nLab
crossed homomorphism

Contents

Contents

Idea

In group theory, and specifically in group cohomology-theory, the notion of crossed homomorphisms (Def. below) is a generalization of that of group homomorphisms where a group action – of the domain-group GG by automorphisms on the codomain group Γ\Gamma – is incorporated.

Crossed homomorphisms are equivalently homomorphic sections of the projection to GG out of the corresponding semidirect product group ΓG\Gamma \rtimes G (Prop. below).

This largely explains their relevance in all contexts of GG-equivariant algebraic topology and equivariant differential topology, such as in equivariant principal bundle-theory (Prop. below) where semidirect product groups appear as GG-equivariant groups.

In particular (Prop. below, which is not made explicit in traditional literature):

This last fact is widely appreciated (see Rem. below) in the special case that Γ\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 internal to any ambient category 𝒞\mathcal{C}, it may not need to strictly coincide with that of cocycles in first group cohomology beyond 𝒞=\mathcal{C} = Set (for reasons discussed at Lie group cohomology, or, for the case of profinite groups, in NSW 2008, p. 24).

Definition

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

α:GAut Grp(Γ) \alpha \;\colon\; G \xrightarrow{\;\;} Aut_{Grp}(\Gamma)

of GG on Γ\Gamma by group automorphisms, hence equivalently a semidirect product group, to be denoted:

1ΓiΓ αGpr 2G1. 1 \to \Gamma \xhookrightarrow{\;i\;} \Gamma \rtimes_\alpha G \xrightarrow{ \;pr_2\; } G \to 1 \,.

Component definition

Definition

A crossed homomorphism from GG to Γ\Gamma is a function (morphism in the ambient category)

ϕ:GΓ \phi \;\colon\; G \xrightarrow{\;} \Gamma

satisfying the following “GG-crossed” homomorphism property:

g 1,g 2Gϕ(g 1g 2)=ϕ(g 1)α(g 1)(ϕ(g 2)). \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) \,.

Remark

Def. appears, already with the non-abelian generality in mind, back in Whitehead 1949, (3.1) (together with the notion of crossed modules, both as tools for analyzing homotopy types with non-trivial fundamental group). For abelian Γ\Gamma the definition is picked up in MacLane 1975, IV (2.1) and many following references on group cohomology, e.g. Brown 1982, p. 45. For general non-abelian Γ\Gamma the notion is later used in Murayama & Shimakawa 1995 for discussion of equivariant principal bundles. In the context of nonabelian group cohomology it is reviewed in Gille & Szamuely, 2006, (1) on p. 25. Another textbook account is in Milne 2017, 15.a (16.a) in the pdf, aimed at application to algebraic groups.

Example

For α\alpha the trivial action, the notion of a crossed homomorphism (Def. ) reduces to that of an ordinary group homomorphism.

Remark

(as 1-cocycles in group cohomology)
In the special case that Γ\Gamma in Def. is an abelian group, crossed homomorphisms are also known as cocycles in group cohomology in degree 1 (e.g. Brown 1982, p. 45). In general they may be understood as 1-cocyles in non-abelian group cohomology (e.g. NSW 2008, p. 16).

Example

The constant function const e:GΓconst_{\mathrm{e}} \,\colon\, G \to \Gamma on the neutral element eΓ\mathrm{e} \in \Gamma is always a crossed homomorphism (Def. ), being the trivial cocycle under the interpretation of Rem. .

Proposition

(as splittings/homomorphic sections of the semidirect product group-projection)
Crossed homomorphisms GΓG \to \Gamma (Def. ) are equivalently homomorphic sections of the projection out of the semidirect product group Γ αGpr 2G\Gamma \rtimes_\alpha G \xrightarrow{pr_2} G (see also at split group extensions):

CrsHom(G,Γ)GrpHom /G(G,Γ αG)GrpHom(G,Γ αG)×GrpHom(G,G){id}. 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\} \,.

