Coxeter Descents and Parabolic Homotopy Colimits: A Hochster-type decomposition and integral Morse reduction

Let $(W,S)$ be a finite Coxeter system and let $\mathcal K\subseteq 2^S$ be a simplicial complex. We define a parabolic bar complex ${\mathcal B}_{\mathcal K}(W)$ and prove a decomposition indexed by $w\in W$ in which the $w$-summand is a relative order-complex chain complex determined by the right