Abstract: 
We describe a unifying approach to a variety of homotopy
decompositions of classifying spaces, mainly of finite groups.
For a group $G$ acting on a poset ${\bf W}$ and an isotropy
presheaf $d:{\bf W}\rightarrow {\cal S}(G)$ we construct a
natural $G$-map $ \mathop {\rm hocolim}\nolimits _{{\cal
W}_d}G/d(-)\rightarrow |{\bf W}|$ which is a (non-equivariant)
homotopy equivalence, hence $ \mathop {\rm hocolim}\nolimits
_{{\cal W}_d}EG\times _GF_d \rightarrow EG\times _G|{\bf W}|$ is
a homotopy equivalence. Different choices of $G$-posets and
isotropy presheaves on them lead to homotopy decompositions of
classifying spaces. We analyze higher limits over the categories
associated to isotropy presheaves ${\cal W}_d$; in some
important cases they vanish in dimensions greater than the
length of ${\bf W}$ and can be explicitly calculated in low
dimensions. We prove a cofinality theorem for functors $F:{\cal
C}\rightarrow {\cal O}(G)$ into the category of $G$-orbits which
guarantees that the associated map $\alpha _F:\mathop {\rm
hocolim}\nolimits _{{\cal C}}EG\times _G F(-)\rightarrow BG$ is
a mod-$p$-homology decomposition.

1. Institute of Mathematics
   Warsaw University
   Banacha 2
   02-097 Warszawa, Poland
   
2. Faculty of Mathematics and Information Sciences
   Warsaw Technical University
   Pl. Politechniki 1
   00-661 Warszawa, Poland