  Twisted diagrams are "diagrams" with components in different categories.
Structure maps are defined using auxiliary data which consists of functors
relating the various categories to each other. Prime examples of the
construction are spectra (in the sense of homotopy theory) and quasi-coherent
sheaves on schemes. We develop the basic theory of twisted diagrams, and
establish various model structures (which are well-known in special cases). We
also introduce a notion of homotopy sheaves, a collection of local data which
is compatible up to weak equivalence, and study basic properties of such
objects. These objects occur in nature; for example, the notion of an
Omega-spectrum fits into this framework. The main purpose of the paper is to
provide a convenient reference for model structures on twisted diagrams, and
for the language of sheaves and homotopy sheaves as defined here.