(e.g. Brown 1982, p. 88, NSW 2008, Ex. 1 on p. 24, Milne 2017, Ex. 15.1 (16.1 in the pdf))
Proof

By immediate unwinding of the definition of the semidirect product group, such a section is an assignment

G Γ αG g (ϕ(g),g) \array{ G &\xrightarrow{\;}& \Gamma \rtimes_\alpha G \\ g &\mapsto& \big( \phi(g), \, g \big) }

such that

g 1,g 2G(ϕ(g 1g 2),g 1g 2)=!(ϕ(g 1),g 1)(ϕ(g 2),g 2)=(ϕ(g 1)α(g 1)(ϕ(g 2)),g 1g 2). \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) \,.

Proposition

(graphs of crossed homomorphisms)
The graph of a function of a crossed homomorphism (Def. ) is a subgroup of the semidirect product group

(1)G^ΓG,such thatpr 2(G^)GandG^i(Γ)={(e,e)}, \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 unique crossed homomorphism.

This is, essentially, considered in tom Dieck 1969, Sec. 2.1, also Guillou, May & Merling 2017, Lem. 4.5.
Proof

The first statement is immediate from the definition.

For the converse, consider a subgroup G^\widehat{G} as in (1). Then the subgroup property implies that

(γ,g),(γ,g)G^(γ,g)(γ,g) 1=(γ,g)(α(g 1)(γ 1),g 1)=(γγ 1,e)G^. (\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 (1) implies that

(γ,g),(γ,g)G^γ=γ. (\gamma,\,g) ,\, (\gamma',\, g) \;\in\; \widehat{G} \;\;\;\; \Rightarrow \;\;\;\; \gamma = \gamma' \,.

Together with the first condition in (1) this implies that G^\widehat{G} is the graph of a function ϕ:GΓ\phi \;\colon\; G \to \Gamma. From this the claim follows by Prop. .

Remark

Subgroups of the form (1) are considered throughout articles by Peter May on equivariant principal bundles (e.g. in Lashof & May 1986, Thm. 10, May 1990, Thm. 7). That these are equivalently (graphs of) crossed homomorphisms may have been one the key observations that lead to Murayama & Shimakawa 1995, though the statement of Prop. was still not explicit there.

Definition

(crossed conjugation)
Two crossed homomorphisms ϕ 1,ϕ 2\phi_1, \phi_2 (Def. ) are adjoint ϕ 1γϕ 2\phi_1 \xrightarrow{\;\gamma\;} \phi_2 if in their incarnation as homomorphisms to ΓG\Gamma \rtimes G (Prop. ) they are related by conjugation with an element
γΓΓG\gamma \in \Gamma \xhookrightarrow{\;} \Gamma \rtimes G, in that:

(2)gGϕ 2(g)=γ 1ϕ 1(g)α(g)(γ). \underset{ g \in G }{\forall} \;\;\;\; \phi_2(g) \;=\; \gamma^{-1} \cdot \phi_1(g) \cdot \alpha(g)(\gamma) \,.

(e.g. Gille & Szamuely 2006, (2) on p. 25, Guillou, May & Merling 2017, Def. 4.11)
Remark

(as 1-coboundaries in group cohomology)
When Γ\Gamma is an abelian group, the conjugates according to Def. of the constant function const 0const_{0} (Exp. ) are of the form

(3)gα(g)(γ)γ g \;\mapsto\; \alpha(g)(\gamma) - \gamma

(writing now in additive notation as befits an abelian group).

These (3) are also known as “principal crossed homomorphisms”.

With crossed homomorphisms understood as 1-cocycles in group cohomology (Rem. ), these elements (3) are the 1-coboundaries.

The definitions here apply verbatim also for non-abelian Γ\Gamma, where we get non-abelian group cohomology (e.g. NSW 2008, p. 16).

As sliced functors of delooping groupoids

All the formulas above may conceptually be understood as follows:

Proposition

(groupoid of crossed homomorphisms is sliced functor groupoid of delooping groupoids)
The groupoid

(4)CrsHom(G,Γ) adΓ(CrsHom(G,Γ)×ΓΓ)Grpd CrsHom(G,\Gamma) \sslash_{\!\! ad} \Gamma \;\; \coloneqq \big( CrsHom(G,\Gamma) \times \Gamma \rightrightarrows \Gamma \big) \;\; \;\;\; \in \; Grpd

of crossed homomorphisms (Def. ) with conjugations between them (Def. ) is isomorphic to the sliced functor groupoid of sections of the delooping groupoid B(ΓG)\mathbf{B}(\Gamma \rtimes G) of the semidirect product group:

(5)CrsHom(G,Γ) adΓFnctr /BG(BG,B(ΓG))=Fnctr(BG,B(ΓG))×Fnctr(BG,BG){id}. 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\} \,.

