Contents

cohomology

# Contents

## Idea

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

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

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

###### Remark

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 $G\to K$ a homomorphism of ∞-groups the problem of factoring a modulating morphism $X\to \mathbf{B}K$ through this morphism, up to a chosen homotopy.

## Definition

We spell out three equivalent definitions.

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

$\array{ K\sslash G &\to& \mathbf{B}G \\ && \downarrow^{\mathrlap{\mathbf{B}\phi}} \\ && \mathbf{B}K }$

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

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

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

### Reduction of the cocycle

The reduction of the structure of the cocycle $k$ is a diagram

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

in $\mathbf{H}$, hence a morphism

$\sigma : k \to \mathbf{B}\phi$

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

### Section of the associated coset-bundle

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

$\sigma \in \Gamma_X(P \times_{K} K\sslash G)$

of the associated $K \sslash G$ fiber ∞-bundle.

### Equivariant map to the coset

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

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

See the discussion at ∞-action.

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

## References

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 09:00:06. See the history of this page for a list of all contributions to it.