### Cocycle categories, by J.F. Jardine

A cocycle category H(X,Y) is defined for objects X and Y in a model category,
and it is shown that the set of morphisms [X,Y] is isomorphic to the set of
path components of H(X,Y) provided the ambient model category is right proper
and satisfies the extra condition that weak equivalences are closed under
finite products. Various applications of this result are displayed, including
the homotopy classification of torsors, abelian cohomology groups, group
extensions and gerbes. The older classification results have simple new proofs
involving canonically defined cocycles.

J.F. Jardine <jardine@uwo.ca>