Formal Lie groupoids
For a smooth space, there are useful refinements of the fundamental groupoid which remember more than just the homotopy class of paths, i.e. whose morphisms are (piecewise, say) smooth paths in modulo an equivalence relation still strong enough to induce a groupoid structure, but weaker than dividing out homotopies relative to endpoints.
Let be a smooth manifold.
For two smooth maps, a thin homotopy is a smooth homotopy, i.e. a smooth map
which is thin in that it doesn’t sweep out any surface: every -form pulled back to it vanishes:
A path has sitting instants if there is a neighbourhood of the boundary of such that is locally constant restricted to that.
The path groupoid is the diffeological groupoid that has
- thin-homotopy classes of paths with sitting instants.
Composition of paths comes from concatenation and reparameterization of representatives. The quotient by thin-homotopy ensures that this yields an associative composition with inverses for each path.
This definition makes sense for any generalized smooth space, in particular for a sheaf on Diff.
Moreover, is always itself naturally a groupoid internal to generalized smooth spaces: if is a Chen space or diffeological space then is itself internal to that category. However, even if is a manifold, will not be a manifold, see smooth structure of the path groupoid for details.
There are various generalizations of the path groupoid to n-groupoids and ∞-groupoids. See
If is a Lie group, then internal (i.e. smooth) functors from the path groupoid to the one-object Lie groupoid corresponding to are in bijection to -valued differential forms on . With gauge transformations regarded as morphisms between Lie-algebra values differential forms, this extends naturally to an equivalence of categories
where on the left the functor category is the one of internal (smooth) functors.
More generally, smooth anafunctors from to are canonically equivalent to smooth -principal bundles on with connection: