Contents

group theory

cohomology

# Contents

## Idea

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

(1)$H_{grp}^{\bullet + 1}(G;\,A) \;\;=\;\; H^{\bullet + 1}(B G;\, A) \xrightarrow{ \;\;\; \tau \;\;\; } H^n \big( \Lambda B G, \, A \big) \;\;\; \simeq \; \underset{ [g] \in G^{ad}/G }{\oplus} H^n_{grp}\big(C_g;\, A \big)$

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

(2)$\Lambda B G \;\simeq\; \underset{ [g] \in G_{ad}/G }{\coprod} B C_g \;\;\;\simeq\;\;\; B G \,\sqcup\, \underset{ [g] \neq [e] }{\coprod} B C_g \,,$

this induces a 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 (1) 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 “$+$”)

(3)$\tau(c)(\gamma, g_{n-1}, \cdots, g_1, g_0) \;=\; \pm \underset{ 0 \leq j \leq n }{\sum} (-1)^j \cdot c( g_{n-1}, \cdots, g_{n-j}, Ad_j(\gamma), g_{n-j-1}, \cdots, g_0 ) \,,$

where

$\gamma \xrightarrow{ g_{n-1} } Ad_{n-1}(\gamma) \xrightarrow{ g_{n-2} } \cdots \xrightarrow{ g_0 } Ad_{0}(\gamma) \;\;\;\;\; \in \;\; \Lambda B G$

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_j)}(\gamma) \\ & \;\coloneqq\; (g_{n-1} \cdots g_j)^{-1} \cdot \gamma \cdot (g_{n-1} \cdots g_j) \,, \end{aligned}

(which restricts to $Ad_j(\gamma) = \gamma$ upon corestriction to the connected component on the right of (2) 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 (3):

Below we mean to spell out a general abstract definition (Def. ) of the transgression map (1) and a full proof of its component formula (3), amplifying that its form is a direct consequence of – besides some basic homotopy theory/homological algebra which we review below – the classical Eilenberg-Zilber theorem, i.e. the Eilenberg-Zilber/Alexander-Whitney deformation retraction (which was partially re-discovered in Willerton 2008, Sec. 1 under the name “Parmesan theorem”).

## Background and Lemmata

### Homotopy and homological algebra

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

###### Definition

(homotopy categories)

We write:

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 simplicial abelian group $\dashv$ underlying simplicial set)-Quillen adjunction (here) by

(4)$Ho(sAb) \underoverset {\underset{frgt}{\longrightarrow}} {\overset{\mathbb{Z}(-)}{\longleftarrow}} {\;\;\;\;\;\;\bot\;\;\;\;\;\;} Ho(sSet)$
• (5)$Ho(Ch_+) \underoverset {\underset{DK}{\longrightarrow}} {\overset{N_\bullet}{\longleftarrow}} {\;\;\;\;\;\;\bot_{\mathrlap{\simeq}}\;\;\;\;\;\;} Ho(sAb)$
• the derived internal-hom-Quillen adjunction (here), for any $X \in$ SimplicialSets, by

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

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

• the derived internal-hom-Quillen adjunction (here) for $\mathbb{Z} \,\in\,$ ConnectiveChainComplexes:

(7)$Ho(Ch_+) \underoverset {\underset{ V_\bullet \mapsto V_{\bullet + 1} }{\longrightarrow}} {\overset{ V_\bullet \mapsto V_{\bullet - 1} }{\longleftarrow}} {\;\;\;\;\;\;\bot\;\;\;\;\;\;} Ho(Ch_+)$

### Simplicial classifying spaces

###### Remark

(delooping groupoid and simplicial classifying space of finite group)
The nerve of the delooping groupoid of a discrete group $G$ is isomorphic to the simplicial classifying space of $G$ (see this Example):

$N \big( G \rightrightarrows \ast \big) \;\simeq\; \overline{W} G \;\;\; \in \; sSet \,.$

For notational brevity we will be referring to $\overline{W}G$ in the following, but it may be helpful to keep thinking of the nerve of the delooping groupoid. From that perspective, an n-simplex in $\overline{W}G$, which is an n-tuple of group elements, is suggestively denoted as a sequence of morphisms:

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

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

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

### Group(oid) cohomology

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

###### Definition

(Eilenberg-MacLane spaces)
For $A \,\in\,$ Ab, and $n \in \mathbb{N}$, we write

$B^n A \,=\, K(A,n) \,\coloneqq\, frgt \circ DK(A[n]) \;\;\; \in \; sAb \xrightarrow{ Loc_{\mathrm{W}}} Ho(sAb) \xrightarrow{frgt} Ho(sSet)$

for (the homotopy type of) the Eilenberg-MacLane space with $A$ in degree $n$.

