crossed complex

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###Context###
#### Higher category theory
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[[!include higher category theory - contents]]
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#### Homological algebra
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[[!include homological algebra - contents]]
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#Contents#
* automatic table of contents goes here
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## Idea

A _crossed complex_ (of [[groupoid|groupoids]]) is a nonabelian and [[horizontal categorification|many object]] generalization of a [[chain complex]] of [[abelian group|abelian groups]].

Crossed complexes are an equivalent way to encode the information contained in [[strict ∞-groupoids]]: the groups appearing in the crossed complex in degree $n \geq 2$ are the source-fibers of the collection of $n$-morphisms of the $\omega$-groupoid.

See also [[homotopy n-type]].

One way to think of a crossed complex is as a chain complex in which the bottom part is a crossed module and the rest is a chain complex of modules over the fundamental group of the crossed complex (that is its cokernel).  This is easy to think of in the case where there is a single object (*crossed complex of groups*), and it is a simple step to extend to the many object case.

Later on we will look in a bit more detail at the fundamental crossed complex of a filtered space, and that is a good example to keep in mind.  For simplicity assume we have a CW-complex, or similar space, together with a filtration by some nice subspaces.  We have the fundamental groupoid, $\Pi_1(X_1,X_0)$, of the '1-skeleton' based at the vertices. For any vertex $x$, we then have $\pi_2(X_2,X_1,x)$, the relative homotopy group of the 2-dimensional stuff relative to the 1-dimensional stuff, based at $x$.  Varying $x$ we get a family of groups which we think of as a groupoid having just vertex groups without any arrows joining distinct vertices. In the next dimension we have $\pi_3(X_3,X_2,x)$, which is the relative homotopy group taking account of the 3-cells modulo the 2-cells, (which is abelian), and so on. Change of base point gives an action of  $\Pi_1(X_1,X_0)$ on all of these. It was studying these groups , actions etc. that gave the abstract definition that follows.

## Definition

### As sequences of groups and groupoids {#SequencesOfGroups}

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

A **crossed complex** ("of groupoids") $C$ is

* a [[groupoid]]  $C_1 \stackrel{\overset{\delta_t}{\to}}{\underset{\delta_s}
{\to}} C_0$

* together with a sequence of [[skeleton|skeletal]] groupoids $(C_k)_{k = 2}^\infty$ over $C_0$, i.e. of  bundles $C_k = \coprod_{x \in C_0} (C_k)_x$ of groups over $C_0$, [[abelian group|abelian]] for $k \geq 3$,  sitting in a diagram

$$
 \begin{array}{c}
   \cdots &\to& C_3 &\stackrel{\delta}{\to}& C_2
   &\stackrel{\delta}{\to}&
   C_1 &
  \stackrel{\overset{\delta_t}{\to}}{\underset{\delta_s}{\to}} &
  C_0
  \\
  && \downarrow && \downarrow && \downarrow^{\mathrlap{\delta_s}}
  && \downarrow^{\mathrlap{=}}
   \\
 \cdots &\to&
 C_0
 &\stackrel{=}{\to}&
 C_0
 &\stackrel{=}{\to}&
 C_0
 &\stackrel{=}{\to}&
  C_0
 \end{array}
$$

* together with an [[action]] of $C_1$ on $C_k$ for $k \geq 0$

such that

* the morphisms $\delta_k$ for $k \geq 2$ are morphisms of groupoids over $C_0$, compatible with the action by $C_1$

* $im(\delta_2) \subset C_1$ acts by conjugation on $C_2$ and trivially on $C_k$ for $k \geq 3$

* $\delta_{k-1} \circ \delta_k = 0$ for $k \geq 3$.

There is an obvious notion of [[morphism|morphisms]] $f : C \to D$ of crossed complexes, being sequences of maps $(f_k : C_k \to D_k)$ preserving all the above structure. The resulting [[category]] is often denoted $Crs$ or $CrsCpx$.

=--

### From strict $\infty$-groupoids

While the above definition of a crossed complex may seem slightly 'baroque', it can naturally be understood as being precisely the data obtained from a [[globular set|globular]] [[strict ∞-groupoid]] by retaining for $k \geq 2$ precisely only those [[k-morphism|k-morphisms]] whose source is a an identity $k-1$-morphisms on an object.

+-- {: .un_defn}
###### Definition
**(crossed complex associated to a strict $\infty$-groupoid)**

For $\mathcal{G}$ a [[globular set|globular]] [[strict ∞-groupoid]]

$$
  \cdots
  \stackrel{\overset{t}{\to}}{\underset{s}{\to}}
  \mathcal{G}_3
  \stackrel{\overset{t}{\to}}{\underset{s}{\to}}
  \mathcal{G}_2
  \stackrel{\overset{t}{\to}}{\underset{s}{\to}}
  \mathcal{G}_1
  \stackrel{\overset{t}{\to}}{\underset{s}{\to}}
  \mathcal{G}_0
$$

the corresponding crossed complex $[\mathcal{G}]$ is defined as follows:

* the groupoid $[\mathcal{G}]_1 \stackrel{\to}{\to} [\mathcal{G}]_0$
  is just the groupoid
    $\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0$;
  underlying $\mathcal{G}$ by forgetting all
  [[k-morphism|k-morphisms]] for $k \geq 2$

* for $k \geq 2$ the bundle of groups $[\mathcal{G}]_k$
  is over
  $x \in \mathcal{G}_0$ the group of [[k-morphism|k-morphisms]] of $\mathcal{G}$
  whose source is the the identity on $x$:

  $$
    [\mathcal{G}]_k := \coprod_x s_k^{-1}(Id_x)
    \,,
  $$

  where the group operation is given by the horizontal composition of
  [[k-morphism|k-morphisms]] (along objects). By the [[Eckmann-Hilton argument]] this
  is indeed an [[abelian group]] structure for $k \geq 3$.

* The [[action]] of $[\mathcal{G}]_1$ on $[\mathcal{G}]_k$ is given by [[whiskering]]/[[conjugation]] of [[k-morphism|k-morphisms]] by 1-morphisms in $\mathcal{G}$.

* The boundary maps $\delta := t : [\mathcal{G}]_{k} \to [\mathcal{G}]_{k-1}$ are the restrictions of the target maps $t : \mathcal{G}_k \to \mathcal{G}_{k-1}$, sending a $k$-morphisms with source an identity on an object to its target $k-1$-morphism.

=--

Write $Str \infty Grpd$ for the 1-[[category]] of [[globular set|globular]] [[strict ∞-groupoids]]. The above construction defines an evident [[functor]]

$$
  [-] : Str \infty Grpd \to CrsCplx
  \,.
$$

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

The functor

$$
 [-] : Str \infty Grpd \to CrsCplx
$$

is an [[equivalence of categories]].

=--

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

The idea of the proof is that a [[strict ∞-groupoid]] may completely be reconstructed from its objects, 1-morphisms and those $(k \geq 2)$-morphisms that start at an identity by using the action of the 1-morphisms on the higher morphisms induced by conjugation.

For instance a [[2-morphism]]

$$
  \array{
    & \nearrow  \searrow^{\mathrlap{f}}
    \\
    x &\Downarrow& y
    \\
    & \searrow \nearrow_{\mathllap{g}}
  }
$$

is, by the [[exchange law]], equal to the horizontal composite of the 2-morphism

$$
  \array{
    & \nearrow  \searrow^{\mathrlap{f}}
    \\
    x &\Downarrow& y & \stackrel{f^{-1}}{\to} & x
    \\
    & \searrow \nearrow_{\mathllap{g}}
  }
$$

(whose source is the identity on $x$) with the 1-morphisms $f$.

A detailed proof is in

* [[Ronnie Brown]], [[Philip Higgins]], _The equivalence of $\infty$-groupoids and crossed complexes_ , Cah. Top. Géom. Diff. 22, 371-386, 1981. ([pdf](http://www.bangor.ac.uk/~mas010/pdffiles/x-comp.pdf))

Notice that this article says "$\infty$-groupoid" for _strict globular $\infty$-groupoid_ and "$\omega$-groupoid" for _strict cubical $\infty$-groupoid_ .

=--

This is a nonabelian and globular version of the [[Dold-Kan correspondence]].

### From chain complexes of modules

We describe a functorial construction of a crossed complex starting with a [[chain complex]] of [[modules over a groupoid]] $(A_n, \mathcal{H})$. As a special case it in particular gives an functor sending ordinary [[chain complex|chain complexes]] of [[abelian group|abelian groups]] into the category of [[crossed complex|crossed complexes]], and hence into [[strict ∞-groupoid|strict ∞-groupoids]]. See also [[Nonabelian Algebraic Topology]].

Recall the definition of the [semidirect product groupoid](#AdjointModule)  $\mathcal{H} \ltimes A_n$.

+-- {: .un_def}
###### Definition
**(crossed complex from a chain complex)**

For $A$ a [[chain complex]] of [[modules over a groupoid]] $\mathcal{H}$, let $\Theta A \in Crs$ be the [[crossed complex]]

$$
  \Theta A := \kappa^* \Theta' A
  \,,
$$

where

$$
  \Theta' A :=
  \left[
    A_n
      \stackrel{\partial_n}{\to}
    A_{n-1}
     \stackrel{}{\to}
     \cdots
     \stackrel{}{\to}
    A_{3}
    \stackrel{\partial_3}{\to}
    A_2
    \stackrel{(0,\partial_2)}{\to}
    \mathcal{H}\ltimes A_1
  \right]
$$

and where

$$
  \kappa : P(A_0, \mathcal{H}) \to \mathcal{H} \ltimes A_0
$$

is the [canonical covering morphism](#AdjointModule) from above.

$$
  \array{
    \cdots \to &
    (\Theta A)_3
    &\to&
    (\Theta A)_2
    &\to&
    (\Theta A)_1
    &\to&
    P(A_0, \mathcal{H})
    \\
    & \downarrow && \downarrow && \downarrow && \downarrow
    \\
    \cdots \to &
    A_3
    &\stackrel{\partial_3}{\to}&
    A_2
    &\stackrel{(0,\partial_2)}{\to}&
    \mathcal{H} \ltimes A_1
    &\stackrel{(1, \partial_1)}{\to}&
    \mathcal{H} \ltimes A_0
  }
  \,.
$$

Here $\mathcal{H} \ltimes A_1$ acts on $A_n$ for $n \geq 2$ via the projection $\mathcal{H} \ltimes A_1 \to \mathcal{H}$, i.e. $A_1$ acts trivially.
(...)

Finally set $\Theta(A)_0 := A_0$.

=--

We spell out what this boils down to explicitly.

**Explicit description**

Let $A_\bullet$ be a [[chain complex]] of [[modules over a groupoid|modules over the groupoid]] $\mathcal{H}$. Then the [[crossed complex]] $\Theta(A)$ is the following.

* Its set of objects is $\Theta(A)_0 = A_0$.

  Remember that $A_0$ itself is a module over $\matcal{H} = (\mathcal{H}_1 \stackrel{\to}{\to} \mathcal{H}_0)$, so that $A_0 = \corpdod_{p \in \mathcal{H}_0} (A_0)_p$.

* For $x \in (A_0)_p$ and $y \in (A_0)_q$ a morphism in $\Theta(A)_1$ from $x$ to $y$ is labeled by $h \in \mathcal{H}_1$ and $a \in (A_1)_q$

  $$
    x \stackrel{(h,a)}{\to} (y = \rho(h)(x) - \partial a)
    \,,
  $$

  where $\rho$ denotes the [[action]] of $\mathcal{H}$ on $A_0$.

  The composition law is given by

  $$
    \array{
       && y
       \\
       & {}^{\mathllap{(h_1, a_1)}}\nearrow && \searrow^{\mathrlap{(h_2,a_2)}}
       \\
       x &&\stackrel{(h_1 \circ h_2, \rho(h_2)(a_1) + a_2)}{\to}&& z
    }
    \,.
  $$

* For $k \geq 2$ the family of groups $\Theta(A)_k$ is over $x \in (A_0)_p$ the group $(A_k)_q$

  $$
    \Theta(A)_{k \geq 2} = \coprod_{p \in \mathcal{H}_0} \coprod_{x\in (A_0)_q} (A_k)q
  $$

* The boundary maps and actions are the obvious ones...

+-- {: .un_example}
###### Example
**(ordinary abelian chain complex as crossed complex)**

Let $C_\bullet$ be an ordinary [[chain complex]] of abelian groups, i.e. a chain complex of [[module over a groupoid|modules over the trivial groupoid]].

Then $(\Theta C)_1$ is the groupoid with objects $C_0$ and morphisms $\{x \stackrel{b}{\to} (x + \partial b)\}$. And for $n \geq 2$ we have that $(\Theta C)_n$ is $\coprod_{x \in C_0} C_n$.

=--

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

These form a pair of [[adjoint functor|adjoint functors]]

$$
  (\nabla \dashv \Theta)
  :
  Chn
  \stackrel{\overset{\nabla}{\leftarrow}}{\underset{\Theta}{\to}}
  Crs
$$

where...

=--

This is proposition 7.4.29.

(...)

## Examples

### In low degree

Say that a crossed complex $C$ is $n$-[[truncated]] if $C_k$ is trivial for $k \gt k$.

Then

* 0-[[truncated]] crossed complexes are canonically equivalent to [[set|sets]], equivalent to [[homotopy n-type|homotopy 0-types]].

* 1-[[truncated]] crossed complexes are canonically equivalently to [[groupoid|groupoids]], equivalent to [[homotopy n-type|homotopy 1-types]]).

* 2-[[truncated]] crossed complexes are equivalent to strict [[2-groupoid|2-groupoids]] equivalent to [[homotopy n-type|homotopy 2-types]].

  2-[[truncated]] and 0-[[connected]] crossed complex, i.e. a 2-truncated one for which $C_0 = *$ is the point is the same as a [[crossed module]] of groups. The equivalence of these to [[strict 2-group|strict 2-groups]] is due to

  R. Brown and C.B. Spencer, _G-groupoids, crossed modules and the fundamental groupoid of a  topological group_, Proc. Kon. Ned. Akad. v. Wet, 79, (1976), 296 – 302.)

A discussion of the kind of homotopy types generally modelled by crossed complexes, namely a _linear_ model is in [[homotopy n-type]].

