  Let V be a cofibrantly generated closed symmetric monoidal model category and
M a model V-category. We say that a weighted colimit W*D of a diagram D
weighted by W is a homotopy weighted colimit if the diagram D is pointwise
cofibrant and the weight W is cofibrant in the projective model structure on
[C^op,V]. We then proceed to describe such homotopy weighted colimits through
homotopy tensors and ordinary (conical) homotopy colimits. This is a homotopy
version of the well known isomorphism W*D=\int^C(W\tensor D). After proving
this homotopy decomposition in general we study in some detail a few special
cases. For simplicial sets tensors may be replaced up to weak equivalence by
conical homotopy colimits and thus the weighted homotopy colimits have no added
value. The situation is completely different for model dg-categories where the
desuspension cannot be constructed from conical homotopy colimits. In the last
section we characterize those V-functors inducing a Quillen equivalence on the
enriched presheaf categories.