###### Definition

(ordinary cohomology)
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

$H^n(X;\, A) \;=\; Ho(sSet)(X, \, B^n A) \,,$

where on the right we have the Eilenberg-MacLane space from Def. .

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

###### Definition

(group cohomology)
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

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

###### Proposition

(group cohomology is ordinary cohomology of classifying space)
For $G \,\in\,$ Groups and $A \,\in\,$ AbelianGroups, the group cohomology (Def. ) of $G$ with coefficient in $A$ is naturally isomorphic to the ordinary cohomology of the simplicial classifying space of $G$ with coefficients in $B^n A$:

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

###### Proof

By the hom-isomorphisms of the above 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 \, frgt \circ DK(A[n]) \big) \\ & \;=\; Ho(sSet) \big( B G \, B^n A \big) \\ & \;=\; H^n(B G;\, A) \,. \end{aligned}

This Prop. , in view of Rem. justifies the following definition:

###### Definition

(groupoid cohomology)
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. ) of the homotopy type of the nerve of $\mathcal{G}$, regarded in the classical homotopy category:

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

### Products of simplices

Some fundamental fact about products of simplicial sets:

###### Proposition

(non-degenerate $(p+q)$-simplices in $\Delta[p] \times \Delta[q]$)
For $p,q \,\in\, \mathbb{N}$ the non-degenerate simplices in the Cartesian product (Prop. )

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

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

(8)$\Delta[p+q] \xrightarrow{\;\; \sigma \;\;} \Delta[p] \times \Delta[q]$

which satisfy the following equivalent conditions:

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:

###### Proof

From this Prop. it is clear (see this Remark) 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 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.

### Nerve of the inertia groupoid

Some basic facts about the nerve of an inertia groupoid:

###### Proposition

(inertia groupoid of delooping groupoid is adjoint action groupoid)
The inertia groupoid $\Lambda \mathbf{B} G$ is isomorphic to the action groupoid of the adjoint action of $G$ on itself:

$\Lambda \mathbf{B}G \;\simeq\; G_{ad} \sslash G \;=\; \left( G \times G \underoverset {Ad_{(-)}(-)} {pr_2} {\rightrightarrows} G \right)$

This follows by immediate inspection. For more discussion see at free loop space of a classifying space the section Examples – For finite groups.

###### Proposition

The groupoid convolution algebra of the inertia groupoid of the delooping groupoid $\mathbf{B}G$ is the Drinfeld double of the group convolution algebra of $G$.

###### Definition

(minimal simplicial circle)
Write

$S \;\coloneqq\; \Delta/\partial\Delta \;\;\; \in \; sSet$

for the simplicial set with exactly two non-degenerate cells,

• one of which in degree 0, which we denote by $ = $,

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

###### Proposition

(normalized chain complex of minimal simplicial circle)
The normalized chain complex of the free simplicial abelian group of the minimal simplicial circle $S$ (Def. ) has the group of integers in degrees 0 and 1, and all differentials are zero:

$N_\bullet \circ \mathbb{Z}(S) \;\simeq\; \left[ \array{ \vdots \\ \big\downarrow \\ 0 \\ \big\downarrow \\ \mathbb{Z} \\ \big\downarrow {}^{\mathrlap{ 0 }} \\ \mathbb{Z} } \;\; \right] \;\simeq\; \mathbb{Z} \,\oplus\, \mathbb{Z} \,.$

The following proposition follows on abstract grounds, but the explicit component-based proof we give is necessary in order to understand the transgression-formula for cocycles in the group cohomology of $G$ to cocycles on the inertia groupoid.

###### Proposition

The nerve of the inertia groupoid of a delooping groupoid of a finite group $G$ is isomorphic to the simplicial hom complex out of the minimal simplicial circle $S$ (Def. ) into the simplicial classifying space $\overline{W}G$ (Rem. ):

$N\big( \Lambda \mathbf{B}G\big)_\bullet \;\simeq\; [S,\overline{W}G]_\bullet \,.$

###### Proof

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)

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

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

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

As a consequence:

###### Proposition
$[S,\overline{W}G] \times S \xrightarrow{\;\;} \overline{W}G$

(out of the product of the simplicial hom complex out of $S$ with $S$) takes any non-degenerate $n+1$-simplex of $[S,\overline{W}G] \times S$ of the form (still in the path notation discussed at product of simplices)