We take this statement and the following proof from SS21.
Proof

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:

from SS21

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 GΓGG \to \Gamma \rtimes G whose composition with ΓGpr 2G\Gamma \rtimes G \xrightarrow{pr_2} G is the identity morphisms – hence are bijective to crossed homomorphisms, by Prop. .

Moreover, natural transformations FFF \Rightarrow F' fitting into this diagram (indicated by the dashed double arrow on the left) must be such that their whiskering with Bpr 2\mathbf{B}pr_2 is the identity natural transformation, which means that their single component (as shown on the right) is of the form

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

where eG\mathrm{e} \in G denotes the neutral element.

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

(γ,e) g (ϕ 1(g),g) (ϕ 2(g),g) (γ,e) AAAAAAAAAAAB(ΓG). \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

(6)(ϕ 2(g),g)=(γ 1,e)(ϕ 1(g),g)(γ,e)ΓG, \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. .

Remark

(conjugacy classes of crossed homomorphisms are non-abelian first group cohomology)
In view of the identification of crossed homomorphisms with 1-cocycles in non-abelian group cohomology, we may identify their conjugacy classes (Def. ), hence, by Prop. , the connected components of the sliced functor groupoid (5) with the group cohomology sets (at least for discrete groups):

CrsHom(H,Γ) /π 0(Fnctr /BG(BG,B(ΓG)))H Grp 1(G,Γ). 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) \,.

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

Proposition

(automorphism group of crossed homomorphisms)
For ϕ:GΓ\phi \colon G \xrightarrow{\;} \Gamma a crossed homomorphism (Def. ), its automorphism group in the conjugation groupoid (4), hence the subgroup

Aut(ϕ)Aut CrsHom(G,Γ) adΓ(ϕ)={γΓ|ϕ()=γ 1ϕ()α()(γ)}Γ 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. ) that fix it, is, equivalently, the intersection of i(Γ)={(γ,e)|γΓ}i(\Gamma) = \{ (\gamma, \mathrm{e}) \,\vert\, \gamma \in \Gamma \} with

  1. the centralizer C ΓG(Graph(ϕ))C_{{}_{\Gamma \rtimes G}}\big(Graph(\phi)\big)

  2. the normalizer N ΓG(Graph(ϕ))N_{{}_{\Gamma \rtimes G}}\big(Graph(\phi)\big)

of the graph of ϕ\phi inside the semidirect product group:

Aut(ϕ) =i(Γ)C ΓG(Graph(ϕ)) =i(Γ)N ΓG(Graph(ϕ)). \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}

(compare Lashof & May 1986, below Thms. 10, 11)
Proof

The first statement is immediate from the re-formulation (6) of crossed conjugation (2).

From this, the second statement follows by observing that the action in the form (6) manifestly fixes the GG-component under projection along pr 2pr_2, while the graph of a function GΓG \to \Gamma has at most one element with a given GG-component.

Properties

Restriction to subgroups

Proposition

(Weyl group acts on non-abelian group 1-cohomology of subgroup)
For HGH \subset G a subgroup, the set of crossed homomorphisms HΓH \to \Gamma (Def. ), with respect to the restricted action of HH on Γ\Gamma, carries a group action of the normalizer subgroup N G(H)GN_G(H) \,\subset\, G, given by

(7)N G(H)×CrsHom(H,G) CrsHom(H,G) (n,ϕ) ϕ nα(n)(ϕ(n 1()n)). \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. ), hence on first non-abelian group cohomology, this action descends to an action of the Weyl group W G(H)N G(H)/HW_G(H) \coloneqq N_G(H)/H:

H Grp 1(H,Γ)W G(H)Act(Sets). H^1_{Grp}(H,\,\Gamma) \;\;\; \in \; W_G(H) Act(Sets) \,.