### Abelian chain complexes

The notion of crossed complex generalizes the notion of [[chain complex]] of [[abelian group|abelian groups]]. Clearly in degree $k \geq 3$ a crossed complex with $C_0 = *$ _is_ a chain complex of abelian groups. To regard the first 2 degrees $A_1 \stackrel{\delta}{\to} A_0$ of a chain complex of abelian groups as a crossed module, form the groupoid

$$
  A_0 \times A_1 \stackrel{\overset{p_1 + \delta}{\to}}{\underset{p_1}{\to}}
  A_0
$$

and take the action of this groupoid on all $C_k$ to be trivial. This yields a functor

$$
  \theta : ChainCplx(Ab) \to CrsCpl
$$

that embeds chain complexes of abelian groups into crossed complexes.

### Complex of modules over a groupoid

The embedding of chain complexes of abelian groups into crossed complexes generalizes to an embedding of chain complexs of [[modules over a groupoid]]

$$
  \theta : ChainCplx(GrpdMod) \to CrsCplx
  \,.
$$

For details see _[[Nonabelian Algebraic Topology]]_, [section 7.4.v](#http://ncatlab.org/nlab/show/Nonabelian+Algebraic+Topology#CrsFromCh).

### Fundamental crossed complex

If $X_*$ is a [[filtered space]], there is a crossed complex $\Pi X_*$ -- the [fundamental crossed complex](http://ncatlab.org/nlab/show/Nonabelian+Algebraic+Topology#FundamentalCrossedComplex) which corresponds to a (filtered and) [[strict ∞-groupoid]] version of the [[fundamental ∞-groupoid]] of $X$. In degree 1 it is the subgroupoid $\Pi_1(X_1,X_0)$ of the [[fundamental groupoid]] $\Pi_1(X_1)$ of $X_1$ on objects in $X_0$.
In degree $n \gt 1$ it is the family of [[relative homotopy group|relative homotopy groups]] $\{\pi_n(X_n,X_{n-1},p) : p\in X_0\}$.

This gives a functor $\Pi$ from filtered spaces to crossed complexes, which may be used to construct the generalisation of the [[Dold-Kan correspondence]], which in this case goes between crossed complexes and [[simplicial T-complex|simplicial T-complexes]].

### Fundamental crossed complex of the $n$-simplex

An important special case of the above is when the filtered space is a [[CW-complex]] and the filtration is by skeleta.  Particularly useful instances of this are the $n$-cubes and $n$-simplices, with their CW-filtration.  We obtain $\Pi(I^n)$ and $\Pi(\Delta^n)$.  These are used to define cubical and simplicial [[nerve|nerves]] of a crossed complex and these in turn define the [[Dold-Kan correspondence]] mentioned above.  For instance if $C$ is a crossed complex, then its simplicial nerve is the simplicial set with $Ner(C)_n =
Crs(\Pi(\Delta^n),C)$ in dimension $n$.

+-- {: .un_example}
###### Example
**(fundamental crossed complex of the $n$-simplex)**

The topological $n$-[[simplex]] $\Delta^n$ is canonically a [[filtered space]] with $(\Delta^n)_k$ being the union of its $k$-faces.

Then we have that $\Pi_1((\Delta^n)_1, (\Delta^n)_0)$ is the groupoid whose objects are the $n+1$ vertices of $\Delta^n$ and which has precisely one morphism $x_i \to x_j$ for each ordered pair $x_i,x_j \in (\Delta^n)_0$
(all of them being [[isomorphism|isomorphisms]])

$$
  \Pi_1((\Delta^2)_1,(\Delta^2)_0) =
  \left\{
     \array{
     && x_1
     \\
     & \nearrow\swarrow && \searrow \nwarrow
     \\
    x_0 &&\stackrel{\leftarrow}{\to}&& x_2
    }
  \right\}
  \,.
$$

At any $x_i$ the relative homotopy group $\pi_2((\Delta^n)_2,(\Delta^n)_1, x_i)$ is a group on the set of 2-faces that have $x_i$ as a 0-face: there is a unique homotopy class of disks in $\Delta^n$ that sits in the 2-faces $(\Delta^n)_2$, whose base point is at $x_j$ and whose boundary runs along the boundary of a given 2-face of $\Delta^n$.

So (using the equivalence of crossed complexes with strict $\omega$-groupoids) for instance $\Pi \Delta^2$ is generated from $\Pi_1((\Delta^2)_1,(\Delta^2)_0)$ as above and a 2-cell

$$
  \array{
     && x_1
     \\
     & \swarrow &\Downarrow& \nwarrow
     \\
    x_0 &&\to&& x_2
    }
$$

under whiskering and composition. For instance whiskering this with $x_1 \to x_2$ yields the 2-morphism

$$
  \array{
     && x_1
     \\
     & \swarrow &\swArrow& \searrow
     \\
    x_0 &&\to&& x_2
    }
  \,.
$$

One sees that $\Pi \Delta^2$ is the strict groupoidification of the second [[oriental]].

Generally, $\Pi \Delta^n$ is the $n$-groupoid freely generated from $k$-morphisms for each $k$-face of $\Delta^n$.

=--

## Crossed complexes as Moore complexes

Crossed complexes (of groups) correspond to [[group T-complex|group T-complexes]]. Any group $T$-complex is a simplicial group and in the entry for them it is mentioned that a simplicial group has a group $T$-complex structure if and only if $\mathcal{N}G\cap D$ is the trivial graded subgroup, where
$D = (D_n)_{n\geq 1}$ is the graded subgroup of $G$ generated by the degenerate elements. If $G$ is such a group $T$-complex then its [[Moore complex]] has a natural structure of a crossed complex. In general the obstruction to a given simplicial group to have  Moore complex which is a crossed complex is exactly that graded subgroup, $\mathcal{N}G\cap D$. (The [[Whitehead product|Whitehead products]] for $G$ live in this graded subgroup, so this provides one way of showing that the homotopy types representable by crossed complexes have trivial  Whitehead products.)

Conversely, for any crossed complex, $C$, there is a simplicial group, $K(C)$, constructed using an analogue of the inverse in the [[Dold-Kan correspondence]], which is a group $T$-complex and whose Moore complex is isomorphic to $C$.

The generalisation to general crossed complexes (of groupoids) and simplicially enriched groupoids
is quite easy to do. We will usually state results below for the group case, leaving the general case to the 'reader'.

## From simplicial group(oid)s to crossed complexes

It is fairly clear that crossed complexes / group(oid) $T$-complexes correspond to [[simplicial group]](oid)s in which certain  equations hold.  It therefore is reasonable that they are equivalent to a variety / [[reflective subcategory]] in the category, $SSet Grpd$, of simplicially enriched groupoids.  (The discussion in the entry on [[group T-complex]] is relevant here.)

There is a functor $C(-)$ from simplicial groups to crossed complexes given by

$${C}(G)_{n+1} = \frac{\mathcal{N}G_n}{(\mathcal{N}G_n\cap D_n)d_0(\mathcal{N}G_{n+1}\cap D_{n+1})},$$

in higher dimensions with at its 'bottom end', the crossed module,

 $$\frac{\mathcal{N}G_1}{d_0(\mathcal{N}G_2\cap D_2)} \to \mathcal{N}G_0$$

with $\partial$ induced from the boundary in the Moore complex.

The category of [[crossed complexes]] form a variety in the category of all [[hypercrossed complexes]]. Alternatively, groupoid T-complexes (the groupoid version of [[group T-complex]]) form
a variety in the category of all simplicial groups.

## Related concepts

* [[group]]

* [[2-group]], [[crossed module]], [[differential crossed module]]

* [[3-group]], [[2-crossed module]] / [[crossed square]], [[differential 2-crossed module]]

* [[n-group]]

* [[∞-group]], [[simplicial group]], **crossed complex**, [[hypercrossed complex]]

* [[model structure on crossed complexes]]

## References

Crossed complexes were defined by Blakers in 1948 (following a suggestion of [[Samuel Eilenberg]]) and developed by Whitehead in 1949 and 1950 (but these authors used different terminology). They were applied by [[Johannes Huebschmann]] to [[group cohomology]] in 1980. They were further developed in  series of articles by Ronnie Brown and collaborators in the context of [[nonabelian algebraic topology]], and partly because they were found equivalent to a form of (strict) cubical $\omega$-groupoid with _connections_. This equivalence enabled a number of new results, including van Kampen type theorems and monoidal closed structures for crossed complexes.

Textbook treatment is in

* [[Ronnie Brown]], [[Philip Higgins]], [[Rafael Sivera]], _[[Nonabelian Algebraic Topology]]_

* [[Tim Porter]], _[[timporter:crossed menagerie|Crossed Menagerie]]_ .

A survey of the use of crossed complexes is in

* [[Ronnie Brown]], _Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems_,  to appear in Michiel Hazewinkel (ed.), Handbook of Algebra, volume 6, Elsevier, 2008/2009. ([arxiv:math.AT/0212274 v7](http://arxiv.org/abs/math.AT/0212274)).

The equivalence of [[strict omega-groupoid|strict omega-groupoids]] and crossed complexes is discussed in

* [[Ronnie Brown]], [[Philip Higgins]], _The equivalence of $\infty$-groupoids and crossed complexes_ , Cah. Top. Géom. Diff. 22, 371-386, 1981. ([pdf](http://www.bangor.ac.uk/~mas010/pdffiles/x-comp.pdf))

Notice that this article says "$\infty$-groupoid" for _strict globular $\infty$-groupoid_ and "$\omega$-groupoid" for _strict cubical $\infty$-groupoid_ .

For the relation to [[group cohomology]] see

* [[Johannes Huebschmann]], _Crossed $n$-fold extensions and group cohomology_ ([web](http://dx.doi.org/10.1007/BF02566688))

[[!redirects crossed complexes]]

[[!redirects crossed complex of groupoids]]
[[!redirects crossed complexes of groupoids]]


crossed G-algebra

# Crossed G-algebra
* table of contents
{: toc}

##Idea

(Here $G$ will be a group (a discrete one for the moment).)

A crossed $G$-algebra is a type of $G$-[[graded algebra]], with an inner product and a 'crossed' or 'twisted' multiplication. They arise as the analogues of [[Frobenius algebras]] for 2d-[[HQFTs]], [[equivariant TQFTs]] and in slight generality in the study of symmetries of singularities.  They were introduced by [[Turaev]] in 1998. The generalised structures have been studied by [[R. Kaufmann]].

##Nomenclature
[[Greg Moore|Moore]] and [[Graeme Segal|Segal]] (see references below) do not like the term 'crossed algebra' and suggest the alternative name 'Turaev algebra'.

##Definitions:

We will lead up to the definition of crossed $G$-algebra through various stages.

We need a particular form of [[graded algebra]] in which the summands are projective modules, so we give that form first.

###Graded $G$-algebra

A **graded $G$-algebra** or **$G$-algebra** over a field (or more generally a commutative ring), $\mathbb{k}$ is an associative algebra, $L$, over $\mathbb{k}$ with a decomposition,

$$L = \bigoplus_{g\in G} L_g,$$

as a direct sum of projective $\mathbb{k}$-modules of finite type such that

(i) $L_g L_h \subseteq L_{gh}$ for any $g,h \in G$
(so, if $\ell_1$ is graded $g$, and $\ell_2$ is graded $h$, then $\ell_1\ell_2$ is graded $gh$),

and

(ii) $L$ has a unit $1 = 1_L\in L_1$ for 1, the identity element of $G$.

###Example:###

(i)  The [[group algebra]], $\mathbb{k}[G]$, has an obvious $G$-algebra structure in which each summand of the decomposition is free of dimension 1.

(ii)  For any associatve $\mathbb{k}$-algebra, $A$, the algebra,   $A[G]= A\otimes_\mathbb{k}\mathbb{k}[G]$, is also $G$-algebra.  Multiplication in $A[G]$ is given by $(ag)(bh) = (ab)(gh)$ for $a,b \in A$, $g,h \in G$, in the obvious notation.

(iii) If $G$ is the trivial group, then a $G$-graded algebra is just an algebra (of finite type), of course.

###Frobenius $G$-algebra

A **Frobenius $G$-algebra** is a $G$-algebra, $L$, together with a symmetric $\mathbb{k}$-bilinear form,

$$\rho : L\otimes L \to \mathbb{k}$$

such that

(i)  $\rho(L_g\otimes L_h) = 0$ if $h \neq g^{-1}$;

(ii) the restriction of $\rho$ to $L_g \otimes L_{g^{-1}}$ is non-degenerate for each $g\in G$;

and

(iii)  $\rho(ab,c) = \rho(a,bc)$ for any $a,b,c \in L$.

\

We note that (ii) implies that $L_{g^{-1}} \cong  L_g^*$, the dual of $L_g$.

###Examples continued:

(i) The group algebra, $L = \mathbb{k}[G]$, is a Frobenius $G$-algebra with $\rho(g,h) = 1$ if $gh = 1$, and 0 otherwise, and then extending linearly. (Here we write $g$ both for the element of $G$ labelling the summand $L_g$, and the basis element generating that summand.)

(iii) For $G$ trivial, a Frobenius 1-algebra is a [[Frobenius algebra]].

###Crossed $G$-algebra

Finally the notion of  crossed $G$-algebra combines the above with an action of $G$ on $L$, explicitly:

A **crossed $G$-algebra** over $\mathbb{k}$ is a Frobenius $G$-algebra, $L$, over $\mathbb{k}$ together with a group homomorphism,

$$\varphi: G \to Aut(L)$$

satisfying:

(i) if $g\in G$ and we write $\varphi_g = \varphi(g)$ for the corresponding automorphism of $L$, then $\varphi_g$ preserves $\rho$, (i.e., $\rho(\varphi_ga,\varphi_gb) = \rho(a,b)$) and

$$\varphi_g(L_h) \subseteq L_{ghg^{-1}}$$

for all $h\in G$;

(ii) $\varphi_g|_{L_g} = id$ for all $g\in G$;

(iii) (twisted or crossed commutativity) for any $g,h \in G$, $a\in L_g$, $b\in L_h$, $\varphi_h(a)b = ba$;

(iv) for any $g,h \in G$ and $c \in L_{ghg^{-1}h^{-1}}$,

$$Tr(c\varphi_h : L_g \to L_g) = Tr(\varphi_{g^{-1}}c : L_h \to L_h),$$

where $Tr$ denotes the $\mathbb{k}$-valued trace of the endomorphism. (The homomorphism $c\varphi_h$ sends $a\in L_g$ to $c\varphi_h(a) \in L_g$, whilst $(\varphi_{g^{-1}}c)(b) = \varphi_{g^{-1}}(cb)$ for $c \in L_h$. This is sometimes called the 'torus condition'.)