$\array{ (\gamma, ) &\xrightarrow{ (g_{n-1}, id) }& ( Ad_{n-1}(\gamma),  ) &\xrightarrow{ (g_{n-2}, id) }& \cdots &\xrightarrow{ (g_{j}, id) }& \big( Ad_{j}(\gamma),  \big) \\ && && && \big\downarrow {}^{\mathrlap{ (e, [ , ]) }} \\ && && && (Ad_j(\gamma), ) &\xrightarrow{ (g_{n-j},id) } & \cdots &\xrightarrow{ (g_0,id) }& (Ad_0(\gamma), ) \mathrlap{\,,} }$

where we are abbreviating

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

to the following $n+1$ simplex of $\overline{W}G$:

$\array{ \bullet &\xrightarrow{ g_{n-1} }& \bullet &\xrightarrow{ g_{n-2} }& \cdots &\xrightarrow{ g_{j} }& \bullet \\ && && && \big\downarrow {}^{\mathrlap{ Ad_j(\gamma) }} \\ && && && \bullet &\xrightarrow{ g_{n-j} } & \cdots & \xrightarrow{ g_0 } & \bullet \mathrlap{\,.} }$

### Transgression in cohomology

###### Definition

(transgression) For $\mathcal{A} \,\in\, sAb$, hence with
free loop space of its classifying space given (by this Prop.) by

(9)$\Lambda \mathbf{B}\mathcal{A} \;\simeq\; \mathcal{A} \times \mathbf{B}\mathcal{A}$

we say that transgression in $\mathcal{A}$-cohomology is

$H \big( -;\, \mathbf{B}\mathcal{A} \big) \xrightarrow{ [S, -] } H \big( \Lambda(-);\, \mathcal{A} \times \mathbf{B}\mathcal{A} \big) \xrightarrow{\;\; pr_2 \;\; } H \big( \Lambda(-);\, \mathcal{A} \big) \,.$

###### Proposition

(free loop space of classifying space identified via Eilenberg-Zilber map)
For $n \in \mathbb{N}_+$ and for $A \,\in\, Ab$ the integers or the circle group, the following composite of is a simplicial weak equivalence

(10)\begin{aligned} \big[ S, \, B^n A \big]_\bullet & \;=\; \big[ S, \, frgt \circ DK(A[n]) \big]_\bullet \\ & \;\simeq\; sSet \big( S \times \Delta[\bullet], \, frgt \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( frgt \circ DK \big( [ \underset{ \mathbb{Z} \oplus \mathbb{Z} }{ \underbrace{ N \circ \mathbb{Z}(S) } }, \, A[n] ] \big) \Big)_\bullet \\ & \;\simeq\; \Big( frgt \circ DK \big( A[n] \oplus A[n-1] \big) \Big)_\bullet \;=\; B^n A \times B^{n-1} A \end{aligned}

Here all isomorphisms are hom-isomorphisms of the above adjunctions, the step denoted $EZ_S$ is pre-composition with the Eilenberg-Zilber map, and under the brace we are using Prop. .

###### 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 retracts the homotopy groups of $B^n A \times B^{n-1}A$ onto those of $\big[S, B^n A \big]$. But, by (9), 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.

## Proof of the component formula

We prove that the formula (3) indeed expresses transgression in group cohomology.

###### Proof

Consider the following sequence of natural isomorphisms of hom-sets of the above homotopy categories (Def. ):

(11)\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, \, frgt \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, frgt \circ DK\big( A[n] \big)\big] \Big) \\ & \;\simeq\; Ho(sSet) \Big( [S,\overline{W}G] \times S, \, frgt \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} \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

Chasing a group cocycle (Def. ) through this sequence, and using Prop. when it gets sent through the Eilenberg-Zilber map, manifestly outputs the sum formula (3) to be proven.

Hence it only remains to see that this concrete composite (11) is equal to the abstractly defined transgression map (Def. ).

This follows by Prop. . In detail, since:

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

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

the chain of hom-isomorphisms of derived adjunctions in (11) is covered (through the quotient by left homotopy that defines the homotopy category of a model category according to this Lemma) by “the same” chain of hom-isomorphisms of plain adjunctions. Using here that every simplicial set $X$ is the colimit over its elements $\Delta[k] \in el(X)$ (this Prop.) and that the 1-category-theoretic hom functors turn this colimit, in their first argument, into a limit (this Prop.), we find that the composite in (11) is covered by

\begin{aligned} sSet \Big( \big[S, \overline{W}G \big], \, \big[S, frgt \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, frgt \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, frgt \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], \, frgt \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], \, frgt \circ DK \big( A[n] \oplus A[n-1] \big) \Big) \\ & \;\simeq\; sSet \Big( [S,\overline{W}G], \, frgt \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 (10) from Prop. .

The transgression map is alluded to in

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

• 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)

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

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