nLab reduction and lift of structure groups





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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.

A GG-reduction of the frame bundle of a smooth manifold is called a G-structure.


As one passes to higher differential geometry, the (epi, mono) factorization system dissolves into the infinite tower of (n-epi, n-mono) factorization systems, and hence the distinction between reduction and lift of structure groups blurs. One may just consider generally for GKG\to K a homomorphism of ∞-groups the problem of factoring a modulating morphism XBKX\to \mathbf{B}K through this morphism, up to a chosen homotopy.


We spell out three equivalent definitions.

Let H\mathbf{H} be the ambient (∞,1)-topos, let G,KGrp(H)G,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.



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

Last revised on March 30, 2019 at 13:00:06. See the history of this page for a list of all contributions to it.