Homotopy classification of categorical torsors

The long-known results of Schreier on group extensions are here raised to a categorical level by giving a factor set theory for torsors under a categorical group (G,circle times) over a small category beta. We show a natural bijection between the set of equivalence classes of such torsors and [B({beta}),B(G,circle times)], the set of homotopy classes of continuous maps between the corresponding classifying spaces. These results are applied to algebraically interpret the set of homotopy classes of maps from a CW-complex X to a path-connected CW-complex Y with pi (i)(Y)=0 for all i not equal1,2.