A note on the homotopy type of posets

Abstract

For any poset P let J( P) denote the complete lattice of order ideals in P. J( P) is a contravariant functor in P. Any order-reserving map f: P→ Q can be regarded as an isotone (=order-preserving) map of either P ∗ into Q or P into Q ∗. The induced map of J( Q) to J( P ∗)(resp. J( Q ∗) into J( P)) will be denoted by J l ( f)(resp. J r ( f)). Our first result asserts that if f: P→ Q, g: Q→ P are maps of a Galois connection, then (a) J r( f): J( Q ∗)→ J( P) ∗, J l ( g): J( P ∗)→ J( Q ∗) and (b) J l ( f): J( Q) ∗→ J( P ∗), J r( g): J( P ∗)→ J( Q) ∗ are Galois connections. For any lattice L, we denote the poset L - {0,1} by L̄. We analyse conditions which will imply that J r(f) (J (Q∗)) ∋ J (P)∗ and J l(g) (J (P)∗) ∋ J (Q)∗ . Under these conditions, from Walker's results [3] it will follow that J r(f)/ (J (Q∗)): (J (Q∗))→ (J (P)∗ is a homotopy equivalence with J l(g) (J (P)∗: (J (P)∗→ (J (Q∗)) as its homotopy inverse. Given an isotone map f: P→ Q it is easy to find the necessary and sufficient conditions for J( f) to satisfy J(f) (J (Q))∋ J (P) . When these conditions are fulfilled, we also find a sufficient condition that ensures that J(f)/ J (Q): J (Q)→ J (P) is a homotopy equivalence. We give examples to show that the homotopy type of P neither determines nor is determined by the homotopy type of J (P) .