contents
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## Definition

The _translation group_ in [[dimension]] $n$ is the [[Cartesian space]] $\mathbb{R}^n$ equipped with its canonical additve [[group]] structure (essentially its [[vector space]]-structure).

## Related concepts

* [[reflection]], [[rotation]], [[boost]]

* [[circle group]]

* [[translation Lie algebra]]

* [[affine group]]

* [[super translation group]]

* [[rotation group]]

* [[Galilei group]], [[Poincare group]]

[[!redirects translation groups]]

[[!redirects translation Lie group]]
[[!redirects translation Lie groups]]

[[!redirects translation]]
[[!redirects translations]]


translation Lie algebra

#Contents#
* table of contents
{:toc}

## Definition

Given a [[vector space]] $V$, the abelian [[Lie algebra]] on $V$, hence the Lie algebra with vanishing [[Lie bracket]], may be thought of as the Lie algebra of $V$ regarded as an additive [[abelian group|abelian]] [[Lie group]]. As such it is the _translation Lie algebra_ of $V$.

## Examples

* [[line Lie algebra]]

## Related concepts

* [[translation group]]

* [[Poincaré Lie algebra]]

* [[super translation Lie algebra]]

[[!redirects translation Lie algebras]]