In Chapter 5, we proved that the bar construction gives a uniform way to construct homotopy limits and colimits of diagrams of any shape, defined to be derived functors of the appropriate limit or colimit functor in a simplicial model category. But, on account of Corollary 7.6.4, such categories admit a richer class of limits and colimits – the weighted limits and colimits for any simplicially enriched weight, computed as enriched functor cotensor or tensor products.

In this chapter we will prove that the enriched version of the two-sided bar construction can be used to construct a derived functor of the enriched functor tensor product. This provides a notion of weighted homotopy colimit that enjoys the same formal properties of our homotopy colimit functor. The construction and proofs closely parallel those of Chapter 5, though more sophisticated hypotheses will be needed to guarantee that the two-sided bar construction is homotopically well behaved. These results, due to Shulman [79], partially motivated our earlier presentation.

The notion of derived functor used here is precisely the one introduced in Chapter 2. This is to say, a derived functor of a ν-functor between ν-categories whose underlying categories are homotopical is defined to be a derived functor of the underlying unenriched functor. A priori, and indeed in many cases of interest, the point-set-derived functor defining the weighted homotopy colimit will not be a ν-functor. However, under reasonable hypotheses, its total derived functor inherits a natural enrichment – just not over ν.