Maps between objects in a homotopy category can be identified with path components of a suitably defined category of cocycles, in great generality. Here are some applications: 1) This device leads to a simple demonstration of the identification of isomorphism classes of torsors for sheaves of groups G with maps in the homotopy category of simplicial sheaves. 2) The concept of torsors for a sheave of groups can be be generalized to torsors for sheaves of groupoids, and more generally to torsors for arbitrary sheaves of index categories A. In general, torsors are diagrams of weak equivalences which have trivial homotopy colimits. There an identification of path components in the category of A-torsors with morphisms [*,BA] in the homotopy category of simplicial sheaves. 3) Path components of categories of gerbes can be identified with the path components of cocycle categories taking values in the 2-groupoid of isomorphisms of sheaves of groups.