  This paper contains two results on how homotopy limits of topological spaces
interact with connectivity. The first is a formula for the connectivity of the
homotopy limit of diagrams shaped over suitably finite categories, in terms of
the connectivity of the spaces in the diagram. The second result shows that the
homotopy fiber of the restriction map induced on homotopy limits by a functor
of indexing categories is itself a homotopy limit, indexed over a Grothendieck
construction.