###Note:

a) We note that the usage of terms differs between Turaev's book (2010) and here, as we have taken 'crossed $G$-algebra' to include the Frobenius condition. We thus follow Turaev's original convention (preprint 1999) in this.

b) The useful terminology 'twisted sector' in a crossed $G$-algebra refers to a summand, $L_g$, for and index, $g$, which is  not the identity element of $G$, of course, then $L_1$ is called  the 'untwisted sector'

##Operations on crossed $G$-algebras

(To be added)

##Related notions

* crossed $C$-algebra for a [[crossed module]] of groups;

*[[twisted G-algebra]]

##References

* V. [[Turaev]],  _Homotopy Quantum Field Theory_ ,
EMS Tracts in Math.10, European Math. Soc. Publ. House, Zurich 2010. (for the detailed development and the links with low dimensional topology and [[HQFT]]s).

* [[Greg Moore]], [[Graeme Segal]], _D-branes and K-theory in 2D topological field theory_ ([arXiv hep-th 0609042](http://arxiv.org/abs/hep-th/0609042))

[[!redirects crossed G-algebras]]
[[!redirects Turaev algebra]]


crossed G-set

#Contents#
* table of contents
{:toc}

## Definition

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

Let $G$ be a [[group]]. A _crossed G-set_ consists of the following data.

1. A [[G-set]], that is to say, a set $X$ together with an [[action]] $\cdot : U(G) \times X \rightarrow X$ of $G$ upon it, where $U$ is the [[forgetful functor]] from [[Grp]] to [[Set]].

2. A map of sets $\left| - \right|: X \rightarrow U(G)$.

We require that $\left| g \cdot x \right| = g \left| x \right| g^{-1}$ for all $g \in G$ and $x \in X$.

=--

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

Let $\underline{X}$ and $\underline{Y}$ be crossed $G$-sets. A _morphism_ from $\underline{X}$ to $\underline{Y}$ is a map of sets $f : X \rightarrow Y$ which is a morphism of $G$-sets, that is to say, $f(g \cdot x) = g \cdot f(x)$ for all $g \in G$ and $x \in X$, and which has the property that $\left| f(x) \right| = \left| x \right|$ for all $x \in X$.

=--

Crossed $G$-sets and morphisms between them assemble into a category. The identity morphism on a crossed $G$-set $\underline{X}$ is defined by the identity map $X \rightarrow X$.

##Braided monoidal structure on the category of crossed G-sets

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

Let $\underline{X}$ and $\underline{Y}$ be crossed G-sets. The _tensor product_ of $\underline{X}$ and $\underline{Y}$ is the product of the underlying $G$-sets, namely the product of sets $X \times Y$ equipped with the action $g \cdot (x,y) = (g \cdot x, g \cdot y)$, equipped with the map $X \times Y \rightarrow U(G)$ given by $(x,y) \mapsto \left| x \right| \left| y \right|$.

=--

Extending in the evident way to morphisms, the tensor product of crossed G-sets equips the category of crossed $G$-sets with the structure of a (not strict, and not symmetric) monoidal category. The unit is the set with one element equipped with its unique crossed $G$-set structure.

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

Let $\underline{X}$ and $\underline{Y}$ be crossed $G$-sets. The _braiding_ of $\underline{X}$ and $\underline{Y}$ is the morphism of crossed $G$-sets $X \times Y \rightarrow Y \times X$ given by $(x,y) \mapsto (\left| x \right| \cdot y, x)$.

=--

The braiding of crossed $G$-sets gives rise, together with the monoidal structure of Definition \ref{MonoidalStructure}, to a braided monoidal structure on the category of crossed $G$-sets.

##References

* [[Peter Freyd]] and [[David Yetter]], _Braided compact closed categories with applications to low dimensional topology_, Adv. Math. **77** (1989), 156--182. (Section 4.2)

* [[André Joyal]] and [[Ross Street]], _Braided tensor categories_ , Adv. Math. **102** (1993), 20--78.  (Example 5.1)


crossed group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **crossed group** is a [[presheaf]] with additional symmetry given by a fiberwise [[group]] structure and actions on the morphisms of the underlying [[category]].

## Definition

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

For $S$ a [[small category]], a **crossed $S$-group** is a [[presheaf]] $G : S^{op} \to Set$ equipped with

1. for each object $s \in S$ a group structure on $G_s$;

1. for all $s, r\in S$ a left $G_r$-[[action]] on the [[hom-set]] $S(s,r)$ ;

such that for all morphisms $\alpha : s \to r$ and $\beta : t \to s$ in $S$ and $g,h \in G_r$ we have

1. $g_*( \alpha \circ \beta)  = g_*(\alpha) \circ \alpha^*(g)_* \beta$;

1. $g_* (id_r) = id_r$;

1. $\alpha^* (g \cdot h) = h_*(\alpha)^*(g)\cdot \alpha^*(h)$;

1. $\alpha^*(e_r) = e_s$;

where $g_*$, $h_*$ denotes the group action and

$\alpha^* : G_r \to G_s$ the presheaf map.

The **total category** $S G$ of an crossed $S$-group $G$ is the category with the same objects as $S$, and with morphisms $r \to s$ being pairs $(\alpha, g) \in S(s,r)\times G_r$ and with composition defined by

$$
  (\alpha, g) \circ (\beta, h)
  =
  (\alpha \cdot g_*(\beta), \beta^*(g) \cdot h)
  \,.
$$

=--

([Ber-Moe, def. 2.1](#BergerMoerdijk)).

## Examples

* In case, $G$ is constant, i.e. $G_s=G$ and $\alpha^*(g)=g$ for all $s,\alpha \in S$, $S G$ reduces to the [[Grothendieck construction]] for the functor $G\to Cat$ that picks out $S$.

* Let $\Omega_{pl}$ be the category of finite planar [[tree|trees]]. Then the category $\Omega$ of non-planar finite rooted [[tree|trees]] arises as the total category of an $\Omega_{pl}$-crossed group which to a _planar_ tree $T$ assigns its group of non-planar automorphisms. This example is somewhat typical, since $\Omega_{pl}$ is a strict [[Reedy category]] where all automorphisms are trivial which gets enhanced with non-trivial symmetries to $\Omega$, now a [[generalized Reedy category]] (cf. below).

## Properties

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

If $S$ is equipped with a generalized Reedy structure, then an $S$-crossed group $G$ is called **compatible** with that generalized Reedy structure if

1. the $G$-action respects $S^+$ and $S^-$;

1. if $\alpha : r \to s$ is in $S^-$ and $g \in G_s$ such that $\alpha^* (g) = e_r$ and $g_*(\alpha) = \alpha$, then $g = e_s$.

=--

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

Let $S$ be a strict [[Reedy category]] and let $G$ be a compatible $S$-crossed group. Then there exists a unique dualizabe generalized Reedy structure on $S G$ for which the embedding $S \hookrightarrow S G$ is a morphism of generalized Reedy categories.

=--

([Ber-Moe, prop. 2.10](#BergerMoerdijk)).

## Remarks

* A related notion is "thickening" (cf. [Cisinski (8.5.8; 2006)](#Cisinski), [Isaacson (2010)](#Isaacson2010)).

## Related entries

* [[skew-simplicial set]]

* [[generalized Reedy category]]

* [[crossed homomorphism]]

* [[cyclic set]]

* [[symmetric set]]

* [[quaternionic set]]

## References

* [[Michael Batanin]], [[Martin Markl]], _Crossed interval groups and operations on the Hochschild cohomology_, arXiv:0803.2249 (2008). ([abstract](https://arxiv.org/abs/0803.2249))

* [[Clemens Berger]], [[Ieke Moerdijk]], _On an extension of the notion of Reedy category_ (2008) ([arXiv:0809.3341](http://arxiv.org/abs/0809.3341))
 {#BergerMoerdijk}

* [[Denis-Charles Cisinski]], _[[joyalscatlab:Les préfaisceaux comme type d'homotopie|Les préfaisceaux comme modèles des types d'homotopie]]_, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages ([pdf](http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf))
 {#Cisinski}

* [[Tobias Dyckerhoff]], [[Mikhail Kapranov]], _Crossed simplicial groups and structured surfaces_, arXiv:1403.5799 (2014). ([abstract](https://arxiv.org/abs/1403.5799))

* B. L. Feigin, B. L. Tsygan, _Additive K-theory_, pp.67-203 in LNM 1289 Springer Heidelberg 1986.

* Zbigniew Fiedorowicz, [[Jean-Louis Loday]], _Crossed simplicial groups and their associated homology_, Trans. AMS **326** pp.57-87 (1991).

* {#Isaacson2010} [[Samuel Isaacson]], _Symmetric cubical sets_, Journal of Pure and Applied Algebra **215** (2011). ([arXiv](https://arxiv.org/abs/0910.4948), [doi:10.1016/j.jpaa.2010.08.001](http://dx.doi.org/10.1016/j.jpaa.2010.08.001))

* R. Krasauskas, _Skew-simplicial groups_, Lithuanian Math. J. **27** pp.47-54 (1987).

* [[Jean-Louis Loday]], _Cyclic Homology_, Springer Heidelberg 1998$^2$.

* Walker H. Stern, _Structured topological field theories via crossed simplicial groups_, arXiv:1603.02614 (2016). ([abstract](https://arxiv.org/abs/1603.02614))

* [[Jun Yoshida]], _A general method to construct cube-like categories and applications to homotopy theory_, arXiv:1502.07539. ([abstract](https://arxiv.org/abs/1502.07539))

* [[Jun Yoshida]], _Colimits and limits of crossed groups_, arXiv:1802.06644 (2018). ([abstract](https://arxiv.org/abs/1802.06644))

[[!redirects crossed groups]]


crossed homomorphism

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In [[group theory]], and specifically in [[group cohomology]]-theory, the notion of *crossed homomorphisms* (Def. \ref{CrossedHomomorphism} below) is a generalization of that of [[group homomorphisms]] where a [[group action]] -- of the [[domain]]-group $G$ by [[automorphisms]] on the [[codomain]] group $\Gamma$ -- is incorporated.

Crossed homomorphisms are equivalently [[group homomorphism|homomorphic]] [[sections]] of the [[projection]] to $G$ out of the corresponding [[semidirect product group]] $\Gamma \rtimes G$ (Prop. \ref{CrossedHomomorphismsAreSplittingsOfTheSemidirectProductGroup} below).

This largely explains their relevance in all contexts of $G$-[[equivariant algebraic topology]] and [[equivariant differential topology]], such as in [[equivariant principal bundle]]-theory (Prop. \ref{EquivariantConnectedComponentsOfEquivariantClassifyingSpaces} below) where [[semidirect product groups]] appear as $G$-[[equivariant groups]].

In particular (Prop. \ref{GroupoidOfCrossedHomorphismsIsSlicedFunctorGroupoid} below, which is not made explicit in traditional literature):

* the [[local coefficient bundle]] for [[non-abelian cohomology|non-abelian]] [[group cohomology]] in degree-1 is the image of the projection out of the semidirect product group under forming [[delooping groupoids]],

* the [[strict (2,1)-category]]-theoretic [[sections]] of this bundle form precisely the [[groupoids]] of crossed homomorphisms with "crossed conjugations" (Def. \ref{AdjointActionOnCrossedHomomorphisms} below) between them;

* and the [[connected components]] of these groupoids is the [[non-abelian cohomology|non-abelian]] [[group cohomology]] $H^1(G,\Gamma)$ (Rem. \ref{AsGroup1Cocycles}, \ref{AsCoboundariesInGroupCohomology}).

This last fact is widely appreciated (see Rem. \ref{ReferencesForNonabelianCrossedHomomorphisms} below) in the special case that $\Gamma$ is an [[abelian group]], where it serves to define ordinary [[group cohomology]] with coefficients in degree 1.

Beware that, while the notion of crossed homomorphisms makes sense for [[group objects]] [[internalization|internal]] to any ambient [[category]] $\mathcal{C}$, it may not need to strictly coincide with that of [[cocycles]] in first [[group cohomology]] beyond $\mathcal{C} = $ [[Set]] (for reasons discussed at *[[Lie group cohomology]]*, or, for the case of [[profinite groups]], in [NSW 2008, p. 24](#NSW08)).

