The higher algebra of weighted colimits

We develop a theory of weighted colimits in the framework of weakly bienriched $\mathrm{\infty }$-categories, an extension of Lurie’s notion of enriched $\mathrm{\infty }$-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 $\mathrm{\infty }$-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 $\mathrm{\infty }$-category. Via the latter we construct for every presentably ${\mathbb{E}}_{k+1}$-monoidal $\mathrm{\infty }$-category $\mathcal{V}$ for $1\le k\le \mathrm{\infty }$ and set $\mathcal{H}$ of weights a presentably ${\mathbb{E}}_k$-monoidal structure on the $\mathrm{\infty }$-category of $\mathcal{V}$-enriched $\mathrm{\infty }$-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 $\mathrm{\infty }$-categories and the relative tensor product for $\mathrm{\infty }$-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 $\mathrm{\infty }$-categories and the relative tensor product on $\mathrm{\infty }$-categories presentably left tensored over $\mathcal{V}\mathrm{.}$ As key applications we construct for every $n\ge 1$ and set $\mathcal{K}$ of $(\mathrm{\infty },n)$-categories a tensor product for $(\mathrm{\infty },n)$-categories that admit $\mathcal{K}$-indexed (op)lax colimits, a tensor product for Cauchy-complete $\mathcal{V}$-enriched $\mathrm{\infty }$-categories and tensor products for (Cauchy complete) $n$-stable, $n$-additive and $n$-preadditive $(\mathrm{\infty },n)$-categories.