nLab higher spin structure - table

smooth ∞-group	Whitehead tower of smooth moduli ∞-stacks	G-structure/higher spin structure	obstruction
	$\vdots$
	$\downarrow$
ninebrane 10-group	$\mathbf{B}Ninebrane$	ninebrane structure	third fractional Pontryagin class
	$\downarrow$
fivebrane 6-group	$\mathbf{B}Fivebrane \stackrel{\tfrac{1}{n} p_3}{\to} \mathbf{B}^{11}U(1)$	fivebrane structure	second fractional Pontryagin class
	$\downarrow$
string 2-group	$\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)$	string structure	first fractional Pontryagin class
	$\downarrow$
spin group	$\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)$	spin structure	second Stiefel-Whitney class
	$\downarrow$
special orthogonal group	$\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2$	orientation structure	first Stiefel-Whitney class
	$\downarrow$
orthogonal group	$\mathbf{B}O \stackrel{\mathbf{w}_1}{\to} \mathbf{B}\mathbb{Z}_2$	orthogonal structure/vielbein/Riemannian metric
	$\downarrow$
general linear group	$\mathbf{B}GL$	smooth manifold

(all hooks are homotopy fiber sequences)