  We look at homotopy-coherent diagrams of spaces (after Segal, Leitch, Vogt,
Mather, Cordier) over a Grothendieck site; we call these ``flexible
presheaves''. After some preliminary materiel, we define the ``flexible sheaf''
condition. This descent condition (known to Thomason) is the same as what
Jardine called being ``flasque'' with respect to the presheaves representable
by objects in the site; and it is more recently known as the condition of being
an $n$-stack. We construct the flexible sheaf associated to a flexible presheaf
in the $n$-truncated case, as an application of a certain natural operation
$n+2$ times. We prove an analogue of Vogt's theorem for the case where the
Grothendieck topology is nontrivial, identifying the set of morphisms in
Illusie's derived category as the set of homotopy classes of homotopy-coherent
morphisms between flexible sheaves. The homotopy-coherent point of view allows
one easily to define the flexible mapping sheaf $Hom (R,T)$ between two
flexible sheaves.
  This revision fills major gaps in the bibliography. References to the
additional items are inserted in the text. A new introduction and abstract are
added (the old ones are retained as comments in the source file). A few other
minor changes in the exposition include arrangement of internal references.
