nLab
Sandbox

Every wiki needs a sandbox! Just test between the horizontal rules below (*** in the source) and don't worry about messing things up.

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	local prequantum field theory	homotopy theory		local action functional / prequantum n-bundle
1)	1-dimensional Dijkgraaf-Witten theory	1-groupoids/homotopy 1-types	$B♭G$	$-$group character$\to$	$B♭U(1)$
	$⋮$
2)	n-dimensional Dijkgraaf-Witten theory	n-groupoids/homotopy n-types	$B♭G$	$-$cocycle in group cohomology$\to$	$B^n♭U(1)$
			$\downarrow$	embed flat ∞-connections in all principal ∞-connections	$\downarrow$
3)	n-dimensional Chern-Simons theory	n-stacks/smooth homotopy n-types	$B{G}_{\mathrm{conn}}$	$-$cocycle in differential cohomology$\to$	$B^{n}U(1{)}_{\mathrm{conn}}$
			$\downarrow$	forget connection, remember principal ∞-bundle	$\downarrow$
	(underlying instanton sectors)		$BG$	$-$cocycle in smooth ∞-group-cohomology$\to$	$B^{n}U(1)$
	(transgression/fiber integration)		$[{\Sigma }_k,B{G}_{\mathrm{conn}}]$	$\stackrel{\exp (2\pi \mathrm{in}{\int }_{{\Sigma }_k}[{\Sigma }_k,\nabla ])}{\to }$	$B^{n-k}U(1)$
				(Chern-Simons invariant)

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category: meta