Weak equivalences of simplicial presheaves
  The usual way of defining weak equivalences for simplicial presheaves is to require an isomorphism on all sheaves of homotopy groups. We unravel some of the machinery here, and give a more concrete description in terms of local homotopy lifting properties. This characterization is used to prove some basic facts about the local homotopy theory of simplicial presheaves. 
Daniel Dugger, Daniel C. Isaksen