Let p be a prime number. We prove that if G is a compact Lie group with a non-trivial p-subgroup, then the orbit space $\left(B_p\left(G\right)\right)/G$ of the classifying space of the category associated to the G-poset ${}_p\left(G\right)$ of all non-trivial elementary abelian p-subgroups of G is contractible. This gives, for every G-CW-complex X each of whose isotropy groups contains a non-trivial p-subgroup, a decomposition of X/G as a homotopy colimit of the functor $X^{Eₙ}/(NE₀\cap ...\cap NEₙ)$ defined over the poset $\left(s{d}_p\left(G\right)\right)/G$, where sd is the barycentric subdivision. We also...

Let be an abelian category, or more generally a weakly idempotent complete exact category, and suppose we have two complete hereditary cotorsion pairs $(,\tilde{})$ and $(\tilde{},)$ in satisfying $\tilde{}\subseteq$ and $\cap \tilde{}=\tilde{}\cap$. We show how to construct a (necessarily unique) abelian model structure on with (resp. $\tilde{}$) as the class of cofibrant (resp. trivially cofibrant) objects, and (resp. $\tilde{}$) as the class of fibrant (resp. trivially fibrant) objects.