  In this paper, we try to realize the unbounded derived category of an abelian
category as the homotopy category of a Quillen model structure on the category
of unbounded chain complexes. We construct such a model structure based on
injective resolutions for an arbitrary Grothendieck category, as has apparently
also been done by Morel. In particular, this works for sheaves on a ringed
space, and for quasi-coherent sheaves on a quasi-compact, quasi-separated
scheme. However, this injective model structure is not well suited to studying
the derived tensor product, so we investigate other model structures. The most
successful of these is the flat model structure on complexes of sheaves over a
ringed space. This is based on flat resolutions, and is compatible with the
tensor product. As a corollary, we get model categories of differential graded
algebras of sheaves and differential graded modules over a given differential
graded algebra of sheaves.
  This is the author's first attempt to understand sheaves, so comments from
those more experienced with the subject are welcome.
