Varifolds generalize the idea of a rectifiable current, and are studied in geometric measure theory.
[1][2] Frederick J. Almgren Jr. slightly modified the definition in his mimeographed notes (Almgren 1965) and coined the name varifold: he wanted to emphasize that these objects are substitutes for ordinary manifolds in problems of the calculus of variations.
The Grassmannian is used to allow the construction of analogs to differential forms as duals to vector fields in the approximate tangent space of the set
Replacing M with more regular sets, one easily see that differentiable submanifolds are particular cases of rectifiable manifolds.
Due to the lack of orientation, there is no boundary operator defined on the space of varifolds.