  We provide a necessary and sufficient condition for the derived
self-intersection of a smooth subscheme inside a smooth scheme to be a
fibration over the subscheme. As a consequence we deduce a generalized HKR
isomorphism. We also investigate the relationship of our result to path spaces
in homotopy theory, Buchweitz-Flenner formality in algebraic geometry, and draw
parallels with similar results in Lie theory and symplectic geometry.
