Special and general types
For a monomorphism of groups, a -structure on a -principal bundle is a reduction of the structure group from to .
Alternatively, for an epimorphism of groups, a -structure on a -principal bundle is a lift of the structure group from to .
A -reduction of the frame bundle of a smooth manifold is called a G-structure.
We spell out three equivalent definitions.
Let be the ambient (∞,1)-topos, let be two ∞-groups and let be a homomorphism, hence the morphism in between their deloopings. Write
for the corresponding fiber sequence, with the homotopy fiber of the given morphism. By the discussion at ∞-action this exhibits the canonical -∞-action on the coset object .
Let furthermore be a -principal ∞-bundle in . By the discussion there this is modulated essentially uniquely by a cocycle morphism such that there is a fiber sequence
Reduction of the cocycle
The reduction of the structure of the cocycle is a diagram
in , hence a morphism
in the slice (∞,1)-topos .
Section of the associated coset-bundle
By the discussion at associated ∞-bundle such a diagram is equivalently a section
of the associated fiber ∞-bundle.
Equivariant map to the coset
The above is the categorical semantics of what in the homotopy type theory internal language of is given by the syntax
See the discussion at ∞-action.
This expresses the fact that the reduction of the structure group along is equivalently a -equivariant map .
reduction of tangent bundle along orthogonal group inclusion : vielbein, orthogonal structure,
reduction of tangent bundle along symplectic group inclusion : almost symplectic structure;
subsequent lift to the metaplectic group : metaplectic structure
induced lift over Lagrangian submanifolds to the metalinear group : metalinear structure;
reduction of tangent bundle along inclusion of complex general linear group : almost complex structure;
further reduction to the unitary group : almost Hermitian structure;
reduction of generalized tangent bundle along : generalized complex geometry,
reduction of generalized tangent bundle along : G2-structure;
reduction of generalized tangent bundle along : generalized vielbein, type II geometry;
reduction of exceptional tangent bundle along maximal compact subgroup of exceptional Lie group : exceptional generalized geometry
reduction of exceptional tangent bundle along : N=1 11d sugra compactification on
In the generality of principal infinity-bundles, reductions/lifts of structure groups are discused in section 4.3 of