## Definition

In all of the following, let $G \,\in\, $ [[Grp]] be a [[group]] (or [[group object]] [[internalization|internal]] to an ambient [[category]] with [[finite products]]) and let $\Gamma \,\in\, G Act$ be an $G$-[[equivariant group]], hence a [[group]] equipped with a [[group action]]

$$
  \alpha \;\colon\; G \xrightarrow{\;\;} Aut_{Grp}(\Gamma)
$$

of $G$ on $\Gamma$ by [[group homomorphism|group]] [[automorphisms]], hence equivalently a [[semidirect product group]], to be denoted:

$$
  1
  \to
  \Gamma
  \xhookrightarrow{\;i\;}
  \Gamma \rtimes_\alpha G
  \xrightarrow{ \;pr_2\; }
  G
  \to
  1
  \,.
$$

### Component definition
 {#ComponentDefinition}

\begin{definition}\label{CrossedHomomorphism}
A *crossed homomorphism* from $G$ to $\Gamma$ is a [[function]] ([[morphism]] in the [[internalization|ambient]] category)

$$
  \phi \;\colon\; G \xrightarrow{\;} \Gamma
$$

satisfying the following "$G$-crossed" homomorphism property:

$$
  \underset{g_1, g_2 \in G}{\forall}
  \;\;\;
  \phi(g_1 \cdot g_2)
  \;=\;
  \phi(g_1) \cdot \alpha(g_1)\big( \phi(g_2) \big)
  \,.
$$
\end{definition}
\begin{remark}
\label{ReferencesForNonabelianCrossedHomomorphisms}
Def.
\ref{CrossedHomomorphism}
appears, already with the non-abelian generality in mind, back in [Whitehead 1949, (3.1)](#Whitehead49) (together with the notion of [[crossed modules]], both as tools for analyzing [[homotopy types]] with non-trivial [[fundamental group]]).
For [[abelian group|abelian]] $\Gamma$ the definition is in [Eilenberg & MacLane (1947) §3.1](#EilenbergMacLane47), [MacLane (1975) IV (2.1)](#MacLane75) and many following references on [[group cohomology]], e.g. [Brown 1982, p. 45](#Brown82).
For general [[non-abelian group|non-abelian]] $\Gamma$ the notion is later used in [Murayama & Shimakawa 1995](#MurayamaShimakawa95) for discussion of [[equivariant principal bundles]].
In the context of [[nonabelian group cohomology]] it is reviewed in [Gille & Szamuely, 2006, (1) on p. 25](#GilleSzamuely06).
Another textbook account is in [Milne 2017, 15.a (16.a) in the pdf](#Milne17), aimed at application to [[algebraic groups]].
\end{remark}

\begin{example}
  For $\alpha$ the [[trivial action]],
  the notion of a crossed homomorphism (Def. \ref{CrossedHomomorphism})
  reduces to that of an ordinary [[group homomorphism]].
\end{example}

\begin{remark}\label{AsGroup1Cocycles}
**(as 1-[[cocycles]] in [[group cohomology]])**
\linebreak
  In the special case that $\Gamma$ in Def. \ref{CrossedHomomorphism} is an [[abelian group]], crossed homomorphisms are also known as *[[cocycles]] in [[group cohomology]]* in degree 1 (e.g. [Brown 1982, p. 45](#Brown82)). In general they may be understood as 1-cocyles in [[non-abelian cohomology|non-abelian]] group cohomology (e.g. [NSW 2008, p. 16](#NWS08)).
\end{remark}

\begin{example}\label{ConstantFunctionOnNeutralElement}
  The [[constant function]]
  $const_{\mathrm{e}} \,\colon\, G \to \Gamma$
  on the [[neutral element]] $\mathrm{e} \in \Gamma$ is always a crossed homomorphism (Def. \ref{CrossedHomomorphism}), being the trivial cocycle under the interpretation of Rem. \ref{AsGroup1Cocycles}.
\end{example}

\begin{proposition}
  \label{CrossedHomomorphismsAreSplittingsOfTheSemidirectProductGroup}
**(as splittings/[[homomorphism|homomorphic]] [[sections]] of the [[semidirect product group]]-projection)**
\linebreak
  Crossed homomorphisms $G \to \Gamma$ (Def. \ref{CrossedHomomorphism}) are equivalently [[group homomorphism|homomorphic]] [[sections]] of the [[projection]] out of the [[semidirect product group]] $\Gamma \rtimes_\alpha G \xrightarrow{pr_2} G$ (see also at [split group extensions](group+extension#SplitExtensionsAndSemidirectProductGroups)):

$$
  CrsHom(G,\,\Gamma)
  \;\simeq\;
  GrpHom_{{}_{/G}}
  \big(
    G
    ,\,
    \Gamma \rtimes_\alpha G
  \big)
  \;\coloneqq\;
  GrpHom(G, \, \Gamma \rtimes_\alpha G)
  \underset{
    GrpHom(G,\,G)
  }{\times}
  \{id\}
  \,.
$$
\end{proposition}
(e.g. [Brown 1982, p. 88](#Brown82), [NSW 2008, Ex. 1 on p. 24](#NSW08), [Milne 2017, Ex. 15.1 (16.1 in the pdf)](#Milne17))
\begin{proof}
  By immediate unwinding of the definition of the [[semidirect product group]], such a section is an assignment
  $$
    \array{
      G &\xrightarrow{\;}& \Gamma \rtimes_\alpha G
      \\
      g
      &\mapsto&
      \big(
        \phi(g), \, g
      \big)
    }
  $$
  such that
  $$
   \underset{
     g_1, g_2 \in G
   }{\forall}
   \;\;\;\;\;
    \big(
      \phi(g_1 \cdot g_2), \, g_1 \cdot g_2
    \big)
    \;\overset{!}{=}\;
    \big(
      \phi(g_1), g_1
    \big)
    \cdot
    \big(
      \phi(g_2), g_2
    \big)
    \;=\;
    \Big(
      \phi(g_1)
        \cdot
      \alpha(g_1)\big(\phi(g_2)\big)
      ,\,
      g_1 \cdot g_2
    \Big)
    \,.
  $$
\end{proof}

\begin{proposition}\label{GraphOfACrossedHomomorphism}
**([[graph of a function|graphs]] of [[crossed homomorphisms]])**
\linebreak
  The [[graph of a function]] of a [[crossed homomorphism]] (Def. \ref{CrossedHomomorphism})
  is a [[subgroup]] of the [[semidirect product group]]
  \[
    \label{MayStyleCrossedHomomorphism}
    \widehat G
    \;\subset\;
    \Gamma \rtimes G
    \,,
    \;\;\;\;
    \text{such that}
    \;\;\;\;
    \mathrm{pr}_2\big(\widehat G\big) \simeq G
    \;\;\;
    \text{and}
    \;\;\;
    \widehat{G} \cap i(\Gamma)
      \;=\;
    \big\{
      (\mathrm{e},\,\mathrm{e})
    \big\}
    \,,
  \]
  and every subgroup of this form arises as the [[graph of a function|graph]] of a unique crossed homomorphism.
\end{proposition}
This is, essentially, considered in [tom Dieck 1969, Sec. 2.1](#tomDieck69),
also [Guillou, May & Merling 2017, Lem. 4.5](#GuillouMayMerling17).
\begin{proof}
  The first statement is immediate from the definition.

  For the converse, consider a subgroup $\widehat{G}$
  as in (eq:MayStyleCrossedHomomorphism).
  Then
  the subgroup property implies that
  $$
    (\gamma,g), (\gamma',g) \,\in\, \widehat{G}
    \;\;\;\;\;
    \Rightarrow
    \;\;\;\;\;
    (\gamma' ,\, g) \cdot (\gamma ,\, g)^{-1}
    \;=\;
    (\gamma' ,\, g)
    \cdot
    \big(
      \alpha(g^{-1})(\gamma^{-1})
      ,\,
      g^{-1}
    \big)
    \;=\;
    \big(
      \gamma' \cdot \gamma^{-1}
      ,\,
      \mathrm{e}
    \big)
    \;\;\;
    \in
    \;
    \widehat{G}
    \,.
  $$
  From this the second condition in (eq:MayStyleCrossedHomomorphism) implies that
  $$
    (\gamma,\,g) ,\, (\gamma',\, g)
    \;\in\;
    \widehat{G}
    \;\;\;\;
    \Rightarrow
    \;\;\;\;
    \gamma = \gamma'
    \,.
  $$
  Together with the first condition in (eq:MayStyleCrossedHomomorphism) this implies that $\widehat{G}$ is the [[graph of a function]] $\phi \;\colon\; G \to \Gamma$. From this the claim follows by Prop. \ref{CrossedHomomorphismsAreSplittingsOfTheSemidirectProductGroup}.
\end{proof}

\begin{remark}
  Subgroups of the form (eq:MayStyleCrossedHomomorphism)
  are considered throughout articles by [[Peter May]] on [[equivariant principal bundles]] (e.g. in [Lashof & May 1986, Thm. 10](equivariant+bundle#LashofMay86), [May 1990, Thm. 7](equivariant+bundle#May90)).
That these are equivalently ([[graph of a function|graphs]] of) [[crossed homomorphisms]] may have been one the key observations that lead to [Murayama & Shimakawa 1995](#MurayamaShimakawa95), though the statement of Prop. \ref{GraphOfACrossedHomomorphism} was still not explicit there.
\end{remark}

\begin{definition}\label{AdjointActionOnCrossedHomomorphisms}
**(crossed conjugation)**
\linebreak
  Two crossed homomorphisms
  $\phi_1, \phi_2$ (Def. \ref{CrossedHomomorphism}) are [[adjoint action|adjoint]] $\phi_1 \xrightarrow{\;\gamma\;} \phi_2$ if in
  their incarnation as homomorphisms to $\Gamma \rtimes G$  (Prop. \ref{CrossedHomomorphismsAreSplittingsOfTheSemidirectProductGroup}) they are related by conjugation with an element
  $\gamma \in \Gamma \xhookrightarrow{\;} \Gamma \rtimes G$, in that:
  \[
    \label{CrossedConjugationRelation}
    \underset{
      g \in G
    }{\forall}
    \;\;\;\;
    \phi_2(g)
    \;=\;
    \gamma^{-1}
      \cdot
    \phi_1(g)
      \cdot
    \alpha(g)(\gamma)
    \,.
  \]
\end{definition}
(e.g. [Gille & Szamuely 2006, (2) on p. 25](#GilleSzamuely06), [Guillou, May & Merling 2017, Def. 4.11](#GuillouMayMerling17))
\begin{remark}\label{AsCoboundariesInGroupCohomology}
**(as 1-[[coboundaries]] in [[group cohomology]])**
\linebreak
  When $\Gamma$ is an [[abelian group]], the conjugates according to Def. \ref{AdjointActionOnCrossedHomomorphisms}
of the constant function $const_{0}$ (Exp. \ref{ConstantFunctionOnNeutralElement}) are of the form
\[
  \label{GroupCoboundaryInSpecializationOfConjugation}
  g \;\mapsto\; \alpha(g)(\gamma) - \gamma
\]
(writing now in additive notation as befits an [[abelian group]]).

These (eq:GroupCoboundaryInSpecializationOfConjugation) are also known as "principal crossed homomorphisms".

With crossed homomorphisms understood as 1-[[cocycles]] in [[group cohomology]]
(Rem. \ref{AsGroup1Cocycles}), these elements (eq:GroupCoboundaryInSpecializationOfConjugation) are the *1-[[coboundaries]]*.

The definitions here apply verbatim also for [[non-abelian group|non-abelian]] $\Gamma$, where we get [[non-abelian cohomology|non-abelian]] [[group cohomology]] (e.g. [NSW 2008, p. 16](#NSW08)).
\end{remark}

### As sliced functors of delooping groupoids

All the formulas [above](#ComponentDefinition) may conceptually be understood as follows:

\begin{proposition}
  \label{GroupoidOfCrossedHomorphismsIsSlicedFunctorGroupoid}
**([[groupoid]] of [[crossed homomorphisms]] is [[slice category|sliced]] [[functor groupoid]] of [[delooping groupoids]])**
\linebreak
  The [[groupoid]]
  \[   \label{ConjugationGroupoidOfCrossedHomomorphismsInPropRelatingToSlicedFunctor}
     CrsHom(G,\Gamma) \sslash_{\!\! ad} \Gamma
     \;\;
       \coloneqq
     \big(
       CrsHom(G,\Gamma) \times \Gamma
       \rightrightarrows
       \Gamma
     \big)
     \;\;
     \;\;\;
     \in
     \;
     Grpd
  \]
  of crossed homomorphisms (Def. \ref{CrossedHomomorphism}) with conjugations between them (Def. \ref{AdjointActionOnCrossedHomomorphisms}) is [[isomorphism|isomorphic]] to the [[slice category|sliced]] [[functor groupoid]] of [[sections]] of the [[delooping groupoid]] $\mathbf{B}(\Gamma \rtimes G)$ of the [[semidirect product group]]:

\[
  \label{SlicedFunctorGroupoid}
  CrsHom(G,\Gamma) \sslash_{\!\! ad} \Gamma
  \;\;
    \simeq
  \;\;
  Fnctr_{{}_{/\mathbf{B}G}}
  \big(
    \mathbf{B}G
    ,\,
    \mathbf{B}(\Gamma \rtimes G)
  \big)
  \;=\;
  Fnctr
  \big(
    \mathbf{B}G
    ,\,
    \mathbf{B}(\Gamma \rtimes G)
  \big)
  \underset{
    Fnctr
    \big(
      \mathbf{B}G
      ,\,
      \mathbf{B}G
    \big)
  }{\times}
  \{id\}
  \,.
\]
\end{proposition}
We take this statement and the following proof from [SS21](https://ncatlab.org/schreiber/show/TED+cohomology).
\begin{proof}
\label{ProofOfCrossedHomomorpismsAreSlicedFunctors}
By definition, a morphism in the groupoid on the right is a
[[commuting diagram]] of [[functors]] and [[natural transformations]]
as shown on the left of the following:

\begin{imagefromfile}
    "file_name": "CrossedHomomorphismsAsSlicedFunctors210831.jpg",
    "width": 800,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "from [SS21](https://ncatlab.org/schreiber/show/TED+cohomology)"
\end{imagefromfile}

Unwinding the definitions, it is clear that functors fitting into this diagram (indicated by the dashed single arrows on the left) are bijective (under passage to their component functions shown on the right) to [[group homomorphisms]] $G \to \Gamma \rtimes G$ whose [[composition]] with $\Gamma \rtimes G \xrightarrow{pr_2} G$ is the [[identity morphisms]] -- hence are bijective to crossed homomorphisms, by Prop. \ref{CrossedHomomorphismsAreSplittingsOfTheSemidirectProductGroup}.

Moreover, [[natural transformations]] $F \Rightarrow F'$ fitting into this diagram (indicated by the dashed double arrow on the left) must be such that their [[whiskering]] with $\mathbf{B}pr_2$ is the [[identity natural transformation]], which means that their single component (as shown on the right) is of the form

$$
  \eta(\bullet) \;=\; \big( \gamma, \mathrm{e}\big)
  \;\;\;
  \in
  \;
  \Gamma \rtimes G
  \,,
$$

where $\mathrm{e} \in G$ denotes the [[neutral element]].

