The higher algebra of weighted colimits

We develop a theory of weighted colimits in the framework of weakly bienriched $\infty$-categories, an extension of Lurie's notion of enriched $\infty$-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 $\infty$-category of enriched presheaves as the free cocompletion under weighted colimits, prove a Bousfield-Kan formula for weighted colimits and an enriched adjoint functor theorem and develop a theory of universally adjoining weighted colimits to an enriched $\infty$-category. Via the latter we construct for every presentably $\mathbb{E}_{k+1}$-monoidal $\infty$-category $\mathcal{V}$ for $1 \leq k \leq \infty$ and set $\mathcal{H}$ of weights a presentably $\mathbb{E}_k$-monoidal structure on the $\infty$-category of $\mathcal{V}$-enriched $\infty$-categories that admit $\mathcal{H}$-weighted colimits. Varying $\mathcal{H}$ this $\mathbb{E}_k$-monoidal structure interpolates between the tensor product for $\mathcal{V}$-enriched $\infty$-categories and the relative tensor product for $\infty$-categories presentably left tensored over $\mathcal{V}$. Studying functoriality in $\mathcal{H}$ we deduce that taking $\mathcal{V}$-enriched presheaves is $\mathbb{E}_k$-monoidal with respect to the tensor product on small $\mathcal{V}$-enriched $\infty$-categories and the relative tensor product on $\infty$-categories presentably left tensored over $\mathcal{V}.$ As key applications we construct for every $n \geq 1 $ and set $\mathcal{K}$ of $(\infty, n)$-categories a tensor product for $(\infty,n)$-categories that admit $\mathcal{K}$-indexed (op)lax colimits, a tensor product for Cauchy-complete $\mathcal{V}$-enriched $\infty$-categories and tensor products for (Cauchy complete) $n$-stable, $n$-additive and $n$-preadditive $(\infty,n)$-categories.