Proof

1. Action property. It is clear that (7) is a group action if only ϕ n\phi_n is indeed a crossed homomorphism. This follows by a direct computation:

ϕ n(h 1h 2) =α(n)(ϕ(n 1h 1h 2n)) definition ofϕ n =α(n)(ϕ(n 1h 1nn 1h 2n)) group property ofN G(H)G =α(n)(ϕ(n 1h 1n)α(n 1h 1n)(ϕ(n 1h 2n))) crossed homomorphism property ofϕ =α(n)(ϕ(n 1h 1n))α(h 1n)(ϕ(n 1h 2n)) action property ofα =α(n)(ϕ(n 1h 1n))α(h 1)(α(n)(ϕ(n 1h 2n))) action property ofα =ϕ n(h 1)α(h 1)(ϕ n(h 2)) definition ofϕ n. \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 (7) descends to group cohomology, we need to show for

ϕ()=γ 1ϕ()α()(γ) \phi'(-) \;=\; \gamma^{-1} \cdot \phi(-) \cdot \alpha(-)(\gamma)

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

ϕ n(h) =α(n)(γ 1ϕ(n 1hn)α(n 1hn)(γ)) assumption with definition ofϕ n =α(n)(γ 1)α(n)(ϕ(n 1hn))α(h)(α(n)(γ)) action property ofα =(α(n)(γ)) 1ϕ n(h)α(h)(ϕ(n)(γ)) definition ofϕ n. \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. of the conjugation groupoid of crossed homomorphisms ϕ:HΓ\phi \colon H \to \Gamma with that of homomorphic sections (ϕ(),()):HΓH \big( \phi(-),\, (-) \big) \,\colon\, H \to \Gamma \rtimes H. In this latter incarnation, the action by nn is simply the “conjugation action by the adjoint action”, in that:

n:(ϕ(),())(ϕ n(),())=(e,n)(ϕ(n 1()n),(n 1()n))(e,n 1). 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 HB G(H)H \subset B_G(H) is trivial on conjugacy classes, hence that for nHN(H)n \in H \subset N(H) there is a crossed conjugation between ϕ n\phi_n and ϕ\phi. The following direct computation shows that this is given by crossed conjugation with ϕ(n)\phi(n) (which is well-defined, by the assumption that nHn \in H):

ϕ n(h) =α(n)(ϕ(n 1hn)) definition ofϕ n =α(n)(ϕ(n 1)α(n 1)(ϕ(h)α(h)(ϕ(n)))) crossed homomorphism property ofϕ =α(n)(ϕ(n 1))ϕ(h)α(h)(ϕ(n)) action property ofα =(ϕ(n)) 1ϕ(h)α(h)(ϕ(n)) crossed homomorphism property ofϕ. \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}

Examples

In relation to crossed modules

Example

(a relation to crossed modules)
Let (ΓδG)(\Gamma \xrightarrow{\delta} G) be a crossed module, with the corresponding strict 2-group

𝒢(ΓG(δ())()pr 2G). \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 1-morphism on 𝒢\mathcal{G} are in bijection to the crossed homomorphisms GΓG \to \Gamma:

(8)CrsHom(G,Γ){id 𝒢F|FStr2Grp(𝒢,𝒢)}. 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 η:id 𝒢F\eta \colon id_{\mathcal{G}} \Rightarrow F of endofunctors of the underlying action groupoid, whose component function

η 0:Obj(𝒢)Mor(𝒢) \eta_0 \,\colon\, Obj(\mathcal{G}) \xrightarrow{\;} Mor(\mathcal{G})

is a group homomorphism

η 0:GΓG \eta_0 \,\colon\, G \xrightarrow{\;} \Gamma \rtimes G

such that

s(η 0(g))=(id 𝒢) 0(g)=g, s \big( \eta_0(g) \big) \;=\; (id_{\mathcal{G}})_0(g) \;=\; g \,,

where

spr 2:Γ×GG s \,\coloneqq\, pr_2 \;\colon\; \Gamma \times G \xrightarrow{\;} G

is the source map of the groupoid 𝒢\mathcal{G}.

This means that the admissible η 0\eta_0 are precisely the homomorphic sections of ΓGG\Gamma \rtimes G \xrightarrow{\;} G. Conversely, by the invertiblity of all morphisms involved, every such η 0\eta_0 is the component homomorphism of some 2-morphism η\eta out of id 𝒢id_{\mathcal{G}}.

