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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
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For G a Lie group, can I make sense of G/G as a derived manifold in a nice way?
The point of derived manifolds is to fix up limits, not colimits, of manifolds. A standard "fix" for what you bring up is to instead derive the notion of $G$-invariants, rather than to change the quo …