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Algebra

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Proof

The sections in question are diagrams in Grpd of the form

BG ⟶σ S//G id↘ ⇙ ≃ ↙ p ρ BG, \array{ \mathbf{B}G && \stackrel{\sigma}{\longrightarrow} && S//G \\ & {}_{\mathllap{id}}\searrow &\swArrow_{\mathrlap{\simeq}}& \swarrow_{\mathrlap{p_\rho}} \\ && \mathbf{B}G } \,,

hence the groupoid which they form is equivalently the hom-groupoid

Grpd /BG(id BG,p ρ)∈Grpd Grpd_{/\mathbf{B}G}(id_{\mathbf{B}G}, p_\rho) \in Grpd

in the slice of Grpd over BG\mathbf{B}G. As in the proof of this proposition, with the fibrant presentation (p ρ) •(p_\rho)_\bullet of this proposition, this is equivalently given by strictly commuting diagrams of the form

(BG) • ⟶σ • (S//G) • id •↘ = ↙ (p ρ) • (BG) •. \array{ (\mathbf{B}G)_\bullet && \stackrel{\sigma_\bullet}{\longrightarrow} && (S//G)_\bullet \\ & {}_{\mathllap{id_\bullet}}\searrow &=& \swarrow_{\mathrlap{(p_\rho)_\bullet}} \\ && (\mathbf{B}G)_\bullet } \,.

These σ\sigma now are manifestly functors that are the identity on the group labels of the morphisms

σ •:(* ↓ g *)↦(σ(*) ↓ g σ(*) =ρ(σ(*)(g))). \sigma_\bullet \;\colon\; \left( \array{ \ast \\ \downarrow^{\mathrlap{g}} \\ \ast } \right) \;\; \mapsto \;\; \left( \array{ \sigma(\ast) \\ \downarrow^{\mathrlap{g}} \\ \sigma(\ast) & = \rho(\sigma(\ast)(g)) } \right) \,.

This shows that they pick precisely those elements σ(*)∈S\sigma(\ast) \in S which are fixed by the GG-action ρ\rho.

Moreover, since these functors are identity on the group labels, there are no non-trivial natural isomorphisms between them, and hence the groupoid of sections is indeed a set, the set of invariant elements.

Proposition

Given an associated bundle P× GV→XP \times_G V\to X modulated, as in this proposition, by a morphism of smooth groupoids of the form g:X⟶BGg \colon X \longrightarrow \mathbf{B}G, then its set of sections is equivalently the groupoid of diagrams

X ⟶σ S//G g↘ ⇙ ≃ ↙ p ρ BG, \array{ X && \stackrel{\sigma}{\longrightarrow} && S//G \\ & {}_{\mathllap{g}}\searrow &\swArrow_{\mathrlap{\simeq}}& \swarrow_{\mathrlap{p_\rho}} \\ && \mathbf{B}G } \,,

hence the groupoid of sections is the slice hom-groupoid

Γ X(P× GV)≃Grpd /BG(g,p ρ). \Gamma_X(P\times_G V) \simeq Grpd_{/\mathbf{B}G}(g, p_\rho) \,.

Proof

Since the ground field has characteristic zero, group averaging exists and provides a linear map

V • ⟶p V • G x ↦ 1|G|∑g∈Gg(x) \array{ V_\bullet & \overset{p}{\longrightarrow} & V_\bullet^G \\ x &\mapsto& \frac{1}{{\vert G \vert}} \underset{g \in G}{\sum} g(x) }

onto the GG-invariants.

Now for a chain homology-class [x]∈H •((V •,∂))[x] \in H_\bullet((V_\bullet,\partial)) being GG-invariant means that g[x]≔[g(x)]=[x]g[x] \coloneqq [g(x)] = [x] for all g∈Gg \in G, which implies that [x]=[p(x)][x] = [p(x)]. This means that each invariant homology class has an invariant representative, hence that the map from invariant cycles to invariant chain homology-classes

Z((V • G,∂))⟶H •((V •,∂)) Z((V_\bullet^G,\partial)) \longrightarrow H_\bullet((V_\bullet,\partial))

is an epimorphism.

Next consider the kernel of this map, which a priori is Z((V • G,∂))∩B((V •,∂))Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial)). It is now sufficient to show that this coincides with the space of GG-invariant boundaries:

Z((V • G,∂))∩B((V •,∂))≃B((V • G,∂)). Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial)) \;\simeq\; B((V_\bullet^G, \partial)) \,.

It is clear that there is an inclusion

B((V • G,∂))↪Z((V • G,∂))∩B((V •,∂)) B((V_\bullet^G, \partial)) \hookrightarrow Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial))

so it only remains to see that this is also a surjection.

To that end, consider any

x∈Z((V • G,∂))∩B((V •,∂)). x \in Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial)) \,.

Since in particular x∈B((V •,∂))x \in B((V_\bullet,\partial)), there is y∈V •y \in V_\bullet with x=∂yx = \partial y; and since moreover x∈V •(G)x \in V_\bullet(G), the above implies that

x=p(x)=p(∂y)=∂(py) x = p(x) = p(\partial y) = \partial(p y)

and hence that

x∈B((V • G,∂)). x \in B((V_\bullet^G,\partial)) \,.