Collapsing along monotone poset maps

We introduce the notion of nonevasive reduction, and show that for any monotone poset map $\phi:P\to P$, the simplicial complex $\Delta(P)$ {\tt NE}-reduces to $\Delta(Q)$, for any $Q\supseteq{\text{\rm Fix}}\phi$. As a corollary, we prove that for any order-preserving map $\phi:P\to P$ satisfying $\phi(x)\geq x$, for any $x\in P$, the simplicial complex $\Delta(P)$ collapses to $\Delta(\phi(P))$. We also obtain a generalization of Crapo's closure theorem.