In conclusion, the [[naturality square]] in $\mathbf{B}(\Gamma \rtimes G)$
that all these components must make [[commuting square|commute]] is of the form

$$
  \array{
    \bullet
    &\;\;\;\;\mapsto\;\;\;\;&
    \bullet
    &\xrightarrow{\;\;\;\;\;\;(\gamma,\mathrm{e})\;\;\;\;\;\;}&
    \bullet
    \\
    {}^{\mathllap{ g }}
    \big\downarrow
    &&
    \big\downarrow
    {}^{\mathrlap{ \big( \phi_1(g),\, g\big) }}
    &&
    \big\downarrow
    {}^{\mathrlap{ \big( \phi_2(g),\, g\big) }}
    \\
    \bullet
    &\;\;\;\;\mapsto\;\;\;\;&
    \bullet
    &\xrightarrow{\;\;\;\;\;\;(\gamma,\mathrm{e})\;\;\;\;\;\;}&
    \bullet
  }
  {\phantom{AAAAAAAAAAA}}
  \in
  \;
  \mathbf{B}(\Gamma \rtimes G)
  \,.
$$

Since [[composition]] in the [[delooping groupoid]] is given by the group product, this is equivalently the condition that

\[
  \label{CrossedConjugationAsConjugationInTheSemidirectProduct}
    \big(
      \phi_2(g)
      ,\,
      g
    \big)
      \;=\;
    \big(
      \gamma^{-1}
      ,\,
      \mathrm{e}
    \big)
      \cdot
    \big(
      \phi_1(g)
      ,\,
      g
    \big)
      \cdot
    \big(
      \gamma
      ,\,
      \mathrm{e}
    \big)
    \;\;\;\;\;
    \in
    \;
    \Gamma \rtimes G
    \,,
\]

which is evidently equivalent to the defining relation from Def. \ref{AdjointActionOnCrossedHomomorphisms}.
\end{proof}

\begin{remark}
**([[conjugacy classes]] of [[crossed homomorphisms]] are [[non-abelian cohomology|non-abelian]] first [[group cohomology]])**
\linebreak
  In view of the identification of crossed homomorphisms with 1-cocycles in [[non-abelian cohomology|non-abelian]] [[group cohomology]], we may identify their conjugacy classes (Def. \ref{AdjointActionOnCrossedHomomorphisms}), hence, by Prop. \ref{GroupoidOfCrossedHomorphismsIsSlicedFunctorGroupoid}, the [[connected components]] of the sliced functor groupoid (eq:SlicedFunctorGroupoid)
with the group cohomology sets (at least for [[discrete groups]]):
$$
  CrsHom(H,\Gamma)_{/\sim}
  \;\simeq\;
  \pi_0
  \Big(
    Fnctr_{{}_{/\mathbf{B}G}}
    \big(
      \mathbf{B}G
      ,\,
      \mathbf{B}(\Gamma \rtimes G)
    \big)
  \Big)
  \;\simeq\;
  H^1_{Grp}(G,\Gamma)
  \,.
$$
\end{remark}

But we also get higher homotopy information, beyond the cohomology set:

\begin{proposition}\label{AutomorphismGroupOfCrossedHomomorphism}
**([[automorphism group]] of [[crossed homomorphisms]])**
\linebreak
  For $\phi \colon G \xrightarrow{\;} \Gamma$
  a crossed homomorphism (Def. \ref{CrossedHomomorphism}),
  its [[automorphism group]] in the conjugation groupoid (eq:ConjugationGroupoidOfCrossedHomomorphismsInPropRelatingToSlicedFunctor), hence the [[subgroup]]
$$
  Aut(\phi)
  \;\coloneqq\;
  Aut_{{}_{
    CrsHom(G,\Gamma) \sslash_{\! ad} \Gamma
  }}
  \big(
    \phi
  \big)
  \;=\;
  \Big\{
  \,
    \gamma \,\in\, \Gamma
    \,\vert\,
    \phi(-)
      \,=\,
    \gamma^{-1} \cdot \phi(-) \cdot \alpha(-)(\gamma)
  \,
  \Big\}
  \;\;\;
  \subset
  \;
  \Gamma
$$
of crossed conugations (Def. \ref{AdjointActionOnCrossedHomomorphisms}) that [[fixed point|fix]] it, is, equivalently, the [[intersection]] of $i(\Gamma) = \{ (\gamma, \mathrm{e}) \,\vert\, \gamma \in \Gamma \}$ with

1. the [[centralizer]] $C_{{}_{\Gamma \rtimes G}}\big(Graph(\phi)\big)$

1. the [[normalizer]] $N_{{}_{\Gamma \rtimes G}}\big(Graph(\phi)\big)$

of the [[graph of a function|graph]] of $\phi$ inside the [[semidirect product]] group:
$$
  \begin{aligned}
    Aut(\phi)
    &
    \;\;=\;\;
    i(\Gamma)
      \,\cap\,
    C_{{}_{\Gamma \rtimes G}}\big(Graph(\phi)\big)
    \\
    &
    \;\;=\;\;
    i(\Gamma)
      \,\cap\,
    N_{{}_{\Gamma \rtimes G}}\big(Graph(\phi)\big)
    \,.
  \end{aligned}
$$
\end{proposition}
(compare [Lashof & May 1986, below Thms. 10, 11](equivariant+bundle#LashofMay86))
\begin{proof}
  The first statement is immediate from the re-formulation (eq:CrossedConjugationAsConjugationInTheSemidirectProduct) of
crossed conjugation (eq:CrossedConjugationRelation).

From this, the second statement follows by observing that the action in the form (eq:CrossedConjugationAsConjugationInTheSemidirectProduct) manifestly fixes the $G$-component under projection along $pr_2$, while the [[graph of a function|graph]] of a function $G \to \Gamma$ has at most one element with a given $G$-component.
\end{proof}

## Properties

### Restriction to subgroups

\begin{proposition}\label{WeylGroupActsOnNonAbGroup1CohomOfSubgroup}
**([[Weyl group]] [[group action|acts]] on [[non-abelian group cohomology|non-abelian group 1-cohomology]] of [[subgroup]])**
\linebreak
For $H \subset G$ a [[subgroup]], the set of crossed homomorphisms $H \to \Gamma$ (Def. \ref{CrossedHomomorphism}), with respect to the restricted action of $H$ on $\Gamma$, carries a [[group action]] of the [[normalizer subgroup]] $N_G(H) \,\subset\, G$,
given by
\[
  \label{ActionOfNormalizerOnCrossedHomomorphismsFromSubgroup}
  \array{
    N_G(H) \times CrsHom(H,\,G)
    &\xrightarrow{\;\;}&
    CrsHom(H,\,G)
    \\
    (n, \phi)
    &\mapsto&
    \phi_{n}
    \mathrlap{
    \;\coloneqq\;
    \alpha(n)
    \Big(
      \phi
      \big(
        n^{-1} \cdot (-) \cdot n
      \big)
    \Big)
    \,.
    }
  }
\]
Moreover, on crossed-conjugation classes (Def. \ref{AdjointActionOnCrossedHomomorphisms}), hence on first [[non-abelian group cohomology]], this action descends to an action of the [[Weyl group]] $W_G(H) \coloneqq N_G(H)/H$:
$$
  H^1_{Grp}(H,\,\Gamma)
  \;\;\;
  \in
  \;
  W_G(H) Act(Sets)
  \,.
$$
\end{proposition}
\begin{proof}

**1. Action property.** It is clear that (eq:ActionOfNormalizerOnCrossedHomomorphismsFromSubgroup) is a group action if only $\phi_n$ is indeed a crossed homomorphism. This follows by a direct computation:
$$
  \begin{array}{lll}
    \phi_n( h_1 \cdot h_2 )
    &
    \;=\;
    \alpha(n)
    \Big(
      \phi
      \big(
        n^{-1} \cdot h_1 \cdot h_2 \cdot n
      \big)
    \Big)
    &
    \text{definition of}\; \phi_n
    \\
    &
    \;=\;
    \alpha(n)
    \Big(
      \phi
      \big(
        n^{-1} \cdot h_1 \cdot n \cdot n^{-1} \cdot h_2 \cdot n
      \big)
    \Big)
    &
    \text{group property of}\; N_G(H) \subset G
    \\
    & \;=\;
    \alpha(n)
    \Big(
      \phi
      \big(
        n^{-1} \cdot h_1 \cdot n
      \big)
      \cdot
      \alpha(n^{-1} \cdot h_1 \cdot n)
      \big(
        \phi( n^{-1} \cdot h_2 \cdot n )
      \big)
    \Big)
    &
    \text{crossed homomorphism property of} \; \phi
    \\
    & \;=\;
    \alpha(n)
    \Big(
      \phi
      \big(
        n^{-1} \cdot h_1 \cdot n
      \big)
    \Big)
      \cdot
    \alpha(h_1 \cdot n)
    \Big(
      \phi\big( n^{-1} \cdot h_2 \cdot n \big)
    \Big)
    &
    \text{action property of} \; \alpha
    \\
    & \;=\;
    \alpha(n)
    \Big(
      \phi
      \big(
        n^{-1} \cdot h_1 \cdot n
      \big)
    \Big)
      \cdot
    \alpha(h_1)
    \bigg(
      \alpha(n)
      \Big(
        \phi\big( n^{-1} \cdot h_2 \cdot n \big)
      \Big)
    \bigg)
    &
    \text{action property of} \; \alpha
    \\
    & \;=\;
    \phi_n(h_1)
    \cdot
    \alpha(h_1)
    \big(
      \phi_n(h_2)
    \big)
    &
    \text{definition of}\; \phi_n
    \mathrlap{\,.}
  \end{array}
$$

**2. Descent to crossed-conjugation classes.**
To see that the action (eq:ActionOfNormalizerOnCrossedHomomorphismsFromSubgroup) descends to group cohomology, we need to show for

$$
  \phi'(-)
  \;=\;
  \gamma^{-1} \cdot \phi(-) \cdot \alpha(-)(\gamma)
$$

a crossed conjugation between some $\phi'$ and $\phi$, that there also exists a crossed conjugation between $\phi'_n$ and $\phi_n$. The following direct computation shows that this is given by crossed conjugation with $\alpha(n)(\gamma)$:

$$
  \begin{array}{lll}
    \phi'_n(h)
    &
    \;=\;
    \alpha(n)
    \Big(
      \gamma^{-1}
        \cdot
      \phi\big( n^{-1} \cdot h \cdot n \big)
      \cdot
      \alpha
        \big( n^{-1} \cdot h \cdot n \big)
        (\gamma)
    \Big)
    &
    \text{assumption with definition of} \; \phi_n
    \\
    &
    \;=\;
    \alpha(n)
    \big(
      \gamma^{-1}
    \big)
      \cdot
    \alpha(n)
    \Big(
      \phi\big(
        n^{-1} \cdot h \cdot n
      \big)
    \Big)
    \cdot
    \alpha(h)
    \Big(
      \alpha(n)(\gamma)
    \Big)
    &
    \text{action property of} \; \alpha
    \\
    & \;=\;
    \big(
      \alpha(n)(\gamma)
    \big)^{-1}
      \cdot
    \phi_n(h)
      \cdot
    \alpha(h)
    \big(
      \phi(n)(\gamma)
    \big)
    &
    \text{definition of} \; \phi_n
    \,.
  \end{array}
$$

**Alternative argument for 1. & 2.**
Alternatively, the previous two statements also follow more immediately, by using the identification from Prop. \ref{GroupoidOfCrossedHomorphismsIsSlicedFunctorGroupoid} of the conjugation groupoid of crossed homomorphisms $\phi \colon H \to \Gamma$ with that of homomorphic sections $ \big( \phi(-),\, (-) \big) \,\colon\, H \to \Gamma \rtimes H$. In this latter incarnation, the action by $n$ is simply the "conjugation action by the adjoint action", in that:

$$
  n
  \;\colon\;
  \big(
    \phi(-),\, (-)
  \big)
  \;\;
  \mapsto
  \;\;
  \big(
    \phi_n(-),\, (-)
  \big)
  \;\;
  =
  \;\;
  \big(
    \mathrm{e} ,\, n
  \big)
  \cdot
  \Big(
    \phi
    \big(
      n^{-1}\cdot(-)\cdot n
    \big)
    ,\,
    \big(
      n^{-1} \cdot(-) \cdot n
    \big)
  \Big)
  \cdot
  \big(
    \mathrm{e} ,\, n^{-1}
  \big)
  \,.
$$

In this formulation it is manifest that homomorphisms and conjugation are preserved, and the only point to check is that the section-property is also respected, which is immediate.

**3. Descent to action of Weyl group.**
To conclude, we need to show that the action of $H \subset B_G(H)$ is trivial on conjugacy classes, hence that
for $n \in H \subset N(H)$ there is a crossed conjugation between $\phi_n$ and $\phi$. The following direct computation shows that this is given by crossed conjugation with $\phi(n)$ (which is well-defined, by the assumption that $n \in H$):
$$
  \begin{array}{lll}
    \phi_n(h)
    &
    \;=\;
    \alpha(n)
    \Big(
      \phi
      \big(
        n^{-1} \cdot h \cdot n
      \big)
    \Big)
    &
    \text{definition of} \; \phi_n
    \\
    &
    \;=\;
    \alpha(n)
    \bigg(
      \phi(n^{-1})
      \cdot
      \alpha(n^{-1})
      \Big(
         \phi(h)
         \cdot
         \alpha(h)
         \big(
           \phi(n)
         \big)
      \Big)
    \bigg)
    &
    \text{crossed homomorphism property of} \; \phi
    \\
    & \;=\;
    \alpha(n)
    \big(
      \phi(n^{-1})
    \big)
    \cdot
    \phi(h)
    \cdot
    \alpha(h)
    \big(
      \phi(n)
    \big)
    &
    \text{action property of} \; \alpha
    \\
    & \;=\;
    \big(
      \phi(n)
    \big)^{-1}
    \cdot
    \phi(h)
    \cdot
    \alpha(h)
    \big(
      \phi(n)
    \big)
    &
    \text{crossed homomorphism property of} \; \phi
    \mathrlap{\,.}
  \end{array}
$$
\end{proof}

## Examples

### In relation to crossed modules

\begin{example}\label{ARelationToCrossedModules}
**(a relation to [[crossed modules]])**
\linebreak
  Let $(\Gamma \xrightarrow{\delta} G)$ be a [[crossed module]], with the corresponding [[strict 2-group]]

$$
  \mathcal{G}
    \;\coloneqq\;
  \big(
    \Gamma \rtimes G
    \underoverset
      { (\delta(-))\cdot(-) }
      {pr_2}
      {\rightrightarrows}
    G
  \big)
  \,.
$$

Then in the [[strict (2,1)-category]] of [[strict 2-groups]], the [[2-morphisms]]
out of the [[identity morphism|identity]] [[1-morphism]] on $\mathcal{G}$ are in [[bijection]] to the crossed homomorphisms $G \to \Gamma$:

\[
  \label{CrossedHomomorphismsBijectiveTo2Group2MorphismsOutOfIdG}
  CrsHom(G,\Gamma)
  \;\simeq\;
  \big\{
    id_{\mathcal{G}} \Rightarrow F
    \,\vert\,
    F \in Str2Grp(\mathcal{G}, \mathcal{G})
  \big\}
  \,.
\]

Namely, such a 2-morphism is a [[natural transformation]] $\eta \colon id_{\mathcal{G}} \Rightarrow F$ of [[endofunctors]] of the underlying [[action groupoid]], whose component function
$$
  \eta_0 \,\colon\, Obj(\mathcal{G}) \xrightarrow{\;} Mor(\mathcal{G})
$$
is a [[group homomorphism]]
$$
  \eta_0 \,\colon\, G \xrightarrow{\;} \Gamma \rtimes G
$$
such that
$$
  s
  \big(
    \eta_0(g)
  \big)
  \;=\;
  (id_{\mathcal{G}})_0(g)
  \;=\;
  g
  \,,
$$
where
$$
  s \,\coloneqq\, pr_2 \;\colon\; \Gamma \times G \xrightarrow{\;} G
$$
is the [[source]] map of the [[groupoid]] $\mathcal{G}$.

