  Albeverio, Kondratiev, and R\"{o}ckner have introduced a type of differential
geometry, which we call lifted geometry, for the configuration space $\Gamma_X$
of any manifold $X$. The name comes from the fact that various elements of the
geometry of $\Gamma_X$ are constructed via lifting of the corresponding
elements of the geometry of $X$. In this note, we construct a general algebraic
framework for lifted geometry which can be applied to various ``infinite
dimensional spaces'' associated to $X$. In order to define a lifted geometry
for a ``space'', one dose not need any topology or local coordinate system on
the space. As example and application, lifted geometry for spaces of Radon
measures on $X$, mappings into $X$, embedded submanifolds of $X$, and tilings
on $X$, are considered. The gradient operator in the lifted geometry of Radon
measures is considered. Also, the construction of a natural Dirichlet form
associated to a Random measure is discussed. It is shown that Stokes' Theorem
appears as ``differentiability'' of ``boundary operator'' in the lifted
geometry of spaces of submanifolds. It is shown that (generalized) action
functionals associated with Lagrangian densities on $X$ form the algebra of
smooth functions in a specific lifted geometry for the path-space of $X$.
