nLab
reduction and lift of structure groups

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

A special case of G-structure.

For GKG \to K a monomorphism of groups, a GG-structure on a KK-principal bundle is a reduction of the structure group from KK to GG.

Alternatively, for GKG \to K an epimorphism of groups, a GG-structure on a KK-principal bundle is a lift of the structure group from KK to GG.

Remark

The distinction between epimorphism and monomorphism, hence between reduction and lift of structure groups is, however, not quite “homotopy correct”. For instance in the first steps of the Whitehead tower of the orthogonal group

Spin(n)SO(n)O(n)(GL(n)) Spin(n) \to SO(n) \hookrightarrow O(n) (\hookrightarrow GL(n))

the morphism Spin(n)SO(n)Spin(n) \to SO(n) is epi, while SO(n)O(n)SO(n) \to O(n) is mono.

See (NSS, section 4.3).

Definition

We spell out three equivalent definitions.

Let H\mathbf{H} be the ambient (∞,1)-topos, let G,KGrp(HG,K \in Grp(\mathbf{H} be two ∞-groups and let ϕ:GK\phi : G \to K be a homomorphism, hence Bϕ:BGBK\mathbf{B}\phi : \mathbf{B}G \to \mathbf{B}K the morphism in H\mathbf{H} between their deloopings. Write

KG BG Bϕ BK \array{ K\sslash G &\to& \mathbf{B}G \\ && \downarrow^{\mathrlap{\mathbf{B}\phi}} \\ && \mathbf{B}K }

for the corresponding fiber sequence, with KGK \sslash G the homotopy fiber of the given morphism. By the discussion at ∞-action this exhibits the canonical KK-∞-action on the coset object KGK\sslash G.

Let furthermore PXP \to X be a KK-principal ∞-bundle in H\mathbf{H}. By the discussion there this is modulated essentially uniquely by a cocycle morphism k:XBKk : X \to \mathbf{B}K such that there is a fiber sequence

P X BK. \array{ P &\to& X \\ && \downarrow \\ && \mathbf{B}K } \,.

Reduction of the cocycle

The reduction of the structure of the cocycle kk is a diagram

X σ BG k σ˜ BK \array{ X &&\stackrel{\sigma}{\to}&& \mathbf{B}G \\ & {}_{\mathllap{k}}\searrow &\swArrow_{\tilde\sigma}& \swarrow \\ && \mathbf{B}K }

in H\mathbf{H}, hence a morphism

σ:kBϕ \sigma : k \to \mathbf{B}\phi

in the slice (∞,1)-topos B /BK\mathbf{B}_{/\mathbf{B}K}.

Section of the associated coset-bundle

By the discussion at associated ∞-bundle such a diagram is equivalently a section

σΓ X(P× KKG) \sigma \in \Gamma_X(P \times_{K} K\sslash G)

of the associated KGK \sslash G fiber ∞-bundle.

Equivariant map to the coset

The above is the categorical semantics of what in the homotopy type theory internal language of H\mathbf{H} is given by the syntax

( x:BKPKG):Type. \vdash (\prod_{x : \mathbf{B}K} P \to K\sslash G) : Type \,.

See the discussion at ∞-action.

This expresses the fact that the reduction of the structure group along ϕ\phi is equivalently a KK-equivariant map PKGP \to K\sslash G.

Examples

References

In the generality of principal infinity-bundles, reductions/lifts of structure groups are discused in section 4.3 of

Revised on December 14, 2012 17:19:44 by Urs Schreiber (71.195.68.239)