This means that the admissible $\eta_0$ are precisely the [[group homomorphism|homomorphic]] [[sections]] of $\Gamma \rtimes G \xrightarrow{\;} G$.
Conversely, by the [[invertible morphism|invertiblity]] of all morphisms involved, every such $\eta_0$ is the component homomorphism of some 2-morphism $\eta$ out of $id_{\mathcal{G}}$.

Therefore the statement (eq:CrossedHomomorphismsBijectiveTo2Group2MorphismsOutOfIdG) follows by Prop. \ref{CrossedHomomorphismsAreSplittingsOfTheSemidirectProductGroup}.

The analogous statement for general 2-morphisms is indicated in [Noohi 07, p. 12](#Noohi07).
\end{example}

### In relation to equivariant bundles

Let $G \,\in\, Grp(Top)$ and $\Gamma \,\in\, Grp\big( G Act(Top) \big)$ be [[topological groups]] which are [[compact Lie groups]] (to be on the safe side). Then:

\begin{proposition}
\label{EquivariantConnectedComponentsOfEquivariantClassifyingSpaces}
**(equivariant connected components of equivariant classifying spaces)**
\linebreak
A [[topological G-space|G-equivariant]] [[classifying space]] $B_G \Gamma$ for $G$-[[equivariant bundle|equivariant]] $\Gamma$-[[principal bundles]] exists, and the [[connected components]] of its $H$-[[fixed loci]], for [[compact subgroups]] $H \subset G$, are in [[bijection]] to the [[conjugacy classes]] (Def. \ref{AdjointActionOnCrossedHomomorphisms}) of crossed homomorphisms (Def. \ref{CrossedHomomorphism}) from $H$ to $\Gamma$ (with respect to the [[restricted action]] of $H$ on $\Gamma$):

\[
  \label{EquivariantConnectedComponentsOfEquivariantClassifyingSpaces}
  \pi_0
  \left(
    \left(
      B_G \Gamma
    \right)^{H}
  \right)
  \;\;
  \simeq
  \;\;
  CrsHom(H,\Gamma)_{/\sim}
  \,.
\]
\end{proposition}
\begin{proof}
After a little reformulation via
Prop. \ref{GraphOfACrossedHomomorphism}
and
Prop. \ref{CrossedHomomorphismsAreSplittingsOfTheSemidirectProductGroup}, this is the statement of [Lashof & May 1986, Thm. 10](equivariant+bundle#LashofMay86), [May 1990, Thm. 7](#May90).
\end{proof}

In view of the explicit construction of universal equivariant principal bundles in [Murayama & Shimakawa 1995](#MurayamaShimakawa95), one may understand the statement of Prop. \ref{EquivariantConnectedComponentsOfEquivariantClassifyingSpaces} on elementary grounds, as follows:

Consider the following $G$-[[action objects]] [[internalization|internal]] to [[Groupoids]]:

* the [[delooping groupoid]]

  $$
    \mathbf{B}\Gamma
      \,\coloneqq\,
    \big( \Gamma \rightrightarrows \ast\big)
    \;\;\;
    \in
    \;
    G Act\big( Groupids \big)
  $$

  via the $G$-action on $\Gamma$ by [[group homomorphism|group]] [[automorphisms]];

* the [[pair groupoid]]

  $$
    \mathbf{E}G
      \,\coloneqq\,
    \big( G \times G \rightrightarrows G\big)
  $$

  via the left multiplication action of $G$ on all three copies of $G$;

* the [[functor groupoid]]

  \[
    \label{EquivariantFunctorGroupoidFromEGToBGamma}
    Fnctr
    \big(
      \mathbf{E}G
      ,\,
      \mathbf{B}\Gamma
    \big)
    \;\;\;
    \in
    \;
    G Act\big( Groupids \big)
  \]

  with the induced [[conjugation action]] on component functions of  [[functors]] and [[natural transformations]]:

  $$
    g \,\colon\, F(-) \,\mapsto\,  g \cdot F\big( g^{-1}\cdot (-) \big)
    \,.
  $$

\begin{proposition}
\label{HFixedLociOFMappingGroupoidFromEGToBGamma}
**(crossed homomorphisms are [[fixed loci]] in [[functor groupoid]] from [[pair groupoid|$\mathbf{E}G$]] to [[delooping groupoid|$\mathbf{B}\Gamma$]])**
\linebreak
  If $G$ a [[discrete group]], then for each [[subgroup]] $H \subset G$, there is an [[equivalence of groupoids]] (in fact of [[topological groupoids]])

$$
  Fnctr
  \big(
    \mathbf{E}G
    ,\,
    \mathbf{B}\Gamma
  \big)^H
  \;\simeq\;
  CrsHom(H,\Gamma)
    \sslash_{\!\!ad}
  \Gamma
$$

between

* the $H$-[[fixed locus|fixed]] sub-groupoid of the functor groupoid (eq:EquivariantFunctorGroupoidFromEGToBGamma) and

* the conjugation groupoid (eq:ConjugationGroupoidOfCrossedHomomorphismsInPropRelatingToSlicedFunctor) of crossed homomorphisms $H \to \Gamma$.

\end{proposition}
This is essentially the statement of [Guillou, May & Merling 2017, Thm. 4.14, Cor. 4.15](#GuillouMayMerling17).
We take the following detailed proof from [SS21](https://ncatlab.org/schreiber/show/TED+cohomology). Its ingredients are needed below in the proof of Prop. \ref{ResidualWeylGroupActionOnFixedLocuOfEquivariantClassifyingSpace}.
\begin{proof}
  By Prop. \ref{GroupoidOfCrossedHomorphismsIsSlicedFunctorGroupoid} the statement is equivalently that
$$
  Fnctr
  \big(
    \mathbf{E}G
    ,\,
    \mathbf{B}G
  \big)^H
  \;\simeq\;
  Fnctr_{{}_{/\mathbf{B}H}}
  \big(
    \mathbf{B}H
    ,\,
    \mathbf{B}(\Gamma \rtimes H)
  \big)
  \,.
$$

In the special case $H = G$ there is in fact an [[isomorphism]],
evidently exhibited by the following functor:

\begin{imagefromfile}
    "file_name": "GFixedMappingGroupoidFromEGToBGamma210901.jpg",
    "width": 660,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

For general $H \subset G$, choose a section of the [[coset space]]-projection

\[
  \label{SectionOfTheCoprojectionToTheCosetSpace}
  \sigma \,\colon\, G/H \xrightarrow{\;} G
  \,,
  \;\;\;\;
  \text{such that}
  \;
  \sigma([\mathrm{e}]) \,=\, \mathrm{e}
  \,,
\]

which exists and is continuous by the assumption that $G$ is discrete.

Observe that then $\mathbf{E}G$ is generated, under

1. composition,

1. taking inverses,

1. acting with elements of $H$

by the following two classes of morphisms:

\[
  \label{GeneratingMorphisms}
  \big\{ (\mathrm{e} \to h) \,\vert\, h \in H \big\}
  \,,
  \;\;\;
  \big\{
    \mathrm{e} \to \sigma([g]) \,\vert\, [g] \in G/H
  \big\}
  \;\;\;\;\;
  \subset
  \;
  G \times G
  \,.
\]

Using this, consider the following expression for a pair of comparison functors:

\begin{imagefromfile}
    "file_name": "HFixedMappingGroupoidFromEGToBGamma210905.jpg",
    "width": 660,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Here

* $L$ is the restriction along $(H \times H \rightrightarrows H) \xhookrightarrow{\;} (G \times G \rightrightarrows G)$ of the previous isomorphism for $H = $G;

* $R$ is given in terms of the above generating morphisms (eq:GeneratingMorphisms) as follows:

\[
  \label{DefinitionOfFunctorR}
  \;\;
\]

\begin{imagefromfile}
    "file_name": "CompHFixedMappingGroupoidFromEGToBGamma210901.jpg",
    "width": 700,
    "unit": "px",
    "margin": {
        "top": -57,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

One readily sees that this is well-defined, and that $L \circ R \,=\, id$.

Therefore it is now sufficient to give a [[natural transformation]] $id \xRightarrow{\eta} R \circ L$, hence for each functor $F$ a natural transformation

$$
  \eta_F \,\colon\, F \Rightarrow R \circ L(F)
  \,.
$$

This may be taken as follows, again stated in terms of the generating morphisms (eq:GeneratingMorphisms):

\begin{imagefromfile}
    "file_name": "Nat1HFixedMappingGroupoidFromEGToBGamma210901.jpg",
    "width": 490,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

It just remains to check that this is indeed natural in the functors $F$, which amounts, for each $H$-equivariant natural transformation $G \xRightarrow{\beta} F'$,
to the [[commuting square|commutativity]] of the two types of squares shown on the right here:

\begin{imagefromfile}
    "file_name": "Nat2HFixedMappingGroupoidFromEGToBGamma210901.jpg",
    "width": 650,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Indeed, the top square commutes by the $H$-equivariance of $\beta$, while the bottom square commutes by the naturality of $\beta$.
\end{proof}

\begin{prop}\label{ResidualWeylGroupActionOnFixedLocuOfEquivariantClassifyingSpace}
**(residual [[Weyl group]]-[[group action|action]] on [[fixed locus]] of equivariant classifying space)**
\linebreak
Transported through the equivalence of Prop. \ref{HFixedLociOFMappingGroupoidFromEGToBGamma},
the canonical [[group action]] (see [this Prop.](fixed+point+space#PassageToFixedLociIsRightAdjoint)) of the [[Weyl group]] $W_G(H)$ on the $H$-[[fixed locus]]
$
  Fnctr\big(\mathbf{E}G ,\, \mathbf{B}\Gamma\big)^H
$
becomes, on connected components
$
  \pi_0
  \big(
    CrsHom(H,\,\Gamma) \sslash_{\!\!ad} \Gamma
  \big)
  \;\;
  \simeq
  \;\;
  H^1_{Grp}(H,\,\Gamma)
$,
the $W_G(H)$-action on the non-abelian group 1-cohomology of $H$ from Prop. \ref{WeylGroupActsOnNonAbGroup1CohomOfSubgroup}.
\end{prop}
We take this statement and the following proof from [SS21](https://ncatlab.org/schreiber/show/TED+cohomology).
\begin{proof}
  We make explicit use of the functors $L, R$ constructed in the proof of Prop. \ref{HFixedLociOFMappingGroupoidFromEGToBGamma}. Noticing that $R$ is a section of $L$, we need to (1) send a crossed homomorphism up with $R$, (2) there act on it with $n$, (3) send the result back with $L$. The result is the desired induced action.

Explicitly, by the definition of $L$ in the proof of Prop. \ref{HFixedLociOFMappingGroupoidFromEGToBGamma}, this way a crossed homomorpism $\phi \,\colon\, H \to \Gamma$ is sent by $n \in N_G(H)$ to the assignment
\[
  \label{TransportingCanonicalActionOnFixedLocusToGroupCohomology}
  h \,\mapsto\,
    \Big(L\big( n \cdot (R \phi) \big)\Big)(h)
    \;=\;
    \alpha(n)
    \Big(
      (R\phi)
      \big(
        n^{-1} ,\, n^{-1} \cdot h
      \big)
    \Big)
  \,.
\]

It just remains to evaluate the right hand side.

Notice that the definition of $L$ is independent of the choice of $\sigma \,\colon\, G/H \xrightarrow{\;} G$ (eq:SectionOfTheCoprojectionToTheCosetSpace), and that $R$ (whose definition does depend on this choice ) is a section for each choice. Hence we may choose $\sigma$ in a way convenient way *for any given $n$*.