Therefore the statement (8) follows by Prop. .

The analogous statement for general 2-morphisms is indicated in Noohi 07, p. 12.

In relation to equivariant bundles

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

Proposition

(equivariant connected components of equivariant classifying spaces)
A G-equivariant classifying space B GΓB_G \Gamma for GG-equivariant Γ\Gamma-principal bundles exists, and the connected components of its HH-fixed loci, for compact subgroups HGH \subset G, are in bijection to the conjugacy classes (Def. ) of crossed homomorphisms (Def. ) from HH to Γ\Gamma (with respect to the restricted action of HH on Γ\Gamma):

(9)π 0((B GΓ) H)CrsHom(H,Γ) /. \pi_0 \left( \left( B_G \Gamma \right)^{H} \right) \;\; \simeq \;\; CrsHom(H,\Gamma)_{/\sim} \,.

Proof

After a little reformulation via Prop. and Prop. , this is the statement of Lashof & May 1986, Thm. 10, May 1990, Thm. 7.

In view of the explicit construction of universal equivariant principal bundles in Murayama & Shimakawa 1995, one may understand the statement of Prop. on elementary grounds, as follows:

Consider the following GG-action objects internal to Groupoids:

  • the delooping groupoid

    BΓ(Γ*)GAct(Groupids) \mathbf{B}\Gamma \,\coloneqq\, \big( \Gamma \rightrightarrows \ast\big) \;\;\; \in \; G Act\big( Groupids \big)

    via the GG-action on Γ\Gamma by group automorphisms;

  • the pair groupoid

    EG(G×GG) \mathbf{E}G \,\coloneqq\, \big( G \times G \rightrightarrows G\big)

    via the left multiplication action of GG on all three copies of GG;

  • the functor groupoid

    (10)Fnctr(EG,BΓ)GAct(Groupids) 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:F()gF(g 1()). g \,\colon\, F(-) \,\mapsto\, g \cdot F\big( g^{-1}\cdot (-) \big) \,.

Proposition

(crossed homomorphisms are fixed loci in functor groupoid from E G \mathbf{E}G to B Γ \mathbf{B}\Gamma )
If GG a discrete group, then for each subgroup HGH \subset G, there is an equivalence of groupoids (in fact of topological groupoids)

Fnctr(EG,BΓ) HCrsHom(H,Γ) adΓ Fnctr \big( \mathbf{E}G ,\, \mathbf{B}\Gamma \big)^H \;\simeq\; CrsHom(H,\Gamma) \sslash_{\!\!ad} \Gamma

between

  • the HH-fixed sub-groupoid of the functor groupoid (10) and

  • the conjugation groupoid (4) of crossed homomorphisms HΓH \to \Gamma.

This is essentially the statement of Guillou, May & Merling 2017, Thm. 4.14, Cor. 4.15. We take the following detailed proof from SS21. Its ingredients are needed below in the proof of Prop. .
Proof

By Prop. the statement is equivalently that

Fnctr(EG,BG) HFnctr /BH(BH,B(ΓH)). 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=GH = G there is in fact an isomorphism, evidently exhibited by the following functor:

For general HGH \subset G, choose a section of the coset space-projection

(11)σ:G/HG,such thatσ([e])=e, \sigma \,\colon\, G/H \xrightarrow{\;} G \,, \;\;\;\; \text{such that} \; \sigma([\mathrm{e}]) \,=\, \mathrm{e} \,,

which exists and is continuous by the assumption that GG is discrete.

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

  1. composition,

  2. taking inverses,

  3. acting with elements of HH

by the following two classes of morphisms:

(12){(eh)|hH},{eσ([g])|[g]G/H}G×G. \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:

Here

  • LL is the restriction along (H×HH)(G×GG)(H \times H \rightrightarrows H) \xhookrightarrow{\;} (G \times G \rightrightarrows G) of the previous isomorphism for H=H = G;

  • RR is given in terms of the above generating morphisms (12) as follows:

(13) \;\;

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

Therefore it is now sufficient to give a natural transformation idηRLid \xRightarrow{\eta} R \circ L, hence for each functor FF a natural transformation

η F:FRL(F). \eta_F \,\colon\, F \Rightarrow R \circ L(F) \,.

