If Y is a diagram of spectra indexed by an arbitrary poset C together with a
specified sub-poset D, we define the total cofibre \Gamma (Y) of Y as the
strict cofibre of the map from hocolim_D (Y) to hocolim_C (Y). We construct a
comparison map from the homotopy limit of Y to a looping of a fibrant
replacement of Gamma (Y), and characterise those poset pairs (C,D) for which
this comparison map is a stable equivalence. The characterisation is given in
terms of stable cohomotopy of spaces related to C and D. For example, if C is a
finite polytopal complex with underlying space an m-ball with boundary sphere
D, then holim_C (Y) and \Gamma(Y) agree up to m-fold looping and up to stable
equivalence. As an application of the general result we give a spectral
sequence for the homotopy groups of \Gamma(Y) with E_2-term involving higher
derived inverse limits of \pi_* (Y), generalising earlier constructions for
space-valued diagrams indexed by the face lattice of a polytope.Comment: 11 pages; LaTeX source code uses Paul Taylor's "diagrams" and "QED"
macro packages; some diagrams may not display correctly with DVI viewers;
paper also available at http://nyjm.albany.edu:8000/j/2005/11-16.htm