Now if $n \in H \subset N_G(H)$ then its canonical action on the $H$-fixed locus is trivial, and also the claimed induced action is trivial, so that in this case there is nothing further to be proven. Therefore we assume now that $n$ is not in $H$, and then we choose $\sigma$ such as to pick $n^{-1}$ as the representative in its $H$-coset:
$$
  \sigma\big( \big[n^{-1}\big] \big) \;\coloneqq\; n^{-1}
  \,.
$$

Notice that with this choice, $R\phi$ (eq:DefinitionOfFunctorR) assigns  the [[neutral element]]  to the morphism between the neutral element and $n^{-1}$ in the [[pair groupoid]]:

\[
  \label{RPhiOnNeutralTonInverse}
  (R\phi)\big(\mathrm{e}, n^{-1}\big)
  \;=\;
  \mathrm{e}
  \,,
  \;\;\;\;
  (R\phi)\big(n^{-1}, \mathrm{e}\big)
  \;=\;
  \mathrm{e}
  \;\;\;\;\;
  \,.
\]

{#ThisWay} This way, the right hand side of (eq:TransportingCanonicalActionOnFixedLocusToGroupCohomology) is evaluated as follows:

$$
  \begin{array}{lll}
    \Big(L\big( n \cdot (R \phi) \big)\Big)(h)
    & \;=\;
    \alpha(n)
    \Big(
      (R\phi)
      \big(
        n^{-1} ,\, n^{-1} \cdot h
      \big)
    \Big)
    &
    \text{by}\;\text{(eq:TransportingCanonicalActionOnFixedLocusToGroupCohomology)}
    \\
    & \;=\;
    \alpha(n)
    \big(
      (R\phi)(\mathrm{e},\, n^{-1} \cdot h)
    \big)
    &
    \text{by}\;\text{(eq:RPhiOnNeutralTonInverse)}
    \\
    &
    \;=\;
    \alpha\big(n^{-1} \cdot h \cdot n \cdot n\big)
    \big(
      (R\phi)(n^{-1} \cdot h^{-1} \cdot n,\, n^{-1})
    \big)
    &
    \text{by} \; H\text{-equivariance of} \; R\phi
    \\
    &
    \;=\;
    \alpha\big(n^{-1} \cdot h \cdot n  \cdot n\big)
    \big(
      (R\phi)(n^{-1} \cdot h^{-1} \cdot n,\, \mathrm{e})
    \big)
    &
    \text{by}\;\text{(eq:RPhiOnNeutralTonInverse)}
    \\
    &
    \;=\;
    \alpha(n)
    \big(
      (R\phi)(\mathrm{e},\, n^{-1} \cdot h \cdot n)
    \big)
    &
    \text{by} \; H\text{-equivariance of} \; R\phi
    \\
    & \;=\;
    \alpha(n)
    \big(
      \phi(n^{-1} \cdot h \cdot n)
    \big)
    &
    \text{by definition of} \; R \; \text{(eq:DefinitionOfFunctorR)}
    \mathrlap{\,.}
  \end{array}
$$

This is indeed the claimed formula (eq:ActionOfNormalizerOnCrossedHomomorphismsFromSubgroup).
\end{proof}

## Related concepts

* [[group extension]]

* [[group cohomology]]

* [[semidirect product group]]

* [[Fox derivative]]

## Literature

### To abelian groups

Discussion for the special case that $\Gamma$ is an [[abelian group]]:

in the context of [[homological algebra]]:

* {#MacLane75} [[Saunders Mac Lane]], Section IV.2 of: _Homology_ (1975) reprinted as Classics in Mathematics. Springer-Verlag, Berlin, 1995. x+422 pp. ISBN 3-540-58662-8 ([doi:10.1007/978-3-642-62029-4](https://link.springer.com/book/10.1007/978-3-642-62029-4))

in the context of [[group cohomology]]:

* {#EilenbergMacLane47} [[Samuel Eilenberg]], [[Saunders MacLane]], §3.1 in: *Cohomology Theory in Abstract Groups. I*, Annals of Mathematics, Second Series **48** 1 (1947) 51-78 [[doi:10.2307/1969215](https://doi.org/10.2307/1969215)]

* {#Brown82} [[Kenneth Brown]],  p. 45 of: _Cohomology of Groups_, Graduate Texts in Mathematics, **87**, Springer 1982 ([doi:10.1007/978-1-4684-9327-6](https://link.springer.com/book/10.1007/978-1-4684-9327-6))

### To general groups

The general [[non-abelian group|non-abelian]] notion:

In discussion of [[homotopy theory]] (together with [[crossed modules]]):

* {#Whitehead49} [[J. H. C. Whitehead]], Section 3 of: _Combinatorial Homotopy II_, Bull. Amer. Math. Soc., 55, (1949), 453 – 496 ([euclid:1183513797](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-55/issue-5/Combinatorial-homotopy-II/bams/1183513797.full))

A brief textbook account in this generality

* {#NSW08} [[Jürgen Neukirch]], Alexander Schmidt, Kay Wingberg, pages 16 & 24 in: *Cohomology of Number Fields*, Springer Grundlehren der mathematischen Wissenschaften **323**, Springer 2008 ([doi:10.1007/978-3-540-37889-1](https://link.springer.com/book/10.1007/978-3-540-37889-1), [webpage](https://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/))

As 1-[[cocycles]] in [[non-abelian group cohomology]]:

* {#GilleSzamuely06} [[Philippe Gille]], [[Tamás Szamuely]], Def. 2.3.2 in: *Central Simple Algebras and Galois Cohomology*, Cambridge University Press  2006 ([doi:10.1017/CBO9780511607219](https://doi.org/10.1017/CBO9780511607219), [pdf](http://www.math.ens.fr/~benoist/refs/Gille-Szamuely.pdf))

* {#Milne17} [[James Milne]], Sec. 15.a-b & 3.k (16.a-b & 27.a in the [pdf](https://www.jmilne.org/math/CourseNotes/iAG200.pdf)) of: *Algebraic Groups*, Cambridge University Press 2017  ([doi:10.1017/9781316711736](https://doi.org/10.1017/9781316711736), [webpage](http://www.jmilne.org/math/Books/iag.html), [pdf](https://www.jmilne.org/math/CourseNotes/iAG200.pdf))

* [[Groupprops]], *[First cohomology set with coefficients in a non-abelian group](https://groupprops.subwiki.org/wiki/First_cohomology_set_with_coefficients_in_a_non-abelian_group)*

In view of [[homotopy fixed points]]:

* R. Virk, Section 3 of: *On fixed point stacks* ([arXiv:2112.06871](https://arxiv.org/abs/2112.06871))

In discussion of [[equivariant principal bundles]]:

* {#tomDieck69} [[Tammo tom Dieck]], Sec. 2.1 in: _Faserbündel mit Gruppenoperation_, Arch. Math 20, 136–143 (1969) ([doi:10.1007/BF01899003](https://doi.org/10.1007/BF01899003))

  > (not using the terminology "crossed homomorphism", but already characterizing their graphs, as in Lem. \ref{GraphOfACrossedHomomorphism})

* {#MurayamaShimakawa95} [[Mitutaka Murayama]], [[Kazuhisa Shimakawa]], p. 2 of: _Universal equivariant bundles_, Proc. Amer. Math. Soc. 123 (1995), 1289-1295 ([doi:10.1090/S0002-9939-1995-1231040-9](https://doi.org/10.1090/S0002-9939-1995-1231040-9))

* {#GuillouMayMerling17} [[Bertrand Guillou]], [[Peter May]], [[Mona Merling]], around Def. 4.1 in: _Categorical models for equivariant classifying spaces_, Algebr. Geom. Topol. 17 (2017) 2565-2602 ([arXiv:1201.5178](https://arxiv.org/abs/1201.5178), [doi:10.2140/agt.2017.17.2565](https://doi.org/10.2140/agt.2017.17.2565))

Discussion for [[finite groups]]:

* Tsunenobu Asai, Yugen Takegahara, *On the number of crossed homomorphisms*, Hokkaido Math. J. **28** 3 (1999) 535-543 ([doi:10.14492/hokmj/1351001235](https://projecteuclid.org/journals/hokkaido-mathematical-journal/volume-28/issue-3/On-the-number-of-crossed-homomorphisms/10.14492/hokmj/1351001235.full))

Discussion for [[Lie groups]]:

* [[Karl-Hermann Neeb]], Def. 2.3 in: *Lie group extensions associated to projective modules of continuous inverse algebras*, Archivum Mathematicum, **44** 5 (2008) 465-489 ([dml:127115](https://dml.cz/handle/10338.dmlcz/127115))

Discussion in relation to [[crossed modules]]:

* {#Noohi07} [[Behrang Noohi]], *Notes on 2-groupoids, 2-groups and crossed-modules*, Homotopy, Homology, and Applications, 9, (2007), no. 1, 75--106 ([arXiv:math/0512106](https://arxiv.org/abs/math/0512106))

[[!redirects crossed homomorphisms]]

[[!redirects crossed group homomorphism]]
[[!redirects crossed group homomorphisms]]


Crossed Menagerie

[[!redirects Menagerie]]
This entry is about the document

* [[Tim Porter]], _The Crossed Menagerie: an introduction to crossed gadgetry and cohomology in algebra and topology_. (Notes, in part, prepared for the  _XVI Encuentro Rioplatense de Algebra y Geometría Algebraica_, in Buenos Aires, 12-15 December 2006, and for course MATH5312, Spring-Summer term 2007,
University of Ottawa) ([[menagerie14.pdf|pdf:file]])

This is an ongoing set of notes outlining an approach to [[nonabelian cohomology]], [[stack]]s, etc., and Grothendieck's conjectured extension of 'Galois-Poincaré theory'. The title refers to the array of strange beasties that occur as generalisations of [[crossed modules]]. (The present version is over 1004 pages long, and is changing quite often. A fairly stable version (but up-dates are planned shortly!) is available as a download,  [[menagerie12a.pdf|here:file]].

These notes, or at least the first few chapters of them, acted as course notes for a 4 lecture course in Buenos Aires, and later on for a course on cohomology in Ottawa.

  Some idea of the content can be gleaned from the Table of Contents.

##Table of Contents ##

(This gives an indication of the Chapter and section headings, although these have changed in detail for the latest version available. Subsection headings give too long a list to be that useful, so if you want more detail (for the moment) look at the pdf file.)

* Introduction

* 1 Preliminaries

  * 1\.1 Groups and Groupoids

  * 1\.2 A very brief introduction to cohomology

  * 1\.3 Simplicial things in a category

* 2 Crossed modules - definitions, examples and applications

  * 2\.1 Crossed modules

  * 2\.2 Group presentations, identities and 2-syzyzgies

  * 2\.3 Cohomology, crossed extensions and algebraic 2-types

* 3 Crossed complexes and (Abelian) Cohomology

  * 3\.1 Crossed complexes: the Definition

  * 3\.2 Crossed complexes and chain complexes: I

  * 3\.5 Simplicial groups and crossed complexes

  * 3\.6 Cohomology and crossed extensions

  * 3\.7 2-types and cohomology

  * 3\.8 Re-examining group cohomology with Abelian coefficients

* 4 Syzygies, and higher generation by subgroups

* 5 Beyond 2-types

  * 5\.1 Crossed squares

  * 5\.2 2-crossed modules and related ideas

  * 5\.3 Cat$^n$ -groups and crossed $n$-cubes

  * 5\.4 Loday's Theorem and its extensions

  * 5\.5 Crossed N-cubes

* 6 Classifying spaces, and extensions

  * 6\.1 Non-Abelian extensions revisited

  * 6\.2 Classifying spaces

  * 6\.3 Simplicial Automorphisms and Regular Representations

  * 6\.4 Simplicial actions and principal fibrations

  * 6\.5 $\overline{W}$, $W$,  and twisted Cartesian products

  * 6\.6 More examples of Simplicial Groups

* 7 Non-Abelian Cohomology: Torsors, and Bitorsors

  * 7\.1 Descent: Bundles, and Covering Spaces

  * 7\.2 Descent: simplicial fibre bundles

  * 7\.3 Descent: Sheaves

  * 7\.4 Descent: Torsors

  * 7\.5 Bitorsors

  * 7\.6 Relative $M$-torsors

* 8 Hypercohomology and exact sequences

  * 8\.1 Hyper-cohomology

  * 8\.2 Mapping cocones and Puppe sequences

  * 8\.3 Puppe sequences and classifying spaces

* 9 Non-Abelian Cohomology: Stacks

  * 9\.1 Fibred Categories

  * 9\.2 The Grothendieck construction

  * 9\.3 Prestacks: sheaves of local morphisms

  * 9\.4 From prestacks to stacks

* 10 Non-Abelian Cohomology: Gerbes

  * 10\.1 Gerbes

  * 10\.2 Geometric examples of gerbes

  * 10\.3 Cocycle description of gerbes

* 11 Homotopy Coherence and Enriched Categories

  * 11\.1 Case study: examples of homotopy coherent diagrams

  * 11\.2 Simplicially enriched categories

  * 11\.3 Structure

  * 11\.4 Nerves and Homotopy Coherent Nerves

  * 11\.5 Useful examples

  * 11\.6 Two nerves for 2-groups

  * 11\.7 Pseudo-functors between 2-groups

* 12 Other enrichments, other versions of homotopy coherence

  * 12\.1 Other enrichments`?

  * 12\.2 From simplicially enriched to chain complex enriched

    * [[Eilenberg-Zilber map]], [[shuffle]], [[products of simplices]]

    * [[monoidal Dold-Kan correspondence]]

* 13 More simplicial constructions!

* 14 Indexed / weighted limits and colimits

(This gives the list for the first few chapters. There are more, but they are in a more fluid state.  Anyone who would like a `fluid copy' is welcome to contact the author.)

category: reference
[[!redirects crossed menagerie]]

[[!redirects Crossed Menagerie]]


crossed module

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Content#
* table of contents
{:toc}

## Idea

The concept of _crossed modules_  of groups ([Whitehead 41](#Whitehead41), [Whitehead 49](#Whitehead49)) is a basic concept in [[homotopical algebra]] and [[homological algebra]]: It is (from the [[nPOV]]) a convenient way of encoding a [[strict 2-group]] $G$ in terms of a [[homomorphism]] $\partial : G_2 \to G_1$ of two ordinary [[groups]].

From other points of view it is:

*  like the inclusion of a [[normal subgroup]], but isn\'t an inclusion in general;

*  like a [[module]] with a twisted 'multiplication';

*  like an [[action]] by [[automorphisms]] on a group;

*  a [[crossed complex]] concentrated in degrees $1$ and $2$;

*  a nonabelian [[chain complex|chain-complex]] of length 2;

*  a [[Moore complex]] of certain [[simplicial groups]].

Historically, crossed modules were among the first examples of [[higher dimensional algebra]] to be studied.

