# Pro-isomorphisms of homology towers

** ABSTRACT.**
We find criteria which ensure that two towers of spaces
have pro-isomorphic mod p homology towers. In particular, this gives
criteria for determining when the homology of an inverse limit is isomorphic to
the inverse limit
of the homology tower. As a consequence, we establish new conditions for the
convergence of the generalized Eilenberg-Moore spectral sequence (i.e. the
homology spectral sequence of a cosimplicial space), which computes the
homology of inverse limits such as mapping spaces or fibers of fibrations.