This may be taken as follows, again stated in terms of the generating morphisms (12):

It just remains to check that this is indeed natural in the functors FF, which amounts, for each HH-equivariant natural transformation GβFG \xRightarrow{\beta} F', to the commutativity of the two types of squares shown on the right here:

Indeed, the top square commutes by the HH-equivariance of β\beta, while the bottom square commutes by the naturality of β\beta.

Proposition

(residual Weyl group-action on fixed locus of equivariant classifying space)
Transported through the equivalence of Prop. , the canonical group action (see this Prop.) of the Weyl group W G(H)W_G(H) on the HH-fixed locus Fnctr(EG,BΓ) H Fnctr\big(\mathbf{E}G ,\, \mathbf{B}\Gamma\big)^H becomes, on connected components π 0(CrsHom(H,Γ) adΓ)H Grp 1(H,Γ) \pi_0 \big( CrsHom(H,\,\Gamma) \sslash_{\!\!ad} \Gamma \big) \;\; \simeq \;\; H^1_{Grp}(H,\,\Gamma) , the W G(H)W_G(H)-action on the non-abelian group 1-cohomology of HH from Prop. .

We take this statement and the following proof from SS21.
Proof

We make explicit use of the functors L,RL, R constructed in the proof of Prop. . Noticing that RR is a section of LL, we need to (1) send a crossed homomorphism up with RR, (2) there act on it with nn, (3) send the result back with LL. The result is the desired induced action.

Explicitly, by the definition of LL in the proof of Prop. , this way a crossed homomorpism ϕ:HΓ\phi \,\colon\, H \to \Gamma is sent by nN G(H)n \in N_G(H) to the assignment

(14)h(L(n(Rϕ)))(h)=α(n)((Rϕ)(n 1,n 1h)). 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 LL is independent of the choice of σ:G/HG\sigma \,\colon\, G/H \xrightarrow{\;} G (11), and that RR (whose definition does depend on this choice ) is a section for each choice. Hence we may choose σ\sigma in a way convenient way for any given nn.

Now if nHN G(H)n \in H \subset N_G(H) then its canonical action on the HH-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 nn is not in HH, and then we choose σ\sigma such as to pick n 1n^{-1} as the representative in its HH-coset:

σ([n 1])n 1. \sigma\big( \big[n^{-1}\big] \big) \;\coloneqq\; n^{-1} \,.

Notice that with this choice, RϕR\phi (13) assigns the neutral element to the morphism between the neutral element and n 1n^{-1} in the pair groupoid:

(15)(Rϕ)(e,n 1)=e,(Rϕ)(n 1,e)=e. (R\phi)\big(\mathrm{e}, n^{-1}\big) \;=\; \mathrm{e} \,, \;\;\;\; (R\phi)\big(n^{-1}, \mathrm{e}\big) \;=\; \mathrm{e} \;\;\;\;\; \,.

This way, the right hand side of (14) is evaluated as follows:

(L(n(Rϕ)))(h) =α(n)((Rϕ)(n 1,n 1h)) by(14) =α(n)((Rϕ)(e,n 1h)) by(15) =α(n 1hnn)((Rϕ)(n 1h 1n,n 1)) byH-equivariance ofRϕ =α(n 1hnn)((Rϕ)(n 1h 1n,e)) by(15) =α(n)((Rϕ)(e,n 1hn)) byH-equivariance ofRϕ =α(n)(ϕ(n 1hn)) by definition ofR(13). \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{(14)} \\ & \;=\; \alpha(n) \big( (R\phi)(\mathrm{e},\, n^{-1} \cdot h) \big) & \text{by}\;\text{(15)} \\ & \;=\; \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{(15)} \\ & \;=\; \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{(13)} \mathrlap{\,.} \end{array}

This is indeed the claimed formula (7).

Literature

To abelian groups

Discussion for the special case that Γ\Gamma is an abelian group:

in the context of homological algebra:

in the context of group cohomology:

To general groups

The general non-abelian notion:

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

A brief yextbook account in this generality

As 1-cocycles in non-abelian group cohomology:

In discussion of equivariant principal bundles:

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)

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)

Discussion in relation to crossed modules:

Last revised on September 6, 2021 at 12:21:08. See the history of this page for a list of all contributions to it.