## Definition

### Diagrammatic definition

\begin{defn}\label{DiagrammaticDefinition}
A **crossed module of groups** is

* a [[pair]] of [[groups]] $G_2, G_1$,

* a [[group homomorphism]]

  $$
    G_2
      \overset{
        \delta
       }{
         \longrightarrow
       }
    G_1
  $$

* a [[group homomorphism]] from $G_1$ to the [[automorphism group]] of $G_2$:

  $$
    G_1
      \overset{
        \alpha
      }{
        \longrightarrow
      }
      Aut(G_2)
      \,,
  $$

  which we may equivalently regard as a [[function]]

  $$
    \alpha
     \;\colon\;
     G_1 \times G_2
       \longrightarrow
     G_2
   $$

   (out of the [[Cartesian product]] of [[underlying]] [[sets]]/[[objects]])

   that satisfies the $G_1$-[action property](action#eq:ActionProperty) and is such that for every $g_1 \in G_1$ it is a group [[automorphism]] of $G_2$;

such that the following [[commuting diagram|diagrams commute]]:

$$
  \array{
    G_2 \times G_2
    &&
      \overset{
        \delta \times Id
      }{
        \longrightarrow
      }
    &&
    G_1 \times G_2
    \\
    &
    {}_{\mathllap{Ad}}
    \searrow
    &&
    \swarrow_{\mathrlap{\alpha}}
    \\
    &&
    G_2
  }
$$

and

$$
  \array{
    G_1 \times G_2
    &
      \stackrel{
        \alpha
      }{
        \longrightarrow
      }
    &
    G_2
    \\
    \big\downarrow\mathrlap{{}^{
      {Id \times \delta}
    }}
    &&
    \big\downarrow\mathrlap{{}^{
      {\delta}
    }}
    \\
    G_1 \times G_1
    &
      \stackrel{
        Ad
      }{
        \longrightarrow
      }
    &
    G_1
    \,,
  }
$$
where $Ad$ denotes the [[adjoint action]] of $G_2$ on itself.
\end{defn}

\begin{remark}
The diagrammatic Def. \ref{DiagrammaticDefinition} makes sense  [[internalization|internal]] to any [[cartesian monoidal category]] $\mathcal{C}$:

* for $\mathcal{C}=$ [[Sets]] we get bare crossed modules of [[discrete groups]];

* for $\mathcal{C}=$ [[SmoothManifolds]] we get crossed modules of [[Lie groups]];

  if one allows [[infinite-dimensional manifolds]] here then this subsumes crossed modules of [[Kac-Moody groups]] such as appear in models for the [[String 2-group]].

\end{remark}

Alternatively, one can take another tack, and define crossed module objects in categories that support enough structure without using internal groups, the most general case of which, in practice, are [[semiabelian categories]]. There one considers the objects to behave 'like groups' in the sense that the category they form looks very much like the category of groups. Janelidze ([Janelidze 2003](#Janelidze_03)) defined the notion of internal crossed module in a semiabelian category (so that in the prototypical example of the category of groups, they reduce to the above notion).

A key result, also due to ([Janelidze 2003](#Janelidze_03)) and generalising the Brown-Spencer theorem from the case of ordinary crossed modules, is the following:

+-- {: .num_theorem}
###### Theorem
**(Janelidze's Brown-Spencer theorem).**
Let $C$ be a [[semiabelian category]]. Then the category $XMod(C)$ of crossed modules in $C$ is equivalent to the category $Gpd(C)$ of internal groupoids in $C$.
=--

Here the notion of [[internal groupoid]] is the usual diagrammatic notion.

### Definition in terms of equations

The two [[diagrams]] can be translated into [[equations]], which may often be helpful.

*   If we write the effect of acting with $g_1\in G_1$ on $g_2\in G_2$ as ${}^{g_1}g_2$, then the second diagram translates as the equation:
    $$\delta({}^{g_1}g_2) = g_1\delta(g_2)g_1^{-1}.$$
In other words, $\delta$ is equivariant for the action of $G_1$.

*   The first diagram is slightly more subtle.  The group $G_2$ can act on itself in two different ways, (i) by the usual conjugation action, ${}^{g_2}g^\prime_2=g_2g^\prime_2g_2^{-1} $ and (ii) by first mapping $g_2$ down to $G_1$ and then using the action of that group back on $G_2$.  The first diagram says that the two actions coincide. Equationally this gives:
$${}^{\delta(g_2)}g^\prime_2 = g_2g^\prime_2g_2^{-1}.$$
This equation is known as the **Peiffer rule** in the literature.
Another way to interpret it is to rewrite it slightly:
$${}^{\delta(g_2)}g^\prime_2 g_2 = g_2g^\prime_2$$
The Peiffer rule can thus be seen as a 'twisted commutativity law' for $G_2$.

### Morphisms {#Morphisms}

For $G$ and $H$ two [[strict 2-group | strict 2-groups]] coming from crossed modules $[G]$ and $[H]$, a morphism of strict 2-groups $f : G \to H$, and hence a morphism of crossed modules $[f] : [G] \to [H]$ is a [[2-functor]]

$$
  \mathbf{B}f : \mathbf{B}G \to \mathbf{B}H
$$

between the corresponding [[delooping|delooped]] [[2-groupoid | 2-groupoids]]. Expressing this in terms of a diagram of the ordinary groups appearing in $[G]$ and $[H]$ yields a diagram called a [[butterfly]]. See there for more details.

## Examples

* For $H$ any group, its [[automorphism]] crossed module is

  $$
    AUT(H)
    \coloneqq
    (G_2 = H, G_1 = Aut(H), \delta = Id, \alpha = Ad)
   \,.
  $$

  Under the equivalence of crossed modules with [[strict 2-group | strict 2-groups]] this corresponds to the [[automorphism 2-group]]

  $$
    Aut_{Grpd}(\mathbf{B}H)
  $$

  of [[automorphism | automorphisms]] in the category [[Grpd]] of [[groupoid | groupoids]] on the one-object [[delooping]] [[groupoid]] $\mathbf{B}H$ of $H$.

*  Almost the canonical example of a crossed module is given by a group $G$ and a [[normal subgroup]] $N$ of $G$.  We take $G_2 = N$, and $G_1 = G$ with the action $\alpha$ being the [[conjugation action]], whilst $\delta$ is the given inclusion, $N \hookrightarrow G$.

  This is 'almost canonical', since if we replace the groups by simplicial groups $G_.$ and $N_.$, then $(\pi_0(G_.),\pi_0(N_.),\pi_0(inc))$ is a crossed module, and given any crossed module, $(C,P,\delta)$, there is a simplicial group $G_.$ and a normal subgroup $N_.$, such that the construction above gives the given crossed module up to isomorphism.

* Another standard example of a crossed module is $M \to ^0 P$ where $P$ is a group and $M$ is a $P$-module. Thus the category of modules over groups embeds in the category of crossed modules.

* If $\mu: M \to P$ is a crossed module with cokernel $G$, and $M$ is abelian, then the operation of $P$ on $M$ factors through $G$. In fact such crossed modules in which both $M$ and $P$ are abelian should not be sneezed at! A good example is $\mu: C_2 \times C_2 \to C_4$ where $C_n$ denotes the cyclic group of order $n$, $\mu$ is injective on each factor, and $C_4$ acts on the product by the twist.  This crossed module has a [[classifying space]] $X$ with fundamental and second homotopy groups $C_2$ and non trivial $k$-invariant in $H^3(C_2, C_2)$, so $X$ is not a product of [[Eilenberg-MacLane space | Eilenberg-MacLane spaces]].  However the crossed module is an algebraic model and so one one can do algebraic constructions with it. It  gives in some ways a better feel for the space than the $k$-invariant. The [[higher homotopy van Kampen theorem]] implies that   the above $X$ gives the 2-type of the [[mapping cone]] of the map of [[classifying space | classifying spaces]] $BC_2 \to BC_4$.

* Suppose $F\stackrel{i}{\to}E\stackrel{p}{\to}B$ is a [[fibration sequence]]

  of [[pointed object|pointed spaces]], thus $p$ is a [[fibration]] in the [[model structure on topological spaces|topological sense]] (lifting of paths and homotopies of paths will suffice), $F = p^{-1}(b_0)$, where $b_0$
  is the basepoint of $B$.  The fibre $F$ is pointed at $f_0$, say, and $f_0$
  is taken as the basepoint of $E$ as well.

    There is an induced map on [[homotopy group | homotopy groups]]

    $$\pi_1(F) \stackrel{\pi_1(i)}{\longrightarrow} \pi_1(E)$$

    and if $a$ is a loop in $E$ based at $f_0$, and $b$ a loop in $F$ based at $f_0$, then the composite path corresponding to $a b a^{-1}$ is [[homotopy|homotopic]] to one wholly within $F$.  To see this, note that $p(a b a^{-1})$ is [[null homotopy|null homotopic]].  Pick a [[homotopy]] in $B$ between it and the constant map, then lift that homotopy back up to $E$ to one starting at $a b a^{-1}$.  This homotopy is the required one and its other end gives a well defined element ${}^a b \in \pi_1(F)$ (abusing notation by confusing paths and their homotopy classes).  With this action $(\pi_1(F), \pi(E), \pi_1(i))$ is a crossed module.  This will not be proved here, but is not that difficult.  (Of course,  secretly, this example is 'really' the same as the previous one since a fibration of [[simplicial group | simplicial groups]] is just morphism that is an [[epimorphism]] in each degree, and the [[fibration sequence|fibre]] is thus just a [[simplicial group|normal simplicial subgroup]]. What is fun is that this generalises to 'higher dimensions'.)

* A particular case of this last example can be obtained from the inclusion of a subspace $A\to X$ into a pointed space $(X,x_0)$, (where we assume $x_0\in A$).  We can replace this inclusion by  a homotopic fibration, $\overline{A}\to X$ in 'the standard way', and then find that the fundamental group of its fibre is $\pi_2(X,A,x_0)$.

A deep theorem of J.H.C. Whitehead is that the crossed module

 $$\delta: \pi_2(A \cup \{e^2_\lambda\}_{\lambda \in \Lambda},A,x) \to \pi_1(A,x)$$

 is the [[free crossed module]]  on the characteristic maps of the $2$-cells. One utility of this is that it enables the  expression of  nonabelian chains and boundaries ideas in dimensions $1$ and $2$: thus for the standard picture of a Klein Bottle formed by identifications from a square $\sigma$ the formula

$$\delta \sigma = a+b-a +b $$

makes sense with $\sigma$ a generator of a free crossed module; in the usual abelian chain theory we can write only $\partial \sigma =2b$, thus losing information.

Whitehead's proof of this theorem used knot theory and transversality. The theorem is also a consequence of the $2$-dimensional Seifert-van Kampen Theorem, proved by Brown and Higgins, which states that the functor

$\Pi_2$: (pairs of pointed spaces) $\to$ (crossed modules)

preserves certain colimits (see reference below).

This last example was one of the first investigated by Whitehead and his proof appears also in a little book by [[Hilton]]; see also [[Nonabelian algebraic topology]], however the more general result of Brown and Higgins determines also the group $\pi_2(X \cup CA,X,x)$ as a crossed $\pi_1(X,x)$ module, and then Whitehead's result is the case with $A$ is a wedge of circles.

## Related concepts

* [[group]]

* [[2-group]], **crossed module**, [[differential crossed module]]

* [[3-group]], [[2-crossed module]] / [[crossed square]], [[differential 2-crossed module]]

* [[n-group]]

* [[∞-group]], [[simplicial group]], [[crossed complex]], [[hypercrossed complex]]

## References
 {#References}

The second axiom for a crossed module first appeared as footnote 35 on p. 422 of Whitehead's paper:

* {#Whitehead41} [[J.H.C. Whitehead]], *On adding relations to homotopy groups*, Ann. of Math. **42** 2 (1941) 409–428 [[doi:10.2307/1968907](https://doi.org/10.2307/1968907)];

A key result on "Free crossed modules".

* {#Whitehead49} [[J. H. C. Whitehead]],  Section 16 of: *Combinatorial Homotopy II*, Bull. Amer. Math. Soc., {\bf 55} (1949) 453-496 [[euclid](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-55/issue-5/Combinatorial-homotopy-II/bams/1183513797.full)]

An exposition of this proof is in

* [[R. Brown]], _On the second relative homotopy group of an adjunction space: an exposition of a theorem of J.H.C. Whitehead_,   J. London Math. Soc._ (2) 22 (1980) 146-152 ([doi:10.1112/jlms/s2-22.1.146](https://doi.org/10.1112/jlms/s2-22.1.146))

see also

* [[Peter J. Hilton]], _An Introduction to Homotopy Theory_, Cambridge University Press (1953) [[doi:10.1017/CBO9780511666278](https://doi.org/10.1017/CBO9780511666278)]

Note that the geometric core of the proof uses [[knot theory]] and transversality arguments which come from the "previous paper" of Whitehead:

* [[J. H. C. Whitehead]], *On adding relations to  homotopy groups*, Annals of Math., **42** (1941) 400-428 [[doi:10.2307/1968907](https://doi.org/10.2307/1968907)]

Textbook account of crossed modules and their relation to [[strict 2-groups]]:

* [[Saunders MacLane]], §XII.8 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

further exposition:

*  {#BaezLauda03} [[John Baez]], [[Aaron Lauda]], pp. 45 of: *HDA V: 2-Groups*, Theory and Applications of Categories **12** (2004) 423-491. [[arXiv:math.QA/0307200](http://arxiv.org/abs/math.QA/0307200)]

The following paper

* [[Ronnie Brown]], [[Philip Higgins]], _On the connections between the second relative homotopy groups of some related spaces_,  _Proc. London Math.  Soc._ (3) 36 (1978) 193-212.

showed that the theorem of Whitehead on free crossed modules from CH II Sec 16 is a special case of a 2-dimensional Van Kampen type Theorem for the homotopy crossed modules  (over groupoids) of open unions of "connected" triples $(X,A,S$ of spaces where $S$ is a set of base points. However the proof of the main theorem uses the relation of crossed modules not to cat$^1$-groups but to "double groupoids with connections", also proved with Spencer. Full details and references are  in Part I of:

* [[Ronnie Brown]], [[Philip Higgins]], and R. Sivera, _[[Nonabelian Algebraic Topology]]: Filtered spaces, Crossed Complexes, Cubical Homotopy Groupoids_, EMS Tracts in Mathematics, Vol. 15, (2011).

See also

* [[Ronnie Brown]], _Groupoids and crossed objects in algebraic topology_,  Homology, Homotopy and Applications, 1 (1999) 1-78.

* {#Janelidze_03} [[George Janelidze]], _Internal crossed modules_, Georgian Mathematical Journal **10** (2003) pp 99-114. ([EuDML](https://eudml.org/doc/51553))

For wider uses of crossed modules in other algebraic contexts see for example

* A. S-T. Lue, _Cohomology of groups relative to a variety_, J. Algebra 69 (1) (1981) 155–174.

[[!redirects crossed modules]]