The higher algebra of weighted colimits

We develop a theory of weighted colimits in the framework of weakly bi-enriched -categories, an extension of Lurie's notion of enriched -categories. We prove an existence result for weighted colimits, study weighted colimits of diagrams of enriched functors, express weighted colimits via enriched coends, characterize the enriched -category of enriched presheaves as the free cocompletion under weighted colimits and develop a theory of universally adjoining weighted colimits to an enriched -category. We use the latter technique to construct for every presentably E₊+₁-monoidal -category V for 1 k and class H of V-weights, with respect to which weighted colimits are defined, a presentably Eₖ-monoidal structure on the -category of V-enriched -categories that admit H-weighted colimits. Varying H this Eₖ-monoidal structure interpolates between the tensor product for V-enriched -categories and the relative tensor product for -categories presentably left tensored over V. As an application we prove that forming V-enriched presheaves is Eₖ-monoidal, construct a V-enriched version of Day-convolution and give a new construction of the tensor product for -categories presentably left tensored over V as a V-enriched localization of Day-convolution. As further applications we construct a tensor product for Cauchy-complete V-enriched -categories, a tensor product for (, 2) -categories with (op) lax colimits and a tensor product for stable (, n) -categories.