  We construct algebraic and algebro-geometric models for the spaces of
unparametrized paths. This is done by considering a path as a holonomy
functional on indeterminate connections.
  For a manifold X, we construct a Lie algebroid P which serves as the tangent
space to X (punctual paths) inside the space of all unparametrized paths. It
serves as a natural receptacle of all "covariant derivatives of the curvature"
for all bundles with connections on X.
  If X is an algebraic variety, we integrate P to a formal groupoid G which can
be seen as the formal neighborhood of X inside the space of paths. We establish
a relation of G with the stable map spaces of Kontsevich.
